Wavefield tomography using extended images

Abstract

Estimating an accurate velocity model is crucial for seismic imaging to obtain a good understanding of the subsurface structure. The objective of this thesis is to investigate methods of velocity analysis by optimizing seismic images. A conventional seismic image is obtained by zero-lag crosscorrelation of wavefields extrapolated from a source wavelet and recorded data on the surface using a velocity model. The velocity model provides the kinematic information needed by the imaging algorithm to position the reflectors at correct locations and to focus the image. In complex geology, wave-equation migration is a powerful tool for accurately imaging the earth's interior; the quality of the output image, however, depends on the accuracy of the velocity model. Given such a dependency between the image and model, analyzing the velocity information from the image is still not intuitive and often ambiguous. If the nonzero space- and time-lags information are preserved in the crosscorrelation, the output are image hypercube defined as extended images. Compared to the conventional image, the extended images provide a straightforward way to analyze the image quality and to characterize the velocity model accuracy. Understanding the reflection moveout is the key to developing velocity model building methods using extended images. In the extended image space, reflections form coherent objects which depend on space (lags) and time (lags). These objects resemble cones which ideally have their apex at zero space and time lags. The symmetry axis of the cone lies along the time-lag axis. The apex of the cone is located at zero lags only if the velocity model is accurate. This corresponds to the situation when reflection energy focuses at origin in both the space- and time-lag common-image gathers (the slices at zero time and space lags, respectively). When the velocity model is inaccurate, the cone shifts along the time-lag axis. This results in residual moveout in space-lag gathers (zero time-lag slice) and defocusing in time-lag gathers (zero space-lag slice). These phenomena are correlated, and they are a rich source of information for velocity model updates. The extended image distortions caused by velocity model errors can be used to design velocity model building algorithms. When the extended image cones shift, the distance and direction of their apex away from zero time lag constrain model errors. This information can be used to construct an image perturbation, from which a slowness perturbation is inverted under the framework of linearized wave-equation migration velocity analysis. Alternatively, one can formulate a non-linear optimization problem to reconstruct the model by minimizing this image error. This approach requires the adjoint-state method to compute the gradient of the objective function, and iteratively update the model in the steepest-descent direction. The space-lag subset of extended images has been used to reconstruct the velocity model by differential semblance optimization for a decade. The basis of the method is to penalize the defocusing in the gathers and to focus the reflection energy at zero lags by optimizing the model. The assumption that defocusing is caused by velocity model error is violated where the subsurface illumination is uneven. To improve the robustness and accuracy of the technique, the illumination compensation must be incorporated into the model building. The illumination compensation effectively isolates the defocusing due to uneven illumination or missing data. The key is to construct an illumination-based penalty operator by illumination analysis. Such a penalty automatically downweights the defocusing from illumination effects and allows the inversion to suffer less from the effects of uneven illumination and to take into account only the image error due to inaccurate velocity models. One major issue for differential semblance optimization with space-lag gathers is the cost of computing and storing the gathers. To address the problem, extended space- and time-lag point gathers can be used as an alternative to the costlier common-image gathers. The point gathers are subsets of extended images constructed sparsely in subsurface on reflectors. The point gathers share similar reflection moveout characteristics with space-lag gathers, and thus differential semblance optimization can be implemented with such gathers. The point gathers reduce the computational and storage cost required by space-lag gathers especially in 3-D applications. Furthermore, the point gathers avoid the dip limitation in space-lag gathers and more accurately characterize the velocity information for steep reflections.