Traditional methods of synthesizing seismograms in the near or regional distance range (0 to 500 km) describe the earth model with planar homogeneous layers. In the high-frequency band (0.2 to 10 Hz), in which the seismic ground motion is primarily observed, uniformly asymptotic solutions to the depth eigenfunctions can, instead, allow a radially symmetric earth model to be described by inhomogeneous spherical layers. Ground displacement u is calculated in the frequency domain by 
where Γ is a contour in the complex ray parameter (p) plane, and M (ω, p, ϕ, φ) a point representation of the earthquake or explosion source including the effect of horizontal propagation to Δ. The response of the earth model f(ω, p), calculated from the propagator matrix equation for a source in a radially inhomogeneous sphere, includes all possible body and surface waves. Airy functions are chosen to define the inhomogeneous layer matrices. Numerical difficulties usually encountered in the calculation of f(ω, p) in layered media are avoided by the calculation of subdeterminants of the fundamental matrix solution and by the decomposition of the propagator matrix in a layer into a sum of matrices of differing numerical order whenever the Airy functions behave exponentially. The contour integral can be evaluated by the residue theorem or by numerical integration, the time-domain response obtained by Fast Fourier transform. Most applications require an earth model to be described by no more than four to five inhomogeneous layers.

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