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ABSTRACT

Enhancement of diffractions and their use to resolve finescale details in a seismic image is increasingly important. Diffractions carry useful information about small-scale characteristics of the subsurface associated with features such as faults, pinch-outs, stratigraphic variations, and other geologic features linked to hydrocarbon reservoirs. Extracting the information content of diffractions and forming an image of the features they illuminate are not trivial tasks. The conventional approach relies on seismic migration, or modifications of seismic migration, and is thus limited by the Rayleigh criterion. The limited aperture and finite bandwidth make it difficult to extract all the potential information content. An alternative approach that overcomes such limitations is to turn to signal processing approaches, which extract information from the structure of the data with the aim of detecting and characterizing a finite number of desired events. Our interest is in diffractions, so we used a windowed or steered version of the MUltiple SIgnal Classification method. Use of this method allowed diffractions to be imaged at resolutions finer than the Rayleigh limit.

INTRODUCTION

The early use of seismic diffractions dates back to the pioneering works of Krey (1952), Haagedorn (1954), and Kunz (1960). Later, Harlan et al. (1984) proposed a technique to identify diffractions to be used for velocity estimation. More recently, much attention has been paid to the enhancement and separation of diffractions from reflection data for many imaging purposes. Several approaches have

been proposed, among them the use of antistationary filtering (Moser and Howard, 2008), plane-wave destruction filters (Fomel et al., 2007), multifocusing (MF) (Berkovitch et al., 2009), or the common reflection surface (CRS) (Asgedom et al., 2011a, 2011b, 2012a, 2012b) stacking methods. Because the diffracted events are known to carry high-resolution information about the subsurface, their optimal use in imaging is an important topic.

By means of illustrative examples, we verify that the use of a standard migration type of method will not be able to fully honor this potential information content. As a way to overcome such limitations, we propose to use a variant of the high-coherency MUltiple SIgnal Classification (MUSIC) method (Schmidt, 1986). Typical applications of classical MUSIC are within high-resolution estimation of direction of arrivals (Schmidt, 1986; DeGroat et al., 1993; Aliyazicioglu et al., 2008; Hislop and Craeye, 2011), as well as superresolution imaging of point scatterers (Miwa and Arai, 2004; Agarwal and Chen, 2008). MUSIC is a sparsity technique, which implies that the number of features to be resolved must be much less than the number of sources and/or receivers. For such a case, the experimental covariance matrix can be decomposed in two orthogonal spaces: signal space and nil space. Classical MUSIC is robust with respect to ambient noise, but it requires narrowband and uncorrelated sources. In recent years, a modified version of classical MUSIC denoted “time-reversal MUSIC” has been introduced. Time-reversal MUSIC is a monochromatic technique, which, in the same way as its classical counterpart, has the potential of superresolving a smaller subset of pointlike scatterers (Lehman and Devaney, 2003; Devaney et al., 2005; Agarwal and Chen, 2008; Gelius and Asgedom, 2011). The main difference is that the multistatic response matrix (i.e., the matrix connecting monochromatic source fields and corresponding measured fields, thus describing propagation and scattering effects) replaces the role of the covariance matrix. Time-reversal MUSIC can handle correlated sources but is

sensitive to noise for some acquisition geometries (Gelius and Asgedom, 2011). To make time-reversal MUSIC more robust, multifrequency versions have been developed (Asgedom et al., 2011a; Solimene et al., 2012). In this work, a windowed or steered type of classical MUSIC is used, with window being defined by the migration operator in use.

The diffraction-enhanced seismic data will in general contain a high number of diffractions, and MUSIC does not directly apply. We show that, by introducing such a window, the sparsity condition is again fulfilled. Moreover, by steering the data window, wideband data, such as seismic data, can be handled well. The concept of steered MUSIC is used by Kirlin (1992) to further improve the velocity analysis. The work accounted for here can be regarded as a practical demonstration of the conceptual ideas presented by Gelius and Asgedom (2011). Note, however, in that paper, timereversal MUSIC was considered, and not classical MUSIC as addressed here. The window-steered MUSIC imaging technique discussed in this paper has already been used by the authors with success in a series of publications related to diffraction enhancement (Asgedom et al., 2012a, 2012b). However, the details of the actual method have not been described. The main purpose of this paper is, thus, to introduce the basics of the proposed technique in a clear and concise manner. The actual performance of the method is also demonstrated using two different data sets. First, we review the concepts of resolution and diffraction limits using a simple numerical example. Then, we introduce the window-steered MUSIC algorithm. Using the simple synthetic example, we demonstrate the method’s ability to surpass the Rayleigh resolution limit. Finally, we show the method applied to a real data set.

