Mathias Rafler

Ill-posed retrieval of aerosol extinction coefficient profiles from Raman lidar data by regularization

In the analysis of Raman lidar measurements of aerosol extinction, it is necessary to calculate the derivative of the logarithm of the ratio between the atmospheric number density and the range-corrected lidar-received power. The statistical fluctuations of the Raman signal can produce large fluctuations in the derivative and thus in the aerosol extinction profile. To overcome this difficult situation we discuss three methods: Tikhonov regularization, variational, and the sliding best-fit (SBF). Three methods are performed on the profiles taken from the European Aerosol Research Lidar Network lidar database simulated at the Raman shifted wavelengths of 387 and 607 nm associated with the emitted signals at 355 and 532 nm. Our results show that the SBF method does not deliver good results for low fluctuation in the profile. However, Tikhonov regularization and the variational method yield very good aerosol extinction coefficient profiles for our examples. With regard to, e.g., the 532 nm wavelength, the L2 errors of the aerosol extinction coefficient profile by using the SBF, Tikhonov, and variational methods with respect to synthetic noisy data are 0.0015(0.0024), 0.00049(0.00086), and 0.00048(0.00082), respectively. Moreover, the L2 errors by using the Tikhonov and variational methods with respect to a more realistic noisy profile are 0.0014(0.0016) and 0.0012(0.0016), respectively. In both cases the L2 error given in parentheses concerns the second example.

Reconstruction method for inverse Sturm–Liouville problems with discontinuous potentials

In this paper, we present a method for solving inverse Sturm–Liouville problems by generalizing a Rundell–Sacks algorithm. The method is extended to deal with a general reference potential which can be adapted, e.g., to estimations of the jump-discontinuity points of the unknown potential. Moreover, its convergence properties are investigated. Numerical examples show that this modification can achieve more precise results from a given data set than the earlier method in using only the null reference potential and, therefore, the L2- and L∞-error can be reduced significantly.

The Pólya sum process: a Cox representation

In [1], Zessin constructed the so-called Pólya sum process via partial integration. Here we use the technique of integration by parts to the Pólya sum process to derive representations of the Pólya sum process as an infinitely divisible point process and a Cox process directed by an infinitely divisible random measure. This result is related to the question of the infinite divisibilty of a Cox process and the infinite divisibility of its directing measure. Finally we consider a scaling limit of the Pólya sum process and show that the limit satisfies an integration by parts formula, which we use to determine basic properties of this limit.

The Pólya Sum Process: Limit Theorems for Conditioned Random Fields

Zessin (J. Contemp. Math. Anal. 44(1):36–44, 2009) constructed the so-called Pólya sum process via partial integration technique. This process shares some important properties with the Poisson process such as complete randomness and infinite divisibility. This work discusses H-sufficient statistics for the Pólya sum process as was done for the Poisson process by Nguyen and Zessin (Z. Wahrscheinlichkeitstheor. Verw. Geb. 37(3):191–200, 1976/77).