Uniform-Penalty Inversion of Multiexponential Decay Data

NMR relaxation data and those from many other physical measurements are sums of exponentially decaying components, combined with some unavoidable measurement noise. When decay data are inverted in order to give quasi-continuous distributions of relaxation times, some smoothing of the distributions is normally implemented to avoid excess variation. When the same distribution has a sharp peak and a much broader peak or a “tail,” as for many porous media saturated with liquids, an inversion program using a fixed smoothing coefficient may broaden the sharp peak and/or break the wide peak or tail into several separate peaks, even if the coefficient is adaptively chosen in accord with the noise level of the data. We deal with this problem by using variable smoothing, determined by iterative feedback in such a way that the smoothingpenaltyis roughly constant. This uniform-penalty (UP) smoothing can give sharp lines, not broadened more than is consistent with the noise, and in the same distribution it can show a tail decades long without breaking it up into several peaks. The noise level must be known approximately, but it can be determined more than adequately by a preliminary inversion. The same iterative procedure is used to implement constraints such as non-negative (NN) or monotonic-from-peak (MT). The significance of an additional resolved peak may be tested by finding the cost of using MT to force a unimodal solution. A bimodal constraint can be applied. Decay data representing sharp lines in contact with broad features can require substantial computing time and some controls to stabilize the iterative sequence. However, UP can be made to function smoothly for a very wide variety of decay curves, which can be processed without adjustment of parameters, including the dimensionless smoothing parameters. Extensive testing has been done with artificial data. Examples are shown for artificial data, biological tissues, ceramic technology, and sandstones. Expressions are given relating noise level to line width and for significance of increase or decrease in error of fit.