Here we devise a simple mathematical model for deep water bottom multiple reflections. There are two unknown waveforms, the source waveform and the ocean-floor reflection .The water-bottom primary reflection is the convolution of the source waveform with the water-bottom response; so .The first multiple reflection sees the same source waveform, the ocean floor, a minus one for the free surface, and the ocean floor again. Thus the observations and as functions of the physical parameters are
(1) | ||
(2) |
(3) | ||
(4) |
These solutions can be computed in the Fourier domain by simple division. The difficulty is that the divisors in equations (3) and (4) can be zero, or small. This difficulty can be attacked by use of a positive number to stabilize it. For example, multiply equation (3) on top and bottom by and add to the denominator. This gives
(5) |
Functions that are rough in the frequency domain will be long in the time domain. This suggests making a short function in the time domain by local smoothing in the frequency domain. Let the notation denote smoothing by local averaging. Thus, to specify filters whose time duration is not unreasonably long, we can revise equation (5) to
(6) |
These frequency-duration difficulties do not arise in a time-domain formulation. Unlike in the frequency domain, in the time domain it is easy and natural to limit the duration and location of the nonzero time range of and .First express (3) as
(7) |
To imagine equation (7) as a fitting goal in the time domain, instead of scalar functions of ,think of vectors with components as a function of time. Thus is a column vector containing the unknown sea-floor filter, contains the ``multiple'' portion of a seismogram, and is a matrix of down-shifted columns, each column being the ``primary''.
(8) |
To minimize ,we could use the conjugate-direction subroutine cgmeth() , but we would remove its call to the matrix multiply subroutine and replace it by a convolution subroutine with boundary conditions of our choice.