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We evaluate the Hamiltonian particle methods (HPM) and the Nambu discretization applied to shallow-water equations on the sphere using the test suggested by Galewsky et al. (2004). Both simulations show excellent conservation of energy and are stable in long-term simulation. We repeat the test also using the ICOSWP scheme to compare with the two conservative spatial discretization schemes. The HPM simulation captures the main features of the reference solution, but wave 5 pattern is dominant in the simulations applied on the ICON grid with relatively low spatial resolutions. Nevertheless, agreement in statistics between the three schemes indicates their qualitatively similar behaviors in the long-term integration.
The novel space-borne Global Navigation Satellite System Reflectometry (GNSS-R) technique has recently shown promise in monitoring the ocean state and surface wind speed with high spatial coverage and unprecedented sampling rate. The L-band signals of GNSS are structurally able to provide a higher quality of observations from areas covered by dense clouds and under intense precipitation, compared to those signals at higher frequencies from conventional ocean scatterometers. As a result, studying the inner core of cyclones and improvement of severe weather forecasting and cyclone tracking have turned into the main objectives of GNSS-R satellite missions such as Cyclone Global Navigation Satellite System (CYGNSS). Nevertheless, the rain attenuation impact on GNSS-R wind speed products is not yet well documented. Evaluating the rain attenuation effects on this technique is significant since a small change in the GNSS-R can potentially cause a considerable bias in the resultant wind products at intense wind speeds. Based on both empirical evidence and theory, wind speed is inversely proportional to derived bistatic radar cross section with a natural logarithmic relation, which introduces high condition numbers (similar to ill-posed conditions) at the inversions to high wind speeds. This paper presents an evaluation of the rain signal attenuation impact on the bistatic radar cross section and the derived wind speed. This study is conducted simulating GNSS-R delay-Doppler maps at different rain rates and reflection geometries, considering that an empirical data analysis at extreme wind intensities and rain rates is impossible due to the insufficient number of observations from these severe conditions. Finally, the study demonstrates that at a wind speed of 30 m/s and incidence angle of 30 degrees, rain at rates of 10, 15, and 20 mm/h might cause overestimation as large as approximate to 0.65 m/s (2%), 1.00 m/s (3%), and 1.3 m/s (4%), respectively, which are still smaller than the CYGNSS required uncertainty threshold. The simulations are conducted in a pessimistic condition (severe continuous rainfall below the freezing height and over the entire glistening zone) and the bias is expected to be smaller in size in real environments.
The success of the ensemble Kalman filter has triggered a strong interest in expanding its scope beyond classical state estimation problems. In this paper, we focus on continuous-time data assimilation where the model and measurement errors are correlated and both states and parameters need to be identified. Such scenarios arise from noisy and partial observations of Lagrangian particles which move under a stochastic velocity field involving unknown parameters. We take an appropriate class of McKean–Vlasov equations as the starting point to derive ensemble Kalman–Bucy filter algorithms for combined state and parameter estimation. We demonstrate their performance through a series of increasingly complex multi-scale model systems.
Particle filters contain the promise of fully nonlinear data assimilation. They have been applied in numerous science areas, including the geosciences, but their application to high-dimensional geoscience systems has been limited due to their inefficiency in high-dimensional systems in standard settings. However, huge progress has been made, and this limitation is disappearing fast due to recent developments in proposal densities, the use of ideas from (optimal) transportation, the use of localization and intelligent adaptive resampling strategies. Furthermore, powerful hybrids between particle filters and ensemble Kalman filters and variational methods have been developed. We present a state-of-the-art discussion of present efforts of developing particle filters for high-dimensional nonlinear geoscience state-estimation problems, with an emphasis on atmospheric and oceanic applications, including many new ideas, derivations and unifications, highlighting hidden connections, including pseudo-code, and generating a valuable tool and guide for the community. Initial experiments show that particle filters can be competitive with present-day methods for numerical weather prediction, suggesting that they will become mainstream soon.
Nonlinear data assimilation
(2015)
This book contains two review articles on nonlinear data assimilation that deal with closely related topics but were written and can be read independently. Both contributions focus on so-called particle filters.
The first contribution by Jan van Leeuwen focuses on the potential of proposal densities. It discusses the issues with present-day particle filters and explorers new ideas for proposal densities to solve them, converging to particle filters that work well in systems of any dimension, closing the contribution with a high-dimensional example. The second contribution by Cheng and Reich discusses a unified framework for ensemble-transform particle filters. This allows one to bridge successful ensemble Kalman filters with fully nonlinear particle filters, and allows a proper introduction of localization in particle filters, which has been lacking up to now.
Data assimilation
(2019)
Data assimilation addresses the general problem of how to combine model-based predictions with partial and noisy observations of the process in an optimal manner. This survey focuses on sequential data assimilation techniques using probabilistic particle-based algorithms. In addition to surveying recent developments for discrete- and continuous-time data assimilation, both in terms of mathematical foundations and algorithmic implementations, we also provide a unifying framework from the perspective of coupling of measures, and Schrödinger’s boundary value problem for stochastic processes in particular.
