Wave - equation migration velocity analysis . II . Subsalt imaging examples

Subsalt imaging is strongly dependent on the quality of the velocity model. However, rugose salt bodies complicate wavefield propagation and lead to subsalt multipathing, illumination gaps and shadow zones, which cannot be handled correctly by conventional traveltime-based migration velocity analysis (MVA). We overcome these limitations by the wave-equation MVA technique, introduced in a companion paper, and demonstrate the methodology on a realistic synthetic data set simulating a salt-dome environment and a Gulf of Mexico data set. We model subsalt propagation using wave paths created by one-way wavefield extrapolation. Those wave paths are much more accurate and robust than broadband rays, since they inherit the frequency dependence and multipathing of the underlying wavefield. We formulate an objective function for optimization in the image space by relating an image perturbation to a perturbation of the velocity model. The image perturbations are defined using linearized prestack residual migration, thus ensuring stability, relative to the first-order Born approximation assumptions. Synthetic and real data examples demonstrate that wave-equation MVA is an effective tool for subsalt velocity analysis, even when shadows and illumination gaps are present. I N T R O D U C T I O N Depth imaging of complex structures depends on the quality of the velocity model. However, conventional migration velocity analysis (MVA) procedures often fail when the wavefield exhibits complex multipathing caused by strong lateral velocity variations. Imaging under rugged salt bodies is an important case when ray-based MVA methods are not reliable. Sava and Biondi (2004) have presented the theory and the methodology of an MVA procedure based on wavefield extrapolation with the potential of overcoming the limitations of ray-based MVA methods. In this paper, we present the application of the proposed procedure to Sigsbee 2A, a realistic and challenging 2D synthetic data set created by the SMAART JV (Paffenholz Paper presented at the EAGE/SEG Summer Research Workshop, Trieste, Italy, August/September 2003. ∗E-mail: [email protected] et al. 2002) and to a 2D line extracted from a 3D real data set from the Gulf of Mexico. Many factors determine the failure of ray-based MVA in a subsalt environment. Some of them are successfully addressed by our wave-equation MVA (WEMVA) method, whereas others, for example the problems that are caused by essential limitations of the recorded reflection data, are only partially solved by WEMVA. An important practical difficulty encountered when using rays to estimate velocity below rugose salt bodies is the instability of ray tracing. Rough salt topology creates poorly illuminated areas, or even shadow zones, in the subsalt region. The spatial distribution of these poorly illuminated areas is very sensitive to the velocity function. Therefore, it is often extremely difficult to trace rays connecting a given point in the poorly illuminated areas with a given point at the surface (two-point ray tracing). Wavefield extrapolation methods are robust with respect to shadow zones and they always provide wave paths suitable for velocity inversion. C © 2004 European Association of Geoscientists & Engineers 607 608 P. Sava and B. Biondi A related and more fundamental problem with ray-based MVA is that rays poorly approximate actual wave paths when a band-limited seismic wave propagates through a rugose top of the salt. Figure 1 illustrates this issue by showing three bandlimited (1–26 Hz) wave paths, also known in the literature as fat rays or sensitivity kernels (Woodward 1992; Pratt 1999; Dahlen, Hung and Nolet 2000). Each of these three wave paths is associated with the same point source located at the surface, but corresponds to a different subsalt ‘event’. The top panel in Fig. 1 shows a wave path that could reasonably be approximated using the method introduced by Lomax (1994) to trace fat rays using asymptotic methods. In contrast, the wave paths shown in both the middle and bottom panels in Fig. 1 cannot be well approximated using Lomax’s method. The amplitude and shapes of these wave paths are significantly more complex than a simple fattening of a geometrical ray could ever describe. The bottom panel illustrates the worst-case-scenario situation for ray-based tomography because the variability of the top salt topology is on the same scale as the spatial wavelength of the seismic wave. The fundamental reason why true wave paths cannot be approximated using fattened geometrical rays is that they are frequency dependent. Figure 2 illustrates this dependence by depicting the wave path shown in the bottom panel of Fig. 1 as a function of the temporal bandwidth: 1–5 Hz (top), 1–16 Hz (middle) and 1–64 Hz (bottom). The width of the wave path decreases as the frequency bandwidth increases, and the focusing/defocusing of energy varies with the frequency bandwidth. The limited and uneven ‘illumination’ of