Adjoint operators

Adjoint operators

Jon Claerbout

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 maps) 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
	convolve		crosscorrelate
	truncate		zero pad
	replicate, scatter, spray		sum or stack
	spray into neighborhood		sum in 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 on a curve		sum along a curve
	stretch		squeeze
	upward continue		downward continue
	hyperbolic modeling		normal moveout and CDP stack
	diffraction modeling		imaging by migration
	ray tracing		tomography

The left column above is often called ``modeling,'' and the adjoint operators on the right are often used in ``data processing.''

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). For complex-valued operators, the transpose goes together with a complex conjugate. In Fourier analysis, taking the complex conjugate of $\exp(i\omega t)$ 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.

Later in this chapter we relate adjoint operators to inverse operators. Although inverse operators are more well known than adjoint operators, the inverse is built upon the adjoint so the adjoint is a logical place to start. Also, computing the inverse is a complicated process fraught with pitfalls whereas the computation of the adjoint is easy. It's a natural companion to the operator itself.

Much later in this chapter is a formal definition of adjoint operator. Throughout the chapter we handle an adjoint operator as a matrix transpose, but we hardly ever write down any matrices or their transposes. Instead, we always prepare two subroutines, one that performs $\bold y =\bold A \bold x$ and another that performs $\tilde{\bold x} =\bold A' \bold y$. So we need a test that the two subroutines really embody the essential aspects of matrix transposition. Although the test is an elegant and useful test and is itself a fundamental definition, curiously, that definition does not help construct adjoint operators, so we postpone a formal definition of adjoint until after we have seen many examples.

Adjoint operators