BASIC CONCEPT OF RESOLUTION AND A DIFFRACTION-LIMITED SYSTEM

According to Sheriff (1991), resolution is “the ability to separate two features that are very close together; the minimum separation of two bodies before their individual properties are lost.” In case of a finite receiver array, it is well known that the directivity degrades causing a wider main lobe (loss of resolution) and increasing side lobes (loss of signal-to-noise ratio) (Berkhout, 1984). This loss in resolution can be approximately described by the classical Rayleigh criterion:

where l is the spatial resolution limit, D is the aperture or length of the array, L is the distance to the target, and λ is the (dominant) wavelength. If two nearby features (scatterers) cannot be resolved from the acquisition data due to violation of the Rayleigh criterion, such a system is denoted as “diffraction limited.” The same phenomenon can be analyzed from an imaging point of view by backpropagating scattered waves associated with point scatterers. Consider now a simple numerical example as an illustration. The geometry is shown in Figure 1 and involves two nearby point scatterers, embedded in a homogeneous model and illuminated by a point source, and with the scattered data measured at receivers distributed over a surface S.

For definiteness, we assume the vectors rs ¼ 0 (origin), rksc ðk ¼ 1; 2Þ, r, and r0 to designate the source, scatterers, image,

and receiver points, respectively. Our analysis is carried out in the frequency domain, the frequency variable being denoted by ω. As a reasonable approximation, we assume that the scattered field, pscðr0; ωÞ observed at a receiver, has the form (Born-type and unit-strength point scatterers)

the distances between the kth scatterer and the source and the receiver, respectively, with τsk ¼ Rsk∕c and τk0 ¼ Rk0 ∕c being their corresponding traveltimes. Also, sðωÞ represents the spectrum of the source pulse, for convenience chosen to be the zero-phase Ricker wavelet:

where ω0 represents the peak frequency. For computation simplicity, we assume that the measurement boundary is flat and characterized by an infinite aperture crossline (y-direction) and a finite aperture (D) inline (x-direction). Moreover, we restrict the receivers to lie on a single horizontal (x-direction) line and the source, scatterers, and receiver line all to lie on a single vertical z-plane. Such conditions define a so-called 2.5D configuration for which 3D wavefield computations on each receiver can be approximated by a single spatial (x-coordinate) integral (see, e.g., Bleistein,

1986). Under the above conditions, we have rksc ¼ ðxksc; 0; zkscÞ, r ¼ ðx; 0; zÞ, and r0 ¼ ðx0; 0; 0Þ ð−D < x0 < DÞ. As a consequence,

we find

In the simulations, we used a medium velocity c ¼ 2000 m∕s, an aperture D ¼ 3200 m, and an equal depth L ¼ 2000 m of the two nearby scatterers. Moreover, the center frequency was chosen as f0 ¼ 25 Hz. This corresponds to a center wavelength λ0 ¼ 80 m. Replacing λ by the center wavelength λ0 in equation 1 gives the smallest distance that can be resolved according to the Rayleigh criterion: approximately l ≈ 5λ0∕8. With the above configuration, we obtain the data set of Figure 2. We now use the conventional approach of backpropagation to image data with the aim of retrieving the scatterer positions in depth (i.e., prestack depth migration). Following, e.g., Schneider (1978), Esmersoy and Oristaglio (1988), Schleicher et al. (2007), and Gelius and Asgedom (2011), the backpropagated wavefield pbpðr; ωÞ can be expressed by an integral of the form

pbpðr;ωÞ

¼−Z Z ∂ G ðr;r0;ωÞp ðr0;ωÞ−G ðr;r0;ωÞ ∂ p ðr0;ωÞ dr0;

