Aerosol backscatter profiles from ceilometers: validation of water vapor correction in the framework of CeiLinEx2015

With the rapidly growing number of automated single-wavelength backscatter lidars (ceilometers), their potential benefit for aerosol remote sensing received considerable scientific attention. When studying the accuracy of retrieved particle backscatter coefficients, it must be considered that most of the ceilometers are influenced by water vapor absorption in the spectral range around 910 nm. In the literature methodologies have been proposed to correct for this effect; however, a validation was not yet performed. In the framework of the ceilometer intercomparison campaign CeiLinEx2015 in Lindenberg, Germany, hosted by the German Weather Service, it was possible to tackle this open issue. Ceilometers from Lufft (CHM15k and CHM15kx, operating at 1064 nm), from Vaisala (CL51 and CL31) and from Campbell Scientific (CS135), all operating at a wavelength of approximately 910 nm, were deployed together with a multi-wavelength research lidar (RALPH) that served as a reference. In this paper the validation of the water vapor correction is performed by comparing ceilometer backscatter signals with measurements of the reference system extrapolated to the water vapor regime. One inherent problem of the validation is the spectral extrapolation of particle optical properties. For this purpose AERONET measurements and inversions of RALPH signals were used. Another issue is that the vertical range where validation is possible is limited to the upper part of the mixing layer due to incomplete overlap and the generally low signal-to-noise ratio and signal artifacts above that layer. Our intercomparisons show that the water vapor correction leads to quite a good agreement between the extrapolated reference signal and the measurements in the case of CL51 ceilometers at one or more wavelengths in the specified range of the laser diode\u27s emission. This ambiguity is due to the similar effective water vapor transmission at several wavelengths. In the case of CL31 and CS135 ceilometers the validation was not always successful. That suggests that error sources beyond the water vapor absorption might be dominant. For future applications we recommend monitoring the emitted wavelength and providing “dark” measurements on a regular basis

Finite-Elemente-Mortaring nach einer Methode von J. A. Nitsche für elliptische Randwertaufgaben

Viele technische Prozesse führen auf Randwertprobleme mit partiellen Differentialgleichungen, die mit Finite-Elemente-Methoden näherungsweise gelöst werden können. Spezielle Varianten dieser Methoden sind Finite-Elemente-Mortar-Methoden. Sie erlauben das Arbeiten mit an Teilgebietsschnitträndern nichtzusammenpassenden Netzen, was für Probleme mit komplizierten Geometrien, Randschichten, springenden Koeffizienten sowie für zeitabhängige Probleme von Vorteil sein kann. Ebenso können unterschiedliche Diskretisierungsmethoden in den einzelnen Teilgebieten miteinander gekoppelt werden. In dieser Arbeit wird das Finite-Elemente-Mortaring nach einer Methode von Nitsche für elliptische Randwertprobleme auf zweidimensionalen polygonalen Gebieten untersucht. Von besonderem Interesse sind dabei nichtreguläre Lösungen (u \in H^{1+\delta}(\Omega), \delta>0) mit Eckensingularitäten für die Poissongleichung sowie die Lamé-Gleichung mit gemischten Randbedingungen. Weiterhin werden singulär gestörte Reaktions-Diffusions-Probleme betrachtet, deren Lösungen zusätzlich zu Eckensingularitäten noch anisotropes Verhalten in Randschichten aufweisen. Für jede dieser drei Problemklassen wird das Nitsche-Mortaring dargelegt. Es werden einige Eigenschaften der Mortar-Diskretisierung angegeben und a-priori-Fehlerabschätzungen in einer H^1-artigen sowie der L_2-Norm durchgeführt. Auf lokal verfeinerten Dreiecksnetzen können auch für Lösungen mit Eckensingularitäten optimale Konvergenzordnungen nach gewiesen werden. Bei den Lösungen mit anisotropen Verhalten werden zusätzlich anisotrope Dreiecksnetze verwendet. Es werden auch hier Konvergenzordnungen wie bei klassischen Finite-Elemente-Methoden ohne Mortaring erreicht. Numerische Experimente illustrieren die Methode und die Aussagen zur Konvergenz

Nitsche type mortaring for singularly perturbed reaction-diffusion problems

The paper is concerned with the Nitsche mortaring in the framework of domain decomposition where non-matching meshes and weak continuity of the finite element approximation at the interface are admitted. The approach is applied to singularly perturbed reaction-diffusion problems in 2D. Non-matching meshes of triangles being anisotropic in the boundary layers are applied. Some properties as well as error estimates of the Nitsche mortar finite element schemes are proved. In particular, using a suitable degree of anisotropy of triangles in the boundary layers of a rectangle, we derive convergence rates as known for the conforming finite element method in presence of regular solutions. Numerical examples illustrate the approach and the results

Nitsche type mortaring for singularly perturbed reaction-diffusion problems

The paper is concerned with the Nitsche mortaring in the framework of domain decomposition where non-matching meshes and weak continuity of the finite element approximation at the interface are admitted. The approach is applied to singularly perturbed reaction-diffusion problems in 2D. Non-matching meshes of triangles being anisotropic in the boundary layers are applied. Some properties as well as error estimates of the Nitsche mortar finite element schemes are proved. In particular, using a suitable degree of anisotropy of triangles in the boundary layers of a rectangle, we derive convergence rates as known for the conforming finite element method in presence of regular solutions. Numerical examples illustrate the approach and the results