CWP-716

Increasing illumination and sensitivity of reverse-timemigration with internal multiples

Clement FleuryCenter for Wave Phenomena, Colorado School of Mines

ABSTRACTReverse-time migration is a two-way time-domain finite-frequency techniquethat accurately handles the propagation of complex scattered waves and pro-duces a band-limited representation of the subsurface structure that is conven-tionally assumed to be linear in the model parameters. Because of this underly-ing linear single-scattering assumption, most implementations of this method donot satisfy energy conservation and do not optimally use illumination and modelsensitivity of multiply scattered waves. Migrating multiply scattered waves re-quires preserving the nonlinear relation between image and model parameters.I modify the extrapolation of source and receiver wavefields to more accuratelyhandle multiply scattered waves. I extend the concept of the imaging condi-tion in order to map into the subsurface structurally coherent seismic eventsthat correspond to the interaction of both singly and multiply scattered waves.This results in an imaging process here referred to as nonlinear reverse-timemigration. It includes a strategy that analyzes separated contributions of singlyand multiply scattered waves to the final nonlinear image. The goal is to pro-vide a tool suitable for seismic interpretation and potentially migration velocityanalysis that benefits from increased illumination and sensitivity from multiplyscattered seismic waves. It is noteworthy that this method applies to migratinginternal multiples, a clear advantage for imaging challenging complex subsur-face features, e.g., in salt and basalt environments. The results of syntheticseismic imaging experiments, one of which includes a subsalt imaging example,illustrate the technique.

Key words: Imaging, Interpretation, Seismics, Velocity analysis, Multiples

1 INTRODUCTION

Complex subsurface structures, such as sub-basalt andsubsalt structures, generate strong scattering whichmakes them challenging to image with conventionaltechniques (e.g., Martini and Bean, 2002; Leveille et al.,2011). For such geologic environments, multiply scat-tered waves, such as internal and surface multiples,contain useful information about the subsurface butthese waves are traditionally suppressed, e.g., with mul-tiple suppression (Foster and Mosher, 1992; ten Kroode,2002) or surface-related multiple elimination (Verschuuret al., 1992; Dragoset et al., 2010). Alternatively, multi-ply scattered waves are usually just ignored, e.g., withmigration of primaries (Claerbout, 1985; Esmersoy andOristaglio, 1988), least-square migration (Nemeth et al.,1999) or linearized inversion (Symes, 2008a), when lin-earizing the relation between model and data during

the imaging/inversion process. The motivations for us-ing multiply scattered waves and their nonlinear relationto the model are to preserve amplitudes, provide extraillumination, account for energy conservation, and in-crease redundancy and sensitivity to model parameters.

Reverse-time migration (RTM) is now a standardimaging technique in the industry (Baysal et al., 1983;McMechan, 1989). The method provides high-qualityimages for oil and gas exploration and is of special inter-est for complex subsurface geologies (e.g., Farmer et al.,2006; Leveille et al., 2011). Yet, conventional RTM mi-gration does not resolve the problem of imaging wavesmultiply scattered by complex subsurface structures.The underlying single-scattering assumption of mostmigration techniques does not account for the funda-mental nonlinear relation between model and multiplyscattered data (Fleury and Vasconcelos, 2012). To uti-lize the energy, illumination and sensitivity contained

158 Clement Fleury

in multiply scattered data, Fleury and Snieder (2011,2012) have proposed a nonlinear RTM migration algo-rithm has been proposed. In this paper, I expand thedescription of this technique and define a clear strategyfor using multiply scattered waves in seismic interpre-tation and migration velocity analysis.

Advances have been made in using multiples to per-form imaging (Brown and Guitton, 2005; Jiang et al.,2007; Malcolm et al., 2009; Verschuur and Berkhout,2011). These methods apply redatuming techniques(Schuster et al., 2004; Malcolm and de Hoop, 2005;Berkhout and Verschuur, 2006) for extrapolating seis-mic data and reconstructing scattered wavefields withkinematically correct multiples, but they do not mod-ify the imaging condition to account for the nonlin-earity of the resulting wavefields with respect to themodel. As with these methods, the method presentedin this document takes into account multiply scatteredwaves in the extrapolation procedure. It takes advan-tage of computationally affordable reverse-time migra-tion engines to handle complex scattering wave phenom-ena and adapts the extrapolation procedure to includesources of scattering in the migration velocity model,which improves the reconstruction of scattered waves(Vasconcelos et al., 2010). Unlike the previous meth-ods, the method presented here also extends the con-cept of the imaging condition in order to map into thesubsurface structurally coherent seismic events that cor-respond to reflections of both singly and multiply scat-tered waves. In that sense, this method is in agreementwith Verschuur and Berkhout (2011) who call for a newimaging principle, defined as a minimization problem intheir analysis. Fully exploiting multiple-scattering in-teractions results in images of multiples as well as pri-maries. The imaging condition here is nonlinear in themodel parameters (Fleury and Vasconcelos, 2012) andproduces four sub-images that I analyze as separate con-tributions to the final nonlinear image. This strategyleads to a new tool suitable for seismic interpretationand with the potential to improve migration velocityanalysis.

