Emmanuel Cosme

Smoothing Problems in a Bayesian Framework and Their Linear Gaussian Solutions

AbstractSmoothers are increasingly used in geophysics. Several linear Gaussian algorithms exist, and the general picture may appear somewhat confusing. This paper attempts to stand back a little, in order to clarify this picture by providing a concise overview of what the different smoothers really solve, and how. The authors begin addressing this issue from a Bayesian viewpoint. The filtering problem consists in finding the probability of a system state at a given time, conditioned to some past and present observations (if the present observations are not included, it is a forecast problem). This formulation is unique: any different formulation is a smoothing problem. The two main formulations of smoothing are tackled here: the joint estimation problem (fixed lag or fixed interval), where the probability of a series of system states conditioned to observations is to be found, and the marginal estimation problem, which deals with the probability of only one system state, conditioned to past, present, and ...

Efficient Parameterization of the Observation Error Covariance Matrix for Square Root or Ensemble Kalman Filters: Application to Ocean Altimetry

Abstract In the Kalman filter standard algorithm, the computational complexity of the observational update is proportional to the cube of the number y of observations (leading behavior for large y). In realistic atmospheric or oceanic applications, involving an increasing quantity of available observations, this often leads to a prohibitive cost and to the necessity of simplifying the problem by aggregating or dropping observations. If the filter error covariance matrices are in square root form, as in square root or ensemble Kalman filters, the standard algorithm can be transformed to be linear in y, providing that the observation error covariance matrix is diagonal. This is a significant drawback of this transformed algorithm and often leads to an assumption of uncorrelated observation errors for the sake of numerical efficiency. In this paper, it is shown that the linearity of the transformed algorithm in y can be preserved for other forms of the observation error covariance matrix. In particular, quit...

A Reduced-Order Kalman Filter for Data Assimilation in Physical Oceanography

A central task of physical oceanography is the prediction of ocean circulation at various time scales. Mathematical techniques are used in this domain not only for the modeling of ocean circulation but also for the enhancement of simulation through data assimilation. The ocean circulation model of concern here, namely, HYCOM, is briefly presented through its variables, equations, and specific vertical coordinate system. The main part of this paper focuses on the Kalman filter as a data assimilation method, and especially on how this mathematical technique, usually associated with a prohibitively high computing cost for operational sciences, is simplified in order to make it applicable to the simulation of realistic ocean circulation models. Some practical issues are presented, such as a brief explanation about ocean observation systems, together with examples of data assimilation results.

Free and laterally nudged Antarctic climate of an atmospheric general circulation model

Because many of the synoptic cyclones south of the 60°S parallel originate from 60°S and lower latitudes, nudging an atmospheric general circulation model (AGCM) with meteorological analyses at the periphery of the Antarctic region may be expected to exert a strong control on the atmospheric circulation inside the region. Here, the ECMWF reanalyses are used to nudge the atmospheric circulation of the Laboratoire de Meteorologie Dynamique Zoom (LMDZ) stretched-grid AGCM in a 15-yr simulation spanning the 1979–93 period. The horizontal resolution (grid spacing) in the model reaches ∼100 km south of 60°S. Nudging is exerted along the 60°S parallel, and this is called lateral nudging for the Antarctic region. Nudging is also performed farther north, near 50° and 40°S, but this is not essential for the results discussed here. Surface pressure and winds in the atmospheric column are nudged without relaxation to maximize control by the meteorological analyses, at the expense of some “noise” confined to the latitudes where nudging is exerted. The performances of lateral nudging are evaluated with respect to station observations, the free (unnudged) model, the ECMWF reanalyses, and in limited instances with respect to nudging the surface pressure only. It is shown that the free model has limited but persistent surface pressure and geopotential defects in the Antarctic region, which are efficiently corrected by lateral nudging. Also, the laterally nudged simulations confirm, and to some extent correct, a geopotential deficiency of the ECMWF reanalyses over the east Antarctic continent previously identified by others. The monthly mean variability of surface climate at several stations along a coast-to-pole transect is analyzed. A significant fraction of the observed variability of surface pressure and temperature is reproduced. The fraction is often less than in the reanalyses. However, the differences are not large considering that the nudged model is forced at distances of hundreds to thousands of kilometers whereas the reanalyses are forced at much shorter distances, in principle right at each station site by the very station data. The variability of surface wind is significantly less well reproduced than that of pressure and temperature in both the nudged model and the reanalyses. Carefully adjusted polar physics in the LMDZ model seems to compensate for a distant observational constraint in the cases when the nudged model results appear similar or even superior to the reanalyses. Lateral nudging is less computationally intensive than global nudging, and it induces realistic variability and chronology while leaving full expression of the model physics in the region of interest. Laterally nudging an AGCM with meteorological analyses can offer complementary value over the analyses themselves, not only by producing additional atmospheric information not available from the analyses, but also by correcting possible regional defects in the analyses.

Efficient Adaptive Error Parameterizations for Square Root or Ensemble Kalman Filters: Application to the Control of Ocean Mesoscale Signals

Abstract In Kalman filter applications, an adaptive parameterization of the error statistics is often necessary to avoid filter divergence, and prevent error estimates from becoming grossly inconsistent with the real error. With the classic formulation of the Kalman filter observational update, optimal estimates of general adaptive parameters can only be obtained at a numerical cost that is several times larger than the cost of the state observational update. In this paper, it is shown that there exists a few types of important parameters for which optimal estimates can be computed at a negligible numerical cost, as soon as the computation is performed using a transformed algorithm that works in the reduced control space defined by the square root or ensemble representation of the forecast error covariance matrix. The set of parameters that can be efficiently controlled includes scaling factors for the forecast error covariance matrix, scaling factors for the observation error covariance matrix, or even a...

