Incorporating gain into inversion

  In the inversions considered so far, the signal $\sv s$ and noise $\sv n$ have been characterized only by the filters $\st S$ and $\st N$,limiting the description of the data by the filters to the spectrum. For some applications, more information must be specified. In particular, the amplitudes of the seismic data should be considered. Recordings of seismic data show rapid weakening of the signal with time caused by a combination of wavefront spreading and the attenuation of the earth. If this weakening of the signal is not compensated for when an inversion is attempted, most of the inversion's effort will be expended on the strong shallow portion of the records. The weaker deeper portion of the records, which appears small in the least-squares sense, will be almost ignored by a least-squares solver. To equalize the treatments of the shallow and deep portions of the seismic records, this amplitude difference must be accounted for.

Another reason to account for the amplitudes is to take advantage of the extra information contained in the differences of the expected amplitudes of the noise and signal. Much noise originates from the surface and will either be of constant amplitude or weaken at a slower rate than does the signal. For example, in chapter  the noise is expected to be of constant amplitude, whereas the signal is expected to weaken as t², where t is sample time.

One method of accounting for the weakening of the signal would be to gain the input by t², but this would strengthen the noise at depth. A better method would be to account for the amplitude differences in the inversion itself. These amplitude differences may be taken advantage of by using them as part of the characterization of the signal and noise. As an example, suppose the noise amplitude falls off as t and the signal amplitude falls off as t². Systems () and  () may be modified to become  

$$0 \approx \st S t^2 \sv s$$ (56)

 

$$0 \approx \st N t \sv n,$$ (57)

then by substituting $\sv d-\sv n$ for $\sv s$,system () becomes  

$$\left( \begin{array} {c} \st S t^2 \sv d \\ 0 \end{array} ... ...{c} \st S t^2 \\ \epsilon \st N t \end{array} \right) \sv n.$$ (58)

The factors t and t² might be considered as weights that control the distribution of the expected signal and noise, as well as being factors that equalize the contributions of the signal and noise to the output. The factors t and t² in system () are in fact matrices that have the values of t and t² along the diagonal that correspond to the time values of the samples of $\sv n$ and $\sv s$.For simplicity, I will represent these matrices with t and t² here and later in this thesis.

Notice that the assumption of stationarity for $\sv s$ and $\sv n$ has been violated somewhat by the time scaling. This is not necessarily a problem since the filters $\st S$ and $\st N$involve only the spectrum of the signal and noise. This spectrum could be assumed to be constant. The functions t and t² that balance the contributions of $\st S$ and $\st N$ in () would presumably be applied to the data from which $\st S$ and $\st N$ are calculated so the scaled $\sv s$ and n would be stationary. In system () it is assumed that the filters $\st S$ and $\st N$are small enough to ignore the variation of $\sv s$ and $\sv n$ within the filter caused by the t and t² scaling. If this is not true, there will be a difference between applying the scaling before the filters and applying the scaling after the filters.