First, I present the fundamentals of my nonlin-ear reverse-time migration (NLRTM) method and thenshow how this method leads to increased illuminationand sensitivity for reverse-time migration with multiplyscattered waves. Finally, we see the efficacy of this newseismic interpretation tool with application to syntheticdata, including a subsalt imaging example.

2 THEORY OF NONLINEARREVERSE-TIME MIGRATION

2.1 Wavefield extrapolation

The division of Earth properties into subdomains ofshort and long wavelengths justifies the use of a smoothmodel m0 perturbed by a rough model ∆m in seismic

migration (Jannane et al., 1989). In the acoustic as-sumption, the model is given by m = m0+∆m = (ρ, c),where ρ and c are density and velocity, respectively.Model m0 provides a kinematically correct wavefield w0

and is used as a conventional migration velocity model.Model perturbation ∆m acts as a scattering source forthe scattered wavefield ws superimposed on referencewavefield w0. Model ∆m is the part of the Earth modelthat one tries to represent with the seismic image i.

For each shot of an active seismic survey, referencewavefield w0 is obtained by propagating the estimatedsource signature s in migration model m0:

H0 ·w0 = s, (1)

where the acoustic wave operator is given by H0 =

ρ0∇ · (1

ρ0∇) +

1

c20

∂2

∂t2. The dot symbol · denotes the

tensor contraction between matrix operators and vec-tor fields throughout this paper. Because model ∆mis unknown in the imaging process, scattered wavefieldws is not obtained by forward modeling but approxi-mated by the receiver-side extrapolated wavefield ws,rec

which is conventionally obtained by backpropagatingthe recorded scattered data ds in background modelm0,

H†0 ·ws,rec = ds, (2)

where the adjoint acoustic wave operator is given by

H†0 =

1

ρ0∇ · (ρ0∇) +

1

c20

∂2

∂t2.

The perturbation of operator H0 describing scat-tering caused by model ∆m is denoted by the operatorP and is a function of models m0 and ∆m (e.g., Kato,1995). The perturbed operator is defined as H0 − P.The relationship between wavefield ws (equal to thescattered seismic data ds at the receiver locations) andperturbation P is nonlinear (e.g., Weglein et al., 2003):

ws =1

s(w0 ·P ·w0) +

1

s2(w0 ·P ·w0 ·P ·w0) + ... (3)

This nonlinear relationship between wavefield and per-turbation fundamentally explains multiply scatteredwaves, i.e, waves that have been reflected, diffracted,and more generally scattered more than once, such asinternal multiples. To preserve the nonlinear relation be-tween wavefield and perturbation, I modify conventionalRTM extrapolation. Given an estimate Pest of the per-turbation operation, the scattered wavefield ws,rec isextrapolated with a more exact equation than equation(2) (Vasconcelos et al., 2010):

(H†0 −P†

est) ·ws,rec = ds + P†est ·w0. (4)

This is equivalent to including scattering sources in theextrapolation of scattered wavefield ws,rec and results inbetter reconstruction of scattered waves. Additionally,let us introduce the source-side extrapolated scattered

Nonlinear reverse-time migration 159

wavefield ws,sou defined in the following relationship.

(H0 −Pest) ·ws,sou = Pest ·w0. (5)

Wavefield ws,sou is originally missing in conventionalRTM migration. In the conventional extrapolation pro-cedure, this wavefield would be zero.

