Reflectivity blurring in subsurface extended image gathers

Abstract

We explain how a reverse time migration (RTM) subsurface extended-image gather (EIG) relates to the reflection function for a finite-contrast interface via a blurring function. For a plane interface between locally homogeneous media, the reflection function contains the plane-wave reflection coefficients, and so we determine how the EIG relates to amplitude versus angle. The EIG and reflection function are multidimensional; hence, the blurring function in their linear relationship is higher dimensional. We explain how it may be computed and show that it describes spatial blurring and blurring over angle of the plane-wave reflection coefficients. We determine explicitly how a slant stack of the EIG at one slowness depends on the plane-wave coefficients at nearby slownesses. This angle blurring stems from the spatial nonstationarity of the blurring function, so it should be the most significant where the illumination changes most rapidly in space. To evaluate the theory, we use finite-difference modeling in the Sigsbee 2a model to generate synthetic survey data, RTM EIGs, blurring functions, and modeled gathers for a deep reflector. Two example image points are chosen. One has good illumination, with blurring over the angle less than 5°. The other point is just under the salt body, with poorer illumination and angle blurring of almost 10°. The description and examples are for two dimensions, but the extension to three dimensions is conceptually straightforward, as is an interface that dips relative to the EIG datum level. The computational cost of blurring functions implies their targeted use for the foreseeable future, for example, in reservoir characterization. The extension to elasticity and more-complicated scatterers is also foreseeable, and we emphasize the separation of the overburden and survey-geometry blurring effects from target properties.