Geophysical Data Assimilation Research Articles

An Evaluation of Non-Gaussian Data Assimilation Methods in Moist Convective Regimes

Abstract Obtaining a faithful probabilistic depiction of moist convection is complicated by unknown errors in subgrid-scale physical parameterization schemes, invalid assumptions made by data assimilation (DA) techniques, and high system dimensionality. As an initial step toward untangling sources of uncertainty in convective weather regimes, we evaluate a novel Bayesian data assimilation methodology based on particle filtering within a WRF ensemble analysis and forecasting system. Unlike most geophysical DA methods, the particle filter (PF) represents prior and posterior error distributions nonparametrically rather than assuming a Gaussian distribution and can accept any type of likelihood function. This approach is known to reduce bias introduced by Gaussian approximations in low-dimensional and idealized contexts. The form of PF used in this research adopts a dimension-reduction strategy, making it affordable for typical weather applications. The present study examines posterior ensemble members and forecasts for select severe weather events between 2019 and 2020, comparing results from the PF with those from an ensemble Kalman filter (EnKF). We find that assimilating with a PF produces posterior quantities for microphysical variables that are more consistent with model climatology than comparable quantities from an EnKF, which we attribute to a reduction in DA bias. These differences are significant enough to impact the dynamic evolution of convective systems via cold pool strength and propagation, with impacts to forecast verification scores depending on the particular microphysics scheme. Our findings have broad implications for future approaches to the selection of physical parameterization schemes and parameter estimation within preexisting data assimilation frameworks. Significance Statement The accurate prediction of severe storms using numerical weather models depends on effective parameterization schemes for small-scale processes and the assimilation of incomplete observational data in a manner that faithfully represents the probabilistic state of the atmosphere. Current generation methods for data assimilation typically assume a standard form for the error distributions of relevant quantities, which can introduce bias that not only hinders numerical prediction, but that can also confound the characterization of errors from the model itself. The current study performs data assimilation using a novel method that does not make such assumptions and explores characteristics of resulting model fields and forecasts that might make such a method useful for improving model parameterization schemes.

A Consistent Stochastic Large-Scale Representation of the Navier–Stokes Equations

In this paper we analyze the theoretical properties of a stochastic representation of the incompressible Navier–Stokes equations defined in the framework of the modeling under location uncertainty (LU). This setup built from a stochastic version of the Reynolds transport theorem incorporates a so-called transport noise and involves several specific additional features such as a large scale diffusion term, akin to classical subgrid models, and a modified advection term arising from the spatial inhomogeneity of the small-scale velocity components. This formalism has been numerically evaluated in a series of studies with a particular interest on geophysical flows approximations and data assimilation. In this work we focus more specifically on its theoretical analysis. We demonstrate, through classical arguments, the existence of martingale solutions for the stochastic Navier–Stokes equations in LU form. We show they are pathwise and unique for 2D flows. We then prove that if the noise intensity goes to zero, these solutions converge, up to a subsequence in dimension 3, to a solution of the deterministic Navier–Stokes equation. similarly to the grid convergence property of well established large-eddies simulation strategies, this result allows us to give some guarantee on the interpretation of the LU Navier–Stokes equations as a consistent large-scale model of the deterministic Navier–Stokes equation.

