Citation
Borehole Seismic Imaging

Material Information

Title:
Borehole Seismic Imaging A Full Waveform Inversion Approach
Creator:
Jiang, Pengxiang
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (119 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Civil Engineering
Civil and Coastal Engineering
Committee Chair:
HILTUNEN,DENNIS R
Committee Co-Chair:
MCVAY,MICHAEL C
Committee Members:
ROQUE,REYNALDO
VU,LOC QUOC
Graduation Date:
12/13/2013

Subjects

Subjects / Keywords:
Geometric planes ( jstor )
Imaging ( jstor )
Modeling ( jstor )
Multiscale modeling ( jstor )
Parametric models ( jstor )
S waves ( jstor )
Spatial models ( jstor )
Time windows ( jstor )
Velocity ( jstor )
Waveforms ( jstor )
Civil and Coastal Engineering -- Dissertations, Academic -- UF
borehole -- inversion -- seismic
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Civil Engineering thesis, Ph.D.

Notes

Abstract:
Site characterization for the design of deep foundations is crucial for ensuring a reliable and economic substructure design, as unanticipated site conditions can cause significant problems and disputes during construction. Traditional invasive exploration methods sample a small volume of material and insufficiently assess spatial variation in subsurface conditions. Established and emerging surface-based geophysical exploration methods may identify large-scale spatial variability, but fail to provide a detailed picture of the rock quality at depths where a socket is required for the design of a drilled shaft foundation. In order to compensate for the shortcomings of these methods, a new borehole-based characterization method has been developed, which creates images of the shear wave velocity profile along and around the borehole to provide credible socket material analyses and detect nearby anomalies. The proposed imaging technique is based on the time-domain full waveform inversion of elastic waves generated inside a borehole, which are captured by a string of sensors placed vertically along the borehole wall. This approach has the ability to simulate all possible wave types of seismic wavefields, and then compare these simulations with observed data to infer complex subsurface properties. This method formulates and solves the forward model of elastic wave propagation within a borehole using ABAQUS, a commercially available finite element package. The inversion is cast as a least-squares optimization problem solved using the regularized Gauss-Newton method. To test the proposed imaging technique, the present study performed comprehensive numerical studies. First, the accuracy of the forward model and the effectiveness of the inversion scheme was validated. Then, the capability of the proposed imaging technique was evaluated by inverting a series of three-dimensional (3-D) synthetic data sets, including a homogeneous model, a horizontally layered model with high impedance contrast, a cylindrically layered model that mimicked borehole preparation, and simplified earth models containing ring-type anomalies and isolated anomalies. Good models were recovered for each case presented herein. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: HILTUNEN,DENNIS R.
Local:
Co-adviser: MCVAY,MICHAEL C.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-12-31
Statement of Responsibility:
by Pengxiang Jiang.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Embargo Date:
12/31/2014
Resource Identifier:
907780622 ( OCLC )
Classification:
LD1780 2013 ( lcc )

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1 B OREHOLE S EISMIC IMAGING : A F ULL W AVEFORM INVERSION APPROACH By PENGXIANG JIANG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2013

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2 2013 Pengxiang Jiang

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3 To my beloved f amily

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4 ACKNOWLEDGMENTS First of all, I would like to express my sincere gratitude to my advisor Dr. Dennis R. Hiltunen H is vision, passion, guidance and support ma de this work possible. Working with him benefitted me tremendously throughout my graduate life here at the University of Florida. I also would like to thank my supervisory committee members Dr. Mike Mc Vay, Dr. Ronald Roque, and Dr. Vu Quoc for their valuable comments and suggestions for the completion of my dissertation. I am in debt to my parents. T heir unconditional love is what drives me forward. I am grateful to my wife for the love and sense of home that she offers. I thank my friends, w ith whom I have shared many wonderful memories.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF FIGURES ................................ ................................ ................................ .......... 7 ABSTRACT ................................ ................................ ................................ ................... 10 C HAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 12 1.1 Problem Statement ................................ ................................ ........................... 12 1.2 Hypothesis ................................ ................................ ................................ ........ 14 1 .3 Objectives ................................ ................................ ................................ ......... 14 1.4 Scope ................................ ................................ ................................ ................ 15 1.5 Organization of Dissertation ................................ ................................ .............. 15 2 SITE CHARACTERIZATION USING SEISMIC WAVES ................................ ......... 16 2.1 Introduction ................................ ................................ ................................ ....... 16 2.2 Seismic Waves ................................ ................................ ................................ 17 2.3 Surface based Seismic Methods ................................ ................................ ...... 18 2.3.1 Seismic Refle ction Method ................................ ................................ ...... 18 2.3.2 Seismic Refraction Method ................................ ................................ ...... 19 2.3.3 Surface Wave Method ................................ ................................ ............. 21 2.4 Borehole based Seismic Methods ................................ ................................ .... 22 2.4.1 Crosshole Met hod ................................ ................................ ................... 22 2.4.2 Downhole Method ................................ ................................ ................... 23 2.4.3 Suspension P S Velocity Logging ................................ ........................... 24 2.4.4 Full Waveform Sonic Logging ................................ ................................ .. 25 2.5 Initiation of t he Current Research ................................ ................................ ..... 26 3 FULL WAVEFORM INVERSION WITHIN A BOREHOLE ................................ ...... 36 3.1 Introduction of Full Waveform Inversion ................................ ............................ 36 3.2 Forward Problem ................................ ................................ .............................. 38 3.2.1 Theoretical Derivation ................................ ................................ .............. 38 3.2.2 Finite Element Modeling ................................ ................................ .......... 39 3.2.2.1 Spatial temporal discretization ................................ ....................... 40 3.2.2.2 Numerical Implementation ................................ ............................. 41 3.3 Inverse Problem ................................ ................................ ................................ 43 3.3.1 Introduction ................................ ................................ .............................. 43 3.3.2 Gauss Newton Method ................................ ................................ ............ 44 3.3.3 Regularized Gauss Newton Method ................................ ........................ 46 3.4 Practical S trategies for FWI ................................ ................................ .............. 48

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6 3.4.1 Frequency Filtering ................................ ................................ .................. 48 3.4.2 Time Windowing ................................ ................................ ...................... 49 4 VALIDATION OF FORWARD MODEL AND FULL WAVEFORM INVERSION ...... 56 4.1 Introduction ................................ ................................ ................................ ....... 56 4.2 Validation of the Forward Model ................................ ................................ ....... 56 4.3 Inversion of Synthetic Data Generated by the Forward Model .......................... 57 4.3.1 Horizontally Layered Model ................................ ................................ ..... 59 4.3.2 Cylindrically L ayered Model ................................ ................................ ..... 61 4.4 Inversion of Synthetic Data Generated from the 3 D Borehole Model .............. 64 4.4.1 Homogeneous Model ................................ ................................ .............. 65 4.4.2 Homogeneous Models with Ring Anomalies ................................ ........... 65 4.4.3 Hor izontally Layered Models with a Ring Anomaly ................................ .. 67 4.5 Summary ................................ ................................ ................................ .......... 68 5 LOCATING ISOLATED ANOMALIES NEAR A BOREHOLE ................................ .. 86 5.1 Introduction ................................ ................................ ................................ ....... 86 5.2 Overview of Strategies ................................ ................................ ...................... 86 5.2.1 Inversion with Multi component Data ................................ ....................... 86 5.2.2 Inversion with Multi bandwidth Sources ................................ .................. 87 5.2.3 Inversion from Multiple Planes and with Multiple Shots ........................... 87 5.3 Synthetic Model Studies ................................ ................................ ................... 88 5.3.1 Synthetic Model 1 ................................ ................................ .................... 88 5.3.2 Synthetic Model 2 ................................ ................................ .................... 91 5.3.3 Synthetic Model 3 ................................ ................................ .................... 92 5.3.4 Synthetic Model 4 ................................ ................................ .................... 93 5.3.5 Synthetic Model 5 ................................ ................................ .................... 95 5.4 Summary ................................ ................................ ................................ .......... 96 6 CLOSURE ................................ ................................ ................................ ............ 110 6.1 Summary of Findings ................................ ................................ ...................... 110 6.1.1 Forward Modeling ................................ ................................ .................. 110 6.1.2 Inversion System ................................ ................................ ................... 110 6.1.3 Inversion Results ................................ ................................ ................... 111 6.2 Conclusions ................................ ................................ ................................ .... 111 6.3 Borehole Logging Tool ................................ ................................ .................... 112 6.4 Recommendations ................................ ................................ .......................... 113 LIST OF REFERENCES ................................ ................................ ............................. 114 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 119

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7 LIST OF FIGURE S Figure page 2 1 Body waves: a) Compressional waves and b) Shear waves (from Stokoe and Santamarina, 2000) ................................ ................................ ............................ 29 2 2 Surface waves: a) Rayleigh waves and b) Love waves (from Stokoe and Santamarina, 2000) ................................ ................................ ............................ 29 2 3 Field arrangement used in the seismic reflection method: a) No rmal moveout, b) Common offset, and c) Common depth point (from Stokoe and Santamarina, 2000) ................................ ................................ ............................ 30 2 4 Interpretation of a seismic reflection test: a) Time migrated cross section and b) Interpreted geologic profile (from NRC, 2000) ................................ ................ 30 2 5 Seismic refraction met hod: a) Test configuration, b) Travel times, and c) Reconstructed P wave velocity tomogram ................................ ........................ 31 2 6 Surface wave method: a) Test c onfiguration, b) Rayleigh wave dispersion, and c) Inverted shear wave velocity profile ................................ ......................... 32 2 7 Crosshole and downhole methods: a) Crosshole testing, b) Downhole testing, and c) Crosshole tomography (from Stokoe and Santamarina, 2000) ... 33 2 8 Test setup for the suspension P S velocity logging (from Stokoe and Santamarina, 2000) ................................ ................................ ............................ 34 2 9 Test setup for the full waveform sonic logging (from Chabot, 2003) ................... 34 2 10 Schematics of test setup for the proposed borehole based FWI: a) Cross sectional view and b) Top view with eight planes to be scanned ........................ 35 3 1 Dispersion of an axially propagating surface wave in a borehole with (from Kalinski, 1998) ................................ ...................... 51 3 2 Discretization of an axisymmetric borehole model ................................ .............. 51 3 3 A simplified flowchart for the proposed inversion scheme ................................ .. 52 3 4 Concept of frequency filtering: a) Raw data and the frequency spectrum, b) First level filtering, and c) Second level filtering ................................ ................. 54 3 5 Concept of time windowing: gradually increasing the length of the window to facilitate convergence ................................ ................................ ......................... 55 4 1 Waveform comparison for boreholes with varying radii: a) Axial displacement and b) Radial displacement ................................ ................................ ................ 69

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8 4 2 Waveform comparison for a borehole with a 1 km radius and plane strain flat ground solution: a) Tangential displacement and b) Normal displacement ........ 70 4 3 Triangular wavelet sources: a) High frequency source and b) Low frequency source ................................ ................................ ................................ ................. 71 4 4 Borehole synthetic data: a) Radial displacement and b) Axial displacement ...... 72 4 5 The process for conducting the synthetic experiment inside a borehole (axisymmetric model) ................................ ................................ ......................... 73 4 6 Horizontally layered model: a) True model and b) Initial model .......................... 74 4 7 Inversion of the horizontally layered model using the multiscale approach: a) Short time window with low pass filter, b) Short time window without filter, c) Full time window with low pass filter, and d) Full time window without filter ....... 76 4 8 Cylindrically layered model: a) True model and b) Initial model .......................... 76 4 9 Inversion of the cylindrically layered model using the multiscale approach: a) Short time window with low pass filter, b) Short time window without filter, c) Full time window with low pass filter, and d) Full time window without filter ....... 78 4 10 The process for conducting the synthetic experiment inside a borehole (3 D model) ................................ ................................ ................................ ................ 79 4 11 Homogeneous model: a) True model, b) Inverted model, and c) Waveform match and convergence curve ................................ ................................ ............ 80 4 12 Homogeneous model with ring type anomaly (near): a) True model, b) Inverted model, and c) Waveform match and convergence curve ...................... 81 4 13 Homogeneous model with ring type anomaly (far): a) True model, b) Inverted model, and c) Waveform match and convergence curve ................................ .... 82 4 14 Homogeneous model with ring type anomaly (farther): a) True model, b) Inverted model, and c) Waveform match and convergen ce curve ...................... 83 4 15 Horizontally layered model with ring type anomaly: a) True model, b) Inverted model, c) Inversion for short time window, d) Inversion for medium time window, and e) Inversion for full time window ................................ ..................... 85 5 1 Synthetic model 1: true model and veloci ty section ................................ ............ 98 5 2 Synthetic model 1: inversion result of 0 degree plane ................................ ........ 98 5 3 Synthetic model 1: inversion result of 90 degree plane ................................ ...... 99 5 4 Synthetic model 1: inversion result of 180 degree plane ................................ .... 99

