|Basic operators and adjoints|
A great many of the calculations we do in science and engineering are really matrix multiplication in disguise. The first goal of this chapter is to unmask the disguise by showing many examples. Second, we see how the adjoint operator (matrix transpose) back projects information from data to the underlying model.
Geophysical modeling calculations generally use linear operators that predict data from models. Our usual task is to find the inverse of these calculations; i.e., to find models (or make images) from the data. Logically, the adjoint is the first step and a part of all subsequent steps in this inversion process. Surprisingly, in practice the adjoint sometimes does a better job than the inverse! This is because the adjoint operator tolerates imperfections in the data and does not demand that the data provide full information.
Using the methods of this chapter, you will find that once you grasp the relationship between operators in general and their adjoints, you can obtain the adjoint just as soon as you have learned how to code the modeling operator.
If you will permit me a poet's license with words, I will offer you the following table of operators and their adjoints:
|matrix multiply||conjugate-transpose matrix multiply|
|replicate, scatter, spray||sum or stack|
|spray into neighborhoods||sum within bins|
|derivative (slope)||negative derivative|
|causal integration||anticausal integration|
|add functions||do integrals|
|assignment statements||added terms|
|plane-wave superposition||slant stack / beam form|
|superpose curves||sum along a curve|
|scalar field gradient||negative of vector field divergence|
|upward continue||downward continue|
|diffraction modeling||imaging by migration|
|hyperbola modeling||CDP stacking|
The left column above is often called ``modeling,'' and the adjoint operators on the right are often used in ``data processing.''
When the adjoint operator is not an adequate approximation to the inverse, then you apply the techniques of fitting and optimization explained in Chapter . These techniques require iterative use of the modeling operator and its adjoint.
The adjoint operator is sometimes called the ``back projection'' operator because information propagated in one direction (earth to data) is projected backward (data to earth model). Using complex-valued operators, the transpose and complex conjugate go together; and in Fourier analysis, taking the complex conjugate of reverses the sense of time. With more poetic license, I say that adjoint operators undo the time and phase shifts of modeling operators. The inverse operator does this too, but it also divides out the color. For example, when linear interpolation is done, then high frequencies are smoothed out, so inverse interpolation must restore them. You can imagine the possibilities for noise amplification. That is why adjoints are safer than inverses. But nature determines in each application what is the best operator to use, and whether to stop after the adjoint, to go the whole way to the inverse, or to stop partway.
The operators and adjoints above transform vectors to other vectors. They also transform data planes to model planes, volumes, etc. A mathematical operator transforms an ``abstract vector'' which might be packed full of volumes of information like television signals (time series) can pack together a movie, a sequence of frames. We can always think of the operator as being a matrix but the matrix can be truly huge (and nearly empty). When the vectors transformed by the matrices are large like geophysical data set sizes then the matrix sizes are ``large squared,'' far too big for computers. Thus although we can always think of an operator as a matrix, in practice, we handle an operator differently. Each practical application requires the practitioner to prepare two computer programs. One performs the matrix multiply and another multiplys by the transpose (without ever having the matrix itself in memory). It is always easy to transpose a matrix. It is less easy to take a computer program that does and convert it to another to do . In this chapter are many examples of increasing complexity. At the end of the chapter we will see a test for any program pair to see whether the operators and are mutually adjoint as they should be. Doing the job correctly (coding adjoints without making approximations) will reward us later when we tackle model and image estimation applications.
Mathematicians often denote the transpose of a matrix by . In physics and engineering we often encounter complex numbers. There the adjoint is the complex-conjugate transposed matrix denoted . In this book, the adjoint is simply written as . What this book calls the adjoint is more properly called the Hilbert adjoint.
|Basic operators and adjoints|