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  

module scaletrace {
integer, private                :: n1, n2
real, dimension( :, :), pointer :: data
#%  _init( data)
n1 = size( data, 1)
n2 = size( data, 2)
#% _lop( scale( n2), sdata( n1, n2))
integer i1,i2
do i2= 1, n2
do i1= 1, n1
        if( adj) 
                scale(   i2) += sdata(i1,i2) * data(i1,i2)
	else
                sdata(i1,i2) += scale(   i2) * data(i1,i2)
}

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.