
Citation 
 Permanent Link:
 https://ufdc.ufl.edu/UF00098912/00001
Material Information
 Title:
 Tsunami interaction with coastlines and elevation predictions
 Creator:
 Houston, James R ( James Robert ), 1947 ( Dissertant )
Millsaps, Knox ( Thesis advisor )
Hammack, Joseph L. ( Thesis advisor )
Sheppard, D. M. ( Reviewer )
Eisenberg, M. A. ( Reviewer )
Thomas, Michael E. ( Reviewer )
 Place of Publication:
 Gainesville, Fla.
 Publisher:
 University of Florida
 Publication Date:
 1978
 Copyright Date:
 1978
 Language:
 English
 Physical Description:
 ix, 105 leaves : ill., graphs, maps ; 28 cm.
Subjects
 Subjects / Keywords:
 Amplitude ( jstor )
Coasts ( jstor ) Earthquakes ( jstor ) Modeling ( jstor ) Oceans ( jstor ) Topographical elevation ( jstor ) Tsunamis ( jstor ) Water depth ( jstor ) Waveforms ( jstor ) Waves ( jstor ) Dissertations, Academic  Electrical Engineering  UF ( local ) Electrical Engineering thesis Ph. D ( local ) Tsunamis  Mathematical models ( lcsh )
 Genre:
 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Abstract:
 The development is described of two numerical models that accurately
simulate the propagation of tsunamis to nearshore regions and the
interaction of tsunamis with coastlines. One of these models is a
finite element model which uses a telescoping numerical grid to cover
a section of the Pacific Ocean, including all eight islands of the
Hawaiian Islands. The second model is a finite difference scheme that
uses four rectilinear grids to cover most of the west coast of the
continental United States. The finite element model solves linear and
dissipationless longwave equations. Such equations govern nearshore
propagation in the Hawaiian Islands since the short continental shelf
of the islands limits the time available for nonlinearities and
dissipation to cause significant effects. The finite difference
model solves nonlinear longwave equations that include bottom stress
terms which are important for the long continental shelf of the west coast of the United States. Both models are verified by hindcasting
actual historical tsunamis and comparing the numerical model calculations
with tidegage recordings. A frequency of occurrence analysis
of tsunami elevations at the shoreline in the Hawaiian Islands is
described. This analysis is based upon local historical data with the
finite element model used to interpolate between historical data recorded
during the period of accurate survey measurements since 1946.
Historical data recorded at Hilo, Hawaii, and dating to 1337 is used
in conduction with data recorded since 1946 in Hilo and throughout the
Hawaiian Islands to reconstruct elevations at locations in the islands
lacking data prior to 1946. Frequency of occurrence curves are determined
for the entire coastline of the Hawaiian Islands using these
reconstructed elevations. Since most of the west coast of the United
States lacks local data of tsunami activity, a frequency of occurrence
analysis of tsunami elevations is based upon historical data of tsunami
occurrence in tsunamigenic regions in addition to numerical model calculations.
A generation and deepocean propagation numerical model is
used to propagate tsunamis with varying intensities from locations
throughout the AleutianAlaskan and PeruChile regions to a water depth
of 500 meters off the west coast. The nearshore finite difference model
propagates these tsunamis from the 500 meter depth to shore. The
frequency of occurrence of combined tsunami and astronomical tide
elevations is determined by an analysis involving the numerical superposition
of tsunamis and local tides.
 Thesis:
 ThesisUniversity of Florida.
 Bibliography:
 Bibliography: leaves 99104.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by James Robert Houston.
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 022934837 ( AlephBibNum )
05313024 ( OCLC ) AAK0598 ( NOTIS )

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Full Text 
TSUNAMI INTERACTION WITH COASTLINES
AND ELEVATION PREDICTIONS
By
JAMES ROBERT HOUSTON
A Dissertation Presented to the Graduate Council of
The University of Flori'da
In Partial Fulfillment of the Requlrerents for the
Degree of Doctor of Philosophy
UNIVERSITY OF FLORIDA
1978
ACKNOWLEDGMENTS
The writer wishes to express his sincere gratitude to his
supervisory committee chairman, Professor Knox Millsaps, and co
chairman, Professor Joseph Hammack, for their support and efforts to
provide him the opportunity to complete his studies at the University
of Florida.
A special debt of gratitude is owed to Dr. R. W. hhalin, Chief of
the Wave Dynamics Division, Hydraulics Laboratory, U. S. Army Waterways
Experiment Station for his encouragement and active support during
the research and preparation periods of this dissertation.
The writer wishes to thank Professor Umit Un)uata for the insight
he provided and for organizing a supervisory committee prior to his
departure from ths University. Appreciation also is extended to
Dr. D. L. Durham for many helpful discussions at the Waterways Experi
ment Station.
The research upon which this dissertation is based was funded by
the Pacific Ocean Division of the U. S. Corps of Engineers and the
Federal Insurance Administration of the Department of Housing and
Urban Development through the Office of the Chief of Engineers.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ...... ................. i
KEY TO SYMBOLS . . . .. v
ABSTRACT . ................... .viii
CHAPTER I: INTRODUCTION .. ........ ..... . 1
Tsunamis ... . ..... .. 1
Objective and Scope of Study . ... . ....... 2
CHAPTER II: TSUNAMI INTERACTION WITH FTHE HAWAIIAN ISLANDS 4
Introduction .... ... .. ...... . .... 4
Governing Equations . . . .. . ..... 7
Finite Element Model . . ............ 12
Numerical Grid . .......... ......... 21
Model Verification ................. 25
CHAPTER III: TSUNAMI INTERACTION WITH CONTINENTAL
COASTLINES . .. .. ... .... ...... . 43
Introduction.. .. . .............. . 43
Governing Equations .. .... ... . . . . 45
Finite Difference Model .... . . .. . . 48
Model Verification ...... . . . . . 51
CHAPTER IV: ELEVATION PREDICTIONS . . . . . 58
Hawaiian Island Predictions . . . ..... .. . 58
Introduction .. .. . . . . . 58
Interpolation of Recent Historical Data . . 61
Time Period Analysis . .. ... . . .. 63
Reconstruction of Historical Data . ... .. . 65
Frequency of Occurrence Distribution . . . 69
Results .................. 71
Page
West Coast of the United States Predictions . . .. 75
Introduction . ... .... . .. ... 75
Tsunami Occurrence Probabilities .. . .... 77
Use of Deterministic Numerical Models . . . 80
Effect of the Astronomical Tides . . . . 85
Comparison with Local Observation Predictions . 88
CHAPTER V: CONCLUSIONS ...... . . . . . 95
APPENDIX: SUPERPOSITION OF LARGE tSUNAMIS AND
ASTRONOMICAL 'ITDES ................. 97
REFERENCES .................. .. 99
BIOGRAPHICAL SKETCH .... ............. . 105
KEY TO SYMBOLS
a Wave amplitude
b Incident wave amplitude
0
b(m) Amplitude of frequency component j
C Chezy's coefficient
th
C Amplitude of m tidal coefficient
D Period of time
f Frequency of occurrence
>
F External force vector
F( ) Functional
g Acceleration of gravity
h W'ater depth
S Average runup, meters
avg
H Hankel function of first kind of order m
n
i (1)1/2
I Tsunami intensity
j Integer
k Wave number
[K] Coefficient matrix
L Wavelength
m Integer
n Unit normal vector
n Manning's coefficient
ma
n( ) Frequency of occurrence
N Interpolation function
P Pressure
?
q Velocity vector
{Q} Total load vector
r Spherical coordinate
t Time
T Wave period
u Velocity component in xdirection
v Velocity component in ydirection
w Velocity component in zdirection
x Cartesian coordinate
Ax Small increment in x
y Cartesian coordinate
Ay Small increment in y
z Cartesian coordinate (vertical direction)
a Constant
a Constant coefficient
m
B Constant coefficient
m
S Area of element
cE Coefficient of horizontal eddy viscosity
E2 Coefficient of vertical eddy viscosity
n Freesurface elevation
P
SM
p
p(u)
2
o
Tb
0
Velocity potential
Velocity potential
Velocity potential
Velocity potential
Angular frequency
Gradient operator
in region A
in region B
of incident wave
of scattered wave
Spherical coordinate
Dynamic viscosity constant
Response to arbitrary tsunami
Mass density
Phase angle
Tidal variance
Bottom stress
Velocity potential
Abstract of Dissertation Presented to the Graduate Council of
The University of Florida in Partial Fulfillment of the Requirements
For the Degree of Doctor of Philosophy
TSUNAMI INTERACTION WITH COASTLINES
AND ELEVATION PREDICTIONS
By
James Robert Houston
June 1978
Chairman: K. Millsaps
Cochairman: J. L. Hammack
Major Department: Engineering Science
The development is described of two numerical models that accurately
simulate the propagation of tsunamis to nearshore regions and the
interaction of tsunamis with coastlines. One of these models is a
finite element model which uses a telescoping numerical grid to cover
a section of the Pacific Ocean, including all eight islands of the
Hawaiian Islands. The second model is a finite difference scheme that
uses four rectilinear grids to cover most of the west coast of the
continental United States. The finite element model solves linear and
dissipationless longwave equations. Such equations govern nearshore
propagation in the Hawaiian Islands since the short continental shelf
of the islands limits the time available for nonlinearities and
dissipation to cause significant effects. The finite difference
model solves nonlinear longwave equations that include bottom stress
terms which are important for the long continental shelf of the west
viii
coast of the United States. Both models are verified by hindcasting
actual historical tsunamis and comparing the numerical model calcu
lations with tidegage recordings. A frequency of occurrence analysis
of tsunami elevations at the shoreline in the Hawaiian Islands is
described. This analysis is based upon local historical data with the
finite element model used to interpolate between historical data re
corded during the period of accurate survey measurements since 1946.
Historical data recorded at Hilo, Hawaii, and dating to 1837 is used
in conjuction with data recorded since 1946 in Hilo and throughout the
Hawaiian Islands to reconstruct elevations at locations in the islands
lacking data prior to 1946. Frequency of occurrence curves are deter
mined for the entire coastline of the Hawaiian Islands using these
reconstructed elevations. Since most of the west coast of the United
States lacks local data of tsunami activity, a frequency of occurrence
analysis of tsunami elevations is based upon historical data of tsunami
occurrence in tsunamigenic regions in addition to numerical model cal
culations. A generation and deepocean propagation numerical model is
used to propagate tsunamis with varying intensities from locations
throughout the AleutianAlaskan and PeruChile regions to a water depth
of 500 meters off the west coast. The nearshore finite difference model
propagates these tsunamis from the 500 meter depth to shore. The
frequency of occurrence of combined tsunami and astronomical tide
elevations is determined by an analysis involving the numerical super
position of tsunamis and local tides.
CHIAPT',R I
INTRODUCTION
Tsunamis
Of all water waves that occur in nature, or.e of the most destruc
tive is the tsunami. The term "tsunami," originating from the Japanese
words "tsu" (harbor) and "nami" (wave), is used to describe sea waves
of seismic origin. Tectonic earthquakes, ie., earthquakes that cause
a deformation of the sea bed, appear to be the principal seismic mech
anism responsible for the generation of tsunamis. Coastal and submarine
landslides and volcanic eruptions also have triggered tsunamis.
Tsunamis are principally generated by undersea earthquakes of the
dipslip type with magnitudes greater than 6.5 on the Richter scale and
focal depths less tha n 50 kilometers. They are very long period waves
(5 minutes to several hours) of low height (a few feet or less) when
transversing water of oceanic depth. Consequently, they are not dis
cernible in the deep ccean and go unnoticed by ships. Tsunamis travel
at the shallowwater wave celerity equal to the square root of the
acceleration due to gravity times water depth even in the deepest oceans
because of their very long wavelengths. This speed of propagation can
be in excess of 500 miles per hour in the deep ocean.
When tsunamis approach a coastal region where the water depth
decreases rapidly, wave refraction, shoaling, and bay or harbor reso
nance may result in significantly increased wave heights. The great
period and wavelength of tsunamis preclude their dissipating energy
as a breaking surf; instead, they are apt to appear as rapidly rising
water levels and only occasionally as bores.
Over 500 tsunamis have been reported within recorded history.
Virtually all of these tsunamis have occurred in the Pacific Basin.
This is because most tsunamis are associated with earthquakes, and most
seismic activity beneath the oceans is concentrated in the narrow
fault zones adjacent to the great oceanic trench systems that are
predominantly confined to the Pacific Ocean.
The loss of life and destruction of property due to tsunamis have
been immense. 'Ihe Great Ioei TokaidoNanhaido tsunami of Japan killed
30,000 people in 1707. In 1868, the Great Peru tsunami caused 25,000
deaths and carried the frigate U.S.S. Waterlee 1,300 feet inland.
The Great Meiji Sanriku tsunami of 1896 killed 27,122 persons in Japan
and washed away over 10,000 houses.
In recent times, three tsunamis have caused major destruction in
areas of the United States. The Great Aleutian tsunami of 1946 killed
173 persons in Hawaii, where runup heights as great as 55 feet were
recorded. The 1960 Chilean tsunami killed 330 people in Chile, 61 in
Hawaii, and 199 in distant Japan. The most recent major tsunami to
affect the United States, the 1964 Alaskan tsunami, killed 107 people in
Alaska, 4 in Oregon, and 11 in Crescent City, California, and caused
over 100 million dollars in damage on the west coast of North America.
Objective and Scope of Study
The objective of the present study is to determine tsunami eleva
tion frequencies of occurrence at the shoreline for the Hawaiian Islands
and the west coast of the continental United States. Such elevation pre
dictions are required by the Pacific Ocean Division of the U. S. Corps
of Engineers for use in tsunami flood hazard evaluations for floodplain
management and the Federal Insurance Administration of the Department of
Housing and Urban Development for flood insurance rate calculations.
The current trend of general population migration to coastal
regions and the growth of the tourist industry makes it essential to
know possible tsunami inundation levels even for coastal regions that
are currently undeveloped. There are only a few isolated locations in
the United States that have sufficient local historical data of tsunami
activity to allow accurate predictions solely based upon historical
data. This lack of historical data of tsunami activity on most of the
coastline of the United States makes it necessary to use numerical
models in addition to existing historical data to predict tsunami
elevation frequency of occurrence. However, numerical models have not
been successfully applied in the past to the simulation of actual
tsunami nearshore propagation and interaction with coastlines. There
fore, two numerical models are developed in the present study to
accurately simulate tsunami nearshore activity. Calculations of these
deterministic numerical models are combined with probabilistic analyses
based upon historical data to predict frequencies of occurrence of
tsunami elevations for the Hawaiian Islands and the west coast of the
United States.
CHAPTER II
TSUNAMI INTERACTION WITH THE HAWAIIAN ISLANDS
Introduction
Early studies of tsunami interaction with islands involved analytic
investigations of the scattering of monochromatic plane waves by a
single island of simple geometric shape. Oner and Hall (1949) deter
mined the theoretical diffraction pattern for longwave scattering by
a circular cylinder in water of constant depth by adopting results from
acoustic theory. They found qualitative agreement between this dif
fraction pattern and smoothed runup heights observed for the 1946
tsunami on the island of Kauai in the Hawaiian Islands. Homma (1950)
determined the diffraction pattern for long wave scattering off a circu
lar cylinder surrounded by a parabolic bathymetry that extended to a
prescribed distance offshore, beyond which the depth was constant.
Hydraulic models have been used to study tsunami interaction with
single islands that are realistically shaped and surrounded by variable
bathymetry. Van Dorn (1970) studied tsunami interaction with Wake
Island using a 1:57000 undistorted scale model. Jordaan and Adams
(1968) studied tsunami interaction with the island of Oahu in the
Hawaiian Islands using a 1:20000 undistorted scale model. They
found poor agreement between historical measureents of tsunami runup
and the hydraulic model data. Scale effects (e.g. viscous effects)
and the effects of the arbitrary boundaries that confine the nodel are
problems that are inherent in hydraulic modeling of tsunamis. The small
heights of tsunamis relative to deep ocean depths is also difficult to
simulate. Jordaan and Adams (1968) modeled waves to a vertical scale
of 1:2000; thus, the waves had heights ten times the normal proportion.
Even with this distortion, waves typically had heights in the model of
only 0.3 millimeters.
Vastano and Reid (1966) studied the problem of the response of
a single island to monochromatic plane waves of tsunami period using
a finite difference numerical model. A transformation of coordinates
allowed a mapping of the island shore as a circle in the image plane.
The finite difference solution employed a grid which allowed greater
resolution in the vicinity of the island than in the deep ocean.
Vastano and Bernard (1973) extended the techniques developed by
Vastano and Reid (1966) to multipleisland systems. However, the
transformation of coordinates technique allows high resolution only
in the vicinity of one island of a multipleisland system. Thus, for
the threeisland system of Kauai, Oahu, and Niihau in the Hawaiian
Islands, the two islands of Oahu and Niihau had to be represented by
cylinders with vertical walls whose cross sections were truncated
wedges. Kauai was represented by a circular cylinder with the sur
rounding bathymetry increasing linearly in depth with distance radially
from the island until a constant depth was attained. A single Gaussian
shaped plane wave was used as input to the model. No comparisons were
made with historical tsunami data for the three islands.
Hwang and Divoky (1975) presented a simulation of the 1964 tsunami
at Hilo, Hawaii, using a finite difference numerical model. The 1964
Alaskan tsunami was generated and propagated across the deep ocean using
a finite difference numerical model employing a coarse grid. The
Hawaiian Islands were too small to be represented by this numerical grid.
A waveform was calculated at a point in water 5000 meters deep that was
approximately 200 kilometers from where Hilo, Hawaii, would have been if
the Hawaiian Islands could have been modeled. This waveform was used as
input to a finite difference grid with fine grid cells that covered an
area approximately 22 kilometers by 67 kilometers in the immediate
vicinity of Hilo. The water depth along the input boundary of the fine
grid varied from approximately 250 meters to 1500 meters. Hwang's
approach neglected the influence of the Hawaiian Islands and of the
varying bathymetry and depth changes between the deep ocean point and the
fine grid input boundary on the tsunami arriving in Hilo. The numerical
model calculations for the 1964 tsunami were in poor agreement with
the Hilo tide gage recording of this tsunami.
A finite difference model employing a grid covering the eight
major islands of the Hawaiian Island chain was used by Bernard and
Vastano (1977) to study'the interaction of a plane Gaussian pulse with
the Hawaiian Islands. The square grid cells were 5.5 kilometers on a
side and close to the minimum feasible (because of computer time and
cost limitations) size for a rectangular cell finite difference grid
covering the major islands of Hawaii. Yet historical data indicate that
significant variations of tsunami elevations occur over distances much
less than 5.5 kilometers. The islands of Hawaii are relatively small
and very poorly represented by a 5.5 kilometer grid. For example,
Oahu has a diameter of only approximately 30 kilometers and the land
water boundary of the island has characteristic direction changes
that occur over distances much less than 5.5 kilometers. The off
shore bathymetery of the islands also varies rapidly, with depth changes
of more than 1500 meters frequently occurring in distances of 5.5 kil
ometers. Furthermore, if a resolution of eight grid cells per tsunami
wavelength is maintained for tsunami periods as low as 15 minutes, a
5.5 kilometer grid cannot be used to propagate waves into depths less
than approximately 150 meters. The processes that cause significant
wave modifications and subsequent rapid variations of elevations along
the coastline (that are known from historical observations to occur
during tsunami activity in the Hawaiian Islands) probably occur within
this region extending from water at a depth of 150 meters to the
shoreline.
In the present study, a finite element numerical model is used to
propagate tsunamis from the deep ocean to the shoreline of the Hawaiian
Islands. Since finite element techniques allow dramatic changes in
element sizes and shapes, this model has great advantages relative to
finite difference numerical models in accurately representing land
shapes, ocean bathymetry, and tsunami waveforms. Element size can be
large in deep water where bathymetric variations are gradual and wave
lengths are long. As a wave enters shallow water its length decreases
and the elements of the grid can be telescoped to smaller sizes with no
accompanying loss of resolution.
Governing Equations
Typical tsunamis have lengths much greater than the water depths
over which they propagate; for such waves, fluid motions are
approximately twodimensional, i.e., vertical fluid velocities and
accelerations are small in comparison with horizontal velocities and
accelerations, respectively.
Consider a small element of fluid as shown in Figure 1. The rate
of increase of volume of the element is given by
S{n+h) dxdy = dxdy (2.1)
where,
n = a freesurface elevation,
h = the water depth,
t = time,
and (x,y) are Cartesian coordinates.
7,4
dy x
Figure 1. Element of Fluid
In Equation (2.1), it is assumed that h=h(x,y), i.e. h has no time
dependence. The net rate of volume flux is given in difference form by
{u(n+h)} x + dx dy {u(n+h)}xdy
+ (v(n+h)}y+dy dx {v(n+h) }dx
+ vn+h)}I + 0(dx dy2
= dxdy { u(n+h) + y v
= dxdy V q(n+h) (2.2)
where u,v are horizontal velocity components, w is the vertical
velocity component (w=0 here), and q = (u,v,w) is a velocity vector
(a Taylor expansion is used in Equation (2.2) and terms of order
dx2dy2 have been dropped). Equating the rate of increase of volume to
the rate of volume flux yields the following continuity equation:
S+ q (Tl+h) = 0 (2.3)
The governing momentum equation for incompressible and constant
viscosity fluid flow is the NavierStokes equation in the form
2q V, V2(
p t + q V = pF Vp + q (2.4)
where
p = the fluid pressure,
p = the mass density,
p = the dynamic viscosity constant,
I = an external force vector.
Since vertical fluid velocities and accelerations are small and
assuming that gravity is the only external force and the fluid is in
viscid so that the last term of Equation (2.4) can be dropped, Equation
(2.4) becomes in component form
uu u + u 1 u2.
xu + v .. (2.5)
wt X Sy p x
__ yv av I arp
S+ u + v Ip (2.6)
3t 3x Uy p S
0 = g (2.7)
p 1(
WIhen integrated, Equation (2.7) yields (with atmospheric pressure
equal to zero)
p = pg(Sn) (2.8)
i.e., the pressure is hydrostatic, and
Vp = pgVn (2.9)
Equations (2.5) and (2.6) can be recombined to yield
t + (q V)q = gn (2.10)
Since along most of the path of propagation of a tsunami the
wave amplitude is much less than the water depth, Equation (2.3) can be
approximated as
+ V (hq) = 0 (2.11)
It can also be shown that the second term in Equation (2.10) is
generally less than the other terms by a factor of the order of n/h
Therefore, for n/h small, Equation (2.10) is approximated by
>
aq
= gVn (2.12)
Taking a time derivative of Equation (2.12) and substituting Equation
(2.11) into the righthand side of the subsequent equation yields
2>
t =2 gV(V (hq)) (2.13)
dt
Equation (2.13) is known as the linear long wave equation. It is a
governing equation for small amplitude long waves in an inviscid and
incompressible variable depth fluid and is valid when the assumptions
made in its derivation remain valid.
Assuming that the fluid is irrotational q can be expressed
as a gradient of a scalar function t(x,y,t) known as the
velocity potential, i.e,
q= V (2.14)
Assuming all motions to be simple harmonic in time yields
5(x,yt) = Re [(x,y) e t] (2.15)
where Re [ ] indicates the real part of the braced quantity, and a
is the frequency of the motion. Substituting Equations (2.14) and
(2.15) into (2.13) yields
2
7 (h7V) + 0 = 0 (2.16)
Equation (2.16) is the well known generalized Helmholtz equation.
The wave amplitude n can be related to the velocity potential D
by evaluating the Bernoulli equation at the free surface. The Bernoulli
equation is given by
p g7 + V = 0 (2.17)
however; the last term in Equation (2.17) is generally smaller than the
others by a factor of the order of n/h and can be neglected. Also,
taking p=0 at the free surface where =n Bernoulli's equation
becomes to the same order of approximation as Equation (2.16)
n = a (2.18)
2 = 0
Several assumptions are made in the derivation of Equations (2.16)
and (2.18) and the adequacy of these assumptions for the particular
problem of tsunami interaction with the Hawaiian Islands must be con
sidered. An important factor in this consideration is the fact that the
continental shelf of the Hawaiian Islands is exceptionally short.
Thus, during most of the time of tsunami propagation, fluid velocities
are much less than wave propagation velocities since wave amplitudes
are much less than water depths and bottom stress forces as well as
nonlinearities have little time to cause significant effects.
Ursell (1953) and Hammack (1972) have shown that nonlinear effects
grow with time. However, for a typical large tsunami, such as the 1964
Alaskan tsunami; Hammack and Segur (1977) have shown that neither non
linearity nor frequency dispersion have any significant effect on the
lead wave as it propagates across the ocean. In addition, when the lead
wave reaches a continental shelf a certain time must pass before non
linearity or frequency dispersion become significant. For a typical
large tsunami, Hammack and Segur (1977) show that linear and nondis
persive theory holds for the lead wave over a distance of approximately
2h
a ( a is a wave amplitude). Taking the average depth of the region
from the edge of the continental shelf (600 foot depth) to shore as 300
feet and the average amplitude as 3 feet (the amplitude becomes large
2h2
only near shore) yields  = 12 miles. This distance is much greater
a
than the length of the continental shelf of the Hawaiian Islands
(usually less than 1 nile). Since the lead wave is followed by waves
which have been reflected by land areas in the generation region, linear
and nondispersive theory should be adequate, at least for the major
waves of a tsuna i.
Finite Element Model
Figure 2 shows the problem of tsunami interaction with islands
with the process divided into two domains. One domain (A) includes the
islands and the surrounding variable depth region. The second domain
(B) is a constant depth region extending to infinity. This constant
/ BOUNDARY B
(AT INFINITY)
/
/
/
/
/
\
K
REGION B
Figure 2. Regions of Computation.
depth region is realistic since the Pacific Ocean has a remarkably
constant depth beyond the immediate region of the Hawaiian Islands.
The boundary value problem expressed in ter s of these domains becomes
the following:
2
V (, (hV) + = 0 in Region A (2.19)
g
2
V (VI) + O = 0 in Region B (2.20)
SA = B on 3A (2.21)
(h nA ) (h )B on .A (2.22)
A A
= 0 on 3C = 3C1+3C2.. .3C (2.23)
1/2 a
lim (r)112 (yik)%S = 0 (2.24)
r >
where
j = number of land masses,
r = a spherical coordinate,
k = the wave number,
S = the scattered wave velocity potential,
i = (1)1/2
Equations (2.21) an4 (2.22) express the continuity of the velocity
potential and its deriva:ive along the boundary separating the two
domains and Equation (2.23) expresses the impermeability of solid bound
aries. Equation (2.24) is the Sommerfeld radiation condition that
requires the scattered wave to be an outgoing wave at infinity.
A calculus of variations approach can be used to obtain an Euler
Lagrange formulation of boundary value problems. This variational
formulation is based upon the principle that of all possible displace
ment configurations a body can assume that satisfy compatibility and
the constraints of cinematic boundary conditions, the configuration
satisfying equilibrium makes the potential energy a minimum. The Euler
Lagrange formulation of boundary value problems has often been used in
classical dynamics (Hamilton's principle of least action) and solid
mechanics (structural, solid, and rock mechanics). Of course, it is not
possible to obtain analytical solutions for many engineering problems.
