There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

A new sequential data assimilation method is discussed. It is based on forecasting the error statistics using Monte Carlo methods, a better alternative than solving the traditional and computationally extremely demanding approximate error covariance equation used in the extended Kalman filter. The unbounded error growth found in the extended Kalman filter, which is caused by an overly simplified closure in the error covariance equation, is completely eliminated. Open boundaries can be handled as long as the ocean model is well posed. Well-known numerical instabilities associated with the error covariance equation are avoided because storage and evolution of the error covariance matrix itself are not needed. The results are also better than what is provided by the extended Kalman filter since there is no closure problem and the quality of the forecast error statistics therefore improves. The method should be feasible also for more sophisticated primitive equation models. The computational load for reasonable accuracy is only a fraction of what is required for the extended Kalman filter and is given by the storage of, say, 100 model states for an ensemble size of 100 and thus CPU requirements of the order of the cost of 100 model integrations. The proposed method can therefore be used with realistic nonlinear ocean models on large domains on existing computers, and it is also well suited for parallel computers and clusters of workstations where each processor integrates a few members of the ensemble.

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Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics

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.

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The Ensemble Kalman Filter: theoretical formulation and practical implementation

The Twentieth Century Reanalysis (20CR) project is an international effort to produce a comprehensive global atmospheric circulation dataset spanning the twentieth century, assimilating only surface pressure reports and using observed monthly sea-surface temperature and sea-ice distributions as boundary conditions. It is chiefly motivated by a need to provide an observational dataset with quantified uncertainties for validations of climate model simulations of the twentieth century on all time-scales, with emphasis on the statistics of daily weather. It uses an Ensemble Kalman Filter data assimilation method with background ‘first guess’ fields supplied by an ensemble of forecasts from a global numerical weather prediction model. This directly yields a global analysis every 6 hours as the most likely state of the atmosphere, and also an uncertainty estimate of that analysis.

The 20CR dataset provides the first estimates of global tropospheric variability, and of the dataset's time-varying quality, from 1871 to the present at 6-hourly temporal and 2° spatial resolutions. Intercomparisons with independent radiosonde data indicate that the reanalyses are generally of high quality. The quality in the extratropical Northern Hemisphere throughout the century is similar to that of current three-day operational NWP forecasts. Intercomparisons over the second half-century of these surface-based reanalyses with other reanalyses that also make use of upper-air and satellite data are equally encouraging.

It is anticipated that the 20CR dataset will be a valuable resource to the climate research community for both model validations and diagnostic studies. Some surprising results are already evident. For instance, the long-term trends of indices representing the North Atlantic Oscillation, the tropical Pacific Walker Circulation, and the Pacific–North American pattern are weak or non-existent over the full period of record. The long-term trends of zonally averaged precipitation minus evaporation also differ in character from those in climate model simulations of the twentieth century. Copyright © 2011 Royal Meteorological Society and Crown Copyright.

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The Twentieth Century Reanalysis Project

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

Statistical definitions.- Analysis scheme.- Sequential data assimilation.- Variational inverse problems.- Nonlinear variational inverse problems.- Probabilistic formulation.- Generalized Inverse.- Ensemble methods.- Statistical optimization.- Sampling strategies for the EnKF.- Model errors.- Square Root Analysis schemes.- Rank issues.- Spurious correlations, localization, and inflation.- An ocean prediction system.- Estimation in an oil reservoir simulator.

Data Assimilation: The Ensemble Kalman Filter

This paper discusses an important issue related to the implementation and interpretation of the analysis scheme in the ensemble Kalman filter. It is shown that the observations must be treated as random variables at the analysis steps. That is, one should add random perturbations with the correct statistics to the observations and generate an ensemble of observations that then is used in updating the ensemble of model states. Traditionally, this has not been done in previous applications of the ensemble Kalman filter and, as will be shown, this has resulted in an updated ensemble with a variance that is too low. This simple modification of the analysis scheme results in a completely consistent approach if the covariance of the ensemble of model states is interpreted as the prediction error covariance, and there are no further requirements on the ensemble Kalman filter method, except for the use of an ensemble of sufficient size. Thus, there is a unique correspondence between the error statistics from the ensemble Kalman filter and the standard Kalman filter approach.

/pdf/analysis-scheme-in-the-ensemble-kalman-filter-23wq55uwgs.pdf

Geir Evensen

Papers

Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics

The Ensemble Kalman Filter: theoretical formulation and practical implementation

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

Data Assimilation: The Ensemble Kalman Filter

Analysis Scheme in the Ensemble Kalman Filter