https://hal.archives-ouvertes.fr/hal-00457602/document

A global overview of drought and heat-induced tree mortality reveals emerging climate change risks for forests

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

This chapter assesses long-term projections of climate change for the end of the 21st century and beyond, where the forced signal depends on the scenario and is typically larger than the internal variability of the climate system. Changes are expressed with respect to a baseline period of 1986-2005, unless otherwise stated.

Chapter 12 - Long-term climate change: Projections, commitments and irreversibility

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.

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Analysis Scheme in the Ensemble Kalman Filter

It is formally proved that the general smoother for nonlinear dynamics can be formulated as a sequential method, that is, observations can be assimilated sequentially during a forward integration. The general filter can be derived from the smoother and it is shown that the general smoother and filter solutions at the final time become identical, as is expected from linear theory. Then, a new smoother algorithm based on ensemble statistics is presented and examined in an example with the Lorenz equations. The new smoother can be computed as a sequential algorithm using only forward-in-time model integrations. It bears a strong resemblance with the ensemble Kalman filter. The difference is that every time a new dataset is available during the forward integration, an analysis is computed for all previous times up to this time. Thus, the first guess for the smoother is the ensemble Kalman filter solution, and the smoother estimate provides an improvement of this, as one would expect a smoother to do. The method is demonstrated in this paper in an intercomparison with the ensemble Kalman filter and the ensemble smoother introduced by van Leeuwen and Evensen, and it is shown to be superior in an application with the Lorenz equations. Finally, a discussion is given regarding the properties of the analysis schemes when strongly non-Gaussian distributions are used. It is shown that in these cases more sophisticated analysis schemes based on Bayesian statistics must be used.

/pdf/an-ensemble-kalman-smoother-for-nonlinear-dynamics-5en0ictzpl.pdf

An ensemble Kalman smoother for nonlinear dynamics

The weak constraint inverse for nonlinear dynamical models is discussed and derived in term of a probabilistic formulation. The well-known result that for Gaussian error statistics the minimum of the weak constraint inverse is equal to the maximum-likelihood estimate is rederived. Then several methods based on ensemble statistics that can be used to find the smoother (as opposed to the filter) solution are introduced and compared to traditional methods. A strong point of the new methods is that they avoid the integration of adjoint equations, which is a complex task for real oceanographic or atmospheric applications. They also avoid iterative searches in a Hilbert space, and error estimates can be obtained without much additional computational effort. The feasibility of the new methods is illustrated in a two-layer quasigeostrophic ocean model.

Data assimilation and inverse methods in terms of a probabilistic formulation

The ring-shedding process in the Agulhas Current is studied using the ensemble Kalman filter to assimilate Geosat altimeter data into a two-layer quasigeostrophic ocean model. The properties of the ensemble Kalman filter are further explored with focus on the analysis scheme and the use of gridded data. The Geosat data consist of 10 fields of gridded sea surface height anomalies separated 10 days apart that are added to a climatic mean field. This corresponds to a huge number of data values, and a data reduction scheme must be applied to increase the efficiency of the analysis procedure. Further, it is illustrated how one can resolve the rank problem occurring when a too large dataset or a small ensemble is used.

Assimilation of Geosat Altimeter Data for the Agulhas Current Using the Ensemble Kalman Filter with a Quasigeostrophic Model

Between 1995 and 2000, on average 4 eddies per year are observed from satellite altimetry to propagate southward through the Mozambique Channel, into the upstream Agulhas region. Further south, these eddies have been found to control the timing and frequency of Agulhas ring shedding. Within the Mozambique Channel, anomalous SSH amplitudes rise to 30 cm , in agreement with in situ measured velocities. Comparison of an observed velocity section with GCM model results shows that the Mozambique Channel eddies in these models are too surface intensified. Also, the number of eddies formed in the models is in disagreement with our observational analysis. Moored current meter measurements observing the passage of three eddies in 2000 are extended to a 5-year time series by referencing the anomalous surface currents estimated from altimeter data to a synoptic LADCP velocity measurement. The results show intermittent eddy passage at the mooring location. A statistical analysis of SSH observations in different parts of the Mozambique Channel shows a southward decrease of the dominant frequency of the variability, going from 7 per year in the extension of the South Equatorial Current north of Madagascar to 4 per year south of Madagascar. The observations suggest that frequency reduction is related to the Rossby waves coming in from the east.

http://www.met.reading.ac.uk/~xv901096/research/Schouten-2003.pdf

Peter Jan van Leeuwen

Papers

Analysis Scheme in the Ensemble Kalman Filter

An ensemble Kalman smoother for nonlinear dynamics

Data assimilation and inverse methods in terms of a probabilistic formulation

Assimilation of Geosat Altimeter Data for the Agulhas Current Using the Ensemble Kalman Filter with a Quasigeostrophic Model

Eddies and variability in the Mozambique Channel