The finite element method is a numerical method first introduced by
Turner et al. (1956) (although some of the underlying ideas were dis
cussed by Courant (1943)) which has found wide application in solid
mechanics for problems based upon an EulerLagrange formulation involving
complex material properties or boundary conditions (see Zienkiewicz,
1971 and Desai and Abel, 1972). The finite element method is a discrete
approximation procedure applicable whenever a variational principle can
be formed. This method has a very short history in fluid mechanic
applications despite advantages over conventional numerical schemes such
as finite difference methods. The first application of the finite
element method to fluid mechanic problems involved a study by
Zienkiewicz and Cheung (1966) of seepage through porous media.
Chen and Mei (1974) studied the problem of the forced oscillation
of a small basin protected by a breakwater and containing floating
nuclear power plants by using the finite element method. They dis
cretized a small constant depth region using approximately 250 triangular
elements that were all about the same size and shape. Since small ampli
tude long waves were used as a forcing function, Equation (2.20) was the
governing equation (the water depth being constant in their application)
In the present study an approach similar to that of Chen and Mei is
used to solve the problem of tsunami interaction with islands in a
variable depth region.
For the finite element method to be useful for tsunami problems, it
is necessary to take advantage of two important properties of 'this
algorithm. First, the finite element grid which discretizes a domain
can have grid cells (elements) of arbitrary size and shape. Thus, the
grid can telescope from a large cell size in one section of a variable
depth domain to a very fine cell size in another. Secondly, if the
form of the spatial variation of a physical parameter is known a priori,
the finite element method can employ a very small number of elements in
the discretization used to estimate the exact variation. For example, a
single finite element can be used to solve the problem of the seiching
of a lake for a constant depth rectangular lake if the basic form
(unknown coefficients) of the solution is known. The finite element
method degenerates into the wellknown RayleighRitz variational method
in this case. This property of the finite element method will be im
portant in modeling the infinite region B shown in Figure 2.
The variational principle for the boundary value problem given
by Equations (2.19) through (2.24) requires a certain functional
F(4) to be stationary with respect to arbitrary first variation of .
The first variation of the functional F(4) for this problem is well
known and can be expressed as
2 ,6
,F (j) f.[ (hVi) + C ] ; h; a B
S/[(I A h @1 DB A C A h 'A (2.25
A B 'O 
The functional can he derived from this expression for its first vari
ation (Chen and Mei, 1974) and is
2 2 13 5 )
F(,) = fI [h(VD) + h' ) 3n
A A 2 I mA
Sa( BI) a I
h, i h 
MA A 3A A A
+ / ho (2.26)
A D nA
where 0 is the velocity potential of the incident waves (forcing
function).
The finite element method (unlike the finte difference method)
discretizes the domain itself rather than the governing equations.
The region under consideration is partitioned into small regions known
as finite elements and the assemblage of all such elements represents
the original region. Instead of solving the problem for the entire body
in one operation, the solutions are formulated for each constituent
element and combined to obtain the solution for the original region.
Thus, for example, the first term of Equation :2.26) is evaluated on the
element level and the results for all elements are summed. This sum is
equivalent to the original integral over the complete region A.
In order to evaluate the first term in Equation (2.26) on the
element level, it is necessary to approximate the field variable 6
within each element by a polynomial N called a trial or interpolation
function, in the coordinates x and y
For each element
o = {N} {C) (2.27)
where
{0} = (2.28)
is a column vector with ., 0j, k0 the unknown values of the field
3 k
variable at the nodal points (vertices) of a triangular element and
{N} = [N.,N ,Nk] (2.29)
is a row vector with
N. = (a. + bix + c.y)/26ri=1,2,3) (2.30)
ai = xYj xjyk (2.31)
bi = y j (2.32)
ci = xk (2.33)
and A is the area of the element. The coordinates of the nodal point
i are (xi,yi) and the N. and N' are similarly expressed by the
cyclical permutation of i, j, and k .
A linear interpolation function can be used for tsunamis propa
gating within region A since elements within this region will always be
much smaller than local wavelengths (long waves are adequately repre
sented as linear variations over such short distances). The convergence
of the solution is assured as the elements are made infinitesimal
(Strang and Fix, 1973). Since each element has its own polynomial
expansion independent of all other elements, the treatment of the entire
domain is systematically handled by surmming the contributions from each
element.
The expression g 1 in Equation '2.26) is the scattered wave
velocity potential %S Since region B has a constant depth, :S can
be solved analytically. The governing equation for this case is the
Ilelmholtz equation (Equation (2.20)). The boundary condition at in
finity requires the scattered wave obey the Sonr.erfeld radiation
condition given by Equation (2.24). The ellkr.nown solution to this
boundary value problem is as follows:
s = Zi H(kr) (a cos I: sin ) (2.34)
s m=0 m m n
m=0
whore
11 are Hankel functions of the first kind of order m,
m
a and P are constant coefficients,
m m
and 8 is a spherical coordinate.
In the finite element formulation, the infinite region B can
be considered to be a single element with the interpolation function
given by Equation (2.34). Thus, within region A the interpolation
functions are linear, but within region B the interpolation function
is an infinite series involving unknown coefficients. The terms in
Equation (2.34) are oscillatory but have a ronotonically decreasing
modulation; hence, the series can be truncated after a finite number of
terms depending upon the desired accuracy (see next section).
The required stationarity of the functional F(P) with respect to
arbitrary variations of P at each nodal point yields a set of linear
algebraic equations when expressed on the element level that can be
solved using matrix methods. For each specified incident wave period,
the velocity potential 0 can be determined at all nodal points. The
linearized dynamic Free surface boundary condition (Equation (2.18)) can
then be used to calculate the surface elevation r .
The set of linear algebraic equations evolving from the element
level can be written in the simple matrix forn
[K] {4} = (Q} (2.35)
where [K] is the large coefficient matrix, {f} is a column vector
representing a combination of nodal unknowns for the velocity potential
and the coefficient unknowns from the single element covering region B,
and {Q} is the total load column vector related to the forcing function.
In the finite element solution of structural mechanics problems, one
obtains a similar equilibrium relation with [K] being a stiffness matrix,
{)} a nodal displacement vector, and {Q) a nodal force vector (Desai
and Abel, 1972). The coefficient matrix [K] is symmetric, banded, and
complex (complex notation was used to represent waveforns, e.g., the
Hankel functions are complex). For the case of tsunami interaction with
the Hawaiian Islands, [K] will be exceptionally large and sparse (many
zero terms). Of course, only half of the banded part of the matrix
alone needs to be manipulated in the solution process. The banded part
of the matrix [K] is solved using standard Gaussian elimination methods.
The matrix solution used by Chen and Mei (1974) was modified in this
study to take advantage of the sparseness of the matrix in order to re
duce computational time. Also, the elimination method was modified so
that calculations involved only small blocks of terms at a time with the
remainder of the matrix kept in peripheral storage. This matrix parti
tioning was necessary for the finite element grid shown in the next
section since the banded part of the matrix had '.er Sno,000 terms.
Partitioning is possible because the syi.r.eric eSfficient matrix is
positive definite; hence, a solution is possible b.; elimination methods
without pivoting. Without pivoting, cli,.inatior. erforned using one
row, affects only the triangle of elements ithin the band below that
row. Thus, only two triangular submatrices neecde to be stored in
computer memory at a given time.
Numerical Grid
The finite element grid covering the variable depth region A
is shown in Figure 3. Elements of the grid telescope from large sizes
in the deep ocean to triangles with areas as snall as 0.5 square
kilometers in shallow coastal waters. The grid covers an area of
approximately 450,000 square kilometers. In general, finite element
techniques allow elements to be any arbitrary shape (e.g. quadrilateral)
provided that the ratio of the lengths of the shortest and longest
sides of an element is not extreme. Triangular shapes were used
for convenience in the present study.
The geometric shapes of the eight islands co.prising the Hawaiian
Island chain and the rapid bathynetric variations surrounding the islands
are modeled very precisely by the finite element grid of Figure 3.
Also, the number of node points along the shorelines of the islands
is very dense. Since wave heights are calculated at node points by
the finite element model, the model can adequately represent the rapid
waveheight variations along coastlines that are known to occur
during tsunami activity in the Hawaiian Islands.
Figure 3. Finite Eleiment Grid.
The finite element grid was drawn by hand with the element sizes
dictated by the local water depth. The lengths of element sides were
allowed to be no greater than oneeighth of the local wavelength for the
shortest period waves considered. Such a cell size results in a wave
resolution error of no more than one percent ('ei 1978) There are
computer programs (Cole and Rjech, 1976) that generate finite element
grids and number the nodes to minimize bandwidths; however, such pro
grams only can be applied to fairly homogeneous domains. The rapid
depth variations in region A require a rapidly telescoping grid that is
too complicated for automatic grid generation techniques.
Each node of the finite element grid must be numbered and this
numbering is extremely important since it determines the bandwidth of
the large coefficient matrix discussed in the last section. The band
width of this matrix is controlled by the largest difference (among all
elements) in the nodal numbers between two nodes of the same element.
The computational ti:e of the numerical model is proportional to the
cube of the bandwidth of the coefficient matrix.
For a simple region covered by homogeneous elements, the bandwidth
is minimized by numbering nodal points in cyclic sequence following a
spiraling pattern :here one spiral is shielded by the following spiral
from a third spiral. The grid pictured in Figure 3, however, is so
large and complicated that considerable judgement and effort is involved
in determining a numbering pattern that minimizes the bandwidth, thus
resulting in reasonable computational times for the numerical model.
The numerical model also requires the spatial coordinates of all
node points and information concerning how the elements are connected
relative to each other (connectiveness). ]he coordinates were deter
mined using an electronic digitizer and a simple computer program
developed to relate the internal coordinate system of the digitizer
to a prototype coordinate system through coordinate rotation and
scaling. The elements are numbered in an arbitrary fashion and then
connectiveness information assembled for the numerical model by listing
the three nodes of each element.
The nodal numbering, digitizing, and connectiveness assembly
necessarily introduce errors. For example, the connectiveness assembly
involves determining over 17,000 numbers and placing them on computer
cards. Many connectiveness mistakes do not result in error conditions
which terminate the numerical model's computations. Instead, incorrect
answers are produced by the model. To eliminate all errors in the
input data, computer plotting programs were developed to draw the
grid and number nodes and elements based upon the connectiveness and
node coordinate information used as input to the inuerical model. Such
plots make errors readily apparent.
The number of terms retained in the truncation of Equation (2.34)
was determined by trial and error. The final number of terms was such
that addition of further terns produced negligible effects. It was
found that this occurs for terms approximately 105 times smaller than
the initial terms.
The computational time requirements of the finite element model for
the grid shown in Figure 3 are very modest, making it economically
feasible to determine the interaction of arbitrary tsunamis with the
Hawaiian Islands (discussed in next section). The reason that the
computationtal time required by the finite element model is reasonably
small is that the grid uses small elements only in areas where they
are necessary. The grid of Figure 3 has approximately 2500 points,
whereas the finite difference grid of Bernard and Vastano (1977) had
26,000 grid points. Even so, some of the elements of Figure 3 are as
much as sixty times smaller than the finite difference grid cells used
by Bernard and Vastano.
Model Verification
The finite element model is verified in the present study by
comparing numerical simulations of the 1960 Chilean and 1964 Alaskan
tsunamis with tide gage recordings of these tsunamis in the Hawaiian
Islands. These two tsunamis are the only major tsunamis for which
reliable information exists concerning characteristics of the ground
uplift which generated the waves (much more information exists for the
Alaskan source).
A deep ocean recording of a tsunami near the Hawaiian Islands
has never been .sad. Therefore, prototype wave records in the deep
ocean of the 1960 Chilean and 1964 Alaskan tsunamis are not available
for use as input to verify the finite element model. However, a finite
difference numerical model which solves the linear long wave equations
employing a spherical coordinate grid has been developed (Hiwang et al.,
1972), and used in several studies (Houston and Garcia, 1974; Houston
and Garcia, 1975; and Houston et al., 1975b) to generate tsunamis and
propagate them across the deep ocean. This deep ocean numerical model
is used in this study to determine deep ocean waveforms of the 1960 and
1964 tsunamis that were needed to verify the finite element model.
The deepocean numerical model simulates the uplift deformation of
the ocean water surface caused by the permanent vertical displacement of
the ocean bottom during an earthquake and the subsequent propagation of
the resulting tsunami across the deep ocean. The permanent deformation
(permanent in the sense that the time scale associated with it is much
longer than the period of the tsunami) and not the transient movements
within the timehistory of the ground motion is considered to be the
important parameter governing farfield wave characteristics. Tran
sient movements occur for a period of the order of tens of seconds,
whereas, tsunami wave periods are of the order of tens of minutes.
Experimental investigations of tsunami generation by !lammack (1972) show
that the transient ground movements do not influence farfield char
acteristics of resulting tsunamis for spatially large ground displace
ments occurring over a short period of time.
Hwang et al. (1972) verified the deep ocean model by simulating
the 1964 Alaskan tsunami. The permanent ground deformation in the
source region was taken from Plafker (1969) and a 1/40 by 1/6 grid was
used for the simulation. Since this grid was much too coarse to allow
accurate modeling of shallowwater propagation, a comparison was made
between the numerical model calculations and a gage recording of the
1964 tsunami at Wake Island where Van Dorn (1970) had a pressure trans
ducer in relatively deep water (800 foot depth) some distance offshore.
The amplitude and length of the first wave of this tsunami were shown
to be in good agreement. (Actually, the grid used by Hwang et al.
(1972), did not extend as far as Wake Island. The deep ocean model
propagated the tsunami to a point in the Pacific Ocean and then simple
refraction techniques were used to estimate the change in the waveform
during propagation from this point to ;ake Island.)
The simulation of the 1960 and 1964 tsunamis performed in this
study uses a 1/30 by 1/30 grid. Prior simulations of the 1964 tsunami
using both a 1/4 by 1/60 grid and a 1/3' by 1/30 grid demonstrated
negligible differences between use of the two grid spacings (Houston et
al., 1975a). The permanent ground deformation used as input is also
taken from Plafker (1969). The numerical grid was large enough to cover
a substantial portion of the Pacific Ocean including Wake Island. The
numerical model calculations at W'ake Island were found to be similar to
those presented by Hwang et al. (1972). Figure 4 shows a comparison of
the numerical model calculations of this study and the Wake Island
recording of the 1964 tsunami. The lack of agreement in the trailing
wave region may be due to local effects (e.g. wave trapping by the Wake
Island seamount) or reflections from the nearby transmission boundaries
of the numerical grid (these boundaries are not perfectly transmitting).
Figures 5 and 6 show timehistories of the 1960 Chilean and 1964
Alaskan tsunamis in deep water near the Hawaiian Islands calculated
using the deep ocean propagation model. The permanent ground defor
mation in the source region of the Chilean earthquake of 1960 is taken
from Plafker and Savage (1970). Calculations for the Alaskan tsunami
involve a grid with approximately 30,000 cells and the calculations for
the Chilean tsunami a grid with approximately 90,000 cells.
The finite element numerical model is a timeharmonic solution of
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the boundary value problem. The response of a groupp of islands to an
arbitrary tsunami can easily be determined within the framework of a
linearized theory using the theory of superposition. For example, an
arbitrary tsunami in the deep ocean can be Fourier decomposed as
follows:
b (t) = i b(a) ei(t+p[l)] d. (2.36)
where
b = incident wave amplitude
b(w) = amplitude of frequency component L
p(m) = phase angle
If n(x,y,u) is the response amplitude at any point (x,y) along the
island coasts due to an incident plane wave of unit amplitude and
frequency w then the response of the islands to the arbitrary tsunami
time history b (t) is given by
S(x,y,t) = Re [ fb(j)n(x,vy,w) eit' ) )d,] (2.37)
Therefore, when n(x,y,') is known for all u the island response to
an arbitrary tsunami can be calculated. Of course, it is not feasible
to calculate the integrals of Equations (2.36) and (2.37) over all
frequencies. Instead, the frequency range must be discretized and the
integrals replaced by sums over a frequency range containing most of the
energy of the tsunami.
Equation (2.36) involves a Fourier decomposition of a time series.
This decomposition is accomplished for the timehistories of the 1960
Chilean and 1964 Alaskan tsunamis (Figures S and 6) using a least
squares harmonic fitting procedure. The timehistory of the Alaskan
tsunami in deep water was decomposed into 18 components with periods
ranging from 14.5 minutes to the time length of the record (260
minutes). The variance of the residual (difference between the actual
record and a recomposition of the 18 components) was approximately
0.2 percent of the variance of the record. Therefore, virtually all the
energy of the wave record was contained in the 18 components. The
timehistory of the Chilean tsunami in deep water was decomposed into
11 components with periods ranging from 15.5 minutes to the time length
of the record (170 minutes). The variance of the residual was less than
0.1 percent of the variance of the record. For both cases, the original
timehistory and a timehistory constructed from a recomposition of the
components were virtually indistinguishable.
Equations (2.36) and (2.37) take the following form when discre
tized:
m (2.3
b (t) i b(n ) e )} (2.38)
boft) = I b(,n)
n=]
(x,y,t) = Re b(mn) n(x,yn) ei nt+p(n}] (2.39)
n=l
where
n = number of components,
th
a = frequency of n component
The b(n ) tern is determined by the leastsquares harmonic
fitting procedure and n(x,y,Yn) is determined by the finite element
numerical model for each frequency n and at each location (x,y).
Therefore, the timehistory E(x,y,t), which represents the response of
location (x,y) to the deepwater tsunami timehistory given by bo(t),
can be calculated for any location along the coastline of the Hawaiian
Islands.
Figures 7, S, and 9 present comparisons between tide gage re
cordings and numerical model calculations of the 196 Alaskan tsunami at
Kahului, Maui; Honolulu, Oahu; and Hilo, Hawaii, respectively. The
largest waves recorded at each of these sites are shown. Waves
arriving at later times are all much smaller than those shown. When
ever tide gage limits were encountered, the recordings were linearly
extended (the tide gage locations are shown in Figure 10).
The wave records shown in these figures are in remarkable agree
ment, especially considering the fact that the ground displacement of
the 1964 earthquake was not precisely known. Since the Hilo breakwater
was not included in the numerical model, the numerical model calcula
tions for Hilo are probably too large; however, this breakwater was
undoubtedly highly permeable to the 1964 tsunami. The numerical results
appear to be too large in Hilo by some constant factor, since the tide
gage recording and the numerical model calculations have the same form
with the first wave crest and trough having approximately the same
amplitude and being proportionately smaller than the second crest.
Figure 11 shows a comparison between the tide gage recording of the
1960 Chilean tsunami at Honolulu and the numerical model calculations.
Again the largest waves recorded are shown (the Honolulu gage was the
only tide gage in the Hawaiian Islands not destroyed by the 1960
tsunami). The permanent ground motion of the 1960 earthquake is not
known nearly as well as the ground motion for the 1964 earthquake.
~
; // 1 \\
I 1
I1 \
GAGE LIMIT GAGE LIIT
'\ i 'I
7 I I
S
4(
8.45 9:00 10:00
APPROXIMATE HOURS GREENWICH MEAN TIME
Figure 7. 1964 Tsina'mi from \Aaska Recorded at Kahului, Mui
B:45 9:00 10:00
APPROXIMATE HOURS GREENWICH MEAN TIME
Figure 8. 1964 Tsunami from Alaska Rcorded at .Ionolulu, Oabu.
U 0
1
Z
l'
2
3
4
LEGEND
S  TIDE GAGE RECORD
5
SI NUMERICAL MODEL
CALCULATIONS
6
7
e
9:00 10.00
APPROXIMATE HOURS GREENWICH MEAN TIME
Figure 9. 1961 Tsunami from Alaska Recorded at iilo, Hawaii.
HILO
HAWAII
1:igirc 10. (;;e Lo locations.
S
i '33NV1SIC
N N NO m
I I
Hence, the agreement indicated in Figure 11 is quite good considering
the uncertainty of the uplift in the source region.
Tide gages, of course, do not record tsunamis perfectly since they
are nonlinear devices with responses which depend upon both the period
and amplitude of the disturbances they measure. However, simple
calculations based upon the paper of Noye (1970) show that the distortion
is small for the tsunamis shown in Figures 7 through 9 and Figure 11.
Figure 12 shows the response amplitude as a function of wave period
at the locations of tide gages in Kahului, Honolulu, and Hilo. The
response versus wave period is greatly dependent upon the precise
location of recording for bays such as Hilo. For example, the Hilo
response (Figure 12) at the tide gage location shown in Figure 10 has
a single large resonant peak at 28.9 minutes. This peak occurs at the
fundamental period of oscillation of Hilo Bay and, consequently, all
locations in the bay have a resonant peak at this incident wave period
(the amplitude of the peak depends on location in the bay). Other
locations in Hilo Bay also have significant peaks at lower periods.
Loomis (1970) calculated a free oscillation fundamental period (node
assumed at bay mouth) of 26.2 minutes for Hilo Bay using a finite
difference grid spacing of approximately onehalf a kilometer.
Bernard and Vastano (1977) show a contour nap of a normalized
energy parameter versus wave period and location around the island of
Hawaii. The map shows Hilo displaying an energy peak from 10 to 16
minutes centered at approximately 12.5 minutes; no significant peak at
28.9 minutes is observed. The difference between Bernard and Vastano's
calculations and the results presented in this paper is attributed to
KAHULUI TIDE GAGE
HONOLULU TIDE GAGE
HILO TIDE GAGE
10 20 30 40 50 60
PERIOD,MIN
Figure 12. Amplification Fnctor Versus WI'ave Period.
the poor resolution of the spatial features of Hilo Bay by their finite
difference grid spacing of 5.5 kilometers. HilD Bay has the shape of a
right triangle with sides that are approximately 9 kilometers and 13.5
kilometers long. The grid used by Bernard and '.astano represented Hilo
Bay as a right triangle with sides that were 5.3 kilometers and 11
kilometers long; thus, the surface area of Hilo Bay was approximately
one half its actual area. This distortion of the size of Hilo Bay and
also, perhaps, the distortion of the bathymetry of the bay by the coarse
grid accounts for the lower response periods for Hilo Bay presented by
Bernard and Vastano.
The importance of accurately representing island shapes and off
shore bathymetry and resolving tsunami wavelengths can be illustrated by
considering a study by Adams (1975) that used the numerical model
developed by Bernard and Vastano (197") to interpolate between his
torical measurements of tsunami elevations prod:ceI by the 1946 Aleutian
tsunami. A single Gaussian wave crest was used as input for the
numerical model. If a resolution of four grid cells per crest wave
length is maintained, this numerical model cannot propagate waves into
water with a depth less than about 500 feet. The main processes in
volved in the transformation of the form of a tsunami probably occur in
the region over the continental shelf (depths less than 600 feet).
Consequently, the elevations calculated by Ada:s were small and did not
display the great variation along coastlines known to occur during
historical tsunamis in the Hawaiian Islands.
Adams included the effects of propagation over continental shelves
by multiplying the numerical model elevations by a factor (different
for each island) that forced the best agreement with the historical
412
data. However, even with artifact, agreement with historical data was
poor. For example, on the northeast coast of Oahu, the numerical model
elevations multiplied by the factor resulted in predicted elevations
ranging from 14 to 21.4I feet. Loomis (1976) presents 36 historical
elevations for the 19i6 Aleutian tsunami on the northeast coast of Oahu
and only 7 even fall within this range of elevations. The historical
elevations actually varied from 1 to 37 feet on this coast.
CHAPTER III
TSUNAMI INTERACTION 1'ITH CONTINEN_AL COASTLINES
Introduction
Experimental investigations of long wave interactions with
continental coastlines have generally been oredimensional and have
involved idealized conditions. Flume tests by Savage (1958) and
Saville (1956) measured runup of periodic wavere on constant slope
beaches ad 'all and Watts (1953), Kaplan (1953), and Kishi and
Saeki (1966) measured runup of solitary waves on constant slope
beaches. All of these flume experiments used waves much steeper
than typical tsunanis due to the difficulties involved in the
generation of small amplitude long waves.
Many theoretical studies of onedimensional propagation of
weakly dispersive long waves have been perfored in recent years.
Tsunamis become weakly dispersive (amplitude and frequency dispersive)
long waves at so.e point during their propagation, although (as was
the case for the Hawaiian Islands) they .ay not be weakly dispersive
for a time period sufficiently long for related effects to develop.
The proper equations for the investigation of weakly dispersive
long wave propagation over a steeply sloping bottom were first
derived by Mei and LeMehaute (1966) and in a slightly different
form by Peregrine (1967) who also gave the first numerical solution
in the case of a solitary wave using an implicit finite difference
scheme. A more complete solution was later given by Madsen and Mel
(1969) who transformed the equations to characteristic form before
they were solved numerically. Svendsen (1974) studied weakly dis
persive wave propagation over a gently sloping bottom. Hammack
(1972) made comparisons of long wave generation by bottom displace
ments in the laboratory and numerical solutions of the Korteweg
DeVries equation using the numerical approach of Peregrine (1966).
Several studies of onedimensional tsunami propagation over
continental shelves of constant slope have been performed. Keller
and Keller (1964) solved nonlinear long wave equations numerically
for a constant slope bathymetry. Takahasi (1964) studied tsunami
propagation over a simple bathymetry through analytic solutions of
the linear long wave equation. Carrier (1966) assumed a tsunami
was generated by a point source and studied nonlinear propagation
over a constant slope bathymetry. Heitner and Housner (1970) used
a finite element numerical method to solve nonlinear equations for
propagation up constant slope beaches. More recently, Mader (1974)
studied tsunami propagation over linear slopes using a finite
difference solution of the complete NavierStokes equations (not
vertically integrated).
Occasional twodimensional studies have been performed for
tsunami propagation in nearshore regions. Grace (1969) studied
tsunamis affecting the reef runway of the Honolulu International
Airport using a hydraulic model. However, the wavemakers used were
only a small fraction of a tsunami wavelength from the region to be
studied and waves reflected by land were reflected again almost
immediately by the wavemakers. This is a common problem in hydraulic
model tests involving long waves. The wavemakers can be moved a
few wavelengths from the region of interest by reducing the model
scale; however, scale effects may then become very significant.
Wave absorber screens may also be placed in front of the wavemakers;
however, it is very difficult to absorb very long waves and such an
approach is often not very helpful. Similar problems with re
reflected waves can occur in numerical studies. The calculations
of Hwang and Divoky (1975) were in poor agreement with the tide gage
recording of the 1964 tsunami at Hilo not only for the reasons
presented in Chapter II, but also because their algorithm allowed
waves to leave the numerical grid efficiently only for waves approach
ing the input boundary normally. Since Hilo Bay is triangularly
shaped, the reflected waves approached the input boundary at various
angles and were partially rereflected toward Hilo.