The success of the ensemble Kalman filter has triggered a strong interest in expanding its scope beyond classical state estimation problems. In this paper, we focus on continuous-time data assimilation where the model and measurement errors are correlated and both states and parameters need to be identified. Such scenarios arise from noisy and partial observations of Lagrangian particles which move under a stochastic velocity field involving unknown parameters. We take an appropriate class of McKean-Vlasov equations as the starting point to derive ensemble Kalman-Bucy filter algorithms for combined state and parameter estimation. We demonstrate their performance through a series of increasingly complex multi-scale model systems.
The novel space-borne Global Navigation Satellite System Reflectometry (GNSS-R) technique has recently shown promise in monitoring the ocean state and surface wind speed with high spatial coverage and unprecedented sampling rate. The L-band signals of GNSS are structurally able to provide a higher quality of observations from areas covered by dense clouds and under intense precipitation, compared to those signals at higher frequencies from conventional ocean scatterometers. As a result, studying the inner core of cyclones and improvement of severe weather forecasting and cyclone tracking have turned into the main objectives of GNSS-R satellite missions such as Cyclone Global Navigation Satellite System (CYGNSS). Nevertheless, the rain attenuation impact on GNSS-R wind speed products is not yet well documented. Evaluating the rain attenuation effects on this technique is significant since a small change in the GNSS-R can potentially cause a considerable bias in the resultant wind products at intense wind speeds. Based on both empirical evidence and theory, wind speed is inversely proportional to derived bistatic radar cross section with a natural logarithmic relation, which introduces high condition numbers (similar to ill-posed conditions) at the inversions to high wind speeds. This paper presents an evaluation of the rain signal attenuation impact on the bistatic radar cross section and the derived wind speed. This study is conducted simulating GNSS-R delay-Doppler maps at different rain rates and reflection geometries, considering that an empirical data analysis at extreme wind intensities and rain rates is impossible due to the insufficient number of observations from these severe conditions. Finally, the study demonstrates that at a wind speed of 30 m/s and incidence angle of 30 degrees, rain at rates of 10, 15, and 20 mm/h might cause overestimation as large as approximate to 0.65 m/s (2%), 1.00 m/s (3%), and 1.3 m/s (4%), respectively, which are still smaller than the CYGNSS required uncertainty threshold. The simulations are conducted in a pessimistic condition (severe continuous rainfall below the freezing height and over the entire glistening zone) and the bias is expected to be smaller in size in real environments.
We propose a computational method (with acronym ALDI) for sampling from a given target distribution based on first-order (overdamped) Langevin dynamics which satisfies the property of affine invariance. The central idea of ALDI is to run an ensemble of particles with their empirical covariance serving as a preconditioner for their underlying Langevin dynamics. ALDI does not require taking the inverse or square root of the empirical covariance matrix, which enables application to high-dimensional sampling problems. The theoretical properties of ALDI are studied in terms of nondegeneracy and ergodicity. Furthermore, we study its connections to diffusion on Riemannian manifolds and Wasserstein gradient flows. Bayesian inference serves as a main application area for ALDI. In case of a forward problem with additive Gaussian measurement errors, ALDI allows for a gradient-free approximation in the spirit of the ensemble Kalman filter. A computational comparison between gradient-free and gradient-based ALDI is provided for a PDE constrained Bayesian inverse problem.
Towards the assimilation of tree-ring-width records using ensemble Kalman filtering techniques
(2015)
This paper investigates the applicability of the Vaganov–Shashkin–Lite (VSL) forward model for tree-ring-width chronologies as observation operator within a proxy data assimilation (DA) setting. Based on the principle of limiting factors, VSL combines temperature and moisture time series in a nonlinear fashion to obtain simulated TRW chronologies. When used as observation operator, this modelling approach implies three compounding, challenging features: (1) time averaging, (2) “switching recording” of 2 variables and (3) bounded response windows leading to “thresholded response”. We generate pseudo-TRW observations from a chaotic 2-scale dynamical system, used as a cartoon of the atmosphere-land system, and attempt to assimilate them via ensemble Kalman filtering techniques. Results within our simplified setting reveal that VSL’s nonlinearities may lead to considerable loss of assimilation skill, as compared to the utilization of a time-averaged (TA) linear observation operator. In order to understand this undesired effect, we embed VSL’s formulation into the framework of fuzzy logic (FL) theory, which thereby exposes multiple representations of the principle of limiting factors. DA experiments employing three alternative growth rate functions disclose a strong link between the lack of smoothness of the growth rate function and the loss of optimality in the estimate of the TA state. Accordingly, VSL’s performance as observation operator can be enhanced by resorting to smoother FL representations of the principle of limiting factors. This finding fosters new interpretations of tree-ring-growth limitation processes.