Estimation of angle-dependent reflectivity is a critical step in the larger inverse problem of mapping subsurface material properties from surface seismic reflection data (e.g., Lumley and Beydoun, 1991). The latter is, in turn, important for hydrocarbon exploration and detection, and crucial for reservoir characterization and monitoring in the development and production phases.
Stolt and Weglein (1985) gave a comprehensive review of linearized seismic migration and inversion techniques, which we recommend as useful background information to the angle-dependent reflectivity analysis we present here. Bleistein (1987) addressed the angle-dependent reflectivity problem based on the work of Beylkin (1985). By a suitable choice of a Kirchhoff integrand, Bleistein was able to show that his trial solution reduced to an estimate of angle-dependent reflectivity under the stationary phase approximation. However, it is not clear that the solution is robust in the sense of an lp-norm estimate, nor that it can be derived in a straightforward inverse theory approach, starting with an explicit forward model for reflection and diffraction.
More recently, de Bruin (1992) and Cunha (1992) approached this problem from a finite-difference perspective. de Bruin estimates angle-dependent reflectivity using an -x migration scheme. However, de Bruin processes the data in constant ray parameter sections, and so cannot estimate the reflection angles directly from the data, without a posteriori information on the geologic structure and dip. His suggestion of a second-pass procedure of evaluating the reflection angles after picking migrated reflector horizons may represent an inefficient and unstable process. Cunha (1992) applied a reverse-time migration procedure to estimate angle-dependent reflectivity. As with de Bruin, Cunha needs to invoke and evaluate a set of new imaging conditions to estimate reflectivity, but, unlike de Bruin, Cunha is able to directly estimate reflection angles from the data. Possible disadvantages to Cunha's method are its prohibitive computational expense, and its potential for instability due to its formulation in the common-shot domain.
In this paper we derive a robust least-squares estimate of angle-dependent reflectivity. We start from an elastic representation theorem-based forward modeling theory of generalized diffraction and reflection, and invoke a straightforward application of least-squares inverse theory. The method does not require the arbitrary definition of any particular imaging conditions, and both reflectivity and reflection angle are directly estimated from the data without a priori knowledge of structural dip. The results can be interpreted as a least-squares generalization of Bleistein's approach, and cost little more than a standard Kirchhoff prestack migration to implement.
This paper proceeds as follows. We first derive a forward theory relating angle-dependent seismic reflection data to a generalized reflection model which combines elements of Zoeppritz plane-wave reflection and Rayleigh-Sommerfeld elastic diffraction. On the basis of this forward model, we formulate a least-squares inverse problem to estimate the angle-dependent reflection coefficient as a function of subsurface coordinate and offset. This analysis leads to two coupled normal equations for reflectivity and reflection angle, as a function of the recorded seismic data. We decouple the two equations using a stationary phase approximation. We finally solve the reflectivity and angle equations separately using a standard Gauss-Newton gradient optimization technique and an approximate diagonal Hessian operator.