Governing Equations
The governing equations for long wave propagation over the
continental shelf of the west coast of the United States are Equations
(2.3) and (2.4). Thus, the long wave assumption is made once again
with vertical velocities and accelerations neglected relative to
horizontal components. The continuity equation is given by the
equation
+ V q(n + h) = 0 (3.1)
and the momentum equation is
p t + q v = pF Vp +V (3.2)
If the only external force is gravity and the pressure is hydro
static, then Equation (3.2) becomes
p + q Vq = pgVn + Iv~q (3.3)
It
The west coast of the United States has a much broader continental
shelf than that of the Hawaiian Islands; thus, the nonlinear advective
and bottom stress effects may be more significant, and the nonlinear
and viscous terms of Equation (3.3) must be retained.
The dynamic viscosity p in the viscous term of Equation (3.3)
is replaced by an eddy viscosity constant E for the case of
turbulent flow. Using this approach the viscous term can be expressed
as
2t 2 + 2
Ix ly Iz
where the coefficients s and E2 represent coefficients of
horizontal and vertical eddy viscosity, respectively. The term
7 2
Sx 3y
2
is usually small in comparison with the E 0 term (Dronkers,
z2 2
1964) and will be neglected in this study. If C1 s
az
integrated with respect to z from the bottom to the surface
and divided by the distance from bottom to surface n + h we find
c2 au (3.5)
n+h Ia z=h s z =
The terms e2J zh and c2( ) are components of the
SsZ z=h n Iz 2 zZ=T
tangential stresses at the sea bottom and surface, respectively.
The shear stress at the sea surface is quite important for storm
surge problems where wind velocities are very large but can be
neglected relative to the bottom stress for tsunamis.
From experiments and theoretical considerations for one
dimensional flow in a river, it has been shown (Dronkers, 1964), that
the bottom stress Tb can be expressed as follows
T= pguul (3.6)
b 2
where C is Chezy's coefficient and is given by the expression
C = 8 h1/6 (3.7)
n
where n is Manning's n Thus, Equation (3.3) may be written
>+ 2 2 1/2
p Vq = pgV pgq(uv) (3.8)
C (n+h)
Equation (3.8) contains the nonlinear advective term but neglects
frequency dispersion since vertical accelerations have been neglected.
This equation adequately describes wave propagation for Ursell
numbers much greater than unity (Ursell, 1953). The Ursell number
is proportional to the ratio of nonlinear effects to linear effects
n2
(i.e., frequency dispersion) and is equal to where L is a
1/2 h
typical wavelength. Since L=(gh) 12T for long waves, where T is
2
the wave period, an Ursell number may be chosen as gn()2
The numerical model described in the next section uses Equation
(3.8) to propagate waves from depths of 500 meters to shore. These
waves have periods of the order of 1800 seconds and initial have
amplitudes of the order of a foot. 'Thus, the Ursell number, gn( )2
is approximately 30 and increases as the wave enters shallow water
with h decreasing and n increasing.
Mader (1974) found that the onedimensional form of Equation
(3.8), without the viscous term, produced very similar results to a
onedimensional NavierStokes equation (viscosity neglected) for
propagation of long wavelength tsunamis over linear continental
slopes. The pressure term in Mader's solution was not hydrostatic;
thus, frequency dispersion was permitted.
Finite Difference Model
To solve Equations (3.1) and (3.8) numerically, the differential
equations are discretized and replaced by a system of finite difference
equations using central differences on a spacestaggered grid. The
spacestaggered scheme describes velocities, water levels, and depths
at different grid points. This scheme, first used by Platzman (1958)
for lake surge problems, has the advantage that in the equation for the
variable operated upon in time, there is a centrally located spatial
derivative for the linear term. Leendertse's (1967) implicitexplicit
multioperational method is employed in determining the solution
for q and n as functions of time. The stability and convergence
of this method is described in detail by Leendertse (1967). This
method for solving finitedifference equations has been applied in
studies analyzing the tidal hydraulics of harbors and inlets (Raney,
1976). Houston and Garcia (1975) used the method to calculate the
decay of a tsunami as it enters and spreads throughout San Francisco
Bay.
The nearshore finitedifference numerical model uses the time
history calculated by the generation and deep ocean propagation
model described in Chapter II as input in the following manner. A
tsunami is generated in the AleutianAlaskan area or the west coast
of South America and propagated across the deep ocean to a 500
meter depth off the west coast of the United States. Waveforms
calculated at this depth by the deep ocean numerical model are
recorded all along the west coast. These waveforms are then used
as input to the nearshore numerical model, which propagates the
tsunamis from the 500meter depths across the continental slope and
shelf to shore.
Figure 13 shows outlines of the four numerical grids used to
cover the west coast. The grids have square grid cells two miles
on a side (southern California and some of San Francisco Bay were
not covered by numerical grids because elevation predictions were
made in previous studies (Houston and Garcia, 1974; and Houston and
Garcia, 1975) using simple onedimensional analytical solutions for
propagation over the continental shelf). The offshore bathymetry
was modeled from the 500meter contour to shore. Beyond the 500
meter contour, the ocean was assumed to have a constant 500meter
depth. The input boundary of each grid was located approximately
one and onehalf wavelengths of a 30minute wave from the shore.
Therefore, at least three typical waves could arrive at the shore
before waves reflected from the input boundary became a problem.
OREGON
CALIFORNIA
SCALE IN MILES
50 0 50 100
BAJA
CALIFORNIA
Figure 13. Finite Differcnce Grid L.ocations.
The input boundaries of the grids were oriented approximately
parallel to the shoreline since refraction will bend the wavefronts
to such an orientation before they reach the 500meter contour
(Wilson and Torum, 1968). Lateral boundaries of the grids were
taken to be impermeable vertical walls.
Model Verification
The finite difference nearshore numerical model was also veri
fied by numerical simulations of the 1964 Alaskan tsunami. The
deep ocean propagation model discussed in Chapter II was used to
generate the 1964 tsunami and propagate it across the deep ocean.
Figure 14 shows surface elevation contours 31/2 hours after the
1964 Alaskan earthquake, as calculated by the deepocean propagation
numerical model. This figure illustrates the concentration of energy
on the northern California, Oregon, and Washington coasts due to the
directional radiation of energy from the source region. The upper
section of Figure 15 shows a time history of the 1964 Alaskan tsunami
calculated by the deerocean model near Crescent City, California, in
a water depth of 500 meters. This waveform was used as input to the
finite difference nearshore model.
The lower section of Figure 15 shows a comparison between a tide
gage recording of the 1964 tsunami at Crescent City and the nearshore
model simulation of this tsunami. The numerical model calculations
agree well with the tide gage recording; periods, phases, and
amplitudes are accurately reproduced. The main disagreement occurs
L
o 0
\I ,e '
d P L0
i ~I
! 6
/'i /
N Li"
~ / / L
/ I 2i ,
U) 1 6 C~i
O D 4
1.!i
Z IS/ N
Liii
I, IL "
:1 L! 
'I."':~ Li
J I IL o
o U) U)
U) 0 0 ~ L'S
Un o
Uo t0
I
I 2.0
0
 1.0
O
z 0
 i.o1I
> 1.0
j 0 20 40 60 BO
l TIME, PROTOTYPE MIN
100 120
PROTOTYPE GAGE RECORD
INFERRED PROTOTYPE GAGE
RECORD (FROM WILSON, 1968)
NUMERICAL SIMULATION
8 9 10
APPROX TIME, HRS, GMT
11 12
Figure 15. 1964 Tsunami from Alaska Recorded at Crescent City, California
for the amplitude of the third wave. With regard to this disagree
ment, it should be noted that there is also a lack of agreement
on the amplitudes of the actual historical third and fourth wave
crests. Figure 15 shows a reconstruction of the 1964 tsunami at
Crescent City inferred by Wilson and Torum (196S) from the prototype
gage record and later survey measurements by Magoon (1965). The
elevation of the fourth wave (not shown) was estimated by Wilson
and Torum to have been a little less than 14 feet above mean lower
low water (MLLW) (the zero elevation shown in Figure 15 is MLLW
datum); however, Wiegel (1965) estimated that the elevation of the
third wave was approximately 16 feet above MLL' and that the fourth
wave attained the highest elevation of 18 or 19 feet above MLLW at the
tide gage location. The numerical model predicted a fourth wave
elevation approximately the same as that estimated by Wilson and
Torum (14 feet above MLLIW). However, reflections from the input
boundary were probably growing in importance in the numerical model
calculations during the arrival of the fourth wave.
There may be several reasons why the maximum wave elevation
predicted by the numerical model does not agree with the historical
record as well as the elevations of the initial waves. First, the
later waves in the deepwater waveform are probably waves which have
been reflected from land areas in the source region. The fairly
large grid cell spacing used in the deep ocean propagation model
to represent the landwater boundary in this region probably distorts
the reflected waveform. Total reflection of waves in this region
also is not completely realistic. Furthermore, the nearshore grid may
not have fine enough cells to completely model the resonance
phenomenon leading to the large third or fourth wave.
Since 1964, there has been considerable speculation concerning
the reasons that the effects of the 1964 tsunami were so great at
Crescent City. .The finite difference numerical models show that
both the directional radiation of energy from the source region
(see Figure 14) and a local resonance caused the relatively large
elevations at Crescent City. Actually, large elevations at Crescent
City were not unique since directional radiation of energy from the
source also caused large elevations along the Oregon and northern
California coasts. Runup 10 to 15 feet above the high tide level
occurred all along the Oregon coast south of the Columbia River
(runup at Crescent City was approximately 15 feet above the high
tide level). However, the Oregon coast is very sparsely populated
and there were few damage reports. The severity of structural
damage at Crescent City, which has a large logging industry, also
was apparently due to the impact of logs carried by the tsunami
(Wilson and Torum, 1968). The nearshore numerical model indicates
that the resonant effects at Crescent City were fairly local,
extending over 2 to 4 miles of coastline. This behavior contradicts
Wilson and Torum's (1968) speculation that a bowlshaped section of
the continental shelf with a diameter of approximately 50 miles
experienced a resonant oscillation. Historical data support the
numerical model calculations. At the mouth of the Klamath River,
approximately 15 miles south of Crescent City, elevations observed
during the 1964 tsunami were only 2 to 3 feet above normal high
tide. Wilson and Torum's hypothesis of a shelf oscillation would
predict elevations at the mouth of the Klamath River greater than
those that occurred at Crescent City.
Figure 16 shows a comparison between a tide gage recording of
the 1964 tsunami at Avila Beach, California, and the nearshore
numerical model calculations. The elevations recorded by the Avila
Beach tide gage were larger than those recorded at any tide gage on
the west coast except the Crescent City gage. The historical and
calculated waveforms shown in Figure 16 are in good general agreement.
The tide gage record obviously has higher frequency components which
are not predicted by the numerical model; these components may be local
oscillations of water areas which were too small to be accurately
represented by the numerical grid. Important features such as the wave
amplitudes are in good agreement.
The waveforms recorded for the 1964 tsunami by the Crescent
City and Avila Beach tide gages have elevations larger than those
recorded by any other gage on the west coast. Other tide gage
recordings of this tsunami on the west coast (except at Astoria,
Oregon) were in areas not covered by the grids shown in Figure 13.
A comparison with the tide gage at Astoria, Oregon, was not made
because Astoria is approximately 12 miles away from the coast in
the estuary of the Columbia River, and amplitudes of the 1964
tsunami were small at Astoria.
8
z 2
>0
Lh
J
2
4 GAGE   LIMIT
I LEGEND
6 \ PROTOTYPE GAGE RECORD
"J  NUMERICAL SIMULATION
8 I I
8.5 9 9.5 10 10.5 11
APPROX. TIME, HRS., GMT.
Fitwre 16. 196i Tsunami from Alaska Recordcd at Availa Beach,
Cal ifornia.
CHAPTER IV
ELEVATION PREDICTIONS
Hawaiian Island Predictions
Introduction
The Hawaiian Islands, a chain of eight islands as shown in
Figure 17, have a history of destructive tsunamis generated both in
distant areas and locally. The earliest recording of a severe tsunami
in the Hawaiian Islands was in 1837 when a tsunami from Chile reached
an elevation of 20 feet at Hilo, Hawaii, and killed 46 people in the
Kau District of the big island of Hawaii. Prior to 1837, a number
of severe tsunamis undoubtedly reached the islands but, unfortu
nately no detailed records were kept. Since 1837, there have been 16
tsunamis that have caused significant damage.
Most of the destructive tsunamis in the Hawaiian Islands have
been generated along the coast of South America, the Aleutian Islands,
and the Kanchatkan Peninsula of the Soviet Union. Approximately one
fourth of all the tsunamis recorded in the Hawaiian Islands have
originated along the coast of South America, while more than one
half have originated in the KurilKamchatkaAleutian region of the
north and northwestern Pacific. Tsunamis generated by local seismic
events have caused large runup in the islands, especially on the
southeast coast of the big island of Hawaii. The 1868 tsunami
produced the largest waves of record in the Hawaiian Islands with
A' *
/99 /5 < \ S
:9 :90
,191
/ ? ^
':9
11 4
a ,g g
A. j 
.9;, :9  :
I i h^ I5
:sZ1 I \
^~ ~ Ar
r >= <0
:: 'CI
^h r n\ I
'"
s ;*
:9 'I
z s
.sJA.
4 99'
N i '* 5
? I ^
S J 2 o
0 < .9;
^______________________ ___________)
60foot waves reported on the South Puna coast of the island of
Hawaii. The most recent major tsunami in Hawaiian history occurred
on November 29, 1975, when waves generated by an earthquake with
an epicenter on the South Puna Coast may have reached elevations as
great as 45 feet along the southeast coast. The most destructive
tsunami to ever hit the islands in terms of both loss of life and
property destruction was the Great Aleutian tsunami of 1946, which
killed 173 people and produced waves over 55 feet in elevation. Hilo
incurred S26 million in property damage attributable to this tsunami.
The 1960 Chilean tsunami is the most recent distantly generated
tsunami that produced major effects in the Hawaiian Islands. Sixty
one lives, all at Hilo, were claimed by the tsunami. Damage through
out the state was estimated to be $23.5 million. Inspection of the
damage at Hilo revealed much evidence of the tremendous forces
developed by the waves. Twentyton boulders had been moved hundreds
of feet, asphaltic concrete pavements were peeled from their sub
base, and hundreds of automobiles were moved and crushed.
The following sections discuss the methods used to obtain
sufficient data to predict runup elevations for the entire coastline
of the Hawaiian Islands. The finite element numerical model discussed
in Chapter II is used to interpolate between local historical data
for the years of accurate survey measurements from 1946 through 1977.
Such a method of interpolation is needed so that elevations can be
predicted for the entire coastline even for locations having no
actual historical data. A method is discussed herein which allows
reconstruction of historical data for the period of time from 1837 to
1946 when elevations for the period from 1946 through 1977 are known.
Such an approach is necessary in order to have sufficient data for
accurate elevation predictions since it is shown herein that reason
able predictions cannot be made solely based upon known historical
data for the short time span from 1946 through 1977.
Interpolation of Recent Historical Data
Historical data of tsunami activity in the Hawaiian Islands
are, of course, often limited to certain locations. Information on
tsunami activity in the islands prior to the 1946 tsunami is concentrated
in Ililo (Hawaii) and to a lesser extent in Honolulu (Oahu) and
Kahului and Lahaina (Maui). Even data for tsunamis occurring during
the period of accurate survey measurements from 1946 through the
present are absent or fragmentary (i.e., data exist only for certain
of the events) for most of the coastline of the islands. Therefore,
it is necessary to rely on more than just available historical data
to determine tsunami occurrence frequencies.
The numerical model is used to fill in historical data gaps for
tsunamis from 1946 through 1977 by providing relative responses of
the Hawaiian Islands to tsunamis. For example, although the deepwater
waveform of a tsunami such as that of 1946 is not known, the direction
of approach of this tsunami and its range of wave periods are known.
The average response of the islands to a tsunami similar to the 1946
tsunami is determined by inputting sinusoidal waves of unit amplitude
from the direction of the 1946 tsunami into the numerical model over
a band of wave periods. If historical data exist at one location,
wave elevations at a nearby location for which historical data do
not exist are determined by multiplying the historical data at the
first location by the ratio of the response calculated by the numer
ical model at the second location and the response calculated at the
first. The numerical model takes into account the major processes
that would cause different relative wave elevations at the two
locations. That is, the model calculates shoaling, refraction,
diffraction, reflection, resonance, shielding of the backside of an
island by the front side, and interactions between islands. The
historical data account for the unknown absolute heights of the
deepwater tsunamis. The "correction factors" provided by the
historical data are local instead of global (single factor for all
islands) because there are certain local factors that cannot be
determined by the numerical model. For example, coral reefs extend over
parts of some of the islands and are known to protect against tsunami
attack (e.g., the extensive reefs in Kaneohe Bay, Oahu, which are
powerful tsunami dissipators). The effects of the reefs on tsunanis
are implicitly contained in the historical data itself. The historical
data from 1946 through the present were taken from the most recent
and complete compilation of these data (Loomis, 1976).
As an example of the use of the numerical model to fill in
historical data gaps for tsunamis from 1946 through the present,
consider the 1946 tsunami recorded on the island of Lanai only at
Kaumalapua Harbor and Manele Bay. Wave elevations can be calculated
for the 1946 tsunami at other locations on the coastline of Lanai by
multiplying these historical values by the ratios of the responses
calculated by the numerical model at locations of interest and the
responses calculated at Kaumalapua Harbor and Manele Bay. In this
way, elevations for the 1946 tsunami can be determined for all of
Lanai.
Time Period Analysis
It is necessary to use historical data of tsunami occurrence
covering the greatest possible time span to properly determine
frequency of occurrence of tsunami elevations for the Hawaiian
Islands. Tsunami activity has not been uniform in these islands
during recorded history. For example, the two largest and four of
the ten largest tsunamis striking Hilo from 1837 through 1977 occurred
during the 15year period from 1946 through 1960. Two of the tsunamis
from 1946 through 1960 originated in the Aleutian Islands, one in
Kamchatka and one in Chile. However, six of the ten largest tsunamis
occurred during the 109year period from 1837 to 1945 with three
originating in Chile, two in Kamchatka and one in Hawaii. Therefore,
both the frequency of occurrence and place of origin of tsunamis
have been remarkably variable. Any study (e.g., Towill Corporation,
1975) basing frequency calculations on a short time span that includes
the period from 1946 through 1960 will predict a significantly more
frequent occurrence of large tsunamis than is warranted by historical
data from 1837 through 1977.
The errors introduced in frequency of occurrence calculations
by consideration only of a short time period that includes the
unrepresentative years from 1946 through 1960 will be greater than
the errors resulting from possible observational inaccuracies of the
19th Century. This is easily demonstrated for Hilo, Hawaii, which
is the location in the Hawaiian Islands having the most complete
data of tsunami activity. Using the data compiled by Cox (1961) for
the ten largest tsunamis in Hilo from 1837 through 1977 (his data
were actually through 1964, but no large tsunamis have occurred in
Hilo since 1964), a leastsquares fit of the data was made employing
the logarithmic frequency distribution discussed later in this
chapter. The 100year tsunami elevation for Hilo calculated using
this method was 27.3 feet. If the elevations recorded during the
nineteenth century (five of the ten largest tsunamis), are increased
by 50 percent (which is much larger than possible observational
error), the 100year elevation is found to be 30.4 feet. Similar
calculations just based upon data taken during the period of accurate
survey measurements in Hilo from 1946 through 1977 yields a 100year
elevation of 44.2 feet. Since the largest elevation in Cox's data
for the 141year period from 1I37 through 1977 was just 28 feet, the
100year elevation of 44.2 feet is obviously too large. Similar
overestimates will occur at any location in the Hawaiian Islands if
an analysis is based up3n a short period of time that includes the
years 1946 through 1964 since the exceptionally frequent occurrence
of major tsunamis in Hilo from 1946 through 1964 is a property of
the unusual activity of tsunami generation areas and not of special
properties of Hilo.
The unusual tsunamigenic activity in generation regions in
recent years is reflected in the most uptodate catalog of tsunami
occurrence in the circumpacific area complied by Soloviev and Go
(1969). This catalog shows, for example, that the three greatest
intensity tsunamis generated in the AleutianAlaskan region since
1788 occurred in 1946, 1957, and 1964. The catalog also lists the
1960 Chilean tsunami as the greatest intensity tsunami generated in
South America in recorded history (116year period from 1562 through
1977). Reporting of tsunamis on the west coast of South America is
quite good with,105 tsunamis reported during the 416year period.
Kanamori (1977) also notes the unusual seismic activity that occurred
during a relatively short time span in recent years. He shows that
the four earthquakes having the greatest seismic moments (true
measure of magnitude for great earthquakes) in this century occurred
during a thirteen year period from 1952 through 1964 (1960 Chilean,
1964 Alaskan, 1957 Aleutian, and 1952 namchatkan earthquakes).
Evidence that tsunamis were unusually active in the Hawaiian
Islands from 1946 through 1964 is also apparent from historical data
at Kahului and Lahaina, Maui. Kahului, Lahaina, luilo, and Honolulu
are the only locations in the islands with historical data extending
to 1837. Four of the five largest tsunami elevations recorded at
Kahului occurred during the years from 1946 through 1964. Three of
the four largest tsunami elevations recorded at Lahaina occurred
during the same period. The historical data for Honolulu is not
very useful because the commonly reported elevations for the 1837
and 1841 tsunamis were, in fact, drops in water level (Pararas
Carayannis, 1977) and there are no reports for the 1868 and 1877
tsunamis from Chile.
Reconstruction of Historical Data
In order to have data covering a sufficiently long time span to
make reasonable predictions of tsunami elevations for the Hawaiian
Islands, it is necessary to reconstruct historical data for the
years prior to 1946. To do this, it is assumed in this study that
tsunamis generated in a single source region (Kamchatka or Chile but
not the AleutianAlaskan region) approach the islands from approximately
the same direction and have energy lying within the same band of
wave periods. The difference in wave elevations at the shoreline in
the Hawaiian Islands produced by tsunamis generated at different
times in the same region is attributed mainly to differences in
deepwater wave amplitudes. For example, the 1841 tsunami from
Kamchatka produced a wave elevation in lilo that was approximately
25 percent greater than that of the 1952 tsunami from Kamchatka. It
is assumed that these two tsunamis had the same relative magnitudes
throughout the islands. Since the finite element model allows a
determination of elevations for the 1952 tsunami along the entire
coastline of the islands, the elevations of the 1S41 tsunami and all
ether Kamchatkan tsunamis occurring during the period of record in
Hilo from 1837 to 1946 can be reconstructed for the entire coastline.
Similarly, Chilean tsunami elevations can be reconstructed knowing
elevations for the 1960 Chilean tsunami. No major tsunamis were
generated in the AleutianAlaskan region from 1837 through 1946; hence,
no reconstruction is necessary for tsunamis from this region.
Fortunately, tsunamis from this region since 1946 were representative
of the entire region since these tsunamis were generated in the
western Aleutians (1957), central Aleutians (1946) and eastern
Alaskan area (1964).
The assumption that tsunamis generated in a single source
region (Kamchatka or Chile) approach the Hawaiian Islands from
nearly the same direction is justified in the case of Kamchatka by
the small spatial extent of the known generation area. However, the
tsunamigenic area of Chile is considerably larger than that of
Kamchatka. Still, the direction of approach of different tsunamis
from Chile is nearly the same because the tsunamigenic region in
Chile subtends a relatively small angle with respect to the islands
as a result of the great distance between Chile and the Hawaiian
Islands and the axis of the Chile Trench maintains a constant orienta
tion relative to the islands (thus directional radiation effects are
not important).
The assumption that tsunamis generated in a single source region
have energy lying within the same band of wave periods is supported
by the elevation patterns produced along coastlines in the Hawaiian
Islands by different historical tsunamis. These patterns are such
that it has been concluded that "tsunamis of diverse geographic
origin are strikingly different, whereas those from nearly the same
origin are remarkably similar" (Eaton, et al., 1961). A recent
study by Wybro (1977) shows that even the distributions of normalized
elevations (i.e. the elevation patterns) produced in the Hawaiian
Islands by different AleutianAlaskan tsunamis are nearly the same
yet quite different front the distributions for tsunamis of other
origins. This agreement occurs despite the fact that the relatively
close proximity to Hawaii of the AleutianAlaskan Trench and the
varying orientation of the trench axis relative to Hawaii introduces
important directional effects for tsunamis generated in the Aleutian
Alaskan area. Apparently, these directional effects influence the
magnitude of the elevations in Hawaii but do not greatly alter ele
vation patterns. Therefore, historical observations support the
approach used in this study which estimates the elevations produced
by tsunamis from Chile or Kamchatka prior to 1946 based upon data
for tsunamis from these tsunamigenic regions recorded during the
years of accurate survey measurements from 1946 to 1977.
In this study elevations are calculated along the coastline
of the Hawaiian Islands for the ten tsunamis which produced the
greatest elevations in Hilo, Hawaii, from 1837 through 1977. The
finite element numerical model fills in all data gaps for those tsunamis
of the ten that occurred from 1946 through 1977 and then the Hilo
data (Cox, 1964) is used to determine relative magnitudes to reconstruct
the elevations for those tsunamis that occurred prior to 1946. For
much of the coast of the Hawaiian Islands, the ten largest tsunamis
since 1837 would be the same as the ten largest tsunamis in Hilo for
the same period. That is, the ten largest would be the tsunamis of
1960, 1946, 1923, 1837, 1877, 1841, 1957, 1952, and the two in 1868.
Of course, the order of the ten largest would vary from location to
location. For example, the largest tsunami on the South Puna coast
of the big island of Hawaii was the locally generated tsunami of
1868.
There are a few locations where historical data show that one
of the ten largest tsunamis was not among the ten largest in Hilo
from 1837 through 1977. For example, the 1896 tsunami generated
near Japan produced small elevations in the Hawaiian Islands except
at Keauhou on the Kona coast of the big island of Hawaii. The
observation at Keauhou was, therefore, included in the frequency of
occurrence analysis for that location. Similarly, all other histor
ical observations of significant tsunami elevations at isolated
locations were included in the frequency of occurrence analyses for
those locations,. Historical data for tsunamis prior to 1946 were
taken from PararasCarayannis (1977). Data for the local tsunami of
November 29, 1975, were taken from the compilation by Loomis (1976).
Since the local tsunami of 1868 had on earthquake epicenter very
near that of the 1975 tsunami and the two apparently produced very
similar elevation patterns (Tilling et al., 1976), historical ele
vations for the two tsunamis recorded at the same location were used to
determine a relative magnitude of the tsunamis. The many observations
for the 1975 tsunami were then used to reconstruct elevations for the
1868 tsunami.
Frequency of Occurrence Distribution
After the ten largest tsunamis from 1837 through 1977 were
determined at locations all along the coasts of the Hawaiian Islands
using the methods discussed in the previous sections, elevation
versus frequency of occurrence curves were determined at each location
by leastsquares fitting of the data by the following expression:
h = B A log0 f (4.1)
where
h = elevation of maximum wave at the shoreline,
f = frequency per year of occurrence.
