Modeling light penetration through a slash pine (Pinus elliottii) canopy

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Title:
Modeling light penetration through a slash pine (Pinus elliottii) canopy
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vii, 136 leaves : ill. ; 28 cm.
Language:
English
Creator:
McKelvey, Kevin Scot, 1956-
Publication Date:

Subjects

Subjects / Keywords:
Forest Resources and Conservation thesis Ph. D   ( lcsh )
Dissertations, Academic -- Forest Resources and Conservation -- UF   ( lcsh )
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bibliography   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1990.
Bibliography:
Includes bibliographical references (leaves 107-109).
Statement of Responsibility:
by Kevin Scot McKelvey.
General Note:
Typescript.
General Note:
Vita.

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Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 001603299
oclc - 23245351
notis - AHM7549
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Full Text












MODELING LIGHT PEN
A SLASH PINE (Pinus


ETRATION THROUGH
elliottii)CANOPY


KEVIN


SCOT


MCKELVEY
a


DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY


OF FLORIDA


.,ull(*Y















This


work


is dedicated


wife


Madeleine,


and


daughters


Adrienne


Tara.














ACKNOWLEDGEMENTS


Much


data


this


dissertation


was


the


result


work


many:


Collection


of data


on CO,


assimilation,


foliar


biomass


light


absorption


canopy


were


carried


out


as part


a four-year


study


slash


pine


funded


National


Science


Foundation.


In particular


would


like


thank


Henry


Gholz


Sherry


Vogel


analysis


analysis


biomass


fluctuation,


of assimilation


data,


Wendell


Cropper


Katherine


Ewel


long


hours


of consultation


without


which


this


work


would


not


have


progressed.


there


is good


science


here,


should


receive


a great


deal


credit.
















TABLE


OF CONTENTS


ACKNOWLEDGEMENTS


ABSTRACT


S.


INTRODUCTION


MODELING


HIERARCHICAL


EFFECTS


IN SLASH


PINE


a a. a 14


Characteristic
Evaluation of


of Sla


Model


sh Pine


Approache


* a a. 14


Light


Penetration


Modeling


in Discrete


Space


. 23


A DISCRETE


-SPACE


MODEL


FOR


SLASH


PINE


. . . 29


Data
Model
Model


Acquisition
Structure
Results


. . 29
*. . . a a 45
*. a . a . 53


MODELING

MODELING


SEASONAL

LIGHT PE


BELOW


-CANOPY


NETRATION


LIGHT


WITHIN


PENETRATION

CANOPY


* 62


Sa a *. 88


EVALUATING


MODEL


a a a a a a a a . 95


CONCLUSIONS


a a . a a a a a a a a .a


Discu


ssion


of Light


Penetration


Through


Future


Slash


Research


Pine


Needs


a a a a a a .
. a a a . a a a


LITERATURE


CITED


a a a a a a a a a a a a a a


APPENDIX


COMPUTER


CODE


FOR


DISCRETE


-SPACE


MODELING


OF LIGHT


PENETRATION


BIOGRAPHICAL


SKETCH










Abstract


of Dissertation


he University
Requirements


MODELING
A SLASH P


of Florida


Presented


in Partial


Degree


LIGHT


INE


of Doctor


PENETRATION


(Pinus elliotti


Graduate


School


Fulfillment


of Philosophy


THROUGH
i) CANOPY


Kevin


Scot


McKelvey


August,


1990


Chairman:


Major


Katherine


Department:


Ewel


School


Forest


Resources


and


Conservation


Conifer


canopies


are


a hierarchy


of discrete


objects:


crowns,


shoots,


and


needles.


Light


penetration


through


this


hierarchy


is controlled


size


arrangement,


and


opacity


of objects


at each


level.


If light


penetration


patterns


crown


are


level,


dominated


relationship


organization


between


the


light


shoot


or the


penetration


and


leaf


area


will


be indirect.


The


crowns


of slash


pine


are


conic,


and


the


foliage


is clumped.


A discrete


shoot-level


geometric


model


was


created


to determine


the


importance


this


structure


on light


penetration.


Shoot-level


parameters


were


measured


the


laboratory


using


controlled


light.


Light


penetration


data


was


gathered


from


sensor


arrays


placed


above


below


the


canopy.


Seasonal


foliage


biomass


patterns


stand


were


derived


from


destructive


sampling,


litterfall,


and


needle


elongation








Light


penetration


through


canopy


was


simulated


full


range


sun


angles


azimuths.


Azimuth


was


found


to be


an important


controlling


parameter


light


penetration,


and


model


output


reduced


to a single


multiple


regression


equation


based


on sun


angle


and


biomass


per


shoot.


This


simplified


model


was


used


to simulate


light


penetration


days.


Data


mid-sky


simulated


indicated


on uniformly


assuming


that


light


overcas


that


emanated


t days.


diffuse


primarily


Diffuse


light


light


from


was


emanated


from


45-degree


angles


of elevation.


model


duplicated


observed


patterns


seasonal


variation


in below-canopy


light


penetration


and


explained


the


daily


variation


in below-canopy


light


penetration


during


a 2-month


period


centered


on the


winter


solstice.


Light


penetration


within


canopy,


when


simulated


planar


surfaces,


was


similar


to patterns


generated


vertically


stratified,


horizontally


homogeneous


models.


When


light


penetration


the


simulated


shoot


positions


was


modeled,


however,


the


patterns


diverged


due


to self-shade


within


crown.


The


model


accurately


tracked


pattern


light


penetration


over


time


and,


once


calibrated,


could


used


predict


seasonal


changes


in foliar


biomass.


Leaf


area


could


nont


however


be accurately


estimated


directly


from


liaht









penetration


because


proportion


light


interception


to woody


biomass


was


unknown.















INTRODUCTION


Photosynthesis


only


pathway


which


sun


energy


can


enter


an ecosystem


be stored


in a metabolically


utilizable


form.


Ecosystem


metabolism


is dependent


upon


limited

estimate


photosynthetic


of photosynthetic


production.


input


Obtaining


therefore


an accurate


essential


any


attempt


to quantify


energy


or carbon


flow


through


ecosystem.


The


actual


measurement


gross


primary


productivity


(GPP)


difficult


due


to photorespiration.


For


most


practical


purposes,


however,


rate


of CO,


ass


imilation,


which


measurable


, provides


a good


approximation


carbon


input


into


system.


intensity


photosynthetically


active


radiation


(PAR)


the


leaf


surface


acts


as a direct


control


on assimilation


rates.


Because


canopy-level


assimilation


sum


the


assimilation


rates


of all


leaves


leaf


assimilation


can


be scaled


canopy


level


quantity


foliage


and


light


penetration


patterns


through


the


foliage


are


known.











Over


time,


foliage


position


light


penetration


are


coupled.


shift


A change


light


in foliage


penetration


distribution


pattern.


will


shift


cause


in the


light


penetration


pattern


will


turn


cause


a change


foliage


light


distribution


penetration


component


of either


due


to growth


formulations


static


are


or dynam


abscission.


therefore a nece

ic physiological


Accurate


ssary


models.


Life


is defined


pattern,


pattern


exists


at all


levels


of scale


nature.


In science


in general


modeling


in particular,


pattern


is often


ignored


in favor


mean


values.


Models


based


on mean


values


are


simpler


less data


intensive,


but


there


are


dangers


this


approach


if relationships


between


variables


are


nonlinear.


In light


foliage


penetration


is generally


modeling


assumed


through


to lack


forest


spatial


canopies,


pattern,


mean


values


light


penetration


through


canopy


are


used


indirectly


to derive


a variety


of parameters


such


leaf


area


index


(LAI),


photosynthesis


, and


evapotranspiration.


Because


these


parameters


are


related


to light


penetration


through


nonlinear


functions,


heterogeneous


foliage


distribution


can


affect


calculated


examines


value


the


these


effect


parameters


foliage


. This


distribution


dissertation


patterns


- I.a


- a


a a a a


'and


mm m


.


.











Traditional


foliage


light


distribution


penetration


models


homogeneous


assume


space


and


that


that


leaves


are


opaque.


this


assumption


of opacity


valid,


probability


of a beam


light


penetrating


canopy


is simply


probability


that


it will


pass


through


canopy


without


contacting


any


the


leaves


. If


the


leaves


are


randomly


distributed,


number


of contacts


follows


Poisson


distribution


with


contacts


= exp(-uz)


where


contacts


= the


probability


that


a beam


passes


through


canopy


without


intersecting


any


leaves,


= the


average


leaf


area,


and


= the


length


path


that


beam


must


travel


(Oker-Blom


1986).


a large


number


beams


, Pno


contacts


becomes


the


average


through


proportion


canopy.


of light


This


beams


equation,


penetrating


generally


to depth


known


Beer


s law,


is often


expressed


o exp(-kuz)










where


= the


photon


density


at depth


photon


density


above


canopy,


and


= the


ratio


Oker-Blom


projected


(1983)


argues


to measured


that,


leaf


area.


in conifers,


shoots


should


be considered


to be discrete


entities.


the


shoots


are


randomly


distributed


opaque


then


equation


2 is


valid


shoots


as well


needles.


this


case,


light


penetration


penetration


through


through shoots

needles to t


will


differ


extent


tha


front

t the


light

density


shoots


differs


from


density


needles


(Carter


Smith


1985).


This


concept


can


be expanded


to include


all


potential


distribution


canopies


patterns


consist


scales


of a hierarchy


of patterning.


of discrete


Conifer


objects


distributed


space,


from


individual


crowns


down


opaque


particles


within


needles.


