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Renormalized nonlinear sensitivity kernel and inverse thin-slab propagator in T-matrix
formalism for wave-equation tomography
View the table of contents for this issue, or go to the journal homepage for more
Renormalized nonlinear sensitivity kerneland inverse thin-slab propagator in T-matrixformalism for wave-equation tomography
Ru-Shan Wu1, Benfeng Wang2 and Chunhua Hu3
1University of California, Earth and Planetary Sciences, Modeling and ImagingLaboratory, Santa Cruz, CA, USA2China University of Petroleum, State Key Laboratory of Petroleum Resources andProspecting, Beijing, China3 Tsinghua University, Beijing, China
Received 16 November 2014, revised 10 July 2015Accepted for publication 26 August 2015Published 2 October 2015
AbstractWe derived the renormalized nonlinear sensitivity operator and the related inversethin-slab propagator (ITSP) for nonlinear tomographic waveform inversion basedon the theory of nonlinear partial derivative operator and its De Wolf approx-imation. The inverse propagator is based on a renormalization procedure to theforward and inverse transition matrix scattering series. The ITSP eliminates thedivergence of the inverse Born series for strong perturbations by stepwise partialsummation (renormalization). Numerical tests showed that the inverse BornT-series starts to diverge at moderate perturbation (20% for the given model ofGaussian ball with a radius of 5 wavelength), while the ITSP has no divergenceproblem for any strong perturbations (up to 100% perturbation for test model). Inaddition, the ITSP is a non-iterative, marching algorithm with only one sweep,and therefore very efficient in comparison with the iterative inversion based onthe inverse-Born scattering series. This convergence and efficiency improvementhas potential applications to the iterative procedure of waveform inversion.
(Some figures may appear in colour only in the online journal)
1. Introduction
The existing seismic inversion methods seem clustered as two distinct groups: One is thelinear or quasi-linear inversion based on the linear functional derivative (Fréchet derivative);
the other is the Monte Carlo type nonlinear inversion. There is a vast gap between these twoterritories. In this study we introduce an approach based on the nonlinear sensitivity operator(NLSO) and the renormalization of inverse scattering series (ISS), which may be termed as‘analytical nonlinear inversion’ or ‘direct nonlinear inversion’ in order to distinguish it fromthe Monte-Carlo or iterative linearized inversion.
The gradient method in full waveform inversion (FWI) is based on a linearization of thefull nonlinear functional partial derivative (NLPD) operator (see Tarantola 1984, 2005, Prattet al 1998, Pratt 1999), and can be considered as a quasi-linear inversion. The NLPD can beexpanded into a Taylor series which corresponds to a full scattering series, the Born series(Wu and Zheng 2012, 2014). The convergence problems of the iterative procedure of quasi-linear inversion, such as cycle-skipping, local minima, and starting model dependence, are alldeeply rooted in the well-known convergence problem of the Born series and inverse Bornseries (IBS) (see e.g., Morse and Feshback 1953, Moses 1956, Prosser 1969, Aki andRichards 1980, Weglein et al 1997, 2003, Wu and Zheng 2014). For the real Earth, the waveequation is strongly nonlinear with respect to the medium parameter changes. In order toavoid the divergence of the gradient method (Gauss–Newton method), people have tocarefully chose some ‘good’ starting model so that the weak perturbation condition issatisfied. Many investigators rely on other geophysical methods, such as traveltime tomo-graphy (including ray-based or wave-based, first-arrival traveltime tomography and reflectiontraveltime tomography) and velocity analysis to provide the appropriate large-scale (low-wavenumber) starting models (for a review see Virieux and Operto 2009). Within FWI, longoffsets, multi-scale inversion has been developed to reduce the starting model dependence(Bunks et al 1995, Pratt et al 1996, 1998, Sirgue and Pratt 2004, Plessix et al 2010, Vighet al 2011, Baeten et al 2013). Recent development of low-frequency land source (down to1.5 Hz) has allowed multi-scale FWI to use 1D smooth starting model (Baeten et al 2013) forsome field data sets. However, the ultra-low frequency field sources are very expensive andgenerally not available. Therefore, the starting model dependence is still a major difficulty inpromoting the wide-use of FWI. To overcome this difficulty without relying on the low-frequency seismic source, a recent trend is to combine other seismic data functional (inaddition to the full-waveform itself) into the misfit functional or as inversion constraints. Forexample, traditional migration velocity analysis and focusing analysis can be merged intoFWI by extending the inverted model along the offset axes (see, Symes 2008), or along thetime-lag axis (see, Biondi and Almomin 2014).
In this paper we will try to solve the problem of inversion divergence or low-wave-number starting model recovery without low-frequency source from a totally differentapproach based on the theory of nonlinear sensitivity kernel (NLSK). Wu and Zheng(2012, 2014) introduced the higher order Fréchet derivatives and the theory of nonlinearpartial derivative (NLPD) operator for the acoustic wave equation. Our previous work (Wuand Zheng 2012, 2014, Wu et al 2013) have reported the renormalization procedure using DeWolf series and its approximation to improve the convergence of forward scattering series. Inthis paper we will report the progress in removing the divergence of inverse Born T-series byrenormalization procedure and the derivation of the inverse thin-slab propagator (ITSP). Infact renormalization method (theory) has played a critical role in the development of quantumfield theory and critical phenomena (see, e.g., Zinn-Justin 2007, Zee 2010), and therefore,won twice Nobel prices in physics for its inventors: Tomonaga, Schwinger in 1965, Wilsonin 1982.