Efficient Local Error Parameterizations for Square Root or Ensemble Kalman Filters: Application to a Basin-Scale Ocean Turbulent Flow

Abstract In large-sized atmospheric or oceanic applications of square root or ensemble Kalman filters, it is often necessary to introduce the prior assumption that long-range correlations are negligible and force them to zero using a local parameterization, supplementing the ensemble or reduced-rank representation of the covariance. One classic algorithm to perform this operation consists of taking the Schur product of the ensemble covariance matrix by a local support correlation matrix. However, with this parameterization, the square root of the forecast error covariance matrix is no more directly available, so that any observational update algorithm requiring this square root must include an additional step to compute local square roots from the Schur product. This computation generates an additional numerical cost or produces high-rank square roots, which may deprive the observational update from its original efficiency. In this paper, it is shown how efficient local square root parameterizations can b...

A Consistent Hybrid Variational-Smoothing Data Assimilation Method: Application to a Simple Shallow-Water Model of the Turbulent Midlatitude Ocean

AbstractIn the standard four-dimensional variational data assimilation (4D-Var) algorithm the background error covariance matrix remains static over time. It may therefore be unable to correctly take into account the information accumulated by a system into which data are gradually being assimilated.A possible method for remedying this flaw is presented and tested in this paper. A hybrid variational-smoothing algorithm is based on a reduced-rank incremental 4D-Var. Its consistent coupling to a singular evolutive extended Kalman (SEEK) smoother ensures the evolution of the matrix. In the analysis step, a low-dimensional error covariance matrix is updated so as to take into account the increased confidence level in the state vector it describes, once the observations have been introduced into the system. In the forecast step, the basis spanning the corresponding control subspace is propagated via the tangent linear model.The hybrid method is implemented and tested in twin experiments employing a shallow-wat...

An Efficient Way to Account for Observation Error Correlations in the Assimilation of Data from the Future SWOT High-Resolution Altimeter Mission

AbstractMost data assimilation algorithms require the inverse of the covariance matrix of the observation errors. In practical applications, the cost of computing this inverse matrix with spatially correlated observation errors is prohibitive. Common practices are therefore to subsample or combine the observations so that the errors of the assimilated observations can be considered uncorrelated. As a consequence, a large fraction of the available observational information is not used in practical applications. In this study, a method is developed to account for the correlations of the errors that will be present in the wide-swath sea surface height measurements, for example, the Surface Water and Ocean Topography (SWOT) mission. It basically consists of the transformation of the observation vector so that the inverse of the corresponding covariance matrix can be replaced by a diagonal matrix, thus allowing to genuinely take into account errors that are spatially correlated in physical space. Numerical exp...

A satellite-based method for estimating global oceanic DMS and its application in a 3-D atmospheric GCM

In order to assess in three-dimensional atmospheric models the climate effects of anthropogenic sulphate aerosols, it is necessary not only to compute spatial and temporal distributions of anthropogenic sulphate, but also to simulate spatially and temporally the emission, transport and transformation of natural sulphur gases and aerosols emitted at the Earth’s surface. Jones et al. (2001) recently obtained a value of -1.9 W m-2 for the effect of anthropogenic sulphate aerosol on cloud albedo and on precipitation efficiency (the ‘indirect’ sulphate aerosol forcing effect), and demonstrated in a sensitivity test that doubling oceanic dimethylsulphide (DMS) emission fluxes reduced the indirect effect by over 25%. Thus, changes in marine DMS emissions appear to significantly affect estimates of the magnitude of anthropogenic sulphate forcing.

SWOT Spatial Scales in the Western Mediterranean Sea Derived from Pseudo-Observations and an Ad Hoc Filtering

The aim of this study is to assess the capacity of the Surface Water Ocean Topography (SWOT) satellite to resolve fine scale oceanic surface features in the western Mediterranean. Using as input the Sea Surface Height (SSH) fields from a high-resolution Ocean General Circulation Model (OGCM), the SWOT Simulator for Ocean Science generates SWOT-like outputs along a swath and the nadir following the orbit ground tracks. Given the characteristic temporal and spatial scales of fine scale features in the region, we examine temporal and spatial resolution of the SWOT outputs by comparing them with the original model data which are interpolated onto the SWOT grid. To further assess the satellite’s performance, we derive the absolute geostrophic velocity and relative vorticity. We find that instrument noise and geophysical error mask the whole signal of the pseudo-SWOT derived dynamical variables. We therefore address the impact of removal of satellite noise from the pseudo-SWOT data using a Laplacian diffusion filter, and then focus on the spatial scales that are resolved within a swath after this filtering. To investigate sensitivity to different filtering parameters, we calculate spatial spectra and root mean square errors. Our numerical experiments show that noise patterns dominate the spectral content of the pseudo-SWOT fields at wavelengths below 60 km. Application of the Laplacian diffusion filter allows recovery of the spectral signature within a swath down to the 40–60 km wavelength range. Consequently, with the help of this filter, we are able to improve the observation of fine scale oceanic features in pseudo-SWOT data, and in the estimation of associated derived variables such as velocity and vorticity.