Internal multiple elimination by an extended 3D single

82nd EAGE Annual Conference & Exhibition

Internal multiple elimination by an extended 3D single-sided autofocusing and the application to subsalt imaging Introduction Recently, Zhang et al. (2018, 2019) introduced a method of internal multiple elimination based on the theory of Marchenko redatuming. This is an important step forward since the procedure does not need the information of first-arrival time or the smooth velocity structure for the multiple elimination process. Nevertheless, since the derivation is based on Marchenko redatuming, it should suffer the same restriction as the Marchenko redatuming, which is valid only for layered media with moderately curved interface and may have some severe restrictions when applying to irregular, strong-scattering structures. Rose et al. (2001, 2002) established the link of the kernel functions of GL (Gel'fand-Levitan) equation and Marchenko equation to the focusing process in the TRM (time-reversal mirror) autofocusing (Fink et al., 1989; Prada et al., 1991, 2002) and termed the method as “single-sided autofocusing”. Rose’s derivation is based on some basic physical principles, such as causality, linear superposition and some symmetry properties and is only for 1-D media, i.e. layered media. It is appealing to generalize the concept a to the 3D case. However, to establish a theory and method of 3D autofocusing faces substantial difficulties. Rose et al (1984, 1985) derived a time-domain three-dimensional Marchenko equation based on R. Newton’s frequency-domain formulations (Newton, 1989). In their derivation, a 3600 full acquisition aperture is needed for the exact solution. In 3D single-sided autofocusing the full-aperture requirement cannot be satisfied. In this expanded abstract, we propose an extended 3D single-sided autofocusing guided by inverse-scattering theory for limited-aperture scattering source imaging. This work is the continuation and further development of Wu and Gu (2020). The current status quo in standard 3D TRM is to use only one pulse source in the center plus a TRM array source. As a result, an iterative TRM process ends up to focusing only on the most reflective target point (Prada et al., 1991). In our approach, we extend the standard 3D TRM operation in two aspects. One is to extend the pulse-sources acquisition over the full array and a cumulative update for the focusing array source. The other is to apply the time-truncation to the TRM sources based on causality which is the key element (triangularity) in the GLM (Gel'fand-Levitan Marchenko) inverse-scattering theory. Theory To construct accurately the focusing source, we assume an acquisition system composed of a source-receiver array, in which each element in the array can both receive and emit waves. Assume we have a N element array, then for the array we have N N records. We first discuss the focusing source from the TRM (time-reversal mirror) principle. When we put a point pulse source 0( ) '( )t x x at 0x on the

surface, the impulsive response at 0x on the surface is 0 0( , , )R tx x , which includes both primary

reflections rR and multiples (Internal multiples) MR , i.e. r MR R R . In the derivation, we assume the

acquisition surface is a reflection-free surface on top of a heterogeneous half-space. From all the data recorded on the surface for a given point-source at 0x , we can construct an array source, the “focusing

source” fS composed of a δ-source (point-impulsive source) and a coda-source (the array source) cS ,

0 0 0 0 0( , , ) ( ) '( ) ( , , )f cS t t S t x x x x x x (1)

The focusing coda source radiates convergent waves and focus the scattered waves to subsurface reflectors at local time zero (Rose, 2001). For the δ-source, the received seismograms at any receiver point, 0"x equates the original seismograms 0 0( " , , )R tx x ; while the scattered waves due to the focusing

array source will be equal to the multiples with negative sign, which cancel out the original multiples, including transmission compensation. The resulted seismograms are single-scattered waves only! Here “single-scattered waves” mean single-reflected waves from local reflectors without transmission loss. Now we show an iterative process to construct the focusing source and the associated single-reflection retrieval procedure. In the first iteration we use the time-truncated TRM of the original impulse response across the array as the first-order focusing array source,


(1)0 0 0 0 0 0 0 0( , , ) ( , , ), ,t U

cS t R t x x x x x x D (2)

where 0x and 0x are the source and receiver points on the acquisition surface, and 0t UR is the time-

truncated received records on the surface which are composed only by upgoing waves, 0t is the

truncation operator which truncated the record at time t. The coda array source (time-reversed and truncated) in (2) will radiate convergent wavefronts which automatically focus at each scattering points and then be re-scattered back to the receiver array on the surface. The scattering-focusing-rescattering process is all realized by the physical process. Propagation and backpropagation (reverse-time propagation) are all happened in the real media. At the focusing points, the re-scattering effects are much stronger than that at other points. Therefore, we can consider the re-scattering, including the one along multiple-scattering paths, mainly happens at the original scattering points. In this way, the re-scattering will increase the orders of multiple scattering (reflection). Let us first look at the scattering process. We consider only the strong-scattering case, i.e. the boundary scattering for IME. Point scattering is very weak (by orders) compared to boundary scattering in forming internal multiples and will be neglected in our treatment. The original scattered waves from boundary scattering received at receiver 0x at time t by a delta-source at 0x can be formulated in this case under

the small incident-angle approximation as (Wu and Chen, 2018; Wu, 2020)