For forward problems of wave scattering (modeling), Sams and Kouri (1969) carried outa renormalization transformation of the Lippmann–Schwinger equation into a Volterraequation which guarantees an absolutely convergence of the forward series. De Wolf
Inverse Problems 31 (2015) 115004 R-S Wu et al
2
(1971, 1985) introduced the forward-scattering renormalized Green’s function to eliminatethe forward-scattering divergence. Wu and his collaborators took use of the De Wolf’srenormalized Green’s function in acoustic and elastic wave propagation and termed therenormalized series as ‘De Wolf series’ and the first order approximation as ‘De Wolfapproximation’ (Wu 1994, 1996, 2003, Wu and Huang 1995, De Hoop et al 2000, Xie andWu 2001, Wu et al 2007, 2011). Furthermore, in the above mentioned work, they developedan implementation procedure for the De Wolf series: the thin-slab propagator (TSP) and thegeneralized screen propagator, which are both efficient and stable (ibid). Jakobsen (2012)used the renormalization method to remove the singular integral and obtained a modifiedBorn series which improves the convergence of the original Born series.
For inverse scattering problems, Weglein and his group have been promoting the ISSapproach based on Moses (1956), Prosser (1969, 1976, 1980) and Razavy (1975), and madeimportant contributions (Weglein et al 1997, 2003, 2006). Kouri’s group applied the renor-malization method to transform the Lippmann–Schwinger equation in 1D media from aFredholm integral to a Volterra integral which has absolute and uniform convergence (Kouriand Vijay 2003, 2004, Yao et al 2014). Lesage et al (2013) also used the renormalizedVolterra integral for inverse acoustic scattering series in 1D media with reflection andtransmission data.
In this paper, we apply the renormalization method and the De Wolf approximation to theISS in the transition matrix (T-matrix) formalism, resulting in an ITSP. The renormalizationprocedure, embedded into ITSP serves to remove the divergence of the IBS. This inversepropagator can serve as a nonlinear inverse sensitivity operator (ISO) in FWI. Numerical testsproved that ITSP has no divergence and is very efficient in implementation. We give anintroduction in section 1. In section 2 we present some background on scattering and ISSsolutions for acoustic wave equation Then we introduce the T-matrix formalism to the for-ward and ISS, The NLSK and its De Wolf approximation are introduced in section 3 thoughoperator spit. In section 4 we develop an efficient implementation of the De Wolf kernelintroduced in section 3, the ITSP and show some results of numerical tests in section 5. Theconclusion is given in section 6.
2. Forward and inverse NLSOs in T-matrix formalism
We write the forward problem into an operator form
d A m , 1( ) ( )=
where d is the data vector (pressure field generated by the modeling), m is the model vector,and A is the forward modeling operator. Assume an initial model m0, we want to quantify thesensitivity of the data change δd (also called ‘data residual’) to the model perturbation δm.
For seismic scattering problems, the mapping from md to dd in general is a nonlineardifferential operator defined as
d F m A m m A m d d , 2m 0 0 00 ( )( ) ( ) ( )d d d= = + - = -
where Fm0is the NLSO relating data residual δd to model perturbations md at the current
model (background model) m .0 We can call it the forward sensitivity operator. In the linearinversion theory, the ISO, which predicts md from the data residual d,d is simply a hermitiantranspose of the forward sensitivity operator (the Fréchet derivative). However, in thenonlinear case, it is not as direct as in the linear case. On the contrary, it needs a nonlinear
Inverse Problems 31 (2015) 115004 R-S Wu et al
3
operator to estimate md from d.d The NLSO has been discussed in detail in Wu andZheng (2014).
Consider the inversion process. Assume we know the data residual d,d and try to back-map it to a model perturbation m.d To predict the model perturbation from a measured dataresidual is also a mapping operator, a nonlinear ISO. The kernel of the operator is the NLSK,which plays a similar role as the linear sensitivity kernel, i.e. the gradient, in the linearinversion. However, unlike the linear case in which the ISO is just an adjoint operator of theforward sensitivity operator, while in the nonlinear case the ISO can be very different fromthe forward operator.
First, we can say that the ISO must be a nonlinear one to recover the model perturbationsfrom the data residual d.d This is because the forward model is a nonlinear operator, asillustrated in figure 1.