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9 5 5 Synthetic model 2: true model an d velocity section ................................ .......... 100 5 6 Synthetic model 2: inversion result of 0 degree plane ................................ ...... 100 5 7 Synthetic model 2: inversion result of 90 degree plane ................................ .... 101 5 8 Synthetic model 2: inversion result of 180 degree plane ................................ .. 101 5 9 Synthetic model 3: true model and velocity section ................................ .......... 102 5 10 Synthetic model 3: inversion result of 0 degree plane ................................ ...... 102 5 11 Synthetic model 3: inversion result of 90 degree plane ................................ .... 103 5 12 Synthetic model 3: inversion result of 180 degree plane ................................ .. 103 5 13 Synthetic model 4: true model ................................ ................................ .......... 104 5 14 Synthetic model 4: velocity section of 0 180 degree plane .............................. 104 5 15 Synthetic model 4: inversion result of 0 degree plane with one shot ................ 105 5 16 Synthetic model 4: inversion result of 90 degree plane with one shot .............. 105 5 17 Synthetic model 4: inversion result of 180 degree plane with one shot ............ 106 5 18 Synthetic model 4: inversion result of 0 degree plane with two shots ............... 106 5 19 Synthetic model 4: inversion result of 90 degree plane with two shots ............. 107 5 20 Synthetic model 4: inversion result of 180 degree plane with two shots ........... 107 5 21 Synthetic model 5: true model and velocity section ................................ .......... 108 5 22 Synthetic model 5: inversion result of 0 degree plane ................................ ...... 108 5 23 Synthetic model 5: inversion resu lt of 90 degree plane ................................ .... 109 5 24 Synthetic model 5: inversion result of 180 degree plane ................................ .. 109

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10 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy BOREHOLE SEISMIC IMAGING : A FULL WAVEFORM INVERSION APPROACH By Pengxiang Jiang December 2013 Chair: Dennis R. Hiltunen Major: Civil Engineering Site characterization for the design of deep foundations is crucial for ensuring a reliable and economic substructure design as unanticipated site conditions can cause significant problems and disputes during construction. Traditional invasive exploration methods sample a small volume of material and insufficiently assess spatial variation in subsurface conditions. E stablished and emerging surface based geophysical exploration methods may identify large scale spatial variability but fail to provide a detailed picture of the rock quality at depth s where a socket is required for the design of a drilled shaft foundation. In order to compensate for the shortcomings of these methods a new borehole based characterization method has been developed which creates images of the shear wave velocity profile along and around the borehole to provide credible socket material analyses and detect nearby anomalies The proposed imaging technique is based on the time domain full waveform inversion of elastic waves generated inside a borehole which are captured by a string of sensors placed vertically along the borehole wall. Th is ap proach has the ability to and then compare these simulations with observed data to infer complex subsurface properties. Th is method

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11 formulates and solves the forward model of elastic wave p ropagation within a borehole using ABAQUS a commercially available finite element package. The inversion is cast as a l east s quare s optimization problem solved using the regularized Gauss Newton method. To test the proposed imaging technique, the present study performed comprehensive numerical studies First, the accuracy of the forward model and the effectiveness of the inversion scheme was validated. Then the capability of the propose d imaging technique was evaluated by inverting a series of three dimensional ( 3 D ) synthetic data sets, including a homogeneous model, a horizontally layered model with high impedance contrast, a cylindrically layered model that mimicked borehole preparation, and simplified earth models containing ring type anomalies and isolated anomalies. Good models were recovered f or each case presented herein

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12 CHAPTER 1 INTRODUCTION 1.1 Problem Statement Unanticipated site conditions are the most com mon cause of significant problems and disputes that occur during the construction of deep foundations and other geotechnical structures. The problem is particularly acute in karst terrain where subsurface conditions are often highly variable. Therefore, site characterization appears to be crucial for ensuring a reli able and economic substructure design. Traditional invasive exploration methods (e.g., SPT and CPT ) probe and sample a small volume of material often insufficiently assessing spatial variation in subsurface conditions. Standard surface seismic methods (e.g., seismic refraction and surface wave dispersion ), which are routinely used for subsurface investigation in civil engineering, are limited in the char acterization of challeng ing geological profiles that include low velocity anomalies or sharp impedance contrasts In addition, these methods, like all surface based geophysical methods, have limited resolving capability below surface level To obtain credible information regarding material below surface level borehole based geophysical methods have to be us ed Conventional crosshole and downhole methods provide one dimensional (1 D) vertical velocity prof iles based on direct arrivals, which are often inadequate for design purpose s Borehole surface wave method (Kalinski, 1998) can be used to derive 1 D lateral S wave velocity profile based on surface wave dispersion, but the investigation depth is only a few inches into the formation. On the other hand, c rosshole tomography is capable of producing two dimensional ( 2 D ) velocity tomograms between boreholes using travel time inversion.

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13 However, if 3 D material characterization is required when locating a drilled shaft in karst terrain for instance the crosshole technique can be too costly as it requires multiple receiver holes around the source hole Recent advance s in borehole geophysics reveal that acoustic logging in a fluid filled borehole can create 1 D P and S wave velocity profiles (Kitsunezaki, 1980) This acoustic logging method can also use reflectivity to characterize the structural propert ies of the geologic formation s behind the borehole wall (Chabot, 2003). The challenge remains as to how the derived wave velocities can be associated with the migr ated structural image By taking advantage of the ever increasing power of computers the full waveform inversion (FWI) technique can now extract information from complete ground surface wavefield s in order to provide high resolution 2 D velocity image s of the subsurface (Virieux and Operto 2009 ). Th e current research explores the possibility of extending FWI techniques to the borehole level in order to quantitatively characterize the spatial variation s in rock formations Th is possibility is of particular advantage when evaluating karst terrain as those subsurface conditions are often highly variable. Given the increased use in Florida of single, large diameter, non redundant drilled shafts as foundation elements a comprehensive evaluation technique is imperative Although surface based FWI has proven superior over state of practice seismic methods in characterizing spatial variability, the inherent tradeoff between resolution and depth is unavoidable Most rock socket ed drilled shafts extend deep beneath the ground ; therefore, borehole based FWI could potentially lead to a more reliable and economic

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14 design solution for drilled shafts in karst terrain since this technique is capable of assessing potential sockets in more detail. 3 D FWI within a borehole would be ideal for assessing drilled shaft sockets; however, 3 D surface based FWI is still in its infancy with numerous uncertainties to be addressed, making 3 D borehole based FWI beyond the scope of this study. Therefore two and a half dimensional ( 2.5 D ) borehole based FWI which is based on an axisymmetric forward model, i s proposed instead in order to evalu ate the feasibility of characterizing spatial variation s in rock formations for drilled shaft foundation design T h is study evaluated the capability of the proposed imaging technique by inverting a series of 3 D synthetic data sets, including a homogeneous model, a horizontally layered model with high impedance contrast, a cylindrically layered model that mimicked borehole preparation, and simplified earth models containing ring type anomalies and isolated anomalies. Good models were recovered for each case 1.2 Hypothesis The b orehole based FWI technique can be used to c haracterize the elastic properties of material along and around a borehole and is capable of producing images with hig her resolution than any other available site characterization method 1.3 Objectives The primary object ive of this research is to evaluate the feasibility of 2.5 D borehole based FWI techniqu e for characterizing the elastic properties of material along and around a borehole. Specific objectives of this research include the following: 1. Develop an accurate and efficient borehole model that can be used as the forward engine for generating synthetic waveforms 2. Develop a reliable and efficient inversion algorithm for borehole b ased FWI in the time domain.

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15 3. Perform synthetic studies for a wide range of geological settings to develop a reasonable array design, including receiver type, number, spacing, and source characteristics. 4. Perform synthetic studies to investigate the feasibility of the proposed imaging technique for locating isolated anomalies in the vicinity of a borehole. 1.4 Scope The forward model used in the inversion is based on a 2.5 D axisy mmetric borehole model. The synthetic data sets used in the inversion are obtained from truly 3 D borehole models. This study used ABAQUS and MATLAB in the implementation of the proposed imaging technique. 1.5 Organization of Dissertation Chapter 2 provides an overview of site characterization methods using seismic waves, both surface based and borehole based. Pros and cons of each method are discussed. Chapter 3 elaborates on the algorithm of FWI within a borehole, including a theoretical derivation of wave propagation in a cylindrical cavity This study built the forward model using ABAQUS and implemented the regularized Gauss Newton method for the inversion Chapter 4 validates the forward model and presents applications of FWI within a borehole. Optimum array design is recommended. Chapter 5 investigates the feasibility of the proposed imaging technique to locate isolated a nomalies (i .e., void or cavity) in the vicinity of a borehole. Several strategies are proposed and adopted for this purpose. Chapter 6 summarizes the findings of this study, followed by conclusions and recommendations.

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16 CHAPTER 2 SITE CHARACTERIZATION USING SEISMIC WAVES 2.1 Introduction Exploration geophysics is the study of the subsurface using quantitative physical methods. These methods are based on extensive theoretical and experimental foundations, some dating back more than a century (Stokoe and Santamarina, 2000). For example, Rayleigh (188 5 ), Love (1892), and Lamb (1904) conducted pioneering studies in the propagation of stress waves Much of the progress in this area has been driven by hydrocarbon exploration and global seismology. Over the past 50 years, geotec hnical, geoenvironmental and earthquake engineering applications in civil engineering have stimulated further development in geophysical exploration methods, among which the seismic methods appear to be the most extensively used site characterization techniques Site characterization is a fundamental step in the proper design, construction and long term performance of all types of civil engineering projects, including foundations, excavation s, earth dams, embankments, seismic hazards, environmental issues, tunnels, and offshore structures. Seismic methods of site characterization involve generating and recording seismic waves to investigate the subsurface Each method is based on the propagation of waves from an artificial source to a set of receivers, followed by an analysis of the subsurface properties of the recorded wavefield Seismic methods are nondestructive by nature, because they are conducted with a strain level so small that the material conditions are not altered These methods can be further classified as noninvasive if performed at the surface level, or invasive if boreholes are involved.

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17 Seismic methods can be used to infer engineering design parameters as strong links exist between the wave propagation characteristi cs and the mechanical properties of the medium to be characterized as rep resented by the following equations: (2 1) (2 2) (2 3) where and are compressional and shear wave velocities, is mass density, is shear modulus, As these equations show the compressional and shear wave velocity profiles derived from a typical seismic survey can be used to estimate the elastic modul i of subsurface materials. Therefore, the broad goal of seismic surveys is to assess these parameters and their spatial distribution. 2.2 Seismic Waves In physics a wave is a disturbance that travels through space and time, accompanied by a transfer of energy (Aki and Richards, 1980) A wave can be transverse or longitudinal depending on the direction of particle oscillation in relation to the direction of wave propagation. Transverse waves occur when the oscillations are perpendicular to the direction of propagation. Longitudinal waves occur when the oscillations are parallel to the direction of propagation. Body waves For stress waves propagating far from any boundaries in a uniform medium, two fundamental modes of propag ation exist: compressional waves ( also called P waves )

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18 and shear waves ( also called S waves ) These waves are known as body waves, since they propagate within the body of the medium P waves are longitudinal while S waves are transverse as illustrated by the schematics of body waves (Figure 2 1) Surface waves I nterfaces alter particle motion, instigating other modes of propagation. Rayleigh waves are generated w hen a free surface interacts with the reflected and refracted body waves Love waves ar e created when the multiple total reflections of horizontally polarized shear waves (SH waves) interact with the surface and underlying stiffer layers As illustrated by the schematics of these two surface waves, the particle motion of Rayleigh waves is a combination of vertical and horizontal motions while the particle motion of Love waves is similar to that of SH waves (Figure 2 2) Therefore, Rayleigh waves are mixed and Love waves are transverse. 2.3 Surface based Seismic Methods Seismic tests are routinely conducted on the ground surface as the acquisition geometry is readily accessible and the t est procedures are relatively cost effective 2.3.1 Seismic Reflection Method The seismic reflection method is one of the oldest seismic methods and is well documented in ASTM standards (ASTM D7128) and many geophysics textbooks (e.g., Burger, 1992). Seismic reflection is founded on the presence of impedance contrasts in the subsurface which occur as a result of variation s in mass density seismic velocity or both. The basic field arrangement used in the seismic reflection method places both source an d receivers on the ground surface (Figure 2 3) Typically, P wave measurements are performed using either vertically impacting sources or explosives.

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19 Waves reflected from interfaces at depth are monitored with vertically sensitive geophones. Depending on the specific application, varying patterns of the source receiver layout can be used in the reflection survey in order to optimize the measurements (Stokoe and Santamarina, 2000). The normal moveout is used to estimate the average velocity of the formation ( Figure 2 3a) Detection of reflectors is usually obtained using the common offset ( Figure 2 3b). The common depth point is used to enhance the si gnal to noise ratio at a specific location ( Figure 2 3c). Advanced data processing algorithms (e.g., reverse time migration), mostly developed by exploration geophysicists, are available and are becoming more widely used in civil engineering. T ypical subsurface cross section s are interpreted from a seismic reflection survey for identifying the alluvium bedrock interface (Figure 2 4) Advantages Seismic reflection is effective regardless of depth related variance in the velocity at which waves propagate through the Earth The subsurface can be qualitatively imaged even for complex geologic profiles as the full reflected wavefield is used for data processing Disadvantages The s eismic reflection survey is costly to conduct, as it requires many source and receiver arrangements in order to produce meaningful subsurface image s Data processing of seismic reflection requires a high level of expertise. Resolution of the migrated image decreases as depth increases. 2.3.2 Seismic Refraction Method The seismic refraction method is another well documented (ASTM D5777) geophysical method for noninvasively identifying the stiffness and layer interface of

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20 sediment s at shallow depth s Conventional refraction methods are used for simple geologic profiles consisti ng of a few constant velocity layers with linear interfaces (Palmer, 1980) Using the ever increasing power of computers seismic refraction tomography can now produce 2 D velocity tomograms of the subsurface by conducting multiple shot gathers along the receiver array (White, 1989; Zhang and Toksoz, 1998) In principle, the seismic refraction method is based on the ability to detect wave energy that is critically refracted from a higher velocity layer that underlies lower velocity sediment P wave measurements are typically performed using either vertically impacting sources or explosives, while direct and refracted waves are monitored with vertically sensitive geophones on the ground surface ( Figure 2 5a ) The travel times are derived by manually selecting the first arrival signals in the recorded wavefield The P wave velocity tomogram is then reconstructed ( Figure 2 5b) by matching the synthetically generated travel times with the manually derived travel times ( Figure 2 5c). Advantages Seismic refraction can provide high resolution velocity images of normal profiles at shallow depth s including mild lateral variability. The method is well established, and the survey is r elatively easy to conduct. M any types of commercial software are available for tomographic reconstruction of the subsurface based on the matching of travel times. Disadvantages Seismic refraction is only effective if the velocity at wh ich waves propagate through the Earth increases as depth increases In other words, velocity reversals and low velocity anomalies cannot be detected using this method. Only the first arrival signals are used in data processing, and manually selecting the se signals can be time consuming and tedious.