At each


level


the


distribution


canopy


of foliage


is contained


is discontinuous.


within


All


individual


the


crowns


foliage


which


are


generally


crowns,


foliage


conic


or paraboloid


is confined


the


in form.


regions


Within


adj acent


actively


growing


meristems


(shoots),


and


within


these


shoots,


foliage


confined


to volumes


defined


individual


needles.


These


levels


u .


hierarchy


ars


i t srrata










in which


foliage


can


exist


and


where


the


probability


existence


volume


zero.


defined


For


branch


crowns


foliage


length


is confined


inclination.


Moving


down


through


the


hierarchy,


each


level


is spatially


bounded


higher


levels


. Shoot


position


is confined


within


crown,


needle


position


within


shoot.


For


any


discrete


array


of n objects


space


, the


proportion


light


penetrating


through


array


K
array


= Po


+ PiK,


+ P2K,


PK,


where


Karray


-= the


total


light


penetrating


the


array,


= the


probability


that


no object


encountered,


p1. pn


= the


probabilities


that


objects


are


encountered,


K,.. .K


= the


proportion


light


penetrating


the


objects


given


that


objects


are


encountered.


If it is


assumed


that


objects


are


identical,


then


defined


average


proportion


of light


penetrating


through


an individual


object.


Equation


then


eQmnlI fo


KobJect


l...n


l...n












K
array


+ C PI.


object


If Kobject


small


(the


objects


are


opaque),


then


Karray


approximately


equal


to P


entirely


dependent


stribution


internal


and


structure


density


. If


objects


dominates,


and


then


not


light


their


penetration


through


canopy


will


be controlled


the


distribution


pattern


of objects


canopy.


Because

discrete ob


conifer


jects


canopies


, equation


are

can


composed o

be applied


f a hierarchy


repeatedly


each


level


to define


overall


light


penetration


through


canopy:


-0 Po + Pi.crown


crown


Crown


P shoo
i.shoot


Ki
shoot


0 + Pi,1eaf


leaf


leaf


~lcel1


When


the


canopy


is viewed


in terms


these


equations


it i


possible


to define


the


precise


effect


of each


level


cell


i; P


Kcanopy


K
ahoot











canopy


potential


errors


involved


in assuming


homogeneity


any


level.


Any


level


hierarchy


homogeneous


density


distributed


objects


. If


that


average


define


density


level


, Object


evenly


of integer


order


, then


homogeneity


can


be defined


as a


special


case


of equation


: all


light


beams


encounter


equal


number


of objects


. As


a system


approaches


homogeneity


, all


the


terms


series


except


the


term


defining


mean


number


of encounters


vanish,


Pmean


approaches


, and


the


system


collapses


the


next


lower


level


the


hierarchy.


If Kobject


approximately


equal


zero


any


level,


then


effects


of all


lower


level


will


vanish.


has


traditionally


been


assumption


light


penetration


through


leaves


. Equation


Therefore,


simply


represents


special


case


: Kleaf
- "leaf


, leaves


are


randomly


distributed,


homogeneity


assumed


higher


level


(shoot


and


crown)


hierarchy.


patterns


above


leaf


level


and


are


ignored,


then


light


penetration


through


canopy


will


always


underestimated


primarily


. The


extent


be a function


underestimation


magnitude


will


the


terms


because


wi th


n P hnmnnnna4n i -nt


lQQiimnI-i nn


C~hOh


Ch rm ~


I 1 *"











dominate


their


respective


series,


errors


of omi


ssion


can


large


. If


= 0.5


instance


, with


a stand


random


has


a measured


distribution


= 4.0


leaf


level


and


homogeneity


calculated


light


shoot


penetration


crown


(equation


level


vertical


light


.135*1


. If


however


foliage


is held


randomly distributed

4.0 and a projected


shoots

shoot a


with


an internal


rea


then


leaf

from


area


equation


, the


Kah~


anticipated


light


2 0.37 + 0.37*0


penetration


18*0


0.42


.0182


case


, three


times


as much


light


expected


penetrate


through


random


shoots


even


though


the


mean


projected


largely


leaf


due


area

the


constant


magnitude


in both


the


cases.


term


Thi

.37).


equal


to 1


.0 in


homogeneous


case,


only


0.37


the


canopy


composed


randomly


placed


shoots


In a field


crowns,


crown


structure


and


position


will


dominate


light


penetration


crowns


are


widely


spaced,


no crown


will


approach


ratio


space


between


crowns


crown


diameter


becomes


large.


For


this


reason,


discrete


and


crown


orchards


model


(Mann


have


et al


been


1980


used


to simulate


Norman


and


row


Well


crops


1983;











point


therefore


crown c

shifts


losure

with


and

high


increases

t. For ve


with


rtical


height.

light


crown


entering


an idealized


closed


stand


composed


of uniform


conic


crowns,


Pno crown


equals


point


crown


closure


and


converges


rapidly


on 1


.0 with


increasing


height


(Figure


In species


with


conic


or parabolic


crowns,


light


penetration


patterns


in the


upper


crown


may


therefore


dominated


crown


form


even


if homogeneity


is a reasonable


assumption


light


penetration


the


forest


floor.


Leaf


area


magnitude


an important


canopy


level


parameter


physiological


determining


processes,


is difficult


measure


directly.


common


practice


sample


light


above


below


canopy


and,


assuming


equation


to be valid,


to compute


leaf


area


(Lang


and


Yuequin,


1986;


Pierce


and


Running,


1988;


Campbell


and


Norman


1989)


inverting


the


function.


The


problem


with


this


approach


that,


there


canopy


structure


the


shoot


crown


levels,


then


k loses


definition


ratio


of projected


to measured


leaf


area.


leaves,


k is generally


close


to 0.5,


the


theoretical


value


angular


distribution


leaves


random


(Campbell


and


Norman


1989)


. This


figure,


or one


very


close,


often


used


in equation


to determine


leaf


area


AI U U -


w v


~,~,,, irrr


r


* ~





































































r* %o wn


-4-


Cod
"4

H O

C
4-'O
ui V
HO)


4~VC
CI~r




>vrO
."LCW











distribution


and


shape


clusters


and


will


be only


indirectly


related


leaf


area.


In general,


on light


the


penetration


effects

are di


of various


fficult


scal


to separate


of

at


patterning

the


forest


floor,


and


may


have


no clear


geometrical


interpretation.


does


not


necessarily


mean


that


Beer


doesn


'work


' for


stand,


simply


that


there


may


uncertainty


as to which


level


of hierarchy


controlling


parameters


Mistakes


in understanding


these


hierarchical


effects


can


lead


to a variety


errors


when


using


inver


sion


techniques.


instance,


example


listed


above


(equation


leaf


area


was


unknown


and


the


was


assumed


to be 0.5


calculated


leaf


area


stand


based


on light


penetration


forest


-In(O


floor


.42)/0


would


= 1.74


rather


than


the


actual


value


of 4.0.


Predictions


based


on inversion


are


prone


to similar


errors.


The


effects


new


foliage


growth


on light


penetration,


instance,


depend


impact


new


leaf


production


terms


at each


level


the


hierarchy











it enters


form


new


shoots


of constant


dimension


then


shoot


area


will


be related


to leaf


area


through


a constant


SIf cluster


location


is random,


slightly


modified


Beer


will


approximate


light


penetration


leaf


densities


=1


o exp(


-kuzq)


where


= a clustering constant

interpretation of the


with

ratio


geometrical


of measured


leaf


area


to projected


cluster


area


(Campbell


and


Norman


1989)


Spatial


independence


new


foliage


in conifers


however,

foliage


biologically


production


improbable.


occur


Because


exclusively


both


the


branch


branch


and


tips,


new


foliage


will


enter


system


at spatial


locations


determined


largely


the


position


the


existing


foliage.


means


that


the


position


the


new


foliage


space


will


be conditioned


foliage,


conditional


position


probabilities


the

will


exi


sting


distort


Poisson


distribution.


If constant


shoot


density


assumed,


increases


in shoot


size


will


translated


into


changes


a S. a


m


..


* *


i d










assumed


that


shoots


are


spherical


in form,


then


ratio


of projected


cluster


area


to projected


leaf


area


will


2/ 4r3


= 3/4r


where


= the


radius


the


shoot


Other


geometrical


forms


will


have


more


complex


relationships,


and


proj ected


area


will


change


with


sun


angle


and


azimuth,


but,


regardless


the


shape


shoot


or crown,


equation


cannot


be considered


to be


valid


model


foliage


growth


associated


with


shoot


elongation


or enlargement


Spatial


heterogeneity


of foliage


can


influence


both


the


quantity


Because


pattern


clustering


of light


can


occur


penetrating


the


level


canopy.


crown,


shoot,


or leaf,


it i


necessary


to determine


the


extent


which


heterogeneity


at each


these


levels


affects


light


penetration


before


measured


light


below


the


canopy


can


reliably


used


to estimate


leaf


area.















MODELING


HIERARCHICAL


EFFECTS


IN SLASH


PINE


Characteristics


Slash


Pine


In slash


pine,


carbon


assimilation


rate


individual


needles


largely


dependent


on the


light


that


those


needles


receive


. Nearly


measured


variance


of CO2


assimilation


in needles


can


be accounted


regressing


ass


imilation


against


photosynthetically


active


radiation


(PAR)


(Figure


The


crowns


of slash


pine


are


conic


, and


distribution


of foliage


spatially


limited


the


volume


defined


branch


length


crown


form


(Figure


, which


. The


turn


canopy


controlled


therefore


collection


needle

needles


conic


distribution

exclusively


units

. Slash

within


each

pine

the


having

s (and


area


own


other

branc


individual


pines)


or stem


produce


wood


created


during


current


season


s growth.


