SCALING THE ADJOINT

First I remind you of a rarely used little bit of mathematical notation. Given a vector $\bold m$ with components (m₁,m₂,m₃), the notation ${\bf diag\ }\bold m$ means

$${\bf diag\ }\bold m \quad =\quad \left[ \begin{array} {ccc... ... & 0 & 0 \\ 0 &m_2 & 0 \\ 0 & 0 & m_3 \end{array} \right]$$ (38)

Given the usual linearized fitting goal between data space and model space, $\bold d \approx \bold F \bold m$,the simplest image of the model space results from application of the adjoint operator $\hat \bold m = \bold F' \bold d$.Unless $\bold F$ has no physical units, however, the physical units of $\hat \bold m$ do not match those of $\bold m$,so we need a scaling factor. The theoretical solution $\bold m_{\rm theor} = (\bold F'\bold F)^{-1}\bold F'\bold d$suggests that the scaling units should be those of $(\bold F'\bold F)^{-1}$.We could probe the operator $\bold F$or its adjoint with white noise or a zero-frequency input. Bill Symes suggests we probe with the data $\bold d$ because it has the spectrum of interest. He proposes we make our image with $\hat \bold m = \bold W^2 \bold F'\bold d$where we choose the weighting function to be  

$$\bold W^2 \quad =\quad { {\bf diag\ } {\bold F' \bold d} \over {\bf diag\ } {\bold F'\bold F\bold F'\bold d} }$$ (39)

which obviously has the correct physical units. (The mathematical function ${\bf diag}$ takes a vector and lies it along the diagonal of a square matrix.) The weight $\bold W^2$can be thought of as a diagonal matrix containing the ratio of two images. A problem with the choice (39) is that the denominator might vanish or might even be negative. The way to stabilize any ratio is suggested at the beginning of Chapter ; that is, we revise the ratio a/b to  

$$\bold W^2 \quad =\quad { {\bf diag} < ab \gt \over {\bf diag} < b^2 + \epsilon^2 \gt }$$ (40)

where $\epsilon$ is a parameter to be chosen, and the angle braces indicate the possible need for local smoothing.

To go beyond the scaled adjoint we can use $\bold W$ as a preconditioner. To use $\bold W$ as a preconditioner we define implicitly a new set of variables $\bold p$by the substitution $\bold m=\bold W\bold p$.Then $\bold d \approx \bold F\bold m=\bold F\bold W\bold p$.To find $\bold p$ instead of $\bold m$,we do CD iteration with the operator $\bold F\bold W$ instead of with $\bold F$.As usual, the first step of the iteration is to use the adjoint of $\bold d\approx \bold F\bold W\bold p$ to form the image $\hat\bold p=(\bold F\bold W)'\bold d$.At the end of the iterations, we convert from $\bold p$ back to $\bold m$with $\bold m=\bold W\bold p$.The result after the first iteration $\hat\bold m=\bold W\hat\bold p=\bold W(\bold F\bold W)'\bold d=\bold W^2\bold F'\bold d$turns out to be the same as Symes scaling.

By (39), $\bold W$ has physical units inverse to $\bold F$.Thus the transformation $\bold F\bold W$ has no units so the $\bold p$ variables have physical units of data space. Experimentalists might enjoy seeing the solution $\bold p$with its data units more than viewing the solution $\bold m$with its more theoretical model units.

The theoretical solution for underdetermined systems $\bold m =\bold F' (\bold F \bold F')^{-1} \bold d$suggests an alternate approach using instead $\hat{\bold m} =\bold F' \bold W_d^2 \bold d$.A possibility for $\bold W_d^2$ is  

$$\bold W_d^2 \quad =\quad { {\bf diag\ } {\bold d} \over {\bf diag\ } {\bold F\bold F'\bold d} }$$ (41)

Experience tells me that a broader methodology is needed. Appropriate scaling is required in both data space and model space. We need something that includes a weight for each space, $\bold W_m$ and $\bold W_d$ where $\hat \bold m = \bold W_m \bold F'\bold W_d \bold d$.

I have a useful practical example (stacking in v(z) media) in another of my electronic books (BEI), where I found both $\bold W_m$ and $\bold W_d$ by iterative guessing. But I don't know how to give you a simple strategy that is not iterative. Either this is a major unsolved opportunity for a theorist, or I'll need to write down my iterative guess.

The PhD thesis of James Rickett experiments extensively with data space and model space weighting functions in the context of seismic velocity estimation.