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The Ensemble Kalman Filters (EnKF) employ a Monte-Carlo approach to represent covariance information, and are affected by sampling errors in operational settings where the number of model realizations is much smaller than the model state dimension. To alleviate the effects of these errors EnKF relies on model-specific heuristics such as covariance localization, which takes advantage of the spatial locality of correlations among the model variables. This work proposes an approach to alleviate sampling errors that utilizes a locally averaged-in-time dynamics of the model, described in terms of a climatological covariance of the dynamical system. We use this covariance as the target matrix in covariance shrinkage methods, and develop a stochastic covariance shrinkage approach where synthetic ensemble members are drawn to enrich both the ensemble subspace and the ensemble transformation. We additionally provide for a way in which this methodology can be localized similar to the state-of-the-art LETKF method, and that for a certain model setup, our methodology significantly outperforms it.

Abstract Kinetic integration of large and stiff chemical mechanisms is a computational bottleneck in models of atmospheric chemistry. It requires implicit solution of the coupled system of kinetic differential equations with time‐consuming construction and inversion of the Jacobian matrix. We present here a new version of the Kinetic Pre‐Processor (KPP 3.0.0) for fast integration of chemical kinetics featuring a range of improvements over previous versions in performance, diagnostics, versatility, and community openness. KPP 3.0.0 includes a new adaptive auto‐reduction solver to decrease the size of any mechanism locally and on the fly under conditions where full complexity is not needed, by partitioning species as “fast” or “slow” based on their local production and loss rates. Previous implementations of this adaptive solver suffered from excessive overhead in the repeated construction of the local Jacobian matrix or were hard‐wired to specific mechanisms. Here we retain the general applicability of the method to any mechanism and avoid overhead by using pre‐computed Jacobian matrix terms for the full mechanism and cropping the matrix locally to remove the slow species with no change in memory allocation. We apply this adaptive solver within KPP 3.0.0 to the GEOS‐Chem global 3‐D model of atmospheric chemistry and demonstrate a 32% reduction in solver time while maintaining a mean error lower than 1% for key species in the troposphere.

Data assimilation is a Bayesian inference process that obtains an enhanced understanding of a physical system of interest by fusing information from an inexact physics-based model, and from noisy sparse observations of reality. The multifidelity ensemble Kalman filter (MFEnKF) recently developed by the authors combines a full-order physical model and a hierarchy of reduced order surrogate models in order to increase the computational efficiency of data assimilation. The standard MFEnKF uses linear couplings between models, and is statistically optimal in case of Gaussian probability densities. This work extends the MFEnKF into to make use of a broader class of surrogate model such as those based on machine learning methods such as autoencoders non-linear couplings in between the model hierarchies. We identify the right-invertibility property for autoencoders as being a key predictor of success in the forecasting power of autoencoder-based reduced order models. We propose a methodology that allows us to construct reduced order surrogate models that are more accurate than the ones obtained via conventional linear methods. Numerical experiments with the canonical Lorenz'96 model illustrate that nonlinear surrogates perform better than linear projection-based ones in the context of multifidelity ensemble Kalman filtering. We additionality show a large-scale proof-of-concept result with the quasi-geostrophic equations, showing the competitiveness of the method with a traditional reduced order model-based MFEnKF.

Data assimilation combines information from models, measurements, and priors to obtain improved estimates of the state of a dynamical system such as the atmosphere. Ensemble-based data assimilation approaches such as the Ensemble Kalman filter (EnKF) have gained wide popularity due to their simple formulation, ease of implementation, and good practical results. Many of these methods are derived under the assumption that the underlying probability distributions are Gaussian. It is well accepted, however, that the Gaussianity assumption is too restrictive when applied to large nonlinear models, nonlinear observation operators, and large levels of uncertainty. When the Gaussianity assumptions are severely violated, the performance of EnKF variations degrades. This paper proposes a new ensemble-based data assimilation method, named the sampling filter, which obtains the analysis by sampling directly from the posterior distribution. The sampling strategy is based on a Hybrid Monte Carlo (HMC) approach that can handle non-Gaussian probability distributions. Numerical experiments are carried out using the Lorenz-96 model and observation operators with different levels of non-linearity and differentiability. The proposed filter is also tested with shallow water model on a sphere with linear observation operator. Numerical results show that the sampling filter performs well even in highly nonlinear situations where the traditional filters diverge.

The solution of chemical kinetics is one of the most computationally intensivetasks in atmospheric chemical transport simulations. Due to the stiff nature of the system,implicit time stepping algorithms which repeatedly solve linear systems of equations arenecessary. This paper reviews the issues and challenges associated with the construction ofefficient chemical solvers, discusses several families of algorithms, presents strategies forincreasing computational efficiency, and gives insight into implementing chemical solverson accelerated computer architectures.

Chemical data assimilation is the process by which models use measurements to produce an optimal representation of the chemical composition of the atmosphere. Leveraging advances in algorithms and increases in the available computational power, the integration of numerical predictions and observations has started to play an important role in air quality modeling. This paper gives an overview of several methodologies used in chemical data assimilation. We discuss the Bayesian framework for developing data assimilation systems, the suboptimal and the ensemble Kalman filter approaches, the optimal interpolation (OI), and the three and four dimensional variational methods. Examples of assimilation real observations with CMAQ model are presented.

