Parametric covariance dynamics for the nonlinear diffusive Burgers equation

The parametric Kalman filter (PKF) is a computationally efficient alternative method to the ensemble Kalman filter. The PKF relies on an approximation of the error covariance matrix by a covariance model with a space–time evolving set of parameters. This study extends the PKF to nonlinear dynamics using the diffusive Burgers equation as an application, focusing on the forecast step of the assimilation cycle. The covariance model considered is based on the diffusion equation, with the diffusion tensor and the error variance as evolving parameters. An analytical derivation of the parameter dynamics highlights a closure issue. Therefore, a closure model is proposed based on the kurtosis of the local correlation functions. Numerical experiments compare the PKF forecast with the statistics obtained from a large ensemble of nonlinear forecasts. These experiments strengthen the closure model and demonstrate the ability of the PKF to reproduce the tangent linear covariance dynamics, at a low numerical cost.</p

From the Kalman Filter to the Particle Filter: A Geometrical Perspective of the Curse of Dimensionality

The aim of this contribution is to provide a description of the difference between Kalman filter and particle filter when the state space is of high dimension. In the Gaussian framework, KF and PF give the same theoretical result. However, in high dimension and using finite sampling for the Gaussian distribution, the PF is not able to reproduce the solution produced by the KF. This discrepancy is highlighted from the convergence property of the Gaussian law toward a hypersphere: in high dimension, any finite sample of a Gaussian law lies within a hypersphere centered in the mean of the Gaussian law and of radius square-root of the trace of the covariance matrix. This concentration of probability suggests the use of norm as a criterium that discriminates whether a forecast sample can be compatible or not with a given analysis state. The contribution illustrates important characteristics that have to be considered for the high dimension but does not introduce a new approach to face the curse of dimensionality.</jats:p

Combined assimilation of IASI and MLS observations to constrain tropospheric and stratospheric ozone in a global chemical transport model

Accurate and temporally resolved fields of free-troposphere ozone are of major importance to quantify the intercontinental transport of pollution and the ozone radiative forcing. We consider a global chemical transport model (MOdèle de Chimie Atmosphérique à Grande Échelle, MOCAGE) in combination with a linear ozone chemistry scheme to examine the impact of assimilating observations from the Microwave Limb Sounder (MLS) and the Infrared Atmospheric Sounding Interferometer (IASI). The assimilation of the two instruments is performed by means of a variational algorithm (4D-VAR) and allows to constrain stratospheric and tropospheric ozone simultaneously. The analysis is first computed for the months of August and November 2008 and validated against ozonesonde measurements to verify the presence of observations and model biases. Furthermore, a longer analysis of 6 months (July–December 2008) showed that the combined assimilation of MLS and IASI is able to globally reduce the uncertainty (root mean square error, RMSE) of the modeled ozone columns from 30 to 15% in the upper troposphere/lower stratosphere (UTLS, 70–225 hPa). The assimilation of IASI tropospheric ozone observations (1000–225 hPa columns, TOC – tropospheric O₃ column) decreases the RMSE of the model from 40 to 20% in the tropics (30° S–30° N), whereas it is not effective at higher latitudes. Results are confirmed by a comparison with additional ozone data sets like the Measurements of OZone and wAter vapour by aIrbus in-service airCraft (MOZAIC) data, the Ozone Monitoring Instrument (OMI) total ozone columns and several high-altitude surface measurements. Finally, the analysis is found to be insensitive to the assimilation parameters. We conclude that the combination of a simplified ozone chemistry scheme with frequent satellite observations is a valuable tool for the long-term analysis of stratospheric and free-tropospheric ozone

Balance conditions in variational data assimilation for a high-resolution forecast model

This paper explores the role of balance relationships for background error covariance modelling as the model's grid box decreases to convective-scales. Data assimilation (DA) analyses are examined from a simplified convective-scale model and DA system (called ABC-DA) with a grid box size of 1.5km in a 2D 540km (longitude), 15km (height) domain. The DA experiments are performed with background error covariance matrices B modelled and calibrated by switching on/off linear balance (LB) and hydrostatic balance (HB), and by observing a subset of the ABC variables, namely v, meridional wind, r', scaled density (a pressure-like variable), and b', buoyancy (a temperature-like variable). Calibration data are sourced from two methods of generating proxies of forecast errors. One uses forecasts from different latitude slices of a 3D parent model (here called the `latitude slice method'), and the other uses sets of differences between forecasts of different lengths but valid at the same time (the National Meteorological Center method). Root-mean-squared errors computed over the domain from identical twin DA experiments suggest that there is no combination of LB/HB switches that give the best analysis for all model quantities. It is frequently found though that the B-matrices modelled with both LB and HB do perform the best. A clearer picture emerges when the errors are examined at different spatial scales. In particular it is shown that switching on HB in B mostly has a neutral/positive effect on the DA accuracy at `large' scales, and switching off the HB has a neutral/positive effect at `small' scales. The division between `large' and `small' scales is between 10 and 100km. Furthermore, one hour forecast error correlations computed between control parameters find that correlations are small at large scales when balances are enforced, and at small scales when balances are not enforced (ideal control parameters have zero cross correlations). This points the way to modelling B with scale-dependent balances

