Fast multi-parameter anisotropic full waveform

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Fast multi-parameter anisotropic full waveform inversion with irregular shot sampling
Chao Wang∗ , David Yingst, John Brittan, Paul Farmer, and Jacques Leveille, ION Geophysical
SUMMARY
leads to slow convergence.
The goal of full waveform inversion (FWI) is to derive highfidelity earth models for seismic imaging by fitting the acquired data. One of the major drawback of FWI is that it is
highly compute-intensive. In this paper, we propose a fast
multi-parameter FWI to dramatically reduce the computation
cost. Considering the often significant numbers of sources
(and receivers) in 3D seismic data acquisition, we propose a
method to largely reduce the simulation time per iteration by
using a reasonably small subset of sources instead of the full
volume. We choose the subset by randomly picking a number of sequential sources, designed to follow an irregular sampling pattern. This subset will be re-selected and different at
each iteration and needs to be sufficiently sampled to avoid artifacts. The results achieved from this subsampling approach
are comparable to the conventional method, with a highly reduced computation cost at each iteration.
In 2013, van Leeuwen and Hermann (2013) proposed to use
randomly chosen sequential sources instead of encoded sources
to eliminate the source cross-talk for frequency domain FWI.
In their paper, they showed that we do not need to use simultaneous sources to reap the benefits of stochastic optimization.
In order to eliminate the source cross-talk and make it easily
fit into a general acquisition framework, we select a subset of
sequential sources to reduce the simulation cost at each iteration. Our picked sources follow an irregular sampling pattern. This subset of sources will be re-selected and different at
each iteration and needs to be appropriately sampled to avoid
artifacts. We apply our source selection scheme to our time
domain multi-parameter FWI with our optimization and regularization techniques.
Our forward modeling and its adjoint computation are based
on the acoustic wave equation in vertical transversely isotropic
(VTI) media and our target model includes three parameters
that are P-wave velocity and Thomsen’s anisotropy parameters
(epsilon and delta). During the iterative process of the multiparameter FWI, we update three parameters simultaneously at
each iteration. Multi-parameter VTI FWI does not require any
extra wavefield computation than mono-parameter VTI FWI.
Hense it is more efficient and speeds up the process if our target
model includes more than one parameter.
This paper presents the time domain implementation for a fast
multi-parameter VTI FWI. This approach will be illustrated
on 3D marine data from the Green Canyon area of the Gulf of
Mexico.
INTRODUCTION
For the conventional FWI, forward and back propagation are
performed for each source individually, which means the cost
of conventional FWI for simulating the wavefields is proportional to the number of sources. Taking into account of the
excessive growth in the number of sources (and receivers) in
large-scale 3D data acquisition, one of the main challenges of
conventional FWI is the high computation cost. Recently, various researchers have investigated different methods to speed
up FWI. The source-encoding technique has been proposed
and studied for the purpose of cost saving in FWI (Krebs et al.
(2009); van Leeuwen et al. (2011)). While replacing the sequential sources by a number of simultaneous sources, a sourceencoding technique can significantly reduce the computation
cost per iteration. However, simultaneous sources will introduce noisy cross-talk in the gradient and can thus damage the
model update. Therefore, the cross-talk must be suppressed
during the iterations. However averaging out the cross-talk
© 2014 SEG
SEG Denver 2014 Annual Meeting
As considering that including the effects of anisotropy often
helps to improve FWI results, our forward modeling and its adjoint computation are based on the time domain VTI acoustic
wave equations. The definition of a suitable parameterization
is a crucial issue for multi-parameter FWI. We choose to parameterize our VTI FWI by the P-wave velocity and anisotropy
parameters ε and δ . Our sensitivity analysis of acoustic VTI
FWI has shown that velocity can be updated successfully with
a rough guess of ε and δ . However ε and δ updates rely on
a relatively good starting velocity model. Based on the sensitivity analysis, we designed a two-step practical workflow. It
consists of one step of mono-parameter inversion for velocity
only, followed by another step of multi-parameter inversion for
velocity, ε and δ simultaneously. This joint inversion increases
the convergence rate for updating three parameter simultaneously at no extra wavefield cost.
This paper presents the objective function, its gradient, and
model update for fast multi-parameter VTI FWI. It also discusses the randomized techniques using irregular shot sampling and compares the results with uniform shot sampling.
This approach will be illustrated on 3D marine data from the
Green Canyon area of the Gulf of Mexico.
