We extend recent discussions of singularity avoidance in quantum gravity from isotropic to anisotropic cosmological models. The investigation is done in the framework of quantum geometrodynamics (Wheeler-DeWitt equation). We formulate criteria of singularity avoidance for general Bianchi class A models and give explicit and detailed results for Bianchi I models with and without matter. The singularities in these cases are big bang and big rip. We find that the classical singularities can generally be avoided in these models. I. INTRODUCTIONA major issue in any quantum theory of gravity is the fate of the classical singularities. So far, such a theory is not available in final form, although various approaches exist in which this question can be sensibly addressed [1,2]. It is clear that such an investigation cannot yet be done at the level of mathematical rigor comparable to the singularity theorems in the classical theory (see e.g. [3]). Nevertheless, focusing on concrete approaches and concrete models, one can state criteria of singularity avoidance and discuss their implementation. This is what we shall do here.We restrict our analysis of singularity avoidance to quantum geometrodynamics, with the Wheeler-DeWitt equation as its central equation [1]. Although this may not be the most fundamental level of quantum gravity, it is sufficient for addressing the issue of singularity avoidance. Quantum geometrodynamics follows directly from general relativity by rewriting the Einstein equations in Hamilton-Jacobi form and formulating quantum equations that yield the Hamilton-Jacobi equations in the semiclassical (WKB) limit. It thus makes as much sense to addressing singularity avoidance here than it does to addressing it in quantum mechanics at the level of the Schrödinger equation. Singularity avoidance has also been discussed in loop quantum gravity [2,4,5], with various results, but we will not consider this here.Singularity avoidance was already addressed by DeWitt in his pioneering paper on canonical quantum gravity [6]. He suggested to impose the condition Ψ → 0 for the quantum-gravitational wave functional Ψ when approaching the region of a classical singularity. The wave functional is effectively defined on the configuration space of all threedimensional geometries, also called superspace [1,7]. The "DeWitt criterion" of vanishing wave function then means that Ψ must approach zero when approaching a singular three-geometry (which itself is not part of superspace, but can be envisaged as its boundary). It is important to emphasize that this criterion is a sufficient but not a necessary one: singularities can be avoided for non-vanishing or even diverging Ψ (recall the ground state solution of the Dirac equation for the hydrogen atom, which diverges).DeWitt had in mind cosmological singularities such as big bang or big crunch. The DeWitt criterion applies, of course, also to the singularities that classically arise from gravitational collapse. In simple models of quantum geometrodynamics, their avoidance can be...

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Volcanic ash clouds are a threat to air traffic security and, thus, can have significant societal and financial impact. Therefore, the detection and monitoring of volcanic ash clouds to enhance the safety of air traffic is of central importance. This work presents the development of the new retrieval algorithm VACOS (Volcanic Ash Cloud properties Obtained from SEVIRI) which is based on artificial neural networks, the thermal channels of the geostationary sensor MSG/SEVIRI and auxiliary data from a numerical weather prediction model. It derives a pixel classification as well as cloud top height, effective particle radius and, indirectly, the mass column concentration of volcanic ash clouds during day and night. A large set of realistic one-dimensional radiative transfer calculations for typical atmospheric conditions with and without generic volcanic ash clouds is performed to create the training dataset. The atmospheric states are derived from ECMWF data to cover the typical diurnal, annual and interannual variability. The dependence of the surface emissivity on surface type and viewing zenith angle is considered. An extensive dataset of volcanic ash optical properties is used, derived for a wide range of microphysical properties and refractive indices of various petrological compositions, including different silica contents and glass-to-crystal ratios; this constitutes a major innovation of this retrieval. The resulting ash-free radiative transfer calculations at a specific time compare well with corresponding SEVIRI measurements, considering the individual pixel deviations as well as the overall brightness temperature distributions. Atmospheric gas profiles and sea surface emissivities are reproduced with a high agreement, whereas cloudy cases can show large deviations on a single pixel basis (with 95th percentiles of the absolute deviations > 30 K), mostly due to different cloud properties in model and reality. Land surfaces lead to large deviations for both the single pixel comparison (with median absolute deviations > 3 K) and more importantly the brightness temperature distributions, most likely due to imprecise skin temperatures. The new method enables volcanic ash-related scientific investigations as well as aviation security-related applications.

Volcanic ash clouds can damage aircrafts during flight and, thus, have the potential to disrupt air traffic on a large scale, making their detection and monitoring necessary. The new retrieval algorithm VACOS (Volcanic Ash Cloud properties Obtained from SEVIRI) using the geostationary instrument MSG/SEVIRI and artificial neural networks is introduced in a companion paper. It performs pixelwise classifications and retrieves (indirectly) the mass column concentration, the cloud top height and the effective particle radius. VACOS is comprehensively validated using simulated test data, CALIOP retrievals, lidar and in situ data from aircraft campaigns of the DLR and the FAAM, as well as volcanic ash transport and dispersion multi model multi source term ensemble predictions. Specifically, emissions of the eruptions of Eyjafjallajökull (2010) and Puyehue-Cordón Caulle (2011) are considered. For ash loads larger than 0.2g/□m and a mass column concentration-based detection procedure, the different evaluations give probabilities of detection between 70% and more than 90% at false alarm rates of the order of 0.3–3%. For the simulated test data, the retrieval of the mass load has a mean absolute percentage error of ~40% or less for ash layers with an optical thickness at 10.8m of 0.1 (i.e., a mass load of about 0.3– 0.7g/□m, depending on the ash type) or more, the ash cloud top height has an error of up to 10% for ash layers above 5km, and the effective radius has an error of up to 35% for radii of 0.6– 6m. The retrieval error increases with decreasing ash cloud thickness and top height. VACOS is applicable even for overlaying meteorological clouds, for example, the mean absolute percentage error of the optical depth at 10.8m increases by only up to ~30%. Viewing zenith angles > 60∘ increase the mean percentage error by up to ~20%. Desert surfaces are another source of error. Varying geometrical ash layer thicknesses and the occurrence of multiple layers can introduce an additional error of about 30% for the mass load and 5% for the cloud top height. For the CALIOP data, comparisons with its predecessor VADUGS (operationally used by the DWD) show that VACOS is more robust, with retrieval errors of mass load and ash cloud top height reduced by >10% and >50%, respectively. Using the model data indicates an increase in detection rate in the order of 30% and more. The reliability under a wide spectrum of atmospheric conditions and volcanic ash types make VACOS a suitable tool for scientific studies and air traffic applications related to volcanic ash clouds.