0 0 0 0 0 0 0 0 0( ) ( )ˆ( , , ) 2 ( ) ( , , ) ( , ) ( , , ) 2 ( , , ) ( , , ), U

i i i i i n i e iV t V tR t ds x G t n G t dx G t s t

n

x x x x x x x x x x x x D (3)

where ( )ids x is the boundary element of the reflecting boundary located at ix , G is the full Green's

function, and 0G is the Green's function for the smooth background without scattering potentials, V(t)

is the volume truncated by a curved surface ( )t which is the envelope surface defined by the first-arrival time. ˆ( , )i n x is the local reflection coefficient of the boundary element with dip n̂ ,

0 0 0ˆ( , , ) ( , ) ( , , )e i i is t n G tx x x x x is the equivalent scattering source of a scattering element at ix generated by

a delta-source at 0x and “ ” represents the time-convolution. The time-truncation is based on the

causality principle. Here we use a linear boundary element approximation in above equation. Boundary element scattering has a dipole scattering pattern. If the element is horizontal, then there is no energy loss in the acquisition surface. However, for a dip boundary element, some radiation cannot reach the acquisition surface. Nevertheless, if the aperture-limited radiation energy can be correctly recovered at the imaging point, then it is a faithful imaging, although not a true-reflection imaging. For IME, if we can recover the aperture-limited multiples which are accurately match the same aperture-limited multiple in the data, then an exact IME can be achieved. In this way, the limited aperture for reflectors of complex structures will not impede solving the IME problem. To the contrast, any task involving the Green's function retrieval will face fundamental difficulty due to the limited aperture problem. The extended autofocusing procedure avoid the need of Green's function retrieval. By the TRM principle, the time-truncated TRM operator can change the original equivalent source into an equivalent source of first order estimate at any specified boundary element ix ,

0

0

(1) *0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

ˆ( , , ) {[ ( , , )] ( , ) ( , , )] ( , , )}( )

{ ( ) ( , ; ) ( , , )}( ) { }( , , )

t Ue i i i i

t U t U t t Ui i

s t d G t n G t R t t

d R R t t R t

D

D

x x x x x x x x x x

x x x x x x R x x (4)

where G and 0G are the time-reversed G and 0G , R stands for the time-space correlation operator. In

the above derivation, we modify the TRM focusing operator by adding a factor 0G , which is the time-

reversal of the background Green’s function, to make the local scattering source radiation starting at time zero. In 1D problem, it is only a time-shift; in the 3D case, it has a geometric spreading. However, in the sub sequential stage the spreading can be compensated by a source-side focusing. We recognized that the term inside the square-brackets in (4) is the time-reversed reflection records 0 0( , ; )U

iR x x x by a

boundary element at ix , so that the TRM focusing operation can be implemented by a time-space

correlation integral with its reflection record as the kernel. After the scattered waves are refocused to the local scatterers, the waves will be re-scattered back to the surface. Any receiver in the receiver array, e.g. the receiver at 0x , will record the re-scattered waves. The received seismogram is the sum of


reflected waves from all the local reflectors, and can be formulated by a convolution integral in the data domain as follows,

0 0

(1) (1) (1)0 0 0 0 0 0 0 0( ) ( )

( , , ) ( , , ) ( , , ) ( , , ) ( , , ; ) ( , , )t UM i r i i r e i r i r i e iV t V t

R t d d G t G t s t d d R t s t

D D

x x x x x x x x x x x x x x x x x (5)

where rx is the position of any receiver in the array receiver reflected waves excited by the point source

at 0x . By reciprocity, we know that 0 0( , , ; ) ( , , ; )U Ur i r iR t R t x x x x x x . In (5), “ t ” is the shifted truncated

time to exclude the reflection pulse which is already included in the equivalent source (1)0( , , )e is tx x . In

this way, the data 0 0 0( , ' , )t UR t x x in(5) is used only as propagation operator. Inserting (4) into (5) we

have an operator form as (1)

0 0 0 0 0 0 0 0 0( , , ) { }( , , ), t t t UMR t R t x x R R x x x D (6)

where R is the time-space convolution operator and 0t is the time-shifted truncation operator to avoid

the “double count”. This has been discussed in detail by Zhang et al., (2019). To improve the accuracy of the multiple estimate, we use an iterative process of cumulative addition

*0 0 0

1 1

( )t t n t UM n

n n

R R R

R R . (7)

MR in above equation is a series summation. The nth term is

*0 0 0 0 0 0 0( , , ) [( ) ]( , , ), ,t t n t U

n n n nR t R t x x R R x x x x D . (8)

