Maximal spaces for approximation rates in $$\ell ^1$$-regularization

We study Tikhonov regularization for possibly nonlinear inverse problems with weighted $$\ell ^1$$ ℓ 1 -penalization. The forward operator, mapping from a sequence space to an arbitrary Banach space, typically an $$L^2$$ L 2 -space, is assumed to satisfy a two-sided Lipschitz condition with respect to a weighted $$\ell ^2$$ ℓ 2 -norm and the norm of the image space. We show that in this setting approximation rates of arbitrarily high Hölder-type order in the regularization parameter can be achieved, and we characterize maximal subspaces of sequences on which these rates are attained. On these subspaces the method also converges with optimal rates in terms of the noise level with the discrepancy principle as parameter choice rule. Our analysis includes the case that the penalty term is not finite at the exact solution (’oversmoothing’). As a standard example we discuss wavelet regularization in Besov spaces $$B^r_{1,1}$$ B 1 , 1 r . In this setting we demonstrate in numerical simulations for a parameter identification problem in a differential equation that our theoretical results correctly predict improved rates of convergence for piecewise smooth unknown coefficients.

Evaluation of the COSMO model (v5.1) in polarimetric radar space – Impact of uncertainties in model microphysics, retrievals, and forward operator

Sensitivity experiments with a numerical weather prediction (NWP) model and polarimetric radar forward operator (FO) are conducted for a long-duration stratiform event over northwestern Germany, to evaluate uncertainties in the partitioning of the ice water content and assumptions of hydrometeor scattering properties in the NWP model and FO, respectively. Polarimetric observations from X-band radar and retrievals of hydrometeor classifications are used for comparison with the multiple experiments in radar and model space. Modifying two parameters (Dice and Tgr) responsible for the production of snow and graupel, respectively, was found to improve the synthetic polarimetric moments and simulated hydrometeor population, while keeping the difference in surface precipitation statistically insignificant at model resolvable grid scales. However, the model still exhibited a low bias in simulated polarimetric moments at lower levels above the melting layer (−3 to −13 °C) where snow was found to dominate. This necessitates further research into the missing microphysical processes in these lower levels (e.g., fragmentation due to ice-ice collisions), and use of more reliable snow scattering models to draw valid conclusions.

Understanding the model representation of clouds based on visible and infrared satellite observations

There is a rising interest in improving the representation of clouds in numerical weather prediction models. This will directly lead to improved radiation forecasts and, thus, to better predictions of the increasingly important production of photovoltaic power. Moreover, a more accurate representation of clouds is crucial for assimilating cloud-affected observations, in particular high-resolution observations from instruments on geostationary satellites. These observations can also be used to diagnose systematic errors in the model clouds, which are influenced by multiple parameterisations with many, often not well-constrained, parameters. In this study, the benefits of using both visible and infrared satellite channels for this purpose are demonstrated. We focus on visible and infrared Meteosat SEVIRI (Spinning Enhanced Visible InfraRed Imager) images and their model equivalents computed from the output of the ICON-D2 (ICOsahedral Non-hydrostatic, development version based on version 2.6.1; Zängl et al., 2015) convection-permitting, limited area numerical weather prediction model using efficient forward operators. We analyse systematic deviations between observed and synthetic satellite images derived from semi-free hindcast simulations for a 30 d summer period with strong convection. Both visible and infrared satellite observations reveal significant deviations between the observations and model equivalents. The combination of infrared brightness temperature and visible reflectance facilitates the attribution of individual deviations to specific model shortcomings. Furthermore, we investigate the sensitivity of model-derived visible and infrared observation equivalents to modified model and visible forward operator settings to identify dominant error sources. Estimates of the uncertainty of the visible forward operator turned out to be sufficiently low; thus, it can be used to assess the impact of model modifications. Results obtained for various changes in the model settings reveal that model assumptions on subgrid-scale water clouds are the primary source of systematic deviations in the visible satellite images. Visible observations are, therefore, well-suited to constrain subgrid cloud settings. In contrast, infrared channels are much less sensitive to the subgrid clouds, but they can provide information on errors in the cloud-top height.

A model reduction approach for inverse problems with operator valued data

We study the efficient numerical solution of linear inverse problems with operator valued data which arise, e.g., in seismic exploration, inverse scattering, or tomographic imaging. The high-dimensionality of the data space implies extremely high computational cost already for the evaluation of the forward operator which makes a numerical solution of the inverse problem, e.g., by iterative regularization methods, practically infeasible. To overcome this obstacle, we take advantage of the underlying tensor product structure of the problem and propose a strategy for constructing low-dimensional certified reduced order models of quasi-optimal rank for the forward operator which can be computed much more efficiently than the truncated singular value decomposition. A complete analysis of the proposed model reduction approach is given in a functional analytic setting and the efficient numerical construction of the reduced order models as well as of their application for the numerical solution of the inverse problem is discussed. In summary, the setup of a low-rank approximation can be achieved in an offline stage at essentially the same cost as a single evaluation of the forward operator, while the actual solution of the inverse problem in the online phase can be done with extremely high efficiency. The theoretical results are illustrated by application to a typical model problem in fluorescence optical tomography.

