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Building Subsurface Velocity Models with Sharp Interfaces Using Interface-Guided Seismic

Full-Waveform Inversion

YOUZUO LIN1 and LIANJIE HUANG

1

Abstract—Reverse-time migration has the potential to image

complex subsurface structures, including steeply-dipping fault

zones, but the method requires an accurate velocity model.

Acoustic- and elastic-waveform inversion is a promising tool for

high-resolution velocity model building. Because of the ill-posed-

ness of acoustic- and elastic-waveform inversion, it is a great

challenge to obtain accurate velocity models containing sharp

interfaces. To improve velocity model building, we develop an

acoustic- and elastic-waveform inversion method with an interface-

guided modified total-variation regularization scheme to improve

the inversion accuracy and robustness, particularly for models with

sharp interfaces and steeply-dipping fault zones with widths much

smaller than the seismic wavelength. The new regularization

scheme incorporates interface information into seismic full-wave-

form inversion. The interface information of subsurface interfaces

is obtained iteratively using migration imaging during waveform

inversion. Seismic migration is robust for subsurface imaging. Our

new acoustic- and elastic-waveform inversion takes advantage of

the robustness of migration imaging to improve velocity estima-

tion. We use synthetic seismic data for a complex model containing

sharp interfaces and several steeply-dipping fault zones to validate

the improved capability of our new acoustic- and elastic-waveform

inversion method. Our inversion results are much better than those

produced without using interface-guided regularization. Acoustic-

and elastic-waveform inversion with an interface-guided modified

total-variation regularization scheme has the potential to accurately

build subsurface velocity models with sharp interfaces and/or steep

fault zones.

Key words: Interface information, acoustic- and elastic-

waveform inversion, total-variation regularization, regularization

techniques, velocity estimation.

1. Introduction

It is challenging to obtain accurate subsurface

velocity models containing sharp interfaces (e.g.

basalt body) and/or fracture/fault zones for many

applications, such as geothermal exploration. Fault

zones may provide pathways for efficient geothermal

fluid flow and heat transfer (Goyal and Kassoy 1980;

Corbel et al. 2012), or confine boundaries of

geothermal reservoirs (Knipe 1992; Barnicoat et al.

2009). Therefore, imaging fault zones plays an

important role in geothermal energy exploration and

other applications.

Reverse-time migration (RTM) can image com-

plex subsurface structures (Baysal et al. 1983). RTM

solves the full scalar-wave equation in heterogeneous

media for forward propagation of source wavefields

and backward propagation of recorded seismic

reflection data from receivers. Chen and Huang

(2013, 2014) and Huang et al. (2011) further

demonstrated that reverse-time migration has the

potential to image steeply-dipping fault zones in

geothermal fields (Huenges and Patrick 2010).

However, RTM usually requires an accurate velocity

model to yield high-resolution images (Chen and

Huang 2013). Therefore, it is essential to obtain

accurate subsurface velocity models for high-resolu-

tion imaging of steeply dipping fault zones.

Acoustic- and elastic-waveform inversion

(AEWI) is a quantitative method for estimating sub-

surface geophysical properties. It is challenging to

use AEWI for practical applications (Tarantola 1984;

Sirgue and Pratt 2004; Virieux and Operto 2009).

Many different methods have been developed to

alleviate the ill-posedness problem of AEWI caused

by limited data coverage, including regularization

techniques (Hu et al. 2009; Burstedde and Ghattas

2009; Ramirez and Lewis 2010; Guitton 2012; Lin

and Huang 2015), joint-inversion techniques (Ma and

Hale 2013; Zhang and Chen 2014), preconditioning

approaches (Guitton et al. 2012; Tang and Lee 2010),1 Geophysics Group, Los Alamos National Laboratory, Los

Alamos, NM 87545, USA. E-mail: [email protected]

Pure Appl. Geophys.

� 2017 Springer-Verlag (outside the USA)

DOI 10.1007/s00024-017-1628-5 Pure and Applied Geophysics

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dimensionality reduction methods (Moghaddam and

Herrmann 2010; Habashy et al. 2011; Abubakar et al.

2011), and using a priori information (Ma et al. 2012;

Ma and Hale 2012; Zhang and Huang 2013).