METHOD: OBJECTIVE FUNCTION
The full misfit is a slightly modified misfit function from Tarantola (1987). Our objective function is an approximation to
the full misfit that depends on the chosen subset of sequential
sources with additional well constraints.
min
m
s.t.
J[m] =
1 X
Φi [m]
|S|
Pm = m0 ,
(1)
xi ∈S
2
pred
where Φi [m] = 12 T diobs − ζi di [m] is the misfit for
2
source xi that belongs to the chosen subset S. Sources that
are not in the subset S will not be included in the objective
DOI http://dx.doi.org/10.1190/segam2014-0234.1
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Fast multi-parameter anisotropic FWI
(a)
(b)
(c)
(d)
Figure 1: (a) initial velocity, (b) inverted velocity with 25% uniform picking, (c) inverted velocity with 25% irregular picking, and
(d) inverted velocity with all the sources
pred
function. diobs is the observed seismic data and di [m] is the
predicted data for the model m at source locations xi . The predicted data are obtained by sampling the extrapolated wavefield p generated by a high-order finite difference scheme to
the receiver locations, based on the acoustic VTI wave equations (3). T is a data preconditioner and ζi is a normalization
scalar. m0 is the extended model generated from well logs
based on the neighboring structure. P is the projection operator that maps the model m to m0 ’s grid.
This approximation relies on the selection of sequential sources
based on a desired irregular pattern. In order to avoid insufficient sampling, we need to constrain the picking criteria with
a maximum lag between sources. Then we choose the sequential source randomly within the area that satisfy the constraints.
For the next iteration, we re-select a different subset of sources.
We can solve the constrained problem (1) by minimizing the
following unconstrained objective function with respect to m
using an Augmented Lagrangian Method (Hestenes, 1969; Powell, 1969; Li et al., 2013):
L [m] =
1 X
µ
Φi [m] − hλ , Pm − m0 i + kPm − m0 k22 ,
|S|
2
xi ∈S
(2)
where λ is a Lagrange multiplier and µ is a penalty scalar.
The major advantage of the method is that unlike the penalty
method, it is not necessary to take µ → ∞ in order to solve the
original constrained problem and avoids numerical issue for
large µ.
© 2014 SEG
SEG Denver 2014 Annual Meeting
METHOD: GRADIENT COMPUTATION
In 2000, Alkhalifah (2000) derived pseudo-acoustic wave equations for anisotropic media that kinematically model the compressional wave propagation. A number of variations of pseudoacoustic wave equations have been developed since then (Zhou
and Bloor, 2006). Here we use a VTI system of two coupled
second-order partial differential equations in terms of P-wave
vertical velocity v, Thomsen parameters, ε and δ , assuming
a constant density and zero shear velocity, with initial and
boundary conditions:
q
p
1 + 2ε
1 + 2δ
1 2
∂
v2 t
=
1
1
∂x2 + ∂y2
0
0
∂z2
q
p
+
0
f
,
(3)
where f is the input source wavelet, p is the forward-propagated
wavefield and q is the auxiliary wavefield. We then solve the
adjoint equations of the forward equations (3) and obtain the
back-propagated wavefields p+ and q+ by back-propagating
the residual.
In this case, the approximated misfit gradient ∇J with respect
to model m that includes three parameters (velocity v, Thomsen parameters ε and δ ), are given by

X
X
1

∇J(x) =
|S|
xi ∈S
t
2
(∂ 2 pp+ + ∂t2 qq+ )(x,t; xi )
v3 (x) t
2((∂x2 q + ∂y2 q)q+ )(x,t; xi )
2((∂x2 q + ∂y2 q)p+ )(x,t; xi )

.
(4)
DOI http://dx.doi.org/10.1190/segam2014-0234.1
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Fast multi-parameter anisotropic FWI
The total gradient is given by
∇L (x) = ∇J(x) − (P∗ λ )(x) + µ(P∗ (Pm − m0 ))(x).
Our forward and adjoint equations provide a very simple form
of gradient calculation. Convergence can be accelerated using
gradient preconditioning. The current preconditioning normalizes the gradient by the amplitude of the forward propagated
wave with a whitening factor.