Similar to the Taylor expansion of forward sensitivity operator, a straightforward way toexpress the nonlinearity of the inverse sensitivity is to expand it into a Taylor series. With thesame normalization as in the forward expansion, we can write the ISO as a series,
B d B d d B d d B d d , 3nn
1 0 2 02
0( ) ( ) ( )( ) ( ) ( ) ( )d d d d= + + + +
where d0 is the wavefield data from the background medium. Now the critical procedure is tofind the coefficients in (3), and change the perturbation series, i.e. the IBS into a convergentseries. The data are generated by multiple-scattering process, but the gradient method (linearFrechet derivative) treats it as a single scattering data and applies the back-mappingaccordingly to the model space. As we mentioned above, the series may converge veryslowly, or not converge at all (Morse and Feshback 1953, Prosser 1976, Newton 1982). Inthis paper, we will apply the renormalization procedure and use the De Wolf series expansionof the NLSO (Wu and Zheng 2012, 2014) to the inverse series. To simplify the treatment ofdirect inversion using the NLSO, we first reduce the problem into the T-matrix formulism forhandling the multiple scattering.
The above IBS is formulated directly in the data space. We see from (3) that there aremany repeated operations of back mapping (B d1 )d and forward scattering, which are com-putationally expensive and unnecessary. In addition, when the linear inversion has errors, theerrors will pass to the higher order data and then to the higher order corrections. An alter-native way, which is more convenient and computationally efficient, is to formulate theinverse scattering in the image space (model space). This is the contrast-source approach orthe T-matrix approach. In this section, we will apply the T-matrix approach to the NLSO.
T-matrix formalism is widely used in scattering and inverse scattering theory, especiallyin quantum scattering theory (Taylor 1972, Newton 1982). It was introduced to seismic oracoustic scattering since the fifties (Moses 1956, Prosser 1969, 1976, 1980, Razavy 1975,Stolt and Jocobs 1981, Weglein et al 1981, for a review see Weglein et al 2003). T-matrix Tis defined through the following equation (Taylor 1972)
Figure 1. Nonlinear forward and inverse filters.
Inverse Problems 31 (2015) 115004 R-S Wu et al
4
T V VGV, 4( )= +
or
VG TG , 50 ( )=
where V is the scattering potential, G and G0 are the actual and background Green’s operatorsrespectively. These two definitions are identical, since (5) can be derived by multiplying (4)with G0 from the right and invoking the Lippmann–Schwinger equation. Equation (5) has asymmetric form
GV G T. 60 ( )=
From the definition, we see that T-matrix is the intrinsic scattering property of themedium depending only on V. T-matrix is a convenient tool for scattering theory. However,T-matrix has also a direct connection with the scattering experiments:
pd x x x x x G VG x x G TG x, , , 7r s s r s r 0 s r 0 0 s( ) ( ) ( )d = = =
where pd sd = is the scattered field on the observation surface. Scattering potential V x x,( )¢ isdefined as (in the scalar wave case)
kc
cVV x x x x x x
xx x, , 1, , , 8v v
2 02
2( ) ( ) ( ) ( )
( )( )e d e¢ = ¢ - ¢ ¢ =
¢- ¢ Î
where εν is the velocity perturbation function and c, c0 are the local velocity and backgroundvelocity respectively. In the derivation, we still use the operator form so that we can use theintegral form in the derivation. Here we also use the Dirac’s ‘bra’ and ‘ket’ notation foroperator representation, where xr and xs are spatial locations of receiver and source,respectively. In this paper, we do not distinguish the T-operator from its discretized version,T-matrix. Operator application is realized by a matrix multiplication, which is equivalent to adiscretized volume integral. T-matrix includes all the interactions between the incident fieldand the medium, and transforms the incident wave into the scattering data. It peels off theincident field calculation and the final propagation to the receivers, so that the effort can beconcentrated to the multiple-scattering treatment inside the medium. Note that in terms ofmatrix operation, the scattering potential V x x,( )¢ is a diagonal matrix; however, the T-matrixbecomes non-diagonal due to scattering.
In the following we formulate the forward scattering and inverse scattering in terms ofT-matrix. First we discuss the forward problem. From (4) and (6), we obtain
T I VG V. 901[ ] ( )= - -
Equation (9) is in a form of integral equation similar to the Lippmann–Schwingerequation. In the case of weak scattering, in which the norm VG 1,0 < an iterative pro-cedure can be used to get a Born series of T-matrix for the forward scattering solution.However, if the scattering is not weak, the Born series of T-matrix may diverge. Then theexact integral equation can be solved for the solution. In matrix form, the inverse of matrix in(9) may be obtained numerically for small-size model (Jakobsen and Ursin 2011, 2012,Jakobsen 2012).
Now we discuss the inversion through T-matrix. After obtaining the T-matrix based ondata of experiments, we can invert for the scattering potential and perturbation function εν.The scattering data are kept in the T-matrix and stay in the model space (image space), and theacquisition process is peeled off. Of course, the knowledge of acquisition process is needed toestimate the T-matrix by inversion. In the same way of deriving (9), we derive the operatorequation for V
Inverse Problems 31 (2015) 115004 R-S Wu et al
5
V T I G T . 1001[ ] ( )= + -
For a small model, the matrix inverse in above equation can be implemented directly, butnot for large model. Traditionally the above integral equation of the inverse problem is solvedby a perturbation series method. Assuming G T 1,0 < (10) can be expanded into pertur-bation series. By taking the first order estimate V T,1 = a series solution ISS is obtained by aniterative process
V T TG T T G T T G T T G T
T I G T
1
1 .