S ∂n 0 sc 0 ∂n sc

(5)

where the time-advanced Green’s function G0 is given explicitly as

Following, e.g., Bleistein (1986), the above integral admits a station- ary-phase, approximate evaluation of the crossline (y-coordinate) integral, taken as an inner integral in equation 5, so that we are left with an integration over the inline x-coordinate only. If we divide the back-propagated field by the source field incident at each image point (i.e., we apply the U/D imaging condition), then summing over the available bandwidth ωB will give the final migrated image IðrÞ ¼ Iðx; 0; zÞ in the vertical cross section defined by y ¼ 0:

>

>

:

(7)

where the notation ¼ Rk0 − R 0 has been used. Figure 3a shows the migrated image of the two scatterers when separated by a distance of l ¼ 5λ0∕8 (only a selected part of the image surrounding the two scatterers shown for convenience). It can easily be seen that migration is not able to resolve the two features. This example demonstrates that the Rayleigh criterion is an empirical measure only able to give an approximate estimate of the actual diffraction limit of a system. By increasing the scatterer separation to λ0, it follows from Figure 3b that the two scatterers now separate as expected. For completeness, wiggle plots of Figure 3a and 3b are also provided by Figure 4a and 4b, respectively. In the next section, we will discuss an alternative approach to image scattering events. The basis of this technique is to select the data on a local window steered by the migration operator and apply to it a high-resolution coherency measure, denoted MUSIC, of Schmidt (1986). As shown below, this type of reconstruction technique has the potential of imaging scatterers beyond the

classical diffraction limit just discussed.

DESCRIPTION OF WINDOW-STEERED MUSIC

Before the actual MUSIC imaging algorithm can be introduced, it is important to understand how the input data are conditioned. Such an input can be a stacked seismic section or prestack data (commonoffset [CO] section or source gather as two examples). First, diffractions or scattered events are separated from the reflections by using a proper diffraction-enhancement technique. This is due to the fact that the amplitudes of diffracted waves are in general much weaker than those of the reflected ones. Different approaches have been proposed to properly utilize diffractions. Khaidukov et al. (2004) separate diffractions from reflections using a transformation that focuses reflections (but not diffractions) to small neighborhoods (nearly points). Muting such neighborhoods and applying the inverse transform attenuates the reflections, leaving diffractions unaltered. Similarly, Fomel et al. (2007) use plane-wave-destruction filters to perform diffraction separation. From an imaging point of view, Moser and Howard (2008) use antistationary filtering to separate reflective and diffractive components during the migration, while Zhang (2004) performs prestack depth diffraction imaging

using the idea of the Fresnel aperture. Diffraction enhancement can also be carried out using MF (Berkovitch et al., 2009) or the CRS (e.g., Asgedom et al., 2011b; Dell and Gajewski, 2011) stacking methods. In this work, the CRS approach has been applied, in which a modified version of the CRS moveout, better tailored for diffractions, is used. For more details and examples, the reader is referred to Asgedom et al. (2012a, 2012b). Note, however, that the MUSIC imaging method advocated in this paper works with any diffraction separation technique. The authors chose to use the modified CRS technique based on a long record of successful applications of this method. In case of a homogeneous earth model, the modified CRS technique is based on a Taylor approximation of the double-square-root equation. However, this approximation is still very accurate in a paraxial sense. Also note that in diffraction enhancement, we are first of all interested in recovering the most energetic part of a scattering response associated with the apex and its nearest neighborhood. Finally, it should be mentioned that

Figure 1. Sketch of scattering geometry.

Offset (m)

Figure 2. Common-shot section of two nearby diffractions. The solid and dashed gray curves indicate the travel times for the scatterers located, respectively, at (–20, 2000ÞT and (30, 2000ÞT . The coordinates are given by ðx; zÞT in meters.

horizontally displayed. This procedure is referred to as a steeredwindow operation. If a diffraction event falls inside that steered window and in case of an accurate velocity model, this event will now

Figure 4. Migrated image (wiggle plots) of two nearby scatterers separated by (a) 5λ0∕8 and (b) λ0 (λ0 being the center wavelength).

