Multidimensional autoregression

SOURCE WAVEFORM, MULTIPLE REFLECTIONS

Here we devise a simple mathematical model for deep water bottom multiple reflections.There are two unknown waveforms, the source waveform $S(\omega )$ and the ocean-floor reflection $F(\omega )$ . The water-bottom primary reflection $P(\omega )$ is the convolution of the source waveform with the water-bottom response; so $P(\omega )=S(\omega )F(\omega )$ . The first multiple reflection $M(\omega )$ sees the same source waveform, the ocean floor, a minus one for the free surface, and the ocean floor again. Thus the observations $P(\omega )$ and $M(\omega )$ as functions of the physical parameters are

$$P(\omega ) = S(\omega ) F(\omega )$$ (1)

$$M(\omega ) = -S(\omega ) F(\omega )^2$$ (2)

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

$$F(\omega ) = - M(\omega )/P(\omega )$$ (3)

$$S(\omega ) = - P(\omega )^2/M(\omega )$$ (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 $\epsilon$ to stabilize it. For example, multiply equation (3) on top and bottom by $P(\omega )\T$ and add $\epsilon >0$ to the denominator. This gives

$$F(\omega )\eq - \frac{M(\omega ) P\T(\omega )}{P(\omega )P(\omega )\T + \epsilon}$$ (5)

where $P\T(\omega )$ is the complex conjugate of $P(\omega )$ . Although the $\epsilon$ stabilization seems nice, it apparently produces a nonphysical model. For $\epsilon$ 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 $< \cdots >$ denote smoothing by local averaging. Thus, to specify filters whose time duration is not unreasonably long, we can revise equation (5) to

$$F(\omega )\eq - \frac{<M(\omega ) P\T(\omega )>}{<P(\omega )P\T(\omega ) >}$$ (6)

where instead of deciding a size for $\epsilon$ we need to decide how much smoothing. I find that smoothing has a simpler physical interpretation than choosing $\epsilon$ . The goal of finding the filters $F(\omega )$ and $S(\omega )$ 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 $F(\omega )$ and $S(\omega )$ . First express (3) as

$$0 \eq P(\omega )F(\omega ) +M(\omega )$$ (7)

To imagine equation (7) as a fitting goal in the time domain, instead of scalar functions of $\omega$ , think of vectors with components as a function of time. Thus $\bold f$ is a column vector containing the unknown sea-floor filter, $\bold m$ contains the ``multiple'' portion of a seismogram, and $\bold P$ is a matrix of down-shifted columns, each column being the ``primary''.

$$\bold 0 \quad\approx\quad \bold r \eq \left[ \begin{array}{c} r_1... ...ay}{c} m_1 \ m_2 \ m_3 \ m_4 \ m_5 \ m_6 \ m_7 \ m_8 \end{array} \right]$$ (8)

Multidimensional autoregression