both the reflectivity model and the velocity model in the subsalt region is a challenging problem for both WEMVA and conventional raybased MVA (see Fig. 7 for an example of this problem). For the reflectors under salt, the angular bandwidth is drastically reduced in the angle-domain common-image gathers (ADCIGs). This phenomenon is due to a lack of oblique wave paths in the subsalt, causing a reduction in the ‘sampling’ of the velocity variations in the subsalt. Consequently, the velocity inversion is more poorly constrained in the subsalt sediments than in the sediments on the side of the salt body. Uneven illumination of subsalt reflectors is even more of a challenge than reduced angular coverage. It makes the velocity information present in the ADCIGs less reliable by causing discontinuities in the reflection events and creating artefacts. MVA methods assume that when the migration velocity is correct, events are flat in ADCIGs along the aperture-angle axis. Velocity updates are estimated by minimizing the curvature of events in ADCIGs. MVA methods may provide biased estimates where uneven illumination creates events that bend along the aperture-angle axis, even where the image is created with the correct velocity. We address this issue by weighting the image perturbations before inverting them into velocity perturbations. Our weights are functions of the ‘reliability’ of the moveout measurements in the ADCIGs. WAV E E Q U AT I O N M VA A L G O R I T H M In this section, we briefly summarize the theory of waveequation migration velocity analysis (WEMVA). In contrast with the companion paper (Sava and Biondi 2004), we avoid mathematical detail and concentrate on the principles on which WEMVA is developed. Therefore, this section complements the theory presented in Sava and Biondi (2004), and is designed as a quick introduction to WEMVA for the reader less interested in mathematical detail. The computation of the velocity updates from the results of migrating the data with the current (background) velocity model comprises three main components that are summarized by the flow chart in Fig. 3. The three components are labelled A, B and C on the chart. Box A corresponds to the computation of the background wavefield, based on the surface data and background slowness. Boxes B and C correspond respectively to the forward and adjoint WEMVA operator. Using wavefield extrapolation, the data recorded at the surface (D) are downward continued to all depth levels, using the background slowness (S) to generate a background wavefield (U). The known background slowness (S) can incorporate lateral variations. Extrapolation can be carried out with kernels corresponding to such methods as Fourier finite difference (Ristow and Rühl 1994), or generalized screen propagator (Rousseau, Calandra and de Hoop 2003). From the extrapolated wavefield, we can construct the background image (R) by applying a standard imaging condition, for example, a simple summation over frequencies. The background wavefield (U) is an important component of the WEMVA operator. This wavefield plays a role analogous to the one played in traveltime tomography by the ray-field obtained by ray tracing in the background model. The wavefield is the carrier of information and defines the wave paths along which we spread the velocity errors measured from the migrated images obtained using the background slowness function. The wavefield is band-limited, unlike a ray-field, which describes propagation of waves with an infinite frequency band. Therefore, the background wavefield provides a more accurate description of wave propagation through complex media than the corresponding ray-field (Figs 1 and 2). Typical examples are salt bodies characterized C © 2004 European Association of Geoscientists & Engineers, Geophysical Prospecting, 52, 607–623 Wave-equation migration velocity analysis II 609 Figure 1 Wave paths for frequencies between 1 and 26 Hz for various locations in the image and a point on the surface. Each panel is an overlay of three elements: the slowness model, the wavefield corresponding to a point source on the surface at x = 16 km, and wave paths from a point in the subsurface to the source. C © 2004 European Association of Geoscientists & Engineers, Geophysical Prospecting, 52, 607–623 610 P. Sava and B. Biondi Figure 2 Frequency dependence of wave paths between a location in the image and a point on the surface. Each panel is an overlay of three elements: the slowness model, the wavefield corresponding to a point source on the surface at x = 16 km, and wave paths from a point in the subsurface to the source. The different wave paths correspond to frequency bands of 1–5 Hz (top), 1–16 Hz (middle) and 1–64 Hz (bottom). The larger the frequency band, the narrower the wave path. The end member for an infinitely wide frequency band corresponds to an infinitely thin geometrical ray. C © 2004 European Association of Geoscientists & Engineers, Geophysical Prospecting, 52, 607–623 Wave-equation migration velocity analysis II 611