Data Assimilation: A Mathematical Introduction

These notes provide a systematic mathematical treatment of the subject of data assimilation

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

Generalised Latent Assimilation in Heterogeneous Reduced Spaces with Machine Learning Surrogate Models

Reduced-order modelling and low-dimensional surrogate models generated using machine learning algorithms have been widely applied in high-dimensional dynamical systems to improve the algorithmic efficiency. In this paper, we develop a system which combines reduced-order surrogate models with a novel data assimilation (DA) technique used to incorporate real-time observations from different physical spaces. We make use of local smooth surrogate functions which link the space of encoded system variables and the one of current observations to perform variational DA with a low computational cost. The new system, named Generalised Latent Assimilation can benefit both the efficiency provided by the reduced-order modelling and the accuracy of data assimilation. A theoretical analysis of the difference between surrogate and original assimilation cost function is also provided in this paper where an upper bound, depending on the size of the local training set, is given. The new approach is tested on a high-dimensional CFD application of a two-phase liquid flow with non-linear observation operators that current Latent Assimilation methods can not handle. Numerical results demonstrate that the proposed assimilation approach can significantly improve the reconstruction and prediction accuracy of the deep learning surrogate model which is nearly 1000 times faster than the CFD simulation

Design and implementation of data assimilation methods based on Cholesky decomposition

In Data Assimilation, analyses of a system are obtained by combining a previous numerical model of the system and observations or measurements from it. These numerical models are typically expressed as a set of ordinary differential equations and/or a set of partial differential equations wherein all knowledge about dynamics and physics of, for instance, the ocean and or the atmosphere are encapsulated. We treat numerical forecasts and observations as random variables and therefore, error dynamics can be estimated by using Bayes’ rule. For the estimation of hyper-parameters in error distributions, an ensemble of model realizations is employed. In practice, model resolutions are several order of magnitudes larger than ensemble sizes, and consequently, sampling errors impact the quality of analysis corrections and besides, models can be highly non-linear and well-common Gaussian assumptions on prior errors can be broken. To overcome these situations, we replace prior errors by a mixture of Gaussians and even more, precision covariance matrices intra-clusters are estimated by means of the modified Cholesky decomposition. Four different methods are proposed, namely the Posterior EnKF with its deterministic and stochastic variations, a Non-Gaussian method and a MCMC filter, which used the Bickel-Levina estimator; these methods are based on a modified Cholesky decomposition and tested with the Lorenz 96 model. Their implementations are shown to provide equivalent solutions compared to another EnKF methods like the LETKF and the EnSRF.DoctoradoDoctor en Ingeniería de Sistemas y Computació

Multivariate data assimilation in snow modelling at Alpine sites

The knowledge of snowpack dynamics is of critical importance to several real-time applications such as agricultural production, water resource management, flood prevention, hydropower generation, especially in mountain basins. Snowpack state can be estimated by models or from observations, even though both these sources of information are affected by several errors