Bandlimited Functions

An objective of this series of experiments is to provide a preliminary assessment of the tomographic reconstruction approach for combustion diagnostics. In order to minimize experimental difficulties introduced by combustion, a simple flow configuration has been chosen for this initial study. It is mathematical truism that any bandlimited function can be accurately reconstructed from its projections if both the number and the signal to noise ratio of these projections approach infinity. In any real combustion situation, both of these conditions will be severely limited. The present results provide insights into the measurement capabilities of the tomographic reconstruction approach under such limitations. 

If, in practice, a Fourier transformation shall be carried out, it is meaningful to search for functions that are not only bounded, but, which even vanish when taking limits s —> oo or r —> °o. Such functions are called bandlimited. Let us consider the function h(r) = (H (s)). Then the reciprocal space image H (s) is bandlimited if its... 

The Bleistein method helps to determine not only the anomalous slowness distribution, but also another important characteristic of a seismic model, a reflectivity function. In a 1-D case this function is a spike train with spikes located at the positions of reflectors in the seismic model, whose heights are proportional to the reflection coefficients of the corresponding reflectors. It was demonstrated by Bleistein et al. (2001) that the reflectivity function can be represented as a combination of bandlimited delta-functions. 

Mathematically the bandlimited data can be introduced by filtering the theoretical signals. We consider, for example, the effect of this filtering on a delta-function. There is a well-known integral representation for a delta-function ... 

Using a simple 1-D model, we will illustrate below the basic principles of the reflectivity function definition based on bandlimited data. Let us assume first that the anomalous square slowness distribution As (z) is described by a piecewise constant function as shown in Figure 15-1. This means that the wavespeed is constant within each layer, equal, say, to Co within the first layer, ci within the second layer, etc. We can describe, for example, the first step on this curve by the Heaviside step function... 

Figure 15-1 A model with piecewise constant distribution of slowness (left panel). The corresponding bandlimited reflectivity function (right panel) is represented by a combination of the bandlimited delta-functions with the positions of extremum at the reflecting boundaries, and the sizes of the peaks proportional to the corresponding reflection coefficients...

Differentiating function Hb (z — h) with respect to variable 2, we arrive at the bandlimited delta-function (15.65)... 

In the case of a multilayered model, this function represents a combination of the bandlimited delta-functions with the positions of extremum at the reflecting... 

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