Tu G Nonlinear Sensitivity Operator and Its De Wolf Approximation in T-matrix Formalism

Transcription

1 Tu G13 7 Nonlinear Sensitivity Operator and Its De Wol Approximation in T-matrix Formalism R.S. Wu* (University o Caliornia), C. Hu (Tsinghua University) & M. Jakobsen (University o Bergen) SUMMARY We derived the Nonlinear sensitivity operator in nonlinear tomographic waveorm inversion based on the theory o nonlinear partial derivative operator. We apply the renormalization procedure (De Wol approximation) to the orward and inverse T-matrix series. Numerical tests proved that the renormalized inverse scattering series has much better convergence property than the inverse Born series. This convergence improvement may be applied to the iterative procedure o waveorm inversion.

2 Introduction Current gradient method in ull waveorm inversion (FWI) is based on a linearization o the ull nonlinear unctional partial derivative (NLPD) operator, and can be considered as a quasi-linear inversion. NLPD can be expanded into a Taylor series which corresponds to a ull scattering series, the Born series. The convergence problem o the iterative procedure o quasi-linear inversion, problem o local minima, and the starting model dependence, are all deeply rooted in the well-known convergence problem o the Born series and inverse Born series (see e.g., Morse and Feshback, 1953; Prosser, 1969; Weglein et al., 23). For the real Earth, the wave equation is strongly nonlinear with respect to the medium parameter changes. Wu and Zheng (212, 214) introduced the higher order Fréchet derivatives and the theory o nonlinear partial derivative (NLPD) operator or the acoustic wave equation. Our precious work (Wu and Zheng, 212, 213; Wu et al., 213) have reported the renormalization procedure using De Wol series and approximation to improve the convergence o orward scattering series. In this work we will irst introduce the De Wol approximation to the T- matrix series and report the progress in improving the convergence o inverse Born series by renormalization procedure in the orm o T-matrix series. Summary on nonlinear partial derivative operator. Assume an initial model m, we want to quantiy the sensitivity o the data change δd (also called data residual ) to the model perturbation δm, ddd Fm ( m ) A ( m m ) A ( m ) (1) I we deine a nonlinear partial derivative operator A ( m, m ) based on the nonlinear dierential operator Fm (, m) through A ( m, m) m F( m, m), then we have, ' 1 1 ( ) n A m m A m A m m A n1 m m. (2) 2! n! where A ', A, and A n are the irst, second, and the nth order Fréchet derivatives. Note that A is m( x) -dependent because o the nonlinear mutual interactions) between perturbations b I we split the scattering operator into orward scattering and backscattering parts, SS S and substitute it into the Fréchet series, we can have all combinations o higher order orward and backward derivatives. The De Wol approximation corresponds to neglecting multiple backscattering (reverberations), i.e. dropping all the terms containing two or more backscattering operators but keeping all the orward scattering terms untouched (De Wol, 1985; Wu, 23). Forward and inverse scattering series in T-matrix approach IBS (inverse Born series) originally is ormulated directly in the data space. An alternative way, which is more convenient and computational eicient, is to ormulate the inverse scattering in the image space (model space). This is the approach o contrast-source approach and T-matrix approach (e.g., Prosser, 1969; Weglein et al., 23; Jakobsen, 212; Jakobsen and Ursin, 212). Here, we will apply the T-matrix approach to the NLSO (nonlinear sensitivity operator). T-matrix (transition matrix) T is deined through the ollowing equation d ( xr, xs) ps( xr, xs) Gxr; xv( x, x') g( x', xs) Gxr; xt( x, x') g( x', xs) (3) where ps is the scattered ield, G is the background Green s operator, g pis the total iled, g pis the incident ield, and Vxx (, ') is the scattering potential deined as (in the scalar wave case) 2 3 Vxx, ' SVxx, ' v( x') k d x' ( ') ',, ' V ' v x xx xx V (4) In the ollowing we ormulate the orward scattering and inversion scattering in terms o T-matrix. First we discuss the orward problem. From the relation Tg Vg VgVGTgwe have T1VG 1 VPV (5) where P is the propagator. This is in a orm o integral equation similar to the Lippmann-Schwinger equation. From the above equations, we can write T = V + VG T (5)

