I demonstrate a method for computing wave-equation Hessian operators, also known as resolution functions or point-spread functions, under the Born approximation. The proposed method modifies the original explicit Hessian formula, enabling efficient computation of the operator. A particular advantage of this method is that it reduces or eliminates storage of Green''s functions on the hard disk. The modifications, however, also introduce undesired crosstalk artifacts. I introduce two different phase-encoding schemes, namely, plane-wave phase encoding and random phase encoding, to suppress the cross-talk. I applied the Hessian operator obtained by using random phase encoding to the Sigsbee2A synthetic data set, where a better subsalt image with higher resolution is obtained.
Migration is an important tool for imaging subsurface structures using reflection seismic data. The classic imaging principle (Claerbout, 1971) for shot-based migration states that reflectors are located where the forward-propagated source wavefield correlates with the backward-propagated receiver wavefield. However, it has been found that such an imaging principle is only the adjoint of the forward Born modeling operator (Lailly, 1983), which provides reliable structural information of the subsurface but distorts the amplitude of the reflectors because of the non-unitary nature of the Born modeling operator. To improve relative amplitude behavior, the imaging problem can be formulated as an inverse problem based on the minimization of a least-squares functional. The inverse problem can be formulated either in the data space (Lailly, 1983; Tarantola, 1984) or in the model space (Beylkin, 1985; Chavent and Plessix, 1999; Plessix and Mulder, 2004; Valenciano et al., 2006; Yu et al., 2006). The data-space approach can be solved iteratively using the gradient-based method (Nemeth et al., 1999; Clapp, 2005) without explicit construction of the Hessian, the matrix of the second derivatives of the error functional with respect to the model parameters. The iterative solving, however, is relatively costly and converges very slowly.
On the other hand, the model-space approach requires the explicit construction of the Hessian, and its pseudo-inverse is applied to the migrated image. The full Hessian is too big and expensive to be computed in practice; hence Chavent and Plessix (1999); Plessix and Mulder (2004) approximate it by a diagonal matrix. In the case of high-frequency asymptotics, and with an infinite aperture, the Hessian is diagonal in most cases (Beylkin, 1985). For a finite range of frequencies and limited acquisition geometry, however, the Hessian is no longer diagonal and not even diagonally dominated (Plessix and Mulder, 2004; Valenciano et al., 2006). It has been shown by Valenciano et al. (2006) that, in areas of poor illumination, e.g., subsalt regions, the Hessian''s main diagonal energy is smeared along its off-diagonals. Therefore, the migrated image pre-multiplied by a diagonal matrix cannot perfectly recover the amplitude information, especially in poorly illuminated areas. That''s why Valenciano et al. (2006) suggest computing a limited number of the Hessian off-diagonals to compensate for poor illumination and improve the inversion result. However, computing the Hessian off-diagonals, even for a limited number, is very expensive by directly implementing the explicit Hessian formula.