There are two major regularization techniques

used in seismic full-waveform inversion: L2-norm-

based regularization such as the Tikhonov regular-

ization (Hu et al. 2009), and L1-norm-based

regularization, such as the total-variation (TV) regu-

larization (Anagaw 2010; Guitton 2012; Lin and

Huang 2015), Cauchy function regularization (Guit-

ton 2012), and compressive sensing (Li et al. 2012).

AEWI with Tikhonov regularization techniques are

computationally efficient, but usually yield inversions

with smooth interfaces. On the other hand, AEWI

with TV regularization techniques provide inversions

with sharp interfaces, thus is more appropriate for

estimating subsurface geophysical properties than

AEWI with Tikhonov regularization when subsurface

velocity models contains sharp interfaces. Despite

their interface perseverance capability, AEWI with

TV regularization techniques face some challenges.

First, the TV regularization term makes the numerical

methods for solving the inverse problem extremely

unstable because of the existence of the singular

points in the solution space (Rudin et al. 1992).

Second, the convergence of the TV-based AEWI is

highly sensitive to the smoothing parameter that

makes the TV regularization term differentiable at the

origin point (Vogel 2002).

A priori information has been shown to success-

fully reduce the ambiguity and improve the accuracy

of various seismic inversions (Chasseriau and Chou-

teau 2003; Dewaraja et al. 2010; Zhang and Huang

2013). The most commonly used a priori information

include: location of the regions of interest (Zhang

and Huang 2013), velocity boundary (Li and Olden-

burg 2003), variance of noise (Chasseriau and

Chouteau 2003), and structural orientation (Lelievre

and Oldenburg 2009; Ma et al. 2012; Ma and Hale

2012). In particularly, we are mostly interested in

using the interfaces of geologic formations as a priori

information. It has been proven in other research

areas that interfaces can be useful for image recon-

structions. For instance, Dewaraja et al. (2010) used

the boundary information to improve the recon-

struction quality for SPECT imaging; Guo and Yin

(2012) encoded the interface information in MRI

reconstructions, which also yields improved imaging

results; Baritaux and Unser (2010) applied the inter-

face information to the fluorescence diffuse optical

tomography, and obtained enhanced reconstructions

compared to results produced without using the

interface information.

We develop acoustic- and elastic-waveform

inversion methods with interface-guided regulariza-

tion to improve the accuracy of velocity model

building. The interface-guided regularization can be

combined with any other regularization techniques.

We reported some preliminary results in Lin and

Huang (2014). In this paper, we provide a compre-

hensive study of the improved capability of acoustic-

and elastic-waveform inversion by combining the

interface-guided regularization with a modified total-

variation regularization (MTV) (Lin and Huang

2015). We obtain the interface information using

RTM imaging and then use the interface information

to further improve the AEWI accuracy.

We employ an alternating-direction minimization

method to solve our new acoustic- and elastic-

waveform inversion problems (Bauschke et al.

2006). We decouple the minimization problem of the

misfit function for AEWI with the interface-guided

MTV regularization into two standard subproblems:

an AEWI subproblem with the L2-norm-based

Tikhonov regularization, and an interface-guided L2-

TV subproblem. We further use the nonlinear con-

jugate gradient (NCG) scheme (Nocedal and Wright

2000) to solve the first subproblem, and use the split-

Bregman iterative method to solve the second sub-

problem, an L2-TV problem (Osher et al. 2005;

Goldstein and Osher 2009). There are two major

benefits for using our computational methods. First,

we are able to decouple the TV regularization term

and the data misfit term to separate the nonlinearity of

data misfit from that of the TV regularization term.

This simplifies the complexity of the inverse prob-

lem. Second, the use of split-Bregman iterative

method avoids the selection of the smoothing

parameter in the TV term, which significantly

improves the algorithm robustness and computational

efficiency.

We use synthetic data generated using both

acoustic- and elastic-velocity models to validate the

Y. Lin and L. Huang Pure Appl. Geophys.

improved capability of our new acoustic- and elastic-

waveform inversion method for building subsurface

velocity models containing sharp interfaces and

steeply-dipping fracture/fault zones. We build the

acoustic- and elastic-velocity models using geologic

features found at the Soda Lake geothermal field in

Nevada. The models contain sharp interfaces of a

basalt body and several steeply-dipping fault zones

with widths much smaller than the seismic wave-

length. Our inversion results demonstrate that our

new AEWI method produces more accurate velocity

estimations in the basalt body in the models and in

steeply-dipping fault zones compared to those

obtained without using interface-guided

regularization.