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21 Resolution of the reconstructed velocity tomograms decreases as depth increases. 2.3.3 Surface Wave Method The surface wave method, as the name suggests, uses Rayleigh and Love waves, although Rayleigh waves are used more widely in the field. Two of the most popular methods are the spectral analysis of surface waves (SASW) (Nazar ian and Stokoe, 1984; Stokoe et al., 1994) and the multi channel analysis of surface waves (MASW) (Park, el al., 1999; Zywicki, 1999). These methods involve actively exciting Rayleigh wave energy at a fixed point and then measuring the resulting vertical motion on the surface at various distances (offsets) from the source. The surface wave method primarily characterizes the dispers ion of Rayleigh waves that propagate in a horizontally layered system. The phase velocity, primarily depends ratio) in a sample approximately one wavelength deep Wave s with different wave lengths sample different depths. As a result of the varying shear stiffness of the layers, waves with different wavelengths travel at different phase velocities. Phase velocity varies with wavelength and frequency and this variation is characterized as a surface wave dispersion curve This phase velocity variation is an important site characteristic that is evaluated in the field. The field dispersion curve can be extracted by using 2 D FFT (Foti, 2000) or slant stack analysis (McMechan and Yedlin, 1981) to transform the raw data set from the time space domain in to the frequency wave number domain Once a credible field dispersion curve has been established, it is then inverted to obtain the shear wave velocity profile with depth Figure 2 6 illustrates a typical

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22 MASW test configuration, field dispersion image, and inverted shear wave velocity profile Advantages The energy of Rayleigh waves is predominant in the recorded wavefield because these waves represent a high percentage of the energy generated by a vertically impacting source In addition, since the wavefront is cylindrical, the geometrical attenuation of Rayleigh waves is low as opp osed to the higher geometrical attenuation caused by the spherical wavefronts of body waves. Theoretically, the surface wave method is capable of identifying softer layers beneath stiffer materials. The field test can be done rapidly and cost effectively. Disadvantages Assuming the E arth model is horizontally layered, it can only obtain 1 D shear wave velocity profile s Thin layers can be missed if they exhibit high impedance contrasts ( e.g., are much stiffer or much softer than the surrounding material ) Resolution of the inverted shear wave velocity profile decreases as depth increases. 2.4 Borehole based Seismic Methods Borehole methods are often employed when inclusions and anomalies may not be properly resolved using surface based geophysics or when a higher resolution image is needed at a target zone 2.4.1 Crosshole Method The crosshole method (ASTM D4428; Stokoe and Woods, 1972) measures the one way travel time of seismic energy transmitted between boreholes to determine t he elastic moduli of the intervening materials ( Figure 2 7a ) This method measures t he travel times from the source to the receivers ( direct travel times ) and the travel times between receivers ( interval travel times ) By measuring the space s between each

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23 borehole the depth of the seismic source and the depth of each receiver, the apparent compressional and shear wave velocities can be calculated. The test is then repeated at multiple depths to obtain a 1 D profile of compressional and shear wave velocities at various depths Crosshole tomography measures the travel times of seismic raypaths between two or more boreholes in order to create velocity images of the intermediate materials ( Pratt and Worthington, 1990; Pratt, 1990; Fernandez and Santamarina, 2003) These travel times are collected along the length of the receiver borehole s for each shot position providing more spatial cover age than the standard crosshole test. By varying the depth of the seismic source, a dense network of overlapping raypaths is obtained ( Figure 2 7c). The composite travel times are then used to reconstruct a highly accurate velocity tomog ram of the space between the boreholes This tomogram can then be used to identify anomalies and individual velocity layers. Advantages Source and receivers are placed close to the material to be evaluated, thus enhancing resolution where inclusions and anomalies may not be properly resolved using surface based geophysical methods. Crosshole tomogr aphy is able to produce high resolution 2 D velocity tomograms between boreholes for P SV and SH waves, although not simultaneously Disadvantages The main disadvantage of crosshole testing is the time and cost associated with drilling boreholes. 2.4.2 Downhole Method The downhole method measures the travel time of compressional and shear waves between a source on the surface and receivers within a borehole (ASTM D7400; Mok et al., 198 8) In this method, a geophone or a string of geophones receives energy

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24 from waves that are measured on a seismograph. The downhole method is based on the assumption that the first arrival signal at a given depth is from the direct wave, since the waves travel almost vertically. Once travel distances have been determined, w ave velocities are then calculated from the corresponding travel times Tra vel distances are typically based on the assumption that raypaths between the source and receivers are straight although advanced analysis can account for curved r aypaths as well Figure 2 7b presents a conventional test setup used in a downhole seismic survey. Advantages The downhole metho d requires only one borehole, and thus is more cost effective than the crosshole method. The downhole method can create both P and SH wave ve locity profiles. This method does not require that the layer velocities increase with depth. Disadvantages This method obtains o nly 1 D seismic wave velocit y profiles Resolution of the inverted wave velocities decreases as depth increases. 2.4.3 Suspension P S Velocity Logging Suspension P S velocity logging is a relatively new method for determining seismic wave velocity profiles in both soil and rock formations (Kitsunezaki, 1980; Nigbor and Imai, 1994) The wave velocities are usually measured in a single, fluid filled borehole. A typical logging system uses a probe, consisting of a pressure source and two receivers suspended by a cable ( Figure 2 8 ) The probe is lowered into the borehole to a specified depth, where the pressure source generates a pressure wave in the borehole fluid. The pressure wave is converted to seismic waves (P and S waves) at

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25 the borehole wall. Along the wall at each receiver location, the P and S waves are converted back to pressure waves in the fluid and detected by the hydrophones. The average compressional and shear wave velocities of the surrounding material can be det ermined from the travel time s between the two receivers. The pressure source generates pressure waves at each depth interval until waves are measured along the entire length of the borehole. Advanta ges The wire line allows for penetration to depths of hundreds of meters. This method can reliably obtain b oth P and SH wave velocity profiles in either cased or uncased borehole s Th is method can resolve t hin and soft layers Disadvantages This method obtains only 1 D seismic wave velocity profiles 2.4.4 Full Waveform Sonic Logging In a full w aveform sonic logging survey the data acquisition method is similar to the suspension logger as both the pressure source and the receivers are placed in the same borehole (Hornby, 1989; Chabot, 2003) However, the full waveform sonic logging tool uses a string of receivers located at different offsets along the body of the well logging tool ( Figure 2 9 ) As the tool is moved up wards along the bore hole, it repeatedly logs the formations surrounding the borehole The complete signal of acoustic pressure is monitored by the hydrophone at fixed sampling rate s for a certain length of time T he resulting recorded signal is called a full waveform which is recorded at each receiver. Each full waveform contains several pressure sig nifiers, such as direct or fluid waves, P and S head waves, pseudo

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26 Rayleigh waves, Stoneley waves, normal modes and converted modes. Full waveforms are then processed and analyzed to investigate the structural properties of the formations surrounding the borehole Advantages Data processing involves analysis of full waveforms as opposed to the first arrival signals used in P S suspension logging. Disadvantages Migration type analysis results in only a qualitative description of the formation s surrounding the borehole. 2.5 Initiation of the C u rrent R esearch As discussed above, characteriz ing spatial variation s in rock formations is essential for the design of drilled shaft foundations in karst terrain where subsurface conditions are often highly variable. Crosshole tomography is well known for its capability to produce reliable 2 D velocity tomograms of the material between boreholes ( Pra tt and Worthington 1990; Pratt, 199 0 ) However, reconstructed velocity tomograms fail to characterize the existence of isolated anomal ies such as void s or low velocity inclusion s, in planes outside of where the boreholes are located In order to detect such anomalies multiple receiver holes must be drilled around the source hole, a nd then crosshole tomography must be applied between each source hole and the individual receiver hole s This task substantially increases the cost of conducting the crosshole test. Therefore building on the work of Tran et al. (2011), the current study aimed to use only one borehole while still characteriz ing 2 D and 3 D velocity variation s in the surrounding materia l Previous studies have employed the suspension logging method for obtaining reliable 1 D S wave velocity profile s versus depth Kalinski (1998) attempted to apply

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27 the SASW inside a borehole in order t o obtain lateral variation measurements Cheng (1997) found that the cylindrical geometry of the borehole significantly affected the dispersive ch aracteristics of surface waves and that appropriate numerical models were needed to accurately simulate the experimental data. By assuming that the formation surrounding the borehole was cylindrically multilayered, 1 D S wave velocity profile s of a few inches into the radius could be inverted. Thus, as an alternative to crosshole tomography, Kalinski ( 1998 ) proposed that the borehole SASW method be used in conjun ction with suspension logging to derive a high res olution S wave velocity profile of the subsurface However, it must be noted that 1 D vertical variation plus 1 D lateral variation is not equ al to 2 D variation. More importantly, an isolated anomaly cannot be characterized using this combined method. Recent advances in borehole acoustic logging use the acquired full waveforms to look deep into the borehole walls. Hornby (1989), Fortin et al. (1991), Coates et al. (2000) and Chabot (2003) used an acoustic well logging tool, equipped with monopole/dipole sources and a string of receivers suspe nded in a fluid filled borehole Th ese tool s combined with some advanced signal processing flows were used to image scattered energy originating from acoustic impedance contrasts from beyond the borehole wall Those contrasts could be interpreted as structural changes, providing improved knowledge of subsurface spatial variations not easily captured through surface seismic analysis. However, it must be noted that the migrated image does not provide any quantitative correlation between impedance contrasts and engineering design parameters of interest (e.g., S wave velocity).

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28 FWI a combination of migration and tomography (Mora, 1989) offers a new angle on imaging as it uses only one borehole to characterize 2 D and 3 D velocity variation s in rock formations The proposed experimental setup uniformly mounts a linear array of three component transducers along the vertical borehole wall (Figure 2 10) S imilar to the MASW or seismic refraction conducted on the ground surface this method uses an appropriate seismic source either mechanical, piezoelectric or electromagnetic to generate a tri axial shot gather With an array of receiver s fully coupled to the wall, the seismic source can be moved up and down to acquire multiple shot gathers. Ide ally, this method creates a 3 D scan of material along and around the borehole by rotating the receiver array circumferentially inside the borehole Alternatively, a 3 D receiver array can be placed at the beginning of the test Chapter 3 to Chapter 5 will present d etails of numerical formulations and applications

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29 a) b) Figure 2 1. Body waves : a ) C ompressional wave s and b ) S hear wave s ( from Stokoe and Santamarina, 2000) a) b) Figure 2 2. Surface waves : a ) Rayleigh wave s and b ) Love wave s ( from Stokoe and Santamarina, 2000)

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30 Figure 2 3. Field arrangement used in the seismic reflection method : a ) Normal moveout, b ) Common offset, and c ) Common depth point ( from Stokoe and Santamarina, 2000) Figure 2 4. Interpretation of a seismic reflection test : a ) Time migrated cross section and b ) Interpreted geologic profile ( from NRC, 2000)

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31 a) b) c) Figure 2 5. Seismic refraction method : a ) Test configuration b ) T ravel time s and c ) R econstructed P wave velocity tomogram

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32 a) b) c) Figure 2 6. Surface wave method: a) Test configuration b) Rayleigh wave dispersion and c) I nverted shear wave velocity profile

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33 c) Crosshole Tomography Figure 2 7. Crosshole and d ownhole methods: a) Crosshole testing, b) Downhole testing, and c) Crosshole tomography ( from Stokoe and Santamarina, 2000)

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34 Figure 2 8. Test setup for the suspension P S velocity logging ( from Stokoe and Santamarina, 2000) Figure 2 9. Test setup for the full waveform sonic logging ( from Chabot, 2003)

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35 a) b) Figure 2 10. Schematics of test setup for the proposed borehole based FWI : a) C ross sectional view and b) T op view with eight planes to be scanned

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36 CHAPTER 3 FULL WAVEFORM INVERSION WITHIN A BOREHOLE 3.1 Introduction of Full Waveform Inversion As discussed above standard seismic methods use limited wavefield information such as reflectivity (as in the seismic reflection and borehole acoustic logging methods) travel times (as in the seismic refr action, crosshole, and PS suspension logging methods) and phase velocity dispersion (as in the surface wave method ) Therefore the resulting capabilities of these methods are limited. The theory of seismic FWI was formulated in the 1980s (Lailly, 198 3 ; Tarantola, 1984) ; however, it was rarely used as computers at the time had limited power and memory Recent advances in high performance comput ers have generated considerable interest in applying FWI to exploration seismology, mainly for the purpose of reservoir characterization. Given the new capabilities of this met hod, civil engineering projects have recently begun using FWI for the purpose of site characterization (Tran and Hiltunen 20 12 ; Tran and McVay, 2012 ; Romdhane et al., 2011). Seismic inversion is essentially equivalent to conducting migration and reflection tomography simultaneously (Mora, 1989). M ajor advantages of FWI are that it utilizes the amplitude and phase information of seismic wavefield s properly manag ing multipathing, mode conversion and other complex wavefield phenomena (Virieux and Operto 2009). In addition, FWI is largely automated as opposed to requiring intense manual interpretation, and can be formulated in either time or frequency domain (Tarantola, 1984; Pratt e t al., 1998). In a re strictive sense, FWI requires a complete recording of all wave paths specifically reflection, refraction and multiple s In a less restrictive sense, FWI aims to identify Earth models that enable the production of

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37 synthetics that mimic the complete observed data set (Plassix, 2008). Two key elements of FWI are an accurate and efficient forward model and a repeatable and reliable inversion scheme. It must be noted that the majority of seismic full waveforms are generated, recorded, and inverted from the ground surface as this technique is the easiest and presents the most well developed forward models (e.g., 2 D time domain finite difference solution). Although full waveforms are also logged using b orehole acoustic techniques FWI has never been attempted using this technique as wave propagation inside a borehole is significantly more complicated than wave propagation from flat ground. In the case of fluid filled borehole s an omnidirectional pressure source (monopole) creates a compressional wave pulse in the borehole fluid This pulse propagates out into the formation creating a disturbance around the borehole wall and exciting compressional and shear waves in the formation (Schlumberger, 1997). As these waves propagate in the formation, they generate h ead waves in the borehole fluid thus generating refracted arrivals. After the head waves guided borehole waves arrive, followed by the Stoneley wave s The guided borehole waves are generated by reflections of source waves reverberating in the borehole (Schlumberger, 1997). The Stoneley wave s travel more slowly than the fluid waves, and decay as they travels through the fluid borehole interface away from the borehole Far field velocities are associated with geophysical parameters while near field velocities are associated with geomechanical properties (Chabot, 2003).