Midpoint coordinate (m)

Figure 5. Window surrounding migration operator curve.

be horizontally aligned (in the same way as an NMO-corrected event). It is to be observed that, in contrast to NMO correction, no wavelet stretching is involved in the steering process, due to the linear time-shift operation. In a situation in which there is more than one wavefront inside the data window, if the velocity model provided is adequate, one event will be aligned and the other(s) will not be so coherent. The coherency of the aligned event can further be enhanced by simple spatial smoothing of the windowed data along the horizontal direction. It is stressed that MUSIC imaging does not need a more accurate velocity model than normally used in migration. As an example, consider again the case of the two scatterers shown in Figure 1. Figure 6a shows the scattered data associated with these two nearby diffractors measured along the receiver line defined by y ¼ 0 (source-gather). The two solid gray curves superimposed in the same figure define the data window computed in case of an image point (see the gray dot in Figure 6a) coinciding with the location of one of the two scatterers. This window is, as explained above, centered around the migration operator curve (assuming the true velocity). The corresponding steered data window is shown in Figure 6b, where, as expected, the true diffraction event is aligned, but has superimposed weak interference due to the other nearby scatterer. The data within the window can be further processed by applying spatial smoothing (horizontal 2D filter)

and amplitude balancing. The final result is shown in Figure 6c where the horizontal trend has been further enhanced. This smoothing operation is more important in the case of real data, where the optimal window size and the operator length of the smoothing filter need to be tested for. Note that the spatial smoothing operation could alternatively be replaced by a spatial smoothing of the covariance matrix (Biondi and Kostov, 1989; Kirlin, 1992). Now, the question is, what kind of response will be obtained in case the image point considered does not represent the location of a true scatterer? Figure 7 is similar to Figure 6, with the difference that the data window is no longer centered around one of the scatterers but a nearby point (represented by a gray dot in Figure 7a). The steered data window now looks quite different both before and after further processing (Figure 7b and 7c), and it no longer contains near-hori- zontally aligned events. After these considerations, we are now in the position to introduce the details of the MUSIC imaging technique. The idea of MUSIC is sparsity; thus, the number of features to be resolved must be smaller than the total number of sources/ receivers. The use of a steered data window ensures that such a condition is fulfilled. The MUSIC imaging technique should work as follows: In case of a scattered event, it should be resolved beyond the classical diffraction limit and, if the considered image point is not a scatterer, a negligible image value should be output.

Offset (m)

Offset (m)

As demonstrated above, the steered data window (after proper spatial filtering) does look very different for these two scenarios. Only when the image point coincides with a true diffractor location is a horizontally aligned event obtained (assuming a good velocity model is provided). This is the fundamental observation that win- dow-steered MUSIC makes use of.

Consider now the window steered data matrix, D, that corresponds to a well-aligned scattering event (see Figure 6b and 6c). The matrix D can then be written in the form

representing the set of samples recorded by receiver i that falls within the window, with i ¼ 1; 2 · · · ; Nr, and t is the sample rate. Moreover, Nr is the number of traces (aperture) and Nt is the number of time samples within the window, respectively. Normally, Nr > ð>ÞNt. We now introduce the conceptual trace decomposition

is the residual trace contribution, and ni is the noise. The above decomposition readily carries over to the steered data matrix D, namely,

Here, all matrices have the dimension Nr × Nt. Moreover, we used

the matrix decomposition of equation 11 is provided by Figure 8. In terms of the decomposition of equation 11, the sample correlation matrix R (covariance matrix) of the steered matrix, D, can be

computed as

final assumptions, we consider that the residual traces are uncorrelated and that the noise is a white Gaussian realization. As a consequence, and under the consideration that a migration operator is defined by an optimal velocity field, the correlation matrix of equation 12 can be further approximated as

in which the factor ε represents the energy of each residual trace and

with λ being the corresponding eigenvalue.

