Inverse Problems and Data Assimilation

These notes are designed with the aim of providing a clear and concise introduction to the subjects of Inverse Problems and Data Assimilation, and their inter-relations, together with citations to some relevant literature in this area. The first half of the notes is dedicated to studying the Bayesian framework for inverse problems. Techniques such as importance sampling and Markov Chain Monte Carlo (MCMC) methods are introduced; these methods have the desirable property that in the limit of an infinite number of samples they reproduce the full posterior distribution. Since it is often computationally intensive to implement these methods, especially in high dimensional problems, approximate techniques such as approximating the posterior by a Dirac or a Gaussian distribution are discussed. The second half of the notes cover data assimilation. This refers to a particular class of inverse problems in which the unknown parameter is the initial condition of a dynamical system, and in the stochastic dynamics case the subsequent states of the system, and the data comprises partial and noisy observations of that (possibly stochastic) dynamical system. We will also demonstrate that methods developed in data assimilation may be employed to study generic inverse problems, by introducing an artificial time to generate a sequence of probability measures interpolating from the prior to the posterior

Displacement Data Assimilation

We show that modifying a Bayesian data assimilation scheme by incorporating kinematically-consistent displacement corrections produces a scheme that is demonstrably better at estimating partially observed state vectors in a setting where feature information important. While the displacement transformation is not tied to any particular assimilation scheme, here we implement it within an ensemble Kalman Filter and demonstrate its effectiveness in tracking stochastically perturbed vortices.Comment: 26 Pages, 9 figures, 5 table

Data Assimilation by Artificial Neural Networks for an Atmospheric General Circulation Model: Conventional Observation

This paper presents an approach for employing artificial neural networks (NN) to emulate an ensemble Kalman filter (EnKF) as a method of data assimilation. The assimilation methods are tested in the Simplified Parameterizations PrimitivE-Equation Dynamics (SPEEDY) model, an atmospheric general circulation model (AGCM), using synthetic observational data simulating localization of balloon soundings. For the data assimilation scheme, the supervised NN, the multilayer perceptrons (MLP-NN), is applied. The MLP-NN are able to emulate the analysis from the local ensemble transform Kalman filter (LETKF). After the training process, the method using the MLP-NN is seen as a function of data assimilation. The NN were trained with data from first three months of 1982, 1983, and 1984. A hind-casting experiment for the 1985 data assimilation cycle using MLP-NN were performed with synthetic observations for January 1985. The numerical results demonstrate the effectiveness of the NN technique for atmospheric data assimilation. The results of the NN analyses are very close to the results from the LETKF analyses, the differences of the monthly average of absolute temperature analyses is of order 0.02. The simulations show that the major advantage of using the MLP-NN is better computational performance, since the analyses have similar quality. The CPU-time cycle assimilation with MLP-NN is 90 times faster than cycle assimilation with LETKF for the numerical experiment.Comment: 17 pages, 16 figures, monthly weather revie

Multiscale assimilation of Advanced Microwave Scanning Radiometer-EOS snow water equivalent and Moderate Resolution Imaging Spectroradiometer snow cover fraction observations in northern Colorado

Eight years (2002–2010) of Advanced Microwave Scanning Radiometer–EOS (AMSR-E) snow water equivalent (SWE) retrievals and Moderate Resolution Imaging Spectroradiometer (MODIS) snow cover fraction (SCF) observations are assimilated separately or jointly into the Noah land surface model over a domain in Northern Colorado. A multiscale ensemble Kalman filter (EnKF) is used, supplemented with a rule-based update. The satellite data are either left unscaled or are scaled for anomaly assimilation. The results are validated against in situ observations at 14 high-elevation Snowpack Telemetry (SNOTEL) sites with typically deep snow and at 4 lower-elevation Cooperative Observer Program (COOP) sites. Assimilation of coarse-scale AMSR-E SWE and fine-scale MODIS SCF observations both result in realistic spatial SWE patterns. At COOP sites with shallow snowpacks, AMSR-E SWE and MODIS SCF data assimilation are beneficial separately, and joint SWE and SCF assimilation yields significantly improved root-mean-square error and correlation values for scaled and unscaled data assimilation. In areas of deep snow where the SNOTEL sites are located, however, AMSR-E retrievals are typically biased low and assimilation without prior scaling leads to degraded SWE estimates. Anomaly SWE assimilation could not improve the interannual SWE variations in the assimilation results because the AMSR-E retrievals lack realistic interannual variability in deep snowpacks. SCF assimilation has only a marginal impact at the SNOTEL locations because these sites experience extended periods of near-complete snow cover. Across all sites, SCF assimilation improves the timing of the onset of the snow season but without a net improvement of SWE amounts

The analog data assimilation

In light of growing interest in data-driven methods for oceanic, atmospheric, and climate sciences, this work focuses on the field of data assimilation and presents the analog data assimilation (AnDA). The proposed framework produces a reconstruction of the system dynamics in a fully data-driven manner where no explicit knowledge of the dynamical model is required. Instead, a representative catalog of trajectories of the system is assumed to be available. Based on this catalog, the analog data assimilation combines the nonparametric sampling of the dynamics using analog forecasting methods with ensemble-based assimilation techniques. This study explores different analog forecasting strategies and derives both ensemble Kalman and particle filtering versions of the proposed analog data assimilation approach. Numerical experiments are examined for two chaotic dynamical systems: the Lorenz-63 and Lorenz-96 systems. The performance of the analog data assimilation is discussed with respect to classical model-driven assimilation. A Matlab toolbox and Python library of the AnDA are provided to help further research building upon the present findings.Fil: Lguensat, Redouane. Université Bretagne Loire; FranciaFil: Tandeo, Pierre. Université Bretagne Loire; FranciaFil: Ailliot, Pierre. University of Western Brittany. Laboratoire de Mathématiques de Bretagne Atlantique; FranciaFil: Pulido, Manuel Arturo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Nordeste. Instituto de Modelado e Innovación Tecnológica. Universidad Nacional del Nordeste. Facultad de Ciencias Exactas Naturales y Agrimensura. Instituto de Modelado e Innovación Tecnológica; ArgentinaFil: Fablet, Ronan. Université Bretagne Loire; Franci