Preview for inverse theorists

People who are already familiar with ``geophysical inverse theory'' may wonder what new they can gain from a book focused on ``estimation of images.'' Given a matrix relation $\bold d = \bold F \bold m$ between model $\bold m$ and data $\bold d$,common sense suggests that practitioners should find $\bold m$in order to minimize the length $\vert\vert \bold r \vert\vert$ of the residual $\bold r = \bold F \bold m - \bold d$.A theory of Gauss suggests that a better (minimum variance, unbiased) estimate results from minimizing the quadratic form $\bold r' \bold \sigma_{rr}^{-1}\bold r$,where $\bold \sigma_{rr}$ is the noise covariance matrix. I have never seen an application in which the noise covariance matrix was given, but practitioners often find ways to estimate it: they regard various sums as ensemble averages.

Additional features of inverse theory are exhibited by the partitioned matrix  

$$\bold d \eq \left[ \begin{array} {c} \bold d_{\rm incons} ... ...\ \bold m_{\rm null} \end{array} \right] \eq \bold F \bold m$$ (1)

which shows that a portion $\bold d_{\rm incons}$ of the data should vanish for any model $\bold m$,so an observed nonvanishing $\bold d_{\rm incons}$is inconsistent with any theoretical model $\bold m$.Likewise the $\bold m_{\rm null}$ part of the model space makes no contribution to the data space, so it seems not knowable from the data.

Simple inverse theory suggests we should minimize $\vert\vert \bold m \vert\vert$which amounts to setting the null space to zero. Baysian inverse theory says we should use the model covariance matrix $\bold \sigma_{mm}$and minimize $\bold m' \bold \sigma_{mm}^{-1}\bold m$for a better answer although it would include some nonzero portion of the null space. Never have I seen an application in which the model-covariance matrix was a given prior. Specifying or estimating it is a puzzle for experimentalists. For example, when a model space $\bold m$ is a signal (having components that are a function of time) or, a stratified earth model (with components that are function of depth z) we might supplement the fitting goal $\bold 0 \approx \bold r = \bold F\bold m - \bold d$with a ``minimum wiggliness'' goal like $dm(z)/dz\approx 0$.Neither the model covariance matrix nor the null space $\bold m_{\rm null}$seems learnable from the data and equation (0.1).

In fact, both the null space and the model covariance matrix can be estimated from the data and that is one of the novelties of this book. To convince you it is possible (without launching into the main body of the book), I offer a simple example of an operator and data set from which your human intuition will immediately tell you what you want for the whole model space, including the null space. Consider the data to be a sinusoidal function of time (or depth) and take $\bold B=\bold I$ so that the operator $\bold F$ is a delay operator with truncation of the signal shifted off the end of the space. Solving for $\bold m_{\rm fit}$, the findable part of the model, you get a back-shifted sinusoid. Your human intuition, not any mathematics here, tells you that the truncated part of the model, $\bold m_{\rm null}$,should be a logical continuation of the sinusoid $\bold m_{\rm fit}$ at the same frequency. It should not have a different frequency nor become a square wave nor be a sinusoid abruptly truncated to zero $\bold m_{\rm null}=\bold 0$.

Prior knowledge exploited in this book is that unknowns are functions of time and space (so the covariance matrix has known structure). This structure gives them predictability. Predictable functions in 1-D are tides, in 2-D are lines on images (linements), in 3-D are sedimentary layers, and in 4-D are wavefronts. The tool we need to best handle this predictability is the multidimensional ``prediction-error filter'' (PEF), a central theme of this book.