The introduction of operator Pest in the extrapo-lation allows us to account for scattering contrasts inreconstructing scattered wavefields ws,sou and ws,rec.For acoustic media, operator Pest is defined as

Pest = ρ0∇ · (1

ρ0∇)− (ρ0 + ∆ρest)∇ · (

1

ρ0 + ∆ρest∇)

+ (1

c20

− 1

(c0 + ∆cest)2)∂2

∂t2, (6)

where model ∆mest is an estimate of the true scatter-ing contrast model ∆m. There are two main approachesto retrieving model ∆mest. An estimate of model∆m might possibly be obtained in an automatic/semi-automatic algorithm-based manner by essentially solv-ing an inverse problem for model ∆m in the data do-main (e.g., Tarantola, 1984, 1986). The linearization ofthis inverse problem, written as

ds = w0 ·Pest ·w0, (7)

leads to a solution ∆mest that can be computed byleast-squares migration (Nemeth et al., 1999; Plessixand Mulder, 2004) or optimal scaling of conventionalRTM images (Rickett, 2003; Symes, 2008a). The ap-plication of gradient-based methods allows for updat-ing model ∆mest in several linear iterations (e.g., Prattet al., 1998). Therefore, the NLRTM method is fullyintegrable with full waveform inversion (FWI) technol-ogy (e.g., Tarantola, 1984; Pratt, 1999; Plessix, 2006;Virieux and Operto, 2009; Zhu et al., 2009). Alterna-tively, tools in velocity model building based on the in-put of a human interpreter could be used for creatinga model with sharp interfaces. This interpreted modeldefines scattering model ∆mest for the extrapolation ofscattered wavefields ws,sou and ws,rec. The next sectioncontains examples using both approaches.

2.2 Imaging condition

The imaging condition maps into the subsurface the in-teraction of source-side extrapolated wavefields w0 andws,sou with receiver-side extrapolated wavefield ws,rec

to create a representation of the Earth’s physical prop-erties. Following the method of Fleury and Vasconcelos(2012), let us define image i as

i =X

sources

w0 ?ws,rec| {z }i0

+1

2

Xsources

ws,sou ?ws,rec| {z }is

, (8)

where ? denotes either zero-time crosscorrelation or de-convolution. A crosscorrelation imaging condition corre-

sponds to an energy-based representation of the subsur-face and maps an estimate of “energy loss” in scattering(Fleury and Vasconcelos, 2012). A deconvolution imag-ing condition provides a reflectivity-based representa-tion and maps into the subsurface a quantity that ap-proximates reflectivity or scattering amplitude depend-ing on the particular definitions of the imaging condi-tion (e.g., de Bruin et al., 1990). In this paper, I adopta deconvolution imaging condition following the recom-mendations of Schleicher et al. (2008).

This imaging condition is nonlinear with respect toperturbation P and therefore with respect to model ∆m(Fleury and Vasconcelos, 2012), but preserves a linearrelation between image i and the data ds. To emphasizethese properties, let us separately analyze the two con-tributions, images i0 and is, to image i (properties sum-marized in Table 1). Image i0 maps the interaction ofreference wavefield w0 with scattered wavefield ws,rec.The imaging condition that defines image i0,

i0 =X

sources

w0 ?ws,rec, (9)

is similar to conventional imaging conditions for RTMmigration. The only difference from the conventionalimaging condition comes from the modified extrapola-tion of wavefield ws,rec. Image i0 is a function of per-turbation P and the data ds because of its dependencyon scattered wavefield ws,rec. Wavefield ws,rec is a lin-ear function of the data ds. As a result, image i0 islinear with respect to the data ds. Wavefield ws,rec isin general a nonlinear function of perturbation P. Onlyconventional reverse-time extrapolation under a linearsingle-scattering assumption for the scattered data ds

gives a linear estimated wavefield ws,rec with respectto perturbation P. Image i0 thus results from the lin-earization of equation (8) with respect to perturbationP under a linear single-scattering assumption. This as-sumption is, however, not always accurate enough andmore importantly limits the capability of RTM migra-tion to image complex scattering events such as thosethat involve internal multiples (e.g., Malcolm et al.,2007, 2011). Image is maps the interaction of scatteredwavefields ws,sou and ws,rec. The imaging conditionthat defines image is,

is =X

sources

ws,sou ?ws,rec, (10)

takes into account higher orders of scattering neglectedunder a linear single-scattering assumption. Through itsdependencies of scattered wavefield ws,rec, image is is afunction of the data ds, and, similar to image i0, imageis is linear with respect to the data ds. Image is is afunction of perturbation P because of its dependenciesof both scattered wavefields ws,sou and ws,rec. Wave-fields ws,sou and ws,rec are at least linear in pertur-bation P and are in general nonlinear in perturbationP. Image is is therefore at least quadratic in pertur-

