Plane waves in three dimensions

Computing the proper scale factor for a seismogram

With data like Figure 3, rescaling traces to have equal energy would obviously be wrong. The question is, ``How can we determine the proper scale factor?'' As we have seen, a superposition of N plane waves exactly satisfies an N-th order (in $x$ ) difference equation. Given a 2-D wave field, we can find its PEF by minimizing output power. Then we ask the question, could rescaling the traces give a lower output power? To answer this, we set up an optimization goal: Given the leveler (be it a cubic PEF or two planar ones), find the best trace scales. (After solving this, we could return to re-estimate the leveler, and iterate.) To solve for the scales, we need a subroutine that scales traces and the only tricky part is that the adjoint should bring us back to the space of scale factors. This is done by scaletrace user/gee/scaletrace.c

    for (i=i2=0; i2 < n2; i2++) {
	for (i1=0; i1 < n1; i1++, i++) {
	    if( adj) scale[i2] += sdata[i] * data[i];
	    else     sdata[i] += scale[i2] * data[i];
	}
    }

Notice that to estimate scales, the adjoint forms an inner product of the raw data on the previously scaled data. Let the operator implemented by scaletrace be denoted by $\bold D$ , which is mnemonic for ``data'' and for ``diagonal matrix,'' and let the vector of scale factors be denoted by $\bold s$ and the leveler by $\bold A$ . Now we consider the fitting goal $\bold 0\approx \bold A \bold D \bold s$ . The trouble with this fitting goal is that the solution is obviously $\bold s = \bold 0$ . To avoid the trivial solution $\bold s = \bold 0$ , we can choose from a variety of supplemental fitting goals. One possibility is that for the $i$ -th scale factor we could add the fitting goal $s_i\approx 1$ . Another possibility, perhaps better if some of the signals have the opposite of the correct polarity, is that the sum of the scales should be approximately unity. I regret that time has not yet allowed me to identify some interesting examples and work them through.

Plane waves in three dimensions