METHOD: MODEL UPDATE
The basic scheme for updating the model using an iterative
optimization method is
mk+1 = mk + αk Bdk ,
(a)
where αk is the step length computed using line search and dk
is the search direction at the k-th iteration. It uses the gradient
at an initial point for an initial direction estimate and updates
that direction using nonlinear conjugate gradient method.
mk =
vk
εk
δk
!
,B =
bv
0
0
0
bε
0
0
0
bδ
!
.
(5)
mk is the joint model including three parameters at the k-th
iteration. B is a scaling factor that is chosen to correct the
weights for each direction component.
GULF OF MEXICO EXAMPLE
We present an application of fast multi-parameter FWI to 3D
marine data. This deep water survey is located in the Green
Canyon area of the Gulf of Mexico. The acquisition area was
160 km2 and used four-component ocean bottom seismic receivers in deep water over relatively shallow salt bodies with
19901 shots. Maximum offset used is 7000 m. The lowest
frequency observed in the data is about 3 Hz. The source signature was derived from the down-going wavefield on a zero
offset section.
We ran three multi-parameter FWI tests to illustrate the benefits of irregular shot sampling using exactly the same workflow
with same number of iterations. First we chose 25% of sequential sources on a uniform sampling pattern. Second, we picked
25% of sequential sources using our randomized irregular sampling pattern. For the third test, we used all the sources. From
the velocity update shown in Figure 1, we illustrate that the
irregular shot sampling using 25% of sources shown in Figure
1(c) reduced the simulation cost by a factor of four comparing
to the conventional method shown in Figure 1(d) and produced
a comparable result. However, the uniform shot sampling using 25% of sources shown in 1(b) introduced strong artifact
due to insufficient sampling. For this data example, 25% of
sources are the minimum subset required for successful inversion due to frequency and source spacing.
The inverted epsilon and delta models in Figure 2(b) and 3(b)
show reasonable shallow updates up to a depth of 2500 m including some detailed structures above the salt. Therefore 25%
© 2014 SEG
SEG Denver 2014 Annual Meeting
(b)
Figure 2: (a) initial epsilon, (b) inverted epsilon with 25% irregular picking
of sequential sources using irregular shot sampling generate
reliable anisotropy updates as well.
In the FWI workflow for this data, we first applied monoparameter FWI for 33 iterations and then we parameterized the
target model by three parameters for 7 iterations of simultaneous inversion. The initial models, Figure 1(a), 2(a), and 3(a),
were built from anisotropic VTI tomographic inversions. Various strategies such as multi-scale, layer stripping, and offset
weighting have been applied to minimize the risk of converging to local minima.
We validate our results by comparing the flatness of the migrated gathers. Gather 4(b) after randomized FWI shows overall improvement in flatness compared to gather 4(a) before
FWI. To further evaluate randomized FWI results, we generated stack images with the initial models and the inverted models obtained by FWI with irregular shot sampling. The stack
image after FWI in Figure 5(b) shows improvement compared
to the initial stack shown in Figure 5(a) with better focus and
event consistency.
DOI http://dx.doi.org/10.1190/segam2014-0234.1
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Fast multi-parameter anisotropic FWI
(a)
(a)
(b)
Figure 4: Gather using (a) initial models, (b) inverted models
with 25% irregular picking
(b)
Figure 3: (a) initial delta, (b) inverted delta with 25% irregular
picking
CONCLUSION
In this paper, we have presented a methodology and strategies for fast multi-parameter FWI using a randomized irregular shot sampling technique. Picking a small subset of sequential sources helps to reduce the computation cost significantly. These approaches were illustrated on a 3D marine data
set from the Green Canyon area of the Gulf of Mexico. From
the results, we showed that simultaneous inversion for multiple
parameters using a small subset produced good results comparable with using the entire dataset with only 25% simulation
cost of conventional FWI.
(a)
ACKNOWLEDGMENTS
We would like to thank ION-GXT for permission to publish
the results and SINBAD consortium for stimulating the innovation. We also thank our colleagues at ION GeoScience Team
for providing us valuable support in this work, especially Ian
Jones, Helen Delome, Guoquan Chen, Jianyong Bai, and Mohamed Dolliazal.
© 2014 SEG
SEG Denver 2014 Annual Meeting
(b)
Figure 5: Stack image using (a) initial models, (b) inverted
models with 25% irregular picking
DOI http://dx.doi.org/10.1190/segam2014-0234.1
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EDITED REFERENCES
Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2014
SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for
each paper will achieve a high degree of linking to cited sources that appear on the Web.
REFERENCES
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