Equation (4.1) was used as the frequency of occurrence distribution
since it has been found to agree with historical data at several
locations. Cox (1964) found that the logarithm of tsunami frequency
of occurrence was linearly related to tsunami elevations for the ten
largest tsunamis occurring from 1837 through 1964 at Hilo, Hawaii.
Soloviev (1969) .has shown a similar relationship between tsunami
frequency of generation and intensity for moderate to large tsunamis.
Also, earthquake frequency of occurrence and magnitude have been
similarly related by Gutenberg and Richter (1965); hiegel (1965)
found the same relationship for historical tsunamis at San Francisco
and Crescent City, California and Adams (1970) for tsunamis at Kahuku
Point, Oahu. A recent study by Rascon and Villarreal (1975) revealed
the same relationship for historical tsunamis on the west coast of
America, excluding Mexico.
It is possible that other distributions may agree with the
historical data equally well as the logarithmic distribution. For
example, the Gunbel distribution has been used in the past to study
annual streamflow extremes (Gumbel, 1955). Borgman and Resio (1977)
illustrate the use of this distribution to determine frequency curves
for nonannual events in wave climatology. To investigate the sensi
tivity of calculations of 100year elevations on the assumed frequency
distribution, the approach of Borgman and Resio was applied to the
Hilo data of Cox. The Gumbel distribution yields a 100year elevation
of 28.8 feet for data from 1837 through 1977 and an elevation of 42.5
feet for data from 1946 through 1977. This compares with the elevations
of 27.3 feet and 44.2 feet calculated for the same time periods using
the logarithmic distribution (Equation (4.1)). Clearly, the arguments
used earlier concerning the period of time that must be considered for
a valid analysis are not dependent upon the assumed frequency distri
bution. Since ample precedent exists for using the logarithmic distri
bution for analyzing tsunami frequency of occurrence, this distribution
is used in this stud)'.
Results
The methods described in the previous sections allow frequency of
occurrence curves to be constructed for points all along the coast
line of the Hawaiian Islands. The coefficients A and B of Equation
(4.1) contain all necessary information to determine frequency of
occurrence of tsunami elevations at a location. Plots of the coeffi
cients A and B versus location along the coasts were constructed for
all of the islands. Smooth curves were drawn through these coeffi
cients to allow elevation predictions at any location. Figures 18
ard 19 show typical plots of A and B for the southeastern coast of
the big island of Hawaii (Figure 20). For example, at Hoopuloa
(location 11 in Figure 20) A = 6 and B = 4; hence, the 100year
elevation (F = .01) is only 8 feet. The 100year elevations for
most locations in the Hawaiian Islands are considerably larger.
For such situations involving human life, one is less interested
in the mean exceeda:e frequency f (the average frequency per year
of tsunamis of equal or greater elevations) than in the chance of a
given elevation being exceeded in a certain period of time. To
calculate risk, one first notes that tsunamis are usually caused by
earthquakes, and earthquakes are often idealized as a generalized
Poisson process (Newmark and Rosenblueth, 1971 and Der Kiureghian
and Ang, 1977). It was assumed by Wiegel (1965) and Rascon and
72
m
C~
rj
CD
0
% W O7 CD
SS31INOISN3MG~ V
 ) N,
N
m
I N
4  T ~
ii ,U.
0
 
o o u N c o
rc
BAY
S9399
PEE PEE
P A A
HAWA I I
20 NAPOOPOO
Figure 20. Location Map for !awaii.
PT
Villarreal (1975) that tsunamis also follow such a stochastic process.
The probability that a tsunami with an average frequency of occur
rence of f is exceeded in D years, assuming that tsunamis follow
a Poisson process, is given by the following equation:
P = 1 e (4.2)
Thus, if an acceptable risk is one chance in one thousand that an
elevation be exceeded during a 100year period, P = .001 and
D = 100 years. Substituting these values in Equation (4.2) yields
the elevation h which has one chance in one thousand of being
exceeded during the next 100 years.
West Coast of the United States Predictions
Introduction
Unlike the Hawaiian Islands, the west coast of the continental
United States lacks sufficient data to allow tsunami elevation
predictions based upon local historical records of tsunami activity.
Virtually all of the west coast is completely without data of tsunami
occurrence, even for the prominent 1964 tsunami. Only a handful of
locations have historical data for tsunamis other than the 1964 tsunami.
However, the Federal Insurance Administration requires information on
tsunami elevations for the entire west coast of the continental United
States; even for the many locations that have no known historical
data of tsunami activity and for coastal areas that are currently
not developed (since these areas may be developed in the future).
The lack of historical data of tsunami activity on the west coast
necessitates the use of numerical models to predict runup eleva
tions. The AleutianAlaskai area and the west coast of South America
were found by Houston and Garcia (1974) to be the tsunamigenic
regions of concern to the west coast of the United States. Both
regions have sufficient data on the generation of major tsunamis to
allow a statistical investigation of tsunami generation. The genera
tion and deepocean propagation model described in Chapter III can
then be used to generate representative tsunamis and propagate them
across the deep ocean. The nearshore numerical model also described
in Chapter III, can then be used to propagate tsunamis from the deep
ocean over the continental slope and shelf to shore.
In this study, only tsunamis of distant origin are considered
in the analysis. Hammack (1972) has shown that near the generation
region of an impulsively generated tsunami, the waveform is dependent
upon details of the timedependent movement of the ground during the
earthquake. Little is known about the actual timedependent ground
motion during earthquakes generating tsunamis and this motion cannot
be predicted in advance. Also, there is not enough historical data
concerning locally generated tsunamis on the west coast to allow
predictions of tsunami occurrence. Thus, reasonable predictions on
the west coast of the properties of locally generated tsunamis or
their likelihood of occurrence are not possible at this time.
The probability is not considered very great that a destructive,
locally generated tsunami will occur on the west coast of the conti
nental United States. Tsunamis are generally produced by earthquakes
having fault movements that exhibit a pronounced "dipslip," or
vertical component of motion. "Strikeslip," or horizontal displace
ment, fault movements are inefficient generators of tsunamis.
Faults on the west coast of the United States characteristically
exhibit "strikeslip" motion since the Pacific block of the earth's
crust is moving horizontally relative to the North American block.
The west coast of the United States does not share the characteristics
(ocean trenches and island arcs) of knowngenerating areas and, in
fact, has not historically been one. Relatively small locally
generated tsunamis have been known to occur on the west coast, but
there are no reliable reports of major locally generated tsunamis.
Heights of 6 feet in the immediate vicinity of the 1927 Point
Arguello earthquake are the largest authenticated heights produced
by local tsunamis on the west coast). There could be a few locations
on the west coast for which locally generated tsunamis pose a greater
hazard than do distantly generated tsunamis because the elevations
produced by distantly generated tsunamis are small. However, pre
dictions of elevations produced by locally generated tsunamis are
beyond the scope of this study.
Tsunami Occurrence Probabilities
Historical data of tsunami generation must be the basis for an
analysis that considers the probability of tsunami generation in the
two tsunamigenic regions of the Pacific Ocean of concern to the west
coast of the continental United Statesthe Aleutian and PeruChile
Trench regions. A satisfactory correlation between earthquake
magnitude and tsunami intensity has never been demonstrated. Not
all large earthquakes occurring in the ocean even generate noticeable
tsunamis. Furthermore, earthquake parameters of importance to
tsunami generation, such as focal depth, rise time, and vertical
ground displacement, have only been measured for earthquakes occurring
in recent years. Therefore, data of earthquake occurrence cannot be
used to determine occurrence probabilities of tsunamis. Instead,
historical data of tsunami occurrence in generation regions must be
used to determine these probabilities.
In South America, a wealth of information exists concerning
tsunami generation. Reliable data (grouped in intensity increments
of onehalf) exist for tsunamis with intensity greater than or equal
to 0 for a 171year period and greater than or equal to 21/2 for a
416year period (Soloviev and Go, 1969). The intensity scale used
is a modification (by Soloviev and Go) of the standard Imamuralida
tsunami intensity. Intensity is defined as
S I= log2(2 Ha) (4.3)
This definition in terms of an average runup (in meters) over a
coast instead of a maximum runup elevation at a single location
(used for the standard ImamuraIida scale) tends to eliminate any
spurious intensity magnitudes caused by often observed anomalous
responses (due, for example, to local resonances) of single isolated
locations.
Using the most recent and complete catalog of tsunami occurrence.
in the Pacific Ocean (Soloviev and Go, 1969) a relationship between
tsunami intensity and frequency of occurrence was determined for the
tsunamigenic trench running the length of the PeruChile coast.
Tsunamis with intensity greater than or equal to 0 were considered.
It was assumed that the logarithm of the tsunami frequency of occur
rence was linearly related to the tsunami intensity. Earlier in
this chapter, it was shown that such a relationship holds between
both earthquake magnitude and frequency of occurrence and tsunami
intensity and frequency of occurrence.
Letting n(I) equal the probability of a tsunami with an
intensity I being generated during any given year and using
statistics for the entire trench along the PeruChile coast, a
leastsquares analysis results in the following expression:
n(I) = 0.74e0.631 (4.4)
In using statistics for the entire trench area along the PeruChile
coast, it is assumed that the probability of tsunami occurrence is
uniform along the trench. This is a standard assumption for earth
quake frequency analysis (Gutenberg and Richter, 1965). The tectonic
justification of this assumption lies in the fact that a single Sialic
block or plate of the earth's crust or lithosphere is dipping into
the PeruChile Trench (Wilson, 1959). It can reasonably be expected
that the movement of this single plate is similar along its entire
length.
In the Aleutian Trench region, only large tsunamis occurring
in relatively recent years (since 1788) have been recorded due to the
isolation of the area. Assuming an exponential coefficient of 0.71
for this trench area (determined by Soloviev, 1969) as a mean value
for regions of the Pacific with the most data on tsunamis) and using
only the reliable data for large tsunamis (intensity greater than or
equal to 3.5) from Soloviev and Go (1969), the following relationship
is determined by a leastsquares analysis:
n(l) = 0.l e0.711
Again, the probability of tsunami occurrence is assumed to be uniform
along the trench.
Use of Deterministic Numerical Models
To relate the probability distributions of tsunami intensities to
source characteristics, it is assumed that the ratio of the source
uplift heights producing two tsunamis of different intensity (as
defined in the previous section) is equal to the ratio of the average
runup heights produced on the coasts near these tsunami sources. This
ratio is equal to 2 for two tsunamis with intensities I1 and
12
If H is the wave height in the direction parallel to the major
a
axis of length a of a tsunami source with an elliptical shape
(large tsunamis have historically had ellipticallyshaped uplifts)
and Hb is tha wave height in the direction parallel to the minor
axis of length b, then experimental research of tsunami generation
has shown that Hb/Ha is approximately equal to a/b (Hatori,
1963). For a large tsunami, Hb can be larger than H by a factor
b a
of as much as S or 6. Thus, the orientation of the tsunami source
relative to the area where runup is to be determined is very important,
i.e., the runup at a distant site due to the generation of a tsunami
at one location along a trench cannot be considered as being repre
sentative of all possible placements of the tsunami source in the
entire trench region. Hence, the Aleutian and PeruChile Trenches
had to be segmented and runup along the west coast of the United
81
States determined for tsunami sources located at the center of each
of the segments.
The spatial size of a tsunami source was standardized since
there is not an apparent correlation between tsunami intensity and
spatial size of a tsunami source. For example, the 1946 Aleutian
tsunami had an uplift region of very small spatial extent, whereas
the 1957 Aleutian tsunami had an uplift region that covered perhaps
the greatest spatial extent of any known earthquake (Kelleher,
1972). Yet the 1946 tsunami had the greater intensity, producing,
in general, greater runup elevations in the near and distant regions.
The standard source used is discussed in detail by Houston and
Garcia (1974). It is elliptical in plan view with a major axis length
of 600 miles and a minor axis length of 130 miles. The vertical dis
placement increases linearly from a zero elevation on the ellipse
perimeter to a maximum displacement of 30 feet. The vertical dis
placement has a parabolic crest (concave downward) parallel to the
direction of the major axis. The standard source represents a large
tsunami with intensity 4 on the modified Imamuralida scale.
Certainly, tsunamis of low intensity may have smaller spatial extents;
however, large tsunamis pose the greatest threat to a distant area
such as the west coast of the United States. These large tsunamis
can be expected to have similar spatial extents, with any spatial
differences being unimportant in the farfield compared with the effects
of source orientation and vertical uplift. Vertical uplifts are assigned
to different intensities in accordance with the convention discussed
at the beginning of this section.
Figures 21 and 22 show the Aleutian Trench divided into 12 seg
ments and the PeruChile Trench into 3 segments. The segments in the
Aleutian Trench were approximately onequarter he length of the major
axis of the standard source, whereas the segments in the PeruChile
Trench were approximately the length of the major axis of the standard
source. The standard source was centered in each segment such that
the major axis of the source was parallel to the trench axis. Uplift
regions historically have had such an orientation relative to trench
systems. The Aleutian Trench is segmented much finer than the Peru
Chile Trench since the Aleutian Trench is oriented relative to the
west coast such that elevations produced on the west coast are very
sensitive to the exact location of a source along the Trench. Uplifts
along the PeruChile Trench do not radiate energy directly toward the
west coast regardless of their position along the Trench. The Peru and
Chile sections of the FeruChile Trench have constant orientations
relative to the west coast of the United States; therefore, eleva
tions on the west coast are not very sensitive to source location
within these sections.
In each of the segments of the Aleutian and PeruChile Trenches
tsunamis with intensities from 2 to 5 in steps of onehalf intensity
are generated and propagated across the deep ocean using the deep
ocean numerical model discussed in Chapter II. Tsunamis with inten
sity less than 2 are too small to produce significant runup on the
west coast. An upper limit of 5 was chosen because the largest
tsunami intensity ever reported was less than 5 (Soloviev and Go,
1969). Gutenberg and Richter (1965) indicate that there is an upper
[ 
o
O
0 1
H z
1 >
< *4

Jc*
r3
44J
0
o
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6'
ENTURA
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Figure 22. SegmcnLed Chilean Trench.
limit to the strain that car be supporl od by rock before fracture.
Thus, earthquakes only reach certain: s;axJmum inagn i udes and tsunamis
can be expected to have sii ;iar uppi.; limits to intensity. Perkins
(19721 and McGarr (1976) have denor, .fo ':ted thai. future earthquaklce
cannot have seismic moments fmeasuce of earthquake magnitude for
large earthquakes) much larger than hose of earthquakes that occurred
in recorded history.
f'ne waveforms propagated to c.be icst coast by the deepocean
model are used as before as input to Ihe nearshore propagation model.
Each waveform is propagated from a '*: cr depth of 5>00 meters to
shore using the nearslore model and c.o, of the grids shown in FPiure
13. Thus, at each grid location ou the shoreline of the wen.t coast
there is a group of 105 waveforms seven waveforms (for intensities
fro 2 to 5 in onehalf intensity i:nciremerts) for each segment of
the Aleutait'n and PeruChile Trenchi.. Each of ihese waveforms has an
associated probabilityi equal to the 'irrbi)ijty that a certain
intensity tsunami will be generated ;ji a parltcular segment of a
trench region.
Effect of the Astrononical Tides
The maxiun "stillwater" elevation produced during tsunami
activity is the result of a superposition of tsunamis and tides.
Therefore, :he statistical effect of the astronomical tides on
total tsun mi runup must be include i in the predictive scheme pre
sented in this study. Since the ;aveforms calculated by the nearshore
model do not have a simple form (e.g. sinusoidal), the statistical
effect of the astronomical tide or tsunami runup nust be determined
86
through a numerical approach. (The effect of the astronomical tides
was not considered in the section of predictions for the Hawaiian
Islands because the tidal range is quite small for these islands and
the local historical data implicitly contain the effects of the
tides.)
The waveforms calculated by the nearshore numerical model
extend over a period of time of approximately two hours. Three or
four wave crests (the largest waves in the tsunami) arrive during
this time. Smaller waves arriving at later times, however, often
persist for days at a coastal location. An analysis of tide gage
records of the 1960 and 1964 tsunamis on the west coast indicates
that these smaller waves have amplitudes on the average of 40 percent
of the maximum wave amplitude of the tsunami. Therefore, a sinu
soidal group of these smaller waves were added to each of the calculated
waveforms so that the total waveform extended over a 24 hour period.
These smaller waves are important for locations where tsunamis
are fairly small compared with tidal variations. At such locations
the maximum combined tsunami and astronomical elevation occurs
during the maximum tidal elevation.
A computer program was developed to predict timehistories of
the astronomical tides at all grid locations on the west coast of
the United States. The program was based upon the harmonic analysis
methods used in the past by the Coast and Geodetic Survey for mechan
ical tidepredicting machines (Schureman, 1948). Tidal constants
available from the Coast and Geodetic Survey were used as input to
the computer program; and a year of tidal elevations was then predicted
for grid locations all along the west coast. The year 1964 was
w~
selected because all the major tidal components had a node factor
equal to approximately 1.00 during this year making it an average
year (node factors can vary by about 10 percent). The node factor
is associated with the revolution of the moon's node and has an
18.6year cycle. Since a tsunami can arrive at any time during this
18.6year period (arrival at a low of the node factor is equally
likely as an arrival at a high), the statistical effect of the
varying node factor is small and an average value should be used.
The statistical effect of the varying node factor on the predicted
runup elevations can be shown to be a small fraction of an inch,
using the approach discussed in the next section and Appendix A
(with the variance of the nodal variation equal to approximately
0.1 feet squared).
The year of tidal elevations calculated at each of the nearshore
numerical model grid points along the west coast was then subdivided
into 15minute segments. The 24hour waveforms were allowed to
arrive at the beginning of each of these 15minute segments and
then superposed upon the astronomical tide for the 24hour period.
The maximum combined tsunami and astronomical tide elevation over
the 24hour period was determined for tsunamis arriving during each
of these 15minute starting times during a year. All of the maximum
elevations had an associated probability equal to the probability
that a certain intensity tsunami would be generated in a particular
segment of the two trench regions and arrive during a particular 15
minute period of a year.
The many maximum elevations with associated probabilities can
be used to determine cumulative probability distributions of combined
tsunami and astronomical tide elevations. The maximum elevations
are ordered and probabilities summed, starting with the largest
elevations, until a desired probability is obtained. The elevation
encountered when the summed probabilities reach a desired value P
is the elevation that is equaled or exceeded with an average fre
quency of once every 1/P years. Thus, when the summed probabilities
reach the value .01, the elevation associated with the last prob
ability summed is the 100year elevation.
The 100year and 500year elevations are determined at all
grid points of the nearshore numerical grids using techniques des
cribed previously in this chapter. Smooth curves are then used to
connect all discrete elevations so that continuous predictions of
elevation versus coastline location can be made for all the west
coast. Figure 23 shows 100year and 500year elevations in the
Crescent City, California, area.
Comparison with Local Observation Predictions
Crescent City and San Francisco, California, are the only
locations on the portion of the west coast of the United States
considered in this study that have sufficient historical data of
tsunami activity to allow frequency of occurrence predictions based upon
local historical observations. Wiegel (1965) made such predictions of
tsunami height (trough to crest height) for Crescent City based upon
the period from 1900 to 1965. He predicted 100year and 500year
heights at Crescent City of approximately 25.6 feet and 43.2 feet,
respectively. If the crest amplitude is taken to be onehalf the total
height, the 100year and 500year elevations are 12.8 feet and 21.6
feet, respectively. However, the crest amplitude at Crescent City is
typically greater than onehalf the wave height (e.g. the crest ampli
tude of the largest wave of the 1964 tsunami was approximately 60 per
cent of the total height). If Wiegel's analysis is applied to historical
crest elevations instead of heights, the 100year and 500year elevations
are found to be 15.4 and 26.4 feet, respectively. Furthermore, the
analysis can now be applied to the longer time period from 1900 through
1977. The 100year and 500year elevations based upon this longer time
span are 14.5 feet and 25.5 feet respectively.
Figure 23 shows 100year and 500year elevations of 13.1 feet
and 24.9 feet, respectively. These values compare very favorably
with the 14.5 feet and 25.5 feet elevations determined from historical
data for the period of time from 1900 through 1977. The elevations
predicted by the analysis based upon the local historical data are
probably somewhat larger than the elevations predicted in this
report because the short time period (relative to 100 or 500 years)
from 1900 through 1977 includes the exceptionally active years of
tsunami generation from 1946 through 1964 (see Chapter III).
The 13.1foot and 24.9foot elevations predicted for Crescent
City using the techniques discussed in this study are not totally
comparable to the elevations predicted using Wiegel's analysis based
upon local historical data since the effect of the astronomical tide
has been included in the elevations predicted in this study. However,
I' _
I [i K
iiI F\ ii~
I I
Cci
i t" ____ ____ ___
L
b
0
LA
0
4J
0
CD
0
0
L)
0CD
z
'4f Ci
0
I LC
CO
o ci
oo
(
CCC
'ax
0 I
UU
cW c o Nc 0 cO Co N 0 co o
N N N N .
1A 'NOIiVA313
the statistical effect of the astronomical tide on the total ele
vation is not significant at Crescent City due to the large amplitude
of tsunamis there. Appendix A shows that the statistical effect of
the astronomical tide for a location where tsunamis are large is to
2
increase the predicted elevation by an amount equal to 2
where c is the tidal variance and equals E C C is
m m m
m=l
equal to the mth tidal constituent, and a is given by the following
expression (since it was shown earlier in this chapter that elevations
are linearly related to the logarithm of probabilities):
P(Z) = AeaZ
with
Z = the elevation above local mean sea level, and
P(Z) = the cumulative probability distribution for the elevation
at a given site being equal to or exceeding Z due only to the
maximum wave of the tsunami. Based upon the tidal constituents
predicted by the Coast and Geodetic Survey, o = 7.1 for Crescent
City. Using Wiegel's data for the period of time from 1900 through
1977 gives a = .145 Therefore, the astronomical tide contributes
approximately 0.5 feet to the elevations predicted for Crescent
City.
Frequency of occurrence calculations just inside San Francisco
Bay were made in an earlier study (Houston and Garcia, 1975) using a
simple onedimensional analytical solution for nearshore propagation.
However, portions of San Francisco Bay are included in the nearshore
numerical model (see Figure 13) used in this study in order that the
effect of the Bay on elevations outside the Bay can be properly

Full Text 
PAGE 1
TSUNAMI INTERACTION WITH COASTLINES AND ELEVATION PREDICTIONS By JAMES ROBERT HOUSTON A Dissertation Presented to the Graduate Council of The University of Florida In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy UNIVERSITY OF FLORIDA 197S
PAGE 2
ACKNOWLEDGMENTS The writer wishes to express his sincere gratitude to his supervisory committee chairman, Professor Knox Millsaps, and cochairman, Professor Joseph Hammack, for their support and efforts to provide him the opportunity to complete his studies at the University of Florida. A special debt of gratitude is owed to Dr. R. IV. Khalin, Chief of the Wave Dynamics Division, Hydraulics Laboratory, U. S. Army Waterways Experiment Station for his encouragement and active support during the research and preparation periods of this dissertation. The writer wishes to thank Professor limit Unluata for the insight he provided and for organizing a supervisory committee prior to his departure from the University. Appreciation also is extended to Dr. D. L. Durham for many helpful discussions at the Waterways Experiment Station. The research upon which this dissertation is based was funded by the Pacific Ocean Division of the U. S. Corps of Engineers and the Federal Insurance Administration of the Department of Housing and Urban Development through the Office of the Chief of Engineers.
PAGE 3
TABLE OF CONTENTS ACKNOWLEDGMENTS KEY TO SYMBOLS ABSTRACT Page VI 11 CHAPTER I: INTRODUCTION 1 Tsunamis \ Objective and Scope of Study 2 CHAPTER II: TSUNAMI INTERACTION KITH THE HAWAIIAN ISLANDS . 4 Introduction 4 Governing Equations . 7 Finite Element Model 12 Numerical Grid 21 Model Verification 25 CHAPTER III: TSUNAMI INTERACTION WITH CONTINENTAL COASTLINES 43 Introduction . . 43 Governing Equations 45 Finite Difference Model 43 Model Verification 51 CHAPTER IV: ELEVATION PREDICTIONS 58 Hawaiian Island Predictions 58 Introduction 58 Interpolation of Recent Historical Data 61 Time Period Analysis 63 Reconstruction of Historical Data 65 Frequency of Occurrence Distribution 69 Results 71
PAGE 4
Page West Coast of the United States Predictions 75 Introduction 75 Tsunami Occurrence Probabilities 77 Use of Deterministic Numerical Models 80 Effect of the Astronomical Tides 85 Comparison with Local Observation Predictions . . 88 CHAPTER V: CONCLUSIONS 95 APPENDIX: SUPERPOSITION OF LARGE TSUNAMIS AND ASTRONOMICAL TIDES 97 REFERENCES 99 BIOGRAPHICAL SKETCH 105
PAGE 5
KEY TO SYMROLS a Wave amplitude b Incident wave amplitude o t b(co) Amplitude of frequency component u C Chezy's coefficient C Amplitude of m tidal coefficient m r D Period of time f Frequency of occurrence F External force vector F( ) Functional g Acceleration of gravity h Water depth H Average runun, meters avg Hankel function of first kind of order m m i (1J I Tsunami intensity j Integer k Wave number [K] Coefficient matrix L Wavelength m Integer n Unit normal vector n Manning's coefficient
PAGE 6
n( ) Frequency of occurrence N Interpolation function P Pressure > q Velocity vector {Q} Total load vector r Spherical coordinate t Time T Wave period u Velocity component in xdirection v Velocity component in ydirection w Velocity component in zdirection x Cartesian coordinate Ax Small increment in x y Cartesian coordinate Ay Small increment in y z Cartesian coordinate (vertical direction) a Constant a Constant coefficient m 3 Constant coefficient m A Area of element Â£ Coefficient of horizontal eddy viscosity eÂ„ Coefficient of vertical eddy viscosity n Freesurface elevation
PAGE 7
6 Spherical coordinate \\ Dynamic viscosity constant Â£ Response to arbitrary tsunami p Mass density p (u>) Phase angle 2 a Tidal variance t. Bottom stress b $ Velocity potential <Â£> . A Velocity potential m region A $ Velocity potential in region B B $ Velocity potential of incident wave $ Velocity potential of scattered wave to Angular frequency V Gradient operator
PAGE 8
Abstract of Dissertation Presented to the Graduate Council of The University of Florida in Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy TSUNAMI INTERACTION WITH COASTLINES AND ELEVATION PREDICTIONS By James Robert Houston June 197S Chairman: K. Millsaps Cochairman: J. L. Hammack Major Department: Engineering Science The development is described of two numerical models that accurately simulate the propagation of tsunamis to nearshore regions and the interaction of tsunamis with coastlines. One of these models is a finite element model which uses a telescoping numerical grid to cover a section of the Pacific Ocean, including all eight islands of the Hawaiian Islands. The second model is a finite difference scheme that uses four rectilinear grids to cover most of the west coast of the continental United States. The finite element model solves linear and dissipationless longwave equations. Such equations govern nearshore propagation in the Hawaiian Islands since the short continental shelf of the islands limits the time available for nonlinearities and dissipation to cause significant effects. The finite difference model solves nonlinear longwave equations that include bottom stress terms which are important for the long continental shelf of the west
PAGE 9
coast of the United States. Both models are verified by hindcasting actual historical tsunamis and comparing the numerical model calculations with tidegage recordings. A frequency of occurrence analysis of tsunami elevations at the shoreline in the Hawaiian Islands is described. This analysis is based upon local historical data with the finite element model used to interpolate between historical data recorded during the period of accurate survey measurements since 1946. Historical data recorded at Hilo, Hawaii, and dating to 1337 is used in conduction with data recorded since 1946 in Hilo and throughout the Hawaiian Islands to reconstruct elevations at locations in the islands lacking data prior to 1946. Frequency of occurrence curves are determined for the entire coastline of the Hawaiian Islands using these reconstructed elevations. Since most of the west coast of the United States lacks local data of tsunami activity, a frequency of occurrence analysis of tsunami elevations is based upon historical data of tsunami occurrence in tsunamigenic regions in addition to numerical model calculations. A generation and deepocean propagation numerical model is used to propagate tsunamis with varying intensities from locations throughout the AleutianAlaskan and PeruChile regions to a water depth of 500 meters off the west coast. The nearshore finite difference model propagates these tsunamis from the 500 meter depth to shore. The frequency of occurrence of combined tsunami and astronomical tide elevations is determined by an analysis involving the numerical superposition of tsunamis and local tides.
