born waveform inversion via variable projection and shot
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Yin HuangEducation
Ph.D. Candidate, Rice University, Houston, TX, USA, 08/2010 - Present
Dissertation Topic: Extended waveform inversion in shot coordinate
model extension
M.A. with Master Thesis: Transparency property of one dimensional
acoustic wave equations 12/2012
M.S, Shanghai Jiao Tong University, Shanghai, China, 09/2006 - 03/2009
Dissertation topic: Comparison of numerical methods for saddle point
system arising from the mixed finite element method of elliptic
problems with nonsmooth coefficients
Research InterestsExtended Full Waveform Inversion and Born Waveform Inversion
Migration/Inversion Velocity Analysis, Seismic Imaging
High Performance Computing
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Born Waveform Inversionvia Variable Projection and Shot
Record Model Extension
Yin Huang
TRIP annual review meeting
April 30, 2015
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Introduction
Full waveform inversion (Tarantola, 1984, Virieux & Operto, 2009)
J[m] =12‖F [m]−d‖2
d observed data;
F [m] wave propagation operator;
has the ability to invert for fine structure of the earth subsurfacemodel by solving a nonlinear model-based least squares datafitting problem;
initial model needs to be close to true model to avoid localminima problem (Gauthier et al., 1986).
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Born modeling and waveform inversion
Scale separation of model ≈m + δm (long scale background modelplus short scale reflectivity).
Born modeling: DF [m]δm
Born waveform inversion: given data d , find m and δm that minimizes
JBWI[m,δm] =12‖DF [m]δm−d‖2.
easy to fit the data;
suffer from the same local minima problem as FWI.
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Variable projection method
Obtain VP objective by minimizing over reflectivity for fixedbackground model (van Leuwen & Mulder, 2009; Xu et al., 2012)
JVP[m] = minδm
JBWI[m,δm] =12‖DF [m]δm−d‖2.
less likely to be trapped by a local minimizer;
may also exhibit cycle skipping in some cases.
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VP method assisted by model extension
Introduce model extension to VP objective, to permit better data fit(Kern & Symes 1994)
JEVP[m] = minδm̄
JEBWI[m,δm̄] =12‖DF̄ [m]δm̄−d‖2 +
α2
2‖Aδm̄‖2.
δm̄ extended reflectivity;
A annihilator, A =∂
∂xsfor shot record model extension;
‖Aδm̄‖2 differential semblance penalty, the only choice thatleads to smooth objective function for shot record (Stolk &Symes, 2003);
α →+∞, JEVP→ JVP.
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Value of EVP objective and approximate gradient
Evaluation
JEVP[m] = minδm̄
12‖DF̄ [m]δm̄−d‖2 +
α2
2‖Aδm̄‖2.
involves solving a least squares migration (LSM)
(DF̄ [m]T DF̄ [m] + α2AT A)δm̄ = DF̄ [m]T d .
Approximate gradient:
∇JEVP = Λ−1D2F̄ T [δm̄,DF̄ [m]δm̄−d ].
Λ power of Laplacian operator, Λ−1 acts as smoothing operator;
D2F̄ T WEMVA or tomographic operator (Biondi & Sava 2004);
Gradient of JVP[m] is the same, without model extension.
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Main claim
Both extended modeling (EXM) and variable projection (VP) arenecessary to enable convergence to a global best fitting model.
VP without VPEXM X 7
without EXM 7 7
NOTE: VP objective function without model extension works well tosome extent, but suffer from cycle skipping when initial model is toofar away from true model.
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Extended 2D Constant Density Acoustics
Extended Born modeling: DF̄ [m]δm̄ = δu(xr ,xs, t), with m = c2 andδm̄ = δ c̄2(
∂ 2
∂ t2 −c2(x)∆x
)u(x,xs, t) = δ (x−xs)ω(t),
u(x,xs, t) = 0, t � 0.(∂ 2
∂ t2 −c2(x)∆x
)δu(x,xs, t) = δ c̄2(x,xs)∆u(x,xs, t),
δu(x,xs, t) = 0, t � 0.
c velocity of wave propagation,
u acoustic pressure wave field, F [c2] = u(xr ,xs, t),
δu perturbed wave field due to the extended modelperturbation δ c̄2.