Research Spotlights

The first Research Spotlights article in this issue is concerned with filtering, a task of paramount importance in a great many applications such as numerical weather prediction and geophysical data assimilation. Authors Alessio Spantini, Ricardo Baptista, and Youssef M. Marzouk, in their article “Coupling Techniques for Nonlinear Ensemble Filtering,” describe discrete-time filtering as the act of characterizing the sequence of conditional distributions of the latent field at observation times, given all currently available measurements. Despite the existing literature on filtering, issues such as high-dimensional state spaces and sparse (in both space and time) observations still prove formidable in practice. The traditional approach of ensemble-based data assimilation is the ensemble Kalman filter (EnKF), involving a prediction (forecasting) step followed by an analysis step. However, the authors note an intrinsic bias of EnKF due to the linearity of the transformation, estimated under Gaussian assumptions, that is used in the analysis step, which limits its accuracy. To overcome this, they propose two non-Gaussian generalizations of the EnKF---the so-called stochastic and deterministic map filters---using nonlinear transformations derived from couplings between the forecast distribution and the filtering distribution. What is crucial is that the transformations “can be estimated efficiently...perhaps using only convex optimization,” that they “are easy to `localize' in high dimensions,” and that their computation “should not become increasingly challenging as the variance of the observation noise decreases.” Following a comprehensive description of their new approaches, the authors demonstrate numerically the superiority of their stochastic map filter approach over traditional EnKF. The subsequent discussion offers the reader several jumping off points for future research. Recovery of a sparse solution to a large-scale optimization problem is another ubiquitous problem arising in many applications such as image reconstruction, signal processing, and machine learning. The cost functional typically includes a regularization term in the form of an $\ell_1$ norm term on the solution and/or regularized solution to enforce sparsity. Designing suitable algorithms for such recovery problems is the subject of our second Research Spotlights article. In “Sparse Approximations with Interior Point Methods,” authors Valentina De Simone, Daniela di Serafino, Jacek Gondzio, Spyridon Pougkakiotis, and Marco Viola set out to correct the misconception that first-order methods are to be preferred over second-order methods out of hand. Through case studies, they offer evidence that interior point methods (IPMs) which are constructed to “exploit special features of the problems in the linear algebra of IPMs” and which are designed “to take advantage of the expected sparsity of the optimal solution” can in fact be the method of choice for solving this class of optimization problems. The key to their approach is a reformulation of the original sparse approximation problem to one which is seemingly larger but which has properties upon which one can capitalize for computational gain. For each of four representative applications, the authors show how to take computational advantage of the problem-specific structure of the underlying linear systems involved at each iteration. These efforts are complemented by leveraging the expected sparsity: employing heuristics to drop near zero variables, thereby replacing very large, ill-conditioned intermediate systems by better conditioned, smaller systems. Their conclusion is that time invested in tailoring solvers to structure admitted by the reformulated variant, and in taking advantage of expected sparsity, may be well spent, since their demonstrations have shown it is possible for IPMs to have a “noticeable advantage” over state-of-the-art first-order methods for sparse approximation problems.

A local sigma-point unscented Kalman filter for geophysical data assimilation

The prior covariance calculated in the Reduced-rank Sigma-Point Kalman filter (RSPKF) data assimilation method can be suboptimal as a result of finite number of sigma points, effects of sampling error, process error, and other factors. In this study, we analyse the performance of RSPKF by applying a localization scheme that combines local analysis and global generation of sigma points. The analysis at each model grid for the entire domain are updated using only the observations local to the analysis grid. The localization enables a high rank approximation of the background error covariance in a local subspace of greatly lower dimension than the global domain using a small number of sigma points. The global-analysis vector is constructed combining the local analyses at all model grid points. The global analysis-covariance matrix is generated in the ensemble subspace and the global sigma points are constructed following the RSPKF algorithm. Numerical experiments of our method utilized the Lorenz-96 model. The performance of the localization scheme is assessed in the presence of varying parameters such as the number of sigma points (k), inflation factor (ϕ), localization radius (d) and the number of model variables (Nx). When the localization is implemented, the number of sigma points required to achieve the minimum RMSE is significantly reduced compared to a case where no localization is used, for three different cases of model variables (Nx). We also show that the approximate number of sigma points used to obtain optimal estimate is independent of the state dimension Nx of the model. This further highlights the importance of localization in RSPKF, making it a potential candidate for data assimilation in oceanic or atmospheric General Circulation Models (GCMs).

A particle flow filter for high-dimensional system applications.

