Least-squares methods

While measures of the residuals other than the L-2 norm may be used, the least-squares method is used here. Norms between one and two may be obtained with the iteratively re-weighted least-squares (IRLS) methodDarche (1989); Green (1984); Nichols (1994b). Norms other than L-2, especially L-1, may be used in cases where the presence of high-amplitude events overwhelm the least-squares residuals.

In this thesis, two different approaches to high-amplitude errors are taken. In chapter , the problem of high-amplitude noise corrupting the filter calculation is solved by iteratively solving for a signal and a signal annihilation filter. Another approach is taken in chapter , where samples that produce large residuals are removed before the inversion, then, in chapter , the signal and noise are separated simultaneously with the prediction of the missing data caused by the sample removal. It is hoped that these techniques will solve most practical problems.

The calculation of a filter is done by solving $\sv 0 \approx \st D \sv f$,where $\st D$ is a matrix made up of the given data and $\sv f$ is the filter to be solved for. Claerbout1992a shows how to solve for this filter $\sv f$ by expressing the convolution of the filter $\sv f$ and the data $\st D$ as matrix operators and their adjoints. If the operator and its adjoint are available, a conjugate-gradient routine may be used to calculate a least-squares minimization of a systemLuenburger (1984). This approach to computing the filter simplifies the problem considerably. Two- and three-dimensional filters may be calculated as simply as a one-dimensional filter, provided the filter operation and its adjoint are available.