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38 In the case of dry bore hole s an omnidirectional mechanical source creates compressional and shear waves that travel along and into the formation. Dispersive, Rayleigh type surface wave s develop in the borehole annulus The phase and group velocities depend on the ratio between the wavelength and the size of the borehole (Biot, 1952). C hapter 3 details the numerical implementations of FWI within a dry borehole 3.2 Forward Problem The prediction of observations, given the parameters defining the model constitutes the forward problem. In the context of s eismic FWI forward modeling involves solving the governing wave equations, thereby predicting the particle responses at observational points. 3.2.1 Theoretical Derivation Wave propagation in cylindrical cavities is gov erned by the wave equation ( Biot 1952) which can be solved by transforming the data into cylindrical coordinates and treating the equation as an axisymmetric problem This process requires solving the following equations: (3 1) (3 2) where is compressional potential function, is shear potential function, is radial distance from the cavity center, is axial distance, is time, is compressional wave

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39 velocity, and is shear wave velocity. For non attenuated wave s propagating in the axial direction, the solution s to the above equations are as follows : (3 3) (3 4) wh e re and are constants, is wave number ( ), is angular frequency ( ), is modified Bessel function of the second kind of zero order, and is modified Bessel function of the second kind of first order. The parameters and are defined as follows: (3 5) (3 6) where is Rayleigh wave velocity. Given the boundary conditions of zero normal and shear stress at the cavity wall, Equations (3 3) and (3 4) are used to derive an implicit relationship between and : (3 7) where is cavity radius, and Figure 3 1 illustrates this relationship as derived from E quation (3 7) w ith the being 0.25. 3.2.2 Finite Element Modeling Analytical solutions can only be obtained for simple models (e.g., a cylindrical cavity in a homogeneous medium as illustrated above). Numerical approximations are

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40 required to handle complex geometry, loading, boundary conditions, and material properties A number of numerical methods are commonly used to g enerate synthetic seismograms, s uch as the method based on ray theory, the finite difference method, and the finite element method When developing a feasibility study on characterizing the spatial variation of material using FWI within a borehole, it is beneficial to determine the viability of the whole system before developing a spe cialized forward modeling code Therefore, this study used ABAQUS a commercially available, general purpose finite element package to model the propagation of seismic waves inside a borehole. Specifically, an axisymmetric borehole model was formulated as the forward model ( Figure 3 2 ) The model was 4 m long with a 2 m radial extension and a 10 cm diameter The model was uniformly discretized into 36 cells (9 by 4) measuring 0.5 m by 0.5 m each Material properties were assigned modulus node modified quadratic axisymmetric triangular element was used in the explicit dynamics analysis. 3.2.2.1 Spatial temporal discretization In wave propagation problems, element dimensions are chosen with res pect to the highest f requency for the lowest wave velocity. Element dimensions that are too large will filter high frequencies, whereas very small element dimensions can introduce numerical instability as well as require considerable computational resources ( Zerwer et a l., 2002) An approximate element dimension is calculated using the following equation: (3 8)

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41 where (3 9) where is minimum wavelength, is minimum wave velocity in the model, and is maximum frequency of interest. The constant must be less than 0.5 because of the Nyquist limit, and the actual value depends on whether the mass matrices are consistent ( ) or lumped ( ). The time increment must be carefully chosen to maintain numerical accuracy and stability. Numerical instability may cause the solution to diverge if the time increment is too large. Conversely, a very short time increment can cause spurious oscillations, known as the nomenon The calculation of the time increment depends on the element dimension computed with the following expression: (3 10) where is time increment, is element dimension, and is maximum compressional wave velocity in the model. Although the time increment incorporates the spatial Nyquist limit, it must also entail the temporal Nyquist limit, as expressed in the following equation: (3 11) 3.2.2.2 Numerical I mplementation The explicit dynamics analysis procedure in ABAQUS / Explicit is based on the implementation of an explicit integration rule and the use of lumped mass matrices. The

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42 explicit central difference integration rule is used to integrate the following equations of motion: (3 12) (3 13) where is displacement, is velocity, and is acceleration. The superscript refers to the increment number while and refer to mid increment values The central difference integration operator is explicit in that the kinematic state can be advanced using known values from previous increment s The use of lumped element matri ces is key to the computational efficiency of the explicit procedure because the inversion of the mass matrix used to compute the accelerations at the beginning of the increment is triaxial as expressed in the following equation : (3 14) where M is diagonal lumped mass matrix, F is applied load vector, and I is internal force vector The explicit procedure requires no iterations and no tangent stiffness matrix. The mean velocities and require special treatment under initial conditions and certain constraints. The state velocities are stored as a linear interpolation of the mean velocities : (3 15) The initial values of velocity and acceleration are set to zero unless specified otherwise. The following condition is asserted: (3 16)

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43 Substituting this expression for in Equation (3 12) yields the following solution : (3 17) 3.3 Inverse Problem 3.3.1 Introduction The inverse problem refers to using actual measurement s to estimate the model parameters characterizing the system under investigation. The goal is to derive an appropriate model that minimizes the error between measurement and estimation. In the context of seismic FWI measurement refer s to the observed wavefield recorded in each receiver while estimation refers to the synthetic wavefield generated by the forward operator for the most geologically meaningful model. In fitting a function of an independent variable t and a vector of n parameters to a set of data points it is customary and convenient to minimize the sum of the weighted squares of the errors (or weighted residuals) between the measured data and the curve fitting function This scalar valued fitness measure ( obj ective function ) is called the chi squared error criterion, or the L2 norm : (3 18) where is a measure of the error in measurement and the weighting matrix is diagonal with T he minimization of is carried out iteratively. The goal of each iteration is to find a perturbation to the parameter that reduces Inverse problem solving is essentially implemented as a process of numerical optimization. In general, numerical optimization techniques can be classified into two

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44 broad categories: local optimization techniques and global optimization techniques. Global optimization techniques, such as simulated annealing (Ingber, 1989) and genetic algorithm (Sen and Stoffa, 1992), can be used for seismic waveform inversion. As compared to local optimization techniques, these techniques require neither a good starting model for convergence nor the calculation of Jacobian and Hessian matrices for model update s However, since effective sampling of the model parameter space requires a sufficiently large number of forward calculations these optimization techniques are not particularly efficient for 2 D/3 D parameter estimation problems. On the other hand, local optimization techniques, such as the conjugate gradient method (Mora, 1987; Tarantola, 1987) and the Gauss Newton method (Sheen et al., 2006), are historically more widely implemented for seismic waveform inversion. Local optimization te chniques attempt to find a local minimum by searching along the downhill direction Therefore these methods do not guarantee that the local minimum is the needed global minimum unle ss, in rare cases, the starting model is in the vicinity of the global minimum. Nonetheless, in the current implementation of FWI within a borehole, the Gauss Newton method was chosen for the relatively straightforward implementation, fast convergence property, and a record of successful application (Pratt e t a l 1998 ; Sheen et al., 2006 ) The requirement of constructing a good starting model was circumvented by a multistage approach described below 3.3.2 G auss Newton Method The Gauss Newton method is an iterative technique for solving nonlinear least squares problems. This method assumes that the objective function and its first and second partial derivatives with respect to model parameters are continuous. In addi tion,

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45 the objective function is assumed to be approximately quadratic in the parameter space near the optimal solution. Functions evaluated with perturbed model parameters may be locally approximated through a first orde r Taylor series expansion : (3 19) where is the Jacobian matrix, the first partial derivative with respect to model parameters. Substituting the approximation for the perturbed function, for in Equation ( 3 18 ) results in the following equation : (3 20) Equation (3 20) implies that is approximately quadratic in the perturbation and that the Hessian matrix of the chi squared fit criterion, the second partial derivative with respect to model parameters can be approximated by The perturbation that minimizes is found by enforcing : (3 21) The resulting equation for the Gauss Newton perturbation is as follows : (3 22) The model parameters, then update as follows : (3 23) In the current implementation, the Jacobian matrix is numerically approximated using finite differences :

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46 (3 24) where is the Jacobian matrix, is estimation of the perturbed model, is estimation of the current model, and is absolute value of perturbation of the model parameter. 3.3.3 Regularized G auss Newton Method Many inverse problems are mathematically ill posed because they operate with insufficient data. T he inversion if at all possible would unevenly magnify noise in the solution particularly on values of the model parameters that are least constrained by the data (Santamarina and Fratta, 2005). Regularization is a technique for making inverse problems well posed by adding bias to the solution, such as assuming a smooth variation among model parameters. Similarly, in the application of the Gauss Newton method to seismic waveform inversion, regularization is particularly important for stabilizing the system and incorporating a priori information in to the problem (Tarantola, 1987). The current study discards the measurement error (synthetic data) and follows the approach presented by Sheen et al. (2006). The regularized misfit function can be defined as follows: (3 25) where is the data objective function ( as in Equation 3 18), is the model objective function that contains the a priori information of the model, and is a scalar (between 0

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47 and 1) that controls the relative importance of The model objectiv e function can be defined as follows: (3 26) Ignoring the measurement error changes Equation ( 3 23 ) to the following: (3 27) Note that if it represents the damped least squares solution (Levenberg, 1944; Marquardt, 1963), which corresponds to the zeroth order Tikhonov reg u larization where the L2 norm of model parameters is minimized (Aster et al., 2005). If is a discrete spatial differential operator, it resembles 1987), which corresponds to the second order Tikhonov regularization where the spatial variation among model parameters is minimized (Aster et al., 2005). Combined use of these two regularization techniques yields the following regularized Gauss Newton formula : (3 28) where and are used, is a step length, is the identity matrix, and is the roughening matrix the elements of which are determined using the following discrete 2 D Laplacian operator : (3 29) where the superscripts E W N and S refer to four neighbors of the model parameter and refers to the row of the matrix , whose element is either 1, 4, or 0.

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48 A con stant step length of is taken through all iterations for simplicity. However, an optimal value of the step length can be determined by a linearized approach as follows (Sheen et al., 2006) : where (3 30) 3.4 Practical Strategies for FWI I nve rse problems are generally very d ifficult. The difficulties stem from three issues: existence, uniqueness and instability (Aster et al., 2005) First, inverse problems may not have a solution due to inexact physics in the forward model or noise in the data. Second, there may be an infinite number of models that can fit t he data equally well. Lastly inverse problems may be ill posed or ill conditioned, where the solution is very sensitive to small changes in the data. In the synthetic model studies, however, the solution always exists, and the issues of uniqueness and instability can be largely solved using regularization. No netheless, another important issue exists : convergence. Depending on the initial model, local optimization techniques may converge to a local minimum, or may not converge at all. 3.4.1 Frequency Filtering S eismic FWI applied to surface data is highly nonlinear, and tends to converge to a loc al minimum if the starting model is not in the vicinity of the global minimum. The refore, a good initial model is often required to avoid local minima ; however obtaining a suitable initial model is often difficult if no a priori information is available. To mitigate nonlinearity and loosen the initial model requirement a multiscale approach is often utilized in either time domain (Bunks et al., 1995) or frequency domain (Sirgue and Pratt, 200 4) This multiscale approach builds a background model by inverting the low

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49 frequency component and then increases the resolution by gradually adding high frequency component s in the data. The degree of nonlinearity, or the multimodal distribution of the misfit function with respect to the model parameters depends o n the frequency content of seismic d ata. The misfit (objective) function is more linear at low frequencies tha n at high frequencies. For this reason inversion process es that sequentially proceed from low to high frequencies are more likely to reach the global minimum than processes that start with high frequency raw data To illustrate the effect of frequency filtering, a series of low pass filters was applied to raw data ( Figure 3 4 ) Each sub part of the Figure displays the time signal and the associated frequency conte nt. It should be noted that, in Figure 3 4a the frequency spectrum in t he raw data is being dominated by three peaks, approximately centered at 500, 1500, and 2500 Hz After first level filtering, the hi ghest frequency peak disappears and only the first two peaks remain in the spectrum ( Figure 3 4b) Applying another lower pass filter leaves only one peak in the spectrum ( Figure 3 4c) 3.4.2 Time Windowing As discussed below synthetic wavefield s g enerated within a borehole are significantly more complicated than wavefields generated from the ground surface. Thus problem s can become extremely nonlinear when FWI is applied to borehole data To further reduce the nonlinearity, this study proposes starting with a short time window at a low frequency band After some i terations, higher frequencies are incorporated in th e in version with the same time window. Then, the inversion proceeds with long er time windows followed by similar filtering strateg ies, until the full data set has been considered Figure 3 5 illustrates this windowing technique For the waveform shown in Figure 3 4c the first window was clipped at 2.5 ms and the second window was clipped

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50 at 5 ms with the full window correspond ing to the entire length of the signal. Gradually increasing the time window and frequency bandwidth applied to the residual enable d a radial velocity update and reduce d the possibility of the misfit function being stuck at local minima. Implemented in this way, the proposed inversion scheme essentially uses the a priori information continuously gained from previous iterations, and steers the model update towards the global minimum. The p roposed inversion scheme is illustrated in a flowchart in Figure 3 3.

