Inversion and statistics

From a purely mathematical viewpoint, inversion of a linear system is solving for a vector $\sv m$ when a matrix $\st A$ and a vector $\sv d$are supplied in an expression $\st A\sv m=\sv d$.From an applied point of view, we are generally attempting to derive some information about a physical system when, from this system, a quantitative description of the system is built by choosing a set of parameters of interestTarantola (1987). This description of the physical system is referred to as the model $\sv m$and is represented by a vector of numbers that parameterize the physical model. Also available is a set of measurements, or data $\sv d$,collected in the effort to derive some information about model $\sv m$.The relationship between the data $\sv d$ and the model $\sv m$is assumed here to be linear and described by the matrix $\st A$,where $\st A\sv m=\sv d$.The problem of taking a model $\sv m$ and deriving the expected data $\sv d$ is referred to as the forward problem. This forward problem assumes that the relevant physics of the problem is described in the matrix $\st A$.The inverse problem involves calculating the model $\sv m$from a given set of data $\sv d$.As an example of the use of an inverse system in geophysics, a description of the earth is derived from measurements taken at the surface. The measurements from the surface correspond to $\sv d$,and the desired description of the earth corresponds to $\sv m$.Given $\st A$ and $\sv d$, the description of the earth $\sv m$ is to be calculated by inversion.

The measured data $\sv d$ are likely to include some uncertainty, which is generally due to effects not included in the model. For example, when trying to derive an earth model using seismic data, the relevant physical laws to be taken into account would be those governing the propagation of seismic energy through the earth. Noise, or effects not included in the model, would be, for example, wind, local traffic, animals, Brownian motion, and so on. While most extraneous effects are unpredictable, or at least very difficult to predict, these noises can often be assumed to be random. Allowing for these unpredictable effects may then be left to statistical methods where the data $\sv d$ are considered realizations of random variables. Since the data are random variables, the estimates of the model are also random variables.

In the inversion of the expression $\st A\sv m=\sv d$,the noise is considered undesirable and is eliminated as far as possible when calculating the desired $\sv m$.Much of the work here involves a different system in which the noise, or the unpredictable part of the data is of interest. This system is $\sv r = \sv f \ast \sv d$, where $\sv d$ is a recorded data series, $\sv f$ is a filter to be calculated, $\ast$ indicates convolution, and $\sv r$ is the unpredictable reflection series. This can be expressed in terms of the matrix equations $\sv r=\st F\sv d$ or $\sv r=\st D\sv f$, where $\st F$ is the filter $\sv f$ expressed as a filter convolution matrix, and $\st D$ is the data $\sv d$ expressed as a data convolution matrix. The next section addresses the assumptions made about $\sv r$ and $\sv d$.