Analysis of Approximate Inverses in Tomography I. Resolution Analysis of Common Inverses
BERRYMAN, JAMES G.
Primary Author
mixed material
bibliography
Netherlands
Springer Netherlands
2000
monographic
Volume 1, Number 1
en
English
29p
Optimization and Engineering
The process of using physical data to produce images of important physical parameters is an inversion
problem, and these are often called tomographic inverse problems when the arrangement of sources and receivers
makes an analogy to x-ray tomographic methods used in medical imaging possible. Examples of these methods
in geophysics include seismic tomography, ocean acoustic tomography, electrical resistance tomography, etc., and
many other examples could be given in nondestructive evaluation and other applications. All these imaging methods
have two stages: First, the data are operated upon in some fashion to produce the image of the desired physical
quantity. Second, the resulting image must be evaluated in essentially the same timeframe as the image is being used
as a diagnostic tool. If the resolution provided by the image is good enough, then a reliable diagnosis may ensue.
If the resolution is not good enough, then a reliable diagnosis is probably not possible. But the first question in this
second stage is always “How good is the resolution?” The concept of resolution operators and resolution matrices
has permeated the geophysics literature since the work of Backus and Gilbert in the late 1960s. But measures of
resolution have not always been computed as often as they should be because, for very data rich problems, these
computations can actually be significantly more difficult/expensive than computing the image itself.
It is the purpose of this paper and its companion (Part II) to show how resolution operators/matrices can be
computed economically in almost all cases, and to provide a means of comparing the resolution characteristics of
many of the common approximate inverse methods. Part I will introduce the main ideas and analyze the behavior
of standard methods such as damped least-squares, truncated singular value decomposition, the adjoint method,
backprojection formulas, etc. Part II will treat many of the standard iterative inversion methods including conjugate
gradients, Lanczos, LSQR, etc.
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