11
n n
n
n n
0 02
03
0
10
( ) ( ) ( )
( )
( )
( )( )
⎡⎣⎢
⎤⎦⎥å
= - + - + ⋅ ⋅ ⋅ + -
+ ⋅ ⋅ ⋅ = + -=
¥
The perturbation function can be recovered by solving the integral equation or by a seriessolution in (11).
T-matrix can be estimated from the data by a linear inversion:
T F d, 121˜ ( )d= +
where F1+ is a linear pseudo-inverse operator and T̃ is the first order estimate of the T-matrix.
To have a good estimate T̃ is also a critical step in the recovery of the true perturbationfunction. The inversion for T̃ is similar to reconstruct the local scattering matrices from thedata. It is also related to the contrast-source inversion (Tsihrintzis and Devaney 2000a, 2000b,Van den Berg and Abubakar 2001, Abubakar et al 2008) and Green’s function retrieval (see,Wapenaar et al 2008).
We see that higher orders of G T0 ˜ involve higher order data interactions. Invoking thenotation of nonlinear ISO defined in (3), we can write (11) as
V T G T B d1 1 . 13n
n n
10( )˜ ˜( ) ( ) ( )
⎡⎣⎢
⎤⎦⎥å d= + - =
=
¥
The sum inside the braces of (13) is a nonlinear filter representing the multiple-scatteringcorrection. We know V is a diagonal matrix, but the zero-order estimate T̃ is non-diagonal.Through the ISS of the T-matrix we expect to diagonalize the matrix gradually by iteration. Inparallel to the terminology in linear inversion, we call B d( )d as the NLSO and its kernel asNLSK. With the series expression, the nonlinear corrections by multiple scattering can beperformed directly on T̃ without shuttling back and forth between the data and model space.The process is illustrated in figure 2 (top panel). On the other hand, if we substituteT F d B d1 1˜ d d= =+ into (13), we obtain the NLSO in terms of higher order data terms:
B d B d G B d G B d G B d
G B d
B d G B d
1
1
1 1 . 14
n n n
n
n n n
1 0 1 0 12 2
0 13 3
0 1
11
0 1
{}
( ) ( )( )
( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( )⎡⎣⎢
⎤⎦⎥å
d d d d d
d
d d
= - + -
+ ⋅ ⋅ ⋅ + - + ⋅ ⋅ ⋅
= + -=
¥
Inside the braces, it is a nonlinear filter function acting on data before linear inversion, asillustrated by figure 2 (bottom panel). Although the two forms are equivalent, the latter ismore computationally intensive.
The ISS of the Born–Fredholm type, such as (13) or (14), is a highly oscillating serieswithout guarantee of convergence (Morse and Feshback 1953, Newton 1982, Kouri andVijay 2003, 2004). Nevertheless, (11) or (13) is a formal solution which includes all the
Inverse Problems 31 (2015) 115004 R-S Wu et al
6
higher order scattering terms. The current status is to calculate only a few terms and see theimprovement compared with the first order term (in most cases only to 3rd order correction,see Tsihrintzis and Devaney 2000a, 2000b). Some theoretical work using renormalization toderive Volterra series type solution for one-dimensional media has been introduced recentlyby Kouri’s group (Kouri and Vijay 2003, 2004). In the following section, we will apply theDe Wolf series method (De Wolf transform) to reorder and renormalize the ISS, and hencetransform the ISS into a ITSP , which is a Volterra-type series with guaranteed con-vergence. Furthermore, under a smooth perturbation approximation for the transmissiontomography, we derive a simple form for the NLSK, leading to an efficient nonlinearinversion algorithm.
3. NLSK for transmission tomography under the De Wolf approximation
In order to improve the convergence or essentially remove the divergence of the Fredholmtype ISS (11), some approaches have been proposed to reform the series into a Volterra typeseries, which has a guaranteed convergence. Tsihrintzis and Devaney (2000a) applied theRytov transform to the iterative forward solution, so that the resulted series became aVolterra type. Due to the complicated form of the higher order terms, they tried only up tothird order terms in their paper. Kouri and Vijay (2003, 2004) decomposed the scatteringintegral into a Volterra type and an oscillating type (with a sine kernel) and applied arenormalization procedure to the series. The proposed renormalization method needssome extra data (measure total transmitted field) for the procedure and only 1D case istreated in the paper. In this section, we propose to apply the De Wolf series andDe Wolf approximation to split the kernel and re-sum (renormalize) the ISS. It can beproved that the marching algorithm of ITSP forms a Volterra series, therefore has aguaranteed convergence.
To simplify the derivation on the NLSO in inversion, here we assume the T-matrix can beobtained accurately, which corresponds to the case of full acquisition aperture, and derive theinversion theory based on known exact T-matrix. The influence of incomplete acquisitionaperture will be postponed to future treatment.
Figure 2. Two equivalent processes of nonlinear inverse scattering.
