Adjoint operators

ADJOINTS AND INVERSES

Consider a model $\bold m$ and an operator $\bold F$ which creates some theoretical data $\bold d_{\rm theor}$.

$$\bold d_{\rm theor} \eq \bold F \bold m$$ (13)

The general task of geophysicists is to begin from observed data $\bold d_{\rm obs}$ and find an estimated model $\bold m_{\rm est}$ that satisfies the simultaneous equations

$$\bold d_{\rm obs} \eq \bold F \bold m_{\rm est}$$ (14)

This is the topic of a large discipline variously called ``inversion'' or ``estimation''. Basically, it defines a residual $\bold r = \bold d_{\rm obs}-\bold d_{\rm theor}$ and then minimizes its length $\bold r \cdot \bold r$. Finding $\bold m_{\rm est}$ this way is called the least squares method. The basic result (not proven here) is that

$$\bold m_{\rm est} = (\bold F'\bold F)^{-1}\bold F'\bold d_{\rm obs}$$ (15)

In many cases including all seismic imaging cases, the matrix $\bold F'\bold F$ is far too large to be invertible. People generally proceed by a rough guess at an approximation for $(\bold F'\bold F)^{-1}$. The usual first approximation is the optimistic one that $(\bold F'\bold F)^{-1}=\bold I$. To this happy approximation, the inverse $\bold F^{-1}$ is the adjoint $\bold F'$.

In this book we'll see examples where $\bold F'\bold F\approx \bold I$ is a good approximation and other examples where it isn't. We can tell how good the approximation is. We take some hypothetical data and convert it to a model, and use that model to make some reconstructed data $\bold d_{\rm recon} = \bold F \bold F' \bold d_{\rm hypo}$. Likewise we could go from a hypothetical model to some data and then to a reconstructed model $\bold m_{\rm recon} = \bold F' \bold F \bold m_{\rm hypo}$. Luckily, it often happens that the reconstructed differs from the hypothetical in some trivial way, like by a scaling factor, or by a scaling factor that is a function of physical location or time, or a scaling factor that is a function of frequency. It isn't always simply a matter of a scaling factor, but it often is, and when it is, we often simply redefine the operator to include the scaling factor. Observe that there are two places for scaling functions (or filters), one in model space, the other in data space.

We could do better than the adjoint by iterative modeling methods (conjugate gradients) that are also described elsewhere. These methods generally demand that the adjoint be computed correctly. As a result, we'll be a little careful about adjoints in this book to compute them correctly even though this book does not require them to be exactly correct.

Adjoint operators