Given data in the slowness,frequency domain I wish to calculate the response in the time,offset domain P(t,x,y). This response can be calculated by a double inverse slant stack and inverse fourier transform of the input data.
In the discrete modeling scheme the integral over px and py is replaced by a discrete summation and the inverse fourier transform is replaced by an inverse discrete fourier transform.
There are problems with the slowness integration caused by singularities in the modeling operation. Whenever the modeling algorithm is performed for a slowness where pz is zero, or nearly zero, for some wavetype there is a singularity or instability in the modeling algorithm. These points occur when the waves are traveling horizontally. The integration may be performed in a stable manner by choosing to integrate in the complex slowness domain. I can choose some other contour between pmin and pmax along which to evaluate the integral. In my program I choose the contour lying along the line between and .This contour is shown in figure . In practice I choose and I do not bother to sample the contour along the small vertical parts of the contour.
Figures to show the same data examples as before but now they are calculated in the (x,t) domain. The most noticeable feature in the (x,t) domain is the triplication of the shear wave arrival. The amplitudes at the cusps of the triplication are relatively high. This matches well with theory and modeling results using other methods.
The P-wave direct arrival is clear in figure but no direct shear wave is seen because the integration in the slowness domain did not cover a large enough range.
Figure shows the results of modeling a full nine-component shot gather. The shot is oriented at to the symmetry axis of an orthorhombic medium. There are clear shear wave conversions and shear wave splitting in the data. The range of offsets and dips modeled was not as great as the previous examples so the shear wave triplications are not visible. The top row shows the x,y and z velocities for an x-traction. The second rows shows the results for a y-traction and the third row for a z-traction.