Pitfalls in setting parameters

The output dispersion curves computed by disper, typically saved in a file named earth.crv, contains rows, each row for a computed frequency. The other elements in the row are wavenumbers for each mode. This frequency increment is set by the aperture implied by file disper.d. In running gendis, the aperture questions are the first two (sample interval and tmax). The frequency step size is $\Delta f= 1/(N\Delta t)=1/tmax$, where $\Delta t$ is the sample interval. The program waves must use the exact same frequency step size. When running genwav, answer the question for maximum trace time, tmax, using the same value as when running gendis. The sample rate question (really sample interval) should also be answered using the same value as with gendis. This will guarantee that the two programs are consistent.

However, even if one does specify the same aperture, there still may be a problem with the implied model. Program disper uses parameter deltz to subdivide intervals between control points with layers. The smaller deltz, the more layers. In the above example, a single layer over a half space is forced using three control points. The first two (depth=0 and depth=2.0) have the same material properties, making that entire interval produce the same result regardless of the deltz. The discontinuity is represented by a third control point just slightly below the second (at depth=2.001). While abrupt, a finite interval exists between 2.000 and 2.001 meters depth. When running waves, a smaller interval than deltz is required to compute the energy integrals which yield group velocity and the lagrangian. The smaller interval is set by a computed “minimum wavelength”, subdivided by namelist parameter, stepz. If stepz is too large, the implied depth interval (analogous to deltz) may actually subdivide the discontinuity into additional layers. The minimum wavelength is given by

$$\lambda = \dfrac{\beta_{minimum}}{f_{max}}$$ (51)

This means that changes, like extending the maximum frequency computed, may lead to a resampling of the discontinuity into layers. This problem presents itself as a glich in the group velocities and an increase in the lagrangian. The corrective action is to edit the waves.d namelist file, decreasing the stepz parameter. Alternatively, one may wish to re-run disper making the discontinuity thinner. Figure 64 illustrates the problem in (B).

Figure 64: (A). Correct waves computation of dispersion. (B). Illustrates too large a depth difference between top and bottom of the discontinuity. The solution is to make the discontinuity more abrupt in disper.d or decreasing stepz in waves.d to remove the glitches.