MUSIC is a sparsity technique that requires a larger number of observations than features to be resolved. If this requirement is fulfilled, the correlation matrix can be decomposed into two orthogonal spaces (respectively, signal and nil spaces). In the case of steered MUSIC as discussed here, we have Nr multiple measurements of the same single horizontal event. Thus, the fundamental requirement is well satisfied. Note also that a horizontal steering vector is required to generate the MUSIC pseudospectrum (see equation 14), implying frequency independency. This allows us to handle wideband seismic data as discussed by Kirlin (1992), who applies the concept of window-steered MUSIC to further improve the velocity analysis.

The singular-value decomposition of the correlation matrix R, when computed from the actual data window, can now be formally written as

It is assumed that the energy of the traces is well balanced and that the same therefore can be expected for the residual traces (because of the assumed incoherency between the three components). As

Figure 8. Decomposition of steered data matrix ðDÞ, into three parts: average trace contribution ðAÞ, residual trace ð AÞ and noise ðNÞ contributions.

where Pn ¼ VnVTn represents the (steered) noise subspace projection matrix and wsb is the socalled semblance-balancing factor as commented on below.

Regarding the semblance-balancing factor, MUSIC, despite its high-resolution capability, yields amplitudes (pseudospectra) of arbitrary (nonnormalized) values. This is probably also the main reason why the MUSIC type of seismic signal analysis has not been much discussed in

recent years. To condition the MUSIC pseudospectrum to be a normalized quantity, Asgedom et al. (2011b) propose semblancebalanced MUSIC. This idea will also be used in this work, and the factor wsb introduced in equation 16 represents this semblancebalancing factor. The pseudospectrum in equation 16 is expressed by a nil-space projection. Alternatively, that pseudospectrum can be constructed using its counterpart signal-space projection (Biondi and Kostov, 1989; Asgedom et al., 2011a). After spatial smoothing, the steered data window will contain essentially one aligned event in case of a true diffractor. One may assume that the dimension of the signal subspace of R is still one, and that only the largest eigenvector of the correlation matrix will span the signal subspace. However, in this work, we have chosen to use the pseudospectrum

MUSIC pseudospectrum

−100

Horizontal

axis (m)

0

100

200

1900

1950

2100

Figure 9. MUSIC imaging (3D rendered plot) of the two nearby scatterers.

Horizontal axis (m)

Figure 11. Upper part of the Marmousi model (grayscale coded by interval velocity).

Figure 13. (a) Kirchhoff migration of enhanced diffractions.

(b) MUSIC imaging of the same diffracted events of the Marmousi model of Figure 11.

in equation 16, which we believe is slightly more robust than its signal subspace counterpart (although more computer intensive).

Before closing this section, we demonstrate the performance of steered MUSIC using the synthetic scattered data set plotted in Figures 6a and 7a. The result is shown in Figure 9, with the corresponding wiggle plot shown in Figure 10. We can easily see that the two nearby scatterers are well separated (only a selected part of the image surrounding the two scatterers is shown for convenience). The result is certainly different from the migrated image shown in Figure 3a, demonstrating the high-resolution capability of the proposed new technique.

DATA EXAMPLES

The potential of the window-steered MUSIC algorithm will now be further demonstrated using two different data sets. The first set is taken from the Marmousi model and represents a controlled test case. The second set is part of a marine seismic line acquired in offshore Brazil including a well-known fault system.

Marmousi model

The Marmousi data set was generated by the Institut Français du Pétrole using a 2D finite-difference (FD) modeling code (Versteeg, 1994). It is based on a seismic profile running through the North Quenguela trough in the Cuanza Basin in Angola. In this study, we selected the upper part of the original model containing a series of faults as shown in Figure 11. Diffractions were enhanced using the modified CRS technique. For more details about this enhancement as well as the Marmousi data, the reader is referred to Asgedom et al. (2012b).