Inverse Problems 31 (2015) 115004 R-S Wu et al
7
3.1. Iterative solutions for forward and inverse T-matrix series
In operator form, or the discrete counterpart, the matrix form, the forward Born T-series (9)can be written as
T PV
P VG VG
,
1 , 15n
n0
1
00( ) ( ) ( )å
=
= - =-
=
¥
where P is the forward propagator (global scattering operator) which spread the diagonal V-matrix into the off-diagonal elements of the T-matrix due to scattering. The perturbation seriescan be implemented by an iterative procedure, formally written as
P I VG VG VG
I VG I VG I VG . 16
0 0 0
0 0 0( )( )( )( ) ( )
= + + + ¼
= + ¼ + +
We see that the propagator matrix P is close to full due to multiple scattering.For the inverse T-series, we reform (11) as
V TP
P G T
1 G T 1 G T 1 G T
1
, 17
n
Nn n
00
0 0 0( )( )( )
( )
( )
( )
( )
å
=
= -
= - - -
-
-
=
where P- is the inverse propagator (global inverse-scattering operator) which will eliminate thespreading due to scattering and recover the diagonal velocity potential. Therefore, for parameterinversion, we are mainly concerned with the diagonal terms of the recovered V̂-matrix
V T G TDiag 1 . 18n
n n
00( )ˆ ( ) ( )
⎪ ⎪
⎪ ⎪
⎧⎨⎩
⎡⎣⎢
⎤⎦⎥
⎫⎬⎭å= -
=
¥
As shown in Wu and Zheng (2012, 2014), we can split the scattering operator into aforward part and a backward part so that the Born series can be reformed into a De Wolfseries. In this case, we split the velocity potential into
V V V , 19f b ( )= +
where Vf and Vb are the forward and backward scattering potentials, respectively. Now weapply the operator split and forward-scattering renormalization to the inverse T-matrix series.We can split the T-matrix into
T T T , 20f B ( )= +
where Tf is the T-matrixdue to forward-scattering, and TB is the T-matrix involving in anybackscattering. To simplify the treatment in this application, we assume the medium issmooth and therefore can only produce forward scattering. The full treatment will be left forfuture study. From (15) we have
T V G V . 21n
nf
0f 0 f( ) ( )
⎧⎨⎩⎫⎬⎭å=
=
¥
In this case, the forward-scattering T-matrix Tf for any point x in the medium can bedecomposed into one derived from the interaction with the upper half-space velocity per-turbation (up-scattering) and one from the lower half-space velocity perturbation (down-
Remember that our positive z-axis is downward and all T ,u Tz and Td are produced byforward scattering only. Then we can decompose the inverse propagator P- into contributionsfrom T ,u Td and Tz respectively. As we mentioned, we consider only the diagonal terms of therecovered V-matrix, therefore
V kx x x TP
P G T 1 C C C
1 G T G T G T
, Diag
1
1 1 1 , 23
n
n n u d z
n
n du
n
n
n ud
n
n
n zz
n
f2
00 f
1 1 1( ) ( ) ( )
{ }( )
ˆ ( ) ˆ ( )
( )
( ) ( ) ( ) ( )⎡⎣⎢
⎤⎦⎥
å
å å å
e= =
= - = + + +
= + - + - + -
-
-
=
¥
=
¥
=
¥
=
¥
where C C C, ,u d z are the corresponding forward-scattering corrections from T ,u Td and T ,z and
x z x z z zz z
x z x zz zz z
x z x zz zz z
GG
G G
G G
, ;0
, ,, ,
0, ;
, ,, ,
0, ;
, ,, .
24
u
z
d
0 2 2 1 1 2 1
2 1
0 2 2 1 1
2 1
2 1
0 2 2 1 1
2 1
2 1
( )
( )
( ) ( )
⎧⎨⎩⎧⎨⎩⎧⎨⎩
<>
¹=
<>
In the case of limited aperture such as in exploration geophysics, usually Tz is very smallin magnitude and even not possible to obtain and therefore its contribution may be droppedfrom (23). In fact, by reducing the step length in depth, the contribution of Tz will diminish, soits neglect will not cause significant errors, which will be demonstrated later by numericalexamples. The nth term for the correction factor in Cu (23) is
z z z z z
C G T G T x x
G T x x G T x x
1 1 ,
, , ,
. 25
nu n d
un n d
u n n
du n n
du
n n
1
1 2 1 0
1 1 0
( ) ( )( ) ( )
( ) ( ) ( )
( ) ( )
( )
= - = -
´ ¼
= > > ¼ > >
-
- -
-
Above matrix operation in (23) and (24) can be summarized as
C x C x x G T
C x C x x G T
C x C x x G T
, 1 ,
, 1 ,
, 1 26
u u
n
n du
n
d d
n
n ud
n
z z
n
n zz
n
1
1
1
( )
( )
( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
⎡⎣⎢
⎤⎦⎥
⎡⎣⎢
⎤⎦⎥
⎡⎣⎢
⎤⎦⎥
å
å
å
= = -
= = -
= = -
=
¥
=
¥
=
¥
Inverse Problems 31 (2015) 115004 R-S Wu et al
9
and (23) becomes
V kx x x TP T I C C C, Diag Diag . 27u d zf 2 { }{ }ˆ ( ) ˆ ( ) ( )⎡⎣ ⎤⎦e= = = + + +-
4. ITSP for the calculation of inverse sensitivity kernel in smooth media
The direct calculation of the series (23) is both instable and computationally intensive.Following the thin-slab formulism for forward modeling (Wu 1994 2003, Wu et al2007, 2012), which is a renormalization procedure for the forward-scattering perturbationseries, we derive the thin-slab formulation for the ISS (23), and call it the (inverse TSP,or ITSP).
