Particle filters are ensemble-based assimilation schemes that, unlike the ensemble Kalman filter, employ a fully nonlinear and non-Gaussian analysis step to compute the probability distribution function (pdf) of a system’s state conditioned on a set of observations. Evidence is provided that the ensemble size required for a successful particle filter scales exponentially with the problem size. For the simple example in which each component of the state vector is independent, Gaussian and of unit variance, and the observations are of each state component separately with independent, Gaussian errors, simulations indicate that the required ensemble size scales exponentially with the state dimension. In this example, the particle filter requires at least 1011 members when applied to a 200-dimensional state. Asymptotic results, following the work of Bengtsson, Bickel and collaborators, are provided for two cases: one in which each prior state component is independent and identically distributed, and one in which both the prior pdf and the observation errors are Gaussian. The asymptotic theory reveals that, in both cases, the required ensemble size scales exponentially with the variance of the observation log-likelihood, rather than with the state dimension per se. 2

The Maximum Likelihood Ensemble Filter (MLEF) equations are derived without the differentiability requirement for the prediction model and for the observation operators. Derivation reveals that a new non-differentiable minimization method can be defined as a generalization of the gradient-based unconstrained methods, such as the preconditioned conjugate-gradient and quasi-Newton methods. In the new minimization algorithm the vector of first order increments of the cost function is defined as a generalized gradient, while the symmetric matrix of second order increments of the cost function is defined as a generalized Hessian matrix. In the case of differentiable observation operators, the minimization algorithm reduces to the standard gradient-based form. The non-differentiable aspect of the MLEF algorithm is illustrated in an example with one-dimensional Burgers model and simulated observations. The MLEF algorithm has a robust performance, producing satisfactory results for tested non-differentiable observation operators.

The ensemble Kalman Filter (EnKF) and variants derived therefrom have become important techniques in data assimilation problems. One breakdown of the EnKF method is that in the case of sparsely observed, accurate data, least squares properties of the EnKF create posterior ensemble members that are not compatible with the dynamic model. We propose a modification of Kalman and EnKF filters by imposing a constraint either by projection or by a penalty. A two-step ensemble Kalman Filter is proposed that imposes smoothness as a penalized constraint. The smoothing step consists of another application of the same EnKF code with the smoothness constraint as an independent observation. The utility of the method is demonstrated on a nonlinear dynamic model of wildfire.

A wildfire model is formulated based on balance equations for energy and fuel, where the fuel loss due to combustion corresponds to the fuel reaction rate. The resulting coupled partial differential equations have coefficients which can be approximated from prior measurements of wildfires. An Ensemble Kalman Filter technique is then used to assimilate temperatures measured at selected points into running wildfire simulations. The assimilation technique is able to modify the simulations to track the measurements correctly even if the simulations were started with an erroneous ignition location that is quite far away from the correct one.

this article, we will focus on sequential DA methods that constitute the second class. These methods use a probabilistic framework and give estimates of the whole system state sequentially by propagating information only forward in time. This avoids deriving an inverse or an adjoint model and therefore makes sequential methods easier to adapt for all models. Further, the probabilistic framework is more convenient for error estimation and further stochastic analysis such as threshold characterization

This paper introduces a new approximate solution of the optimal nonlinear filter suitable for nonlinear oceanic and atmospheric data assimilation problems. The method is based on a local linearization in a low-rank kernel representation of the state’s probability density function. In the resulting low-rank kernel particle Kalman (LRKPK) filter, the standard (weight type) particle filter correction is complemented by a Kalman-type correction for each particle using the covariance matrix of the kernel mixture. The LRKPK filter’s solution is then obtained as the weighted average of several low-rank square root Kalman filters operating in parallel. The Kalman-type correction reduces the risk of ensemble degeneracy, which enables the filter to efficiently operate with fewer particles than the particle filter. Combined with the low-rank approximation, it allows the implementation of the LRKPK filter with high-dimensional oceanic and atmospheric systems. The new filter is described and its relevance demonstrated through applications with the simple Lorenz model and a realistic configuration of the Princeton Ocean Model (POM) in the Mediterranean Sea. 1.

Particle filter (PF) is a fully non-linear filter with Bayesian conditional probability estimation, compared here with the well-known ensemble Kalman filter (EnKF). A Gaussian resampling (GR) method is proposed to generate the posterior analysis ensemble in an effective and efficient way. The Lorenz model is used to test the proposed method. The PF with Gaussian resampling (PFGR) can approximate more accurately the Bayesian analysis. The present work demonstrates that the proposed PFGR possesses good stability and accuracy and is potentially applicable to large-scale data assimilation problems. 1.

The ensemble Kalman filter (EnKF) is a recursive filter suitable for problems with a large number of variables, such as discretizations of partial differential equations in geophysical models. The EnKF originated as a version of the Kalman filter for large problems (essentially, the covariance matrix is replaced by the sample covariance), and it is now an important data assimilation component of ensemble forecasting. EnKF is related to the particle filter (in this context, a particle is the same thing as an ensemble member) but the EnKF makes the assumption that all probability distributions involved are Gaussian. This article briefly describes the derivation and practical implementation of the basic version of EnKF, and reviews several extensions. 1

A deterministic square root ensemble Kalman filter and a stochastic perturbed observation ensemble Kalman filter are used for data assimilation in both linear and nonlinear single variable dynamical systems. For the linear system, the deterministic filter is simply a method for computing the Kalman filter and is optimal while the stochastic filter has suboptimal performance due to sampling error. For the nonlinear system, the deterministic filter has increasing error as ensemble size increases because all ensemble members but one become tightly clustered. In this case, the stochastic filter performs better for sufficiently large en-sembles. A new method for computing ensemble increments in observation space is proposed that does not suffer from the pathological behavior of the deterministic filter while avoiding much of the sampling error of the stochastic filter. This filter uses the order statistics of the prior observation space ensemble to create an approximate continuous prior probability distribution in a fashion analogous to the use of rank histograms for ensemble forecast evaluation. This rank histogram filter can represent non-Gaussian observation space priors and posteriors and is shown to be competitive with existing filters for problems as large as global numerical weather prediction. The ability to represent non-Gaussian distributions is useful for a variety of applications such as convective-scale assimilation and assimilation of bounded quantities such as relative humidity. 1.

The concept of alternative error models is suggested as a means to redefine estimation problems with non-Gaussian additive errors so that familiar and near-optimal Gaussian-based methods may still be successfully applied. The specific example of a mixed error model including both align-ment errors and additive errors is examined. Using the specific form of a soliton, an analytical solution to the Korteweg-de Vries equation, the total (additive) errors of states following the mixed error model are demonstrably non-Gaussian, and an ensemble of such states is handled poorly by a traditional ensemble Kalman filter, even if position observations are included. Consideration of the mixed error model itself naturally suggests a two-step approach to state estimation where the align-ment errors are corrected first, followed by application of an estimation scheme to the remaining additive errors, the first step aimed at removing most of the non-Gaussianity so the second step can proceed successfully. Taking an ensemble approach for the soliton states in a perfect model sce-nario, this two-step approach shows a great improvement over traditional methods in a wide range of observational densities, observing frequencies, and observational accuracies. In cases where the two-step approach is not successful, it is often attributable to the first step not having suciently removed the non-Gaussianity, indicating the problem strictly requires an estimation scheme that does not make Gaussian assumptions. However, in these cases a convenient approximation to the two-step approach is available that trades obtaining a minimum variance estimate ensemble mean for more physically sound updates of the individual ensemble members.