Datuming from an irregular datum to a planar datum, upward

Let us now increase the degree of complexity and consider datuming from an irregular datum upward to a planar datum. As in the previous example, a scatterer is located at (z=0.15, y=0.32) in a constant velocity v=1 medium. And there is a supposedly irregular earth surface that can be expressed as the function iz = 10 + 5 sin( 2.*3.14*2* iy/ny). The maximum elevation difference is 50 m. The datuming geometry is shown in Figure 3. This kind of geometry is not unusual in land seismic surveys such as those of the mountainous regions in southwest China. The indices iz, iy are the depth and horizontal grid indices. I have chosen this particular surface for the sake of convenience. Any irregular surface can be used, as long as every boundary grid point as defined by the boundary has a recorded trace. No gaps can exist on the boundary grid. I generate the synthetic zero-offset section in Figure 4 at the irregular datum by ray tracing. The diffraction time curve deviates greatly from a perfect hyperbola because of the irregular recording surface.

 

Datum2.H Figure 3 The datuming geometry of Figures 4, 5, and 6. The point S represents the position of the scatterer. The asterisk curve represents the field recording datum. The dotted and dashed lines represent the new datums for upward and downward datuming.

 

d2 Figure 4 The wavefields recorded at the irregular surface datum.

Then I use a grid with nz=20, ny=128, dz=0.005, dy=0.005, and dt=0.001 to run the two-way wave equation forward in time, with the synthetic record as the boundary condition at the irregular recording surface, which is located inside the computational rectangle. The top and the two sides of the rectangle are designed as absorbing boundaries. The bottom boundary can be assigned any condition because it does not affect the computation. The desired wavefields at the new datum surface are the absorbing history wavefields at the top iz=1 of the grid. The results are shown in Figure 5. The datuming is done by an average of 50 m. Now the diffraction time curve has been restored to a hyperbola, and the apex of the hyperbola is at time t=0.2, and lateral position y=0.32. Boundary reflection effects appear at the two sides, because in datuming to a higher datum the wave incidence angle upon the two sides increases, and the effects of absorbing boundary conditions (Clayton and Engquist, 1977) at large incidence angles are not as good as those at small incidence angles. By upward datuming to a planar datum whose elevation equals that of the highest point of the irregular topography, the edge reflection can be decreased. The other way to deal with boundary reflections is to pad with zeroed traces beyond the edge as placeholders for the energy that migrates out of the original grid during the datuming extrapolation process (Berryhill, 1984). There is a light ghost following the main hyperbola, which looks like the lateral reverse of the original diffraction curve. The ghost reflection results from the reverberations at the input datum. The ghost can be removed by implementing the recorded wavefields as a series of point sources instead of simple boundary conditions.

 

d2.up Figure 5 The extrapolated wavefields at an upper planar datum.