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51 Figure 3 1. Dispersion of an axially propagat ing surface wave in a borehole with of 0.25 (from Kalinski, 1998) Figure 3 2. Discretization of an axisymmetric borehole model

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52 Figure 3 3. A simplified flowchart for the proposed inversion scheme Initial model Synthetic data Processed synthetic data Intermediate inverted model Forward operator Multiscale strategies Regularized Gauss Newton inversion Full data incorporated ? No Yes Final inverted model

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53 a) b) ( Figure 3 4 is continued in the next page)

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54 c) Figure 3 4. Concept of frequency filtering: a) R aw data and the frequency spectrum, b) F irst level filtering, and c) S econd level filtering

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55 Figure 3 5. Concept of time windowing: gradually increasing the length of the window to facilitate convergence

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56 CHAPTER 4 VALIDATION OF FORWARD MODEL AND FULL WAVEFORM INVERSION 4.1 Introduction Using the previously developed forward mo del and inversion scheme, C hapter 4 describes how the coupled system works. First, the forward model is validated by comparing the synthetic waveforms with known analytical solut ions. Second, the inversion scheme is tested using synthetic records generated from ax isymmetric forward models. Lastly the inversion scheme is further tested using synthetic records generated from 3 D borehole models. 4.2 Validation of the Forward Model The solution provided by the forward model must be accurate; otherwise the subsequent inversion may be inaccurate or even misleading, as FWI aims at minimizing errors between the observed wav eforms and the synthetic waveforms generated by the forward model. Biot (1952) predicted that the velocity of Rayleigh wave s propagating inside a borehole depends on the ratio between the wavelength and the size of th e borehole To test this theoretical prediction, th e present study investigated homogeneous axisymmetric borehole models with radii of 0.05, 0.25, 1, and 10 m shear wave velocity of 1000 m/s kg/m 3 The particle displacements were recorded at the same location on each borehole and then compared A triangular wavelet source generated the waveforms (Figure 4 3a ) ensuring that the wavelength was fixed for all cases. Axial and radial p article displacements were measured and graphed individually for each borehole radius and the amplitude of individual waveforms w as normalized for ease of comparison ( Figure 4

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57 1 ) The resulting graphs clearly illustrate that Rayleigh wave s travel faster with in borehole s with smaller radi i given the wavelength being the same which confirms the theoretic al prediction depicted in Figure 3 1 On the other hand, it is hypothesized that the plane strain model can be approximated as the radius of the axisymmetric borehole model approaches infinity. T o test this hypothesis this study compared the waveforms generated from the axisymmetric borehole model with a 1 km radius to the plane strain solution. Result s confirm that the normal and tangential components of the corresponding waveforms matched perfectly ( Figure 4 2 ) This set of waveform comparisons demonstrate d though indirectly, that the axisymmetric forward model formulated using ABAQUS was correct and accurate for use in the subsequent waveform inversion s 4.3 Inversion of Synthetic Data Generated by the Forward Model In order to obtain credible velocity profiles, an analysis protocol that systematically and consistently analyze s wavefield data must be developed This protocol can be established using synthetic model studies with known solutions; t herefore, the current project performed such a study to test the effectiveness of th e inversion scheme proposed in Figure 3 3 In this study the synthetic records were generated using the axisymmetric forward model discussed above, and treated as field data The velocity profile was then reconstructed by means of inversion. Theoretically, the inverted velocity profile should be the same as the assumed known model. In the current implementation, only the shear wave velocity was inverted, while the kg/m 3 ) remained constant dur ing the inversion. As shown in Figure 3 2, the inversion model was uniformly discretized into 36

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58 pixels (9 by 4), and shear wave velocity values were assigned independently to each pixel resulting in 36 model para meters to be inverted. T he synthetic waveforms were generated using a triangular wavelet source ( Figure 4 3b ) Note that due to the axisymmetric nature of the model the source was not a point load as it may appear in Figure 2 10 but rather a uniformly distributed ring load that excites perpendicularly to the borehole wall. A low frequency source is desirable because lower frequencies generally allow farther energy penetration and thus deeper characterization For this study, the goal was to obtain a velo city profile that radially extended 5 ft into the borehole wall. Using the ring load source, the ratio of the dominating wavelength to the borehole radius far exceeded the 3.2 cut off value shown in Figure 3 1 This ratio resulted in mode conversion as the surface wave energy convert ed to shear wave energy that propagate d at a velocity equal to The resulting radial and axial particle displacements are graphed in Figure 4 4. Note that the radial component was highly oscillatory while the axial component was relatively well behaved. As discussed above the nonlinearity of the inverse problem depends on the level of oscillation in the seismic data. The use of radial component s inside a borehole is a natural extension of the fact that the vertica l components of seismic data are routinely used in surface waveform inversions However, the radial component in this study was highly oscillatory, which prevented the inversion from converging, despite the use of the proposed multiscale strategy Conversely, convergence did occur when the initial model was very close to the true model. In fact, if convergence is to be achieved, the quality of the initial model must increase as the nonlinearity of the inverse problem increase s T h is is the very reason

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59 why t he multiscale approach was developed to les sen the need for a good initial model by reducing the nonlinearity of the error space. It is therefore concluded that the radial component as generated in the above study is not viable for use in borehole waveform inversion as the level of detail needed in the a priori information is nearly impossible to meet On the othe r hand, the axial component, or the in plane component, of the borehole data is much easier to manage Figure 4 5 presents the schematic used for conducting the synthetic experiment A triangular wavelet source ( Figure 4 3b ) was excited at the center ( the red arrow in Figure 4 5) of the inner wall of an axisymmetric borehole model A 10 channel receiver array was uniformly placed along the wall, with the nearest offset and trace interval being 0.5 m ( the hollow circles in Figure 4 5) Using this configuration, the axial component of the particle displacements was monitored ( the double headed arrows near the receiver array in Figure 4 5) For the synthetic model studies discussed in the following sections inversions were carried out by matching the axial component of the particle displacements The amplitude of the waveforms and the L2 norm of the misfit function were normalized for ease of comparison. 4.3.1 Horizontally Layered Model The horizontally layered model ( Figure 4 6a ) comprised nine distinct ve locity layers each 0.5 m thick The shear wave velocity varied gradually from 450 m/s at the top to 1000 m/s in the middle and back to 450 m/s at the bottom. This model was chosen because a majority of geological materials are multi layered and exhibit spatial variations The shear wave velocity of the initial model was 775 m/s uniformly across all p ixels ( Figure 4 6b ) A total of 4 ms of full waveforms was acquired The inversion

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60 started with a short time window of 2 m s and a low pass filter was applied to the associated waveforms. The first column of pixels in the updated model after 10 iterations had good resolution, while the second and third columns were vague, and the last column exhibited no change in values ( Figure 4 7a ) The associated waveforms matched very well, and the inversion converge d rapidly This model required approximately twenty minutes for 10 iterations on a standard personal computer and the computer time will scale up with the number of iterations Next, using the last updated model as the initial model, the waveforms were inverted under the same time window with no filter applied After an additional 10 iterations, the resolution of the middle columns of pixels improved considerably, while no major change occur red in the last column (Figure 4 7b ) The waveforms were well matched, and the inversion converged very rapidly To increase resolution FWI was applied and the time window was set as the entire length of the acquired waveforms. Using the last updated model as the initial model, a low pass filter was applied to the inversion after 25 iterations which significantly improved the resolution in the last three columns (Figure 4 7c) At this point, the layered structure became identifiable, especially for the middle rows of pixels. Again, the waveforms were well matched a nd the convergence rate was fast The solution illustrated in Figure 4 7c is considered to be a very good initial model for further velocity update s To further increase resolution high frequency component s of the data should be incorporated in to the inversion. Therefore, the last step in this process was to set the inversion to continue for 100 iterations based on the raw data set unless converged

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61 according to the predefined stopping criteria The layered structure of the final inverted model was very well recovered as the first three pixel columns closely approximated the true model (Figure 4 7d) Some smearing is visible in the last column despite the fact that the waveforms were in excellent agreement across all channels. This smearing occurred because th ose model parameters were less constrained by the acquired data and thus could not be inverted with certainty. Two possible solutions to this problem are : 1) to further increase the length of the acquired data in order to look deeper behind the borehole wall, or 2) to collect multiple shot gathers by placing the source at multiple locations along the rec eiver array in order to improve spatial coverage. T he latter approach is more effective particularly when resolving isolated anomalies in the vicinity of a borehole as detailed in C hapter 5. The convergence rate in the last step of the inversion was much slower than in the previous steps (Figure 4 7d) P reviously the L2 norm had decreased by a factor o f 10 in less than 10 iterations while this decrease occurred over approximately 50 iterations in the final step In fact, the L2 norm only decreased by a factor of 2 in the first 10 iterations and it continued decreasing at an extremely slow rate for the last 50 iterations. In the Gauss Newton method convergence tend s to be quadratic once the search approach es the local minimum. However, since misfit function at high frequencies tend to have an extremely nonlinear error space a reduced convergence rate can be expected. It is also possible that limi ted sensitivity of the acquired data discussed above, caused the slow convergence rate 4.3.2 Cylindrically Layered Model The cylindrically layered model ( Figure 4 8a ) consisted of four constant velocity layers each 0.5 m thick The shear wave velocity varied from 1000 m/s to 775 m/s to

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62 450 m/s and back to 1000 m/s This model is capable of producing significant reflections and mode conversions, and was therefore selected to test the effectiveness of the proposed inversion scheme in handling complicated wavefield data A uniform shear wave velocity of 775 m/s was selected as the initial model ( Figure 4 8b ) The multiscale strategy was applied throughout the inversion. First, the inversion ran for 20 iterations with a 2 m s time window and a low pass filter (Figure 4 9a) The resulting first column show ed good pixel recovery, and the resolution in the second column changed slightly as a result of the assigned true velocity values The third column exhibit ed a low velocity zone while no significant changes appear ed in the last column. The wavefo rms were in good agreement, and the inversion converge d fairly rapidly. Next, the inversion continued for another 20 iterations with the same time window and without filter ing ( Figure 4 9b ) The first two layers were near perfect reconstruct ions of the true model, a low velocity l ayer became visible i n the third column, and the last column remain ed unchanged Again, the waveforms were well matched and the convergence rate was fast The next 20 iterations used a full time window of 4 m s and a low pass filter ( Figure 4 9c ) The most noticeable change appeared in the last column of pixels which began to gain high velocity values A more clearly developed low velocity layer was also observed in the third column The waveforms were once again in good agreement ; however, the convergence curve was not as smooth as previous curves To increase the resolution of the recovered image, the raw data set was inverted in the last step for 100 iterations ( Figure 4 9d ) This set of iterations perfectly recovered the first three layers of the true model, whereas some uncertainties appeared in the last

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63 layer As expected the good model recovery caused the waveforms to be in near perfect agreement across all channels ; however, the s hape s of the waveforms were very complicated and difficult to match. Had the inversion started with the raw data set it would have never converged given the chosen initial model. In other words, the final inverted model shown in Figure 4 9d cannot be obtained using the initial model presented in Figure 4 8b. Interestingly, the convergence curve shown in Figure 4 9d exhibited a discontinuity at approximately 75 th iteration after which the convergence rate increased significantly then gradually decreased These two synthetic model studies illustrate that the proposed inversion scheme is indeed capable of reconstructing the assumed true model. Applying the multiscale strategy in the time and frequency domain s can mitigate the nonlinearity of the inverse problem and lessen the need for a good initial model for essentially any gradient based inversion technique. In addition, the tradeoff between penetration depth and resolution must be realized meaning long wavelengths (low frequencies) penetrate further while short wavelengths (high frequencies) provide higher spatial resolutions. It must be noted that imperfect model recovery is always possible, even when using data generated by the forward model as the data may poorly constrain some of the model parameters, resulting in non unique solutions in practice. Therefore, synthetic model studies are valuable for guiding experiment design The viability of a solution and the attainab ility of the resolution depend on two critical aspects: 1) the distribution of measurements for good spatial coverage, and 2) the selection of instruments for gathering high quality data (Santamarina and Fratta, 2005). Three options exist for i mproving model recovery : 1) increase the length of the acquired full waveforms 2)

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64 increase spatial coverage by collecting multiple shot gathers at multiple locations along the borehole, or 3) use a combination of the first two options Again, in Chapter 5 it is demonstrated that multiple shots are effective, particularly when resolving isolated anomalies in the vicinity of a borehole. 4.4 Inversion of Synthetic Data Generated f rom the 3 D Borehole Model To evaluate the compatibility of the proposed inversion scheme with respect to the input data synthetic records in this section were generated from 3 D borehole models using ABAQUS Ideally given the same Earth model, the axisymmetric forward model and the 3 D model should predict the same wave field However, since different types of elements and various meshing techniques were used to formulate the finite element models, the synthetic waveforms are not identical, but are fairly similar Because noisy data were used to infer model parameters true models could not be perfectly recovered Figure 4 10 presents the schematic used for conducting this synthetic experiment inside a 3 D borehole. To approximate a pulse loading a triangular wavelet source ( Figure 4 3b ) was radially excited in the center of the borehole wall (indicated by the red square in Figure 4 10) The axial component of particle displacements ( the double headed arrows in Figure 4 10) was monitored using a 10 channel re ceiver array uniformly placed along the wall, with the nearest offset and trace interval being 0.5 m ( the hollow circles in Figure 4 10) For the synthetic model studies discussed in the following sections, inversions were carried out by matching the predicted waveforms (from the axisymmetric forward model) against the observed waveforms (from the 3 D borehole models ) The amplitude of the waveforms and the L2 norm of the misfit function were normalized for ease of comparison.

