inverse thin-slab propagator based on forward- scattering ...wrs/publication/seg/seg2015-2016/tu...

77th EAGE Conference & Exhibition 2015 IFEMA Madrid, Spain, 1-4 June 2015

1-4 June 2015 | IFEMA Madrid

Tu N104 01Inverse Thin-slab Propagator Based on Forward-scattering Renormalization for Wave-equationTomographyR.S. Wu* (University of California) & B. Wang (UCSC (Visiting from ChinaUniv. of Petroleum))

SUMMARYWe derived the Inverse thin-slab propagator in nonlinear tomographic waveform inversion in T-matrixformalism based on forward-scattering renormalization. The inverse thin-slab propagator eliminates thedivergence of the inverse Born series for strong perturbations by stepwise partial summation(renormalization). It is a non-iterative, marching algorithm with only one sweep, and therefore veryefficient in comparison with the iterative inversion based on the inverse-Born scattering series. Thisconvergence and efficiency improvement has potential applications to the iterative procedure of waveforminversion. Numerical results on the convergence tests and inversion of smooth large-scale structures aregiven to demonstrate the validity of the theory.



Introduction

The gradient method in full waveform inversion (FWI) is based on a linearization of the full nonlinear partial derivative (NLPD) operator (See Tarantola, 1984, 2005; Pratt et al., 1998; Pratt, 1999). The convergence problems of the iterative procedure based on the linear Fréchet derivative, such as cycle-skipping, local minima, and starting model dependence, are all deeply rooted in the well-known convergence problem of the Born series and inverse Born series (see e.g., Moses, 1956; Prosser, 1969; Weglein et al., 2003; Wu and Zheng, 2014). To overcome these difficulties without relying on the low-frequency seismic source, a recent trend is to combine other seismic data functional (in addition to the full-waveform itself) into the misfit functional or as inversion constraints. For example, traditional migration velocity analysis and focusing analysis can be merged into FWI by extending the inverted model along the offset axes (see, Symes, 2008), or along the time-lag axis (see, Biondi and Almomin, 2014). Our precious work (Wu and Zheng, 2012, 2014; Wu et al., 2014) have reported the renormalization procedure using De Wolf series and its approximation to improve the convergence of forward scattering series. Weglein and his group (Weglein et al., 1997, 2003, 2006) have been promoting the inverse scattering series approach based on Moses (1956), Prosser (1969, 1976, 1980) and Razavy (1975). Kouri’s group applied the renormalization method to transform the Lippmann-Schwinger equation in 1D media from a Fredholm integral to a Volterra integral (Kouri and Vijay, 2003; Kouri et al., 2004; Yao et al., 2014). In this paper we will report the progress in removing the divergence of inverse Born T-matrix series in 2D media by renormalization procedure and the derivation of the inverse thin-slab propagator (ITSP) which is an efficient inverse operator without divergence problem. T-matrix approach and the solution of the integral equation T-matrix (transition matrix) T is defined through the following equation

0 0 0( , ) ( , ) ; ( , ') ( ', ) ; ( , ') ( ', )r s s r s r s r sp g gx x x x G x x V x x x x G x x T x x x xd (1)

where sp is the scattered field, 0G is the background Green’s operator, g is the total filed or Green's

function, 0g is the incident field or the background Green's function, and ( , ')V x x is the scattering potential defined as (in the scalar wave case)

2 3

', ' ' ( ') ' , , 'vV

k d VV x x x x x x x x (2)

with v as the velocity perturbation function 2 20 / 1v c cx x . From (1) we see that T-matrix is

obtained though a linear transform of the data set and contains all the information in the data. T-matrix ( , ')T x x can be also transformed into local wave-number domain as a definition of local spectra:

( , ) [ ( , ' )]FTT x K T x x x , which is closely related or equivalent to the local image matrix in the local wave-number domain(Wu and Chen, 2006). From the definition of T-matrix and Lippmann-Schwinger equation we can obtain T from the scattering potential V,

1

0 0 0 0 0,g g g gT V V VG T T 1 VG V (3)

The T-matrix thus obtained corresponds to a full acquisition aperture. In this way, we can derive exact T-matrix from V(scattering potential). On the other hand, if we know the exact T-matrix, the scattering potential can be derived as,

