Far‐field, stationary phase approximations are often used to study the radiation of P‐ and S‐waves from seismic sources located in boreholes, usually with an assumption of low frequency and in application to uncased boreholes. These two assumptions allow explicit analytical results for the radiation patterns to be derived, as boundary condition equations can be solved analytically in fairly simple forms. Applying the same methodology to cased and cemented boreholes, however, is much more difficult because of the increased number of simultaneous boundary condition equations. I circumvent this difficulty by solving the boundary condition equations numerically using propagator matrices, as is generally done in the calculation of synthetic full‐waveform acoustic logs. In this way, the assumption of low frequency is also avoided, and a generalized stationary‐phase solution for sources in general, concentrically layered borehole models is easily obtained. Computation of the radiation patterns for cased and uncased boreholes in various formations shows that the amplitude reduction, because of the introduction of casing, is a function of both source type and of formation velocities. Axial stress sources are less affected by the casing than either radial stress or volume‐injection sources, and as formation velocity decreases, the effect of the casing becomes more significant as the impedance contrast between steel and the formation becomes larger. The new generalized stationary‐phase solution also shows that as frequency approaches 1000 Hz, the results obtained by low‐frequency approximations for stress sources can be inaccurate and that the energy radiated from the source becomes more highly directed in the horizontal directions. The radiation pattern begins to change relatively rapidly as a function of frequency, so that the resulting observations from broad‐band sources will show changes in waveforms that mimic the effects of attenuation. These changes occur because the length of the source becomes important as wavelength decreases, demonstrating the need to consider the influence of frequency, as well as casing and cement, on source radiation.
A stationary phase solution for mountain waves with application to mesospheric mountain waves generated by Auckland Island