3 In the case o weak scattering, in which the norm VG 1, an iterative procedure can be used to get a Born series o T-matrix rom (5) or the orward scattering solution. For strong perturbations, we can apply the orward scattering renormalization, or the De Wol approximation to eliminate the divergence problem. In the ollowing we derive the De Wol approximation or the T-matrix ormulation. Renormalization o the T-matrix series and the De wol approximation We now split the scattering potential V into a orward scattering part V and a backscattering part b V : VV V b (5) T T T. From (5) we have T T1 T2 : T 1= V + V G T; T 2= V b + V bg T (5) Next, we write the T-matrix as the sum o two terms, i.e. 1 2 We see that T 1 and T 2 represent disturbed T-matrices due to orward and backward scattering, including terms related to the interaction between orward and backward scattering. By using the renormalization procedure, one can rewrite the above ormulae or the total T-matrix exactly as T = t + t G T ; T = t + t G T (5) b b 1 where the t-matrices stand or pure multiple-orward and pure multiple-backward scattering (reverbaration) and satisy the ollowing integral equations t =V + V G t ; t = V +V G t (5) b b b b Various approximations can be obtained i one iterates on the above equations. To derive an approximation which accounts or all orward multiple scattering, but only single backward scattering (De Wol approximation), we ignore multiple backward scattering (reverberations) in the second equation in (5), i.e. t b = V b. Then we have T = V + V G T (5) 2 b b 1 To irst order in V b, equation (5) is approximately equivalent to T = t + t G V t G V G t ; T = V +V G t 1 b b 2 b b Thereore, the total T-matrix under single-backscattering approximation is given by T =T T : T t ; T = V +t G V VG t t G VG t (5) DW b b b b b b This orm o De Wol approximation or T-matrix is consistent with the De Wol approximation or the waveield. Now we discuss the inversion through T-matrix. Ater obtaining the T-matrix based on data o experiments, we can invert it or the scattering potential and perturbation unction v. The scattering data are kept in the T-matrix and stay in the model space (image space), and the acquisition process is peeled o. O course, the knowledge o acquisition process is needed to estimate the T-matrix. By assuming the irst order estimate V 1 T, a series solution ISS (inverse scattering series) is obtained by an iterative process: VTVGT (6) n n ( 1) ( ) n... n 1 V 1GT TT GT V VV V (7) Renormalization o ISS (inverse scattering series) under the De Wol approximation Now we apply the operator split and orward-scattering renormalization to the above inverse T-matrix series. As we showed above, the T-matrix is split into a part due to orward scattering and a part due to single backscattering (neglecting multiples). In this work, we treat only the orward scattering problem such as in the case o smooth media, so only T is involved. For T-matrix due to orward scattering, the T-matrix or any point x in the medium can be decomposed into one derived rom the interaction with the upper hal-space velocity potential (up-scattering) and one rom the lower halspace velocity pothential (down-scattering), plus a part rom the same level (5)

4 u d z T (xx, ') T ( xx, ') T ( xx, ') T ( xx, ') T( xx, ') T ( xx, ') T( xx, '), u T ( x, x') T (x, x '), x' z' z z T ( x, x') T (x, x '), x' z' z d T ( x, x') T (x, x '), x' z' z Since only orward scattering is involved, we can recover the velocity potential rom the correspondi ng T d u, T and T z. Substitute the deocmposition (7) into the inverse T-series (7), and apply the orwar d-scattering renormalization, resulting in a De Wol approximation or the inverse T-matrix seriers. We can prove that each term o the inverse series is a Volterra type integral (Tricomi, 1985; Schetze n, 198). Thereore, the inverse T-series is a Volterra series which converges absolutely and uniorm ly (ibid). Numerical tests or renormalized inverse T-matrix series in comparison with the Born T-series. In the ollowing we show some simple examples to demonstrate the convergence improvement o the renormalized orward and inverse T-series. (7) Figure 1 Let: The model o Gaussian ball ( a 5 ) with dierent perturbation strengths; Middel & Right: The ull T-matrix or the model in column-vector representation: Mid: real part; Right: imaginary part. The test model is a Gaussian ball ( a 5 ) with dierent perturbation strengths rom the homogeneous background (Figure 1 on the let). The matrix T (operator matrix) is a complex-value requencydependent matrix o N by N. The whole model space is o 2 by 2 in grid size. The perturbation area is about 5 by 5, and N is about 18. We produce the T-matrix by both exact matrix inverse method and the Born series. In the case o weak scattering, the two are the same. For strong scattering (above 2% perturbation), the Born series ailed to converge and we rely on the exact inverse algorithm to produce the T-matrix. In order to exhibit the physical meaning o the T-matrix, we plot the column vectors rom matrix T column-by-column. In Figure 1 we also plot the sparsely sampled column vectors o the ull T-matrix. In the Figure, each small ball is a representation o the column vector, corresponding to a point spreading unction (a scattering pattern) o the point scatterer. In the middle is the real part and on the right is the imaginary part. This exact T-matrix corresponds to a ullaperture measurement and contains all the inormation in acquired data. Normally, T-matrix is derived by a linear inversion rom the data. In order to test the convergence o the inversion scattering series, we use the exact T-matrix here. Figure 2(a) and 2(b) give the results o convergence tests. Here we only plot the convergence o the velocity value at the center o the ball, inverted rom both the inverse Born series and the renormalized inverse scattering series. From Figure 2(a), we see that IBS converges at a slow rate or 15% perturbation, but diverges ast or 2% perturbation. In comparison, the renormalized ISS converges very ast or 15% and 2% perturbations. It still converges or 25% and 3%, but rather slowly or the latter (Figure 2(b)). We are still working on improving the convergence and to apply to waveorm inversion. Conclusions