In Sect. 2, we briefly describe the fundamentals of

AEWI and the regularization theory. In Sect. 3, we

introduce our new AEWI with the interface-guided

MTV regularization; We then validate our new

AEWI using numerical examples in Sect. 4.

2. Methodology

2.1. Acoustic- and Elastic-Waveform Inversion

The acoustic-wave equation in the time-domain is

given by

1

KðrÞo2

ot2�r � 1

qðrÞ r� ��

pðr; tÞ ¼ sðr; tÞ; ð1Þ

where qðrÞ is the density at spatial location r, KðrÞ isthe bulk modulus, sðr; tÞ is the source term, pðr; tÞ isthe pressure wavefield, and t represents time.

The elastic-wave equation is written as

qðrÞ €uðr; tÞ � r � ½CðrÞ : ruðr; tÞ� ¼ sðr; tÞ; ð2Þ

where CðrÞ is the elastic tensor, and uðr; tÞ is the

displacement wavefield.

The forward modeling problems in Eqs. (1) and

(2) can be written as

P ¼ f ðmÞ; ð3Þ

where P is the either pressure wavefield for the

acoustic case or the displacement wavefields for the

elastic case, f is the forward acoustic- or elastic-wave

modeling operator, and m is the model parameter

vector, including the density and compressional- (P-)

and shear-wave (S-wave) velocities. We use a time-

domain staggered-grid finite-difference scheme to

solve the acoustic- or elastic-wave equation (Tan and

Huang 2014a, b). Throughout this paper, we consider

only constant-density acoustic and elastic media.

The inverse problem of Eq. (3) is usually posed as

a minimization problem

EðmÞ ¼ minm

d � f ðmÞk k22n o

; ð4Þ

where d represents a recorded/field waveform dataset,

f ðmÞ is the corresponding forward modeling result,

d � f ðmÞk k22 is the data misfit function, and jj � jj2stands for the L2 norm. The purpose of the mini-

mization operation in Eq. (4) is to construct a model

m that yields the minimum difference between

measured and synthetic waveforms.

3. Acoustic- and Elastic-Waveform Inversion

with Interface-Guided Regularization

The interface-guided regularization can be incor-

porated into any regularization schemes. We first

provide a general form of the interface-guided regu-

larization, and then we combine the interface-guided

regularization with the modified total-variation (TV)

regularization for acoustic- and elastic-waveform

inversion.

Seismic full-waveform inversion with regulariza-

tion can be written as

EðmÞ ¼ minm

d� f ðmÞk k22þkRðmÞn o

; ð5Þ

where RðmÞ is the regularization term, whose form

depends on the type of regularization used. The

Tikhonov regularization and the TV regularization

are the most commonly used.

To incorporate the interface information, we

reformulate the regularization term RðmÞ as,

RðmÞ ¼ Rðwi;j mÞ; ð6Þ

where the weighting parameter w controls the amount

of regularization among adjacent spatial grid points.

We set the weighting value as the following:

Building Subsurface Velocity Models

wi;j¼0 if pointði;jÞis on interfaces1 if pointði;jÞis not on interfaces

�: ð7Þ

Assigning a zero weight to the points on the inter-

faces frees them from being penalized by the

regularization. Therefore, the weighting parameter w

relies on the detection of the interface locations.

3.1. Acoustic- and Elastic-Waveform Inversion

with Interface-Guided Modified Total-Variation

Regularization

The objective function with a modified TV

regularization is given by (Huang et al. 2008; Lin

and Huang 2015)

Eðm; uÞ ¼ minm;u

d � f ðmÞk k22þk1 m� uk k22þk2 uk kTVn o

;ð8Þ

where k1 and k2 are both positive regularization

parameters, u is an auxiliary vector with a dimension

equal to that of m, and the TV term uk kTV for a 2D

model is defined as the L1 given by

kukTV ¼X

1� i;j� n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijðrxuÞi;jj

2 þ jðrzuÞi;jj2

q; ð9Þ

where ðrxuÞi;j ¼ uiþ1;j � ui;j and ðrzuÞi;j ¼ ui;jþ1 �ui;j are the spatial derivatives at a spatial grid point

(i, j) on a Cartesian coordinate (x, z).

