Synthetic 3-D data

We generate a synthetic 3-D dataset using the formula

$$t_{shift}=t_0+a{\cdot}i_2+b{\cdot}i_3, (i_2=1,2,...,n_2; i_3=1,2,...,n_3)$$

First, a white noise trace is generated and passed through a low-pass filter. Then, according to the position of the trace on the n₂ by n₃ plane, we shift the trace by t_shift. As a special case, if we set a=0 and b=0, we will get a dataset consisting of many horizontal reflections, as shown in Figure 9. Otherwise, we will generate a series of dipping reflections, as shown in Figure 10 (a=1, b=1). Finally, we mix three datasets of different dipping reflections together $(0^\circ, 45^\circ, 135^\circ)$ and get a new dataset, as shown in Figure 11. The size of this cube is n₁=256, n₂=128, and n₃=128.

This dataset mainly consists of coherent component. Therefore, the result from horizontal reflections is nearly perfect. However, with the increase of dipping angle, the extent of coherency decreases and the compression ratio also decreases from several thousand to less than two hundred. This verifies our analysis in last section, i.e., steep dipping reflections are more difficult to compress. This is why we prefer to compress the dataset after NMO correction.

One interesting thing is, the result of the mixing case is better than the case of $45^\circ$ dipping reflections (difference of SNR is 1.17dB). This is because some high-frequency, high-wavenumber coherent noise is introduced into the dataset after compression. In the mixing case, the coherent noise cancels each other. Therefore the influence is not so serious. As shown in Figure 12 and 13, it is very easy to notice the high-frequency, high-wavenumber coherent component in the f-k domain. In the real seismic dataset, there exist reflections with many different dipping angles. This coherent noise usually will not cause big problem.

 

compdip0
Figure 9 Synthetic horizontal reflection section (scale=0.2). In this case, the compression ratio is 3009 and SNR is 46.91dB.
a: Original dataset.
b: Compressed dataset.
c: Difference of the two datasets.

 

compdip45
Figure 10 Synthetic dip reflection section (scale=0.28, dipping angle=$45^\circ$). In this case, the compression ratio is 149 and SNR is 21.49dB.
a: Original dataset.
b: Compressed dataset.
c: Difference of the two datasets.

 

compdipmix
Figure 11 Mixed section consisting of three different dipping events (scale=0.24, dipping angle=$0^\circ, 135^\circ, 45^\circ$). In this case, the compression ratio is 151 and SNR is 22.66dB.
a: Original dataset.
b: Compressed dataset.
c: Difference of the two datasets.

 

ftcompdipmix
Figure 12 Mixed section consisting of three different dipping events in f-k domain.
a: Original dataset.
b: Compressed dataset.

 

ftcompdip45
Figure 13 $45^\circ$ dip reflection section in f-k domain.
a: Original dataset.
b: Compressed dataset.