Review

In inversion we try to estimate some model $\bf m$ given some data $\bf d$ and an operator $\bf L$ that maps between the quantities. If our problem is poorly constrained, we can employ Tikhonov regularization Tikhonov and Arsenin (1977), adding a roughening operator $\bf A$ to our objective function Q. To balance the two components of the objective function we introduce a twiddle parameter $\epsilon$and end up with

$$Q(\bf m) =\vert\vert \bf d- \bf L\bf m\vert\vert^2 + \epsilon^2 \vert\vert \bf A\bf m\vert\vert^2.$$ (1)

The two terms in our objective function serve different purposes. The first deals with data fitting and the second model styling. We can write the minimization in a slightly different form in terms of two fitting goals,

$$\begin{eqnarray} \bf 0&\approx&\bf r_{data} = \bf d- \bf L\bf m\\ \bf 0&\approx&\bf r_{model} = \epsilon \bf A\bf m\nonumber,\end{eqnarray}$$ (2)

where $\bf 0$ is a vector of zeros, $\bf r_{data}$ is the data residual vector, and $\bf r_{model}$ is the model. Our regularization operator, at best, usually only accounts for second order statistics, producing a model that is often unrealistic. In previous papers Clapp (2000, 2001a) I showed how by adding Gaussian random noise to the $\bf r_{model}$ we can add variance to our models and give the a more realistic texture.

If we decorrelate our data residual vector by adding an inverse noise covariance operator $\bf N$,

$$\begin{eqnarray} \bf 0&\approx&\bf r_{data} = \bf N(\bf d- \bf L\bf m) \\ \bf 0&\approx&\bf r_{model} = \epsilon \bf A\bf m\nonumber,\end{eqnarray}$$ (3)

we can account for uncertainty in our data Clapp (2001b). This is similar, but not the same as, using stochastic simulation Isaaks and Srivastava (1989a,b) to create several different datasets. The two most notable differences are that we can handle much more spatially variant and complex covariance descriptions and we have the effect of a model styling goal in our inversion.

 

• Tomography
• Multiple realization methodology