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Extended 2D Constant Density Acoustics
Numerical discretization:
finite difference method: 2-nd order in time, 4-th order in space,reflection boundary condition;
implement the time step function of F̄ [c2];
automatic differentiation tool TAPENADE (Hascöet andPascual, 2004) to generate the time step function of DF̄ [c2],D2F̄ [c2] and their adjoints; DF̄ [c2]T , D2F̄ [c2]T ;
IWAVE framework: provides i/o, job control, and parallelization;
RVL optimization softwarehttps://svn.code.sf.net/p/rsf/code/trunk/trip/
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Example 1: truncated marmousi model
acquisition geometry:110 shots starting from 2km with spacing 64m;481 symmetric receivers for each shot with spacing 16m;
ricker1 wavelet with fpeak=6Hz;
acquire data until 2.6s;
50 steps of conjugate gradient method is used for the LSM;
steepest descent method with line search for backgroundmodel updates.
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Initial modelInitial background model and reflectivities
Common image gathers
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7 steps of EVPBackground model after 7 steps of EVP and reflectivities
Common image gathers
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15 steps of EVPBackground model after 15 steps of EVP and reflectivities
Common image gathers
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At true background modelTrue background model and reflectivities
Common image gathers
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350 steps of EBWIBackground model after 350 steps of EBWI and reflectivities
Common image gathers
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Summary of this example
Figure: Reflectivity model of EVP method
Conclusion from this example: use variable projection method whenupdating more than one parameters.
NOTE: 350 steps of EBWI is roughly equivalent to 7 steps of EVP in
terms of computational cost.
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Example 2: marmousi model
acquisition geometry:110 shots starting from 2km with spacing 64m;481 symmetric receivers for each shot with spacing 16m;
ricker1 wavelet with fpeak=6Hz;
acquire data until 4s;
start with small number of conjugate gradient and increase withbackground model update;
steepest descent method with line search for backgroundmodel updates.
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Initial modelInitial background model and reflectivities
Common image gathers at the initial model
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10 steps of EVPBackground model after 10 steps of EVP and reflectivities
Common image gathers
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18 steps of EVPBackground model after 10 steps of EVP and reflectivities
Common image gathers
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10 steps of VP
Background model after 10 steps of VP and reflectivities
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18 steps of VP
Background model after 18 steps of VP and reflectivities
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True modelTrue model and reflectivities at the true model
Common image gathers at the true model
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Reflectivity of EVP method
Figure: Reflectivity of EVP method
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Reflectivity of VP method
Figure: Reflectivity model of VP method
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Summary of this example
0 0.5 1 1.5 2 2.5 3 3.5 4−0.05
0
0.05trace 120
time(s)
Pre
ssure
real data
VP
EVP
0 0.5 1 1.5 2 2.5 3 3.5 4−0.1
−0.05
0
0.05
0.1trace 240
time(s)
Pre
ssure
real data
VP
EVP
Model extension is necessary to a stable inversion.NOTE: 1 step of VP is roughly equivalent to 1 step of EVP in terms ofcomputational cost.
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Discussion
Compared Born waveform inversion with/without variableprojection and with/without model extension;
Both model extension and variable projection are necessary fora stable Born waveform inversion;
Hundreds of modeling/migration were involved in the inversion⇒ future work.
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Future works
(in progress) apply preconditioning to accelerate theconvergence of the minimization over reflectivity (Tang, 2009;Stolk et al., 2009; Nammour & Symes, 2009)
compare with a similar method that uses full waveform operatoras a prediction operator;
(in progress) inversion velocity analysis for shot record modelextension
minm
‖Aδm̄‖2
with δm̄ = argmin‖DF̄ [m]δm̄−d‖2
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Acknowledgements
Wonderful audience;
Current and former TRIP team members;
Sponsors of The Rice Inversion Project;
Special thanks to Anatoly Baumstein and Yaxun Tang for helpsand inspiring discussions during my internship at Exxonmobil.
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