SCALAR SPACE-TIME WAVES IN THEIR SPECTRAL-DOMAIN 1ST-ORDER AND 2ND-ORDER THIELE APPROXIMATIONS

For the simple case of the scalar wave motion generated by a point source in an unbounded homogeneous medium, it is investigated what the consequences of the first- and second-order Thiele approximations in the spectral domain are in the space-time domain. To this end, the corresponding spectral domain Green's function is transformed back to the space-time domain with the aid of the modified Cagniard method. The exact solution to the problem is a spherical wave with the same wave shape as the source signature and a single wave front. The first-order Thiele, or parabolic, approximation has a wave front in the shape of a double oblate spheroid, combined with a head-wave-like precursor having a cylindrical wave front. The second-order Thiele approximation contains two wave fronts, one associated with a fast body wave and the other with a slow body wave, in combination with a head-wave-like precursor, the latter again having a cylindrical wave front. From the results. it can be concluded in which regions below or above the source the approximations have sufficient accuracy for application in inversion and "true amplitude" depth migration procedures in geophysical prospecting.