Subsurface extended full waveform inversion

Acknowledgments First I would like to express my gratitude to all TRIP members, who have offered invaluable assistance, support, and guidance. Deepest gratitude goes also to all TRIP sponsors, without whose financial assistance this study would not have been a success. I also would like to thank Earth Science department for providing this great opportunity to present our work.

Summary and Prospects •  Extended full waveform inversion requires the extended

modeling operator can/should fit data. •  Conventional seismic data without low frequency components. •  Nested optimization maintains data fit and avoids cycle skip. •  Study the crossing curves when velocity is correct. •  source inversion in EFWI •  Better MPI. (domain decomposition, GPU…) •  3-D •  Field data (Galicia project)

Theory Migration Velocity Analysis (MVA) Reverse time migration imaging: S = source wave field; R = receiver wave field Horizontal space shift RTM image volume: MVA semblance principle: should focus at h=0 for correct c (Claerbout 1971). Inverse problem The relationship between parameters of model c and seismic data d recorded as pressure at receivers is nonlinear and can be summarized as:

d=F[c] in which F denotes the forward modeling operator.

Introduction In exploration seismic experiments, seismic waves are generated by air guns in marine and seismic vibrators on land and propagate into the subsurface. The mechanical response of the earth is measured and recorded by the receivers (geophones or hydrophones) on the surface. The recorded seismograms contain information about the physical properties of the subsurface.

Migration velocity is correct

•  The migrated model perturbation locates at the correct depth. •  The image is focused at h=0. •  Two crossing curves •  Dense sources to remove the noises at edge (h is large)

Lei Fu1,2, William W. Symes1

Objective •  Reconstruct remarkably detailed models of the subsurface

structure. •  Solve local minima problem.

References Sun, Dong. A Nonlinear Differential Semblance Algorithm for Waveform Inversion. Diss. Doctoral Thesis, Rice University. http://hdl. handle. Net/1911/71694, 2013. Symes, William W. "Migration velocity analysis and waveform inversion." Geophysical Prospecting 56.6 (2008): 765-790.

1 The Rice Inverse Project (TRIP); 2 Department of Earth Science, Rice University, Houston, TX 77005-1892, U.S.A. (713) 586-9211 lei.fu.rice@gmail.com

(www.oilspillsolutions.org)

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Cartoon of object function J(c,r) Extended models (c,r)

Based on linearized Born approximation, denotes r(x,z) = model perturbation, it becomes:

d=DF[c]r

The classic seismic waveform inversion (output least square inversion) purports to obtain a velocity model [c,r], which generates synthetic data d that best fits the observed seismic data d0 :

minJOLS = ||DF[c]r-d0||2

Cartoon of object function J(c,r) Physical models (Ar = 0)

0

h

-h (c,r)

Global min

Local min JOLS

JOLS

Extended full waveform inversion: r depends on nonphysical parameters. Example: space shift extended inversion, r = r(x,z,h) - similar to space-shift RTM image volume, physical if concentrated at h=0

minJOLS = ||DF[c]r-d0||2 + α2||Ar||2 (A: annihilator of physical models)

Domain size x: 4 km y: 2 km Source type Ricker wavelet ( fpeak= 15 Hz ) Number of sources 31 ( x = 0.5:0.2:3.5 ) Number of receivers 200 ( x = 0:0.02:4 ) vp(x,z,h=0) of background 3 km/s vp(x,z,h≠0) of background 0 km/s r(x,z,h≠0) 0 Depth of reflection layer 1 km

Examples •  Constant density 2-D acoustic in Born approximation; •  4th order in space 2nd order in time finite-difference method; •  Time-domain scheme

Migrated image I(x,z,h) at c=3.2 km/s

Migrated image I(x,z,h) at c=2.8 km/s

Migrated image I(x,z,h) at c=3.0 km/s (correct)

Key feature of C code: Established: •  Time step function. Easily Applied in IWAVE package. •  Automatically generated its derivative and adjoint. TAPENADE On-line Automatic Differentiation Engine •  Perfectly Matched Layer (PML) •  Parallel computing (MPI) •  Reproducible in Madagascar. •  Source and receiver tapering.

Developing: •  Nested approach Inner optimization over r – optimal r for each c

r[c] = N-1F[c]*d N = F[c]*F[c] + α2A*A

Outer optimization over c – reduced objective: J[c]=||F[c]r[c]–d||2 + α2||Ar[c]||2