Analysis of Approximate Inverses in Tomography I. Resolution Analysis of Common Inverses

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.