PAGE 10
CHAPTER I INTRODUCTION Tsuna mis Of all water waves that occur in nature, ore of the most destructive is the tsunami. The term "tsunami," originating from the Japanese words "tsu" (harbor) and "nami" (wave) , is used to describe sea waves of seismic origin. Tectonic earthquakes, i.e., earthquakes that cause a deformation of the sea bed, appear to be the principal seismic mechanism responsible for the generation of tsunamis. Coastal and submarine landslides and volcanic eruptions also have triggered tsunamis. Tsunamis are principally generated by undersea earthquakes of the dipslip type with magnitudes greater than 6.5 en the Richter scale and focal depths less than 59 kilometers. They are very long period waves (5 minutes to several hoars) of low height (a few feet or less) when transversing water cf oceanic depth, Consequently, they are not discernible in the deep ocean and go unnoticed by ships. Tsunamis travel at the shallowwater wave celerity equal to the square root of the acceleration due to gravity times water depth even in the deepest ocean; because of their very long wavelengths. This speed of propagation can be in excess of 500 miles per hour in the deep ocean. When tsunamis approach a coastal region where the water depth decreases rapidly, wave refraction, shoaling, and bay or harbor resonance may result in significantly increased wave heights. The great
PAGE 11
period and wavelength of tsunamis preclude their dissipating energy as a breaking surf; instead, they are apt to appear as rapidly rising water levels and only occasionally as bores. Over 500 tsunamis have been reported within recorded history. Virtually all of these tsunamis have occurred in the Pacific Basin. This is because most tsunamis are associated with earthquakes, and most seismic activity beneath the oceans is concentrated in the narrow fault zones adjacent to the great oceanic trench systems that are predominantly confined to the Pacific Ocean. The loss of life and destruction of property due to tsunamis have been immense. The Great Iloei TokaidoNanhaido tsunami of Japan killed 30,000 people in 1707. In 186S, the Great Peru tsunami caused 25,000 deaths and carried the frigate U.S.S. Waterlee 1,500 feet inland. The Great Meiji Sanriku tsunami of 1896 killed 27,122 persons in Japan and washed away over 10,000 houses. In recent times, three tsunamis have caused major destruction in areas of the United States. The Great Aleutian tsunami of 1946 killed 173 persons in Hawaii, where runup heights as great as 55 feet were recorded. The 1960 Chilean tsunami killed 330 people in Chile, 61 in Hawaii, and 199 in distant Japan. The most recent major tsunami to affect the United States, the 1964 Alaskan tsunami, killed 10 7 people in Alaska, 4 in Oregon, and 11 in Crescent City, California, and caused over 100 million dollars in damage on the west coast of North America.
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Objectiv e and Scope of Stud}' The objective of the present study is to determine tsunami elevation frequencies of occurrence at the shoreline for the Hawaiian Islands and the west coast of the continental United States, Such elevation predictions are required by the Pacific Ocean Division of the U. S. Corps of Engineers for use in tsunami flood hazard evaluations for floodplain management and the Federal Insurance Administration of the Department of Housing and Urban Development for flood insurance rate calculations. The current trend of general population migration to coastal regions and the growth of the tourist industry makes it essential to know possible tsunami inundation levels even for coastal regions that are currently undeveloped. There arc only a few isolated locations in the United States that have sufficient local historical data of tsunami activity to allow accurate predictions solely based upon historical data. This lack of historical data of tsunami activity on most of the coastline of the United States makes it necessary to use numerical models in addition to existing historical data to predict tsunami elevation frequency of occurrence. However, numerical models have not been successfully applied in the past to the simulation of actual tsunami nearshore propagation and interaction with coastlines. There fore, two numerical models are developed in the present study to accurately simulate tsunami nearshore activity. Calculations of these deterministic numerical models are combined with probabilistic analyses based upon historical data to predict frequencies of occurrence of tsunami elevations for the Hawaiian Islands and the west coast of the United States.
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CHAPTER II TSUNAMI INTERACTION KITH THE HAWAIIAN ISLANDS Int rodu ction Early studies of tsunami interaction with islands involved analytic investigations of the scattering of monochromatic plane waves by a single island of simple geometric shape. Oner and Hall (1949) determined the theoretical diffraction pattern for longwave scattering by a circular cylinder in water of constant depth by adopting results from acoustic theory. They found qualitative agreement between this diffraction pattern and smoothed runup heights observed for the 1946 tsunami on the island of Kauai in the Hawaiian Islands. Homma (1950) determined the diffraction pattern for long wave scattering off a circular cylinder surrounded by a parabolic bathymetry that extended to a prescribed distance offshore, beyond which the depth was constant. Hydraulic models have been used to study tsunami interaction with single islands that are realistically shaped and surrounded by variable bathymetry. Van Porn (1970) studied tsunami interaction with Viake Island using a 1:57000 undistorted scale model. Jordaan and Adams (1968) studied tsunami interaction with the island of Oahu in the Hawaiian Islands using a 1:20000 undistorted scale model. They found poor agreement between historical measurements of tsunami runup and the hydraulic model data. Scale effects (e.g. viscous effects) and the effects of the arbitrary boundaries that confine the model are 4
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problems that are inherent in hydraulic modeling of tsunamis. The small heights of tsunamis relative to deep ocean depths is also difficult to simulate. Jordaan and Adams [1968) modeled waves to a vertical scale of 1:2000; thus, the waves had heights ten times the normal proportion. Even with this distortion, waves typically had heights in the model of only 0.3 millimeters. Vastano and Reid (1966) studied the problem of the response of a single island to monochromatic plane waves of tsunami period using a finite difference numerical model. A transformation of coordinates allowed a mapping of the island shore as a circle in the image plane. The finite difference solution employed a grid which allowed greater resolution in the vicinity of the island than in the deep ocean. Vastano and Bernard (1973) extended the techniques developed by Vastano and Reid (1966) to multipleisland systems. However, the transformation of coordinates technique allows high resolution only in the vicinity of one island of a multipleisland system. Thus, for the threeisland system of Kauai, Oahu, and Niihau in the Hawaiian Islands, the two islands of Oahu and Niihau had to be represented by cylinders with vertical walls whose cross sections were truncated wedges. Kauai was represented by a circular cylinder with the surrounding bathymetry increasing linearly in depth with distance radially from the island until a constant depth was attained. A single Gaussianshaped plane wave was used as input to the model. No comparisons were made with historical tsunami data for the three islands. Hwang and Divoky (1975) presented a simulation of the 1964 tsunami at Hilo, Hawaii, using a finite difference numerical model. The 1964 Alaskan tsunami was generated and propagated across the deep ocean using
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a finite difference numerical model employing a coarse grid. The Hawaiian Islands were too small to be represented by this numerical grid. A waveform was calculated at a point in water 5000 meters deep that was approximately 200 kilometers from where Hilo, Hawaii, would have been if the Hawaiian Islands could have been modeled. This waveform was used as input to a finite difference grid with fine grid cells that covered an area approximately 22 kilometers by b7 kilometers in the immediate vicinity of Hilo. The water depth along the input boundary of the fine grid varied from approximately 250 meters to 1500 meters. Hwang's approach neglected the influence of the Hawaiian Islands and of the varying bathymetry and depth changes between the deep ocean point and the fine grid input boundary on the tsunami arriving in Hilo. The numerical model calculations for the 1964 tsunami were in poor agreement with the Hilo tide gage recording of this tsunami. A finite difference model employing a grid covering the eight major islands of the Hawaiian Island chain was used by Bernard and Vastano (1977) to study 'the interaction of a plane Gaussian pulse with the Hawaiian Islands. The square grid cells were 5.5 kilometers on a side and close to the minimum feasible (because of computer time and cost limitations) size for a rectangular cell finite difference grid covering the major islands of Hawaii. Yet historical data indicate that significant variations of tsunami elevations occur over distances much less than 5.5 kilometers. The islands of Hawaii are relatively small and very poorly represented by a 5.5 kilometer grid. For example, Oahu has a diameter of only approximately 50 kilometers and the landwater boundary of the island has characteristic direction changes
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that occur over distances much less than 5.5 kilometers. The offshore bathymetery of the islands also varies rapidly, with depth changes of more than 1500 meters frequently occurring in distances of 5.5 kilometers. Furthermore, if a resolution of eight grid cells per tsunami wavelength is maintained for tsunami periods as low as 15 minutes, a 5.5 kilometer grid cannot be used to propagate waves into depths less than approximately 150 meters. The processes that cause significant wave modifications and subsequent rapid variations of elevations along the coastline (that are known from historical observations to occur during tsunami activity in the Hawaiian Islands) probably occur within this region extending from water at a depth of 150 meters to the shoreline. In the present study, a finite element numerical model is used to propagate tsunamis from the deep ocean to the shoreline of the Hawaiian Islands. Since finite element techniques allow dramatic changes in element sizes and shapes, this model has great advantages relative to finite difference numerical models in accurately representing land shapes, ocean bathymetry, and tsunami waveforms. Element size can be large in deep water where bathymetric variations are gradual and wave lengths are long. As a wave enters shallow water its length decreases and the elements of the grid can be telescoped to smaller sizes with no accompanying less of resolution. Governing Equations Typical tsunamis have lengths much greater than the water depths over which they propagate; for such waves, fluid motions are
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approximately twodimensional, i.e., vertical fluid velocities and accelerations are small in comparison with horizontal velocities and accelerations, respectively. Consider a small element of fluid as shown in Figure 1. The rate of increase of volume of the element is given by Â— {(n+h) dxdy} = jdxdy where, n = a freesurface elevation, h = the water depth, t = time, and (x,y) are Cartesian coordinates. (2.1) r,4h Figure 1. Element of Fluid In Equation (2.1), it is assumed that h=h(x,y), i.e. h has no time dependence. The net rate of volume flux is given in difference form by {u(n+h)> A , dy {u(n+h)} dy + {v(n+h)> , dx {v(n+h)> dx y+dy )' = dxdy {7Â— u(n+h) + 3y oX = dxdy V Â• q(n+h) v(n+h)l + 0(dx 2 dy ) (2.2)
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where u,v are horizontal velocity components, w is the vertical velocity component (w=0 here), and q = (u,v,w) is a velocity vector (a Taylor expansion is used in liquation (2.2) and terms of order 2 2 dx dy have been dropped). Equating the rate of increase of volume to the rate of volume flux yields the following continuity equation: fÂ£ + V Â• q(mh) =0 (2.3) at The governing momentum equation for incompressible and constant viscosity fluid flow is the NavierStokes equation in the form p + q Â• Vq = pF Vp + yV 2 q (2.4) where p = the fluid pressure, p = the mass density, u the dynamic viscosity constant, F = an external force vector. Since vertical fluid velocities and accelerations are small and assuming that gravity is the only external force and the fluid is inviscid so that the last term of Equation (2.4) can be dropped, Equation (2.4) becomes in component form 3u 3u 3u 1 3p ,. .. Â— + u Â— + v Â— = ^(2.5) 3t 3x 3y p 9x 3v 3v 3v 1 3p ,,,, Â— + u t+ v Â— = ^ (2.6) 3t 3x 3y P 3y
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10 When integrated, Equation (2.7) yields (with atmospheric pressure equal to zero) p = pg(STi) (2.8) i.e., the pressure is hydrostatic, and Vp = pgVn (29) Equations (2.5) and (2.6) can be recombined to yield + (q ' V)q = gVn "(2.10) Since along most of the path of propagation of a tsunami the wave amplitude is much less than the. water depth, Equation (2.3) can be approximated as Â£+ V Â• (hq) = (2.11) It can also be shown that the second term in Equation (2.10) is generally less than the other terms by a factor of the order of n/h . Therefore, for n/h snail, Equation (2.10) is approximated by 8q 9t Taking a time derivative of Equation (2.12) and substituting Equation g 7n (2.12) (2.11) into the righthand side of the subsequent equation yields i = gV(V Â• (hq)) (2.13) dt Equation (2.13) is known as the linear long wave equation. It is a governing equation for small amplitude long waves in an inviscid and incompressible variable depth fluid and is valid when the assumptions made in its derivation remain valid. Assuming that the fluid is irrotational q can be expressed as a gradient of a scalar function (x,y,t) , known as the
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11 velocity potential, i.e, q = VC (2.14) Assuming all motions to be simple harmonic in time yields Hx,y,t) = Re [OCx.yJe 1 ^] (2.15) where Re [ ] indicates the real part of the braced quantity, and w is the frequency of the motion. Substituting Equations (2.1.4) and (2.15) into (2.13) yields 2 7 Â• (hVO) + ~ i (2.16) Equation (2.16) is the well known generalized Helmholtz equation. The wave amplitude r, can be related to the velocity potential $ by evaluating the Bernoulli equation at the free surface. The Bernoulli equation is given by 3 ; P 1 1 1 2 _ = _ g 7 + vol = , (2.17) at p 2 ' ' v ' however; the last term in Equation (2.17) is generally smaller than the others by a factor of the order of n/h and can be neglected. Also, taking p=0 at the free surface where In , Bernoulli's equation becomes to the same order of approximation as Equation (2.16) Several assumptions are made in the derivation of Equations (2.16) and (2.18) and the adequacy of these assumptions for the particular problem of tsunami interaction with the Hawaiian Islands must be considered. An important factor in this consideration is the fact that the continental shelf of the Hawaiian Islands is exceptionally short. Thus, during most of the time of tsunami propagation, fluid velocities are much less than wave propagation velocities since wave amplitudes
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12 arc much loss than water depths and bottom stress forces as well as nonlinearities have little time to cause significant effects. Ursell (1953) and Hammack (1972) have shown that nonlinear effects grow with time. However, for a typical large tsunami, such as the 1964 Alaskan tsunami} Hammack and Segur (1977) have shown that neither nonlinearity nor frequency dispersion have any significant effect on the lead wave as it propagates across the ocean. In addition, when the lead wave reaches a continental shelf a certain time must pass before nonlinearity or frequency dispersion become significant. For a typical large tsunami, Hammack and Segur (1977) show that linear and nondispersive theory holds for the lead wave over a distance of approximately 2 Â±L ^Â— (a is a wave amplitude). Taking the average depth of the region from the edge of the continental shelf (600 foot depth) to shore as 300 feet and the average amplitude as 3 feet (the amplitude becomes large ?h 2 only near shorej yields = 12 miles. This distance is much greater than the length of the continental shelf of the Hawaiian Islands (usually less than 1 mile). Since the lead wave is followed by waves which have been reflected by land areas in the generation region, linear and nondispersive theory should be adequate, at least for the major waves of a tsunami. Finite Element Model Figure 2 shows the problem of tsunami interaction with islands with the process divided into two domains. One domain (A) includes the islands and the surrounding variable depth region. The second domain (B) is a constant depth region extending to infinity. This constant
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15 Figure 2. Regions of Computation
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14 depth region is realistic since the Pacific Ocean has a remarkably constant depth beyond the immediate region of the Hawaiian Islands. The boundary value problem expressed in terms of these domains becomes the following: 2 V ., (hr:) + Â— 5 = in Region A (2.19) 2 V Â• (Vo) + !L co where j = number of land masses, r = a spherical coordinate, k = the wave number, Â§ = the scattered wave velocity potential, i = (D Â• Equations (2.21) and (2.22) express the continuity of the velocity potential and its derivative along the boundary separating the two domains and Equation (2.23) expresses the impermeability of solid boundaries. Equation (2.24) is the Sommerfeld radiation condition that requires the scattered wave to be an outgoing wave at infinity. A calculus of variations approach can be used to obtain an EulerLagrange formulation of boundary value problems. This variational formulation is based upon the principle that of all possible displace
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13 merit configurations a body can assume that satisfy compatibility and the constraints of kinematic boundary conditions, the configuration satisfying equilibrium makes the potential energy a minimum. The EulerLagrange formulation of boundary value problems has often been used in classical dynamics (Hamilton's principle of least action) and solid mechanics (structural, solid, and rock mechanics). Of course, it is not possible to obtain analytical solutions for many engineering problems. The finite element method is a numerical method first introduced by Turner et al. (1956) (although some of the underlying ideas were discussed by Courant (1943)) which has found wide application in solid mechanics for problems based upon an EulerLagrange formulation involving complex material properties or boundary conditions (see Zienkiewicz, 1971 and Desai and Abel, 1972). The finite element method is a discrete approximation procedure applicable whenever a variational principle can be formed. This method has a very short history in fluid mechanic applications despite advantages over conventional numerical schemes such as finite difference methods. The first application of the finite element method to fluid mechanic problems involved a study by Zienkiewicz and Cheung (1966) of seepage through porous media. Chen and Mei (1974) studied the problem of the forced oscillation of a small basin protected by a breakwater and containing floating nuclear power plants by using the finite element method. They discretized a small constant depth region using approximately 250 triangular elements that were all about the same size and shape. Since small amplitude long waves were used as a forcing function, Equation (2.20) was the
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16 governing equation (the water depth being constant in their application) . In the present study an approach similar to that of Chen and Mei is used to solve the problem of tsunami interaction with islands in a variable depth region. For the finite element method to be useful for tsunami problems, it is necessary to take advantage of two important properties of 'this algorithm. First, the finite element grid which discretizes a domain can have grid cells (elements) of arbitrary size and shape. Thus, the grid can telescope from a large cell size in one section of a variable depth domain to a very fine cell size in another. Secondly, if the form of the spatial variation of a physical parameter is known a priori, the finite element method can employ a very small number of elements in the discretization used to estimate the exact variation. For example, a single finite element can be used to solve the problem of the seiching of a lake for a constant depth rectangular lake if the basic form (unknown coefficients) of the solution is known. The finite element method degenerates into the wellknown RayleighRitz variational method in this case. This property of the finite element method will be important in modeling the infinite region B shown in Figure 2. The variational principle for the boundary value problem given by Equations (2.19) through (2.24) requires a certain functional F(}>) to be stationary with respect to arbitrary first variation of . The first variation of the functional F(4') for this problem is wellknown and can be expressed as
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17 2 3<54> ;f co) = //[v Â• (iwo) Â£*] 6? + ; h(i B ; A J H A Â° "A A The functional can be derived from this expression for its first variation (Chen and Mei, 1974) and is ? 3($ $ T ) Ft.) // i[MÂ«) z 1* 2 i * f * Â«VV thHA fe A A 3(VV 3i 1 SA A 3n A 3A A 3n A 3($ R $ T ) + / ho gg T , (2.26) 3A l Sn A where $ is the velocity potential of the incident waves (forcing function) . The finite element method (unlike the. finite difference method) discretizes the domain itself rather than the governing equations. The region under consideration is partitioned into small regions known as finite elements and the assemblage of all such elements represents the original region. Instead of solving the problem for the entire body in one operation, the solutions are formulated for each constituent element and combined to obtain the solution for the original region. Thus, for example, the first term of Equation [2.26) is evaluated on the element level and the results for all elements are summed. This sum is equivalent to the original integral over the complete region A. In order to evaluate the first term in Equation (2.26) on the element level, it is necessary to approximate the field variable
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within each element by a polynomial N , called a trial or interpolation function, in the coordinates x and y . For each element $ = {N} {e(2.27) where {Â£} I
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19 expansion independent of all other elements, the treatment of the entire domain is systematically handled by summing the contributions from each element. The expression $ R * T i n Equation (2. 26 j is the scattered wave velocity potential $Â„ . Since region B has a constant depth, Â£> can be solved analytically. The governing equation for this case is the Helmholtz equation (Equation (2.20)). The boundary condition at infinity requires the scattered wave obey the Sommerfeld radiation condition given by Equation (2.24). The wellknown solution to this boundary value problem is as follows: $ I H (kr) (a cos in? + s sin m9) (2.34) s Â„ m m m m=0 where H are Hankel functions of the first kind of order m, m a and 6 are constant coefficients, m in and 9 is a spherical coordinate. In the finite element formulation, the infinite region B can be considered to be a single element with the interpolation function given by Equation (2.54). Thus, within region A the interpolation functions are linear, but within region B the interpolation function is an infinite series involving unknown coefficients. The terms in Equation (2.54) are oscillatory but have a monotonically decreasing modulation; hence, the series can be truncated after a finite number of terms depending upon the desired accuracy (see next section) . The required stationarity of the functional F(e) with respect to arbitrary variations of $ at each nodal point yields a set of linear algebraic equations when expressed on the element level that can be solved using matrix methods. For each specified incident wave period,
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the velocity potential can be determined at all nodal points. The linearized dynamic free surface boundary condition (Equation (2.18)) can then be used to calculate the surface elevation n . The set of linear algebraic equations evolving from the element level can be written in the simple matrix forn [K] W = {Q} (2.35) where [K] is the large coefficient matrix, {0} is a column vector representing a combination of nodal unknowns for the velocity potential and the coefficient unknowns from the single element covering region B, and {Q} is the total load column vector related to the forcing function. in the finite element solution of structural mechanics problems, one obtains a similar equilibrium relation with [K] being a stiffness matrix, {^} a nodal displacement vector, and {Q} a nodal force vector (Desai and Abel, 1972). The coefficient matrix [K] is symmetric, banded, and complex (complex notation was used to represent waveforms, e.g., the Hankel functions are complex) . For the case of tsunami interaction with the Hawaiian Islands, [K] will be exceptionally large and sparse (many zero terms) . Of course, only half of the banded part of the matrix alone needs to be manipulated in the solution process. The banded part of the matrix [K] is solved using standard Gaussian elimination methods. The matrix solution used by Chen and Mei (1974) was modified in this study to take advantage of the sparseness of the matrix in order to reduce computational time. Also, the elimination method was modified so that calculations involved only small blocks of terms at a time with the remainder of the matrix kept in pe7 % ipheral storage. This matrix partitioning was necessary for the finite element grid shown in the next
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21 section since the banded part of the matrix had over S00,000 terms. Partitioning is possible because the symmetric coefficient matrix is positive definite; hence, a solution is possible by elimination methods without pivoting. Without pivoting, elimination performed using one row, affects only the triangle of elements within the band below that row. Thus, only two triangular submatrices needed to be stored in computer memory at a given time. Numerical G rid The finite element grid covering the variable depth region A is shown in Figure 3. Elements of the grid telescope from large sizes in the deep ocean to triangles with areas as small as 0.5 square kilometers in shallow coastal waters. The grid covers an area of approximately 450,000 square kilometers. In general, finite element techniques allow elements to be any arbitrary shape (e.g. quadrilateral) provided that the ratio of the lengths of the shortest and longest sides of an element is not extreme. Triangular shapes were used for convenience in the present study. The geometric shapes of the eight islands comprising the Hawaiian Island chain and the rapid bathymetric variations surrounding the. islands are modeled very precisely by the finite element grid of Figure 3. Also, the number of node points along the shorelines of the islands is very dense. Since wave heights are calculated at node points by the finite element model, the model can adequately represent the rapid waveheight variations along coastlines that are known to occur during tsunami activity in the Hawaiian Islands.
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Figure 3. Finite I; lenient Grid.