A novel particle filter proposed recently, the particle flow filter (PFF), avoids the long‐existing weight degeneracy problem in particle filters and, therefore, has great potential to be applied in high‐dimensional systems. The PFF adopts the idea of a particle flow, which sequentially pushes the particles from the prior to the posterior distribution, without changing the weight of each particle. The essence of the PFF is that it assumes the particle flow is embedded in a reproducing kernel Hilbert space, so that a practical solution for the particle flow is obtained. The particle flow is independent of the choice of kernel in the limit of an infinite number of particles. Given a finite number of particles, we have found that a scalar kernel fails in high‐dimensional and sparsely observed settings. A new matrix‐valued kernel is proposed that prevents the collapse of the marginal distribution of observed variables in a high‐dimensional system. The performance of the PFF is tested and compared with a well‐tuned local ensemble transform Kalman filter (LETKF) using the 1,000‐dimensional Lorenz 96 model. It is shown that the PFF is comparable to the LETKF for linear observations, except that explicit covariance inflation is not necessary for the PFF. For nonlinear observations, the PFF outperforms LETKF and is able to capture the multimodal likelihood behavior, demonstrating that the PFF is a viable path to fully nonlinear geophysical data assimilation.

A Local Unscented Transform Kalman Filter for Nonlinear Systems

AbstractThis paper proposes an efficient data assimilation approach based on the sigma-point Kalman filter (SPKF). With a potential for nonlinear filtering applications, the proposed approach, designated as the local unscented transform Kalman filter (LUTKF), is similar to the SPKF in that the mean and covariance of the nonlinear system are estimated by propagating a set of sigma points—also referred to as ensemble members—generated using the scaled unscented transformation (SUT), while making no assumptions with regard to nonlinear models. However, unlike the SPKF, the LUTKF can reduce the influence of observations on distant state variables by employing a localization scheme to suppress spurious correlations between distant locations in the error covariance matrix. Moreover, while the SPKF uses the augmented state vector constructed by concatenating the model states, model noise, and measurement noise, the system state for the LUTKF is not augmented with the random noise variables, thereby providing an accurate state estimate with relatively few sigma points. In sensitivity experiments executed with a 40-variable Lorenz system, the LUTKF required only three sigma points to prevent filter divergence for linear/nonlinear measurement models. Comparisons of the LUTKF and the local ensemble transform Kalman filters (LETKFs) reveal the advantages of the proposed filter in situations that share common features with geophysical data assimilation applications. In particular, the LUTKF shows considerable benefits over LETKFs when assimilating densely spaced observations that are related nonlinearly to the model state and that have high noise levels—such as the assimilation of remotely sensed data from satellites and radars.

An Iterative Ensemble Kalman Smoother in Presence of Additive Model Error

Ensemble variational methods are being increasingly used in the field of geophysical data assimilation. Their efficiency comes from the combined use of ensembles, which provide statistics estimates, and a variational analysis, which handles nonlinear operators through iterative optimization techniques. Taking model error into account in four-dimensional ensemble variational algorithms is challenging because the state trajectory over the data assimilation window (DAW) is no longer determined by its sole initial condition. In particular, the control variable dimension scales with the DAW length, which yields a high numerical complexity. This is unfortunate since accuracy improvement is expected with longer DAWs. Building upon the work of [P. Sakov and M. Bocquet, Tellus A, 70 (2018), 1414545], this paper discusses how to algorithmically construct and numerically test an iterative ensemble Kalman smoother with additive model error (IEnKS-Q) which is thought to be the natural weak constraint generalization of the IEnKS [M. Bocquet and P. Sakov, Quart. J. Roy. Meteorol. Soc., 140 (2014), pp. 1521--1535], as well as the generalization of IEnKF-Q [P. Sakov, J. Haussaire, and M. Bocquet, Quart. J. Roy. Meteorol. Soc., 144 (2018), pp. 1297--1309] to general DAWs. The number of model evaluations per cycle of the IEnKS-Q is also examined. Solutions based on perturbation decomposition are proposed to dissociate those numerically costly evaluations from the control variable dimension.