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65 4.4.1 Homogeneous Model A uniform shear wave velocity of 1000 m/s was selected for the true model of this simple homogeneous borehole model ( Figure 4 11a ) The inversion started with an initial shear wave velocity of 775 m/s and followed a similar multiscale approach ( Figure 4 11b ) The true model was well recovered in that the inverted values of shear wave velocity range d from 950 m/s to 1070 m/s The bowl shaped resolvability is due to the limited spatial coverage, as only one shot gather was acquired in the data collection. The observed and predicted wavefor ms were in very good agreement and a typical convergence curve was observed (Figure 4 11c) This synthetic model study illustrates that the proposed inversion scheme is capable of handling 3 D data set because when studying a homogeneous medium the 3 D model and the axisymmetric forward model are compatible with each other. 4.4.2 Homogeneous Models with Ring Anomalies Despite the fact that ring shaped anomalies are rare in practice this study investigated them to test the resolving capability of the proposed inversion scheme in terms of characterization The observed waveforms were generated with a low velocity, ring shaped anomaly ( Vs = 450 m/s ) buried inside a 3 D homogeneou s medium ( Vs = 1000 m/s ). The ring shaped anomalies were designed so that the radial distances into the borehole wall increased continually The near anomaly (Figure 4 12a) had a 0.5 m by 0.5 m square section placed 0.5 m away from the wall. The i nversion started with an initial shear wave velocity of 1000 m/s Only one low pass filter was applied to the full waveforms, and the inversion was set to run for 15 iterations. Figure 4 12b illustrates that the anomaly was successfully detected in the inverted model. The inverted shear wave velocit y value was

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66 approximately 480 m/s for the anomaly and range d from 950 m/s to 1050 m/s for the homogeneous background. The waveforms were well matched and the convergence rate was fast (Figure 4 12c) These results can be attributed to the good initial model used for the inversion, and to the success of the Gauss Newton method The deep anomaly had a 0.5 m by 0.5 m square section placed 1.0 m away from the wall (Figure 4 13a) Using the same initial model and inversion procedure, the inversion was set to stop a fter 50 iterations The anomaly and the homogeneous background were well recovered ( Figure 4 13b ) with the inverted shear wave velocity value at approximately 520 m/s for the anomaly and ranging from 950 m/s to 1050 m /s for the homogeneous background. The waveforms were well matched, and the convergence rate was rapid for the first 10 iterations but slowed considerably over the remaining iterations (Figure 4 13c). The deeper anomaly had a 0.5 m by 0.5 m square section placed 1.5 m away from the wall ( Figure 4 14a ) Following the same initial model and inversion procedure, the inversion was set to stop after 50 iterations. The anomaly and the homogeneous background were well recovered ( Figure 4 14b ) The inverted shear wave velocity value was approximately 550 m/s for the anomaly and range d from 950 m/s to 1100 m/s for the homogeneous background. The waveforms were well matched and the convergence rate was acceptable (Figure 4 14c) The above synthetic model studies based the inversion on an axisymmetric forward model, whereas the input data were taken from 3 D borehole models, and suggest that with a good initial model the inversion technique can consistently detect ring shaped anomalies.

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67 4.4.3 Horizontally Layered Models with a Ring Anomaly To add further complexity, this study investigated a ring shaped anomaly inside a horizontally layered media ( Figure 4 15a ) The anomaly had a 0.5 m by 0.5 m square section placed 0.5 m away from the borehole wall. The shear wave velocity of the anomaly was 450 m/s and the shear wave velocity of the layered structure was the same as in Figure 4 6a. A uniform shear wave velocity of 775 m/s was selected as the initial model ( Figure 4 15b ) Following the proposed multiscale approach the inversion was carried out using three consecutive time windows and a low pass filter was applied to the waveforms at each time window First, the inversion was set to run for 10 iterations with the shortest time window of 2 m s and the last updated model wa s used as the starting model for the next run Next, the inversion was set to run for 15 iterations with the medium time window of 3 m s and again the last updated model was taken as the starting model for the next run. Finally the inversion proceeded until 30 iterations were completed with the full time window of 4 m s The velocity model s were updated radially as illustrated by the intermediately inverted models at each time window presented in Figure s 4 15c through 4 15e The anomaly start ed to become visible even with the shortest time window while the layered structure only became distinguishable as the time window widen ed The radial update s were successful due to the fact that details were successively added to the solution as high frequency component s, and full records were taken into account in the waveform inversion. In the final inverted model ( Figure 4 15e ) the true model was reasonably well recovered as both the layered structure and the anomaly were successfully delineated and detected. It is of note that the recovered shear wave velocity values gradually lost

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68 fidelity as the radial distance increase d which is the inherent tradeoff between penetration depth and resolution discussed above Nonetheless, the observed and predicted waveforms were in good agreement across all channels and the misfit function rapidly converge d as the number of iteration s increase d for all inversion attempts ( Figure s 4 15c through 4 15e ) The veloci ty resolution may be improved by adding multiple shot gathers along the receiver array, as sensitive data would significantly constrain the inverted model 4.5 Summary C hapter 4 validated t he finite element based forward model developed in Chapter 3 and verified that the proposed inversion scheme is effective with synthetic data sets gen erated from both the axisymmetric forward model s and the 3 D borehole models.

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69 a) b) Figure 4 1. Waveform comparison for boreholes with varyin g radii: a) A xial displacement and b) R adial displacement

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70 a) b) Figure 4 2. Waveform comparison for a borehole with a 1 km radius and plane strain flat ground solution: a) T angential displacement and b) N ormal displacement

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71 a) b) Figure 4 3. Triangular wavelet sou rces: a) H igh frequency source and b) L ow frequen cy source

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72 a) b) Figure 4 4. Borehole synthetic data: a) R adial displacement and b) A xial displacement

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73 Figure 4 5. The process for conducting the synthetic experiment inside a borehole (axisymmetric model)

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74 a) b) Figure 4 6. Horizontally layered model: a) T rue model and b) I nitial model a) ( Figure 4 7 is continued in the next page)

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75 b) c) ( Figure 4 7 is continued in the next page)

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76 d) Figure 4 7 Inversion of the horizontally layered model using the multiscale approach : a) S hort time window with low pass filter, b) Short time window without filter, c) Full time window with low pass filter, and d) Full time windo w without filter a) b) Figure 4 8 Cylindrically layered model: a) T rue model and b) I nitial model

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77 a) b) ( Figure 4 9 is continued in the next page)

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78 c) d) Figure 4 9 Inversion of the cylindrically layered model using the multiscale approach : a) Short time window with low pass filter, b) Short time window without filter, c) Full time window with low pass filter, and d) Full time window without filter

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79 Figure 4 10. The process for conducting the synthetic experiment inside a borehole (3 D model)

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80 a) b) c) Figure 4 11. Homogeneous model : a) T rue model, b) I nverted model, and c) W aveform match and convergence curve

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81 a) b) c) Figure 4 12. Homogeneous model with ring type anomaly (near) : a) T rue model, b) I nverted model, and c) Waveform match and convergence curve

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82 a) b) c) Figure 4 13. Homogeneous model with ring type anomaly ( far ) : a) T rue model, b) I nverted model, and c) Waveform match and convergence curve

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83 a) b) c) Figure 4 14. Homogeneous model with ring type anomaly (farther) : a) T rue model, b) I nverted model, and c) W aveform match and convergence curve

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84 a) b) c) ( Figure 4 15 is continued in the next page)

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85 d) e) Figure 4 15. Horizontally layered model with ring type anomaly : a) T rue m odel, b) I nverted model, c) I nversion for short time window, d) I nversion for medium time window, and e) I nversion for full time window

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86 CHAPTER 5 LOCATING ISOLATED ANOMAL IES NEAR A BOREHOLE 5.1 Introduction The d esign and construction of deep foundations in karst terrain require special consideration, as the subsurface conditions are often highly variable, and characterized by sinkholes and cavities, as well as voids filled with water, air, or low velocity materials. These anomalies are typically irregular in shape, variable in composition, and isolated in space Characterizing these anomalies is a 3 D problem and ideally requires a truly 3 D inversion However, as stated above 3 D borehole based FWI is beyond the scope of this work. Therefore this study attempted the proposed inversion scheme based on an axisymmetric forward model as an approximation to identify isolated anomal ies in the vicinity of a borehole. To this end, this research conducted comprehensive synthetic model studies to establish the feasibility of the proposed imaging t echnique for detecting cavities and delineating in the vicinity of a rock socket 5.2 Overview of Strategies As noted above, FWI problems involving borehole data are highly nonlinear and the solutions are generally non unique Therefore, a multiscale strateg y using frequency filtering and time windowing is imperative for a successful inversion that starts with a reasonable initial model and is not prone to getting trapped in the local minima. This research proposes two additional strategies for characterizing isolated anomalies in the vicinity of a borehole. 5.2.1 Inversion with Multi c omponent Data Axial (tangential) component s of particle displacements were inverted throughout this thesis Radial (perpendicular) component s on the other hand, may be added to the

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87 data set to further constrain the inversion as large impedance contrast between the cavity and its surrounding media would create a considerable amount of reflected energy (reflection and backscattering) Theoretically, t his energy can be more readily detected by the radial component if the source is an omnidirectional stress pulse. I n this study, unfortunately, the radial displacements derived from the 3 D models did not compare favorably with those p redicted from the forwar d model, and hence were not included in the I nversion. 5.2.2 Inversion with Multi b andwidth Sources This study used a seismic source with varying frequency content in order to improve model recovery. In general, a low frequ ency source penetrates deeper into the formation, thus providing a more defined skeleton of the model. However, when the dominating wavelength of the propagating waves is larger than or comparable to the size of the anomaly, diffr action allows the waves to pass the anomaly with little or no reflected energy. Since the reflected energy is critical for cavity detection, a high frequency source ( Figure 4 3a) is first applied in the inversion, followed by a low frequency source ( Figure 4 3b). 5.2.3 Inversion from Multiple Planes and with Multiple Shots In blind test s the location of the anomaly is unknown. Therefore, the experiment design ( Figure 2 10) i s critical in identifying the anomaly and characterizing its depth, size, and azimuth The following synthetic model studies use d three viewing planes 0, 90, and 180 degree s with respect to the plane in which the anomaly is embedded to locate the anomaly at a known location near a borehole. Once the anomaly is roughly identified multiple shot gathers can be collected in that particular plane in order to

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88 increase the spatial resolution of the inverted model, as the degree of non uniqueness in the solution can be greatly reduced through the use of sensitive data 5.3 Synthetic Model Studies Synthetic model studies were conducted to evaluate the feasibility of the proposed imaging technique for locating isolated anomalies in the vicinity of a borehole. Anomalies were simplified as cavities (voids filled with air) in the 3 D borehole models, from which synthetic records were generated and treated as data for the inversion s As stated above inversions were based on an axisymmetric borehole model, divided into 36 velocity cells ( Figure 3 2 ) Because an inexact forward model was used to approximate the physical reality, some dis crepancies between the inverted model and the true model were to be expected This section investigated a single anomaly embedded in a homogeneous medium with varying distance away from the borehole wall in order to characters its axial and radial dimensions and azimuth Next, multiple anomalies embedded in the same plane and different planes were examined to evaluate the capability of the proposed imaging technique for locating multi ple isolated anomalies around a borehole Lastly, a single anomaly embedded in a multi layered system was analyzed to test if the proposed imaging technique is capable of characterizing both the anomaly and individual layering. 5.3.1 Synthetic Model 1 First, a homogeneous model w ith an isolated anomaly near the borehole wall was studied. The anomaly was located 0.5 m from the wall and was modeled as a 0.5 m by 0.5 m cavity in the r z plane and 0.5 m into the paper Figure 5 1 shows the 3 D model and its cross section at the 0 degree plane. The shear wave velocity was 1000 m/s for the homogeneous background and 0 m/s for the anomaly.