In slash


pine,


needle


retention


ess


than


years


(Gholz


and


Fisher


1982),


reason,


need


are


restricted


branch


area


immediately


adjacent


the


actively


growing


~n1 nztl


marl


'PHI


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EtPm~


natt8m


rraitag


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C)
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U)
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Co
owk
rio


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000


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foliated


foliage


region


surrounding


distribution


in a slash


stem


pine


(Figure


stand


The


therefore


spatially


heterogeneous.


The

on light


previous


discussion


penetration


suggests


the

that


impact

light


of heterogeneity

penetration


through


a slash


pine


canopy


should


be influenced


canopy


structure.


Given


crown


form


and


patterns


foliage


growth,


several


hypotheses


can


be posed:


H1:Light


penetration


through


a slash


pine


canopy


differs


in both


pattern


and


quantity


from


light


penetration


through


a homogeneous


canopy.


H2: For


canopies


made


crowns


of discrete


dimensions,


differences


in spacing


pattern


will


have


a stronger


influence


on light


than


predicted


homogeneous


canopies.


H3: If


crowns


are


arrayed


rows,


the


orientation


rows


will


have


a significant


impact


on light


penetration


on the


timing


of maximum


penetration


through


Evaluation


stand.


of Model


Approaches


To explore


effects


of various


levels


ar 4-a~ ..A a n ar a 4 J- .. 3 ----!-.


,I I


L


'1


*


I


1


















CO



I




I


r


,-


OHr


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4rt f



od 0
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I 0r

C:r

a)
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4~C0

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Wc,
>4-I
tOWW

010
OCk
9404-4(
4. Z


a2C


f\A Il I L'k-- I I %LkI.\.Il











penetration


through


a canopy


with


detail


at both


crown


cluster


level


. In


most


common


system


modeling


discontinuous


canopies


, sub-canopies


are


represented


as discrete


geometric


solids


. In


type


modeling,


the


sub-canopies


are


usually


defined


level


of individual


plants


(Mann


et al


1979,


1980


Norman


and


Welles


1983


Rook


et al. 1985


Whitfield,


1986


Grace,


1988)


or sometimes


hedgerows


(Charles


-Edwards


and


Thorpe


1976


Cohen


Fuchs


1987)


. In


case


foliage


plant


or hedgerow


assumed


to be confined


within


the


volume


solid


and


equation


generally


assumed


to be valid


within


boundary


. Thi


type


modeling


called


envelope


modeling,


because


the


leaf


area


of each


sub-canopy


strictly


confined


'envelope


generally


defined


as an ellipsoid


rotation.


Envelope


modeling


very


flexible


because


each


envelope


can


be given


individual


transmission


properties,


envelopes


simulate


can


foliated


even


and


nes


non-


inside


foliated


one


portions


another


of a plant


(Norman


and


Welles,


1983).


an array


of canopi


the


light


reaching


a given


point


can


be described


m


P


..










where


= the


extinction


coefficients


sub


-canopi


1... n,


. .zZ


= the


chordal


distances


defined


a beam


of light


as it


passes


through


the


array


from


a point


source


to a target.


Light


penetration


calculating


equation


target


surface


10 repeatedly


computed


a series


of random


locations


on the


surface


and


generating


a statistical


average


(Norman


and


Welles,


1983).


To compute


must


chordal


be geometrically


therefore

simplicity


present

of form


distances

simple. As


enormous


turn


efficiently


lymmetries


computational


leads


the

crown


difficult


to a variety


solids

form


. Thi


of model


artifices


that


may


as severe


those


simple


model


that


they


are


designed


to replace


. Vertical


density


patterns


are


easy


to model


in a homogeneous


canopy


but


are


very


difficult


to model


using


envelopes


. Because


constant


foliage


density


envelope,


generally


vertical


biomass


assumed


profile


throughout


each


a crown


the


cross


-sectional


area


envelope


as a function


height


. For


many


plants


such


as slash


pine


, foliage


concentrated


toward


the


top


and


the


outside


crown


. .kn










longitudinal


(Generally


plants


are


greater


in height


than


in width


To avoid


artifice


it i


necessary


nest


envelopes


envelope.


problem


. Not


with

does


only


lower

not.


does


densities


however


nesting


within


really

present


each


remove

another


sub-canopy

the


level


model

the i


complexity


internal


envelopes


the p

must


roper


position


be based


not


and


density


on their


actual


attributes,


but


rather


way


in which


they


adjust


cross-sectional


area


outer


envelope.


Envelope


modeling


can


be used


to model


canopies


lower


hierarchical


levels


such


individual


shoots


. Many


the


problems


associated


with


representing


crowns


ellipsoids


of rotation


vanish


when


modeling


done


the


shoot


level.


The


effects


clumpiness


on light


penetration


can


be modeled,


transmi


ssion


and


crown


properties


form,


per


shoo


foliage

t can b


placement


e based


and


on data.


Assumptions


of uniformity


in size


density


may


also


more


valid


the


shoot


level.


Shoot


level


modeling


also


has


the


practical


consequence


of concentrating


emphasis


a unit


of vegetation


that


properties


that


are


measurable


either


in a laboratory


or in a chamber


field


A great


deal


current


phys


biological


work


done


-. a. SI


''


m


II


1













work


to be effectively


scaled


the


crown


and


stand


level,


throughout


it is


necessary


canopy.


Ideally,


to integrate


light


these


penetratio


processes

n should


also


modeled


the


same


unit


on which


physiological


measurements


are


taking


place.


Modeling


however,


a stand


requires


of slash


definition


pine


the


the


light


shoot level,

regime at several


hundred


target


surfaces


per


tree


generated


light


penetration


through


several


thousand


potential


shade


objects

several


that

runs


make

will


up the surrounding

be needed to avoid


stand. Fu

artifices


rthermore,


due


particular


patterns


of shoot


placement.


The


problems


associated


with


using


an envelope


model


the


shoot


level


primarily


in computational


expense.


The


modeling


penetration


single


concepts


through


source.


discussed


a canopy


Diffuse


above


light


radiation


compute


emanating


emanates


light

from a


simultaneously


from


points


dome.


Computation


of diffuse


radiation


involves


integrating


the


light


from


entire


sky:


P(X,Y,Z)


jQ(9S) ex
SSg(e


sin( A )nn~( A ~iAa
-- - - N .-. -


sin( )cos


(11)


ded


(Norman


and


Welles


1983)


D(


sin( Al~nlF:(F3I











where


= the


probability


that


diffuse


light


penetrates


a point


(X,Y,Z),


= the


light


intensity


function


and,


9 and


are


angle


of elevation


azimuth


respectively,


s the


density


of foliage,


and


s the


sum


would


pass


the


z distances


through


if its


that

point


a light beam

of origin lay


along


a vector


starting


the


point


(x,Y


with


a direction


defined


8 and l.


Numerically,


this


equation


is difference


into


a series


of direct

Computing


light computations

diffuse radiation


a grid


patterns


sky


a sing


locations.

le point


with


an integration


step


of 5 degrees


therefore


involves


computing


1300


direct


light


readings.


Liaht


Penetration


ModelinQ


in Discrete


Soace


Most


computational


burden


envelope


modeling


lies


the


calculation


chordal


distances


through


each


sub-canopy


. If,


however,


the


sub-canopies


are


small


and


uniform


in dimension


and


light


penetration


properties,


then


each


sub-canopy


can


treated


as if it


were


particulate











canopy


lies


path


of a beam.


If it does,


then


it will


remove


a certain


proportion


light.


To calculate


the


presence


or absence


of a spherical


sub-canopy


in the


path


of a beam


of light,


a vector


drawn


normal


the


beam


path


from


beam


the


center


the


sub-canopy.


magnitude


vector


is less


than


radius


the


sub-canopy,


the


beam


will


pass


through


sub-canopy;


otherwise


will


not.


This


calculation


is much


simpler


than


computation


a chord.


first


calculation can,


step


in computing


however,


further


a chord.)


The


simplified


the


coordinate


system


itself


aligned


normal


the


beam


path,


with


target


point


the


origin,


and


scaled


that


one


unit


equals


radius


the


sub-canopy.


presence


or absence


sub-canopy


in the


beam


path


can


then


be determined


rounding


the


center


the


sub-canopy


the


nearest


integer


node.


after


rounding,


the


center


sub-canopy


coordinates


(0,0)


the


dimensions


normal


the


beam


path


a magnitude


greater


than


0 in the


dimension


parallel


beam,


sub-


canopy


will


intersect


the


beam.


If all


sub-canopies


are


the


same


size,


then


entire


system


can


be scaled


and


rounded


simultaneously,


and


the


sub-canopies


that


S -











space,


light


penetration


is computed


in discrete


integer


space.


If the


target


surface


itself


is composed


a field


system,


of equal-sized


including


simultaneously


sub-canopy


points


be scaled


and


units,


then


representing


rounded.


entire


target,


light


can


penetration


entire


target


can


then


be determined


through


series


row


searches.


In discrete


dimension.


space,


dimension


sub-canopies


only


lose


reflected


explicit


indirectly


scaling


factor.


Sub-canopies


become


points


with


light


penetration


properties


defined


average


light


penetration


per


unit


area


sub-canopy.


targets


(and


therefore


light


beams)


are


also


restricted


integer


space,


light


penetration


target


point


represents


average


light


penetration


to a grid-sized


area


surrounding


target.


