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 **operator**s and their **adjoint**s:

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
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 and another that performs . 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.

- FAMILIAR OPERATORS
- Adjoint derivative
- Zero padding is the transpose of truncation
- Adjoints of products are reverse-ordered products of adjoints
- Nearest-neighbor coordinates
- Data-push binning
- Linear interpolation
- Causal integration

- ADJOINTS AND INVERSES

- About this document ...

Adjoint operators |

2009-03-16