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89 Using the test configuration shown in Figure 4 10, a uniform array of 10 re ceivers was used for data acquisition with the nearest offset and trace interval being 0.5 m The array was placed at 0 90 and 180 degree plane s simultaneously around the borehole wall. The i nversion was carried out sequentially using three consecutive time windows: 2 3 and 4 ms For the first two time windows, a high frequency source ( Figure 4 3a ) was used, and for the last time window, a low frequency source ( Figure 4 3b ) was used. For each time window, only one low pass filter was applied to the waveforms. The initial model was assumed to be homogeneous and having the same background velocity of 1000 m/s The inversion result of 0 degree plane ( Figu re 5 2 ) exhibit ed a sharp velocity contrast between the anomaly and background which indicate s the possible location of the anomaly in terms of axial and radial dimensions The waveforms were reasonably well matched and a good convergence rate was observed Note that the recovered velocity of the anomaly was approximately 700 m/s which is greater than the shear wave velocity of a cavity (0 m/s) This difference was due to the fact that only the adjusted during inversion while the mass density were held constant. As a result, all velocity cells in the inversion model were elastic while the data were generated by modeling the cavity as a boundary in the 3 D simulation. Another cause of the discrepancy was the use of an inexact forward model to approximate the 3 D reality The anomaly shown in Figure 5 2 should be interpreted as a ring while the true model only contains an isolate d cavity embedded in the 0 degree plane. Nevertheless, it appears that an inversion based on the axisymmetric

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90 forward model is able to determine the axial and radial dimensions of an isolated anomaly, as long as the receiver array is in the same plane as the anomaly The inversion results of 90 degree plane and 180 degree plane are presented in Figure s 5 3 and 5 4, respectively. A sharp velocity contrast developed at ex actly the same spot in both inversions which implies that the receiver array can detect an isolated anomaly near a borehole no matter where it is placed on the wall circumference It is arguable that the sharp velocity contra s t was due to the axisymmetric forward model used for the inversion. However, considering the physics of wave propagation inside a borehole with an isolated anomaly, the reflected energy is first detected by the in plane (0 degree plane) receivers, and then by the out of plane receivers (90 degree plane followed by 180 degree plane). Moreover, when waves are traveling quickly inside a small borehole the difference in timing is so small that the reflected energy is detected as if the anomaly were directly in plane This explains why in the out of plane images the v elocity contrast is observed at the same location compared to the in plane image However, a careful examination of the recovered images reveals that the contrast between the anomaly and background weakens as the relative angle between the anomaly and the viewing plane widens For example, the inverted shear wave velocity was approximately 780 m/s for the 18 0 degree plane and approximately 740 m/s for the 9 0 degree plane, as compared to 700 m/s for the 0 degree plane. This relationship is appropriate because the reflected energy attenuates as it propagates away from the anomaly, which in this case is an important indicator of where the anomaly was possibly oriented Therefore, it appears that the proposed

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91 imaging technique has the potential to determine both the axial and radial dimensions and the azimuth of an isolated anomal y in the vicinity of a borehole 5.3.2 Synthetic Model 2 The next model considered in this study was a homogeneous model with an isolated anomaly slightly farther from the borehole wall. In this case, the anomaly was located 1.0 m from the wall. The anomaly was modeled as a 0.5 m by 0.5 m cavity in the r z plane and 0.5 m into the paper Figure 5 5 shows the 3 D model and its cross section at the 0 degree plane. The shear wave velocity was 1000 m/s for the background and 0 m/s for the anomaly. The same acquisition geometry and seismic sources were used as in Synthetic Model 1 The inversion was carried out sequentially using the same initial model. The final inversion results from various viewing planes are presented in Figure s 5 6 through 5 8. The predicted waveforms were in good agreement with the data, and a velocity contrast developed inside a relatively uniform background for all viewing planes. The axial and radial dimensions of the anomaly can be identified as the point at which the contrast developed while the azimuth of the anomaly can be inferred by comparing the contrast level with the background of various viewing planes. The inverted velocity of the anomaly was approximately 820 850 and 880 m/s for the 0 90 and 180 degree plane s respectively, and the color scale ranged from 800 m/s to 1050 m/s. The inverted velocity of the anomaly was greater than it was in the previous case because the anomaly in this case was embedded deeper from the boundary of observation, resulting in weaker reflected/sca ttered energy to the receiver array. Despite the fact that the velocity contrast is not as sharp as desired and that some smearing occur r ed around

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92 the anomaly in the recovered image, the proposed imaging technique still holds the potential to find indications of an isolate d anomaly in a 3 D space 5.3.3 Synthetic Model 3 Th e third model investigated in this study was a homogeneous model with two isolated anomalies embedded in the same plane. Each anomaly was modeled as a 0.5 m by 0.5 m cavity in the r z plane and 0.5 m into the paper. The lower anomaly was 0.5 m away from the wall while the upper anomaly was placed 1.0 m away from the wall. The 3 D model and its cross section at the 0 degree plane are shown in Figure 5 9. Again, the shear wave velocity was 1000 m/s for the background and 0 m/s for the anomaly. The same acquisition geometry and inversion procedure was used for this model. The final inversion results are presented in Figure s 5 10 through 5 12. For the 0 degree plane view, two contrasts occurred ( Figure 5 10 ) The inverted shear wave velocities were approximately 750 m/s for the lower anomaly and 850 m/s for the upper anomaly. Similar to the previous examples, the velocity contrast was relatively sharp for the near (lower) anomaly and weak for the deep (upper) anomaly. The homogeneous background was not recovered perfectly especially in cells surrounding the anomaly In fact, the incompatibility between model and data under the waveform matching algorithm caused the inversion artifact For the 90 degree plane view, only the lower anomaly was visible in the recovered image ( Figure 5 11 ) Moreover, the contrast level was very weak in that the inverted velocity of the anomaly was approximately 875 m/s while the inverted velocity of the background ranged from 950 m/s to 1050 m/s The upper anomaly in this case was not distinguishable despite the fact that the waveforms were reasonably well

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93 matched which suggests that the scattered wavefield was weak in the presence of multiple isolated anomalies and that one shot in the middle of the array was not enough to achieve the desired resolution ( previous research predi ct ed that both anomalies would be visible in the image with weaker contrast ). For the 180 degree plane view, a fairly uniform velocity model was reconstructed ( Figure 5 12 ) The inverted shear wave velocity ranged from 950 m/s to 1050 m/s while the true background velocity was 1000 m/s which implies that the 180 degree out of plane receiver array 0 degree plane anomalies and can only what is within the plane a homogenous background in this case. Th e waveform s comparison also confirm s that the scattered wavefield wa s weak in the presence of multiple isolated anomalies In other words, the data used in the inversion were not sufficiently sensitive to infer the internal structure of the model. Nonetheless, the fact that only the in plane receivers could detect the anomalies simplified the interpretation and characterization processes and demonstrated the value of the proposed multi plane waveform inversion. 5.3.4 Synthetic Model 4 The research next investigated a homogeneous model with two isolated anomalies embedded in the opposite planes. Each anomaly was modeled as a 0.5 m by 0.5 m cavity in the r z plane and 0.5 m into the paper. The lower anomaly was embedded in the 0 degree plane, and the upper anomaly was embedded in the 180 degree plane. Both anomalies were placed 0.5 m away from the wall. The 3 D model and the velocity cross sections are presented in Figure s 5 13 and 5 14. The shear wave velocity was 1000 m/s for the background and 0 m/s for the anomaly.

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94 The same acquisition geome try and inversion procedure was used for this model. The final inversion results are presented in Figure s 5 15 through 5 17 For the 0 degree plane view, the in plane, lower anomaly was clearly visible, while the out of plane upper anomaly was not visible (Figure 5 15). The inverted shear wave velocity of th e anomaly was approximately 725 m/s For the 90 degree plane view ( Figure 5 16 ) both anomalies were visible as well as some artifact in the middle of the image. The inverted shear wave velocity of the anomaly was in the range of 850 875 m/s For the 180 degree plane view, t he in plane, upper anomaly was clearly visible while the out of plane, lower anomaly was not characterized in the image (Figure 5 17) The inverted shear wave velocity of the anomaly was approximately 725 m/s Note that the reconstructed images shown in Figure 5 15 and 5 17 are near perfect reflections of each other due to the symmetry in the true model ( Figure 5 13 ) For this reason, the reconstructed image shown in Figure 5 16 is also symmetric with respect to the horizontal centerline For each case presented herein, the waveforms were in good agreement with the data, and the convergence rate w as reasonable. The se results indicate that the proposed imaging technique can potentially identify isolated anomalies embedded in different planes by performing multi plane waveform inversions. However, previous research indicates that sufficiently sensitive data should characterize even out of plane anomalies so that they are visible in the recovered images Past findings also reveal that the way to infer azimuth is to compare the contrast level with the background of various viewing plane s Therefore, in the next set of experiments two shot gathers were collected and used as data in the inversion to

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95 validate the hypothesis. For the purpose of demonstration, the se two shots were placed directly adjacent to the anomalies of interest i.e., at and m Figure 5 18 presents the final inverted model for the 0 degree plane view using two shots in which both anomalies were characterized The inverted shear wave velocity of the in p lane, lower anomaly was approximately 725 m/s wh ereas the inverted shear wave velocity of the 180 degree out of plane, upper anomaly was approximately 775 m/s Figure 5 19 presents the final inverted model for the 90 degree plane view using two shots. In this case, both anomalies are considered out of plane and are clearly visible in the image. The inverted shear wave velocity was approximately 700 m/s Figure 5 20 presents the final inverted model for the 180 degree plane view using two shots. Once again, both anomalies were well characterized in the image. The inverted shear wave velocity of the in plane, upper anomaly was approximately 725 m/s while that of the 180 degree out of plane, lower anomaly was approximately 800 m/s These results appear to be in good agreement with the hypothesis that the inverted shear wave velocity of the anomalies under the out of plane view s would be greater than that of the anomalies under the in plane views. 5.3.5 Synthetic Model 5 Finally this research analyzed an isolated anomaly embedded in a multi layered system in order to test whether the proposed imaging technique is capable of characterizing the isolated anomaly as well as the individual layering. Note that a ring s haped anomaly was previously investigated in the same layered system, in which the shear wave velocity varie d gradually from 450 m/s at the top to 1000 m/s in the middle and back to 450 m/s at the bottom. The anomaly was located 0.5 m away from the wall

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96 and modeled as a 0.5 m by 0.5 m cavity in the r z plane and 0.5 m into the paper. Figure 5 21 presents the 3 D model and the velocity section at the 0 degree plane. A uniform shear wave velocity of 775 m/s was selected as the initial model. Following the proposed multiscale approach, the inversion w as carried out sequentially using three consecutive time windows : 2 3 and 4 ms A high frequency source ( Figure 4 3a ) was used for the first two time windows and a low frequency source ( Figure 4 3b ) was used for the last time window. For each time window, only one low pass filter was applied to the waveforms. The fi nal inverted models from multiple viewing planes are presented in Figure s 5 22 through 5 24. In general, the layered structure was successfully delineated for all viewing planes The associated waveforms were very well matched in the time domain and a high level of convergence was observed Some smearing is visible around the anomaly which occurred because of the incompatibility between model and data discussed above In addition, data insensitivity prevented good velocit y recovery at the far ends of the model. For the isolated anomal y of interest, the inverted shear wave velocit y was approximately 600 750 and 850 m/s for 0 90 and 180 degree plane s respectively. The fact that t he anomal y is more readily identifiable from the in plane view than the out of plane views is in agreement with previous findings. This example demonstrates that the proposed imaging technique is capable of locating an isolated anomaly embedded in a multi layered system and highlights the importance of the proposed multi plane data acquisition and inversion 5.4 Summary Synthetic model studies demonstrate that the proposed imaging technique is capable of finding indications of isolated anomalies in the vicinity of a borehole. The

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97 number of anomalies and their axial and radial dimensions can be accurately determined from the recovered image s, wh ile the azimuth of anomalies can be inferred by examin ing the level of contrast against the backgro und in the images recovered from multi plane waveform inversion s

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98 Figure 5 1. Synthetic model 1 : t rue model and velocity section Figure 5 2. Synthetic model 1 : i nversion result of 0 degree plane

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99 Figure 5 3. Synthetic model 1 : i nversion result of 90 degree plane Figure 5 4. Synthetic model 1 : i nversion result of 180 degree plane

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100 Figure 5 5. Synthetic model 2 : t rue model and velocity section Figure 5 6. Synthetic model 2 : i nversion result of 0 degree plane

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101 Figure 5 7. Synthetic model 2 : i nversion result of 90 degree plane Figure 5 8. Synthetic model 2 : i nversion result of 180 degree plane

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102 Figure 5 9. Synthetic model 3 : t rue model and velocity section Figure 5 10. Synthetic model 3 : i nversion result of 0 degree plane

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103 Figure 5 11. Synthetic model 3: i nversion result of 90 degree plane Figure 5 12. Synthetic model 3 : i nversion result of 180 degree plane

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104 Figure 5 13. Synthetic model 4 : true model Figure 5 14. Syn thetic model 4 : velocity section of 0 180 degree plane

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105 Figure 5 15. Syn thetic model 4 : i nversion result of 0 degree plane with one shot Figure 5 16. Syn thetic model 4 : i nversion result of 90 degree plane with one shot

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106 Figure 5 17. Syn thetic model 4 : i nversion result of 180 degree plane with one shot Figure 5 18. Syn thetic model 4 : i nversion result of 0 degree plane with two shots

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107 Figure 5 19. Syn thetic model 4 : i nversion result of 90 degree plane with two shots Figure 5 20. Syn thetic model 4 : i nversion result of 180 degree plane with two shots

