Slope estimation tests

Figures 1 and 2 illustrate tests of the my nonlinear two-slope estimation algorithm. The ``textures'' Brown (1999) were computed by constructing nonstationary steering filters (the 9-point Lagrange filter derived by Clapp et al. (1997)) with the estimated slopes, and then using those filters to deconvolve random noise. The textures, used also by Fomel , provide a quick check of the accuracy of the estimated slopes.

In Figure 1, a simple crossing-plane-wave dataset is tested. The slope panels shown have been smoothed with a sliding weighted mean filter (4-by-4 analysis window). The program used to compute the slopes also computes the weights, which are either 1 or 0. If the estimated slopes at a single point in (t,x) are equal, then the result is assumed to be trivial and the weight at that point is set to 0. Otherwise, the weight is set to 1.

The textures in Figure 1 illustrate that the estimated slopes are not totally accurate. The steep positive slope in particular seems smaller than the true positive slopes in the data, while the shallower negative slope seems better represented.

Figure 2 illustrates a more difficult test dataset, a ``CMP gather'' overlain by upward-sloping linear ``noise.'' Notice that some regions of the data contain either one signal or the other. My slope estimation program, through the use of mask operators, allows the user to specify regions where only one slope is present in the data. In those regions, I use Claerbout's 1992 univariate ``puck'' method to estimate that single slope.

We notice some discontinuity in the textures in Figure 2 at one-slope/two-slope boundaries in the data. My two-slope algorithm slightly underestimates the positive ``noise'' slope, while in some sections of the data, it overestimates the magnitude of the ``CMP gather'' slope. Still, the general trend of both slopes honors those present in the data.

At the right and bottom edges of the estimated slopes in Figure 2, notice the constant-valued regions. Because the finite-difference templates of equation (7) run off the right and bottom edges of the data, the slope cannot (easily) be computed in these regions. In this case, the slope remains unchanged from the starting guesses, which in this case were -0.5 and 0.5. We expect the estimated slopes to exhibit some sensitivity to starting guess. I have experimented qualitatitvely, and indeed found some sensitivity, though it is not generally severe.

 

pucknl.dumb2
Figure 1 Test of my nonlinear two-slope estimation algorithm on a simple synthetic test case.

 

pucknl.dumb
Figure 2 Test of nonlinear 2-slope ``puck'' algorithm on a more realistic synthetic test.