In contrast, the standard TRM’s iterative process only uses the nth term of the series, which is composed of multiply reflected/scattered waves. That is why the iterative TRM can only or purposely focus on the strongest scatterer. The resulted equation (7) is same as in Zhang et al. (2019). We rederived it by the extended 3D autofocusing. In the Marchenko multiple elimination (MME) method, the data-updating process is realized by a downward redatuming to the subsurface and a subsequent upward redatuming to the surface. In our derivation, boundary scattering is the mechanism to connect the downward focusing of surface data for updating the incident flied and the upward propagation of the re-scattered waves. Therefore, the use of one-way decomposition and redatuming plane is avoided. As a result, the new derivation implies a broader range of applications, such as the removal of salt multiples in complex structures. Example We test the method with two 2D SEG salt models. The first model has a less irregular salt body. Therefore, redatuming planes are available to retrieve the Green's functions and focusing functions in the subsalt areas far from the salt bottom. The condition for standard Marchenko redatuming and imaging is valid in this case. We show the satisfied imaging results for the fault far from the curved salt bottom in Figure 1(c); However, in the area adjacent to the irregular salt bottom, the fault images are severely distorted and false image and artifacts exist. In comparison, the migrated image using the primary-only data after IME gives good result (Figure 1d), in which the false images due to salt multiples in the standard RTM imaging (see Figure 1b) are effectively removed. The second Model has rugged top salt surface and a large tail intersecting the lower boundary of the model. This large-scale and highly irregular shaped salt body presents a challenge to the standard Marchenko imaging. It is a good model to test our theory and method. Figure 2(a) is the original model and Figure 2(b) is the migration image with original data. Using the retrieved multiple-free data after 25 iterations as input for a conventional reverse-time FD migration, we obtain Figure 2(c) which shows much less artifacts than the image using the original data. Figure 2(d) shows the image difference between Figure 2(b) and 2(c). We see that the image difference between using the original data and the new data after IME in the subsalt region is composed mainly of false images due to salt internal multiples. Conclusions We rederive the internal multiple elimination by an extended 3D single-sided autofocusing method guided by inverse-scattering theory. In our derivation we avoid the use of redatuming plane and the up- and down-going decomposition. The 3D extended autofocusing is obtained by a cumulative TRM (time-


reversal mirror) process with time-truncation operator. In the process, focusing, scattering/reflection from the complex structures and propagation back to receivers, are all realized by physical process in real media without mathematical approximations. The new derivation implies a much broader range of applications to internal multiple removal as demonstrated by the SEG salt model example. Acknowledgements This work was supported by WTOPI (Wavelet Transform On Propagation and Imaging for seismic exploration) Research Consortium and other funding resources at the Modeling and Imaging Laboratory, University of California, Santa Cruz.

Figure 1 The (a) selected region of SEG salt model 1; (b) migrated image using the original data; (c) result of standard Marchenko imaging; (d) migrated image using the retrieved primary-only data.

Figure 2 The (a) SEG salt model 2; (b) Migrated image with original data, (c) Image with data after internal multiple elimination (25 iterations), (d) Image difference between (c) and (b). References Chadan, K. and P.C. Sabatier. [1989] Inverse problems in Quantum scattering theory, second edition”, Springer-Verlag. Fink, M., Prada, C., Wu, F., Cassereau, D. [1989] Self-focusing with time reversal acoustic mirrors. In Proc. IEEE Ultrason. Symp., vol. 2, B. McAvoy, ed., IEEE, New York (1989) p.681. Newton, R.G. [1989] Inverse Schrodinger scattering in three dimensions, Springer-Verlag. Prada, C., Wu, F., and Fink, M. [1991] The iterative time reversal mirror: A solution to self-focusing in the pulse echo mode. The Journal of the Acoustical Society of America, 90(2), 1119-1129. Prada, C., Kerbrat, E., Cassereau, D., and Fink, M. [2002] Time reversal techniques in ultrasonic nondestructive testing of scattering media. Inverse problems, 18(6), 1761. Rose, J. H., Cheney, M., and DeFacio, B. [1984] The connection between time-and frequency-domain three-dimensional inverse scattering methods. Journal of mathematical physics, 25(10), 2995-3000. Rose, J. H., Cheney, M., and DeFacio, B. [1985] Three-dimensional inverse scattering: Plasma and variable velocity wave equations. Journal of mathematical physics, 26(11), 2803-2813. Rose, J. H. [2001] Single-sided focusing of the time-dependent Schrödinger equation. Physical Review A, 65(1), 012707. Wu, R. S., and Chen, G. X. [2018] Multi-scale seismic envelope inversion using a direct envelope Fréchet derivative for strong-nonlinear full waveform inversion. arXiv preprint arXiv:1808.05275. Wu, R. S. [2020] Towards a theoretical background for strong-scattering inversion-Direct envelope inversion and Gel’fand-Levitan-Marchenko theory: Comm in Comput Physics. 28, 41-73. Wu, R. S., and Gu, Z. [2020] Focusing source seismograms and Marchenko multiple elimination applied to salt multiples removal. In: SEG Technical Program Expanded Abstracts 2020. Society of Exploration Geophysicists, 3587-3592. Zhang, L., Slob, E., van der Neut, J., and Wapenaar, K. [2018] Artifact-free reverse time migration. Geophysics, 83(5), A65-A68. Zhang, L., Thorbecke, J., Wapenaar, K., and Slob, E. [2019] Transmission compensated primary reflection retrieval in the data domain and consequences for imaging. Geophysics, 84(4), Q27-Q36.

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Internal multiple elimination by an extended 3D single