ALBERT

Notes: The reflection and diffraction of elastic waves by surfaces in three dimensions and scattering by thin scatterers can be combined in a common formulation. This approach is derived by using an integral formulation of the elastic wavefield together with ray approximations for wave propagation between source or receiver and reflecting surface or scatterer. For a scatterer, first order scattering is assumed and at a reflecting surface, reflection and transmission effects are estimated using the assumption of a locally plane interface. With these approximations, the reflected seismic field can be represented as the convolution of an approximate source function with a weight function for a particular source and receiver configuration. The weight function at a particular time may be evaluated by a contour integral along the isochronal curve for which the total time from source to receiver via points along the curve on the reflecting surface or on the median surface of the scatterer is equal to the specified time. The kernel of this integral contains information on the scattering or reflection coefficients for the incident wave, the angular effects of the incoming and outgoing waves with respect to the surface normal and the speed of advance of the isochron on the surface. By transfering temporal derivatives to the source function in the convolution, a very stable numerical scheme can be formulated for the generation of synthetic seismograms. This method is illustrated by the calculation of P-P reflections for a seismic line oblique to an anticline and for a variety of parameter contrasts for a simple scatterer model. Reflections from multilayered models can be generated for one surface at a time and the effect of a number of scatterers will be additive within the first order approximation employed. The facility to calculate theoretical seismograms for both surface reflections and scattering is exploited to look at the nature of reflection from the crust-mantle boundary in three dimensions. Very similar seismic responses are obtained for the P-waves returned from an irregularly corrugated Moho surface and a model of the crust-mantle transition as a set of scatterers in the lowermost crust with mantle properties. Thus with current resolution there will be inherent ambiguities in interpretations of the character of the crust mantle transition.