skip to main content


The NSF Public Access Repository (NSF-PAR) system and access will be unavailable from 10:00 PM ET on Friday, December 8 until 2:00 AM ET on Saturday, December 9 due to maintenance. We apologize for the inconvenience.

Title: Ensemble Kalman filter updates based on regularized sparse inverse Cholesky factors
Abstract The ensemble Kalman filter (EnKF) is a popular technique for data assimilation in high-dimensional nonlinear state-space models. The EnKF represents distributions of interest by an ensemble, which is a form of dimension reduction that enables straightforward forecasting even for complicated and expensive evolution operators. However, the EnKF update step involves estimation of the forecast covariance matrix based on the (often small) ensemble, which requires regularization. Many existing regularization techniques rely on spatial localization, which may ignore long-range dependence. Instead, our proposed approach assumes a sparse Cholesky factor of the inverse covariance matrix, and the nonzero Cholesky entries are further regularized. The resulting method is highly flexible and computationally scalable. In our numerical experiments, our approach was more accurate and less sensitive to misspecification of tuning parameters than tapering-based localization.  more » « less
Award ID(s):
1934904 1953005 1654083
Author(s) / Creator(s):
Date Published:
Journal Name:
Monthly Weather Review
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    Estimation of an unstructured covariance matrix is difficult because of the challenges posed by parameter space dimensionality and the positive‐definiteness constraint that estimates should satisfy. We consider a general nonparametric covariance estimation framework for longitudinal data using the Cholesky decomposition of a positive‐definite matrix. The covariance matrix of time‐ordered measurements is diagonalized by a lower triangular matrix with unconstrained entries that are statistically interpretable as parameters for a varying coefficient autoregressive model. Using this dual interpretation of the Cholesky decomposition and allowing for irregular sampling time points, we treat covariance estimation as bivariate smoothing and cast it in a regularization framework for desired forms of simplicity in covariance models. Viewing stationarity as a form of simplicity or parsimony in covariance, we model the varying coefficient function with components depending on time lag and its orthogonal direction separately and penalize the components that capture the nonstationarity in the fitted function. We demonstrate construction of a covariance estimator using the smoothing spline framework. Simulation studies establish the advantage of our approach over alternative estimators proposed in the longitudinal data setting. We analyze a longitudinal dataset to illustrate application of the methodology and compare our estimates to those resulting from alternative models.

    This article is categorized under:

    Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data

    Statistical and Graphical Methods of Data Analysis > Nonparametric Methods

    Algorithms and Computational Methods > Maximum Likelihood Methods

    more » « less
  2. Abstract

    For data assimilation to provide faithful state estimates for dynamical models, specifications of observation uncertainty need to be as accurate as possible. Innovation-based methods based on Desroziers diagnostics, are commonly used to estimate observation uncertainty, but such methods can depend greatly on the prescribed background uncertainty. For ensemble data assimilation, this uncertainty comes from statistics calculated from ensemble forecasts, which require inflation and localization to address under sampling. In this work, we use an ensemble Kalman filter (EnKF) with a low-dimensional Lorenz model to investigate the interplay between the Desroziers method and inflation. Two inflation techniques are used for this purpose: 1) a rigorously tuned fixed multiplicative scheme and 2) an adaptive state-space scheme. We document how inaccuracies in observation uncertainty affect errors in EnKF posteriors and study the combined impacts of misspecified initial observation uncertainty, sampling error, and model error on Desroziers estimates. We find that whether observation uncertainty is over- or underestimated greatly affects the stability of data assimilation and the accuracy of Desroziers estimates and that preference should be given to initial overestimates. Inline estimates of Desroziers tend to remove the dependence between ensemble spread–skill and the initially prescribed observation error. In addition, we find that the inclusion of model error introduces spurious correlations in observation uncertainty estimates. Further, we note that the adaptive inflation scheme is less robust than fixed inflation at mitigating multiple sources of error. Last, sampling error strongly exacerbates existing sources of error and greatly degrades EnKF estimates, which translates into biased Desroziers estimates of observation error covariance.

    Significance Statement

    To generate accurate predictions of various components of the Earth system, numerical models require an accurate specification of state variables at our current time. This step adopts a probabilistic consideration of our current state estimate versus information provided from environmental measurements of the true state. Various strategies exist for estimating uncertainty in observations within this framework, but are sensitive to a host of assumptions, which are investigated in this study.

    more » « less
  3. Abstract. Ever since its inception, the ensemble Kalman filter (EnKF) has elicited many heuristic approaches that sought to improve it. One such method is covariance localization, which alleviates spurious correlations due to finite ensemble sizes by using relevant spatial correlation information. Adaptive localization techniques account for how correlations change in time and space, in order to obtain improved covariance estimates. This work develops a Bayesian approach to adaptive Schur-product localization for the deterministic ensemble Kalman filter (DEnKF) and extends it to support multiple radii of influence. We test the proposed adaptive localization using the toy Lorenz'96 problem and a more realistic 1.5-layer quasi-geostrophic model. Results with the toy problem show that the multivariate approach informs us that strongly observed variables can tolerate larger localization radii. The univariate approach leads to markedly improved filter performance for the realistic geophysical model, with a reduction in error by as much as 33 %. 
    more » « less
  4. Spatio-temporal filtering is a common and challenging task in many environmental applications, where the evolution is often nonlinear and the dimension of the spatial state may be very high. We propose a scalable filtering approach based on a hierarchical sparse Cholesky representation of the filtering covariance matrix. At each time point, we compress the sparse Cholesky factor into a dense matrix with a small number of columns. After applying the evolution to each of these columns, we decompress to obtain a hierarchical sparse Cholesky factor of the forecast covariance, which can then be updated based on newly available data. We illustrate the Cholesky evolution via an equivalent representation in terms of spatial basis functions. We also demonstrate the advantage of our method in numerical comparisons, including using a high-dimensional and nonlinear Lorenz model. 
    more » « less
  5. Abstract

    Linear transformations are widely used in data assimilation for covariance modeling, for reducing dimensionality (such as averaging dense observations to form “superobs”), and for managing sampling error in ensemble data assimilation. Here we describe a linear transformation that is optimal in the sense that, in the transformed space, the state variables and observations have uncorrelated errors, and a diagonal gain matrix in the update step. We conjecture, and provide numerical evidence, that the transformation is the best possible to precede covariance localization in an ensemble Kalman filter. A central feature of this transformation in the update step are scalars, which we term canonical observation operators (COOs), that relate pairs of transformed observations and state variables and rank‐order those pairs by their influence in the update. We show for an idealized problem that sample‐based estimates of the COOs, in conjunction with covariance localization for the sample covariance, can approximate well the true values, but a practical implementation of the transformation for high‐dimensional applications remains a subject for future research. The COOs also completely describe important properties of the update step, such as observation‐state mutual information, signal‐to‐noise and degrees of freedom for signal, and so give new insights, including relations among reduced‐rank approximations to variational schemes, particle‐filter weight degeneracy, and the local ensemble transform Kalman filter.

    more » « less