4.1. TSP in T-matrix formalism
For an arbitrary heterogeneous media of a large volume, we can slice the volume intonumerous thin-slabs transversal to the propagation direction (preferred direction), here the z-direction. Within each thin-slab, the Born approximation holds. Assuming the smoothness ofthe perturbations, we concern only the transmission operator. For forward modeling, the thin-slab transmission operator lt is
V G G V G1 , 28l l l l l lf 0 0 f 0( ) ( )t = + = +
where Gl0 is the background propagation operator (thin-slab background Green’s function) for
the current thin-slab, and Vlf is the lth thin-slab forward-scattering potential. The above
operator is a local Born approximation for the local scattering source V .lf If the slab is thin
enough, the interactions between local scattering sources can be neglected. For stability oflong range propagation, the local Born scattering can be reformed into a unitary operator(Wu 2003). It is proved that the TSP in the form of sequential thin-slab transmission operatoris the renormalization to the multiple forward scattering series. Therefore it holds (Wu 2003,Wu et al 2012)
P x x1 V G
V G,1
, 29F Nl
N
ln
N n0
1f
0 0
f0( )( ) ( ) åt= =
-~
= =
¥
which converges even when V G 1.f0 > Here ‘∼’ means asymptotic equivalence when
N→∝. The TSP avoids divergence of the accumulative forward scattering by partialsummation at each marching step. TSP is used in the calculation of total wavefield fromincident field
p z P z z p z, , 30N F N 0 0 0( ) ( ) ( ) ( )=
where p zN( ) is the total wavefield along the thin-slab at depth zN and p0(z0) is the incidentfield at depth z0. From the T-matrix definition, we see it can be calculated by TSP as
P z z g g z z z zT x x V x x x x x x, , , , ; , . 31j i F j i j i i s j s N j i0 0 0( ) ( ) ( ) ( ) ( ) ( ) = >/
Note that in T-matrix formulation, the total field is normalized with the incident field, sothe equivalent source at each point is excited with a unit-amplitude and zero-phase wave. Thestrength of the equivalent source is proportional to the scattering potential.
Inverse Problems 31 (2015) 115004 R-S Wu et al
10
4.2. Inverse thin-slab propagator
Following the derivation of the forward TSP of the De Wolf approximation, we can formulatethe ITSP for the De Wolf approximation of the inverse series. From (27), we have
V
V V V
x x TP T TC TC TC
T
, Diag Diag Diag Diag Diag
Diag .
32
u d z
u d z
f { }{ } { }{ }ˆ ( ) { }
{ }( )
d d d
= = + + +
= + + +
-
First, we recover V .ud We know
TC T G T TG T TG T G T1 1 ,
33
u
n
n du
n du
du
du
1
2( )( ) ( )
( )
⎡⎣⎢
⎤⎦⎥å= - = - + - + ¼
=
¥
To write the kernel operation in the form of integration or summation, we derive the up-scattering correction (upgoing wave inverse scattering) for the mth slab
V m T m m T m i C m i m NTCDiag , , , , 1, , , 34uu
i
m
uu
1
1
{ }( ) ( ) ( ) ( ) ( )⎧⎨⎩
⎫⎬⎭åd = = = ¼=
-
where the kernel
C m i C z z
T m i T z zi m m N
x x
x x
, , ; ; ,
, , ; , ,; 1,... , . 35
u um i
u u m i
( )( )
( )( )
( )¢ ¢
¢ ¢< =
At the entrance of the first slab (m=1), we set:
V 1 0, 36u ( ) ( )d =
and for the mth layer V m ,u ( )d a TSP calculation can be formulated as
V m T m m G m m T m m
G i i T i m
T m m G m m T m m i
, , 1 1,
1 1, ,
, , 1 1, ,
37
ud
u
i m
du
du
i muI
2, 1
1
2, 1
1
{( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( )( )
⎪
⎪ ⎪
⎪
⎪ ⎪
⎡⎣ ⎤⎦⎫⎬⎭
⎧⎨⎩
⎫⎬⎭
d =- - -
´ - +
=- - -
= - -
= - -
where iuI ( )t is the inverse thin-slab transmission operator for the ith slab
i G i i T i m1 1, , , 38uI d
u( ) ( ) ( ) ( )t = - +
with known T-matrix T i m i m, ,u ( ) < .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 renorma-lization serves to remove the divergence of the ISS.
Figure 3 shows a schematic cartoon for ITSP. The red-cross in the nth slab represents thepoint for which we try to recover its velocity perturbation. The real-part of the T-matrix (at thecentral frequency f=20 Hz) for the point is plotted in the cartoon: red color for positivevalues, and blue colors for negative values. As we said, to recover the velocity perturbation at
Inverse Problems 31 (2015) 115004 R-S Wu et al
11
the given point, we need its full T-matrix, which values spread to the whole volume ofperturbed medium.