160 Clement Fleury

``````````Dependency

Field Wavefield Image

w0 ws,sou ws,rec i0 is

w0 Linear None

ws,sou None Linearws,rec Linear Linear

ds None None Linear Linear Linear

P None Nonlinear

(min. linear)

Nonlinear

(min. linear)

Nonlinear

(min. linear)

Nonlinear (min.

quadratic)

Table 1. NLRTM wavefield and image dependencies: mapping linear and nonlinear relation with respect to the data ds and

perturbation P.

bation P and is in general nonlinear in perturbationP. While image is is therefore negligible under a linearsingle-scattering assumption, it is essential when imag-ing general complex scattering subsurface structures.

Years of application in industrial seismic explo-ration have shown that conventional RTM providesfairly accurate results. Apart from intrinsically prob-lematic complicated structures (such as salt bodies andbasalt formations) that break the assumption, linearsingle-scattering resolves most subsurface structures, ormore precisely, it does so in the first order of approxi-mation. In general, nonlinear scattering is not the domi-nant issue, but at the same time, it is rarely a negligibleone. To first order in the perturbation P, image i0 pri-marily contributes to image i, and image is makes asecondary contribution. The examples in the next sec-tion demonstrate that image is nonetheless provides am-plitude compensation, complementary illumination, andincreased sensitivity to the model parameters. To takeadvantage of these qualities of the NLRTM method, letus consider an analysis based on the separate and com-parative interpretation of images i0 and is. To empha-size only structural contrasts in the images, we decom-pose images i0 and is into

i0 =X

sources

(w(d)0 ?w(u)

s,rec + w(u)0 ?w(u)

s,rec

+ w(d)0 ?w(d)

s,rec + w(u)0 ?w(d)

s,rec) (11)

and

is =X

sources

(w(d)s,sou ?w(u)

s,rec + w(u)s,sou ?w(u)

s,rec

+ w(d)s,sou ?w(d)

s,rec + w(u)s,sou ?w(d)

s,rec), (12)

where superscripts (u) and (d) refer to up- and down-going wavefields (w = w(d) + w(u)), respectively, andredefine these images to keep only the interactions ofup- and down-going wavefields:

i0 =X

sources

w(d)0 ?w(u)

s,rec| {z }i(d,u)0

+X

sources

w(u)0 ?w(d)

s,rec| {z }i(u,d)0

(13)

and

is =X

sources

w(d)s,sou ?w(u)

s,rec| {z }i(d,u)s

+X

sources

w(u)s,sou ?w(d)

s,rec| {z }i(u,d)s

.

(14)

The result is the four sub-images i(d,u)0 , i

(u,d)0 , i

(d,u)s ,

and i(u,d)s that contribute to nonlinear image i. For ex-

ample, i(d,u)0 denotes the image of the interaction of

down-going reference wavefield w(d)0 and up-going scat-

tered wavefield w(u)s,rec. These four sub-images are key for

our analysis. The up/down wavefield decomposition inthe imaging condition isolates back-scattered (includingreflected) from forward-scattered (including transmit-ted) energies. The redefinition of images i0 and is sep-arates back-scattering from forward-scattering and re-duces what has been referred to as low-frequency RTMartifacts or “transmission” artifacts (Liu et al., 2011;Fleury and Vasconcelos, 2012). The modified images i0and is do not map exact scattering amplitudes (or en-ergy loss in scattering for crosscorrelation-based imag-ing) because forward-scattering is ignored. The goal isto reveal subsurface scattering contrasts rather than toachieve true amplitude nonlinear scattering-based imag-ing which, like conventional imaging, has some otherintrinsic limitations, such as limited acquisition geom-etry, intrinsic attenuation, anisotropy, or shortcomingsin multi-model parameter estimation (e.g., Gray, 1997;Deng and McMechan, 2007; Virieux and Operto, 2009).The four NLRTM sub-images of equations (13) and (14)prove to be of particular interest for seismic interpreta-tion and model sensitivity analysis.

3 STRATEGIES FOR NONLINEARREVERSE-TIME MIGRATION

3.1 Energy, illumination, and sensitivity ofmultiply scattered waves

The NLRTM method incorporates multiply scatteredwaves into the imaging process by taking the funda-mental nonlinear relation between model and data into

Nonlinear reverse-time migration 161

q

(a) Velocity model c0 + ∆c

q

(b) Velocity model c0 + ∆c

q

(c) Velocity model c0 + ∆c

(d) Density model ρ0 + ∆ρ

Figure 1. Model m0 + ∆m = (ρ0 + ∆ρ, c0 + ∆c) for exper-

iments (a) point-scatterer velocity model, (b) high-velocitylens and point-scatterer velocity model, and (c) low-velocity

zone and point scatterer velocity model. Two-layer densitymodel (d) is the same for all three experiments. The red dots

