Results

At this time I am testing this formulation on synthetic data. I created a ten trace angle gather (Figure 1) using the basic form of Bortfeld's equations. Figure 1 also contains the stacked trace of this angle gather. There are five events with different reflectivities.

 

input
Figure 1 Input angle gather. The bottom trace is the stacked trace.

The results of the inversion are in Figure 2 and are fairly accurate. The answers are:

Event Correct R_O Est. R_O Correct R_sh Est. R_sh Correct R_P Est. R_P

1 0.023 0.0230 0.0 -0.00139 0.023 0.0257

2 0.035 0.0350 -0.01 -0.0132 0.023 0.0251

3 0.01 0.0100 0.01 0.00572 0.03 0.0342

4 -0.03 -0.0300 0.0 -0.00476 0.03 0.0367

5 0.02 0.0200 -0.02 -0.0175 -0.02 -0.0229

 

output
Figure 2 Output reflectivities. Top: R_O, middle: R_sh, bottom: R_P.

The solutions for R_O have been found correctly. There is some error in the solution for R_P and this in turn has produced some error in the calculated values for R_sh. However, the errors are small so it seems that this form can be successfully inverted. The fact that it is not dependent on interval velocities makes it preferable to other forms of the Bortfeld approximation.