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An extended reflectivity method is described by which complete seismograms for a point source in a layered half-space can be calculated. Starting with the differential equations and boundary conditions, the reflection and transmission of plane waves at layered media is treated first, followed by the synthesis of point-source wave fields. The frequency-domain displacements of the half-space surface are expressed as slowness integrals, and the most prominent parts of the integrands are the reflectivities of the layers below and above the point source and a function which is closely related to the transmissivity of the layers above the source. Reflectivities and transmissivities are calculated by recursive methods which are numerically stable for all frequencies and slownesses. Near- and far-field results are given for single-force and moment-tensor point sources. From the general results for the complete medium response, partial responses can easily be extracted, e.g. the original form of the reflectivity method which calculates only the response from the layers below the source. Thus, the extended reflectivity method has a flexibility which is not available if propagator methods are used for the calculation of the integrands. Various other aspects of seismogram calculation are addressed, such as extended sources, an earth-flattening transformation and the inclusion of absorption for constant and frequency-dependent Q. Theoretical seismograms are shown, first for body-wave propagation from explosions in a crustal model and in a model which came from seismic prospecting, and second for surface-wave propagation from a double-couple source. Due to the tutorial nature of this paper the derivations are mostly rather detailed. It is hoped that this will help interested newcomers to the field of theoretical seismograms to get started.
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