Building Subsurface Velocity Models with Sharp Interfaces Using...
Transcript of Building Subsurface Velocity Models with Sharp Interfaces Using...
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
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,
Y. Lin and L. Huang Pure Appl. Geophys.
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
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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
Building Subsurface Velocity Models
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.
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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
Y. Lin and L. Huang Pure Appl. Geophys.
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
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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
Building Subsurface Velocity Models
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
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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
Y. Lin and L. Huang Pure Appl. Geophys.
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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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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
Building Subsurface Velocity Models
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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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
Y. Lin and L. Huang Pure Appl. Geophys.
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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0.8
Dep
th (
km)
0 0.5 1.0 1.5 2.0 2.5 3.0
Horizontal Distance (km)
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1900
2900
3900V
elocity (m/s)
(a)
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th (
km)
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3900
Velocity (m
/s)
(b)
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th (
km)
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Horizontal Distance (km)
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1900
2900
3900
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
Building Subsurface Velocity Models
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
0
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0.8
Dep
th (
km)
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Horizontal Distance (km)
900
1900
2900
3900V
elocity (m/s)
(a)
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)
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1900
2900
3900
Velocity (m
/s)
(b)
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)
900
1900
2900
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
Y. Lin and L. Huang Pure Appl. Geophys.
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
0.2
0.4
0.6
0.8
Dep
th (
km)
2000 2500 3000 3500 4000 4500Velocity (m/s)
(a)
0
0.2
0.4
0.6
0.8
Dep
th (
km)
2000 2500 3000 3500 4000 4500Velocity (m/s)
(b)
0
0.2
0.4
0.6
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
Building Subsurface Velocity Models
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
0.2
0.4
0.6
0.8
Dep
th (
km)
1000 1200 1400 1600 1800 2000 2200
Velocity (m/s)
(a)
0
0.2
0.4
0.6
0.8
Dep
th (
km)
1000 1200 1400 1600 1800 2000 2200
Velocity (m/s)
(b)
0
0.2
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,
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
Y. Lin and L. Huang Pure Appl. Geophys.
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-
0 0.5 1.0 1.5 2.0 2.5 3.0Horizontal Distance (km)
200025003000350040004500
Vel
ocity
(m
/s)
(a)
0 0.5 1.0 1.5 2.0 2.5 3.0Horizontal Distance (km)
200025003000350040004500
Vel
ocity
(m
/s)
(b)
0 0.5 1.0 1.5 2.0 2.5 3.0Horizontal Distance (km)
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,
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 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
among all three methods
Building Subsurface Velocity Models
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.
0 0.5 1.0 1.5 2.0 2.5 3.0Horizontal Distance (km)
1000
1500
2000
Vel
ocity
(m
/s)
(a)
0 0.5 1.0 1.5 2.0 2.5 3.0Horizontal Distance (km)
1000
1500
2000
Vel
ocity
(m
/s)
(b)
0 0.5 1.0 1.5 2.0 2.5 3.0Horizontal Distance (km)
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,
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 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
among all three methods
Y. Lin and L. Huang Pure Appl. Geophys.
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
Horizontal Distance (km)
900
1900
2900
3900V
elocity (m/s)
(a)
0
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th (
km)
0 0.5 1.0 1.5 2.0 2.5 3.0
Horizontal Distance (km)
900
1900
2900
3900
Velocity (m
/s)
(b)
0
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0.8
Dep
th (
km)
0 0.5 1.0 1.5 2.0 2.5 3.0
Horizontal Distance (km)
900
1900
2900
3900
Velocity (m
/s)
(c)
0
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0.8
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th (
km)
0 0.5 1.0 1.5 2.0 2.5 3.0
Horizontal Distance (km)
900
1900
2900
3900
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
Building Subsurface Velocity Models
0
0.2
0.4
0.6
0.8
Dep
th (
km)
0 0.5 1.0 1.5 2.0 2.5 3.0Horizontal Distance (km)
900
1900
2900
3900V
elocity (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 3.0Horizontal Distance (km)
900
1900
2900
3900
Velocity (m
/s)
(b)
0
0.2
0.4
0.6
0.8
Dep
th (
km)
0 0.5 1.0 1.5 2.0 2.5 3.0Horizontal Distance (km)
900
1900
2900
3900
Velocity (m
/s)
(c)
0
0.2
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0.6
0.8
Dep
th (
km)
0 0.5 1.0 1.5 2.0 2.5 3.0Horizontal Distance (km)
900
1900
2900
3900
Velocity (m
/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
Y. Lin and L. Huang Pure Appl. Geophys.
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
Horizontal Distance (km)
900
1900
2900
3900V
elocity (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
Horizontal Distance (km)
900
1900
2900
3900
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
0.2
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0.8
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th (
km)
0 0.5 1.0 1.5 2.0 2.5 3.0
Horizontal Distance (km)
900
1900
2900
3900
Velocity (m
/s)
(a)
0
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0.8
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th (
km)
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Horizontal Distance (km)
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1900
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3900
Velocity (m
/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
Building Subsurface Velocity Models
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)
Building Subsurface Velocity Models
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