On the Efficiency of Covariance Localisation of the Ensemble Kalman Filter Using Augmented Ensembles

Localisation is one of the main reasons for the success of the ensemble Kalman filter (EnKF) in high-dimensional geophysical data assimilation problems. It is based on the assumption that correlations between variables of a dynamical system decrease at a fast rate with the physical distance. In the EnKF, two types of localisation methods have emerged: domain localisation and covariance localisation. In this article, we explore possible implementations for covariance localisation in the deterministic EnKF using augmented ensembles in the analysis. We first discuss and compare two different methods to construct the augmented ensemble. The first method, known as modulation, relies on a factorisation property of the background covariance matrix. The second method is based on randomised singular value decomposition (svd) techniques, which has not been previously applied to covariance localisation. The qualitative properties of both methods are illustrated using a simple one-dimensional covariance model. We then show how localisation can be introduced in the perturbation update step using this augmented ensemble framework and we derive a generic ensemble square root Kalman filter with covariance localisation (LEnSRF). Using twin simulations of the Lorenz-1996 model we show that the LEnSRF is numerically more efficient when combined with the randomised svd method than with the modulation method. Finally we introduce a realistic extension of the LEnSRF that uses domain localisation in the horizontal direction and covariance localisation in the vertical direction. Using twin simulations of a multilayer extension of the Lorenz-1996 model, we show that this approach is adequate to assimilate satellite radiances, for which domain localisation alone is insufficient.

Geophysical and hydrological data assimilation to monitor water content dynamics in the rocky unsaturated zone

In recent years, geophysics is increasingly used to study the flow and transport processes in the vadose zone. Particularly, when the vadose zone is made up of rocks, it is difficult to install sensors in the subsurface to measure hydrological state variables directly. In these cases, the electrical resistivity tomography (ERT) represents a useful tool to monitor the hydrodynamics of the infiltration and to estimate hydraulic parameters and state variables, such as hydraulic conductivity and water content. We propose an integrated approach aimed at predicting water content dynamics in calcarenite, a sedimentary carbonatic porous rock. The uncoupled hydrogeophysical approach proposed consists in 4D ERT monitoring conducted during an infiltrometer test under falling head conditions. Capacitance probes were installed to measure water content at different depths to validate the estimations derived from ERT. A numerical procedure, based on a data assimilation technique, was accomplished by combining the model (i.e., Richards' equation) with the observations in order to provide reliable water content estimations. We have used a new data assimilation method that is easy to implement, based on the ensemble Kalman filter coupled with Brownian bridges. This approach is particularly suitable for strongly non-linear models, such as Richards' equation, in order to take into account both the model uncertainty and the observation errors. The proposed data assimilation approach was tested for the first time on field data. A reasonable agreement was found between observations and predictions confirming the ability of the integrated approach to predict water content dynamics in the rocky subsoil.

Cross validations of radio-frequency interference signature in AMSR-E data using two detection methods

Radio-frequency interference (RFI) detection for low-frequency microwave measurements is an important step before these data are applied to geophysical parameter retrieval or data assimilation. The...

Algorithm for generating cloud‐mask from multi‐channel satellite data

This article presents an algorithm for generating cloud‐mask, over the ocean, from multi‐channel satellite data. The algorithm is based on the assumption that, at any given instant of time, the observed radiances are a mixture of many Gaussians where each Gaussian mixture component is representative of a scene (clear or cloudy). The problem is attempted as one of the unsupervised clustering. The clusters in the data are separated using a Gaussian mixture model. The proposed algorithm is applied on INSAT‐3D imager data and its performance is assessed by comparing the cloud‐mask thus generated against the Moderate Resolution Imaging Spectroradiometer (MODIS) cloud‐mask for forty selected days in the year 2014. The skill of the algorithm developed in this study is about 20% higher than the algorithm currently operational for generating the INSAT‐3D cloud‐mask. Due to the fact that currently geophysical parameter retrieval (e.g. sea‐surface temperature) and data assimilation are performed in clear‐sky regions, the developed algorithm will have large implications in retrieval and data assimilation studies.