and green line indicate the fixed-spread source/receiver ge-

ometry.

account. The NLRTM sub-images result from mappingthe entire linear and nonlinear scattered data into struc-turally coherent seismic events that correspond to theinteraction of both singly and multiply scattered waves.The recorded multiply scattered data contain subsurfaceinformation that conventional RTM migration does notexploit: multiply scattered waves carry additional en-ergy beyond that in singly scattered waves, illuminatethe subsurface with better coverage than do singly scat-tered waves and are more sensitive to the Earth modelthan are singly scattered waves. In this section, I definea strategy for using energy, illumination, and sensitiv-ity of multiply scattered waves in seismic interpretation.We shall also see that these attributes of multiply scat-tered waves are potentially useful for migration velocityanalysis.

Sub-images i(d,u)0 and i

(u,d)0 primarily map single

scattering events into the subsurface. These two sub-images mainly result from the illumination and sensitiv-ity of singly scattered waves. Sub-images i

(d,u)s and i

(u,d)s

map only multiple scattering events into the subsur-face, and thus provide illumination and sensitivity infor-mation from multiply scattered waves. All sub-imagesare nontheless representations of the same subsurfacestructure and must consequently share common struc-tural features. A comparative study of the NLRTM sub-images therefore provides valuable information for inter-pretting subsurface structures. Additionally, the redun-dancy and consistency between NLRTM sub-images arenew appreciable criteria for migration velocity analysis,which I briefly describe in this document and furtherexploit in future studies.

The three synthetic experiments in Figure 1 il-lustrate the fundamental properties of the NLRTMmethod. These experiments, denoted (a-c), emphasizedifferent aspects of the use of NLRTM sub-images. I usethe three models in Figure 1 to synthesize the seismicdata and then migrate all the data with the same iden-tical reference model (Figure 2). Experiment (a) corre-sponds to having a correct migration velocity model. Ex-periments (b) and (c) correspond to having an incorrectmigration velocity model with either a strong localizedanomaly (missing high-velocity lens) or a weak diffusedanomaly (missing low-velocity zone), respectively. De-spite these small velocity anomalies, the conventionalRTM images in all three experiments provide an inter-pretable image of the density-constrast reflector (for ex-ample, see the RTM image in Figure 3 for experiment(a)). After estimating the focus quality at the point scat-terer, one might question the reliability of the interpre-tation of the point scatterer especially for experiments(b) and (c). Based on this interpration of the conven-tional RTM images, I create scattering model estimate∆mest (Figure 4), which contains the interpreted den-sity reflector and leaves out the point scatterer. Model∆mest defines the perturbation operator Pest used inthe modified extrapolation procedure of the NLRTM

162 Clement Fleury

(a) Velocity model c0

(b) Density model ρ0

Figure 2. Migration model m0 = (ρ0, c0): (a) smooth back-

ground velocity model and (b) smooth density model arecommon to the three experiments of Figure 1. The red dots

and green line indicate the fixed-spread source/receiver ge-ometry.

@@I

Figure 3. Conventional RTM image for experiment (a) with

interpreted features indicated with green symbols: the line

identifies the density reflector, the box corresponds to theregion where the point scatterer is located, and the arrow

denotes the presence of a multiple of the point scatterer.

method. For the three experiments, let us focus our at-tention on the reconstruction of the image of the pointscatterer. Figures 5 to 7 show the NLRTM sub-images ofthe point scatterer (region of interest indicated by thesquare box of Figure 3) corresponding to experiments(a) through (c), respectively. In these figures, arrowsprovide a schematic representation of the direction of il-lumination resulting from up- and down-going referenceand scattered wavefields. Sub-images i

(u,d)0 and i

(d,u)s

(a) Velocity model ∆cest

(b) Density model ∆ρest

Figure 4. Model ∆mest = (∆ρest,∆cest): (a) velocity-

contrast estimate model and (b) density-contrast estimatemodel designed based on the interpretation of the RTM im-

age of Figure 3. The red dots and green line indicate thefixed-spread source/receiver geometry.

