Estimation of distribution functions in light scattering: The regularization method and Bayes' Ansatz

An important step in the analysis of dynamic light scattering data is the e stimation of the correlation time distribution given the measurement of the autocorrelation time function. This is an inverse problem, and especially a so-called ill-posed inverse problem: The map from the correlation time di stribution to the autocorrelation time data is singular, a unique inverse o f this map does not exist. Such problems are usually treated by regularizat ion methods. By those an estimator for the relaxation time spectrum is defi ned which differs from the usual Least Squares estimator in conceptual back ground as well as in numerical effort at its implementation. We discuss the regularization method from the Bayesian point of view. The c hoice of the additional prior functional is discussed and also two strategi es for the determination of the so-called regularization parameter. After t his more general introduction two aspects which are more specific for the l ight scattering are addressed: The influence of the model for the experimen tal errors on the quality of the estimation and the generalization of the r egularization method to the multiangle scattering. The size of the experimental errors and their correlation enter significant ly into the mathematical expression for the estimator of the correlation ti me distribution. They can be calculated either from the autocorrelation fun ction using a model derived by Schatzel, or, on the other hand, they could be computed directly from the time series of the scattered light, if such a time series is stored during the experiment. We show by simulations that t he direct method indeed leads to better results than the use of the model b y Schatzel, but that already this use leads to an improvement compared to a n analysis, in which the correlation of the experimental errors is neglecte d at all. The analysis of multi-angle data can easily be incorporated into the framew ork of regularization methods. At first thought one would combine the estim ations of the relaxation time spectrum based on the measurements for the di fferent angles by calculating the mean or some weighted mean of the estimat es. We show that this does not lead to the best results, however. The estim ation of the relaxation time spectrum from all the multi-angle data at once leads to better results than the intuitive combination.