This


discrete


approximation


keeps


fundamental


logic


envelope


model,


modeling


system


as a series


of sub-canopies,


but


allows


application


on a much


finer


scale


or on slower


computers


decrease


in computational


burden.


Using


technique,


modeling


on a cluster


level


can


be accomplished


on micro-computers.


In discrete


space,


homogeneous


medium


of unit











discrete


nature


space


confines


nodes


to integer


locations


, causing


raster


effects


similar


those


seen


low-resolution


computer


graphi


CS.


A plane


diagonal


light


source


will


be represented


model


as a series


of smaller


planes


offset


from


one


another


one


node.


it i


assumed


that


a Beer


s law


model


is appropriate


continuous


shifts,


system,


light


then,


penetration


as the r

through


elative


sun


system


angle

will


decrease


as a function


relative


angle:


= I
i -


o exp(kdz/sin9)


where


= the


relative


angle


between


plane


light


source


If kdz


constant


then


equation


12 simplifies


= I
i


o exp( k/sine)


(13)


where


= a collapsed

penetration


constant


normal


describing


the


light


plane.


- -


-











this


half


one,


same


points


leading


system


will


is modeled


be shaded


to a predicted


as a discrete


points


of 0.252*Io.


entity,


half


this


case


prediction


exponential


error


decline


is less


function


than


Because


is uniformly


convex,


discrete


model


will


always


predict


light


penetration


greater


than


or equal


to a continuous


model.


The


divergence


however,


not


large.


grid


size


decreases,


system


converges


continuous


representation.


Modeling


light


penetration


into


a stand


using


discrete


cluster


level


model


involves


five


steps:


Placing


cluster


into


space


ass


signing


light


penetration


properties


to each


cluster.


Rotating


shoots


to align


system


with


sun


position.


Rounding


clusters


nearest


integer


node.


Searching


presence


of clusters


between


target


nodes


sun.


Quantifying


light


attenuation


caused


shoots


matching


clusters


with


their


specific


light


attenuation


properties.


To summarize


, discontinuous


canopies


can be


modeled


creating


geometric


representations


the


stand


-t S -


a


i


r


.











elipsoids


however,


in real


this


space.


approach


modeling


is extremely


shoot


inefficient.


level,


Discrete


space


models


offer


a more


efficient


alternative


therefore


are


a preferable


approach


shoot


level


modeling.
















A DISCRETE-SPACE


MODEL


FOR


SLASH


PINE


Data


Acquisition


this


study,


a model


was


based


on a stand


plantation-grown


slash


pine.


stand


was


approximately


years.


Total


tree


height


was


approximately


17 m and


crown


depth


was


approximately


Stocking


density


was


approximately


1200


stems*ha-l


parameters


used


to model


location


of foliage


space


were


based


on a data


derived


destructively


sampling


trees


August


1987


trees


in August


1988


. Pertinent


data


obtained


through


sampling


included


number

branch


of branches

length, non


branch


-foliated


position,

branch 1


branch


.ength


angle


total


(length


first


live


side


limb),


number


needle


clusters


(shoots)


per


branch,


biomass


per


branch.


large


one


small


branch


were


removed


from


each


m segment


down


through


crown.


The


segment


began


base


terminal


segments


were


marked


down


stem


base


crown.


Crown


depth


varied


- -


1-I


r


i .











clearly


defined


classes


of branches


exis


within


segment


. If


limbs


were


same


size,


limbs


were


classified


large,


only


one


limb


from


segment


was


sampled


. Large


small


were


therefore


relative


classifications


. The


point


base


terminal


leader


where


top


tree.


segment


terminal,


began

which


was


classified


represented


current


year


s growth,


was


treated


as a branch


was


sampled


. For


each


limb


sampled,


the


position


the


base


limb


tip


limb


relative


tree


were


recorded


. From


these


data


branch


angles


were


determined


using


following


formula


AOEb


= arcsin[(Dtop


- Dbae.)/Lt]


(14)


where


AOEb


= the


branch


angle


from


horizontal,


= the


distance


from


branch


tree,


Dbame


= the


distance


from


base


branch


tree.


- the


total


length


branch.


The


bias


due


sweep


limbs


was


minor


was











Foliage


position


was


determined


assuming


that


foliage


was


branch


restricted


first


region


foliated


from


bifurcation


main


branch.


with


Within


equal


this


region,


foliage


probability,


was


needle


assumed to

biomass was


occur


therefore


uniformly


distributed.


Using


this


technique,


foliage


was


added


to a standardized


stem


composed


segments


to build


a vertical


foliage


profile


stand.


Because


branches


were


angled


between


50 and


degrees

branch


towar

angle


shifted


ertical

the ca


(Figure


Iculated


correcting


position


for

foliage


toward


canopy


(Figure


. For


m long


branch


with


a non-foliated


length


a Db.,e


location


m below


tree


and


an AOEb


of 50


degrees,


this


technique


would


allocate


foliage


m segment,


foliage


to 2


m segment,


only


to 3


m segment


where


branch


originated.


probably


position


Even


placed


. This


after


lower


because


adjustment,


crown


foliage


however,


than


is assumed


foliage


actual


to be evenly


distributed


within


foliated


region


limb,


when,


in fact,


most


foliage


concentrated


toward





















*-0 -


CD
I
NJ


U,




I3


4.


4.


I I p -


1~






















I~


0









in


In
I
4.


-I


O
1


a)




0r)

d4-I











space


was


possible


due


breakage


that


inevitably


accompanied


felling


trees.


each


limb


number


needle


clusters


(apical


growth


meri


stems)


was


counted.


system


used


distribute


foliage


was


also


used


to distribute


clusters,


with


number


of clusters


per


limb


substituted


dry


weight


biomass.


Cluster


distribution


a random


component,


because


number


of clusters


is discrete,


sters


are


placed


with


equal


probability


along


foliated


limb


rather


than


in equal


proportion.


horizontal


distribution


of foliage


around


stem


was


also


determined


using


branch


angle


the


foliated


region


limb.


distance


foliage


from


stem


was


rather


determined


than


sine.


cosine


horizontal


branch


distribution


angle


was


assumed


to be uniform


in regard


to azimuth.


model,


two


techniques


can


be used


to place


foliage

model (


space.


McKelvey,


One,

1988),


which

used


was

the


used


in a preliminary


vertical


distribution


biomass.


this


approach,


needle


clusters


were


assumed


contain


an equal


quantity


of biomass.


Needle


cluster


position


clusters


was


were


therefore


placed


derived


in vertical


from


biomass


space


data,


a weighted


random


--











placement


was


based


on the


branch


length


chosen


height.


second


system


distributing


needle


clusters


uses


vertical


distribution


clusters.


this


approach,

determine


foliage


biomass


extinction


enters


indirectly


coefficient


each


is used


cluster.


This


second


method


was


used


model


presented


here


because


it places


emphasis


on positioning


clusters


accurately


heterogeneity


space.


The


suggests


previous


that


discussion


changes


position


foliage


should


have


more


impact


on light


penetration


than


shifts


in biomass


cluster.


slope


exponential


decline


curve


steepest


values


near


and


progressively


flatter


as values


approach


In a sub-


canopy


model,


where


there


are


gaps


in the


foliage,


model


should


therefore


be most


sensitive


size


gaps


than


properties


foliage


filling


area


between


gaps.


For


purposes


crown


construction,


rules


placing


clusters


space


create


the


gap


structure,


properties


clusters


define


area


between


gaps.


The


priorities


modeling


should


therefore


primarily


to position


needle


sters


accurately


space


a .


I .


.


I


.











direct


light


penetration


is related


to needle


biomass


per


cluster,


whether


orientation


cluster


light


source


affects


either


penetration


or projected


area.


To obtain


these


data


Sa device


was


designed


to allow


controlled


directional


light


to be directed


onto


clusters


from


a variety


of angles


(Figure


. The


device


consisted


of a light


source,


a collimation


tube


a light


sensor


which


was


held


in a fixed


position


relative


light


source.


Cor


A 4


spot


cm parabolic


sensor,


mirror


eliminating


focused


angle


light


coverage


on a Li-


effects


to large


between


objects


such


sensor


as branches.


tube.


Clusters


The


light


were


source


placed


was


mounted


on a stand


with


slip


joints


that


allowed


a full


range


cluster


movement


to remain


three


fixed


dimensions.


light


This


source


allowed


to be moved,


allowing


cluster


to remain


an orientation


similar


natural


orientation


tree.


cluster


was


held


in a movable


stand


(Figure


. By


moving


cluster


vertically


horizontally


through


a series


of fixed


positions


was


possible


to define


a grid


normal


light


source


any


sun


angle.


rotating


cluster


position


clamp


was


possible


to simulate


any


azimuth.

















LIGHT












COLLI MATION
TUBE


SEHSOR


Figure


device


used


to determine


light


penetration


through


clusters.






















































Figure


Apparatus
light pene
horizontal


for positioning needle clusters for
tration measurements. Vertical and
movement of the stand, coupled with











generally


angled


toward


branch


tip.


When


viewed


from


branch


clusters


are


circular,


with


a dense


center


defined


apical


meristem


a maximum


radius


equal


needle


length.


orientation,


clusters


are


circular


with


a radius


of 28


cm regardless


of biomass.


From


side


view,


clusters


are


less


similar


in form.


length


cluster


is a function


growth


rate


over


past


season


or two,


depending


on the


time


year.


