THE SIMULATION

I applied Tieman's method on two different examples of synthetic data. The goal here is to compare the effects of sampling on the quality of the Tieman shot gather slant stacks with quality of standard slant stacks applied to shot gathers. The data sets were generated by an algorithm that generates a given number of regularly spaced constant fold cmp gathers with given offset spacing over a set of seventy five dipping reflectors. The reflectors are chosen to be randomly dipping in ranges from $-45^\circ < dip < +45^\circ$ at randomly chosen depths measured from the center of the midpoint array. I chose a large number of random reflectors to maximize the aliasing effects that occur on the shot gathers in hopes that the limits of either method, the Tieman or the standard slant stack, in distinguishing dips would be tested. Figure 1 is a constant offset section of the first example data set which depicts the geometry of this reflector model.

 

earth Figure 1 Reflector model in time and midpoint for a constant offset of two km

The sampling of the time axis and midpoint axis for both test data sets are 512 an 256 respectively while the sampling of the offset axis in the first test is 80 and 20 for the second test. In the following Figure 2 are examples of a cmp gather and a shot gather of this data set.

 

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Figure 2 Simulated gathers: (left) cmp gather, (right) shot gather

The shots were modeled as being shot offend, a situation commonly encountered in marine surveys, thus the gathers only have positive offsets. The disparity in symmetry between the hyperbolic events in the cmp gather vs. the asymmetric events in the shot gather is clearly evident.