(shown in Figure 5 for experiment (a)) are discarded in

the analysis because wavefields w(u)0 and w

(d)s,sou, which

contribute to sub-images i(u,d)0 and i

(d,u)s , do not illu-

minate the point scatterer. These two sub-images aretherefore identically zero at the point scatterer and donot contribute to its image.

For experiment (a), sub-images i(d,u)0 and i

(u,d)s

(Figures 5a and 5b) show focusing of the scattered en-ergy at the correct location of the point scatterer, thatis at the location of the scatterer (indicated by the bluedot) with the tolerance of a small vertical variation dueto the spatial extent of the point scatterer (modeled bya localized gaussian anomaly). With the correct veloc-ity model, the NLRTM method accurately maps bothsingly and multiply scattered energy. This consistencybetween sub-images i

(d,u)0 and i

(u,d)s is valuable infor-

mation to ensure correct interpretation of the seismicimage of the point scatterer. In sub-image i

(d,u)0 , ref-

erence wavefield w(d)0 illuminates the point scatterer

from above, and, in sub-image i(u,d)s , scattered wave-

field w(u)s,sou illuminates the point scatterer from below.

The NLRTM method provides extra illumination (rep-resented by arrows in Figure 5) of the subsurface. Ex-periment (a) thus illustrates how to utilize energy and il-lumination of multiply scattered waves to provide extra

Nonlinear reverse-time migration 163

@@Rw

(d)0

���w

(u)s,rec

(a) Image i(d,u)0

���w

(u)s,sou

@@R

w(d)s,rec

(b) Image i(u,d)s

���

w(u)0

@@R

w(d)s,rec

(c) Image i(u,d)0

@@R

w(d)s,sou

���w

(u)s,rec

(d) Image i(d,u)s

Figure 5. NLRTM sub-images for experiment (a): (a)

sub-image i(d,u)0 and (b) sub-image i

(u,d)s utilizes extra-

illumination from multiply scattered energy. Because of the

lack of illumination, neither (c) sub-image i(u,d)0 nor (d) sub-

image i(d,u)s contribute to the image of the point scatterer.

information about the subsurface for advanced seismicinterpretation.

For experiment (b), the high-velocity lens causesthe point scatterer to be focused at the incorrect loca-tion in depth in sub-image i

(d,u)0 (Figure 6a) but does

not misposition the reconstructed point scatterer in sub-image i

(u,d)s (Figure 6b) because wavefields w

(d)0 and

@@Rw

(d)0

���w

(u)s,rec

(a) Image i(d,u)0

���w

(u)s,sou

@@R

w(d)s,rec

(b) Image i(u,d)s

Figure 6. NLRTM sub-images for experiment (b): contrary

to (a) sub-image i(d,u)0 , (b) sub-image i

(u,d)s focuses scatter-

ered energy at the correct location of the scatterer and showsalmost no sensitivity to the presence of the velocity lens.

w(u)s,rec, in contrast to wavefields w

(u)s,sou and w

(d)s,rec, are

sensitive to the high-velocity lens. For experiment (c),these observations are reversed. The velocity anomalycaused by the low-velocity zone does not affect sub-image i

(d,u)0 (Figure 7a) but causes the multiply scat-

tered energy by the point scatterer to defocus in sub-image i

(u,d)s (Figure 7b) because, contrary to wavefields

w(d)0 and w

(u)s,rec, wavefields w

(u)s,sou and w

(d)s,rec are sen-

sitive to the low-velocity zone. With an incorrect ve-locity model, the NLRTM method does not simultane-ously map singly and multiply scattered energy withaccuracy. Comparing sub-images i

(d,u)0 and i

(u,d)s thus

provides a diagnostic for interpreting the reliability ofthe reconstruction of the point scatterer. For both ex-periments, sub-images i

(d,u)0 and i

(u,d)s apply different

illumination (represented by arrows in Figures 6 and 7)for mapping the point scatterer in depth. At the pointscatterer location, the incident and scattered wavefieldsthat contribute to either sub-images i

(d,u)0 or i

(u,d)s prop-

agate through different parts of the model and conse-quently do not exhibit the same sensitivity to the model.Experiments (b) and (c) demonstrate how illuminationand sensitivity of multiply scattered waves control thequality of the interpretation of a seismic image. Thegain in sensitivity that comes from the use of multiplyscattered waves in the NLRTM method also providesvaluable information to cross-validate the accuracy of agiven migration velocity model. Discrepancies betweenNLRTM sub-images are potentially usable for velocitymodel building by setting an inverse problem to penal-

164 Clement Fleury

@@Rw

(d)0

���w

(u)s,rec

(a) Image i(d,u)0

���w

(u)s,sou

@@R

w(d)s,rec

(b) Image i(u,d)s