To incorporate the interface information, we

reformulate the TV term given by Eq. (9) as

kukITV ¼ kwruk

¼X

1� i;j� n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwi;j jðrxuÞi;jj

2 þ jðrzuÞi;j

� j2

r;ð10Þ

where w is given by Eq. (7). We then obtain the

interface-guided modified TV regularization

scheme given by

Eðm; uÞ ¼ minm;u

d � f ðmÞk k22þk1 m� uk k22þk2 uk kITVn o

:

ð11Þ

We rewrite the interface-guided modified TV regu-

larization in Eq. (11) as

Eðm; uÞ ¼ minu

minm

d � f ðmÞk k22þk1 m� uk k22n o

þ k2 uk kITVn o

:

ð12Þ

We employ an alternating-minimization algorithm to

solve the double-variable minimization problem in

Eq. (12). Beginning with a starting model uð0Þ,

solving Eq. (12) leads to the solutions of two mini-

mization problems:

mðkÞ ¼ argminm

E1ðmÞf g ¼ argminm

d � f ðmÞk k22þk1 m� uðk�1Þ 22

n o; ð13Þ

uðkÞ ¼ argminu

E2ðuÞf g

¼ argminu

mðkÞ � u 2

2þk2 uk kITV

n o; ð14Þ

for iteration step k ¼ 1; 2; . . .

3.1.1 Choice of the Regularization Parameters k1and k2

Analogous to Guitton (2012), we use the following

formula to select regularization parameters k1:

k1 ¼kd � f ðmÞk22

l m� uðk�1Þk k22; ð15Þ

where l is a dimensionless number, which is

approximately 10 according to Guitton (2012).

We use an unbiased predictive risk estimator

developed by Lin et al. (2010) to select regulariza-

tion parameter k2.

3.2. Interface Detection

During each iteration step of acoustic- and elastic-

waveform inversion, we compute forward propaga-

tion wavefields from sources and backward

propagation wavefields from receivers. Therefore,

we exploit these wavefields to obtain the interfaces of

subsurface geologic structures using reverse-time

migration. Interfaces and their locations in RTM

images are high-wave number features. To further

detect those interface information in RTM images,


we can employ high-pass filtering techniques, such as

the differential-based edge detector or the Canny

edge detector, etc. (Gonzalez and Woods 2008), to

RTM images. Consequently, we gain the interface

information during acoustic- and elastic-waveform

inversion with very little additional computational

costs. After the interfaces are determined, we employ

the interface weighting coefficients according to

Eq. (7).

4. Numerical Results

4.1. Acoustic-Waveform Inversion

We use synthetic surface seismic data for the

model in Fig. 1a to demonstrate the improvement of

our new acoustic-waveform inversion (AWI) or full-

waveform inversion (FWI) method with the interface-

guided modified TV regularization scheme for veloc-

ity model building. We build the model using

geologic features found at the Soda Lake geothermal

field in Nevada. The model contains basalt bodies and

several steeply-dipping fault zones. There are three

basalt regions in Fig. 1a with a high velocity value

and sharp interfaces. We generate two hundreds

common-shot gathers of synthetic seismic data with

1000 receivers at the surface of the model, and use

these data to invert for velocity values of the model.

The shot interval is 20 m and the receiver interval is

5 m. A Ricker wavelet with a center frequency of

25 Hz is used as the source function. We plot two

profiles (one vertical and one horizontal) to visualize

the differences between the inverted velocities and

the true values in the following.

We smooth the original velocity model in Fig. 1a

by averaging the slowness within two wavelengths,

resulting in the model in Fig. 1b. We use this

smoothed model as the starting model, uð0Þ, for full-

waveform inversion.

Figure 2 shows AWI results with TV regulariza-

tion (Fig. 2a), modified TV regularization (Fig. 2b),

and interface-guided modified TV regularization

(Fig. 2c). The AWI result obtained with the inter-

face-guided modified TV regularization in Fig. 2c

give an accurate velocity reconstruction, particularly

in the basalt regions of the model, which cannot be

0

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0.6

0.8

Dep

th (

km)

0 0.5 1.0 1.5 2.0 2.5 3.0

Horizontal Distance (km)

1800

2800

3800

Velocity (m

/s)

(a)

0

0.2

0.4

0.6

0.8

Dep

th (

km)

0 0.5 1.0 1.5 2.0 2.5 3.0


1800

2800

3800

Velocity (m

/s)

(b)

Figure 1An acoustic velocity model built using geologic features found at the Soda Lake geothermal field: a the true model; b a two-wavelength-

smoothed initial model for waveform inversion


accurately reconstructed without using the interface

information. In addition, our new method greatly

improves the inversion of the steeply-dipping fault

zones, as depicted in Fig. 2c.

