Peter Jan van Leeuwen

Analysis Scheme in 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.

An Ensemble Kalman Smoother for Nonlinear Dynamics

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.

Particle Filtering in Geophysical Systems

The application of particle filters in geophysical systems is reviewed. Some background on Bayesian filtering is provided, and the existing methods are discussed. The emphasis is on the methodology, and not so much on the applications themselves. It is shown that direct application of the basic particle filter (i.e., importance sampling using the prior as the importance density) does not work in high-dimensional systems, but several variants are shown to have potential. Approximations to the full problem that try to keep some aspects of the particle filter beyond the Gaussian approximation are also presented and discussed.

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

Abstract 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.

Data Assimilation and Inverse Methods in Terms of a Probabilistic Formulation

Abstract 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.

Global OH trend inferred from methylchloroform measurements

Methylchloroform (MCF) measurements taken at the Atmospheric Lifetime Experiment / Global Atmospheric Gases Experiment (ALE/GAGE) measurement stations are used to deduce the tropospheric OH concentration and its linear trend between 1978 and 1993. Global three-dimensional fields of OH are calculated with a transport model that includes background photochemistry. Despite the large uncertainties in these OH fields, the simulated MGF concentrations at the five ALE/GAGE stations compare reasonably well to the measurements. As a next step, the OH fields are adjusted to fit the measurements optimally. An ensemble (Monte Carlo) technique is used to optimize the OH scaling factor and to derive the linear trend in OH. The optimized OH fields and trend imply a MCF lifetime in the troposphere of 4.7 years in 1978 and of 4.5 years in 1993. For CH4 these lifetimes (due to OH destruction only) are 9.2 and 8.6 years in 1978 and 1993, respectively. Uncertainties in these estimates are discussed using box-model calculations. The optimized OH concentration is sensitive to the strength of other MCF sinks in the model and is constrained to 1.00−0.15+0.09 × 106 × molecules cm−3 in 1978 and to 1.07−0.17+0.09 × 106 molecules cm−3 in 1993. The deduced OH trend is sensitive to the trend in the MCF emissions and is confined to the interval between −0.1 and +1.1% yr−1 with a most likely value of 0.46% yr−1. Possible causes of a global increase in OH are discussed. A positive OH trend is calculated due to stratospheric ozone depletion, declining CO concentrations, increased water vapor abundance, and enhanced NOx emissions. Although the changes in the atmospheric composition are to a large extent unknown, it seems that the observed changes are consistent with significant increases in OH over the past decades.

Impact of interbasin exchange on the Atlantic overturning circulation

Abstract The thermohaline exchange between the Atlantic and the Southern Ocean is analyzed, using a dataset based on WOCE hydrographic data. It is shown that the salt and heat transports brought about by the South Atlantic subtropical gyre play an essential role in the Atlantic heat and salt budgets. It is found that on average the exported North Atlantic Deep Water (NADW) is fresher than the return flows (basically composed of thermocline and intermediate water), indicating that the overturning circulation (OC) exports freshwater from the Atlantic. The sensitivity of the OC to interbasin fluxes of heat and salt is studied in a 2D model, representing the Atlantic between 60°N and 30°S. The model is forced by mixed boundary conditions at the surface, and by realistic fluxes of heat and salt at its 30°S boundary. The model circulation turns out to be very sensitive to net buoyancy fluxes through the surface. Both net surface cooling and net surface saltening are sources of potential energy and impact positi...

Eddies and variability in the Mozambique Channel

Abstract 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.

Upstream control of Agulhas Ring shedding

Rings shed in the Agulhas retroflection region play an important role in the global thermohaline circulation. The sheding of these rings has been considered very irregular. In this paper, we present evidence for remote control of the timing and frequency of these events. This turns out to be a far more regular process, at a frequency of 4-5 cycles per year. The movement of the Agulhas retroflection, and thereby the shedding of rings, is timed by incomming eddies from the upstream regions. Eddies from the Mozambique Channel, and from the East Madagascar current reach the Retroflection region at the frequency of 4-5 times per year. The existence of these eddies can be related to incomming Rossby waves that cross the Indian ocean and reach the Agulhas current system. These may in turn be part of a basin wide oscillation. The irregularity found in ring shedding statistics can be ascribed to processes occuring between the actual shedding and the first unamigouos observation of a seperated ring.

Comment on “Data Assimilation Using an Ensemble Kalman Filter Technique”

In an interesting paper Houtekamer and Mitchell (1998; hereafter HM98) introduce a variant of the ensemble Kalman filter (EnKF) as proposed by Evensen (1994). HM98 point to the hitherto unacknowledged problem that the EnKF has an ‘‘inbreeding’’ problem: in the analysis step the ensemble is updated with a gain calculated from that same ensemble. In their new approach a double ensemble (DEnKF) is used and the gain of each ensemble is used to update the other ensemble in the analysis step. The new approach is argued to be much less sensitive to this inbreeding. They strengthen their argument by a specific example, in which the EnKF shows a variance that is too low for small ensemble sizes (#100), but they fail to give a rigorous justification. The purposes of this comment are twofold. First, a theoretical justification of the inbreeding effect is given, and it is shown that the DEnKF has similar, but smaller, problems. Second, a serious concern about the use of small ensemble sizes is expressed, thus bringing into question the use of the DEnKF over the EnKF for real applications.