Figure 7. NLRTM sub-images for experiment (c): contrary

to (a) sub-image i(d,u)0 , (b) sub-image i

(u,d)s is sensitive to the

low-velocity anomaly and shows severe defocusing of scatter-ered energy at the scatterer location.

ize such discrepancies in the image space (e.g., de Hoopet al., 2006; Symes, 2008b, 2009) in a fashion similarto that in image-domain waveform tomography (e.g.,Sava and Biondi, 2004; Shen and Symes, 2008; Yangand Sava, 2011).

3.2 Target-oriented subsalt imaging

A practical application for the NLRTM method istarget-oriented subsalt imaging. Subsalt imaging is chal-lenging because of the structural complexity of salt bod-ies, which leads to the general lack of illumination belowsalt (e.g., Muerdter and Ratcliff, 2001; Leveille et al.,2011). Energy, illumination, and sensitivity of multi-ply scattered waves are of potential interest for such acomplex scattering geologic environment. The syntheticsubsalt example of this subsection shows the efficacy ofseismic interpretation based on NLRTM sub-images.

For the Sigsbee 2A model (Paffenholz et al., 2002),let us consider synthetic marine data generated usingtime-domain finite-difference modelling with the acqui-sition geometry and stratigraphic model shown in Fig-ure 8. Figure 9 shows the migration velocity model,which contains interpreted hard salt boundaries. Con-ventional RTM migration provides the two images inFigure 10. A crosscorrelation-based imaging conditionyields the image in Figure 10a. A Laplacian filter hasbeen applied to the image in Figure 10a to reduce thelow-frequency RTM artifacts commonly observed on thetop of the salt body. After interpreting this image, let

Figure 8. Sigsbee stratigraphic model c0 + ∆c. The red

and purples lines indicate the constant-offset source/receivergeometry.

Figure 9. Sigsbee migration-velocity model c0. The red and

purples lines indicate the constant-offset source/receiver ge-ometry.

us notice poorly illuminated areas below the salt bodyand select the region of interest indicated by the greenbox in Figure 10a as the target region for improving theimage quality and resolving ambiguities in this subsaltarea after application of the NLRTM method. For com-parison, Figure 10b is a conventional RTM image in theregion of interest using a deconvolution-based imagingcondition. I use a crosscorrelation-based imaging condi-tion for the image in Figure 10a in order to build anestimate of perturbation model ∆m with an automaticalgorithm-based method. Gradient-based data-domainmisfit inversion methods, such as in FWI, require scal-ing of the gradient for model updating at each iteration(e.g., Pratt et al., 1998). Using a scaling function thatis similar to the one used in FWI, I scale the image inFigure 10a to obtain model estimate ∆mest (Figure 11)for use in the NLRTM method.

The NLRTM method produces the sub-images inFigure 12. Sub-image i

(d,u)0 (Figure 12a) exhibits al-

most identical structure and only slightly different am-plitude as compared to the conventional image (Fig-ure 10b). The illumination is relatively poor below the

Nonlinear reverse-time migration 165

(a) Global image

(b) Image of the region of interest

Figure 10. Conventional RTM Sigsbee images: (a) globalcrosscorrelation-based RTM image and (b) deconvolution-

based RTM image in the region of interest. A Laplacian filter

has attenuated the low-frequency artifacts typical of RTM inimage (a). The poorly illuminated subsalt area indicated by

the green box in sub-figure (a) defines the region of interest.

salt body in the central part of sub-image i(d,u)0 (Fig-

ure 12a). Sub-image i(u,d)0 (Figure 12b) does not help to

resolve this part of the image because reference wave-field w

(u)0 carries no up-going energy and therefore does

not illuminate structures below salt. In contrast, sub-images i

(d,u)s and i

(u,d)s provide additional valuable infor-

mation about the targeted subsalt area. Sub-image i(d,u)s

(Figure 12c) maps the interaction of down-going scat-

tered wavefield w(d)s,sou with up-going scattered wave-

field w(u)s,rec. The down-going transmitted energy of di-

rect and multiple waves in wavefield w(d)s,sou illuminates

much the same subsalt region as energy of the directwaves in wavefield w

(d)0 . In this aspect, the information

carried about the subsurface structure is somehow re-dundant. Wavefield ws,sou, however, carries extra infor-mation and is more sensitive to the velocity model be-cause of its higher order of scattering interaction. By ex-tension, for an accurate velocity model, sub-image i