In April


, the


cluster


holds


one


cohort


of needles,


October


it holds


two.


In general,


because


new


needles


grow


a very


acute


angle


from


the


stem,


region


beyond


limb


forms


a hemisphere


defined


the


needle


length.


Below


hemispherical


end,


there


a portion


indeterminate


length


that


is cylindrical.


At its


base,


this


cylinder


can


terminate


in a hemisphere


needles


are


droop,


or in


triangle


needles


are


young


are


still


stiff


angled


toward


branch


tip.


Light


diverge


from


penetration


greatest


(parallel


end


projected


extent


when


limb)


cluster


area


cluster


relative


will


is viewed


side


(normal


limb).


Light


penetration


through


the


cluster


from


other


angles


will


encounter


a form


intermediate


between


these


extremes.


Measuring


light


penetration


-~~~I -


r I


I


mm a












range


of penetration


area


projection


from


potential


sun


angles.


Needle


clusters


were


sampled


at 5


cm intervals


over


grid


of 30x30


cm.


With


cm overlap


caused


parabolic


mirror


area


covered


was


approximately


34x34


cm.


This


area


is large


enough


encompass


the


end


views


needle


clusters


in their


entirety.


It encompassed


side


views


of smaller


clusters,


but


larger


clusters


cm long)


projected


beyond


grid.


Sample


clusters


were


centered


grid


using


apical


meristem


as the


center


end


views


midpoint


cluster


side


views.


Light


penetration


was


calculated


clusters


from


side


results


compared


. The


average


light


penetration


values


unit


area


these


two


views


differed


less


than


among


clusters


(average


less


light


penetration


from


side).


Rounded


nearest


compared


using


chi-square


test


of fit


(Table


Figure


clusters


showed


no significant


difference


< 0.05)


and


four


showed


significant


differences


in pattern.


Significant


differences


between


side


views


were


largely


due


increase


in projected


branch


area


when


viewed


from


side.
















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light


source


changes


. The


larger


clusters


are


similar


density


will


project


as much


as twice


area


when


viewed


from


side


rather


than


from


end.


This


can


lead


to a disparity


projected


area


between


these


modeled


clusters


area


represent


actual


a significant


proportion


total


cluster


population.


To determine


proportion


of proportion


of large


clusters


canopy


lengths


of all


the


clusters


were


measured


on five


trees


with


stem


diameters


ranging


from


15.8


cm to 25.2


cm in diameter


at breast


height


(dbh


1.37


. Of


1585


clusters


sampled,


were


less


than


cm in length


were


less


than


cm in length


(Figure


10).


Based


on this


sample,


the


maximum


error


estimated


cluster


area


due


the


presence


of large


clusters


was


less


than


. At


minimum


biomass


levels,


canopy


is primarily


composed


of clusters


small


enough


to project


equivalent


assumption


that


area


clusters


density


can


from


treated


sun


as points


angles.


with


average


area


uniform


penetration


properties,


necessary


condition


discrete


cluster-level


modeling,


therefore


justified.


A regression


cluster


penetration


as a function


biomass


was


constructed


. For


regression,


light


-~~~' -A-- I -a


^ _m


r r







































O 0
f rn











captures


foliage


regardless


of cluster


length,


because


orientation


clusters


tree


suggests


toward


that


light,


view


limb,


dominates.


which


clusters


represents


grow


change


position


meristem


over


time,


must


roughly


parallel


beams


of light


emanating


from


light


source.)


A log-


linear


(exponential


decline)


curve,


was


chosen


(Figure


because


same


form


exponential


decline


curve


variables


are


analogous


those


used


Beer


s law


modeling.


The


curve


also


stable


properties


when


used


to extrapolate


beyond


edge


data


set.


Model


Structure


Cluster


level


modeling


very


flexible,


a variety


of configurations


can


be used


to model


stand.


For


results


presented


here,


stand


was


modeled


using


following


rules:


trees


were


built


using


same


procedure


(Figure

branch


12).


foliage


angles,


branch


quantity,

lengths


cluster


were


numbers,


identical


trees.


parameters


were


mean


values










a 46





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LOCATE TREE TOP


MOVE DOWN ONE METER


I COMPUTE BRANCH VECTOR


N




(ORE SECTIONS






DONE


Figure


The


rules


building


a tree


in the


cluster


level


PLACE CLUSTER RANDOMLY
ALONG THE FOLIATED
REGION OF THE BRANCH











of clusters


in both


azimuth


angle


was


random.


trees


were


identical


in height


trees


were


evenly


spaced


on a grid


aligned


with


cardinal


compass


directions.


Rows


columns


ran


exactly


north-south


east-west,


respectively.


the


model,


clusters


are


held


as locations


three-dimensional


space.


Light


penetrates


system


clusters


determine


obliquely


light


as a series


penetration


of parallel


through


beams.


system,


discrete


that


approximation


coordinate


technique


system


outlined


be aligned


above


normal


requires


light


source


before


scaling


rounding.


alignment


accomplished


rotating


cluster


positions


around


arbitrary


origin


until


they


occupy


positions


space


that


they


would


have


they


had


remained


fixed


cartesian


coordinate


system


been


rotated.


The


process


starts


with


an initial


orientation


in which


the


axes


axis


define


is height.


a plane


initial


parallel


sun


location


ground


and


is at 0 degrees


parallel


X axis.


Cluster


rotation


is always


relative


this


initial


configuration


and


is inverse


movement


light


source.


light


source


moves


rinht-


I. I


rlu S atsr


nos i t ions


are


rotated


down


.


t~ha


.


.











axes


with


daily


sun


angles


azimuths


can


be generated


through t

available


rigonometric


in the


relationships,


literature


(Campbell,


equations


1977


are


; Walraven,


1978


Gates,


1980).


Rotation


of points


around


an arbitrary


origin


requires


several


transformations


. The


point


set


must


first


centered


on the


origin,


then


scalar


rotations


are


necessary.


first


aligns


axes


with


the


sun


azimuth,


second


with


AOE.


This


process


can


be accomplished


efficiently


through


use


transition


matrices.


Any


rotation


a group


of points


in Cartesian


space


can


be viewed


as a series


of individual


translations


coordinate


point.


vectors,


rotation


each


vector


algorithm


representing


can


therefore


an individual


be determined


tracing


translation


of a single


coordinate


vector,


through


multiplication


with


a series


transition


matrices


first


I*


translation


can


be written


as C


second


' *


so forth


can,


however


, be substituted


in place


of C


second


translation


can


therefore


be expressed


as ,,~


This


process


can


be continued


as many


transition


steps


as are


necessary


to accomplish


desired


rotation.


single


transition


matrix


is created


which


product


,I A J AA


Fri Ff


r ,,',a


- -.4 -


Ir T2


,-- -


LI











vector


to its


final


position


. The


rotation


process


therefore


always


begins


with


same


of initial


coordinates,


there


no cumulative


buildup


errors


during


rotation.


After


rotation,


each


coordinate


vector


multiplied


integer


a scaling


location.


constant


point


rounded


equidistant


nearest


between


integers,


coordinate


is rounded


the


even


integer.


Once


clusters


have


been


rotated


their


positions


rounded,


be computed.


amount


If all


of light


penetrating


clusters


have


target


equal


can


transmission


properties,


light


penetration


point


can


be computed


= Io


exp(-kn)


(15)


where


= the


number


of clusters.


transmission


properties


are


different,


then


light


penetration


is computed


1 I


o exp[


+ k2d,


+...k-d.)]


(16)


where


lr I, 1-ho ~ ~ ~ ah nil- nr- an ran 4 1 an ~c' a 1


kd,


L t


tha


C3V+I nrti an


aan~











individual


needle


cluster,


d,... d.


= the


densities.


light


penetration


cluster


is known


(rather


than


computed


through


a Beer


s law


relationship),


then


this


calculation


collapses


to a product


series:


n
S I I
i=l


(17)


if the


penetration


is identical


clusters:


= Io


(18)


With


Cartesian


coordinate


system


parallel


solar


radiation


along


X axis,


task


of searching


shade


becomes


the


search


points


that


have


same


Z coordinates


target


larger


X values.


This


search


can


potential


coordinate


accomplished


shade


clusters


values.


efficiently


according


array


can


first


to either


then


sorting


or Z


be searched


progressively.


In certain


cases


may


be beneficial


sort


first


X coordinate


to clip


those


points


with


X values


less


than


target,


because


target


is generally


placed


that


maximum


potential


number


shade


clusters


between


it and


the


sun,


this


sort


is not











model


was


run


angles


azimuths


representing


one


quarter


dome.


Light


penetration


was


simulated


AOE


from


to 85 degrees


azimuths


of 185


to 270


degrees


at 5-degree


intervals.


below-canopy


readings,


target


was


a 5x5x1


m rectangular


region


below


canopy


containing


measurement


points


at random


locations.


measurements


within


canopy,


two


target


arrangements

penetration


were


used.


profile,


To simulate


a series


of 5x5


a generalized

xl m planar t


light


argets


were


placed


at 1


m intervals


within


canopy.


To simulate


light


penetration


cluster


surfaces,


a target


tree


was


used


with


each


cluster


acting


as a measurement


point.


vertical


position


of each


cluster


was


used


to determine


a light


penetration


profile


surface


clusters.


modeling


approach,


an evenly


planted


stand,


allows


simulation


from


potential


angles


(above


degrees)


azimuths


. The


other


three


quarters


dome


can


be filled


in by


assuming


symmetry.


