Offset-dependent geometrical spreading in a layered medium

Abstract

Abstract The geometrical spreading for a point source in a horizontally layered medium has been computed by Ursin (1978) and Hubral (1978) as a Taylor series in the offset coordinate. The coefficients in the Taylor series depend on the thicknesses and the velocities of the layers. Here, I start with the exact expression for geometrical spreading and show that it can be expressed as a function of the velocity in the first layer, the offset, and the first- and second-order traveltime derivatives. A shifted hyperbolic traveltime approximation (Castle, 1988) and the usual hyperbolic traveltime approximation are used to derive approximate expressions for geometrical spreading. These expressions can also be derived from a truncated Taylor series by making additional approximations, but this procedure is not so obvious.