To compare the velocity values estimated using

the three AWI methods, we plot two velocity profiles:

a vertical profile and a horizontal profile. The vertical

profile along x ¼ 1250 m and the horizontal profile

along z ¼ 560 m show the inversions of the large

velocity contrast/sharp interfaces between the sedi-

ment and the basalt regions, and between the

dediment and the fractured fault zones.

Figure 3 shows a comparison of the vertical

profiles of AWI results in Fig. 2. AWI with TV

regularization produces an oscillated profile. AWI

with modified TV regularization eliminates some of

the oscillation, but the velocity values in the deep

region of the model (basalt body) are still significant

different from their true values. In contrast, our new

AWI with interface-guided modified TV regulariza-

tion accurately produces both velocity values and the

sharp interfaces.

Figure 4 is a comparison of the horizontal profiles

of the three AWI methods along z ¼ 560 m in Fig. 2.

0

0.2

0.4

0.6

0.8

Dep

th (

km)

0 0.5 1.0 1.5 2.0 2.5 3.0


1800

2800

3800

Velocity (m

/s)

(a)

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0.4

0.6

0.8

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th (

km)

0 0.5 1.0 1.5 2.0 2.5 3.0


1800

2800

3800

Velocity (m

/s)

(b)

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0.8

Dep

th (

km)

0 0.5 1.0 1.5 2.0 2.5 3.0


1800

2800

3800

Velocity (m

/s)

(c)

Figure 2Results of acoustic-waveform inversion with: a TV regularization, b modified TV regularization, and c interface-guided modified TV

regularization for the acoustic model in Fig. 1a. The acoustic-waveform inversion with the interface-guided modified TV regularization yields

the most accurate inversion result among all three methods, particularly for the basalt bodies with sharp interfaces


Our new AWI with interface-guided modified TV

regularization significantly reduces the artifacts and

preserves the sharp interfaces much better than the

other two methods. In addition, AWI with interface-

guided modified TV regularization can clearly dis-

tinguish two fault zones at x ¼ 1600 m, which are

close to each other as depicted in Fig. 2.

We also compare the convergence rates among

the AWI methods with TV, modified TV, and

interface-guided modified TV regularization schemes

in Fig. 5, where Fig. 5a is the convergence of the data

misfit and Fig. 5b is the convergence of the model

misfit. Fig. 5a shows that the data misfits for all three

methods do not change significantly after 25 iteration

0

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0.4

0.6

0.8

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th (

km)

2000 2500 3000 3500 4000 4500Velocity (m/s)

(a)

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th (

km)

2000 2500 3000 3500 4000 4500Velocity (m/s)

(b)

0

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0.4

0.6

0.8

Dep

th (

km)

2000 2500 3000 3500 4000 4500Velocity (m/s)

(c)

Figure 3Comparison of vertical velocity profiles of inversion results at x ¼ 1250 m obtained using acoustic-waveform inversion with: a TV

regularization, b modified TV regularization, and c interface-guided modified TV regularization, with those of the true and initial models in

Fig. 1. In each profile, the blue, red, green lines are the vertical profiles of waveform inversion, the true velocity model, and the initial velocity

model, respectively. Our new acoustic-waveform inversion with the interface-guided modified TV regularization eliminates the oscillation in

the velocity profile of acoustic-waveform inversion with TV regularization, and produces the most accurate velocity values of all layers

among all three methods


steps. The data misfit of the AWI method with

interface-guided modified TV regularization reduces

much more rapidly in the first 20 iteration steps than

those of the AWI methods with the TV and MTV

regularization schemes. Even though the data misfits

of the AWI methods with the TV, modified TV and

interface-guided modified TV regularization schemes

after 30 iteration steps are comparable as shown in

Fig. 5a, their model misfits converge very differently

from one another, as depicted in Fig. 5b. Our new

AWI with the interface-guided modified TV regular-

ization scheme converges much faster than the AWI

0 0.5 1.0 1.5 2.0 2.5 3.0Horizontal Distance (km)

200025003000350040004500

Vel

ocity

(m

/s)

(a)