A Localized Particle Filter for High-Dimensional Nonlinear Systems

Abstract This paper presents a new data assimilation approach based on the particle filter (PF) that has potential for nonlinear/non-Gaussian applications in geoscience. Particle filters provide a Monte Carlo approximation of a system’s probability density, while making no assumptions regarding the underlying error distribution. The proposed method is similar to the PF in that particles—also referred to as ensemble members—are weighted based on the likelihood of observations in order to approximate posterior probabilities of the system state. The new approach, denoted the local PF, extends the particle weights into vector quantities to reduce the influence of distant observations on the weight calculations via a localization function. While the number of particles required for standard PFs scales exponentially with the dimension of the system, the local PF provides accurate results using relatively few particles. In sensitivity experiments performed with a 40-variable dynamical system, the local PF requires only five particles to prevent filter divergence for both dense and sparse observation networks. Comparisons of the local PF and ensemble Kalman filters (EnKFs) reveal advantages of the new method in situations resembling geophysical data assimilation applications. In particular, the new filter demonstrates substantial benefits over EnKFs when observation networks consist of densely spaced measurements that relate nonlinearly to the model state—analogous to remotely sensed data used frequently in weather analyses.

Impact of assimilation window length on diurnal features in a Mars atmospheric analysis

Effective simulation of diurnal variability is an important aspect of many geophysical data assimilation systems. For the Martian atmosphere, thermal tides are particularly prominent and contribute much to the Martian atmospheric circulation, dynamics and dust transport. To study the Mars diurnal variability and Mars thermal tides, the Geophysical Fluid Dynamics Laboratory Mars Global Climate Model with the 4D-local ensemble transform Kalman filter (4D-LETKF) is used to perform an analysis assimilating spacecraft temperature retrievals. We find that the use of a ‘traditional’ 6-hr assimilation cycle induces spurious forcing of a resonantly enhanced semi-diurnal Kelvin waves represented in both surface pressure and mid-level temperature by forming a wave 4 pattern in the diurnal averaged analysis increment that acts as a ‘topographic’ stationary forcing. Different assimilation window lengths in the 4D-LETKF are introduced to remove the artificially induced resonance. It is found that short assimilation window lengths not only remove the spurious resonance, but also push the migrating semi-diurnal temperature variation at 50 Pa closer to the estimated ‘true’ tides even in the absence of a radiatively active water ice cloud parameterisation. In order to compare the performance of different assimilation window lengths, short-term to mid-range forecasts based on the hour 00 and 12 assimilation are evaluated and compared. Results show that during Northern Hemisphere summer, it is not the assimilation window length, but the radiatively active water ice clouds that influence the model prediction. A ‘diurnal bias correction’ that includes bias correction fields dependent on the local time is shown to effectively reduce the forecast root mean square differences between forecasts and observations, compensate for the absence of water ice cloud parameterisation and enhance Martian atmosphere prediction. The implications of these results for data assimilation in the Earth's atmosphere are discussed.