The conventional zero-offset (ZO) stack and the corresponding ZO diffraction-enhanced stack are shown in, respectively, Figure 12a, and 12b. In comparison, it can be easily seen that diffractions now dominate in Figure 12b. The diffracted events fall into two categories: stronger ones corresponding to real faults and local small-scale structures and weaker ones caused by the edges of the gridding used in the data generation (FD-mesh). Thus, a dipping reflector will be represented by a jigsaw pattern. Figure 13a shows the result after Kirchhoff time migration of the separated diffrac-

tions in Figure 12b. The image is fair but with only parts of the fault planes well resolved. The corresponding image obtained using the win- dow-steered MUSIC imaging approach is shown in Figure 13b. The image is of high-resolution with an overall good description of the major fault system including internal weaker faults. Due to the high-resolution characteristics of MUSIC imaging, the fine details associated with the FD discretization artifacts are also revealed (see the right part of Figure 12b).

Figure 14. Seismic line superimposed target zone (poststack-migrated data) of the Jequitinhonha data set.

Field data

A 2D data set acquired over the Jequitinhonha basin (offshore Brazil) (Costa, 2011) was used to test the proposed imaging scheme. The basin has undergone an extensional type of deformation. A very large extensional listric fault formed at the

Figure 15. Jequitinhonha data set: (a) Target zone (poststack migrated) with interpretation. (b) Conventionally stacked section.

(c) Diffraction-enhanced CO section (offset 150 m).

landward edge of the salt basin (Davison, 2007) divides it into two parts characterized by, respectively, shallow and deep water. Compressional deformations present in the basin are due to the salt.

Figure 14 shows the seismic line selected for this study (poststack migrated). It was acquired along a NE direction over the deepwater part of the basin. The rectangle superimposed to the migrated line defines the target zone, which will be in focus here. This area includes a system of major and minor faults that are known to exist from this poststack-migrated reflection data, although not fully resolved.

Figure 15a shows a magnification of the target area defined in Figure 14 including an interpretation of the fault system. The conventionally stacked data before migration are shown in Figure 15b. A diffraction-enhanced stack was computed for this same region based again on the generalized CRS technique as shown in Figure 15c. Note that this stack corresponded to a CO section with a very small offset of 150 m. This diffraction stack was then imaged using both conventional Kirchhoff migration (Figure 16a) and window-steered MUSIC (Figure 16b), as advocated for in this paper. The new high-resolution imaging technique is seen to give a more detailed picture than standard migration as expected. Also, in comparison with the poststack-migrated image in Figure 15a, MUSIC imaging is seen to give information that is highly correlated with the independent interpretation. In addition, MUSIC imaging indicates another fault (encircled with red) that is not clearly visible on the poststack-migrated section.

CONCLUSIONS

This paper has addressed the important question of how to form a high-resolution image of diffracted wave contributions in seismic reflection data. Such diffractions can be observed in the seismic data set, as well as extracted by means of a diffraction-enhancement technique. Straightforward use of migration-type reconstruction methods will not be able to fully capitalize on the resolving power of diffractions, due to the diffraction-limit conditions inherently attached to those approaches. To be able to resolve finer details beyond the classical limits, we propose a new high-resolution imaging technique based on a windowed or steered MUSIC implementation. Application of the method on both synthetic and field data demonstrated a resolving power beyond that of standard migration. Note that the concept of MUSIC imaging applies to all types of data sorting, time and depth domains, as well as prestack and poststack. The only requirement is that data can be steered along the migration operator defined by an optimal velocity field. In three dimensions, the migration operator represents a diffraction surface. Thus, extension of MUSIC imaging to three dimensions implies the use of a steered slice through the data volume instead of a window. The MUSIC pseudospectrum can then be calculated along two orthogonal directions (inline and crossline), and their multiplication can be used as an effective spectrum or coherency measure. The 3D MUSIC algorithm needs to be combined with a three dimensions diffraction-enhancement technique. One possible candidate will be the modified version of 3D CRS.

ACKNOWLEDGMENTS

(FAPESP/Brazil); M. Tygel: National Council for Scientific and Technologic Development (CNPq/Brazil); and A.K. Takahata: Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES/ Brazil) (PDSE program, 5634-11-3). We thank J. Etgen, S. Fomel, and one anonymous reviewer for their contributions that improved the quality of the paper.

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