The


minimum


sun


angle


that


can


modeled


is dictated


distance


from


target


edge


simulated


stand


depth


canopy.


question


is whether


a tree


that


is not


modeled


the


potential


to influence


light


environment


at a target


point.


a field


of conic


crowns


a target


base


canoov.


distance











edge


the


system


necessary


ensure


that


edge


effects


affect


light


penetration


target


(19)


= tan(AOE)/


where


= the


distance


edge


system,


= the


depth


canopy.


This f

breaks


unction

rather


becomes

sharply


infinite


below


as AOE

degrees


approaches


AOE.


0 and


In Florida,


direct


light


emanating


from


AOE


greater


than


30 degrees


dominates

solstice,


daily

sun i


light


s higher


integral.


than


Even


degrees


winter

a 5-hour


period


mid-day.


Florida,


30 degrees


therefore


a reasonable


minimum


angle.


more


northern


latitudes,


much


light


may


emanate


from


AOE


less


than


degrees.


Model


Results


In a three


dimensional


model,


asymmetries


in foliage


distribution


may


make


light


penetration


strongly


dependent


on azimuth.


a row


crop,


highest


penetration


through


a~ ~ ~ ~~~~~~ a1 -. L .-- a-.h --A n --n--.


rt,


.',,,,, ~1


rrnr~~r











penetrates


from


an azimuth


of 270


degrees


(Figure


13).


modeled


tree


configuration,


an azimuth


of 270


degrees


is exactly


parallel


rows


. Thi


phenomenon


however,


only


noticeable


at low


only


occurs


sun


azimuth


precisely


aligned


with


row


. An Azimuth


of 185


with


degrees,


row


instance,


orientation.


is only


Azimuth


degrees


therefore


of line


not


significant


factor


in below-canopy


light


penetration


. Thi


means


that


light


penetration


curves


can


be created


based


sun


angle,


which


simpler


to generate


than


azimuth.


Figure


stand


14 shows


with


average


light


light


penetration


penetration


per


through


cluster


2.88x2


m (1


trees


hectare)


spacing


variance


constant


angles


. For


tree


spacing


biomass


level,


curve


can


be used


directly


generate


light


penetration


patterns


over


time.


The


output


cluster


level


model


, in


case


Collapses


single


curve


that


can


be used


in simple


light


penetration


model


as a replacement


exponential


decline


curves.


Over


course


a year


, needle


growth


abscission


approximately


doubles


halves


foliage


canopy


respectively


. In


model


there


are


three


ways


to simulate


doubling


halving


of foliage


. The


first


to keep


nr A1 A .n 71UT n


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is roughly


equivalent


to dominance


the


side


view


clusters


. The


second


to increase


or decrease


the


penetration

regression


cluster


(Figure


11).


according


This


the


equivalent


developed

t to dominance


view.


third


possibility


to increase


decrease


density


trees.


This


not


a real


possibility


within


stand,


but


halving


number


trees


approximate


equivalent


to a thinning


operation,


leaving


wide


holes


foliage


between


individual

eliminates


crowns


. Doubling


between-tree


gaps


the

and


number


eliminate


trees

s much


effectively


conic


form


canopy


crown


overlap.


Together


these


three


model


runs


tested


importance


cluster


crown


levels


of hierarchy


on light


penetration


through


canopy.


When


these


three


possibilities


were


simulated


There


was


little


difference


between


light


penetration


patterns


generated


shifting


cluster


area


shifting


cluster


density


(Figure


15).


differences


that


exist


occur


at high


AOE


. This


suggests


that,


below-canopy


light


penetration,


sters


were


acting


as if


they


were


randomly


because


located


position


space.


of each


To a large


cluster


extent


was


they


randomly


were,


chosen





























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example


of a Beer


solution


in which


parameters


are


based


on cluster


rather


than


needle


characteristics.


There


were,


however,


significant


differences


between


light


penetration


patterns


due


to changes


of biomass


within


crown


changes


due


to changing


number


crowns


(Figure


15).


These


data


suggest


that


position


trees


is much


more


important


to overall


light


penetration


than


internal


structure


crowns.


Because


model


is insensitive


to whether


foliage


enters


system


as additional


area


or as increased


density


within


same


area,


changing


biomass


within


canopy


was


constant


modeled


changing


keeping the

their light


number


of clusters


penetration


properties.


view


projection


clusters


dominates


over


side


projection


this


view,


area


case,


or if


will


then


clusters


change


keeping


are


with


cluster


small,


shifts


area


cluster


in biomass.


constant


appropriate.


Using


cluster


level


regression


(Figure


11),


light


penetration


potential


penetration


curves


biomass


per


can


created


quantities


cluster


sun


simply


across


shifting


range


angles


light


of potential


values.


This


produces


a family


curves


(Figure


-


i A




















































0J )



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ow


tna I I II Ar a nt~~


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multiple


regression.


equation


to model


output


m spacing


.134


+ 0.130*AOE


- 0.1834*B,


.6E-5*AOE2


(20)


= 0.92


where


= the


fraction


of light


penetrating


base


the


canopy,


= a scalar


producing


varying


between


an average


2.0 and


cluster


light


with


penetration


approximate


median


value


cluster


sample.


A cluster


level


model


was


constructed


slash


pine.


crown

the 1


measured


form


ight


under


was


determined


penetration


controlled


through


through


conditions.


destructive


individual


Model


sampling


clusters


output


was


showed


that


azimuth


was


an important


parameter


controlling


light


behavior


penetration


was


through


captured


canopy.


in a regression


Most


equation


model


based


biomass


AOE.















MODELING


SEASONAL


BELOW-CANOPY


LIGHT


PENETRATION


It is


clear


from


equation


that


most


dynamics


three-dimensional


cluster


level


model


can


collapsed


to a one-dimensional


form.


This


allows


light


penetration


patterns


over


time


to be modeled


using


a much


simpler


model.


Other


than


using


equation


20 rather


than


equation


to a Beer


to predict


s law


light


model


penetration,


a horizontally


a model


identical


homogeneous


canopy


can


be used.


Correct


AOE


can


be generated


through


relatively


simple


entered,


equations,


model


can


biomass


quantities


be validated


can


matching


model


output


with


measured


light


penetration


data


through


the


canopy.


Yearly


biomass


dynamics


were


derived


from


three


sources


of data:


destructive


sampling,


litterfall,


needle


elongation


rates.


Foliage


canopy


any


time


mixture


current


year


s cohort


(new


foliage)


previous


year


s cohort


(old


foliage)


. Over


-year


period,


the


new


foliage


becomes


falls


the


forest


floor


as litterfall.


three


consecutive


years


trees


were


a a a .6-..! a a a I ...fl - ----a-. a... 1.. a... __ 1-a -A--.-


,,,, 1


L,, LL, -I


111


? ,











both


cohorts.


In addition


biomass


estimates


described

at bi-week


earlier,

Iv interv


litterfall


(weekly


was


collected


during


in litter


peak


traps


litterfall


period


in November)


. Needle


growth


was


also


measured


need


at bi-weekly


interval


throughout


season


active


growth


(weekly


during


spring


-break).


these


data


, the


destructive


sampling


was


considered


to be


most


accurate


was


used


as a baseline


biomass


calculation


. From


these


data


biomass


quantities


were


calculated


both


forward


backward


through


time


. The


foliage


dynamics


were


tracked


through


litterfall.


Litterfall


was


assumed


to be entirely


foliage


new


foliage


that


fell


into


traps


due


to storm


damage


was


separated


. New


foliage


dynamics


were


based


on needle


elongation


patterns


. Biomass


was


then


converted


into


sided


leaf


area


using


a geometric


technique


(Gholz


et al.


1990)


Based


this


analysis


was


found


that


the


maximum


foliar


biomass


was


approximately


times


minimum


(Figure


that


biomass


pattern


lagged


the


solar


cycle


approximately


three


months


(Figure


. In


model,


smoothed


biomass


curve


shown


in Figure


was


scaled


a mean


of 1


.0 and


used


as input


for


equation












O
c--0---, -o
-0 -in


0
-in
1-


\%

\J



^/
'-0 I


0-
I


!
-- -"I
I
c-0 I--


0
-0


0
-in


o
Io


0
L-iu


U)

r4

'p>1
'.





+T Ic'





4't
HUV
00)
mxII
00r


Sr Ici I-


Y'~fl I1 kwII


I--
I--


---I
--


n~nlc ilu











(S33 93G)


3J1NV NnS


SA S __


rr~











Light


penetration


through


canopy


was


measured


with


a seri


of fixed


sensor


arrays


. The


arrays


consi


sted


twelve


linear


response


diodes


mounted


on aluminum


beams


m intervals


. Each


array


consisted


beams


placed


right


angel


on narrow


triangular


antenna


towers


. The


arrays


were


level


with


ground


. Two


arrays


were


placed


above


the


canopy,


five


were


placed


immediately


beneath


canopy


were


placed


above


shrub


layer


on the


forest

minute


floor


interval


SReadings


at each


throughout


the


sensor


day


were


taken


at 5-


an averaged


value


recorded


was


each


hour


. The


sensors


were


calibrated


to Li


Cor


spot


sensors


Three


Li-Cor


PAR


spot


sensors


were


also


placed


above


canopy


along


with


Li-Cor


full


spectrum


time


sensors.


regressing


system


hourly


was


above


checked


-canopy


drift


readings


over


from


PAR


spot


sensors


against


above-canopy


diode


arrays


. An


additional

sensors ag


check


ainst


was

the


made

full


regressing


spectrum


sensors


PAR


. The


spot

regression


between


PAR


spot


sensors


above-canopy


diode


arrays


period


starting


on November


, 1987


, through


January


-0.508


1989


was


+ 0.9286


(21)


a 1%











where


= the


mean


above-canopy


diode


arrays,


- = the


mean


above-canopy


spot


sensors


There


no indication


of nonlinear


behavior


any


light


intensity


level


or any


sign


sensor


drift.