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The finite element grid was drawn by hand with the element sizes dictated by the local water depth. The lengths of element sides were allowed to be no greater than oneeighth of the local wavelength for the shortest period waves considered. Such a cell size results in a wave resolution error of no more than one percent f.Mei 197S) . There are computer programs (Cole and Riech, 1976) that generate finite element grids and number the nodes to minimize bandwidths; however, such programs only can be applied to fairly homogeneous domains. The rapid depth variations in region A require a rapidly telescoping grid that is too complicated for automatic grid generation techniques. Each node of the finite element grid must be numbered and this numbering is extremely important since it determines the bandwidth of the large coefficient matrix discussed in the last section. The bandwidth of this matrix is controlled by the largest difference (among all elements) in the nodal numbers between two nodes of the same element. The computational time of the numerical model is proportional to the cube of the bandwidth of the coefficient matrix. For a simple region covered by homogeneous elements, the bandwidth is minimized by numbering nodal points in cyclic sequence following a spiraling pattern '..here one spiral is shielded by the following spiral from a third spiral. The grid pictured in Figure 3, however, is so large and complicated that considerable judgement and effort is involved in determining a numbering pattern that minimizes the bandwidth, thus resulting in reasonable computational times for the numerical model. The numerical model also requires the spatial coordinates of all node points and information concerning how the elements are connected
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relative to each other (connectiveness) . The coordinates were determined using an electronic digitizer and a simple computer program developed to relate the internal coordinate system of the digitizer to a prototype coordinate system through coordinate rotation and scaling. The elements are numbered in an arbitrary fashion and then connectiveness information assembled for the numerical model by listing the three nodes of each element. The nodal numbering, digitizing, and connectiveness assembly necessarily introduce errors. For example, the connectiveness assembly involves determining over 17,000 numbers and placing them on computer cards. Many connectiveness mistakes do not result in error conditions which terminate the numerical model's computations. Instead, incorrect answers are produced by the model. To eliminate all errors in the input data, computer plotting programs were developed to draw the grid and number nodes and elements based upon the connectiveness and node coordinate information used as input to the numerical model. Such plots make errors readily apparent. The number of terms retained in the truncation of Equation (2.34) was determined by trial and error. The final number of terms was such that addition of further terms produced negligible effects. It was found that this occurs for terms approximately 10 " times smaller than the initial terms. The computational time requirements of the finite element model for the grid shown in Figure 3 are very modest, making it economically feasible to determine the interaction of arbitrary tsunamis with the Hawaiian Islands (discussed in next section) . The reason that the
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computationtal time required by the finite element model is reasonably small is that the grid uses small elements only in areas where they are necessary. The grid of Figure 3 has approximately 2500 points, whereas the finite difference grid of Bernard and Vastano (1977) had 26,000 grid points. Even so, some of the elements of Figure 3 are as much as sixty times smaller than the finite difference grid cells used by Bernard and Vastano. Model Verification The finite element model is verified in the present study by comparing numerical simulations of the 1960 Chilean and 1964 Alaskan tsunamis with tide gage recordings of these tsunamis in the Hawaiian Islands. These two tsunamis are the only major tsunamis for which reliable information exists concerning characteristics of the ground uplift which generated the waves (much more information exists for the Alaskan source) . A deep ocean recording of a tsunami near the Hawaiian Islands has never been made. Therefore, prototype wave records in the deep ocean cf the 1960 Chilean and 1964 Alaskan tsunamis are not available for use as input to verify the finite element model. However, a finite difference numerical model which solves the linear long wave equations employing a spherical coordinate grid has been developed (Hwang et al., 1972), and used in several studies (Houston and Garcia, 1974; Houston and Garcia, 1975; and Houston et al . , 1975b) to generate tsunamis and propagate them across the deep ocean. This deep ocean numerical model
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26 is used in this study to determine deep ocean waveforms of the 1960 and 1964 tsunamis that were needed to verify the finite element model. The deepocean numerical model simulates the uplift deformation of the ocean water surface caused by the permanent vertical displacement of the ocean bottom during an earthquake and the subsequent propagation of the resulting tsunami across the deep ocean. The permanent deformation (permanent in the sense that the time scale associated with it is much longer than the period of the tsunami) and not the transient movements within the timehistory of the ground motion is considered to be the important parameter governing farfield wave characteristics. Transient movements occur for a period of the order of tens of seconds, whereas, tsunami wave periods are of the order of tens of minutes. Experimental investigations of tsunami generation by Ilammack (1972) show that the transient ground movements do not influence farfield characteristics of resulting tsunamis for spatially large ground displacements occurring over a short period of time. Hwang et al , (1972) verified the deep ocean model by simulating the 1964 Alaskan tsunami. The permanent ground deformation in the source region was taken from Plafker (1969) and a 1/4Â° by 1/6Â° grid was used for the simulation. Since this grid was much too coarse to allow accurate modeling of shallowwater propagation, a comparison was made between the numerical model calculations and a gage recording of the 1964 tsunami at Wake Island where Van Dorn (1970) had a pressure transducer in relatively deep water (800 foot depth) some distance offshore. The amplitude and length of the first wave of this tsunami were shown to be in good agreement. (Actually, the grid used by Hwang et al.
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27 (1972), did not extend as far as Wake Island. The deep ocean model propagated the tsunami to a point in the Pacific Ocean and then simple refraction techniques were used to estimate the change in the waveform during propagation from this point to Wake Island.) The simulation of the 1960 and 1964 tsunamis performed in this study uses a 1/3Â° by 1/3Â° grid. Prior simulations of the 1964 tsunami using both a 1/4Â° by 1/6Â° grid and a 1/5 D by 1/3Â° grid demonstrated negligible differences between use of the two grid spacings (Houston et al . , 1975a). The permanent ground deformation used as input is also taken from Plafker (1969). The numerical grid was large enough to cover a substantial portion of the Pacific Ocean including Wake Island. The numerical model calculations at Wake Island were found to be similar to those presented by Hwang et al. (1972). Figure 4 shows a comparison of the numerical model calculations of this study and the Wake Island recording of the 1964 tsunami. The lack of agreement in the trailing wave region may be due to local effects (e.g. wave trapping by the Wake Island seamount) or reflections from the nearby transmission boundaries of the numerical grid (these boundaries are not perfectly transmitting) . Figures 5 and 6 show timehistories of the 1960 Chilean and 1964 Alaskan tsunamis in deep water near the Hawaiian Islands calculated using the deep ocean propagation model. The permanent ground deformation in the source region of the Chilean earthquake of 1960 is taken from Plafker and Savage (1970). Calculations for the Alaskan tsunami involve a grid with approximately 30,000 cells and the calculations for the Chilean tsunami a grid with approximately 90,000 cells. The finite element numerical model is a timeharmonic solution of
PAGE 37
28 J,
PAGE 38
1 I I I I C^T ~ Â— ^ ^Â— Â« Â„. I C 5 o z O 2 c Â•.Â— * c r. Â— JLJ 3d^lOXOad 'SNOIXVA313
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Â— _ o U O li BdAlOlOHd sno;xva3~i^
PAGE 40
the boundary value problem. The response of a group of islands to an arbitrary tsunami can easily be determined within the framework of a linearized theory using the theory of superposition. For example, an arbitrary tsunami in the deep ocean can be Fourier decomposed as follows: b (tO = f b(w) e" l(a)t+pfu) ^ du (2.36) where b = incident wave amplitude o r b(w) = amplitude of frequency component tu p (u) ~ phase angle If n(x,y,(j) is the response amplitude at any point (x,y) along the island coasts due to an incident plane wave of unit amplitude and frequency cj , then the response of the islands to the arbitrary tsunami time history b ft) is given by o f(x,y,t) = Re [ /b(w)n(x,y,oO e" 1 ^^ ^ *du] (2.37) Therefore, when n(x,y,co) is known for all u , the island response to an arbitrary tsunami can be calculated. Of course, it is not feasible to calculate the integrals of Equations (2.56) and (2.37) over all frequencies. Instead, the frequency range must be discretized and the integrals replaced by sums over a frequency range containing most of the energy of the tsunami. Equation (2.36) involves a Fourier decomposition of a time series. This decomposition is accomplished for the timehistories of the 1960 Chilean and 1964 Alaskan tsunamis (Figures 5 and 6) using a least squares harmonic fitting procedure. The timehistory of the Alaskan
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tsunami in deep water was decomposed into IS components with periods ranging from 14.5 minutes to the time length of the record (260 minutes) . The variance of the residual (difference between the actual record and a recomposition of the 18 components) was approximately 0.2 percent of the variance of the record. Therefore, virtually all the energy of the wave record was contained in the 18 components. The timehistory of the Chilean tsunami in deep water was decomposed into 11 components with periods ranging from 15.5 minutes to the time length of the record (170 minutes). The variance of the residual was less than 0.1 percent of the variance of the record. For both cases, the original timehistory and a timehistory constructed from a recomposition of the components were virtual ly indistinguishable. Equations (2.56) and (2.57) take the following form when discretized : b o (t) = I b(u) n ) e i{ V +p (V } ( 2 5g ) C(x,y,t) = Re[ E b(u ) n(x,y,ai ) e" i{u n t+pCw n 5} ] (2.59) n=l n where m = number of components, th to = frequency or n component The M" ) term is determined by the leastsquares harmonic fitting procedure and n(x,y, u ) is determined by the finite element numerical model for each frequency to and at each location (x,y). Therefore, the timehistory 5(x,y,t), which represents the response of location (x,y) to the deepwater tsunami timehistory given by b (t) ,
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,s5 can be calculated for any location along the coastline of the Hawaiian Islands. Figures 7, S, and 9 present comparisons between tide gage recordings and numerical model calculations of the 1964 Alaskan tsunami at Kahului, Maui; Honolulu, Oahu; and Hilo, Hawaii, respectively. The largest waves recorded at each of these sites are shown. Waves arriving at later times are all much smaller than those shown. Whenever tide gage limits were encountered, the recordings were linearly extended (the tide gage locations are shown in Figure 10) . The wave records shown in these figures are in remarkable agreement, especially considering the fact that the ground displacement of the 1964 earthquake was not precisely known. Since the Hilo breakwater was not included in the numerical model, the numerical model calculations for Hilo are probably too large; however, this breakwater was undoubtedly highly permeable to the 1964 tsunami. The numerical results appear to be too large in Hilo by some constant factor, since the tide gage recording and the numerical model calculations have the same form with the first wave crest and trough having approximately the same amplitude and being proportionately smaller than the second crest. Figure 11 shows a comparison between the tide gage recording of the 1960 Chilean tsunami at Honolulu and the numerical model calculations. Again the largest waves recorded are shown (the Honolulu gage was the only tide gage in the Hawaiian Islands not destroyed by the 1960 tsunami) . The permanent ground motion of the 1960 earthquake is not known nearly as well as the ground motion for the 1964 earthquake.
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LEGEND NUMERICAL MODEL CALCULATIONS ESTIMATED (BEYOND GAGE LIMIT) APPROXIMATE HOURS GREENWICH MEAN TIME Fig ur c 7. 1964 Tsunami from Alaska Recorded at Kahului, Maui
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:45 9:00 10:00 APPROXIMATE HOURS GREENWICH MEAN TIME : igure 8. 1964 Tsunami from Alaska Recorded at Honolulu, Oahu.
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TIDE GAGE RECORD NUMERICAL MODEL CALCULATIONS 9:00 1000 APPROXIMATE HOURS GREENWICH MEAN TIME Figure 9. 1961 Tsunami from Alaska Recorded at fiiio, Hawaii.
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Figure 10. Gage Location:
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39 Hence, the agreement indicated in Figure 11 is quite good considering the uncertainty of the uplift in the source region. Tide gages, of course, do not record tsunamis perfectly since they are nonlinear devices with responses which depend upon both the period and amplitude of the disturbances they measure. However, simple calculations based upon the paper of Noye (1970) show that the distortion is small for the tsunamis shown in Figures 7 through 9 and Figure 11. Figure 12 shows the response amplitude as a function of wave period at the locations of tide gages in Kahului, Honolulu, and Hilo. The response versus wave period is greatly dependent upon the precise location of recording for bays such as Hilo. For example, the Hilo response (Figure 12) at the tide gage location shown in Figure 10 has a single large resonant peak at 2S.9 minutes. This peak occurs at the fundamental period of oscillation of Hilo Bay and, consequently, all locations in the bay have a resonant peak at this incident wave period (the amplitude of the peak depends on location in the bay). Other locations in Hilo Bay also have significant peaks at lower periods. Loomis (1970) calculated a free oscillation fundamental period (node assumed at bay mouth) of 26.2 minutes for Hilo Bay using a finite difference grid spacing of approximately onehalf a kilometer. Bernard and Vastano (1977) show a contour map of a normalized energy parameter versus wave period and location around the island of Hawaii. The map shows Hilo displaying an energy peak from 10 to 16 minutes centered at approximately 12.5 minutes; no significant peak at 28.9 minutes is observed. The difference between Bernard and Vastano 's calculations and the results presented in this paper is attributed to
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KAHULUI TIDE GAGE 15 CE
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41 the poor resolution of the spatial features of Hilo Pay by their finite difference grid spacing of 5.5 kilometers. Hi Id Bay has the shape of a right triangle with sides that are approximately 9 kilometers and 13.5 kilometers long. The grid used by Bernard and Vastano represented Hilo Bay as a right triangle with sides that were 5.5 kilometers and 11 kilometers long; thus, the surface area of Hilo Bay was approximately one half its actual area. This distortion of the sir.e of Hilo Bay and also, perhaps, the distortion of the bathymetry of the bay by the coarse grid accounts for the lower response periods for Hi lo Bay presented by Bernard and Vastano. The importance of accurately representing island shapes and offshore bathymetry and resolving tsunami wavelengths can be illustrated by considering a study by Adams (1975) that used the numerical model developed by Bernard and Vastano (19""") to interpolate between historical measurements of tsunami elevations produced by the 1946 Aleutian tsunami. A single Gaussian wave crest was used as input for the numerical model. If a resolution of four grid cells per crest wavelength is maintained, this numerical model cannot propagate waves into water with a depth less than about 500 feet. The main processes involved in the transformation of the form of a tsunami probably occur in the region over the continental shelf (depths less than 600 feet). Consequently, the elevations calculated by Adams were small and did not display the great variation along coastlines known to occur during historical tsunamis in the Hawaiian Islands. Adams included the effects of propagation over continental shelves by multiplying the numerical model elevations by a factor (different for each island) that forced the best agreement with the historical
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42 data. However, even with artiface, agreement with historical data was poor. For example, on the northeast coast of Oahu, the numerical model elevations multiplied by the factor resulted in predicted elevations ranging from 14 to 21.4 feet. Loomis (1976) presents 36 historical elevations for the 1946 Aleutian tsunami on the northeast coast of Oahu and only 7 even fall within this range of elevations. The historical elevations actually varied from 1 to 37 feet on this coast.
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CHAPTER III TSUNAMI INTERACTION WITH CONTINENTAL COASTLINES In troduc tion Experimental investigations of long wave interactions with continental coastlines have generally been onedimensional and have involved idealized conditions. Flume tests by Savage [1358) and Saville (1956) measured runup of periodic waves on constant slope beaches and Mail and Watts (1953), Kaplan (1935), and Kishi and Saeki (1966) measured runup of solitary waves :>n constant slope beaches. All of these flume experiments used waves much steeper than typical tsunamis due to the difficulties involved in the generation of small amplitude long waves. Many theoretical studies of onedimensional propagation of weakly dispersive long waves have been performed in recent, years. Tsunamis become weakly dispersive (amplitude and frequency dispersive) long waves at some point during their propagation, although (as was the case for the Hawaiian Islands) they may not be weakly dispersive for a time period sufficiently long for related effects to develop. The proper equations for the investigation of weakly dispersive long wave propagation over a steeply sloping bottom were first derived by Mei and LeMehaute (1966) and in a slightly different form by Peregrine (1967) who also gave the first numerical solution in the case of a solitary wave using an implicit finite difference 43
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44 scheme. A more complete solution was later given by Madsen and Mei (1969) who transformed the equations to characteristic form before they were solved numerically. Svendsen (1974) studied weakly dispersive wave propagation over a gently sloping bottom. Hammack (1972) made comparisons of long wave generation by bottom displacements in the laboratory and numerical solutions of the KortewegDeVries equation using the numerical approach of Peregrine (1966). Several studies of onedimensional tsunami propagation over continental shelves of constant slope have been performed. Keller and Keller (1964) solved nonlinear long wave equations numerically for a constant slope bathymetry. Takahasi (1964) studied tsunami propagation over a simple bathymetry through analytic solutions of the linear long wave equation. Carrier (1966) assumed a tsunami was generated by a point source and studied nonlinear propagation over a constant slope bathymetry. Heitner and Housner (1970) used a finite element numerical method to solve nonlinear equations for propagation up constant' slope beaches. More recently, Mader (1974) studied tsunami propagation over linear slopes using a finite difference solution of the complete NavierStokes equations (not vertically integrated). Occasional twodimensional studies have been performed for tsunami propagation in ncarshore regions. Grace (1969) studied tsunamis affecting the reef runway of the Honolulu International Airport using a hydraulic model. However, the wavemakers used were only a small fraction of a tsunami wavelength from the region to be studied and waves reflected by land were reflected again almost
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45 immediately by the wavemakers. This is a common problem in hydraulic model tests involving long waves. The wavemakers can be moved a few wavelengths from the region of interest by reducing the model scale; however, scale effects may then become very significant. Wave absorber screens may also be placed in front of the wavemakers; however, it is very difficult to absorb very long waves and such an approach is often not very helpful. Similar problems with rereflected waves can occur in numerical studies. The calculations of Hwang and Divoky (1975) were in poor agreement with the tide gage recording of the 1964 tsunami at Hilo not only for the reasons presented in Chapter II, but also because their algorithm allowed waves to leave the numerical grid efficiently only for waves approaching the input boundary normally. Since Hilo Bay is triangularly shaped, the reflected waves approached the input boundary at various angles and were partially rereflected toward Hilo. Governing Equations The governing equations for long wave propagation over the continental shelf of the west coast of the United States are Equations (2.5) and (2.4). Thus, the long wave assumption is made once again with vertical velocities and accelerations neglected relative to horizontal components. The continuity equation is given by the equation ~ + V Â• qCn + h) = (3.1) and the momentum equation is 8q > v 2>yf + q Â• Vq = pF Vp + pV q (3.2)
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46 If the only external force is gravity and the pressure is hydrostatic, then liquation (3.2) becomes p S+ q Â• Vc i = ps Vr i + ^ iV " c i ( 3 ^) c)t The west coast of the United States has a much broader continental shelf than that of the Hawaiian Islands; thus, the nonlinear advective and bottom stress effects may be more significant, and the nonlinear and viscous terms of Equation (3.3) must be retained. The dynamic viscosity u in the viscous term of Equation (3.3) is replaced by an eddy viscosity constant e for the case of turbulent flow. Using this approach the viscous term can be expressed as *, c^4 ^4) + b 2 M C3.4) 1 3x" 3y ^z where the coefficients e. and e_ represent coefficients of horizontal and vertical eddy viscosity, respectively. The term 2 2 3x 3y 8 2 is usuallv small in comparison with the e Â— Â§ term (Dronkers, 1964) and will be neglected in this study. If e, Â—  is 9z integrated with respect to z from the bottom to the surface and divided by the distance from bottom to surface n + h , we find [(ItK.k (It) . J VV n+h L 9z z=h 3z z = n The terms Â£:_(Â—) _ . and E 2 ( '^z=n &re cora P onents of the
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47 tangential stresses at the sea bottom and surface, respectively. The shear stress at the sea surface is quite important for storm surge problems where wind velocities are very large but can be neglected relative to the bottom stress for tsunamis. From experiments and theoretical considerations for onedimensional flow in a river, it has been shown (Dronkers, 1964), that the bottom stress t, can be expressed as follows T =lÂ£SHM (3.6) b c 2 where C is Chezy's coefficient and is given by the expression 1,486 h l/6 (37) n m where n is Manning's n . Thus, Equation (3.3) may be written m *, 2 2,1/2 3q Â•*" ^ n p gq(u +v ) f ~ R1 p Â—x + q Â• Vq = pgVr. ~^ Â— ' ' U6 J CT(n+h) Equation (3.8) contains the nonlinear advective term but neglects frequency dispersion since vertical accelerations have been neglected, This equation adequately describes wave propagation for Ursell numbers much greater than unity (Ursell, 1953). The Ursell number is proportional to the ratio of nonlinear effects to linear effects nL 2 (i.e., frequency dispersion) and is equal to Â— =Â— , where L is a 1/2 h typical wavelength. Since L=(gh) T for long waves, where T is T 2 the wave period, an Ursell number may be chosen as gn(^) The numerical model described in the next section uses Equation (3.8) to propagate waves from depths of 500 meters to shore. These
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48 waves have periods of the order of 1800 seconds and initial wave T 2 amplitudes of the order of a foot. Thus, the Ursell number, gnGr) is aproximately 50 and increases as the wave enters shallow water with h decreasing and n increasing. Mader (1974) found that the onedimensional form of Equation (3.8), without the viscous term, produced very similar results to a onedimensional N'avierStokes equation (viscosity neglected) for propagation of long wavelength tsunamis over linear continental slopes. The pressure term in Mader' s solution was not hydrostatic; thus, frequency dispersion was permitted. Finite Difference Model To solve Equations (5.1) and (5.8) numerically, the differential equations are ciscretized and replaced by a system of finite difference equations using central differences on a spacestaggered grid. The spacestaggered scheme describes velocities, water levels, and depths at different grid points. This scheme, first used by Platzman (195S) for lake surge problems, has the advantage that in the equation for the variable operated upon in time, there is a centrally located spatial derivative for the linear term. Leendertse's (1967) implicitexplicit multioperational method is employed in determining the solution for q and n as functions of time. The stability and convergence of this method is described in detail by Leendert.se (1967) . This method for solving finitedifference equations has been applied in studies analyzing the tidal hydraulics of harbors and inlets (Raney,
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49 1976). Houston and Garcia (1975) used the method to calculate the decay of a tsunami as it enters and spreads throughout San Francisco Bay. The nearshore finitedifference numerical model uses the timehistory calculated by the generation and deep ocean propagation model described in Chapter II as input in the following manner. A tsunami is generated in the AleutianAlaskan area or the west coast of South America and propagated across the deep ocean to a 500meter depth off the west coast of the United States. Waveforms calculated at this depth by the deep ocean numerical model are recorded all along the west coast. These waveforms are then used as input to the nearshore numerical model, which propagates the tsunamis from the 500meter depths across the continental slope and shelf to shore. Figure 13 shows outlines of the four numerical grids used to cover the west coast. The grids have square grid cells two miles on a side (southern California and some of San Francisco Bay were not covered by numerical grids because elevation predictions were made in previous studies (Houston and Garcia, 1974; and Houston and Garcia, 1975) using simple onedimensional analytical solutions for propagation over the continental shelf). The offshore bathymetry was modeled from the 500meter contour to shore. Beyond the 500meter contour, the ocean was assumed to have a constant 500meter depth. The input boundary of each grid was located approximately one and onehalf wavelengths of a 30minutc wave from the shore. Therefore, at least three typical waves could arrive at the shore before waves reflected from the input boundary became a problem.
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'QGruc. WASHINGTON ' x "baja CALIFORNIA Figure 15. Finite Difference Grid Locations,
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51 The input boundaries of the grids were oriented approximately parallel to the shoreline since refraction will bend the wavefronts to such an orientation before they reach the 500meter contour (Wilson and Torum, 1968). Lateral boundaries of the grids were taken to be impermeable vertical walls. Model Verification The finite difference nearshcre numerical model was also verified by numerical simulations of the 1964 Alaskan tsunami. The deep ocean propagation model discussed in Chapter II was used to generate the 1964 tsunami and propagate it across the deep ocean. Figure 14 shows surface elevation contours 31/2 hours after the 1964 Alaskan earthquake, as calculated by the deepocean propagation numerical model. This figure illustrates the concentration of energy on the northern California, Oregon, and Washington coasts due to the directional radiation of energy from the source region. The upper section of Figure 15 shows a time history of the 1964 Alaskan tsunami calculated by the deepocean model near Crescent City, California, in a water depth of 500 r.eters. This waveform was used as input to the finite difference nearshore model. The lower section of Figure 15 shows a comparison between a tide gage recording of the 1964 tsunami at Crescent City and the nearshore model simulation of this tsunami. The numerical model calculations agree well with the tide gage recording; periods, phases, and amplitudes are accurately reproduced. The main disagreement occurs
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52
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40 60 80 TIME, PROTOTYPE MIN LEGEND PROTOTYPE GAGE RECORD INFERRED PROTOTYPE GAGE RECORD (FROM WILSON, 196B) NUMERICAL SIMULATION APPROX TIME, HRS, GMT Figure 15. 1964 Tsunami from Alaska Recorded at Crescent City, California
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54 for the amplitude of the third wave. With regard to this disagreement, it should be noted that there is also a lack of agreement on the amplitudes of the actual historical third and fourth wave crests. Figure 15 shows a reconsti action of the 1964 tsunami at Crescent City inferred by Wilson and Torum (1963) from the prototype gage record and later survey measurements by Magoon (1965). The elevation of the fourth wave (not shown) was estimated by Wilson and Torum to have been a little less than 14 feet above mean lower low water (MLLW) (the zero elevation shown in Figure 15 is MLLW datum); however, Wiegel (1965) estimated that the elevation of the third wave was approximately 16 feet above MLLW and that the fourth wave attained the highest elevation of 18 or 19 feet above MLLW at the tide gage location. The numerical mode] predicted a fourth wave elevation approximately the same as that estimated by Wilson and Torum (14 feet above MLLW) . However, reflections from the input boundary were probably growing in importance in the numerical model calculations during the arrival of the fourth wave. There may be several reasons why the maximum wave elevation predicted by the numerical model does not agree with the historical record as well as the elevations of the initial waves. First, the later waves in the deepwater waveform are probably waves which have been reflected from land areas in the source region. The fairly large grid cell spacing used in the deep ocean propagation model to represent the landwater boundary in this region probably distorts the reflected waveform. Total reflection of waves in this region also is not completely realistic. Furthermore, the nearshore grid may
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55 not have fine enough cells to completely model the resonance phenomenon leading to the large third or fourth wave. Since 1964, there has been considerable speculation concerning the reasons that the effects of the 1964 tsunami were so great at Crescent City. .The finite difference numerical models show that both the directional radiation of energy from the source region (see Figure 14) and a local resonance caused the relatively large elevations at Crescent City. Actually, large elevations at Crescent City were not unique since directional radiation of energy from the source also caused large elevations along the Oregon and northern California coasts. Runup 10 to 15 feet above the high tide level occurred all along the Oregon coast south of the Columbia River (runup at Crescent City was approximately 15 feet above the high tide level). However, the Oregon coast is very sparsely populated and there were few damage reports. The severity of structural damage at Crescent City, which has a large logging industry, also was apparently due to the impact of logs carried by the tsunami (Wilson and Torum, 1968) . The nearshore numerical model indicates that the resonant effects at Crescent City were fairly local, extending over 2 to 4 miles of coastline. This behavior contradicts Wilson and Torum' s (1968) speculation that a bowlshaped section of the continental shelf with a diameter of approximately 50 miles experienced a resonant oscillation. Historical data support the numerical model calculations. At the mouth of the Klamath River, approximately 15 miles south of Crescent City, elevations observed during the 1964 tsunami were only 2 to 3 feet above normal high
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56 tide. Wilson and Torum's hypothesis of a shelf oscillation would predict elevations at the mouth of the Klamath River greater than those that occurred at Crescent City. Figure 16 shows a comparison between a tide gage recording of the 1964 tsunami at Avila Beach, California, and the nearshore numerical model calculations. The elevations recorded by the Avila Beach tide gage were larger than those recorded at any tide gage on the west coast except the Crescent City gage. The historical and calculated waveforms shown in Figure 16 are in good general agreement. The tide gage record obviously has higher frequency components which are not predicted by the numerical model; these components may be local oscillations of water areas which were too small to be accurately represented by the numerical grid. Important features such as the wave amplitudes are in good agreement. The waveforms recorded for the 1964 tsunami by the Crescent City and Avila Beach tide gages have elevations larger than those recorded by any other gage on the west coast. Other tide gage recordings of this tsunami on the west coast (except at Astoria, Oregon) were in areas not covered by the grids shown in Figure 13. A comparison with the tide gage at Astoria, Oregon, was not made because Astoria is approximately 12 miles away from the coast in the estuary of the Columbia River, and amplitudes of the 1964 tsunami were small at Astoria.
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GAGE \ .' PROTOTYPE GAGE RECORD \S NUMERICAL SIMULATION 8.5 9 9.5 10 10.5 APPROX. TIME, HRS., GMT. Pi "ure J 6. 'sunami from Alaska Recorded at Avaiia Beach. .alii ornia .
