Assumptions about the data and the errors

In this thesis, the emphasis will be on signal processing aspects of inversion. The problems to be dealt with will be time series or collections of time series. A single time series, referred to as a trace, generates a one-dimensional problem. A collection of traces generates multi-dimensional problems. The data recorded $\sv d$ will be analyzed to produce some information about the reflection series $\sv r$.The series $\sv r$ may also be considered to be the error in the expression $\sv 0=\st F\sv d$, where $\st F$ is a filter that is designed to remove predictable information. Therefore, the reflection series $\sv r$ is assumed to be the unpredictable part of the series $\sv d$.The data is assumed to be stationary, Gaussian, and have a zero mean. Stationarity means that the the statistical character of the data does not change in time. This means that any statistical measure is expected to be unchanged if the trace is shifted. This assumption can be enforced by windowing the data so there are no large character changes within a window. The assumption of a Gaussian distribution might be more difficult to confirm but is still reasonable, since errors that are the result of some kind of summation tend to have Gaussian distributions. The zero mean is generally not a problem, since seismic data normally have had some kind of filter applied, and there is no reason to assume that the errors have a bias.

The reflection series, or error $\sv r$,produced from the filtering operation is also assumed to be stationary, Gaussian, and have a zero mean, but in addition, the samples of the errors are assumed to be independent from the other samples in $\sv r$.These assumptions may be better described by expectations.

The errors may be considered to be realizations of a random process drawn from a population, or ensemble. The expectation of a function of a random variable x is expressed as

$$E[f(x)] = \int_{-\infty}^\infty f(x) \varphi(x) dx$$ (17)

in the continuous case, where $\varphi(x) dx$ is the probability densityKorn and Korn (1968). In the discrete case

$$E[f(x_i)] = \sum_i f(x_i) p_{x_i},$$ (18)

where p_xi is the probability function of x_i. The summation is over all x_i, and p_xi sums to unity. For Gaussian distributions with zero mean, the expectation may considered to be an average as the number of samples goes to infinity, so that the expectation of the sample values is zero.

For the errors $\sv r$ and the data $\sv d$,the assumption of a zero mean is expressed as $E[\sv r]=0$ and $E[\sv d]=0$.The assumption that the samples of $\sv r$ are uncorrelated becomes $E[\sv r \sv r^{\dagger}] = \sigma_r^2 \st I$, where $\st I$ is the identity matrix, and $\dagger$ indicates the conjugate transpose, or adjoint. The dependence of the data values on each other is expressed as $E[\sv d \sv d^{\dagger}] = \st V$,where $\st V$ is the covariance matrix. The Gaussian distribution of $\sv r$ and $\sv d$ may be expressed as $p(\sv r) \propto e^{-\sv r^{\dagger} \sigma_r^{-2} \st I \sv r / 2 }$and $p(\sv d) \propto e^{-\sv d^{\dagger} \st V^{-1} \sv d /2 }$,where $\sigma_r$ is a scalar and $\st I$ is the identity matrix. Notice that these probability functions are distributions that satisfy the zero mean assumption. Also note that the independence of the samples in the errors is seen in the $\sigma_r^{-1} \st I$ factor in $p(\sv r)$,whereas the dependence of the data samples is seen in the inverse covariance matrix $\st V^{-1}$ in $p(\sv d)$.