Virtually all regional phases can be strongly affected by vertical velocity gradients. The best known effects are on the Pn and Sn, in which small changes in the vertical velocity gradient beneath the Moho produce large changes in the decay of Pn and Sn with distance. Methods of synthesizing complete regional seismograms often inadvertently ignore the effect of crustal gradients by parameterizing the Earth model with thick, planar homogeneous layers. To address this problem, we have modified the locked mode method of synthesizing complete regional seismograms to include the Langer uniform asymptotic approximation to vertical wavefunctions within layers having linear vertical velocity gradients. Synthesis of complete regional seismograms using the Langer-locked mode confirm that the Pn and Sn phases are strongly affected by the magnitude of the velocity gradients beneath the Moho, but that Lg is only weakly affected by the details of crustal layering.
Tests were made to quantify the error in the use of the Langer approximation as the magnitude of the vertical gradient increases and/or frequency decreases. At sufficiently small magnitude of gradient and/or high frequency, good argeement can be obtained between synthetics computed using the Langer locked mode method, the colocation method, and the conventional locked mode method in models parameterized by thin homogeneous layers. Errors introduced by the use of the Langer approximation in calculated pole positions, residues, and eigenfunctions are bounded by 5 per cent for frequencies f ≥ 5 | ∇ V |. An upper bound to the error in the time domain can be estimated from this inequality using either the peak frequency in a narrow pass band or the lowest frequency in a broad pass band. When 10 or more thin homogeneous layers are required to represent accurately the seismic wavefield in a gradient layer, it is usually more efficient to represent the gradient layer by continuously varying functions in the vertical direction and employ the Langer approximation, provided the errors in the Langer approximation remain within acceptable limits. By reducing the number of parameters needed to describe an earth model, the Langer locked mode method simplifies the inverse problem of determining structure using observed and synthetic complete seismograms. It also facilitates the use of known relations for the effects of continuously varying pressure and temperature on elastic moduli and density.