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CHAPTER IV ELEVATION PREDICTIONS Hawaiian Island Predictions Introduction The Hawaiian Islands, a chain of eight islands as shown in Figure 17, have a history of destructive tsunamis generated both in distant areas and locally. The earliest recording of a severe tsunami in the Hawaiian Islands was in 1S37 when a tsunami from Chile reached an elevation of 20 feet at Hilo, Hawaii, and killed 46 people in the Kau District of the big island of Hawaii. Prior to 1S37, a number of severe tsunamis undoubtedly reached the islands but, unfortunately no detailed records were kept. Since 1S37, there have been 16 tsunamis that have caused significant damage. Most of the destructive tsunamis in the Hawaiian Islands have been generated along the coast of South America, the Aleutian Islands, and the Kamchatkan Peninsula of the Soviet Union. Approximately onefourth of all the tsunamis recorded in the Hawaiian Islands have originated along the coast of South America, while more than onehalf have originated in the KurilKamchatkaAleutian region of the north and northwestern Pacific. Tsunamis generated by local seismic events have caused large runup in the islands, especially on the southeast coast of the big island of Hawaii. The 1868 tsunami produced the largest waves of record in the Hawaiian Islands with 58
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59
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60 60foot waves reported on the South Puna coast of the island of Hawaii. The most recent major tsunami in Hawaiian history occurred on November 29, 1975, when waves generated by an earthquake with an epicenter on the South Puna Coast may have reached elevations as great as 45 feet along the southeast coast. The most destructivetsunami to ever hit the islands in terms of both loss of life and property destruction was the Great Aleutian tsunami of 1946, which killed 175 people and produced waves over 55 feet in elevation. Hilo incurred $26 million in property damage attributable to this tsunami. The 1960 Chilean tsunami is the most recent distantly generated tsunami that produced major effects in the Hawaiian Islands. Sixtyone lives, all at Hilo, were claimed by the tsunami. Damage throughout the state was estimated to be $25.5 million. Inspection of the damage at Hilo revealed much evidence of the tremendous forces developed by the waves. Twentyton boulders had beer, moved hundreds of feet, asphaltic concrete pavements were peeled from their subbase, and hundreds of automobiles were moved and crushed. The following sections discuss the methods used to obtain sufficient data to predict runup elevations for the entire coastline of the Hawaiian Islands. The finite element numerical model discussed in Chapter II is used to interpolate between local historical data for the years of accurate survey measurements from 1946 through 1977. Such a method of interpolation is needed so that elevations can be predicted for the entire coastline even for locations having no actual historical data. A method is discussed herein which allows reconstruction of historical data for the period of time from 1857 to
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61 1946 when elevations for the period from 1946 through 1977 are known. Such an approach is necessary in order to have sufficient data for accurate elevation predictions since it is shown herein that reasonable predictions cannot be made solely based upon known historical data for the short time span from 1946 through 1977. Interp ol ation of Recent Historical Pat a Historical data of tsunami activity in the Hawaiian Islands are, of course, often limited to certain locations. Information on tsunami activity in the islands prior to the 1946 tsunami is concentrated in Hilo (Hawaii) and to a lesser extent in Honolulu (Oahu) and Kahului and Lahaina (Maui). Even data for tsunamis occurring during the period of accurate survey measurements from 194 6 through the present are absent or fragmentary (i.e., data exist only for certain of the events) for most of the coastline of the islands. Therefore, it is necessary to rely on more than just available historical data to determine tsunami occurrence frequencies. The numerical model is used to fill in historical data gaps for tsunamis from 1946 through 1977 by providing relative responses of the Hawaiian Islands to tsunamis. For example, although the deepwater waveform of a tsunami such as that of 1946 is not known, the direction of approach of this tsunami and its range of wave periods are known. The average response of the islands to a tsunami similar to the 1946 tsunami is determined by inputting sinusoidal waves of unit amplitude from the direction of the 1946 tsunami into the numerical model over a band of wave periods. If historical data exist at one location, wave elevations at a nearby location for which historical data do not exist are determined by multiplying the historical data at the
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b2 first location by the ratio of the response calculated by the numerical model at the second location and the response calculated at the first. The numerical model takes into account the major processes that would cause different relative wave elevations at the two locations. That is, the model calculates shoaling, refraction, diffraction, reflection, resonance, shielding of the backside of an island by the front side, and interactions between islands. The historical data account for the unknown absolute heights of the deepwater tsunamis. The "correction factors" provided by the historical data are local instead of global (single factor for all islands) because there are certain local factors that cannot be determined by the numerical model. For example, coral reefs extend over parts of some of the islands and are known, to protect against tsunami attack (e.g., the extensive reefs in Kaneohe Bay, Oahu, which are powerful tsunami dissipators) . The effects of the reefs on tsunamis are implicitly contained in the historical data itself. The historical data from 1946 through the present were taken from the most recent and complete compilation of these data (Loomis, 1976) . As an example of the use of the numerical model to fill in historical data gaps for tsunamis from 194 6 through the present, consider the 1946 tsunami recorded on the island of Lanai only at Kaumalapua Harbor and Manele Bay. Wave elevations can be calculated for the 1946 tsunami at other locations on the coastline of Lanai by multiplying these historical values by the ratios of the responses calculated by the numerical model at locations of interest and the responses calculated at Kaumalapua Harbor and Manele Bay. In this
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way, elevations for the 1946 tsunami can be determined for all of Lanai . Time Period Analysis It is necessary to use historical data of tsunami occurrence covering the greatest possible time span to properly determine frequency of occurrence of tsunami elevations for the Hawaiian Islands. Tsunami activity has not been uniform in these islands during recorded history. For example, the two largest and four of the ten largest tsunamis striking Hilo from 1857 through 1977 occurred during the 15year period from 1946 through 1960. Two of the tsunamis from 1946 through 1960 originated in the Aleutian Islands, one in Kamchatka and one in Chile. However, six of the ten largest tsunamis occurred during the 109year period from 1857 to 1945 with three originating in Chile, two in Kamchatka and one in Hawaii. Therefore, both the frequency of occurrence and place of origin of tsunamis have been remarkably variable. Any study (e.g., Towill Corporation, 1975) basing frequency calculations on a short time span that includes the period from 1946 through 1960 will predict a significantly more frequent occurrence of large tsunamis than is warranted by historical data from 1857 through 1977. The errors introduced in frequency of occurrence calculations by consideration only of a short time period that includes the unrepresentative years from 1946 through 1960 will be greater than the errors resulting from possible observational inaccuracies of the 19th Century. This is easily demonstrated for Hilo, Hawaii, which is the location in the Hawaiian Islands having the most complete
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61 data of tsunami activity. Using the data compiled by Cox (1964) for the ten largest tsunamis in Hilo from 1857 through 197" (his data were actually through 1964, but no large tsunamis have occurred in Hilo since 1964), a leastsquares fit of the data was made employing the logarithmic frequency distribution discussed later in this chapter. The 100year tsunami elevation for Hilo calculated using this method was 27.3 feet. If the elevations recorded during the nineteenth centry (five of the ten largest tsunamis), are increased by 50 percent (which is much larger than possible observational error), the 100year elevation is found to be 50.4 feet. Similar calculations just based upon data taken during the period of accurate survey measurements in Hilo from 194 6 through 1977 yields a 100year elevation of 44.2 feet. Since the largest elevation in Cox's data for the 141year period from 1S57 through 1977 was just 28 feet, the 100year elevation of 44.2 feet is obviously too large. Similar overestimates will occur at any location in the Hawaiian Islands if an analysis is based upon a short period of time that includes the years 1946 through 1964 since the exceptionally frequent occurrence of major tsunamis in Hilo from 1946 through 1964 is a property of the unusual activity of tsunami generation areas and not of special properties of Hilo. The unusual tsunamigenic activity in generation regions in recent years is reflected in the most uptodate catalog of tsunami occurrence in the circumpacific area complied by Soloviev and Go (1969). This catalog shows, for example, that the three greatest intensity tsunamis generated in the AleutianAlaskan region since
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65 1788 occurred in 1946, 1957, and 1964. The catalog also lists the 1960 Chilean tsunami as the greatest intensity tsunami generated in South America in recorded history [416year period from 1562 through 1977) . Reporting of tsunamis on the west coast of South America is quite good with, 105 tsunamis reported during the 416year period. Kanamori (1977] also notes the unusual seismic activity that occurred during a relatively short time span in recent years. He shows that the four earthquakes having the greatest seismic moments (true measure of magnitude for great earthquakes) in this century occurred during a thirteen year period from 1952 through 1964 (1960 Chilean, 1964 Alaskan, 1957 Aleutian, and 1952 Kamchatkan earthquakes). Evidence that tsunamis were unusually active in the Hawaiian Islands from 1946 through 1964 is also apparent from historical data at Kahului and Lahaina, Maui. Kahului, Lahaina, iiilo, and Honolulu are the only locations in the islands with historical data extending to 1837. Four of the five largest tsunami elevations recorded at Kahului occurred during the years from 1946 through 1964. Three of the four largest tsunami elevations recorded at Lahaina occurred during the same period. The historical data for Honolulu is not very useful because the commonly reported elevations for the 1837 and 1841 tsunamis were, in fact, drops in water level (PararasCarayannis, 1977) and there are no reports for the 1868 and 1877 tsunamis from Chile. Reconstruction of Historic al Data In order to have data covering a sufficiently long time span to make reasonable predictions of tsunami elevations for the Hawaiian Islands, it is necessary to reconstruct historical data for the
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66 years prior to 1946. To do this, it is assumed in this study that tsunamis generated in a single source region (Kamchatka or Chile but not the AleutianAlaskan region) approach the islands from approximately the same direction and have energy lying within the same band of wave periods. The difference in wave elevations at the shoreline in the Hawaiian Islands produced by tsunamis generated at different times in the same region is attributed mainly to differences in deepwater wave amplitudes. For example, the 1841 tsunami from Kamchatka produced a wave elevation in Hilo that was approximately 25 percent greater than that of the 1952 tsunami from Kamchatka. It is assumed that these two tsunamis had the same relative magnitudes throughout the islands. Since the finite element model allows a determination of elevations for the 1952 tsunami along the entire coastline of the islands, the elevations of the 1S41 tsunami and all ether Kamchatkan tsunamis occurring during the period of record in Hilo from 1857 to 1946 can be reconstructed for the entire coastline. Similarly, Chilean tsunami elevations can be reconstructed knowing elevations for the 1960 Chilean tsunami. No major tsunamis were generated in the AleutianAlaskan region from 1857 through 1946; hence, no reconstruction is necessary for tsunamis from this region. Fortunately, tsunamis from this region since 1946 were representative of the entire region since these tsunamis were generated in the western Aleutians (1957), central Aleutians (1946) and eastern Alaskan area (1964) . The assumption that tsunamis generated in a single source region (Kamchatka or Chile) approach, the Hawaiian Islands from
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67 nearly the same direction is justified in the case of Kamchatka by the small spatial extent of the known generation area. However, the tsunamigenic area of Chile is considerably larger than that of Kamchatka. Still, the direction oP approach of different tsunamis from Chile is nearly the same because the tsunamigenic region in Chile subtends a relatively small angle with respect to the islands as a result of the great distance between Chile and the Hawaiian Islands and the axis of the Chile Trench maintains a constant orientation relative to the islands (thus directional radiation effects are not important) . The assumption that tsunamis generated in a single source region have energy lying within the same band of wave periods is supported by the elevation patterns produced along coastlines in the Hawaiian Islands by different historical tsunamis. These patterns are such that it has been concluded that "tsunamis of diverse geographic origin are strikingly different, whereas those from nearly the same origin are remarkably similar" (Eaton, et al., 1961). A recent study by IVybro (1977) shows that even the distributions of normalized elevations (i.e. the elevation patterns) produced in the Hawaiian Islands by different AleutianAlaskan tsunamis are nearly the same yet quite different from the distributions for tsunamis of otherorigins. This agreement occurs dispite the fact that the relatively close proximity to Hawaii of the AleutianAlaskan Trench and the varying orientation of the trench axis relative to Hawaii introduces important directional effects for tsunamis generated in the AleutianAlaskan area. Apparently, these directional effects influence the
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magnitude of the elevations in Hawaii but do not greatly alter elevation patterns. Therefore, historical observations support the approach used in this study which estimates the elevations produced by tsunamis from Chile or Kamchatka prior to 1946 based upon data for tsunamis from these tsunamigenic regions recorded during the years of accurate survey measurements from 1946 to 1977. In this study elevations are calculated along the coastline of the Hawaiian Islands for the ten tsunamis which produced the greatest elevations in Hilo, Hawaii, from 1S57 through 1977. The finite element numerical model fills in all data gaps for those tsunamis of the ten that occurred from 1946 through 1977 and then the Hilo data (Cox, 1964) is used to determine relative magnitudes to reconstruct the elevations for those tsunamis that occurred prior to 1946. For much of the coast of the Hawaiian Islands, the ten largest tsunamis since 1837 would be the same as the ten largest tsunamis in Hilo for the same period. That is, the ten largest wouJd be the tsunamis of 1960, 1946, 1925, lS57,'lS77, 1841, 1957, 1952, and the two in 1868. Of course, the order of the ten largest would vary from location to location. For example, the largest tsunami on the South Puna coast of the big island of Hawaii was the locally generated tsunami of 1868. There are a few locations where historical data show that one of the ten largest tsunamis was not among the ten largest in Hilo from 1837 through 1977. For example, the 1896 tsunami generated near Japan produced small elevations in the Hawaiian Islands except at Keauhou on the Kona coast of the bis island of Hawaii. The
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69 observation at Keauhou was, therefore, included in the frequency of occurrence analysis for that location. Similarly, all other historical observations of significant tsunami elevations at isolated locations were included in the frequency of occurrence analyses for those locations,. Historical data for tsunamis prior to 1946 were taken from PararasCarayannis (1977) . Data for the local tsunami of .November 29, 1975, were taken from the compilation by Loomis (1976). Since the local tsunami of 1868 had an earthquake epicenter very near that of the 1975 tsunami and the two apparently produced very similar elevation patterns (Tilling et al . , 1976), historical elevations for the two tsunamis recorded at the same location were used to determine a relative magnitude of the tsunamis. The many observations for the 1975 tsunami were then used to reconstruct elevations for the 186S tsunami. Frequency of Occurre nce Distribution After the ten largest tsunamis from 1837 through 1977 were determined at locations all along the coasts of the Hawaiian Islands using the methods discussed in the previous sections, elevation versus frequency of occurrence curves were determined at each location by leastsquares fitting of the data by the following expression: h = B A log f (4.1) where h = elevation of maximum wave at the shoreline, f = frequency per year of occurrence. Equation (4.1) was used as the frequency of occurrence distribution
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70 since it has been found to agree with historical data at several locations. Cox (1964) found that the logarithm of tsunami frequency of occurrence was linearly related to tsunami elevations for the ten largest tsunamis occurring from 1837 through 1964 at Hilo, Hawaii. Soloviev (1969] has shown a similar relationship between tsunami frequency of generation and intensity for moderate to large tsunamis. Also, earthquake frequency of occurrence and magnitude have been similarly related by Gutenberg and Richter (1965) ; IViegel (1965) found the same relationship for historical tsunamis at San Francisco and Crescent City, California and Adams (1970) for tsunamis at Kahuku Point, Oahu. A recent study by Rascon and Villarreal (1975) revealed the same relationship for historical tsunamis on the west coast of America, excluding Mexico. It is possible that other distributions may agree with the historical data equally well as the logarithmic distribution. For example, the Gumbel distribution has been used in the past to study annual streamflow extremes (Gumbel, 1955). Borgman and Resio (1977) illustrate the use of this distribution to determine frequency curves for nonannual events in wave climatology. To investigate the sensitivity of calculations of 100year elevations on the assumed frequency distribution, the approach of Borgman and Resio was applied to the Ililo data of Cox. The Gumbel distribution yields a 100year elevation of 28.8 feet for data from 1837 through 1977 and an elevation of 42.5 feet for data from 1946 through 1977. This compares with the elevations of 27.3 feet and 44.2 feet calculated for the same time periods using the logarithmic distribution (Equation (4.1)). Clearly, the arguments used earlier concerning the period of time that must be considered for
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"1 a valid analysis are not dependent upon the assumed frequency distribution. Since ample precedent exists for using the logarithmic distribution for analysing tsunami frequency of occurrence, this distribution is used in this study. Results The methods described in the previous sections allow frequency of occurrence curves to be constructed for points all along the coastline of the Hawaiian Islands. The coefficients A and B of Equation (4.1) contain all necessary information to determine frequency of occurrence of tsunami elevations at a location. Plots of the coefficients A and B versus location along the coasts were constructed for all of the islands. Smooth curves were drawn through these coefficients to allow elevation predictions at any location. Figures 18 and 19 show typical plots of A and B for the southeastern coast of the big island of Hawaii (Figure 20). For example, at Hoopuloa (location 11 in Figure 20) A = 6 and B = 4; hence, the 109year elevation (F = .01) is only 8 feet. The 100year elevations for most locations in the Hawaiian Islands are considerably larger. For such situations involving human life, one is less interested in the mean exceedar.ee frequency f (the average frequency per year of tsunamis of equal or greater elevations) than in the chance of a given elevation being exceeded in a certain period of time. To calculate risk, one first notes that tsunamis are usually caused by earthquakes, and earthquakes are often idealized as a generalized Poisson process (Ncwmark and Rosenblueth, 1971 and Der Kiureghian and Ang, 1977). It was assumed by Wiegel (1965) and Rascon and
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73
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74 Figure 20. Location Map for Hawaii.
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75 Villarreal (1975) that tsunamis also follow such a stochastic process. The probability that a tsunami with an average frequency of occurrence of f is exceeded in D years, assuming that tsunamis follow a Poisson process, is given by the following equation: P = 1 e" fD (4.2) Thus, if an acceptable risk is one chance in one thousand that an elevation be exceeded during a 100year period, P = .001 and D = 100 years. Substituting these values in Equation (4.2) yields the elevation h which has one chance in one thousand of being exceeded during the next 100 years. West Coast of the United States Predictions Introductio n Unlike the Hawaiian Islands, the west coast of the continental United States lacks sufficient data to allow tsunami elevation predictions based upon local historical records of tsunami activity. Virtually all of the west coast is completely without data of tsunami occurrence, even for the prominent 1964 tsunami. Only a handful of locations have historical data for tsunamis other than the 1964 tsunami However, the Federal Insurance Administration requires information on tsunami elevations for the entire west coast of the continental United States; even for the many locations that have no known historical data of tsunami activity and for coastal areas that are currently not developed (since these areas may be developed in the future) . The lack of historical data of tsunami activity on the west coast
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76 necessitates the use of numerical models to predict runup elevations. The AleutianAlaskan area and the west coast of South .America were found by Houston and Garcia (1974) to be the tsunamigenic regions of concern to the west coast of the United States. Both regions have sufficient data on the generation of major tsunamis to allow a statistical investigation of tsunami generation. The generation and deepocean propagation model described in Chapter III can then be used to generate representative tsunamis and propagate them across the deep ocean. The nearshore numerical model also described in Chapter III, can then be used to propagate tsunamis from the deep ocean over the continental slope and shelf to shore. In this study, only tsunamis of distant origin are considered in the analysis. Hammack (1972) has shown that near the generation region of an impulsively generated tsunami, the waveform is dependent upon details of the timedependent movement of the ground during the earthquake. Little is known about the actual timedependent ground motion during earthquakes generating tsunamis and this motion cannot be predicted in advance. Also, there is not enough historical data concerning locally generated tsunamis on the west coast to allow predictions of tsunami occurrence. Thus, reasonable predictions on the west coast of the properties of locally generated tsunamis or their likelihood of occurrence are not possible at this time. The probability is not considered very great that a destructive, locally generated tsunami will occur on the west coast of the continental United States. Tsunamis are generally produced by earthquakes having fault movements that exhibit a pronounced "dipslip," or
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77 vertical component of motion. "Strikeslip/' or horizontal displacement, fault movements are inefficient generators of tsunamis. Faults on the west coast of the United States characteristically exhibit "strikeslip" motion since the Pacific block of the earth's crust is moving, horizontally relative to the North American block. The west coast of the United States does not share the characteristics (ocean trenches and island arcs) of knowngenerating areas and, in fact, has not historically been one. Relatively small locally generated tsunamis have been known to occur on the west coast, but there are no reliable reports of major locally generated tsunamis. Heights of 6 feet in the immediate vicinity of the 1927 Point Arguello earthquake are the largest authenticated heights produced by local tsunamis on the west coast) . There could be a few locations on the west coast for which locally generated tsunamis pose a greater hazard than do distantly generated tsunamis because the elevations produced by distantly generated tsunamis are small. However, predictions of elevations produced by locally generated tsunamis are beyond the scope of this study. Tsu nami Occurrence Probabilities Historical data of tsunami generation must be the basis for an analysis that considers the probability of tsunami generation in the two tsunamigenic regions of the Pacific Ocean of concern to the west coast of the continental United Statesthe Aleutian and PeruChile Trench regions. A satisfactory correlation between earthquake magnitude and tsunami intensity has never been demonstrated. Not all large earthquakes occurring in the ocean even generate noticeable
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78 tsunamis. Furthermore, earthquake parameters of importance to tsunami generation, such as focal depth, rise time, and vertical ground displacement, have only been measured for earthquakes occurring in recent years. Therefore, data of earthquake occurrence cannot be used to determine occurrence probabilities of tsunamis. Instead, historical data of tsunami occurrence in generation regions must be used to determine these probabilities. In South America, a wealth of information exists concerning tsunami generation. Reliable data (grouped in intensity increments of onehalf) exist for tsunamis with intensity greater than or equal to for a 171 year period and greater than or equal to 21/2 for a 416year period (Soloviev and Go, 1969). The intensity scale used is a modification (by Soloviev and Go) of the standard ImamuraIida tsunami intensity. Intensity is defined as I = log (/2 H ) (4.3) This definition in terns of an average runup (in meters) over a coast instead of a maximum runup elevation at a single location (used for the standard ImamuraIida scale) tends to eliminate any spurious intensity magnitudes caused by often observed anomalous responses (due, for example, to local resonances) of single isolated locations. Using the most recent and complete catalog of tsunami occurrence, in the Pacific Ocean (Soloviev and Go, 1969) a relationship between tsunami intensity and frequency of occurrence was determined for the tsunamigenic trench running the length of the PeruChile coast. Tsunamis with intensity greater than or equal to were considered.
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79 It was assumed that the logarithm of the tsunami frequency of occurrence was linearly related to the tsunami intensity. Earlier in this chapter, it was shown that such a relationship holds between both earthquake magnitude and frequency of occurrence and tsunami intensity and frequency of occurrence. Letting n(I) equal the probability of a tsunami with an intensity 1 being generated during any given year and using statistics for the entire trench along the PeruChile coast, a leastsquares analysis results in the following expression: n(I) = 0.74e~Â°6iI (4.4) In using statistics for the entire trench area along the PeruChile coast, it is assumed that the probability of tsunami occurrence is uniform along the trench. This is a standard assumption for earthquake frequency analysis (Gutenberg and Richter, 1965) . The tectonic justification of this assumption lies in the fact that a single Sialic block or plate of the earth's crust or lithosphere is dipping into the PeruChile Trench (Wilson, 1959) . It can reasonably be expected that the movement of this single plate is similar along its entire length. In the Aleutian Trench region, only large tsunamis occurring in relatively recent years (since 17SS) have been recorded due to the isolation of the area. Assuming an exponential coefficient of 0.71 for this trench area (determined by Soloviev, 1969) as a mean value for regions of the Pacific with the most data on tsunamis) and using only the reliable data for large tsunamis (intensity greater than or equal to 3.5) from Soloviev and Go (1969), the following relationship
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80 is determined by a leastsquares analysis: nl n it0.711 n ( I j = . 1 1 3 e Again, the probability of tsunami occurrence is assumed to be uniform along the trench. Use of Deterministic Numerical M odels To relate the probability distributions of tsunami intensities to source characteristics, it is assumed that the ratio of the source uplift heights producing two tsunamis of different intensity (as defined in the previous section) is equal to the ratio of the average runup heights produced on the coasts near these tsunami sources. This l r I 2 l ratio is equal to 2 for two tsunamis with intensities I and l 2 If H is the wave height in the direction parallel to the maior a axis of length a cf a tsunami source with an elliptical shape (large tsunamis have historically had ellipticallyshaped uplifts) and H. is the wave height in the direction parallel to the minor b axis of length b, then experimental research of tsunami generation has shown that H, /H is approximately equal to a/b (Hatori, b a 1963). For a large tsunami. H, can be larger than H by a factor J a k o g of as much as 5 or 6. Thus, the orientation of the tsunami source relative to the area where runup is to be determined is very important, i.e., the runup at a distant site due to the generation of a tsunami at one location along a trench cannot be considered as being representative of all possible placements of the tsunami source in the entire trench region. Hence, the Aleutian and PeruChile Trenches had to be segmented and runup along the west coast of the United
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81 States determined for tsunami sources located at the center of each of the segments. The spatial size of a tsunami source was standardized since there is not an apparent correlation between tsunami intensity and spatial size of a tsunami source. for example, the 1946 Aleutian tsunami had an uplift region of very small spatial extent, whereas the 1957 Aleutian tsunami had an uplift region that covered perhaps the greatest spatial extent of any known earthquake (Kclleher, 1972) . Yet the 1946 tsunami had the greater intensity, producing, in general, greater runup elevations in the near and distant regions. The standard source used is discussed in detail by Houston and Garcia (1974). It is elliptical in plan view with a major axis length of 600 miles and a minor axis length of 130 miles. The vertical displacement increases linearly from a zero elevation on the ellipse perimeter to a maximum displacement of 30 feet. The vertical displacement has a parabolic crest (concave downward) parallel to the direction of the major axis. The standard source represents a large tsunami with intensity 4 on the modified ImamuraIida scale. Certainly, tsunamis of low intensity may have smaller spatial extents; however, large tsunamis pose the greatest threat to a distant area such as the west coast of the United States. These large tsunamis can be expected to have similar spatial extents, with any spatial differences being unimportant in the farfield compared with the effects of source orientation and vertical uplift. Vertical uplifts are assigned to different intensities in accordance with the convention discussed at the beginning of this section.