The efficiency of geophysical adjoint codes generated by automatic differentiation tools

The accuracy of numerical models that describe complex physical or chemical processes depends on the choice of model parameters. Estimating an optimal set of parameters by optimization algorithms requires knowledge of the sensitivity of the process of interest to model parameters. Typically the sensitivity computation involves differentiation of the model, which can be performed by applying algorithmic differentiation (AD) tools to the underlying numerical code. However, existing AD tools differ substantially in design, legibility and computational efficiency. In this study we show that, for geophysical data assimilation problems of varying complexity, the performance of adjoint codes generated by the existing AD tools (i) Open_AD, (ii) Tapenade, (iii) NAGWare and (iv) Transformation of Algorithms in Fortran (TAF) can be vastly different. Based on simple test problems, we evaluate the efficiency of each AD tool with respect to computational speed, accuracy of the adjoint, the efficiency of memory usage, and the capability of each AD tool to handle modern FORTRAN 90–95 elements such as structures and pointers, which are new elements that either combine groups of variables or provide aliases to memory addresses, respectively. We show that, while operator overloading tools are the only ones suitable for modern codes written in object-oriented programming languages, their computational efficiency lags behind source transformation by orders of magnitude, rendering the application of these modern tools to practical assimilation problems prohibitive. In contrast, the application of source transformation tools appears to be the most efficient choice, allowing handling even large geophysical data assimilation problems. However, they can only be applied to numerical models written in earlier generations of programming languages. Our study indicates that applying existing AD tools to realistic geophysical problems faces limitations that urgently need to be solved to allow the continuous use of AD tools for solving geophysical problems on modern computer architectures.

A non-Gaussian analysis scheme using rank histograms for ensemble data assimilation

Abstract. One challenge of geophysical data assimilation is to address the issue of non-Gaussianities in the distributions of the physical variables ensuing, in many cases, from nonlinear dynamical models. Non-Gaussian ensemble analysis methods fall into two categories, those remapping the ensemble particles by approximating the best linear unbiased estimate, for example, the ensemble Kalman filter (EnKF), and those resampling the particles by directly applying Bayes' rule, like particle filters. In this article, it is suggested that the most common remapping methods can only handle weakly non-Gaussian distributions, while the others suffer from sampling issues. In between those two categories, a new remapping method directly applying Bayes' rule, the multivariate rank histogram filter (MRHF), is introduced as an extension of the rank histogram filter (RHF) first introduced by Anderson (2010). Its performance is evaluated and compared with several data assimilation methods, on different levels of non-Gaussianity with the Lorenz 63 model. The method's behavior is then illustrated on a simple density estimation problem using ensemble simulations from a coupled physical–biogeochemical model of the North Atlantic ocean. The MRHF performs well with low-dimensional systems in strongly non-Gaussian regimes.

Evaluating Data Assimilation Algorithms

Abstract Data assimilation leads naturally to a Bayesian formulation in which the posterior probability distribution of the system state, given all the observations on a time window of interest, plays a central conceptual role. The aim of this paper is to use this Bayesian posterior probability distribution as a gold standard against which to evaluate various commonly used data assimilation algorithms. A key aspect of geophysical data assimilation is the high dimensionality and limited predictability of the computational model. This paper examines the two-dimensional Navier–Stokes equations in a periodic geometry, which has these features and yet is tractable for explicit and accurate computation of the posterior distribution by state-of-the-art statistical sampling techniques. The commonly used algorithms that are evaluated, as quantified by the relative error in reproducing moments of the posterior, are four-dimensional variational data assimilation (4DVAR) and a variety of sequential filtering approximations based on three-dimensional variational data assimilation (3DVAR) and on extended and ensemble Kalman filters. The primary conclusions are that, under the assumption of a well-defined posterior probability distribution, (i) with appropriate parameter choices, approximate filters can perform well in reproducing the mean of the desired probability distribution, (ii) they do not perform as well in reproducing the covariance, and (iii) the error is compounded by the need to modify the covariance, in order to induce stability. Thus, filters can be a useful tool in predicting mean behavior but should be viewed with caution as predictors of uncertainty. These conclusions are intrinsic to the algorithms when assumptions underlying them are not valid and will not change if the model complexity is increased.