In the same way, we derive the down-scattering correction (down-going wave inversescattering) with the help of ITSP of the down-going waves
V m T m m G m m T m m i
i G i i T i m i m
, , 1 1,
1 , 1 , , 39
du
di m
N
dI
dI u
d
2, 1
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
⎪ ⎪
⎪ ⎪
⎧⎨⎩
⎫⎬⎭d t
t
= - + +
= - + >
= + +
with known T-matrix T i m m i N, ,d ( ) < .In addition, we need the in-slab correction
V C m m T m m, , . 40zz
z( ) ( ) ( )d = -
However, the correction using Tz(m, m) needs some regularization method and in ourcurrent version of inversion we neglect the contribution of Tz, whose magnitude is propor-tional to the slab thickness. For strong-contrast media, we reduce the step-size accordingly tokeep the accuracy. We will show details later in the numerical tests.
The difference between the forward TSP and ITSP is as follows. The TSP is a scatteringgenerator, 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, a delta-like velocity perturbations. Therefore, to recover theV at one level m, we need all the T-matrix values of Tu(i, m), Td(i, m) and Tz(m, m).
The inverse propagator is still a stepwise forward marching algorithm. However, unlikethe forward scattering propagator where V is a diagonal matrix, here T is a full matrix and theinteraction with the propagator cannot be totally localized.
5. Numerical tests on the ITSP for a transmission tomography problem
In the following, we show some simple examples to demonstrate the convergence property ofthe renormalized inverse T-series: the ITSP. The first test model is a Gaussian ball (a=5λ)with different perturbation strengths from the homogeneous background (figure 4). Thematrix T is a complex-value frequency-dependent matrix of N by N. The whole model space
Figure 3. Schematic cartoon for ITSP (inverse thin-slab propagator). The red-cross inthe nth slab represents the point for which we try to recover its velocityperturbation.
Inverse Problems 31 (2015) 115004 R-S Wu et al
12
is of 200 by 200 in grid size. The perturbation area is about 51 by 51, and N is about 1900.We produce the T-matrix by the matrix inverse method for the given V-matrix (equation (9)).Since T-matrix is defined in the frequency domain, we performed our numerical tests all infrequency domain for f0=20 Hz. In figure 5 we plot the sparsely sampled column vectors ofthe full T-matrix (kernel representation) (f0=20 Hz). In the figure, each small ball is arepresentation of a T-kernel, corresponding to a point spreading function (only real part isshown). On the right side of figure 5 are the two kernels we used in the tests to recover the V(x) at the two corresponding points: one is at the center of the Gaussian ball; the other is forthe point on the right, away from the center of the Gaussian ball.
This exact T-matrix corresponds to a full-aperture measurement and contains all theinformation in full-aperture acquired data. Normally, T-matrix is derived by a linear inversionfrom the data with limited aperture (such as by equation (12). In order to test the convergenceof the ISS, we use the exact T-matrix here. The influence of data aperture will be studied inthe future work.
Figures 6–8 give some results of convergence comparison between the IBS and ourmethod using ITSP. For comparison we only plot the convergence curves of velocity per-turbation value at some fixed points of the model. In figures 6(a) and (b) we plot theconvergence curves from the IBS using the full-T and the T-matrix with missing data (lateralwaves), corresponding to the neglect of self-interaction within a thin-slab. The vertical axis isthe perturbation strength εν, and the horizontal axis is the series summation orders. We seehigh oscillation nature of the IBS in relatively weak scattering (15% perturbation witha=5λ). Note that the full inverse Born T-matrix series converges to a correct value aftermany terms. However, with the missing data (neglect of Tz) , it converges to a wrong value. Infigure 6(c) is the result from the ITSP. For the ITSP calculation, the corrections by Pu
- and Pd-
are computed separately from the top (the curve on the left) and the bottom (the curve on theright), respectively (the horizontal axis is labeled with the slab ordering numbers). Due to thestepwise renormalization of ITSP, the convergence is almost monotonically. Although thereis a minor error due the neglection of Tz compared with the result using the full T-matrix, butthe error is smaller than the result of IBS using the incomplete T-matrix. For 20% perturbation(figure 7), we test the recovery for the point 2. We see that the IBS starts to diverge (on theleft of figure 7), but ITSP (on the right) has a similar convergence as the case of weak
Figure 4. The model of Gaussian ball (a=5λ) with different perturbation strengths(15%, 20%, 50% and 100%).
Inverse Problems 31 (2015) 115004 R-S Wu et al
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Figure 5. T-matrix in the kernel representation. Each small ball is a kernel T x x,( )¢ for afixed x (a point spreading function). On the right are the two kernels we used for the V(x) recovery tests: one is at the cente of the Gaussian ball; The other is for the point onthe right, away from the center of the Gaussian ball.
Figure 6. Convergence comparison for point 1 (the center) in the case of weakperturbation (15%) between inverse Born series (IBS) using full T-matrix (a), IBSwithout Tz (b), and the inverse thin-slab propagator (c). The pink dashed lines are thecorrect value of velocity perturbation εν.
