|Interferometric imaging condition for wave-equation migration|
The main requirement for good-quality imaging is accurate knowledge of the velocity model. Errors in the model used for imaging lead to inaccurate reconstruction of the seismic wavefields and to distortions of the migrated images. In any realistic seismic field experiment the velocity model is never known exactly. Migration velocity analysis estimates large scale approximations of the model, but some fine scale variations always remain elusive. For example, when geology includes complicated stratigraphic structures or complex salt/carbonate bodies, the rapid velocity variations on the scale of the seismic wavelength and smaller cannot be estimated correctly by kinematic methods. Therefore, even if the broad kinematics of the seismic wavefields are reconstructed correctly, the extrapolated wavefields also contain phase and amplitude distortions that lead to image artifacts obstructing the image of the geologic structure under consideration. While it is certainly true that even the recovery of a long-wave background may prove to be a challenge in some circumstances, we do not attempt to address that issue in this paper. Instead, we concentrate solely on the problem of dealing with the effect of a small scale random variations not estimated by conventional methods.
There are two ways in which we can approach this problem. The first option is to improve our velocity analysis methods to estimate the small-scale variations in the model. Such techniques take advantage of all information contained in seismic wavefields and are not limited to kinematic information of selected events picked from the data. Examples of techniques in this category are waveform inversion (Pratt and Worthington, 1990; Pratt, 1990; Sirgue and Pratt, 2004; Tarantola, 1987), wave-equation tomography (Woodward, 1992) or wave-equation migration velocity analysis (Sava and Biondi, 2004a; Shen et al., 2005; Sava and Biondi, 2004b). A more accurate velocity model allows for more accurate wavefield reconstruction. Then, wavefields can be used for imaging using conventional procedures, e.g. cross-correlation. The second option is to concentrate on the imaging condition, rather than concentrate on ``perfect'' wavefield reconstruction. Assuming that the large-scale component of the velocity models is known (e.g. by iterative migration/tomography cycles), we can design imaging conditions that are not sensitive to small inaccuracies of the reconstructed wavefields. Imaging artifacts can be reduced at the imaging condition step, despite the fact that the wavefields incorporate small kinematic errors due to velocity fluctuations.
Of course, the two options are complementary to each other, and both can contribute to imaging accuracy. In this paper, we concentrate on the second approach. For purposes of theoretical analysis, it is convenient to model the small-scale velocity fluctuations as random but spatially correlated variations superimposed on a known velocity. We assume that we know the background model, but that we do not know the random fluctuations. The goal is to design an imaging condition that alleviates artifacts caused by those random fluctuations. Conventional imaging consists of cross-correlations of extrapolated source and receiver wavefields at image locations. Since wavefield extrapolation is performed using an approximation of the true model, the wavefields contain random time delays, or equivalently random phases, which lead to imaging artifacts.
One way of mitigating the effects of the random model on the quality of the resulting image is to use techniques based on acoustic time reversal (Fink, 1999). Under certain assumptions, a signal sent through a random medium, recorded by a receiver array, time reversed and sent back through the same medium, refocuses at the source location in a statistically stable fashion. Statistical stability means that the refocusing properties (i.e. image quality) are independent of the actual realization of the random medium (Fouque et al., 2005; Papanicolaou et al., 2004).
We investigate an alternative way of increasing imaging statistical stability. Instead of imaging the reconstructed wavefields directly, we first apply a transformation based on Wigner distribution functions (Wigner, 1932) to the reconstructed wavefields. We consider a special case of the Wigner distribution function (WDF) which has the property that it attenuates random fluctuations from the wavefields after extrapolation with conventional techniques. The idea for this method is borrowed from image processing where WDFs are used for filtering of random noise. Here, we apply WDFs to the reconstructed wavefields, prior to the imaging condition. This is in contrast to data filtering prior to wavefield reconstruction or to image filtering after the application of an imaging condition.
Our procedure closely resembles conventional imaging procedures where wavefields are extrapolated in the image volume and then cross-correlated in time at every image location. Our method uses WDFs defined in three-dimensional windows around image locations which makes it both robust and efficient. From an implementation and computational cost point of view, our technique is similar to conventional imaging, but its statistical properties are improved. Although conceptually separate, we can lump-together the WDF transformation and conventional imaging into a new form of imaging condition which resembles interferometric techniques (Fouque et al., 2005; Papanicolaou et al., 2004). Therefore, we use the name interferometric imaging condition for our technique to contrast it with the conventional imaging condition.
A related method discussed in the literature is known under the name of coherent interferometric imaging (Borcea et al., 2006b,a,c). This method uses similar local cross-correlations and averaging, but unlike our method, it parametrizes reconstructed wavefields as a function of receiver coordinates. Thus, the coherent interferometric imaging functional requires separate wavefield reconstruction from every receiver position, which makes this technique prohibitively expensive and probably unusable in practice on large-scale seismic imaging projects. In contrast, the imaging technique advocated in this paper achieves similar statistical stability properties as coherent interferometric imaging, but at an affordable computational cost since we apply wavefield reconstruction only once for all receiver locations corresponding to a given seismic experiment, typically a ``shot''.
|Interferometric imaging condition for wave-equation migration|