5 We apply the renormalization procedure and the De Wol approximation to the orward and inverse T-matrix series. Numerical tests proved that the renormalized inverse scattering series has much better convergence property than the inverse Born series. This convergence improvement may be applied to the iterative procedure o waveorm inversion. (a) (b) Figure 2 (a) Convergence comparison between inverse Born series (IBS) and the renormalized inverse scattering series. The red dotted lines are the correct perturbation values o velocity at the central point; the black curves are the convergence curves o the series; (b) Convergence tests o the renormalized inverse scattering series or the case o strong perturbations (25%) Acknowledgements We thank Yingcai Zheng, Lingling Ye, and Bjorn Ursin or discussions and collaborations. This research is supported by the WTOPI Research Consortium at the University o Caliornia, Santa Cruz. Reerences De Wol, D.A., Renormalization o EM ields in application to large-angle scattering rom randomly continuous media and sparse particle distributions. IEEE Trans. Antennas and PropagationsAP-33, Jakobsen, M., 212. T-matrix approach to seismic modelling in the acoustic approximation, Studia Geophysica et Geodaetica, 56, 1-2. Jakobsen, M. and Ursin, B., 212. Nonlinear seismic waveorm inversion using a Born iterative T- matrix method. Extended abstract, 82th SEG Meeting, Las Vegas. Morse, P.M. and Feshback, H., 1953, Methods o theoretical physics, I and II, McGraw-Hill Comparison. Inc.. Prosser, R. T., 1969, Formal solutions o inverse scattering problems, J. Math.Phys., vol. 1, pp Schetzen, M., 198, The Volterra and Wiener Theories o Nonlinear Systems, Wiley. Tarantola, A., 25. Inverse Problem Theory and Methods or Model Parameter Estimation, Society or industrial and Applied Mathematics, Philadelphia, PA. Tricomi, F.G., 1985, Integral Equations, Dover Publications, Inc.. Weglein, A.B., V. Fernanda, P.M. Carvalho, R. H. Stolt, K. H. Matson, R. T. Coates, D. Corrigan, D.J. Foster, S. A. Shaw and H. Zhang, 23, Inverse scattering series and seismic exploration, Inverse Problems 19 (23) R27 R83. Wu, R.S., 23. Wave propagation, scattering and imaging using dual-domain one-way and onereturn propagators, Pure and Appl. Geophys., 16(3/4), Wu, R.S. and Y. Zheng, 212. Nonlinear Fréchet derivative and its De Wol approximation, Expanded Abstracts o Society o Exploration Gephysicists, SI 8.1. Wu, R.S., Ye, L. and Zheng, Y., 213, Nonlinear partial unctional derivative and nonlinear LS seismic inversion, Extended abstract, 75th EAGE Meeting, London. Wu, R.S. and Zheng, Y., 214, Nonlinear partial derivative and its De Wol approximation or nonlinear seismic inversion, Geophys. J. International, doi: 1.193/gji/ggt496.