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82 Figures 21 and 22 show the Aleutian Trench divided into 12 segments and the PeruChile Trench into 3 segments. The segments in the Aleutian Trench were approximately onequarter the length of the major axis of the standard source, whereas the segments in the PeruChile Trench were approximately the length of the major axis of the standard source. The standard source was centered in each segment such that the major axis of the source was parallel to the trench axis. Uplift regions historically have had such an orientation relative to trench systems. The Aleutian Trench is segmented much finer than the PeruChile Trench since the Aleutian Trench is oriented relative to the west coast such that elevations produced on the west coast are very sensitive to the exact location of a source along the Trench. Uplifts along the PeruChile Trench do not radiate energy directly toward the west coast regardless of their position along the Trench. The Peru and Chile sections of the PeruChile Trench have constant orientations relative to the west coast of the United States; therefore, elevations on the west coast are not very sensitive to source location within these sections. In each of the segments of the Aleutian and PeruChile Trenches tsunamis with intensities from 2 to 5 in steps of onehalf intensity are generated and propagated across the deep ocean using the deepocean numerical model discussed in Chapter II. Tsunamis with intensity less than 2 are too small to produce significant runup on the west coast. An upper limit of 5 was chosen because the largest tsunami intensity ever reported was less than 5 (Soloviev and Go, 1969). Gutenberg and Richter (1965) indicate that there is an upper
PAGE 92
rJ kj o o k
PAGE 93
SOUT/y P A i F I C C e <4 4/ S4 COLUMBIA BUEM.ENTURA ECUADOR iGUAYAauti PERU Figure 22. Segmented Chilean Trench (CHILE
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85 limit to the strain that can be supported by rock before fracture. Thus, earthquakes only reach certain maximum magnitudes and tsunamis can be expected to have similar upp . r limits to intensity. Perkins (1972) and McGarr (1976) have demon ^tro Led that future earthquakes cannot have seismic moments (measure of earthquake magnitude for large earthquakes) much larger than, those of earthquakes that occurred in recorded history. The waveforms propagated to the west coast by the deepocean model are used as before as input to the nearshore propagation model. Each waveform is propagated from a water depth of 500 meters to shore using the nearshore model and. one of the grids shown in Figure 13. Thus, at each grid location on the shoreline of the west coast, there is a group of 105 waveforms seven waveforms (for intensities from 2 to 5 in onehalf intensity increments) for each segment, of the Aleutian and PeruChile Trenches. Each of these waveforms has an associated probability equal to the probability that a certain intensity tsunami will be generated in a particular segment of a trench region. Effect of the Astronomic al Tide s The maximum "stillwater" elevation produced during tsunami activity is the result of a superposition of tsunamis and tides. Therefore, she statistical affect of the astronomical tides on total tsunami runup must be include.? in the predictive scheme presented in this study. Since the waveforms calculated by the nearshore model go not have a simple form (e.g. sinusoidal), the statistical effect of the astronomical tide on tsunami runup must be determined
PAGE 95
86 through a numerical approach. (The effect of the astronomical tides was not considered in the section of predictions for the Hawaiian Islands because the tidal range is quite snail for these islands and the local historical data implicitly contain the effects of the tides. ) The waveforms calculated by the nearshore numerical model extend over a period of time of approximately two hours. Three or four wave crests (the largest waves in the tsunami) arrive during this time. Smaller waves arriving at later times, however, often persist for days at a coastal location. An analysis of tide gage records of the 1960 and 1964 tsunamis on the west coast indicates that these smaller waves have amplitudes on the average of 40 percent of the maximum wave amplitude of the tsunami. Therefore, a sinusoidal group of these smaller waves were added to each of the calculated waveforms so that the total waveform extended over a 24 hour period. These smaller waves are important for locations where tsunamis are fairly small compared with tidal variations. At such locations the maximum combined tsunami and astronomical elevation occurs during the maximum tidal elevation. A computer program was developed to predict timehistories of the astronomical tides at all grid locations on the west coast of the United States. The program was based upon the harmonic analysis methods used in the past by the Coast and Geodetic Survey for mechanical tidepredicting machines (Schureman, 1948). Tidal constants available from the Coast and Geodetic Survey were used as input to the computer program; and a year of tidal elevations was then predicted for grid locations all along the west coast. The year 1964 was
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87 selected because all the major tidal components had a node factor equal to approximately 1.00 during this year making it an average year (node factors can vary by about 10 percent). The node factor is associated with the revolution of the moon's node and has an 18.6year cycle. Since a tsunami can arrive at any time during this 18.6year period (arrival at a low of the node factor is equally likely as an arrival at a high], the statistical effect of the varying node factor is small and an average value should be used. The statistical effect of the varying node factor on the predicted runup elevations can be shown to be a small fraction of an inch, using the approach discussed in the next section and Appendix A (with the variance of the nodal variation equal to approximately 0.1 feet squared). The year of tidal elevations calculated at each of the nearshore numerical model grid points along the west coast was then subdivided into 15minute segments. The 24hour waveforms were allowed to arrive at the beginning of each of these 15minute segments and then superposed upon the astronomical tide for the 24hour period. The maximum combined tsunami and astronomical tide elevation over the 24hour period was determined for tsunamis arriving during each of these 15minute starting times during a year. All of the maximum elevations had an associated probability equal to the probability that a certain intensity tsunami would be generated in a particular segment of the two trench regions and arrive during a particular 15minute period of a year.
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The many maximum elevations with associated probabilities can be used to determine cumulative probability distributions of combined tsunami and astronomical tide elevations. The maximum elevations are ordered and probabilities summed, starting with the largest elevations, until a desired probability is obtained. The elevation encountered when the summed probabilities reach a desired value P is the elevation that is equaled or exceeded with an average frequency of once every 1/P years. Thus, when the summed probabilities reach the value .01, the elevation associated with the last probability summed is the 100year elevation. The 100year and 500year elevations are determined at all grid points of the nearshore numerical grids using techniques described previously in this chapter. Smooth curves are then used to connect all discrete elevations so that continuous predictions of elevation versus coastline location can be made for all the west coast. Figure 25 shows 100year and 500year elevations in the Crescent City, California, area. Comparison with Local O bservation Predictions Crescent City and San Francisco, California, are the only locations on the portion of the west coast of the United States considered in this study that have sufficient historical data of tsunami activity to allow frequency of occurrence predictions based upon local historical observations. Wiegel (1965) made such predictions of tsunami height (trough to crest height) for Crescent City based upon the period from 1900 to 1965. He predicted 100year and 500year heights at Crescent City of approximately 25.6 feet and 43.2 feet,
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89 respectively. If the crest amplitude is taken to be onehalf the total height, the 100year and 500year elevations are 12.8 feet and 21.6 feet, respectively. However, the crest amplitude at Crescent City is typically greater than onehalf the wave height (e.g. the crest amplitude of the largest wave of the 1964 tsunami was approximately 60 percent of the total height). If Wiegel's analysis is applied to historical crest elevations instead of heights, the 100year and 500year elevations are found to be 15.4 and 26.4 feet, respectively. Furthermore, the analysis can now be applied to the longer time period from 1900 through 1977. The 100year and 500year elevations based upon this longer time span are 14.5 feet and 25.5 feet respectively. Figure 23 shows 100year and 500year elevations of 13.1 feet and 24.9 feet, respectively. These values compare very favorably with the 14.5 feet and 25.5 feet elevations determined from historical data for the period of time from 1909 through 1977. The elevations predicted by the analysis based upon the local historical data are probably somewhat larger than the elevations predicted in this report because the short time period (relative to 100 or 500 years) from 1900 through 1977 includes the exceptionally active years of tsunami generation from 1946 through 1964 (see Chapter III) . The 13.1foot and 24.9foot elevations predicted for Crescent City using the techniques discussed in this study are not totally comparable to the elevations predicted using Wiegel's analysis based upon local historical data since the effect of the astronomical tide has been included in the elevations predicted in this study. However,
PAGE 99
90 z UJ H h < _J 1J'N011VA313
PAGE 100
91 the statistical effect of the astronomical tide on the total elevation is not significant at Crescent City due to the large amplitude of tsunamis there. Appendix A shows that the statistical effect of the astronomical tide for a location where tsunamis are large is to 2 ceo increase the predicted elevation by an amount equal to Â— xÂ— , 2 Â°Â° 2 where a is the tidal variance and equals E C , C is 1 m m m=l equal to the m tidal constituent, and a is given by the following expression [since it was shown earlier in this chapter that elevations are linearly related to the logarithm of probabilities): P(Z) Ae~ a ~ with Z = the elevation above local mean sea level, and P(Z) = the cumulative probability distribution for the elevation at a given site being equal to or exceeding Z due only to the maximum wave of the tsunami. Based upon the tidal constituents 2 predicted by the Coast and Geodetic Survey, o =7.1 fox Crescent City. Using Wiegel's data for the period of time from 1900 through 1977 gives a = .145 . Therefore, the astronomical tide contributes approximately 0.5 feet to the elevations predicted for Crescent City. Frequency of occurrence calculations just inside San Francisco Bay were made in an earlier study (Houston and Garcia, 1975) using a simple onedimensional analytical solution for nearshore propagation. However, portions of San Francisco Bay are included in the nearshore numerical model (see Figure 13) used in this study in order that the effect of the Bay on elevations outside the Bay can be properly
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92 simulated. Thus, approximate elevation calculations can be made in this study at the location of the Presidio tide gage within San Francisco Bay. Figure 24 shows the 100year and 500year elevation predictions at the Presidio to be 6.1 feet and 10.0 feet, respectively. Local historical data of tsunami activity at the Presidio was compiled by Wiegel (1965). Using a logarithmic distribution and Wiegel's wave height data for San Francisco over the period 1900 through 1977 yields 100year and 500year heights (trough to crest) of 8.0 feet and 14.2 feet respectively. If the crest amplitude is onehalf the height, this yields 100year and 500year elevations of 4.0 feet and 7.1 feet respectively. The same analysis based upon data of crest amplitudes yields 100year and 500year elevations of 4.4 feet and 7.8 feet, respectively. It is difficult to compare the elevations predicted using local historical data with the elevations predicted in this study for San Francisco, since the effect of the astronomical tides (included in the calculations of this study) cannot easily be estimated, as was the case for Crescent City. Tsunamis recorded at the Presidio in San Francisco are known to persist at fairly substantial levels for extended periods of time apparently due to an oscillation phenomenon. For example, during the 1964 tsunami there were six waves of approximately equal or greater amplitude than the initial wave (in addition to smaller waves) during the first 8 hours of the tsunami. Oscillations approximately 40 percent of the amplitude of the initial wave persisted for at least another 24 hours. The tidal range at San Francisco is 5.7 feet with mean higher high water (MHHW) 2.7 feet
PAGE 102
U'NO!JLVA313
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94 above mean sea level (MSL) and mean high water (MHW) 1.0 feet above MSL. At least one of the main waves of a tsunami at San Francisco probably superposes upon either mj iiv' (with, the smaller oscillations then adding to MHHW) or MJIIIW. Thus, the effect of the tides is to contribute 1 to. 2. 7 feet to the total elevation. These elevations when added to the elevations predicted from local historical data yield 100year and 500year elevation:; of 5.4 feet to 7.1 feet and 8.8 feet to 10.5 feet, respectively. These elevations compare favorably with the 6.1 feet and 10.0 feet elevations determined in this study.
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CHAPTER V CONCLUSIONS The major objective of the present study has been to determine tsunami elevation frequencies of occurrence at the shoreline for the Hawaiian Islands and the west coast of the continental United States. Numerical models were developed to simulate nearshore tsunami propagation and interaction with coastlines. A finite element numerical model was used to interpolate between modern historical measurements of tsunami elevations in the Hawaiian Islands. Finite difference numerical models were used to propagate tsunamis from the AleutianAlaskan and PeruChile regions to the west coast of the United States so that frequency of occurrence predictions could be made. The good agreement between tide gage recordings and numerical simulations of the 1960 Chilean and 1964 Alaskan tsunamis in the Hawaiian Islands indicates that linear, nondispersive, and dissipationfree equations govern nearshore propagation of the main leading waves of long period tsunamis for coastal regions having very short continental shelves. A short continental shelf limits the time available for the development of effects governed by the terms neglected in the equation of motion. Long wave equations that include nonlinear and bottom stress effects were shown to govern propagation of the main leading waves of long period tsunamis over the shallow water region of the west coast of the United States. 95
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9u Good agreement between the numerical model simulations and tide gage recordings of historical tsunamis confirms the experimental flume tests by Hammack (1972) that showed that the farfield waveform generated by a large impulsive ground displacement does not depend upon the transient motions accompanying the permanent ground motion. The good agreement also shows that the accepted permanent ground deformations of the 1960 Chilean and 1964 Alaskan tsunamis, which were partly measured and partly estimated by Plafker and Savage (1970) and Plafker (1969) respectively, are fairly accurate. Finally, this study shows that numerical models can be used in conjunction with limited available historical information to make reasonable predictions of tsunami elevations on coastlines. For an area having local historical data of tsunami occurrence, such as the Hawaiian Islands, numerical models can be used to predict elevations at particular locations not having data. For an area lacking almost any local historical data, such as the west coast of the United States, elevation predictions can be made based upon data of tsunami occurrence in tsunamigenic regions and numerical model propagation of tsunamis from generation regions to the area of interest.
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APPENDIX SUPERPOSITION OF LARGE TSUNAMIS AND ASTRONOMICAL TIDES If the largest wave of a tsunami produces the largest total elevation when the tsunami is superposed upon the astronomical tide (regardless of where the largest wave falls in the tidal cycle), the statistical effect of the astronomical tide on the total elevation can be estimated analytically. Tsunamis satisfy this condition when their maximum waves have amplitudes larger than the sum of the maximum tidal elevation and the elevation of one of the smaller waves which follow the major waves and persist over periods of days. Let Z = the elevation at any time above local mean sea level. PÂ„(Z) = the cumulative probability distribution for the elevation at a given site being equal to or exceeding Z due only to the maximum wave of the tsunami. P (Z) = the probability of the elevation at the same location being equal to or exceeding Z clue only to the astronomical tide. P(Z) = the cumulative probability distribution for the elevation at a given site being equal to or exceeding Z due to the maximum wave of the tsunami and the astronomical tide. According to Chandrasekhar (1943), P(Z) can be calculated from the equation P(Z) = / f g (A)P s (ZA)dX (Al) 97
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98 where dP R (Z) 'fr ' dz and f fi^ is the P robabilit >' density for the astronomical tide. Pg(Z) can be approximated by a Gaussian distribution. Therefore, c cn> 1 Z /2a f B ^) Â— e (A2) /2ir c where the variance is 2 Â°Â° 2 a = E C and C equals the m tidal constituent. P S (Z) can be approximated by an exponential function (see Chapter II) as follows: P S C) = Ae~ uZ (A3) Substituting Equations (A2) and (A3) into (Al) and performing the integration yields P(Z) = Ae 2 a(Z =) 7 1 or p(z) = Ae" a " J where z 1 = Z Â°Â°2 Thus, the net effect of the astronomical tide is to produce a P(Z) 2 identical with PÂ„(Z) except for a shift of Z by an amount ^~ o ' 2 2 The variance a can be determined for locations in the United States from tidal harmonic coefficients available from the Coast and Geodetic Survey. The exponential constant a in Equation (A3) is obtained from a leastsquares exponential fit of tsunami elevation data at a location.
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REFERENCES Adams, W. M., 1970: Tsunami effects and risk at Kahuku Point, Oahu, Hawaii. E ngineering Geology Case Histories, Geo. Soc. of Amer., No. 8, 6370. ' """ " "' Adams, W. M. , 1975: Conditional expected tsunami inundation for Hawaii. J. of Water ways, Harbors, and Coa stal Eng. Div., Amer. Soc. Civ. Eng. , 101, No. WW4, 319529. ~ ~ Bernard, E. N. and A. C. Vastano, 1977; Numerical computation of tsunami response for island systems. J. of P lrys. Oceanog r., 7, 389595. Borgman, L. E. and D. T. Resio, 1977: Extremal prediction of wave climatology. Ports 77, 4th Ann. Symp. of the Waterways, Port , Coastal, and Ocean Div., Amer. Soc. Civ. Eng., 1, 394412. Carrier, G. F., 1966: Gravity waves on water of variable depth, J. of F luid Me c h., 24, Pt. 4, 641659. Chandrasekhar, S., 1945: Stochastic problems in physics and astronomy. Rev, of Mod. Phys. 15, 189. Chen, H. S. and C. C. Mei, 1974: Oscillations and wave forces in an offshore harbor. Rept. No. 190, Mass. Inst, of Tech., Cambridge, Mass., 215 pp. Cole, R. A. and P. J. Riech, 1976: Computer program for drawing finite element grids. Tech. Rept. N762, U. S. Army Eng. Waterways Exp. Sta., Vicksburg, Miss. 110 pp. Courant, R., 19^5: Variational methods for the solution of problems of equilibrium and vibration. Bull, of the Amer. Math. Sec, 49, 125. Cox, D. C, 1964: Tsunami heightfrequency relationship of Hilo. Nov. 1964, Hawaii Inst, of Geophys., Univ. of Hawaii, Honolulu, Hawaii, 7 pp. Der Kiureghian, A. and A. HS. Ang, 1977: A faultrupture model for seismic risk analysis. Bull, of the Seis. Soc. of A mer., 64, No. 4, 11731194. " ' "" " " " Desai, C. S. and J. F. Abel, 1972: Introd u ction t o the Finite Element Method; A Numer ical Meth od for En gineering Analysis, Van Nostrand Reinhold" Co., 477 pp. " " 99
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100 Dronkers, J. J. 1964: Tidal Computations in Rivers and C oastal Waters, NorthHolland Publishing Co., ' Amsterd am, 51 S pp . ' Eaton, J. P., D. H. Richter, and W. V. Aults, 1961: The tsunami of May 23, 1960, on the island of Hawaii. Bull, of the Seis. Soc. of Airier . , 51, No. 2, 135157. Grace, R. A., 1969: Reef runway hydraulic model study: Honolulu International Airport, threedimensional tsunami study. Dept. of Transp., St. of Hawaii, Honolulu, 23 pp. Gumbel, E. J., 1955: The calculated risk in flood control, Add. Sci. Res., 5, Sec. A, 273280. ~ ~ Gutenberg, B. and C. F. Richter, 1965: Seismicity of the Earth and Associated Phenomena, 2nd ed., Hafner Publishing Co., 310 pp~ Hall, J. V. and G. M. Watts, 1953: Laboratory investigation of the vertical rise of solitary waves on impermeable slopes; Tech. mem. 35, U. S. Corps of Eng., Beach Erosion Board, 75 pp. Hammack, J. L., 1972: Tsunamis; a model of their generation and propagation. Rept. No. KHR28, Calif. Inst, of Tech. Pasadena, Calif., 261 pp. Hammack, J. L., and H. Segur, 1977: Modeling criteria for long water waves. Univ. of Fla., Gainesville, Fla. (to be published). Hatori, T., 1965: Directivity of tsunamis, B ull, of the Earthq uake Res. Inst., Univ., of Tokoyo, 4T, 6181. Heitner, K. L. and Housner, G. W., 1970: Numerical model for tsunami runup. J. of the Waterways, Harbors, and Coastal F.ng. Div., Amer. Sec. Civ . Eng. 96, No. WW3, 701719. " Homma, S., 1950: On the behavior of seismic sea waves around a circuJar island. Geophys. Mag., 21, 199208. Houston, J. R., R. P. Caxver, and D. G. Markle, 1977: Tsunamiwave elevation frequency of occurrence for the Hawaiian Islands. Tech. Rept. H7716, U. S. Army Eng. Waterways Exp. Sta., Vicksburg, Miss . , 109 pp. Houston, J. R. and A. W. Garcia, 1974: Type 16 flood insurance study: tsunami predictions for Pacific coastal communities . Tech. Rept. H743, U. S. Army Eng. Waterways Experiment Sta., Vicksburg, Miss., 128 pp. Houston, J. R. and. A. W. Garcia, 1975: Type 16 flood insurance study: tsunami predictions for Monterey and San Francisco Bays and Puget Sound. Tech. Rept. H7517, U. S. Army Eng. Waterways Exp. Sta., Vicksburg, Miss. , 265 pp.
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101 Houston, J. R., R. K. Whalin, H. L. Butler, D. Raney, 1975 a: Probable maximum tsunami runup for distant seismic events NORCONP1, PSAR, Fugro Inc., Long Beach, Calif., 253 pp. Houston, J. R., R. IV. Khalin, A. W. Garcia, H. L. Butler, 1975 b: Effect of source orientation and location in the Aleutian Trench on tsunami amplitude along the Pacific coast of the continental United States. Res. Rept . H754, U. S. Army Eng . Waterways Exp. Sta., Vicksburg, Miss., 48 pp. Hwang, L. S., H. L. Butler, and D. J. Divoky, 1975: Tsunami model: generation and opensea characteristics. Bull, of the Seis. Soc. of A mer., 62, No. 6, ^91596. Hwang, L. S. and D. Divoky, 1975: Numerical investigations of tsunami behavior. Tetra Tech, Inc., Pasadena, Calif., 37 pp. Jordaan, J. M. and W. M. Adams, 1968: Tsunami height, Oahu, Hawaii: Model and nature. Proc. of the Eleventh Conf. on C oastal Eng., Inst, of Civ. Eng. and the .Amer. Soc. of Civil Eng. 7 15551574. Kanamori, H., 1977: The energy release in great earthquakes. J. of Geoph ys. Res., 82, No. 20, 29812987. Kaplan, K. , 1955: Generalized laboratory study of tsunami runup. Tech. Memo. No. 60, U. S. Army Corps of Eng., Beach Erosion Board, 30 pp. Kelleher. J., J. Savino, H. Rowlett, and W. McCann, 1974: Why and where great thrust earthquakes occur along island arcs. J. of Geophys. Res., 79, No. 32, 4SS94899. ~ Keller, J. B. and H. B. Keller, 1964: Water wave runup on a beach. Service Bureau Corp., New York, 34 pp. Kishi, J. and H. Saeki, 1966: The shoaling, breaking, and runup of solitary waves on impermeable rough slopes. Proceed, of the Tenth Conf. on Coastal Eng., .Amer. Soc. of Civ. EngT~7 1, 322348. Leendertse, J. J., 196": Aspects of a computational model for longperiod waterwave propagation. Erne. Rm5294PR, Rand Corp., Santa Monica, Calif., 165 pp. Loomis H. G. 1970: A method of setting up the eigenvalue problem for the linear, shallow water wave equation for irregular bodies of water with variable water depth and application to bays and harbors in Hawaii. NOAAJTRE16, Hawaii Inst, of Geophys., Univ. of Hawaii, Honolulu, Hawaii, 77 pp. Loomis, H. B. 1976: The tsunami of November 29, 1975 in Hawaii, HIG7521, Hawaii Inst, of Geophys., Univ. of Hawaii, Honolulu, Hawaii, 95 pp.
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io: McGarr, A., 1976: Upper limit to earthquake size. Nature, 262, No. 5567, 378379. Mader, C. L., 1974: Numerical simulation of tsunamis. J_. o f Phys. , 4, 7482. Madsen, 0. S. and C. C. Mei, 1969: The transformation of a solitary wave over an uneven bottom. J. of Fl uid Mech. , 59, 781791. Magoon, 0. T., 1965: Structural damage by tsunamis. Coastal Eng. Santa Barbara Spec, Con f . , Amer. Soc. of Civ. Eng. 356S. Mei, C. C, 1978: Numerical methods in waterwave diffraction and radiation. An n. Rev, of Fluid Mech ., 10. Mei, C. C. and B. LeMehaute, 1966: Note on the equations of Long waves over an uneven bottom. J. of Geophys. Res., 71, 2. Newmark, N. M. and E. Rosenblueth, 1971: Fundamentals of Earthquake Engineering, PrenticeHall, 528 pp. Noye, B. J., 1970: The frequency response of a tidewell: Proc. of the Third Australasian Conf. on Hyd. and Fluid Mech., Inst, of Fng. , Sydney, Australia, No. 2630. Omer, G. C. and H. H. Hall, 1949: The scattering of tsunamis by a cylinderical island. Bull. Seis. Soc. Ame r., 59 , 257260. ParrarasCarayannis, G., 1977: Catalog of tsunamis in Hawaii. Rept. SE4, World Data Center A for Solid Earth Geophysics, Boulder, Colo., 77 pp. Peregrine, D. H., 1967: .Long waves on a beach. J_. o f Fluid Mechanics, 27, pt. 4, 815827. Perkins, D., 1972: The search for maximum magnitude. Nat. Oceanogr. and Atmos. Admin. Earthquake infor. Bull., July, 1972, 1823. Piafker, G. , 1955: Tectonics of the March 27, 1964, Alaskan earthquake. U. S. Geologica l Survey Professional Pa p er, 5431 , 11174. Piafker, G. and J. C. Savage, 1970: Mechanism of the Chilean earthquake of May 21 and 22, 1960. Bull, of Geo. Soc. of .Amer., 81 No. 4, 10011030. Platzman, G. W., 1958: A numerical computation of the surge of 26 June 1954 on Lake Michigan. Geophysica, 6, 40943S. Raney, D. C., 1976: Numerical analysis of tidal circulation for Long Beach Harborexisting conditions and alternate plans for Pier J completion, Misc. Paper H764, U. S. Army Eng. Waterways Exp. Sta., Vicksburg, Miss., 138 pp.
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104 Vastano, A. C. and E. N. Bernard, 1973: Transient long wave response for a multiple island system. J. Phys . Oceanogr. , 5, 406418. Vastano, A. C. and R. 0. Rcid, 1966: A numerical study of the tsunami response of an island. Ref. 6626T, Dept. of Oceanography, Texas AÂ§M University, 141 pp. Wiegel, R. L., 1965: Protection of Crescent City, California from tsunami waves. Redevelopment Agency of the City of Crescent City, 114 pp. Wilson, B. V,'., 1959: Earthquake occurrence and effects in ocean area. CR 69.027, Naval Civ. Eng. Lab., Port Hueneme, Calif., 118 pp. Wilson, B. W. and A. Torum, 1968: The tsunami of the Alaskan earthquake, 1964: engineering evaluation, Tech. Mem. 25, U. S. Army Coastal Eng. Research Center, Washington, D. C, 401 pp. Wybro. P. C, 1977: M. S. Plan B, Univ. of Hawaii, Honolulu, Hawaii, (to be published) . Zienkiewicz, 0. C. and Y. K. Cheung, 1966: Finite elements in the solution of field problems. The Enginee r, 507510. Zienkiewicz, 0. C, 1971: The finite element method in engineering science, 2d. ed., McGraw Hill, 521 pp.
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BIOGRAPHICAL SKETCH James Robert Houston was born in Berkeley, California, on March 31, 1947. He was awarded a Bachelor of Arts degree in Physics at the University of California at Berkeley in June 1969 and a Masters of Science degree in Physics at the University of Chicago in June 1970. His doctoral studies at the University of Chicago were interrupted in November 1970 when he was inducted into the U. S. Army. For a year and a half he was involved in the study of explosively generated water waves in the Nuclear Weapons Effects Laboratory of the U. S. Corps of Engineers Waterways Experiment Station in Vicksburg, Mississippi. Following an honorable discharge from the U. S. Army in May 1972, he was engaged as a Research Physicist by the Wave Dynamics Division, Hydraulics Laboratory, Waterways Experiment Station. In September 1974, he was awarded a Master of Science degree in Coastal Engineering at the University of Florida. He is the author of several reports and papers dealing with numerical investigations of long wave phenomena. 105
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy % â€¢l J^A&i^fr Knox Mi 11 saps, Chairman^' Professor and Chairman of Engineering Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Joseph L. Hammack, Cochairman Associate Professor of Engineering Sciences I certify that I have read this study and that, in my opinion it conforms tc acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Sheppard Associate Professor of Coastal and Oceanographic Engineering and Engineering Sciences
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. m. A. Eisenberg Professor of Engineering and Engineering Sciences I certify that 1 have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. w *i ?t&r>4L0 Michael E. Thomas Professor and Chairman of Industrial and Systems Engineering This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. June 1978 U^i cu. V 7 Dean, College of Engineering Dean, Graduate School
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