Let us review the big picture. In Chapter we developed adjoints and in Chapter we developed inverse operators. Logically, correct solutions come only through inversion. Real life, however, seems nearly the opposite. This is puzzling but intriguing.

Every time you fill your car with gasoline, it derives much more from the adjoint than from inversion. I refer to the fact that ``practical seismic data processing'' relates much more to the use of adjoints than of inverses. It has been widely known for about the last 15 years that medical imaging and all basic image creation methods are like this. It might seem that an easy path to fame and profit would be to introduce the notion of inversion, but it is not that easy. Both cost and result quality enter the picture.

First consider cost.
For simplicity, consider a data space with *N* values
and a model (or image) space of the same size.
The computational cost of applying a dense adjoint
operator increases in direct proportion to the number
of elements in the matrix, in this case *N ^{2}*.
To achieve the minimum discrepancy between theoretical data
and observed data (inversion) theoretically requires

Consider an image of size .Continuing, for simplicity, to assume a dense matrix of relations between
model and data,
the cost for the adjoint is *m ^{4}* whereas the cost for inversion is

adjoint cost | (2^{9} 2^{9})^{2} |
2^{36} |
||

inverse cost | (2^{6} 2^{6})^{3} |
2^{36} |

These numbers tell us that for applications with dense operators, the biggest images that we are likely to see coming from inversion methods are whereas those from adjoint methods are .For comparison, the retina of your eye is comparable to your computer screen at .We might summarize by saying that while adjoint methods are less than perfect, inverse methods are ``legally blind'' :-)

http://sepwww.stanford.edu/sep/jon/family/jos/gifmovie.html holds a movie blinking between Figures and .

512x512
Jos greets Andrew, ``Welcome back Andrew''
from the Peace Corps.
At a resolution of , this picture
is about the same as the resolution
as the paper it is printed on,
or the same as your viewing screen,
if you have scaled it to 50% of screen size.
Figure 1 |

64x64
Jos greets Andrew, ``Welcome back Andrew'' again.
At a resolution of the pixels are clearly visible.
From far the pictures are the same.
From near, examine their glasses.
Figure 2 |

This cost analysis is oversimplified in that most applications do not require dense operators. With sparse operators, the cost advantage of adjoints is even more pronounced since for adjoints, the cost savings of operator sparseness translate directly to real cost savings. The situation is less favorable and much more muddy for inversion. The reason that Chapter 2 covers iterative methods and neglects exact methods is that in practice iterative methods are not run to their theoretical completion but they run until we run out of patience.

Cost is a big part of the story, but the story has many other parts. Inversion, while being the only logical path to the best answer, is a path littered with pitfalls. The first pitfall is that the data is rarely able to determine a complete solution reliably. Generally there are aspects of the image that are not learnable from the data.

In this chapter we study the simplest, most transparant example
of data insufficiency.
Data exists at irregularly spaced positions in a plane.
We set up a cartesian mesh and we discover that some
of the bins contain no data points.
What then?

- MISSING DATA IN ONE DIMENSION
- WELLS NOT MATCHING THE SEISMIC MAP
- SEARCHING THE SEA OF GALILEE
- INVERSE LINEAR INTERPOLATION
- PREJUDICE, BULLHEADEDNESS, AND CROSS VALIDATION

4/27/2004