Ground-level ozone, which forms photochemically in the atmosphere from precursor emissions of oxides of nitrogen (NO _x ) and volatile organic compounds, is a criteria pollutant that harms human health and public welfare. For a representative summer episode, premature mortality and potential productivity losses (PPLs) of selected crops and trees attributable to ozone exposure have been quantified using ozone fields from the Community Multiscale Air Quality (CMAQ) model. We applied exposure-response models for the increased risk of premature mortality due to long-term exposure to ozone over a theoretical minimum risk exposure level (TMREL) and for the reduced accumulation of vegetative biomass for four crop species and eleven tree species using the W126 metric designed to capture impacts on plants. To elucidate which emissions contributed to these disbenefits, we applied adjoint-based sensitivity analysis, which efficiently estimates sensitivities of concentration-based metrics with respect to numerous emissions parameters simultaneously. The adjoint of CMAQ was applied to the continental US to calculate the influence of spatially-resolved ozone precursor emissions on the annual average, domain-wide daily maximum 8 h average over the TMREL (elevated MDA8), premature mortality attributable to exposure to ozone above the TMREL, and PPLs. These quantities provide the impact in terms of the percent reduction in precursor emissions. Additionally, locations where similar percent reductions in ozone precursor emissions would impact one or more endpoints greater than average have been identified. NO _x emissions were found to contribute most to the three metrics. The distinct spatial patterns of emissions influences on public welfare disbenefits as compared to the elevated MDA8 and premature mortality suggest that the current regulatory averaging time motivates different emissions control strategies than those that could most directly protect public welfare.

This Special Issue presents efficient formulations and implementations of sequential and variational data assimilation methods. The methods address three important issues in the context of operational data assimilation: efficient implementation of localization methods, sampling methods for approaching posterior ensembles under non-linear model errors, and adjoint-free formulations of four dimensional variational methods.

This paper presents a fully non-Gaussian filter for sequential data assimilation. The filter is named the “cluster sampling filter”, and works by directly sampling the posterior distribution following a Markov Chain Monte-Carlo (MCMC) approach, while the prior distribution is approximated using a Gaussian Mixture Model (GMM). Specifically, a clustering step is introduced after the forecast phase of the filter, and the prior density function is estimated by fitting a GMM to the prior ensemble. Using the data likelihood function, the posterior density is then formulated as a mixture density, and is sampled following an MCMC approach. Four versions of the proposed filter, namely C ℓ MCMC , C ℓ HMC , MC- C ℓ HMC , and MC- C ℓ HMC are presented. C ℓ MCMC uses a Gaussian proposal density to sample the posterior, and C ℓ HMC is an extension to the Hamiltonian Monte-Carlo (HMC) sampling filter. MC- C ℓ MCMC and MC- C ℓ HMC are multi-chain versions of the cluster sampling filters C ℓ MCMC and C ℓ HMC respectively. The multi-chain versions are proposed to guarantee that samples are taken from the vicinities of all probability modes of the formulated posterior. The new methodologies are tested using a simple one-dimensional example, and a quasi-geostrophic (QG) model with double-gyre wind forcing and bi-harmonic friction. Numerical results demonstrate the usefulness of using GMMs to relax the Gaussian prior assumption especially in the HMC filtering paradigm.

This paper discusses new simulation algorithms for stochastic chemical kinetics that exploit the linearity of the chemical master equation and its matrix exponential exact solution. These algorithms make use of various approximations of the matrix exponential to evolve probability densities in time. A sampling of the approximate solutions of the chemical master equation is used to derive accelerated stochastic simulation algorithms. Numerical experiments compare the new methods with the established stochastic simulation algorithm and the tau-leaping method.

We investigate a new class of implicit–explicit singly diagonally implicit Runge–Kutta methods for ordinary differential equations with both non-stiff and stiff components. The approach is based on extrapolation of the stage values at the current step by stage values in the previous step. This approach was first proposed by the authors in context of implicit–explicit general linear methods.

The task of providing an optimal analysis of the state of the atmosphere requires the development of dynamic data-driven systems (DDDAS) that efficiently integrate the observational data and the models. Data assimilation, the dynamic incorporation of additional data into an executing application, is an essential DDDAS concept with wide applicability. In this paper we discuss practical aspects of nonlinear ensemble Kalman data assimilation applied to atmospheric chemical transport models. We highlight the challenges encountered in this approach such as filter divergence and spurious corrections, and propose solutions to overcome them, such as background covariance inflation and filter localization. The predictability is further improved by including model parameters in the assimilation process. Results for a large scale simulation of air pollution in North-East United States illustrate the potential of nonlinear ensemble techniques to assimilate chemical observations.

Recent studies have shown that exposure to particulate black carbon (BC) has significant adverse health effects and may be more detrimental to human health than exposure to PM _2.5 as a whole. Mobile source BC emission controls, mostly on diesel-burning vehicles, have successfully decreased mobile source BC emissions to less than half of what they were 30 years ago. Quantification of the benefits of previous emissions controls conveys the value of these regulatory actions and provides a method by which future control alternatives could be evaluated. In this study we use the adjoint of the Community Multiscale Air Quality (CMAQ) model to estimate highly-resolved spatial distributions of benefits related to emission reductions for six urban regions within the continental US. Emissions from outside each of the six chosen regions account for between 7% and 27% of the premature deaths attributed to exposure to BC within the region. While we estimate that nonroad mobile and onroad diesel emissions account for the largest number of premature deaths attributable to exposure to BC, onroad gasoline is shown to have more than double the benefit per unit emission relative to that of nonroad mobile and onroad diesel. Within the region encompassing New York City and Philadelphia, reductions in emissions from large industrial combustion sources that are not classified as EGUs (i.e., non-EGU) are estimated to have up to triple the benefits per unit emission relative to reductions to onroad diesel sectors, and provide similar benefits per unit emission to that of onroad gasoline emissions in the region. While onroad mobile emissions have been decreasing in the past 30 years and a majority of vehicle emission controls that regulate PM focus on diesel emissions, our analysis shows the most efficient target for stricter controls is actually onroad gasoline emissions.