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source waveform
multiple reflection
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) |

Algebraically the solutions of equations
(1) and
(2) are
| |
(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) |

where is the complex conjugate of .Although the stabilization seems nice,
it apparently produces a nonphysical model.
For large or small, the time-domain response
could turn out to be of much greater duration than is physically reasonable.
This should not happen with perfect data, but in real life,
data always has a limited spectral band of good quality.
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) |

where it remains to specify the method and amount of smoothing.
The goal of finding the filters and is to
best model the multiple reflections so that they can
be subtracted from the data,
and thus enable us to see what primary reflections
have been hidden by the multiples.
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) |

** Next:** TIME-SERIES AUTOREGRESSION
** Up:** Multidimensional autoregression
** Previous:** Time domain versus frequency
Stanford Exploration Project

4/27/2004