(d,u)s

must exhibit structure similar to that of sub-image i(d,u)0

and potentially better illumination. The comparison ofthe images in Figures 12a and 12c shows this consistencyand gives confidence for interpreting these images. Sub-image i

(u,d)s (Figure 12d) maps the interaction of up-

going scattered wavefield w(u)s,sou with down-going scat-

tered wavefield w(d)s,rec. The up-going reflected energy in

wavefield w(u)s,sou provides additional illumination and

helps to improve the resolution of subsalt structures.Interestingly, structures that might be masked when il-luminated from above might be observable from below.As shown in the previous sub-section, sub-image i

(u,d)s

also exhibits sensitivity to the velocity model that iscomplementary to the sensitivity of the other three NL-RTM sub-images. New and redundant structural infor-mation contained in sub-image i

(u,d)s facilitates more de-

tailed seismic interpretation of the subsalt target. Thecoherent structures in sub-image i

(u,d)s are similar to

those present in sub-images i(d,u)0 and i

(d,u)s . This consis-

tency is further evidence that migration velocity modelc0 (Figure 9) is accurate. Sub-image i

(u,d)s also shows

better coherence of poorly illuminated reflectors andreveals additional features in the image. For example,the sediment layering across the two faults (shown aslines in Figure 12) is more visible. Comparative studyof NLRTM sub-images thus helps in interpreting subsaltimages in poorly illuminated areas.

4 CONCLUSIONS

The strategy for nonlinear reverse-time migration (NL-RTM) here outlines the potentials of utilizing the en-ergy, illumination, and sensitivity of multiply scatteredwaves in seismic imaging. The nonlinear relation be-tween seismic model and data is key to incorporatingmultiply scattered waves into reverse-time migration. Totake account of this nonlinear relation, the method es-timates a scattering contrast model, modifies wavefieldextrapolation, and extends the concept of the imagingcondition. As a result, the NLRTM method outputs apanel of four nonlinear sub-images that represent thesame subsurface structure from different aspects. It ispossible to combine these four sub-images into a singlenonlinear image of the subsurface, but this operationis nontrivial and must be adaptive to account for theamplitude variations across sub-images.

A comparative analysis of these nonlinear sub-images is a tool for both interpretation and possiblymodel sensitivity analysis. The target-oriented subsaltimaging example here illustrates the application of theNLRTM method: each NLRTM sub-image emphasizesdifferent illumination of the subsurface structure, whichresults in additional information for the description ofgeological subsurface features. The consistency betweenredundant information across different sub-images givesa more confident assessment for seismic interpretation.

This same consistency criterion reveals the gain insensitivity to model parameters that comes from the useof multiply scattered waves. In the future, I intend touse this extra sensitivity to develop new algorithms formigration velocity analysis. Current tools for migration

166 Clement Fleury

Figure 11. Sigsbee velocity model ∆cest: estimate obtained

by amplitude scaling of the conventional RTM image in Fig-ure 10a. The red and purples lines indicate the constant-offset

source/receiver geometry.

velocity analysis rely on assessing the quality of focus ofconventional linear images. These techniques should becomparably applicable to the more sensitive nonlinearsub-images of the NLRTM method and should addition-ally benefit from the introduction of a new consistencycriterion among sub-images.

5 ACKNOWLEDGMENTS

This work was supported by the Consortium Project onSeismic Methods for Complex Structures at the Cen-ter for Wave Phenomena. This research was supportedin part by the Golden Energy Computing Organizationat the Colorado School of Mines using resources ac-quired with financial assistance from the National Sci-ence Foundation and the National Renewable EnergyLaboratory. I thank Professor Roel Snieder for his ad-vice, support, and encouragement that contributed tothe development of this research project. I am gratefulto my CWP colleagues for their comments that helpedimproving this manuscript.

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,,,,,,

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