Multidimensional autoregression |

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

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

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

where instead of deciding a size for we need to decide how much smoothing. I find that smoothing has a simpler physical interpretation than choosing . 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

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''.

Multidimensional autoregression |

2013-07-26