Because


below-canopy


sensors


were


fixed


space,


possibility


that


readings


were


biased


their


specific


locations


. A check


was


therefore


made


regressing


below-canopy


sensors


against


each


other


combinations


on an hourly


basi


(Table


. These


regressions


demonstrate


that


sensors,


both


below


canopy


above


shrub


layer


tracked


each


other


very


well


values


ranged


from


0.93


to 0


. Only


three


36 combinations


had


an r


value


of 1


ess


than


. The


slopes,


ideally


, ranged


from


to 1.31 with


falling


below


1.0 and


20 above


. The


average


slope


was


.01.


intercepts,


PAR/m2*s


. No


ideally


significant


, ranged


from


difference


was


to -7 (

observed


pmol

between


shrub


level


sensors


below-canopy


sensors,


suggesting


that


distance


from


canopy


was


not


important


influence


on light


readings.


determining


below-canopy


light,


below-


a a a -


---- ~- --


~I _I~_ 3


JA


i.












Table


Regression


below-ca
10, 1989
sensors


nopy


constants


sensor


arrays


for hourly re
from November


. BC sensors were immediately
were above the shrub layer.


be


ladings of pairs
6, 1987,to Octob
neath the canopy,


Model


Slope + SE
( pmol/m2*sec)


Intercept + SE
(pmol/m2*sec)


BC1
BC1
BC1
BC1
BC1
BC1
BC1
BC1
BC1
BC2
BC2
BC2
BC2
BC2
BC2
BC2
BC2
BC31
BC31
BC31
BC31
BC31
BC31
BC31
BC32
BC32
BC32
BC32
BC32
BC32
BC4
BC4
BC4
Rr4


= BC2
= BC31
= BC32
= BC4
= S1
= S2
= S31
= S32
= S4
= BC31
= BC32
= BC4
= S1
= S2
= S31
= S32
= S4
= BC32
-= BC4
= Sl1
= S2
= S31
= S32
= S4
= BC4
= Sl1
= S2
= S31
= S32
= S4
= S1
= S2
= S31
--=R


).90
).86
).85
).88
).88
).82
1.89
1.86
).85
1.88
1.88
.76
1.82
'.89
'.93
.90
.75
.89
.81
.81
.85
.92
.89
.77
.82
.83
.89
.92
.90
.80
.86
.81
.85
01'


1.81
'.97
1.86
.01
.16
.86
.01
.96
.17
.19
.99
.97
.31
.05
.22
.18
.08
.80
.93
.05
.80
.97
.91
.05
.02
.21
.96
.12
.07
.18
.06
.76
.93
OR


).005
).006
).005
1.006
1.006
i.006
'.005
.006
.008
.008
.007
.014
.011
.006
.006
.007
.017
.004
.008
.008
.005
.004
.005
.011
.008
.008
.005
.004
.005
.009
.007
.006
.006
rArn'


..79
..61
..60
..40
..36
..80
..35
..53
..50
..35
.36
'.93
;.85
,.02
.75
.05
.90
.25
.68
.85
.56
.99
.20
.81
.82
.84
.50
.11
.25
.86
.44
.69
.47
rA j


)er
S











figures


were


then


divided


into


daily


sum


the


average


above-canopy


sensor


arrays


to obtain


fractional


below-canopy


1987,


light


continuing


penetration


through


daily


March


beginning


1989.


November


A period


August


1988


is missing


to a lightning


strike


several


other


days


are


also


missing


to equipment


failure.


Light


estimates


were


made


days


potential


days.


With


good


data


sets


both


below-canopy


light


penetration


biomass


fluctuation,


light


penetration


over


time


can


modeled


validated.


To obtain


accurate


estimates


of daily


light


integral,


a simple


model


can


used


to generate


sun


angles


approximate


intensities


(Campbell


1977).


sine-corrected


light


impacting


on a


flat


surface,


SinAOE


= sin((


sin(L)


sin(Dec)


cos


cos


(Dec)


cos[15(t-t)] )


where


SinAOE


= the


sine


AOE


= the


time


of day


in hours


hour


clock),


= solar


noon,


-= the


latitude


in degrees,


* -----------


LL ,











Sine


corrected


solar


intensity


on flat


ground


sea


level


can


be approximated


= sin[I.,


* T./SinAOE]


where


= the


solar


intensity


on flat


ground,


-= the


solar


constant,


, = atmospheric


transmissivity.


determining


proportional


light


penetration,


can


to 1.0,


is dimensionless.


is also


dimensionless


varies


from


on very


clear


days


to 0.6


on hazy


polluted


days


(Campbell


1977)


Light


penetration


through


canopy


was


calculated


hourly


intervals


using


equations


and


. To


obtain


daily


time


constant


integrals,


results


duration


were

hour.


assumed


This


to be


integration


technique


is justified


daily


time


integrals


because,


symmetry


noon,


integrated


in angular


light


will


distribution


be underestimate


around s

d before


olar

noon


overestimated


after


noon


approximately


equal


amounts.


time


series


extended


from


November


, 1987


ha- ---

.al ars.n n a -


4 na-


M ~ Pn n I r I u


u L


I


Ckn


nrrrl h~


I.lh n A


Ot~Ah


A 41PS #MI











a tree


cluster


spacing


is assumed


of 2.88x2.88


to be 70%


light


throughout


penetration


time


series,


simulation


matches


data


until


summer


1988


then


diverges


(Figure


. Light


penetration


overestimated


remainder


time


series.


however,


it is assumed


that


light


penetration


per


cluster


represents


minimum


biomass


levels


(March


1988),


biomass


is allowed


to fluctuate


over


time


according


pattern


shown


in Figure


is greatly


improved


(Figure


19).


Diffuse


light


penetration


canopy


differs


from


direct


beam


radiation


because


it emanates


directly


from


portions


dome


simultaneously.


quantity


diffuse


light


reaching


target


point


within


canopy


will


be decreased


dome


that


an amount


is obscured


proportional


surrounding


area


foliage


light


intensity


those


portions


that


are


obscured.


obscured


portion


dome,


turn,


related


distance


that


shade


object


is from


the


target


diffuse


size


modeling


shade


therefore


object.


One


to compute


approach


dome


coverage


each


target


point.


This


approach


used


Norman


Welles


(1983)


envelope


modeling.


Tha r fh Fa


, -


nrhkl arn


ck


ck: n


L~LIA~LII*A~CI~


LL













0
-0
in)


4'0

Vt-
>1.(
-p4C
OWu

'OW


-0
-o
1-


o~m
0 C4 5-I




4.) *4



OkW4
.Com g


vowG
4)J 0 r
4JP-IE
000E
4tQl(
2*4( r
r4v
U) H


4- tO











Diffuse


patterns


are


quite


predictable


when


heavily


overcast


(Figure


become


very


complex


sunny


days


(Grant


1985).


Diffuse


light


patterns


are


therefore


difficult


to simulate


because


they


shift


with


weather


cannot


be generated


through


trigonometric


techniques


. In


clear


conditions,


direct


light


dominates


pattern


of diffuse


light


is strongly


anisotropic.


Under


heavy


overcast,


diffuse


light


is isotropic.


uniformly


overcast


days,


intensity


per


unit


area


dome


is greatest


the


dome


least


horizon.


greatest


area,


quantity


however,


of light


is greatest


therefore


the


emanates


horizon.


from


mid-


region


(Figure


. Because


azimuth


was


unimportant


to light


modeling


penetration


diffuse


into


light


stand,


assume


a very


that


simple


it all


way


emanates


from


mid-sky.


premise


that


diffuse


light


enters


stand


primarily


from


mid-sky


should


have


the


result


that,


winter,


days


dominated


diffuse


light


should


have


higher

days.


light


This


penetration

because, i


through


n North


canopy


Florida,


the


than


clear


maximum


AOE


direct


beam


radiation


winter


solstice


is 38


degrees.


No direct


light


radiation


enters


canopy


from


-I-,-~~~~~F *A t .


* a


L 1


ALI


rr


- .. 1'1


















C
9-4
V
01
0:

ni
0
'C


CJ
0
Ct


.I-


.Cw
nO


0o


a. -a a aL -a a- - -


0


cs)


QGIHO


CSS)


Sg


crm~a,


a~D











be much


more


muted


because


a great


deal


direct


light


emanates


from


mid-sky.


there


a pattern,


it should


reverse


from


winter


pattern


because


-day


direct


light


emanates


from


as high


as 84 degrees.


For


a given


sun


angle,


changes


in above-canopy


PAR


are


to differences


in atmospheric


transmissivity.


Decreased


transmissivity,


proportion


of diffus


to water

e light


or dust,

(Campbell,


increases


1977


the


Spitters


1986).


distributions,


periods


such


with


as exist


similar


sun


around


angle


solstices


, days


with


lowest


integrated


PAR


should


have


highest


proportion


of diffuse


light


those


with


the


highest


PAR


should


have


lowest.


When


a two


month


period


surrounding


each


solstice


was


plotted


against


integrated


PAR,


patterns


supported


hypothesis


that


diffuse


light


emanated


period


solstice,


primarily


extending


there


one


was


from


mid


month


a strong


-sky


on either


negative


(Figure


side


21).


winter


correlation


between


high


above-canopy


PAR


below-canopy


light


penetration.


period


extending


one


month


to either


side


summer


solstice,


there


was


a very


weak


positive


correlation.


