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

• Published in: Macromolecular symposia. 2000-12, Vol.162 (1), p.149-172
• Main Authors: Buttgereit, Ralf, Marth, Michael, Honerkamp, Josef
• Format: Article
• Language: English
• Subjects: Autocorrelation; Correlation; Estimators; Inverse problems; Light scattering; Mathematical models; Regularization; Relaxation time
• ISSN: 1022-1360; 1521-3900
• DOI: 10.1002/1521-3900(200012)162:1<149::AID-MASY149>3.0.CO;2-M

An important step in the analysis of dynamic light scattering data is the estimation 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 distribution to the autocorrelation time data is singular, a unique inverse of this map does not exist. Such problems are usually treated by regularization methods. By those an estimator for the relaxation time spectrum is defined which differs from the usual Least Squares estimator in conceptual background as well as in numerical effort at its implementation. We discuss the regularization method from the Bayesian point of view. The choice of the additional prior functional is discussed and also two strategies for the determination of the so‐called regularization parameter. After this more general introduction two aspects which are more specific for the light scattering are addressed: The influence of the model for the experimental errors on the quality of the estimation and the generalization of the regularization method to the multi‐angle scattering. The size of the experimental errors and their correlation enter significantly into the mathematical expression for the estimator of the correlation time distribution. They can be calculated either from the autocorrelation function using a model derived by Schätzel, 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 the direct method indeed leads to better results than the use of the model by Schätzel, but that already this use leads to an improvement compared to an analysis, in which the correlation of the experimental errors is neglected at all. The analysis of multi‐angle data can easily be incorporated into the framework of regularization methods. At first thought one would combine the estimations of the relaxation time spectrum based on the measurements for the different angles by calculating the mean or some weighted mean of the estimates. We show that this does not lead to the best results, however. The estimation of the relaxation time spectrum from all the multi‐angle data at once leads to better results than the intuitive combination.