CALCULATION OF THE TIME/SPACE DOMAIN RESPONSE

Given data in the slowness,frequency domain $P(\omega,p_x,p_y)$ 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 p_x and p_y 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 p_z 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 p_min and p_max along which to evaluate the integral. In my program I choose the contour lying along the line between $p_{min}(1+\epsilon i)$ and $p_{max}(1+\epsilon i)$.This contour is shown in figure . In practice I choose $\epsilon = 10^{-3}$ and I do not bother to sample the contour along the small vertical parts of the contour.

 

contour
Figure 10 Integration contour is complex slowness domain. The contour avoids the singularities.

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.

 

xt1
Figure 11 Primaries only data in the x-t domain. Traction applied along x-axis, velocities measured along x-axis.

 

xt2
Figure 12 Primaries + direct arrivals in the x-t domain. Traction applied along x-axis, velocities measured along x-axis.

 

xt3
Figure 13 Primaries and first order multiples in the x-t domain. Traction applied along x-axis, velocities measured along x-axis.

Figure  shows the results of modeling a full nine-component shot gather. The shot is oriented at $30^\circ$ 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.

 

gridplot
Figure 14 Synthetic nine component shot gather. Top row is response to x-traction, middle = y-traction, bottom = z-traction. First column = x-displacement, second = y-displacement, third = z-displacement.