Shaping a ghost to a spike

An exasperating problem in seismology is the ``ghost'' problem, in which a waveform is replicated a moment after it occurs because of a strong nearby reflection. In marine seismology the nearby reflector is the sea surface. Because the sea surface is near both the airgun and the hydrophones, it creates two ghosts. Upgoing and downgoing waves at the sea surface have opposite polarity because their pressures combine to zero at the surface. Thus waves seen in the hydrophone encounter the ghost operator $g_t=(1,0,0, \cdots ,-1)$ twice, once for the surface near the source and once for the surface near the hydrophone. The number of zeros is typically small, depending on the depth of the device. The sound receivers can be kept away from surface-water wave noise by positioning them deeper, but that extends the ghost delay; and as we will see, this particular ghost is very hard to eliminate by processing. For simplicity, let us analyze just one of the two ghosts. Take it to be G(Z)=1-Z². Theoretically, the inverse is of infinite duration, namely, $(1,0,1,0,1,0,1,0,1, \cdots)$.

Since an infinitely long operator is not satisfactory, I used the program shaper() above to solve a least-squares problem for an antighost operator of finite duration. Since we know that the least-squares method abhors large errors and thus tends to equalize them, we should be able to guess the result.

The filter (.9, .0, .8, .0,.7 ,.0, .6, .0, .5, .0, .4, .0, .3, .0, .2, .0, .1), when convolved with (1,0,-1), produces the desired spike (impulse) along with equal squared errors of .01 at each output time. Thus, the least-squares filter has the same problem as the analytical one--it is very long. This disappointment can be described in the Fourier domain by the many zeros in the spectrum of (1,0,-1). Since we cannot divide by zero, we should not try to divide by 1-Zⁿ, which has zeros uniformly distributed on the unit circle. The method of least squares prevents disaster, but it cannot perform miracles.

I consider ghosts to be a problem in search of a different solution. Ghosts also arise when seismograms are recorded in a shallow borehole. As mentioned, the total problem generally includes many waveforms propagating in more than one direction; thus it is not as one-dimensional as it may appear in Figures 3 and 1, in which I did not display the wide-offset signals.

EXERCISES:

1. What inputs to subroutine shaper() give the filter $(.9,0,.8, \cdots .1)$mentioned above?
2. Figure 1 shows many seismograms that resemble each other but differ in the x location of the receiver. Sketch the overdetermined simultaneous equations that can be used to find the best-fitting source function S(Z), where $M_x(Z) S(Z) \approx P_x(Z)^2$ for various x.
3. Continue solving the previous problem by defining a contranx() subroutine that includes several signals going through the same filter. In order to substitute your contranx() into shaper() to replace contran() , you will need to be sure that the output and the filter are adjoint (not the output and the input). Suggestion: define real xx(nt,nx), etc.