Inverse Problems 31 (2015) 115004 R-S Wu et al
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scattering. In order to show the detailed behavior, we plot the zoomed figure in figure 7(d) byamplifying the vertical scale. In the same manner, we plot the comparison for the case ofstrong perturbation (50%) in figure 8 (for point 1). We see the fast divergence of IBS, but asimilar convergence curve for the ITSP. In principle, the renormalization procedure of ITSPcan remove the divergence for any strong perturbations.
In figure 9 we plot the recovered velocity distributions and theirs relative errors for thewhole model (perturbations of 15%, 25% and 50%) inverted by our ITSP. We see that theITSP overcomes the divergence problem of IBS for all the cases and can obtain the correctresults in one sweep (two half-sweeps: one from the top, the other from the bottom). It is anaccurate and efficient analytical nonlinear inversion without divergence for this ideal situation(full aperture acquisition and exact reconstruction of T-matrix in the image space). Figure 10shows another inversion result for a double-ellipses model which is composed by two closelylocated smooth anomalies with positive and negative perturbations (perturbation of ±20%and ±50%). The model size is same as the Gaussian ball model, but the perturbation area is101×31. The frequency used is 20 Hz. The T-matrix data are generated in the same way asthe previous example. Similar to the case of Gaussian ball anomalies, the inversion resultshave high accuracy and no divergence.
Now, we discuss the accuracy issue of neglecting the self-interaction of thin-slabs, i.e.neglecting the contribution of Tz. From the definition (4), we have
Figure 7. Convergence comparison for 20% perturbation (at point 2) between inverseBorn series (IBS) using full T-matrix (a), IBS without Tz (b), and the inverse thin-slabpropagator (c). To see the details we plot the zoomed figure in (d) by amplifying thevertical scale. The pink dashed lines are the correct value of velocity.
Inverse Problems 31 (2015) 115004 R-S Wu et al
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T V I G V . 41z z z z( ) ( )= +
We see that Tz depends on scattering potential of the thin-slab Vz and the scattering effectinside the thin-slab. Therefore, the magnitude of Tz is proportional to the product of per-turbation strength and slab-thickness Δz. By reducing Δz, the value of Tz shrinks, so itscontribution will decrease to negligible. For stronger perturbations, if we shrink the step-sizeΔz proportionally, we may keep the accuracy of inversion to be nearly constant. Figure 11shows the deterioration of accuracy if we keep the step-size Δz constant but increase the-perturbation to 75%–100%. We see that for high contrast, e.g. 100% perturbation, theinversion result is not satisfactory, since the interior of the ball (close to it center) is not wellrecovered. However, when we reduce the step-size Δz into half (from original 10 to 5 m), therecovery of velocity distributions become acceptable or comparable to the weak perturbationcases (figure 12).
6. Conclusion and discussions
Based on the NLSO for nonlinear tomographic waveform inversion, we apply an operatorsplit and the forward-scattering renormalization procedure to the forward and inverse T-matrix scattering series. Renormalization of the ISS (in T-matrix form) leads to an ITSP whicheliminates the divergence of the IBS for strong perturbations. Numerical tests proved that the
Figure 8. Convergence tests (at point 1) for strong perturbations (50%). Otherparameters are the same as in figure 6.
Inverse Problems 31 (2015) 115004 R-S Wu et al
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renormalized ISS realized by ITSP has no divergence for any strong perturbations in our testcases. The ITSP is also an accurate and efficient method with only one sweep (no iterationinvolved). Although the geometry of our derivation is for the transmission tomography insmoothly heterogeneous media, this convergence and efficiency improvement may have greatpotential for applications to the general case of FWI.
Figure 9. The recovered velocity distributions and theirs relative errors for theperturbations of 15% (top), 25% (mid), and 50% (bottom) inverted by our ITSP.
Figure 10. The recovered velocity distributions and theirs relative errors for theperturbations of (a) ±20% and (b) ±50% inverted by our ITSP for the double-ellipsesanomalies model.
Inverse Problems 31 (2015) 115004 R-S Wu et al
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The T-matrix data we used correspond to a full-aperture acquisition. The inversionprocedure for the T-matrix from scattering data is a research topic for future study. Also theeffect of imperfect acquisition aperture needs to be investigated. To extend our theory andmethod to the general case including reflections is another major task for future research.Reviewing the development of renormalization theory in quantum field theory and statisticalphysics, we can see that different filtering techniques could be explored to decompose the T-matrix into a forward-scattering part and a backscattering part in the case of complex mediainvolving reflections. Once the decomposition of T-matrix is realized, the ITSP could play acritical role in removing the divergence of the ISS and therefore the divergence of the FWIprocess.
Acknowledgments
We thank the discussions and helps from Yingcai Zheng and Lingling Ye. We alsoacknowledge the helpful discussions with M Jakobsen, T Lay, C Mosher, J Gao, X Xie, BUrsin, and G Newman. The comments from the reviewers, especially the relevant referencesprovided by one reviewer are greatly appreciated. The research is supported by the WTOPIResearch Consortium of Modeling and Imaging Laboratory, University of California,Santa Cruz.
Figure 11. The recovered velocity distributions and theirs relative errors for theperturbations of 75% (top), 85% (mid), and 100% (bottom) inverted byour ITSP.
Inverse Problems 31 (2015) 115004 R-S Wu et al
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