200025003000350040004500

Vel

ocity

(m

/s)

(b)


200025003000350040004500

Vel

ocity

(m

/s)

(c)

Figure 4Comparison of horizontal velocity profiles of inversion results at z ¼ 560 m obtained using acoustic-waveform inversion with: a TV

regularization, b modified TV regularization, and c with interface-guided modified TV regularization, with those of the true and initial models

in Fig. 1. In each profile, the blue, red, green lines are the vertical profiles of waveform inversion, the true velocity model, and the initial

velocity model, respectively. Our new acoustic-waveform inversion with the interface-guided modified TV regularization eliminates the

oscillation in the velocity profile of acoustic-waveform inversion with TV regularization, and produces the most accurate velocity values of all

layers among all three methods


5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Iterations

Dat

a M

isfi

t

AEWI with Edged−MTVAEWI with MTVAEWI with TV

(a)

5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

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0.7

0.8

0.9

1

Iterations

Mo

del

Mis

fit

AEWI with EdgedMTVAEWI with MTVAEWI with TV

(b)

Figure 5Convergence curves of the data misfits (a) and the model misfits (b) of acoustic-waveform inversion with: a TV regularization (in blue),

b modified TV regularization (in green), and c interface-guided modified TV regularization (in red)

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th (

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/s)

(a)

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th (

km)

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1900

2900

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Velocity (m

/s)

(b)

Figure 6Elastic velocity models for the Soda Lake geothermal field in Nevada for validating our new elastic-waveform inversion with the interface-

guided modified TV regularization. a P-wave velocity model; b S-wave velocity model


methods with the TV and modified TV regularization

schemes.

4.2. Elastic-Waveform Inversion

In the next example, we validate the improved

inversion capability of our new elastic-waveform

inversion (EWI) method with the interface-guided

modified TV regularization scheme. We use synthetic

multi-component seismic reflection data for the

elastic model shown in Fig. 6 to validate our new

inversion method. Analogue to the first numerical

example, we build the compressional- and shear-

velocity models using geologic features found at the

Soda Lake geothermal field in Nevada. We use the

same source-receiver configuration as the first numer-

ical example to generate multi-component synthetic

surface seismic reflection data. The Ricker wavelet

with a center frequency of 25 Hz is used as the source

function. Again we compare the inversion results of

three different methods: elastic-waveform inversion

with TV regularization, modified TV regularization,

and interface-guided modified TV regularization.

We smooth the original elastic-velocity models in

Fig. 6a, b by averaging the slowness within two

wavelengths, resulting in models displayed in Fig. 7a,

b. We use these two smoothed models as the starting

models for elastic-waveform inversion.

Figures 8 and 9 show EWI with TV regularization

(Figs. 8a, 9a), modified TV regularization (Figs. 8b,

9b), and the interface-guided modified TV regular-

ization (Figs. 8c, 9c). The EWI results obtained with

the interface-guided modified TV regularization

scheme in Figs. 8c and 9c show accurate velocity

reconstructions, particularly in the basalt regions

(with sharp interfaces) of the models, which are

poorly reconstructed using EWI with TV or modified

TV regularization in Figs. 8a, b 9a, b. In addition, our

new method greatly improves inversion accuracy of

the steeply-dipping fault zones.

To compare of the velocity values estimated using

these three EWI methods, we display two profiles: a

horizontal profile and a vertical profile. The

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th (

km)

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1900

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3900V

elocity (m/s)

(a)

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th (

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900

1900

2900

3900

Velocity (m

/s)

(b)

Figure 7The two-wavelength-smoothed initial elastic models used for elastic-waveform inversion. a The initial P-wave velocity model; b The initial

S-wave velocity model


horizontal profile cuts through two basalt regions and

all five fault zones. The vertical profile is aligned

along the center of the largest basalt region.

Figures 10 and 11 show a comparison of the

vertical profiles of EWI reconstructions in Figs. 8 and

9. EWI with TV regularization produces oscillated

profiles in both compressional- and shear-velocity

results. EWI with modified TV regularization yields

inaccurate velocity values in the deep regions

(basalt). By contrast, our new EWI with interface-

guided modified TV regularization accurately inverts

both compressional- and shear-velocity values as well

as the sharp interfaces.

Figures 12 and 13 display the horizontal profiles

of the three EWI results along the depth of 0.56 km.

