Theory of reflectivity blurring in seismic depth imaging

A subsurface extended image gather obtained during controlled-source depth imaging yields a blurred kernel of an interface reflection operator. This reflectivity kernel or reflection function is comprised of the interface plane-wave reflection coefficients and so, in principle, the gather contains amplitude versus offset or angle information. We present a modelling theory for extended image gathers that accounts for variable illumination and blurring, under the assumption of a good migration-velocity model. The method involves forward modelling as well as migration or back propagation so as to define a receiver-side blurring function, which contains the effects of the detector array for a given shot. Composition with the modelled incident wave and summation over shots then yields an overall blurring function that relates the reflectivity to the extended image gather obtained from field data. The spatial evolution or instability of blurring functions is a key concept and there is generally not just spatial blurring in the apparent reflectivity, but also slowness or angle blurring. Gridded blurring functions can be estimated with, for example, a reverse-time migration modelling engine. A calibration step is required to account for ad hoc band limitedness in the modelling and the method also exploits blurring-function reciprocity. To demonstrate the concepts, we show numerical examples of various quantities using the well-known SIGSBEE test model and a simple salt-body overburden model, both for 2-D. The moderately strong slowness/angle blurring in the latter model suggests that the effect on amplitude versus offset or angle analysis should be considered in more realistic structures. Although the description and examples are for 2-D, the extension to 3-D is conceptually straightforward. The computational cost of overall blurring functions implies their targeted use for the foreseeable future, for example, in reservoir characterization. The description is for scalar waves, but the extension to elasticity is foreseeable and we emphasize the separation of the overburden and survey-geometry blurring effects from the nature of the target scatterer.