Assimilation of Polarimetric Radar Data in Simulation of a Supercell Storm with a Variational Approach and the WRF Model

Polarimetric radar data (PRD) have potential to be used in numerical weather prediction (NWP) models to improve convective-scale weather forecasts. However, thus far only a few studies have been undertaken in this research direction. To assimilate PRD in NWP models, a forward operator, also called a PRD simulator, is needed to establish the relation between model physics parameters and polarimetric radar variables. Such a forward operator needs to be accurate enough to make quantitative comparisons between radar observations and model output feasible, and to be computationally efficient so that these observations can be easily incorporated into a data assimilation (DA) scheme. To address this concern, a set of parameterized PRD simulators for the horizontal reflectivity, differential reflectivity, specific differential phase, and cross-correlation coefficient were developed. In this study, we have tested the performance of these new operators in a variational DA system. Firstly, the tangent linear and adjoint (TL/AD) models for these PRD simulators have been developed and checked for the validity. Then, both the forward operator and its adjoint model have been built into the three-dimensional variational (3DVAR) system. Finally, some preliminary DA experiments have been performed with an idealized supercell storm. It is found that the assimilation of PRD, including differential reflectivity and specific differential phase, in addition to radar radial velocity and horizontal reflectivity, can enhance the accuracy of both initial conditions for model hydrometer state variables and ensuing model forecasts. The usefulness of the cross-correlation coefficient is very limited in terms of improving convective-scale data analysis and NWP.

W-band radar observations for fog forecast improvement: an analysis of model and forward operator errors

The development of ground-based cloud radars offers a new capability to continuously monitor fog structure. Retrievals of fog microphysics are key for future process studies, data assimilation, or model evaluation and can be performed using a variational method. Both the one-dimensional variational retrieval method (1D-Var) or direct 3D/4D-Var data assimilation techniques rely on the combination of cloud radar measurements and a background profile weighted by their corresponding uncertainties to obtain the optimal solution for the atmospheric state. In order to prepare for the use of ground-based cloud radar measurements for future applications based on variational approaches, the different sources of uncertainty due to instrumental, background, and forward operator errors need to be properly treated and accounted for. This paper aims at preparing 1D-Var retrievals by analysing the errors associated with a background profile and a forward operator during fog conditions. For this, the background was provided by a high-resolution numerical weather prediction model and the forward operator by a radar simulator. Firstly, an instrumental dataset was taken from the SIRTA observatory near Paris, France, for winter 2018–2019 during which 31 fog events were observed. Statistics were calculated comparing cloud radar observations to those simulated. It was found that the accuracy of simulations could be drastically improved by correcting for significant spatio-temporal background errors. This was achieved by implementing a most resembling profile method in which an optimal model background profile is selected from a domain and time window around the observation location and time. After selecting the background profiles with the best agreement with the observations, the standard deviation of innovations (observations–simulations) was found to decrease significantly. Moreover, innovation statistics were found to satisfy the conditions needed for future 1D-Var retrievals (un-biased and normally distributed).

Towards operational assimilation of surface based remote sensing temperature and humidity profiler data at MeteoSwiss

<p>The current atmospheric observing systems fail to provide observations of temperature and humidity in the planetary boundary layer (PBL) with satisfactory spatial and temporal resolutions despite their potential positive impact on numerical weather prediction (NWP). This is particularly critical for humidity, which exhibits a very high variability in space and time, and for the vertical profile of temperature, which determines the atmospheric stability. Therefore, the analyzed thermodynamic structure of the PBL can be prone to errors, leading to poor forecasts of warnings for relevant phenomena, such as severe storms due to intense summer convection or winter fog and low stratus.</p><p>One approach to improve the model’s representation of the PBL is to include novel, ground-based remote sensing profiler observations in the data assimilation system to improve the forecast initial conditions. This also improves the quality of downstream applications relying on a good representation of the PBL in the model, such as dispersion modelling for emergency response after nuclear, chemical or biological incidents.</p><p>In this contribution, we present results of the MeteoSwiss effort to include observations from Raman lidar and microwave radiometers into the 1km mesh-size ensemble data assimilation system KENDA-1. To this end, we have developed a forward operator for water vapor mixing ratio and temperature to assimilate profiles from the Raman lidar. Brightness temperatures from the microwave radiometers are assimilated using the RTTOV-gb forward operator. We produced extensive O-B statistics to validate the observations with respect to the model and to derive the error covariance matrices of the observations. Furthermore, we will present results of several data assimilation cycling experiments during summer-time convective situations.</p>

Using visible satellite images for model evaluation and data assimilation

<p>Satellite provide high-resolution information on the state of the atmosphere and thus represent observations are well-suited for data assimilation and model evaluation. So far mainly the thermal infrared channels have been utilized for these purposes. However, there is a rising interest to use also the channels in the solar part of the spectrum, which contain additional, complementary information. Visible channels can provide better information on the water and ice content of clouds than thermal infrared channels, have no problems to detect low clouds and are sensitive to cloud microphysics and the cloud top structure. Moreover, visible reflectances are strongly correlated with the solar irradiation at the surface and thus their assimilation has a clear potential to improve also radiation forecasts.</p><p>So far visible satellite images have not been assimilated directly for operational purposes, as multiple scattering dominates in the visible spectral range and makes radiative transfer (RT) computations with standard methods complex and slow. Only recently, we developed a sufficiently fast and accurate forward operator that relies on a compressed reflectance look-up table (LUT) computed with slow standard RT methods. Here we report on using feed-forward neural networks as an alternative to the look-up table and demonstrate that it is possible to achieve higher speed and accuracy. Moreover, both the amount of training data and the memory required by the operator can be reduced by three orders of magnitude. A further advantage is that tangent-linear and adjoint versions can easily be derived for arbitrary network structures and do not have to be changed when the network is trained with different data.</p><p>We will also discuss two ways to use the forward operator to improve forecasts. First, we show that observed and synthetic visible  Meteosat SEVIRI images can be used to detect systematic errors in the model clouds that can cause severe problems for data assimilation. Second, based on assimilation experiments using the ICON-D2 model and the local ensemble transformation Kalman filter implemented in DWDs data assimilation coding environment (DACE) we demonstrate for test periods of several weeks that errors in the cloud distribution and the surface radiation can be significantly reduced by assimilating visible SEVIRI images.</p>