Boundary Conditions for the Parametric Kalman Filter Forecast

Abstract This paper is a contribution to the exploration of the parametric Kalman filter (PKF), which is an approximation of the Kalman filter, where the error covariances are approximated by a covariance model. Here we focus on the covariance model parameterized from the variance and the anisotropy of the local correlations, and whose parameters dynamics provides a proxy for the full error‐covariance dynamics. For this covariance model, we aim to provide the boundary condition to specify in the prediction of PKF for bounded domains, focusing on Dirichlet and Neumann conditions when they are prescribed for the physical dynamics. An ensemble validation is proposed for the transport equation and for the heterogeneous diffusion equation over a bounded 1D domain. This ensemble validation requires to specify the auto‐correlation time‐scale needed to populate boundary perturbation that leads to prescribed uncertainty characteristics. The numerical simulations show that the PKF is able to reproduce the uncertainty diagnosed from the ensemble of forecast appropriately perturbed on the boundaries, which show the ability of the PKF to handle boundaries in the prediction of the uncertainties. It results that Dirichlet condition on the physical dynamics implies Dirichlet condition on the variance and on the anisotropy

Assimilation des données sur un modèle d'onde diffusante : émulation d'un algorithme de Filtre de Kalman d'Ensemble

International audienceThis study describes the assimilation of synthetically-generated river water level observations in a flood wave propagation model. For this approach to be applied in the framework of real-time flood forecasting, the cost of the data assimilation procedure, mostly related to the estimation of the background error covariance matrix, should be reduced. An Ensemble Kalman Filter (EnKF) algorithm is applied, with a steady observation network, to demonstrate how the assimilation modifies the background correlation function at the observation point. It is shown that an initially Gaussian correlation function turns into an anisotropic function at the observation point, with a shorter correlation length-scale downstream of the observation point than upstream, and that the variance of the error in the water level state is significantly reduced downstream of the observation point. Away from the observation point, an analytical expression describes the evolution of the error variance and the correlation length scale for the water level signal when the distance to the entrance of the domain increases: when the diffusion is small compared to the advection, the covariance function remains gaussian with an increasing correlation length-scale and a decreasing error variance. At the observation point, the reduction of the error variance and correlation length scale can be parametrized as a linear function of the observation error. This parametrization relies on the integration of the EnKF for a given observing network with given error statistics but can be used to fully describe the covariance function when additional observations are available with different error statistics. The background error covariance matrix is thus fully characterized and can be modeled using a diffusion operator with an inhomogenenous diffusion coefficient that relates to the correlation length scale. The resulting covariance matrix is then used as an invariant background error covariance matrix for a series of successive Best Linear Unbiased Estimation (BLUE) algorithms which emulate an EnKF at a lower computational cost. This study shows how the background error covariance matrix can be computed off-line, with an advanced algorithm, and then used with a cheaper algorithm for real-time application