An evolution of the back and forth nudging for geophysical data assimilation: application to Burgers equation and comparisons

We study in this article an improvement to the back and forth nudging (BFN) method for geophysical data assimilation. In meteorology or oceanography, the theoretical equations are usually diffusive free, but diffusion is added into the model equations in order to stabilize the numerical integrations and to take into consideration some subscale phenomena. We propose to change the sign of the diffusion in the backward nudging model, which is physically consistent and stabilizes the backward integration. We apply this method to a Burgers equation, study the convergence properties and report the results of numerical experiments. We compare the quality of the estimated initial condition with two other data assimilation techniques that are close from the algorithmic point of view: a variational method, and the quasi-inverse linear method.

Diffusive Back and Forth Nudging algorithm for data assimilation

In this Note, we propose an improvement to the Back and Forth Nudging algorithm for handling diffusion in the context of geophysical data assimilation. We introduce the Diffusive Back and Forth Nudging, in which the sign of the diffusion term is changed in the backward integrations. We study the convergence of this algorithm, in particular for linear transport equations.

Bayesian design of control space for optimal assimilation of observations. Part I: Consistent multiscale formalism

AbstractIn geophysical data assimilation, the control space is by definition the set of parameters which are estimated through the assimilation of observations. It has recently been proposed to design the discretizations of control space in order to assimilate observations optimally. The present paper describes the embedding of that formalism in a consistent Bayesian framework. General background errors are now accounted for. Scale‐dependent errors, such as aggregation errors (that lead to representativeness errors) are consistently introduced. The optimal adaptive discretizations of control space minimize a criterion on a dictionary of grids. New criteria are proposed: degrees of freedom for the signal (DFS) built on the averaging kernel operator, and an observation‐dependent criterion.These concepts and results are applied to atmospheric transport of pollutants. The algorithms are tested on the European tracer experiment (ETEX), and on a prototype of CO2 flux inversion over Europe using a simplified CarboEurope‐IP network. New types of adaptive discretization of control space are tested such as quaternary trees or factorised trees. Quaternary trees are proven to be both economical, in terms of storage and CPU time, and efficient on the test cases. This sets the path for the application of this methodology to high‐dimensional and noisy geophysical systems. Part II of this article will develop asymptotic solutions for the design of control space representations that are obtained analytically and are contenders to exact numerical optimizations. Copyright © 2011 Royal Meteorological Society

Beyond Gaussian Statistical Modeling in Geophysical Data Assimilation

AbstractThis review discusses recent advances in geophysical data assimilation beyond Gaussian statistical modeling, in the fields of meteorology, oceanography, as well as atmospheric chemistry. The non-Gaussian features are stressed rather than the nonlinearity of the dynamical models, although both aspects are entangled. Ideas recently proposed to deal with these non-Gaussian issues, in order to improve the state or parameter estimation, are emphasized.The general Bayesian solution to the estimation problem and the techniques to solve it are first presented, as well as the obstacles that hinder their use in high-dimensional and complex systems. Approximations to the Bayesian solution relying on Gaussian, or on second-order moment closure, have been wholly adopted in geophysical data assimilation (e.g., Kalman filters and quadratic variational solutions). Yet, nonlinear and non-Gaussian effects remain. They essentially originate in the nonlinear models and in the non-Gaussian priors. How these effects are handled within algorithms based on Gaussian assumptions is then described. Statistical tools that can diagnose them and measure deviations from Gaussianity are recalled.The following advanced techniques that seek to handle the estimation problem beyond Gaussianity are reviewed: maximum entropy filter, Gaussian anamorphosis, non-Gaussian priors, particle filter with an ensemble Kalman filter as a proposal distribution, maximum entropy on the mean, or strictly Bayesian inferences for large linear models, etc. Several ideas are illustrated with recent or original examples that possess some features of high-dimensional systems. Many of the new approaches are well understood only in special cases and have difficulties that remain to be circumvented. Some of the suggested approaches are quite promising, and sometimes already successful for moderately large though specific geophysical applications. Hints are given as to where progress might come from.