Abstract:
Two Ensemble Streamflow Prediction Systems (ESPSs) have been set up at Météo-France. They are based on the French SIM distributed hydrometeorological model. A deterministic analysis run of SIM is used to initialize the two ESPSs. In order to obtain a better initial state, a past discharges assimilation system has been implemented into this analysis SIM run, using the Best Linear Unbiased Estimator (BLUE). Its role is to improve the model soil moisture by using observed streamflows in order to better simulate streamflow. The skills of the assimilation system were assessed for a 569-day period on six different configurations, including two different physics schemes of the model (the use of an exponential profile of hydraulic conductivity or not) and, for each one, three different ways of considering the model soil moisture in the BLUE state variables. Respect of the linearity hypothesis of the BLUE was verified by assessing of the impact of iterations of the BLUE. The configuration including the use of the exponential profile of hydraulic conductivity and the combination of the moisture of the two soil layers in the state variable showed a significant improvement of streamflow simulations. It led to a significantly better simulation than the reference one, and the lowest soil moisture corrections. These results were confirmed by the study of the impacts of the past discharge assimilation system on a set of 49 independent stations.

Abstract:
The use of ensemble streamflow forecasts is developing in the international flood forecasting services. Such systems can provide more accurate forecasts and useful information about the uncertainty of the forecasts, thus improving the assessment of risks. Nevertheless, these systems, like all hydrological forecasts, suffer from errors on initialization or on meteorological data, which lead to hydrological prediction errors. This article, which is the second part of a 2-part article, concerns the impacts of initial states, improved by a streamflow assimilation system, on an ensemble streamflow prediction system over France. An assimilation system was implemented to improve the streamflow analysis of the SAFRAN-ISBA-MODCOU (SIM) hydro-meteorological suite, which initializes the ensemble streamflow forecasts at Météo-France. This assimilation system, using the Best Linear Unbiased Estimator (BLUE) and modifying the initial soil moisture states, showed an improvement of the streamflow analysis with low soil moisture increments. The final states of this suite were used to initialize the ensemble streamflow forecasts of Météo-France, which are based on the SIM model and use the European Centre for Medium-range Weather Forecasts (ECMWF) 10-day Ensemble Prediction System (EPS). Two different configurations of the assimilation system were used in this study: the first with the classical SIM model and the second using improved soil physics in ISBA. The effects of the assimilation system on the ensemble streamflow forecasts were assessed for these two configurations, and a comparison was made with the original (i.e. without data assimilation and without the improved physics) ensemble streamflow forecasts. It is shown that the assimilation system improved most of the statistical scores usually computed for the validation of ensemble predictions (RMSE, Brier Skill Score and its decomposition, Ranked Probability Skill Score, False Alarm Rate, etc.), especially for the first few days of the time range. The assimilation was slightly more efficient for small basins than for large ones.

Abstract:
In recent papers, a few physicists studying Black Hole perturbation theory in General Relativity (GR) have tried to construct the initial part of a differential sequence based on the Kerr metric, using methods similar to the ones they already used for studying the Schwarzschild geometry. Of course, such a differential sequence is well known for the Minkowski metric and successively contains the Killing (order 1), the Riemann (order 2) and the Bianchi (order 1 again) operators in the linearized framework, as a particular case of the Vessiotstructure equations. In all these cases, they discovered that the compatibility conditions (CC) for the corresponding Killing operator were involving a mixture of both second order and third order CC and their idea has been to exhibit only a minimal number of generating ones. Unhappily, these physicists are neither familiar with the formal theory of systems of partial differential equations and differential modules, nor with the formal theory of Lie pseudogroups. Hence, even if they discovered a link between these differential sequences and the number of parameters of the Lie group preserving the background metric, they have been unable to provide an intrinsic explanation of this fact, being limited by the technical use of Weyl spinors, complex Teukolsky scalars or Killing-Yano tensors. The purpose of this difficult computational paper is to provide differential and homological methods in order to revisit and solve these questions, not only in the previous cases but also in the specific case of any Lie group or Lie pseudogroup of transformations. These new tools, which are now available as computer algebra packages, question the mathematical foundations of GR and the origin of gravitational waves.

Abstract:
The search for the generating compatibility conditions (CC) of a given operator is a very recent problem met in general relativity in order to study the Killing operator for various standard useful metrics. Accordingly, this paper can be considered as a natural continuation of a previous paper recently published in JMP under the title Minkowski, Schwarschild and Kerr metrics revisited. In particular, we prove that the intrinsic link existing between the lack of formal exactness of an operator sequence on the jet level, the lack of formal exactness of its corresponding symbol sequence and the lack of formal integrability (FI) of the initial operator is of a purely homological nature as it is based on the long exact connecting sequence provided by the so-called snake lemma in homological algebra. It is therefore quite difficult to grasp it in general and even more difficult to use it on explicit examples. It does not seem that any one of the results presented in this paper is known as most of the other authors who studied the above problem of computing the total number of generating CC are confusing this number with the degree of generality introduced by A. Einstein in his 1930 letters to E. Cartan. One of the motivating examples that we provide is so striking that it is even difficult to imagine that such an example could exist. We hope this paper could be used as a source of testing examples for future applications of computer algebra in general relativity and, more generally, in mathematical physics.