1

01V T G T (4)

Solving V from T is the inverse scattering problem. To simplify the derivation on the nonlinear inverse operator, in this work we assume that the T-matrix can be obtained accurately, and leave the influence of incomplete acquisition aperture to future treatment. For weak perturbations, the norm of (

0G T ) will be smaller than unity, so that a perturbation series can be used to approximate the solution,

1

0 0 0 1 20

1 ( 1) ( ) ... ...n nn

nV T G T T G T V V V V (5)

This is the inverse Born series (IBS) which can be implemented by an iterative procedure. However, if the weak perturbation condition is not satisfied, the IBS will diverge. Wu et al. (2014) has proposed



a renormalization procedure (De Wolf approximation) to change the series into a convergent series. In this paper we will further study the property of the inverse T-series and show some inversion results with the inverse propagator. Figure 1 gives an example of the T-matrix in the kernel representation for a Gaussian ball model (on the left). Each ball in the middle figure is a kernel (column vector) representing the spatial distribution of the point spreading function due to scattering ( , ' )T x x x . On the right is the kernel representation

in the wave-number domain ( , )T x K . From (2) and (4), we see that both V and T are frequency-

dependent, in our example 0 20f Hz . T-matrix contains strong directional information of the scattering process which can be used for efficient recovery of scattering potentials.

Figure 1. The model of a Gaussian ball (left), the T-matrix in kernel representation (space-domain,

real part) ( , ' )T x x x (Mid) and T-matrix in wave-number domain ( , )T x K (right).

Figure 2. The detailed structure of T-matrix kernels in the space-domain ( , ' )T x x x (top) and in

wave-number domain ( , )T x K (bottom) for three individual points

Inverse nonlinear sensitivity operator and the inverse thin-slab propagator The inverse problem in T-matrix formalism is to recover V from T which is the known dataset. From (5) we can define the inverse sensitivity operator (nonlinear) P as

1

0 00

( 1) ( ) ,n n

nP 1 G T G T V TP (6)

Note that in the above equation the series may be divergent and we can apply renormalization procedure to the series. We split the scattering operator into a forward part and a backward part

f bT T T so that the Born series can be reformed into a De wolf series. For T-matrix due to forward scattering, the T-matrix for any point x in the medium can be decomposed into one derived from the



interaction with the upper half-space velocity potential (up-scattering) and one from the lower half-space velocity potential (down-scattering), plus a part from the same level,

( , ') , '), ' '

, ') ( , ') ( , ') ( , '), ( , ') , '), ' '

( , ') , '), ' '

u ff

u d z z f

d f

z zz zz z

, ')f

, ')f , ')f

, ')f

f

T x x T (x x xT (x x T x x T x x T x x T x x T (x x x

T x x T (x x x (7)

Substitute the decomposition (7) into the inverse sensitivity operator (6), and apply the forward-scattering renormalization, resulting in a De Wolf approximation for the inverse T-matrix series. We mainly concern the diagonal terms of the recovered V-matrix (all off-diagonal elements should be zeros due to mutual cancellations by multiple inverse-scattering), therefore,

2ˆ ˆ( , ) ( ) ( )

ff f u d z

f V

f fd d u u z z u d z

V k Diag Diag

Diag Diag Diag Diag Diag V V V

x x x T P T I C C C

T T P T P T P T (8)

We use the up-scattering correction as example, for the mth slab 1

1

( ) ( , ) ( , ) ( , ) , 1,...,m

u uu u

iV m Diag T m m T m i C m i m NTC (9)

For implementation with ITSP (inverse thin-slab propagator), we set (1) 0uV at the entrance of the first slab, then the inverse propagator can be written as

