Dispersion curve

Surface wave dispersion curves are calculated by the method of Saito and Kabasawa (1993), an improved version of the Thomson-Haskell matrix method. As is the case for other methods, it searches dispersion relation in the frequency (f)-phase velocity (c) domain by looking for zeros in a characteristic function. The SEIS_FILO programs involve several parameters that control the search process.

Root search process

The search begins with the evaluation of the characteristic function at (f, c) = (xmin, cmin). This evaluation proceeds with an increment for c by dc until we obtain n_mode+1 zero crossings. Once the desired number of zero crossings occur, a binary search is performed for more tightly constraining the location of the zero-crossing, which is treated as phase velocity estimation at the corresponding frequency. A group velocity is then calculated by numerical differentiation centered on the estimated phase velocity. If c reaches cmax before finding the zero-crossing, the forward calculation terminates with an error message and will not proceed for higher frequencies.

The procedure above are repeated over a given frequency range from xmin to xmax with an interval of dx; but the only difference is initial phase velocity. Instead of using cmin as the starting point, we can start from a point close to the prediction by a dispersion curve slope, . In specific, the start value is given by , where a is an adjustable parameter which we fix at 3.5. Starting from this value, the zero-crossing first encountered is adopted as the target mode.

Full search mode

If one wish to obtain all dispersion curves within a certain region in the f-c domain given by (xmin, xmax) × (cmin, cmax), the full search mode may provide a solution. This mode is activated by setting n_mode<0 and outputs characteristic functions at all grid points with intervals of dx and dc.