Based


this


empirical


support,


diffuse


light


was


aAJ~aA 4n 4-l,


A~ 2 -nA -I


1 .1 A.&. 1 A.


1


--3-'1


*







































































(o o


0CC


(0

0OVC


.C a) )
0)00

S00


ca i


O) ah
.1-


WId AdONVO-3AOBV J0 NOLLDOVY


iVld AdONVO-3AOSV IO NODLLOVUL











days,


randomly


chosen


were


not


sunny;


split


between


direct


diffuse


light


was


purely


random


on those


days


(some


would


be partly


cloudy)


diffuse


light


emanated


from


degrees


AOE.


When


these


rules


were


applied


annual


light


penetration

constancy i


model,


n light


patterns of winter

penetration were d


variance


uplicated


and

with


summer

the


exception


of early


summer


variation


(Figure


22).


early


summer


measured


data


displayed


higher


variation


than


model


predicted.


This


suggests


that


this


approach


to modeling


diffuse


light


captures


a great


deal


the


impact


of cloudy


weather


on canopy


below-canopy


light


penetration,


below-canopy


that


light


pattern


is caused


of variation


differences


observed


in system


sensitivity


to diffuse


radiation


rather


than


differences


weather


pattern.


rules


splitting


days


into


diffuse


direct


light


listed


above


were


largely


arbitrary.


No attempt


was


made


to match


light


penetration


specific


days.


The


proportion


of clear


cloudy


days


was,


however,


reasonably


accurate.


When


total


daily


radiation


days


was


compared


with


theoretical


clear


values,


75.4%


days


were


within


theoretical


values


. The


C 4- ar. r r' av-rn a 4-4 an a4,aa n a 1 A tfltwfA~l fl fn. l


n~a


hnrra 1 aC~nn


khCltnnn


aCrhnn


ChC a 1

















N
N.-
r










I-
NP


FCC

CCQ


0
LO
-0


-0
-0





O
-0
-o


O
-O
r==


N I


Al
4
^Q


0


I











suggests


that


light


penetration


can


be simulated


on a daily


basis if a rule


partitioning


total


light


into


diffuse


direct


components


based


on daily


above-canopy


can


be derived


The


of diffuse


simplest


light


pattern


is a linear


assume


function


that


proportion


of above-canopy


PAR.


period


covered


in Figure


21a,


was


assumed


that


daily


of 100


represented


a uniform


overcast


day


with


100%


diffuse


radiation


of 800


represented


in which


light


entered


stand


as direct


radiation.


When


this


linear


pattern


is modeled,


simulation


tracks


pattern


measured


data,


lower


variation


(Figure


23a)


. This


suggests


that


shift


from


direct


to diffuse


light


is a nonlinear


function


daily

take


PAR

the


In its


form


most


of a switch


extreme


where


form,

the


this

light


nonlinearity


pattern


will


shifts


from


entirely


direct


to entirely


diffuse


at a specific


daily


PAR.


Campbell


(1977)


breaks


diffuse


modeling


into


a clear


model,


which


allocates


to 30%


total


radiation


to diffuse


(depending


on the


amount


haze)


a cloudy


be diffuse.


model


in which


In Campbell


light


s system,


light


is assumed


is assumed
































































dVd AdONVO-3AOa:V JO NOIlOV '


dlVd AdONVD--3AOV JO NOLLDVYL-


cn 1c:m











exist.


If a day


high


atmospheric


transmissivity


partly


cloudy


then,


part


day,


clear


conditions


will


dominate


. Under


these


conditions


there


is a linear


relationship


between


daily


integrated


light


percent


cloud


cover


(Spitters


et al


1986)


. Campbell


models


suggest


that


proportion


of diffuse


radiation


will


also


be linearly


dependent


on the


percent


cloud


cover.


Frontal


weather


will


produce


either


uniformly


cloudy


uniformly


clear


days,


dynamics


should


be much


more


like


with


a switch.


dynamics


A sigmoid


intermediate


a simple


between


nonlinear


a line


function


a switch,


this


pattern


been


shown


to fit


empirical


data


many


sites


(Spitters


et al.


1986)


. When


a sigmoid


function


replaces


linear


dependency,


model


is improved


(Figure


23b).


When


measured


data


are


regressed


simulation,


both


models


account


natural


variation


slope,


ideally


1.0,


is 1


linear


model


sigmoid


model.


Simple


models


diffuse


radiation,


when


coupled


with


regression


duplicate


obtained

observed


through

pattern


cluster

of light


level


modeling,


penetration


through


canopy


on both


a seasonal


a daily


scale.


Because


output


from


cluster


level


model


was


-n A.. A~


4. aI :-a 8. -- i.'


YIIILI CS


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identical


input.


Assuming


constant


foliage,


only


divergence


at high


sun


angles


(Figure


24).


When


biomass


dynamics


were


included


, the


Beer


s law


model


matched


identically


winter


spring


but


was


more


sensitive


biomass


buildup


during


summer


(Figure


. Compared


with


measured


light


data,


Beer


s law


model


provided


than


a poorer


cluster


summer


level


model


light


(Figure


penetration


, in


general,


a sine-corrected


Beer


s model


fits


light


patterns


reasonably


well.


To determine


extent


to which


difference


between


Beer'


on the


model


presence


cluster


of clustering,


level


a modified


model


Beer


was


s law


based


model


was


constructed.


this


model,


was


assumed


that


clusters


rather


than


leaves,


were


the


basic


unit.


clusters


were


randomly


positioned,


light


penetration


through


the


clusters


was


based


regression


derived


through


direct


measurement


(Figure


11).


When


output


from


this


model


is compared


with


output


from


discrete


cluster


model,


patterns


are


very


similar


2.88


m spacing


(Figure


27).


This


provided


further


evidence


that,


in the


cluster-level


model,


clusters


were


projecting


randomly


onto


forest


floor.


close


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representation


provided


a numerically


accurate


picture


light


penetration.















MODELING


LIGHT


PENETRATION


WITHIN


CANOPY


Light


penetration


patterns


through


canopy


are


much


more


difficult


to obtain


than


below-canopy


values


are


more


important


canopy


physiology.


The


cluster-level


model


is capable


of simulating


penetration


either


planar


surface


within


canopy


structure


or to a target


tree


in which


light


penetration


is calculated


surface


of each


cluster.


To obtain


a rough


estimate


light


penetration


series


of planes


within


canopy,


five


profiles


were


run


on a uniformly


overcast


day


(Figure


28).


An overcast


day


was


chosen


to minimize


variation


within


canopy.


profiles


were


created


hanging


a small


triangular


antenna


tower


on the


control


plot


swinging


a line


sensor


in a circle


approximately


m in radius


. Light


readings


were


taken


as much


as possible


at 15 degree


intervals


along


arc.


Each


arc


was


therefore


characterized


individual


readings


which


were


then


averaged


to obtain


a single


reading


representing


arc.


Arcs


were


spaced


at 1-m


intervals


from


approximate


I .- -a- - -5 -.---, -- -


-A


S tl


-~,,~,,
























































U, UIn In
CO In 4-


U




00











to a depth


of 7


The


7-meter


reading


was


entirely


below


canopy.


When


light


profiles


to a series


of planar


surfaces


were


modeled


, the


measured


values


fell


between


simulated


curves


30 and


degrees


AOE


(Figure


29).


model


therefore


data


a mid


-sky


AOE,


as is appropriate


uniform


overcast


conditions.


It is fair


to question


biological


meaning


of light


penetration


profiles


based


on a series


of planar


surfaces


within


the


canopy.


canopy,


line


sensor


was


primarily


sweeping


empty


space ,


measuring


holes


between


crowns.


base


canopy,


was


primarily


measuring


non-foliated


zone


surrounding


stem,


an area


that


presumably


does


provide


a favorable


environment


to support


foliage.


When


was


assumed


that


planes


clusters,


existed


curves


only


at spatial


looked


quite


locations


different


representing


(Figure


strong


self-shade


depression


was


indicated


high


AOE


zone


immediately


below


layer


of maximum


foliar


biomass


(Figure


this


region,


crown


was


still


narrow


(Figure


some


self shading


was


inevitable.


extent


to which


simulated


self-


shade


depression


was


accurate


depends


validity


rules


cluster


nlnmC 1 a n4 --a a -A


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a)

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coS

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t


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presumably


can


grow


of each


other


s way.


A self-shade


depression


should


exist,


is expected


to be smaller


than


was


simulated.


Stratified


Beer


s law


models


were


constructed


both


biomass


profile


adjusted


branch


angle


unadjusted


light


profile


penetration


(Figure


level


6).

the


When

model


below-canopy


based


the


adjusted


biomass


underestimated


light


penetration


the


upper


canopy,


model


based


the


unadjusted


profile


provided


a nearly


perfect


(Figure


. Because


unadjusted


foliage


profile


sted


is clearly


above


incorrect


base


assumed


leader.),


that


this


was


largely


fortuitous.


Returning


discussion


heterogeneity


dense


in the


near


introduction,


canopy.


crowns


no crown


are


small


(equation


will


therefore


should


large


underestimate


computing


(Figure


light


biomass


unadjusted


a homogeneous


penetration.


profile


model


method


shifted


foliage


down


from


actual


position


served


compensate


underestimation


of light


penetration


unit


biomass.