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
		convolve 		crosscorrelate
		truncate 		zero pad
		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
		stretch 		squeeze
		scalar field gradient 		negative of vector field divergence
		upward continue 		downward continue
		diffraction modeling 		imaging by migration
		hyperbola modeling 		CDP stacking
		ray tracing 		tomography       

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 $\exp(i\omega t)$ 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. 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 $\bold y =\bold A \bold x$and another multiplys by the transpose $\tilde{\bold x} =\bold A' \bold y$(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 $\bold y =\bold A \bold x$and convert it to another to do $\tilde{\bold x} =\bold A' \bold y$.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 $\bold A$ and $\bold A'$ 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 problems.

 

• Programming linear operators
• FAMILIAR OPERATORS
  • Adjoint derivative
  • Transient convolution
  • Internal convolution
  • Zero padding is the transpose of truncation
  • Adjoints of products are reverse-ordered products of adjoints
  • Nearest-neighbor coordinates
  • Data-push binning
  • Linear interpolation
  • Spray and sum : scatter and gather
  • Causal and leaky integration
  • Backsolving, polynomial division and deconvolution
  • The basic low-cut filter
  • Nearest-neighbor normal moveout (NMO)
  • Coding chains and arrays
• ADJOINT DEFINED: DOT-PRODUCT TEST
  • Definition of a vector space
  • Dot-product test for validity of an adjoint
  • The word ``adjoint''
  • Matrix versus operator
  • Inverse operator
  • Automatic adjoints