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1 08 Figure 5 21. Synthetic model 5 : t rue model and velocity section Figure 5 22. Synthetic model 5 : i nversion result of 0 degree plane

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109 Figure 5 23. Synthetic model 5 : i nversion result of 90 degree plane Figure 5 24. Synthetic model 5 : i nversion result of 180 degree plane

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110 CHAPTER 6 CLOSURE 6.1 Summary of Findings The technique of borehole based FWI using a regularized Gauss Newton method has been developed to provide high resolution velocity images along and around a borehole at depth and within a socket proposed for use in the design of a drilled shaft foundation. The capability of the proposed imaging technique has been demons trated through comprehensive synthetic model studies and the findings are summarized as follows. 6.1.1 Forward Modeling Wave propagation insid e a cylindrical cavity is complicated and significantly different from that on flat ground, due to a different set of boundary conditions imposed by the borehole geometry. ABAQUS is capable of modeling the borehole geometry and simulating wavefields generated from mechanically impacting sources (either point load or uniformly distri buted ring load). Accurate 3 D borehole model s and axisymmetric borehole model s were both formulated using ABAQUS The axisymmetric model with a ring load was chosen as the forward model to be used in the inversion after considering the computational advantages and assumption limitations of the model. 6.1.2 Inversion System The r egularized Gauss Newton method proved to work well for the time domain FWI within a borehole. An interface written in Python script, was developed to integrate the forward model solved in ABAQUS in to the inversion algorithm implemented in MATLAB High nonlinearity of FWI coupled with a downhill optimization scheme requires a reasonably good starting model to avoid local minima. To lessen the restriction of the initial model, some strategies were adopted during the inversion stage, such as frequency filtering and temporal windowing, which proved to work well in the synthetic model studies

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111 6.1.3 Inversion Results The proposed imaging technique is capable of delineating Earth models that are e ither horizontally or cylindrically multi layered. It is also capable of locating ring type anomalies at various distances away from the borehole wall, embedded inside a homogeneous or a layered background. Despite the fact that the forward model is axisymmetric, detection of isolated anomalies was attempted. For synthetic models in this study, waveforms were collected at 0 90 180 degree planes in relation to the isolated anomaly and th en inverted. It was found that the axial and radial dimensions of a 0.5 m square anomaly within 2 m from a borehole can be accurately determined for all viewing planes. For the azimuth, the inverted velocity corresponding to its viewing plane increase d as the angle between anomaly and viewing plane increase d Also a slight but consistent time shift occurred in the associated wavefields. These two important observations can be explained by the fact that the reflected energy scattered by the anomaly attenuates as it propagates and is detected by the receivers at different timings. The proposed imaging technique appears to be capable of finding indications of isolated anomalies in th e vicinity of a borehole The key is to perform multi plane waveform inversion s around the borehole circumference. In the current implementation on a standard personal computer it takes less than 4 hours per plane to produce the shear wave velocity images. It is recommended that in a n actual test six or eight planes be used for the imaging purpose, as waveforms can be recorded and processed simultaneously if the borehole logging tool is capable of doing so 6.2 Conclusions The findings outlined above have led to following conclusions In order to preform FWI within a borehole, the cylindrical geometry and the associated boundary conditions must be modeled correctly Due to the formidable computer time and uncertainty of 3 D borehole inversion, the axisymmetric forward model was chosen for the proposed imaging effort. The inversion was based on a regularized Gauss Newton method, and the interface between ABAQUS and MATLAB was developed using Python script. Inversions of synthetic data suggest that the proposed imaging technique is capable of delineating E arth models that are either hori zontally or cylindrically multi layered. It is

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112 also capable of locating ring type anomalies at various distances away from the borehole wall. For more realistic anomalies (voids and cavities ), which are often spatially isolated it is proposed that the inversion be done circumferentially on multiple viewing plane s in relation to the anomal ies It was found that the axial and radial dimensions of the anomaly can be accurately determined for all viewing plane s Due to the nature of an axisymmetric forward model, the azimuth of the anomaly can be difficult to determine. However, comparing the velocity value s of the anomaly inverted from multiple viewing planes revealed that the velocity increase d as the angle between the anomaly and the viewing plane increase d This relationship can be explained by the fact that the reflected energy scattered by the anomaly attenuate d as i t propagate d In other words, the in plane view exhibited the str ongest reflection while the 180 degree out of plane view had the weakest. Furthermore, close comparison of the synthetic waveforms recorded on multiple viewing planes revealed that a slight but consistent time shift occurred in the associated wavefields, which can also be explained by the timing of reflection as a result of the anomaly. Therefore, the azimuth of the anomaly can be determined approximately 6.3 Borehole Logging Tool In this research a single well imaging technique based on full waveform inversion was developed The potential of the technique was evaluated by synthetic data generated inside a borehole However, t o be able to implement in the field, a borehole logging tool that can effectively take advantage of the proposed imaging technique must be built The basic design of the tool would include a seismic source and a receiver array, properly instrumented and integrated within a housing that can move up and down a borehole, is waterproof, and will couple to the borehole wall. T he

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113 seismic source should be capable of generating a n omni directional, ring type stress pulse that excites perp endicularly to the borehole wall. The receiver array, comprised of ten to twelve small sized three component accelerometers should be vertically placed at approximately a 15 c m interval along the borehole wall To improve spatial coverage the source should be designed such that it can repeatedly excite at multiple locations along the array In practice the borehole logging tool should be able to create a 3 D scan of material along and around the borehole by rotating the rec eiver array circumferentially inside the borehole. A lternatively a 3 D receiver array consisting of mu l tiple 1 D vertical arrays discussed above, could be placed around the borehole circumference with a uniform azimuthal interval to improve the efficiency of the field testin g 6. 4 Recommendations The following recommendations are suggested after re viewing the findings and conclusions discussed above : A new b orehole logging tool must be developed that is capable of generating and recording full waveforms from an instrumented array deployed vertically and circumferentially along the borehole wall Physical modeling can then validate the proposed imaging technique before it is applied in the field The system of FWI itself needs further development. For example, multiple shots can be added to increase the resolution of velocity recovery. Multi variable inversion needs to be investigated for simultaneous reconstruct ion of bot h compressional and shear wave velocities, as well as density. The integration between ABAQUS and MATLAB offers many opportunities for solving inverse problems in general. The se programs can easily simulate full waveform based seismic crosshole tomography and vertical seismic profiling f or the purpose of site characterization 3 D surface based F WI is still in its in fancy. A successful inversion ultimately requires an extremely efficient forward model coupled with a carefully designed receiver array and a fairly good starting model. 3 D borehole based FWI may be similarly managed where parallel computing is key.

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114 LIST OF REFERENCES Aki, K. and Richards, P. (1980). Quantitative Seismology: Theory and Methods, W. H. Freeman, San Francisco, CA. Aster, R. C., Borchers, B., and Thurber, C. H. (20 05). Parameter Estimation and Inverse Problems. Academic Press, Waltham, MA. ASTM Standard D4428/D4428M 07, 2007, "Standard Test Methods for Crosshole Seismic Testin g," ASTM International, West Conshohocken, PA, 2007, DOI: 10.1520/D4428_D4428M 07, www.astm.org ASTM Standard D7400 08, 2008, "Standard Test Methods for Downhole Seismic Testing," ASTM International, West Conshohocken, PA, 2008, DOI: 10.1520/D7400 08, www.astm.org ASTM Standard D5777 ic Refraction PA, 2011, DOI: 10.1520/D5777 00R11E01, www.astm.org ASTM Standard D7128 05, 2 Reflection Conshohocken, PA, 2010, DOI: 10.1520/D7128 05R10, www.astm.org Biot, M. A Journal of Applied Physics 23(9), 997 1005. Geophysics 60(5), 1457 1473. Burger, H. R. (1992). Exploration Geophysics of shallow subsurface. Prentice Hall, Englewood Cliffs, New Jersey. ingle W ell Imaging U sing F ull W aveform S onic D ata the University of Calgary. Cheng Nondestructive Testing of Multilayered Ph.D. dissertation, the University of Texas at Austin. Coates, R., Kane, M., Chang, C., Esmersoy, C., Fukuhara, M., and Yamamoto, H. (2000) Single well S onic Imaging: High definition Reservoir C ross sections from Horizontal W ells. SPE/CIM International Conference on Horizontal Well Technology, Calgary, Canada Constable, S., Parker, R., and Constable, C i nversion: A practical algorithm for generating smooth models from electromagnetic sounding data. Geophys ics 52(3), 289 300.

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115 Geotechnical Testing Journal 26(4) 410 420. with the EVA full waveform tool. The Log Analyst 32 (3) 271 278. boreh ole structure using full waveform sonic Geophysics 54(6), 747 757. Mathematical and Computer Modeling 12(8), 967 973. K alinski, M. E. (1998). Surface Wave Measurements in Cased and Uncased Boreholes Ph.D. dissertation, the University of Texas at Austin. Geophysics 45(10), 1489 1506. ttering, Theory and Application, Society of Industrial and Applied Mathematics, Expanded Abstracts, 206 220. P Philosophical Transaction s of the Royal Society of London A203, 1 42. A Method for the Solution of Certain Non Linear Problems in The Quarterly of Applied Mathematics 2( 2 ), 164 168. Love, A.E.H. (1892). A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press England. Marquardt, D. W. ( A lgorithm for L east Squares Estimation of Nonlinear Journal of the Society for Industrial and Applied Mathematics 11(2), 431 441. McMechan, G. A. and Yedlin, M. J. ( 1981). Analysis of dispersive waves by wave field transformation. Geophysics 46 (6) 869 874. Earthquake Engineering, 3(1) 65 70, Tokyo, Japan dimensional elastic inversion of multi offset seismic Geophysics 52(9), 1211 1228.

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116 Geophysics 54(12), 1575 1586. Nazarian, S. a nd Transportation Research Record 993, 67 79. Geophysical Characterization of Sites, ISSMFE, Technical Committee 10 for XIII ICSMFE, 57 63, International Science Publishers, New York NRC (2000 ). Seeing into the Earth: Noninvasive Characterization of the Shallow Subsurface for Environment al and Engineering Applications, National Academy Press, Washington D.C. Palmer, D. (1980). The Generalized Reciprocal Method of Seismic Refraction Interpretation. Society of Expl oration Geophysicists, Tulsa Oklahoma. Channel Analysis of Surface Wave Geophysics 64(3), 800 808. Geophysical Prospecting 56 (6) 761 763. source cross hole tomography. Part 1: Acoustic wave Geophysical Prospecting 38(3), 287 310. ry applied to multi source cross hole tomography. Part 2: Elastic wave Geophysical Prospecting 38(3), 311 329. Pratt, R. G., Shin, C., and Hicks, G. J. (1998). Gauss Newton and full Newton methods in frequency space seismic waveform inversion. Geophysical Journal International 133( 2 ), 341 362. Rayleigh, L. (1885 W aves Propagated along the Plane Surface of an Elastic Proceedings of the London Mathematical Society 17 (1) 4 11. Romdhane, A., Grandjean, G., Brossier, R., Rejiba, F., Operto, S., and Virieux, J. structure characterization by 2D elastic full Geophysics 76(3), R81 93. Santamarina, J. C. and Fratta, D. (2005). Discrete Signals and Inverse Problems: An Introduction for Engineers and Scientists, John Wiley & Sons, Chichester. Schlumberger. (1997). DSITM Dipole Shear Imager: Corporate brochure SMP 9200, 36 pages.

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117 Geophysical Journal International 108(1), 281 292. Geophysical Journal International 167(3), 1373 1384. SIMULIA ABAQUS 6.11, Dassault Systems, ABAQUS Inc., 2011. Geophysics 69 (1), 231 248. Stokoe, K. H., II ar Wave Velocity by Cross Hole Journal of the Soil Mechanics and Foundation Division 98(5), 443 460. acteristics of Sites, ISSMFE, Technical Committee 10 for XIII ICSMFE, International Science Publishers, New York, 15 25. Stokoe, K. H., II and Santamarina, J.C. (2000). Wave Based Testing in International Conference on G eotechnical and Geological Engineering, Melbourne, Australia, 1490 1536. Geophysics 49(8), 1259 1266. Tarantola, A. (1987). Inverse Problem Theory: Methods for Data Fitting and Model Parameter Estimation, Elsevier Science Publ. Co., New York. Dimensional Inversion of Full Waveform s Journal of Geotechnical and Geoenvironmental Engineering 1 38(9), 1075 1090. Tran, K. T., Hiltunen, D. R., Sarno, A. I., and Hudyma, N. (2011) FDOT BDK 75 977 01, Florida Department of Transportation, February, 190 pp. Newton inversion of 2 Soil Dynamics and Earthquake Engineering 43 (1) 16 24. Virieux, J. and Ope r waveform inversion in exploration Geophysics 74(6), WCC1 26. Geophysical Journal International 97(2), 223 245.

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118 Zerwer, A., Cascante, G. Journal of Geotechnical and Geoenvironmental Engineering 128(3), 250 261. Geophysics 63(5), 1726 1737. Technology.

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119 BIOGRAPHICAL SKETCH Pengxiang Jiang was born in 198 5 in Zhenjiang, Jiangsu, China, and remained in Zhenjiang until he graduated from high school in 2004. Then he enrolled in the program of Civil and E nvironmental E ngineering at the University of Macau, and received his Bachelor of Science degree in June, 2008. Later the same year he enrolled in the program of Civil and Coastal Engineering at the Unive rsity of Florida, where h e worked as a gradu ated assistant under Dr. Dennis Hiltune n. He obtained his Master of Science degree in May, 2010 and Doctor of Philosophy in December 2013.


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