Out of all three methods, only our new EWI with

interface-guided modified TV regularization yields

the locations of all five fault zones. Furthermore,

EWI with interface-guided modified TV regulariza-

tion significantly improves the reconstruction of the

velocity values within the fault zones compared to

EWI with TV and modified TV regularization. The

0

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th (

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elocity (m/s)

(a)

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/s)

(b)

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km)

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Velocity (m

/s)

(c)

Figure 8P-wave velocity inversion results of elastic-waveform inversion with: a TV, b modified TV, and c interface-guided modified TV

regularization schemes. The elastic-waveform inversion with the interface-guided modified TV regularization yields the most accurate

inversion result among all three methods


true values of the fault zones are 2125 m/s for

compressional velocity and 1062 m/s for shear

velocity. The estimated velocity values obtained

using the EWI with interface-guided modified TV

regularization are approximately 2180 m/s for the

compressional velocity and 1068 m/s for the shear

velocity. However the approximated compressional

and shear velocity values obtained using EWI with

TV or modified TV regularization are very different

from the true values.

To better understand the robustness of our new

elastic-waveform inversion, we test our method for

three different scenarios: initial models with different

smoothness, data with noise, and indication of fault

zones.

To study the robustness of our new elastic-

waveform inversion method on the initial velocity

models, we smooth the true models by averaging the

P- and S-slownesses of the true models within a

region of three and four central wavelengths at each

grid point, and display the resulting smoothed

velocity models in Fig. 14. As the smoothing number

of the wavelength increases, the initial models

become smoother and further deviate away from the

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th (

km)

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elocity (m/s)

(a)

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th (

km)

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Velocity (m

/s)

(b)

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th (

km)

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900

1900

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3900

Velocity (m

/s)

(c)

Figure 9S-wave velocity inversion results of elastic-waveform inversion with: a TV, b modified TV, and c interface-guided modified TV

regularization schemes. The elastic-waveform inversion with the interface-guided modified TV regularization yields the most accurate

inversion result among all three methods


true models. We employ our new elastic-waveform

inversion method with interface-guided modified TV

regularization to invert for the velocity models

starting from these two different initial models, and

depict the corresponding results in Fig. 15. For both

these two fairly-smoothed initial models, our new

EWI method still produces accurate inversion results.

To study the robustness of our new elastic-

waveform inversion method with interface-guided

modified TV regularization for noisy data, we add

20 dB of white noise to our synthetic data and obtain

the inversion results as displayed in Fig. 16. Our new

method still produces high-quality inversion results

of both P- and S-wave velocities, compared to the

0

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0.6

0.8

Dep

th (

km)

2000 2500 3000 3500 4000 4500Velocity (m/s)

(a)

0

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0.8

Dep

th (

km)

2000 2500 3000 3500 4000 4500Velocity (m/s)

(b)

0

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0.8

Dep

th (

km)

2000 2500 3000 3500 4000 4500Velocity (m/s)

(c)

Figure 10Comparison of vertical profiles of P-wave velocity inversion results at x ¼ 1250 m obtained using elastic-waveform inversion with a TV,

b modified TV, and c interface-guided modified TV regularization schemes, with those of the true and initial models in Figs. 6 and 7. In each

profile, the blue, red, green lines are the vertical profiles of elastic-waveform inversion, the true velocity model, and the initial velocity model,

respectively. Our new elastic-waveform inversion with the interface-guided modified TV regularization eliminates the oscillation in the

velocity profile of elastic-waveform inversion with the TV regularization, and produces the most accurate velocity values of all layers among

all three methods


inversion results using noise-free data shown in

Figs. 8 and 9.

To study the robustness of our elastic-waveform

inversion method for the indication of fault zones, we

create a set of initial models by smoothing the true

models in Fig. 6 with all fault zones removed. The

resulting initial models are shown in Fig. 17. The

inversion results obtained using our new elastic-

waveform inversion method are shown in Fig. 18.

Again, our method yields high-quality inversion

results with all fault-zone regions accurately

reconstructed.

