The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

This article describes the governing equations, computational algorithms, and other components entering into the Community Multiscale Air Quality (CMAQ) modeling system. This system has been designed to approach air quality as a whole by including state-ofthe-science capabilities for modeling multiple air quality issues, including tropospheric ozone, fine particles, acid deposition, and visibility degradation. CMAQ was also designed to have multiscale capabilities so that separate models were not needed for urban and regional scale air quality modeling. By making CMAQ a modeling system that addresses multiple pollutants and different spatial scales, it has a “one-atmosphere” perspective that combines the efforts of the scientific community. To implement multiscale capabilities in CMAQ, several issues (such as scalable atmospheric dynamics and generalized coordinates), which depend on the desired model resolution, are addressed. A set of governing equations for compressible nonhydrostatic atmospheres is available to better resolve atmospheric dynamics at smaller scales. Because CMAQ is designed to handle scale-dependent meteorological formulations and a large amount of flexibility, its governing equations are expressed in a generalized coordinate system. This approach ensures consistency between CMAQ and the meteorological modeling system. The generalized coordinate system determines the necessary grid and coordinate transformations, and it can accommodate various vertical coordinates and map projections. The CMAQ modeling system simulates various chemical and physical processes that are thought to be important for understanding atmospheric trace gas transformations and distributions. The modeling system contains three types of modeling components (Models-3): a meteorological modeling system for the description of atmospheric states and motions, emission models for man-made and natural emissions that are injected into the atmosphere, and a chemistry-transport modeling system for simulation of the chemical transformation and fate. The chemical transport model includes the following process modules: horizontal advection, vertical advection, mass conservation adjustments for advection processes, horizontal diffusion, vertical diffusion, gas-phase chemical reactions and solvers, photolytic rate computation, aqueous-phase reactions and cloud mixing, aerosol dynamics, size distributions and chemistry, plume chemistry effects, and gas and aerosol deposition velocity estimation. This paper describes the Models-3 CMAQ system, its governing equations, important science algorithms, and a few application examples. This review article cites 114 references. DOI: 10.1115/1.2128636

Review of the Governing Equations, Computational Algorithms, and Other Components of the Models-3 Community Multiscale Air Quality (CMAQ) Modeling System

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

Numerical Initial Value Problems in Ordinary Differential Equations

Mathematica--A System for Doing Mathematics by Computer.

The two-dimensional Danish Eulerian Model has been developed during the 80'ies. More than 70 chemical reactions (some of them photochemical) are involved in the model. The space domain of the model covers the whole of Europe. It has been discretized by using a (32 × 32) equidistant spatial grid. The concentrations and the species calculated by the model were compared both with measurements taken over land and with measurements taken over sea. The model has also been run by using finer grids; as, for example, a (96 × 96) grid. The experiments indicated that in general the model calculates rather reliable results. However, it is also clear that the results might be improved if a three-dimensional version of the model is developed. Three-dimensional air pollution models are very time-consuming. Therefore the development of a reliable three-dimensional version of an air pollution model is a very challenging task. The efforts to solve some of the numerical problems arising during the development of a large three-dimensional air pollution model (with non-linear chemical reactions) will be discussed in this paper.

Three-Dimensional Version of the Danish Eulerian Model

NO ONE DOUBTS THAT HIGH CONCENTRATIONS OF air pollutants can damage (either directly or indirectly) plants, animals, and people. But at what point do these concentrations go from acceptable to dangerous level-nd just as importantly, from dangerous to acceptable? Recent environmental studies show that in order to prevent ecosystem destruction, it is absolutely necessary to reduce the concentration and deposition of certain dangerous air pollutants, at least in Europe and North America, to acceptable levels, and keep them there. These are urgent tasks: some damaging effects may soon be irreversible. O n the other hand, because lowering pollution levels is expensive, we need to reduce them to safe levels but no further. T h e critical concentration of a pollutant is the highest concentration that will not damage biological systems over a long period of time, say 50 years. Deposition is the physical process by which air pollutants are returned to the surface of the Earth. There are two deposition mechanisms: d y deposition takes place continuously, and met depos.itioii occurs only during precipitation periods. Concentration and deposition are two separate processes, with significant but different effects on people and the environment; much of the discussion below deals with both. However, for the sake of brevity in this discussion, we’ll use the term “concentration” to mean both concentration and deposition. T h e two problems outlined-establishing critical concentration levels of pollutants and developing effective control strategies to keep pollutants just under those levels--are very complicated, for a t least three reasons:

Modeling the long-range transport of air pollutants

Three-dimensional version of the Danish Eulerian Model

Computational challenges in the numerical treatment of large air pollution models

Mathematical models are indispensable tools in different environmental studies. Such models are usually described by systems of partial differential equations (PDE's). The number of equations in the PDE system is equal to the number of the chemical species involved in the model. The application of different discretization and splitting techniques transforms the system of partial differential equations into five very large systems of ordinary differential equations (ODE's). The treatment of the ODE systems leads to the solution of several large systems of linear algebraic equations at every time-step. Sparse matrix technique can be used in order to reduce the number of arithmetic operations. The efficiency can be further increased by applying parallel algorithms. The use of a special sparse matrix algorithm and the parallelization of the computational process will be discussed in this paper.

Krassimir Georgiev

Papers

Three-Dimensional Version of the Danish Eulerian Model

Modeling the long-range transport of air pollutants

Three-dimensional version of the Danish Eulerian Model

Computational challenges in the numerical treatment of large air pollution models

Parallel sparse matrix algorithms for air pollution models