Adaptive observation with drifting platforms

International audienceThe BAMED (Balloons in the Mediterranean) project aims at developping in-situ drifting observing platforms onboard pressurized balloons to be deployed during HyMeX. HyMeX, known as "Hydrological cycle in the Mediterranean Experiment", is an international, multiscale and multidisciplinary experiment including an observing campaign to start in 2012. The BAMED project is lead by the LMD/IPSL (Laboratoire de Météorologie Dynamique), in collaboration with CNES (Centre National d'Etudes Saptiales) and CNRM (Centre National de Recherche en Météorologie). BAMED is supported by the CNES/TOSCA and INSU/LEFE. Within HyMeX, a special attention is dedicated to the predictability of high impact weather events in the Mediterranean basin. Heavy precipitation and wind storms are typical events to focus on. The deployment of specific observing systems during the campaign to observe phenomena with reduced predictability addresses adaptive observation issue. Consequently, BAMED is three-fold: the project includes balloons technical developments at CNES, trajectory modelling at LMD and adaptive observation at CNRM. The objective is to built an efficient and flexible component of the HyMeX observing system. The CNES develops boundary layer pressurized balloons (BLPBs), which can drift above the sea, collecting data that benefit numerical weather prediction systems. Indeed, the prediction of heavy precipitation events lacks of in-situ measurements in the oceanic boundary layer. However, the balloons will be useful if they drift through some so-called sensitive areas. Moreover the control of the flight of such drifting platforms is very limited at a time and location of the launch of the platform. Because the Mediterranean basin is closed and relatively small compared to atmospheric features, the time spent by BLPBs within the basin is expected to be less than 3 days. The dates and the coastal locations of launching these balloons must be thoroughly selected to allow the balloon to drift into the area of interest and prevent the balloon leaving the basin too quickly. Possible launching sites are evaluated through some trajectography and adaptive observation studies on a selection of typical Mediterranean cases. However, a comprehensive adaptive observation system for the Mediterranean basin shall also monitor the predictability of the upstream flow, at larger scales. The only drifting platform that sample the whole troposphere is the CNES-NCAR driftsonde: a stratospheric balloon-carried gondola drop sondes on demand. Such a platform is thought to be helpful, and to benefit also T-NAWDEX, if deployed above the North Atlantic Ocean. A specific targeting guidance tool for drifting platforms has to be set up. This tool is based on the Kalman Filter Sensitivity (KFS) and coupled with accurate trajectory prediction. The KFS predicts areas where additional observations will most benefit the subsequent forecast, accounting for the assimilation of the routine observations. Monitoring of drifting balloons within an adaptive observation approach is a challenge: new tools, new scales, management of the uncertainties related to the balloons' predicted trajectories and anticipation of the cumulative effect of observations being spread over several assimilation cycles. The adaptive observation aspects of BAMED will be described

Adaptive observation with drifting platforms

International audienceThe BAMED (Balloons in the Mediterranean) project aims at developping in-situ drifting observing platforms onboard pressurized balloons to be deployed during HyMeX. HyMeX, known as "Hydrological cycle in the Mediterranean Experiment", is an international, multiscale and multidisciplinary experiment including an observing campaign to start in 2012. The BAMED project is lead by the LMD/IPSL (Laboratoire de Météorologie Dynamique), in collaboration with CNES (Centre National d'Etudes Saptiales) and CNRM (Centre National de Recherche en Météorologie). BAMED is supported by the CNES/TOSCA and INSU/LEFE. Within HyMeX, a special attention is dedicated to the predictability of high impact weather events in the Mediterranean basin. Heavy precipitation and wind storms are typical events to focus on. The deployment of specific observing systems during the campaign to observe phenomena with reduced predictability addresses adaptive observation issue. Consequently, BAMED is three-fold: the project includes balloons technical developments at CNES, trajectory modelling at LMD and adaptive observation at CNRM. The objective is to built an efficient and flexible component of the HyMeX observing system. The CNES develops boundary layer pressurized balloons (BLPBs), which can drift above the sea, collecting data that benefit numerical weather prediction systems. Indeed, the prediction of heavy precipitation events lacks of in-situ measurements in the oceanic boundary layer. However, the balloons will be useful if they drift through some so-called sensitive areas. Moreover the control of the flight of such drifting platforms is very limited at a time and location of the launch of the platform. Because the Mediterranean basin is closed and relatively small compared to atmospheric features, the time spent by BLPBs within the basin is expected to be less than 3 days. The dates and the coastal locations of launching these balloons must be thoroughly selected to allow the balloon to drift into the area of interest and prevent the balloon leaving the basin too quickly. Possible launching sites are evaluated through some trajectography and adaptive observation studies on a selection of typical Mediterranean cases. However, a comprehensive adaptive observation system for the Mediterranean basin shall also monitor the predictability of the upstream flow, at larger scales. The only drifting platform that sample the whole troposphere is the CNES-NCAR driftsonde: a stratospheric balloon-carried gondola drop sondes on demand. Such a platform is thought to be helpful, and to benefit also T-NAWDEX, if deployed above the North Atlantic Ocean. A specific targeting guidance tool for drifting platforms has to be set up. This tool is based on the Kalman Filter Sensitivity (KFS) and coupled with accurate trajectory prediction. The KFS predicts areas where additional observations will most benefit the subsequent forecast, accounting for the assimilation of the routine observations. Monitoring of drifting balloons within an adaptive observation approach is a challenge: new tools, new scales, management of the uncertainties related to the balloons' predicted trajectories and anticipation of the cumulative effect of observations being spread over several assimilation cycles. The adaptive observation aspects of BAMED will be described