0

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0.8

Dep

th (

km)

1000 1200 1400 1600 1800 2000 2200

Velocity (m/s)

(a)

0

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0.8

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th (

km)

1000 1200 1400 1600 1800 2000 2200

Velocity (m/s)

(b)

0

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0.4

0.6

0.8

Dep

th (

km)

1000 1200 1400 1600 1800 2000 2200

Velocity (m/s)

(c)

Figure 11Comparison of vertical profiles of S-wave velocity inversion results at x ¼ 1250 m obtained using elastic-waveform inversion with a TV,


profile, the blue, red, green lines are the vertical profiles of elastic-waveform inversion, the true velocity model, and the initial velocity model,

respectively. Our new elastic-waveform inversion with the interface-guided modified TV regularization eliminates the oscillation in the

velocity profile of elastic-waveform inversion with the TV regularization, and produces the most accurate velocity values of all layers among

all three methods


5. Conclusions

We have developed novel acoustic- and elastic-

waveform inversion methods with interface-guided

modified total-variation regularization. The method

employs the interface information in combination

with a modified total-variation regularization scheme.

We employ an alternating-minimization method to

solve the optimization problem. We have validated

the capability of our new acoustic- and elastic-

waveform inversion methods for accurate building of

velocity models with sharp interfaces and steeply-


200025003000350040004500

Vel

ocity

(m

/s)

(a)


200025003000350040004500

Vel

ocity

(m

/s)

(b)


200025003000350040004500

Vel

ocity

(m

/s)

(c)

Figure 12Comparison of horizontal profiles of P-wave velocity inversion results at z ¼ 560 m obtained using elastic-waveform inversion with a TV,


profile, the blue, red, green lines are the horizontal profiles of elastic-waveform inversion, the true velocity model, and the initial velocity

model, respectively. Our new elastic-waveform inversion with the interface-guided modified TV regularization eliminates the oscillation in

the velocity profile of elastic-waveform inversion with the TV regularization, and produces the most accurate velocity values of all layers



dipping fault zones. Our acoustic- and elastic-wave-

form inversion results of synthetic seismic data for a

Soda Lake geothermal velocity model demonstrate

that our new method can accurately produce not only

velocity values and but also sharp interfaces.

Therefore, our novel acoustic- and elastic-waveform

inversion methods with interface-guided modified

total-variation regularization provide a powerful tool

for accurate velocity model building, particularly for

models with sharp interfaces.


1000

1500

2000

Vel

ocity

(m

/s)

(a)


1000

1500

2000

Vel

ocity

(m

/s)

(b)


1000

1500

2000

Vel

ocity

(m

/s)

(c)

Figure 13Comparison of horizontal profiles of S-wave velocity inversion results at z ¼ 560 m obtained using elastic-waveform inversion with a TV,


profile, the blue, red, green lines are the horizontal profiles of elastic-waveform inversion, the true velocity model, and the initial velocity

model, respectively. Our new elastic-waveform inversion with the interface-guided modified TV regularization eliminates the oscillation in

the velocity profile of elastic-waveform inversion with the TV regularization, and produces the most accurate velocity values of all layers



0

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th (

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(b)

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/s)

(c)

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th (

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Velocity (m

/s)

(d)

Figure 14Two sets of initial elastic models smoothed by averaging the P- and S-slownesses of the true models (Fig. 6a, b) within a region of three (a,

b) and four (c, d) central wavelengths at each grid point


0

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th (

km)


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elocity (m/s)

(a)

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th (

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/s)

(b)

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th (

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/s)

(c)

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th (

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/s)

(d)

Figure 15Inversion results obtained using a, b Fig. 14a, b, and c, d Fig. 14c, d as the initial models for elastic-waveform inversion with interface-guided

modified TV regularization


0

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th (

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elocity (m/s)

(a)

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th (

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Velocity (m

/s)

(b)

Figure 16Inversion results of a P- and b S-wave velocities obtained using the elastic-waveform inversion with the interface-guided modified TV

regularization scheme and synthetic seismic reflection data containing 20 dB of white noise

0

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th (

km)

0 0.5 1.0 1.5 2.0 2.5 3.0


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(a)

0

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th (

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/s)

(b)

Figure 17The initial elastic models smoothed by removing all fault zones and averaging the P- and S-slownesses of the true models within two

wavelengths at each grid point


Acknowledgements

This work was supported by the Geothermal Tech-

nologies Office of the U.S. Department of Energy

through contract DE-AC52-06NA25396 to Los

Alamos National Laboratory. The computation was

performed on super-computers provided by the

Institutional Computing Program of Los Alamos

National Laboratory.

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(Received February 24, 2017, revised May 14, 2017, accepted July 19, 2017)


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