1

2, 1

( ) ( , ) ( , 1) ( 1, ) ( ) ; ( ) ( 1, ) ( , )d I I du u u u u

i m

V m T m m G m m T m m i i G i i T i m1 , (10)

where Gd is the one-way down-going Green’s function. We see that at each marching step, partial summation of multiple inverse-scattering has been done for all the previous slabs (here are the slabs above the mth slab). The renormalization serves to remove the divergence of the inverse scattering series. The forward TSP is a scattering generator, which is a spreading operator scattering the diagonal V-matrix into a full T-matrix; while the ITSP is a scattering eliminator (de-scattering operator), which contracts the full T-matrix into a diagonal V-matrix. Figure 3a is a schematic cartoon showing the process for implementing ITSP; In figure 3b we show the convergence comparison between IBS and ITSP for the case of 50% perturbation for the Gaussian ball model (Figure 1). We see the strong divergence of the inverse Born series and the unconditional convergence of the ITSP. Figure 4 shows the inversion result for a double-ellipses model which is composed by two closely located smooth anomalies with positive and negative perturbations (perturbation of 20% and 50%). The frequency used here is 20Hz. The T-matrix data are generated in the same way as the previous example. Similar to the case of Gaussian ball anomalies, the inversion results have high accuracy and no divergence.

Figure 3 a. (left) Schematic cartoon for ITSP (inverse thin-slab propagator); b. (right) Convergence

tests (central point) for strong perturbations (50%), between inverse Born series (IBS) and the inverse thin-slab propagator (ITSP).

Conclusion



We derive an inverse thin-slab propagator (ITSP) and apply it to the recovery of scattering potential from the known T-matrices. Numerical tests proved that the renormalized inverse scattering series has much better convergence property than the inverse Born series and the inverse thin-slab propagator (ITSP) is an accurate and efficient method using the nonlinear kernel.

Figure 4. The recovered velocity distributions and theirs relative errors for the perturbations of (a)

20% and (b) 50% inverted by our ITSP for the double-ellipses anomalies model.

Acknowledgement We thank the discussions and helps from Chunhua Hu and Lingling Ye. We also acknowledge the helpful discussions with M. Jakobsen and Yingcai Zheng. The research is supported by the WTOPI Research Consortium of Modeling and Imaging Laboratory, University of California, Santa Cruz.

References Biondi, B. and A. Almomin, 2014, Simultaneous inversion of full data bandwidth by tomographic full-waveform inversion, Geophysics, 79, WA129-140. Chen L., Wu R., and Chen Y., 2006. Target-oriented beamlet migration based on Gabor-Daubechies frame decomposition, Geophysics, 71(2): S37-S52. Kouri, D.J. and Vijay A., 2003. Inverse scattering theory: Renormalization of the Lippmann- Schwinger equation for acoustic scattering in one dimension, Phys. Review, E 67. Moses, H. E., 1956, Calculation of scattering potential from reflection coefficients: Phys. Rev., 102, 559-567. Pratt, R. G., 1999. Seismic waveform inversion in the frequency domain, Part I: Theory and verification in a physical scale model, Geophysics, 64, 888–901. Prosser, R. T., 1969, Formal solutions of inverse scattering problems, J. Math.Phys., vol. 10, pp. 1819–1822. Razavy, M., 1975. Determination of the wave velocity in an inhomogeneous medium from reflection data: J. Acoust. Soc. Am., 58, 956-963. Symes, W. W., 2008, Migration velocity analysis and waveform inversion: Geophysical Prospecting, 56, 765–790, doi: 10.1111/j.1365-2478.2008.00698.x. Tarantola, A., 2005. Inverse Problem Theory and Methods for Model Parameter Estimation, Society for Industrial and Applied Mathematics, Philadelphia, PA. Weglein, A.B., Fernanda V., Carvalho P.M., Stolt R. H., Matson K. H., Coates R. T., Corrigan D., Foster D.J., Shaw S. A. and Zhang H., 2003. Inverse scattering series and seismic exploration, Inverse Problems 19 (2003) R27–R83. Wu, R.S. and Zheng Y., 2014. Non-linear partial derivative and its De Wolf approximation for nonlinear seismic inversion, Geophys. J. International, 196, 1827-1843. Wu, R.S., Hu, C. and Wang, B., 2014, Nonlinear sensitivity operator and inverse thin-slab propagator

for tomographic waveform inversion, 85th Annual International Meeting, SEG. Yao J., Lesage A.-C., Bodmann B. G., Hussain F., and Kouri D. J., 2014. One dimensional acoustic direct nonlinear inversion using the Volterra inverse scattering series, Inverse problems, 30(7): 075006.

inverse thin-slab propagator based on forward- scattering ...wrs/publication/seg/seg2015-2016/tu...

Documents