Abstract:
The use of ensemble streamflow forecasts is developing in the international flood forecasting services. Ensemble streamflow forecast systems can provide more accurate forecasts and useful information about the uncertainty of the forecasts, thus improving the assessment of risks. Nevertheless, these systems, like all hydrological forecasts, suffer from errors on initialization or on meteorological data, which lead to hydrological prediction errors. This article, which is the second part of a 2-part article, concerns the impacts of initial states, improved by a streamflow assimilation system, on an ensemble streamflow prediction system over France. An assimilation system was implemented to improve the streamflow analysis of the SAFRAN-ISBA-MODCOU (SIM) hydro-meteorological suite, which initializes the ensemble streamflow forecasts at Météo-France. This assimilation system, using the Best Linear Unbiased Estimator (BLUE) and modifying the initial soil moisture states, showed an improvement of the streamflow analysis with low soil moisture increments. The final states of this suite were used to initialize the ensemble streamflow forecasts of Météo-France, which are based on the SIM model and use the European Centre for Medium-range Weather Forecasts (ECMWF) 10-day Ensemble Prediction System (EPS). Two different configurations of the assimilation system were used in this study: the first with the classical SIM model and the second using improved soil physics in ISBA. The effects of the assimilation system on the ensemble streamflow forecasts were assessed for these two configurations, and a comparison was made with the original (i.e. without data assimilation and without the improved physics) ensemble streamflow forecasts. It is shown that the assimilation system improved most of the statistical scores usually computed for the validation of ensemble predictions (RMSE, Brier Skill Score and its decomposition, Ranked Probability Skill Score, False Alarm Rate, etc.), especially for the first few days of the time range. The assimilation was slightly more efficient for small basins than for large ones.

Abstract:
Two Ensemble Streamflow Prediction Systems (ESPSs) have been set up at Météo-France. They are based on the French SIM distributed hydrometeorological model. A deterministic analysis run of SIM is used to initialize the two ESPSs. In order to obtain a better initial state, a past discharges assimilation system has been implemented into this analysis SIM run, using the Best Linear Unbiased Estimator (BLUE). Its role is to improve the model soil moisture by using streamflow observations in order to better simulate streamflow. The skills of the assimilation system were assessed for a 569-day period on six different configurations, including two different physics schemes of the model (the use of an exponential profile of hydraulic conductivity or not) and, for each one, three different ways of considering the model soil moisture in the BLUE state variables. Respect of the linearity hypothesis of the BLUE was verified by assessing of the impact of iterations of the BLUE. The configuration including the use of the exponential profile of hydraulic conductivity and the combination of the moisture of the two soil layers in the state variable showed a significant improvement of streamflow simulations. It led to a significantly better simulation than the reference one, and the lowest soil moisture corrections. These results were confirmed by the study of the impacts of the past discharge assimilation system on a set of 49 independent stations.

Abstract:
The performance of the joint assimilation in a land surface model of a Soil Wetness Index (SWI) product provided by an exponential filter together with Leaf Area Index (LAI) is investigated. The data assimilation is evaluated with different setups using the SURFEX modeling platform, for a period of seven years (2001–2007), at the SMOSREX grassland site in southwestern France. The results obtained with a Simplified Extended Kalman Filter demonstrate the effectiveness of a joint data assimilation scheme when both SWI and Leaf Area Index are merged into the ISBA-A-gs land surface model. The assimilation of a retrieved Soil Wetness Index product presents several challenges that are investigated in this study. A significant improvement of around 13 % of the root-zone soil water content is obtained by assimilating dimensionless root-zone SWI data. For comparison, the assimilation of in situ surface soil moisture is considered as well. A lower impact on the root zone is noticed. Under specific conditions, the transfer of the information from the surface to the root zone was found not accurate. Also, our results indicate that the assimilation of in situ LAI data may correct a number of deficiencies in the model, such as low LAI values in the senescence phase by using a seasonal-dependent error definition for background and observations. In order to verify the specification of the errors for SWI and LAI products, a posteriori diagnostics are employed. This approach highlights the importance of the assimilation design on the quality of the analysis. The impact of data assimilation scheme on CO2 fluxes is also quantified by using measurements of net CO2 fluxes gathered at the SMOSREX site from 2005 to 2007. An improvement of about 5 % in terms of rms error is obtained.

Abstract:
This study examines whether the assimilation of remotely sensed near-surface soil moisture observations might benefit an operational hydrological model, specifically Météo-France's SAFRAN-ISBA-MODCOU (SIM) model. Soil moisture data derived from ASCAT backscatter observations are assimilated into SIM using a Simplified Extended Kalman Filter (SEKF) over 3.5 years. The benefit of the assimilation is tested by comparison to a delayed cut-off version of SIM, in which the land surface is forced with more accurate atmospheric analyses, due to the availability of additional atmospheric observations after the near-real time data cut-off. However, comparing the near-real time and delayed cut-off SIM models revealed that the main difference between them is a dry bias in the near-real time precipitation forcing, which resulted in a dry bias in the root-zone soil moisture and associated surface moisture flux forecasts. While assimilating the ASCAT data did reduce the root-zone soil moisture dry bias (by nearly 50%), this was more likely due to a bias within the SEKF, than due to the assimilation having accurately responded to the precipitation errors. Several improvements to the assimilation are identified to address this, and a bias-aware strategy is suggested for explicitly correcting the model bias. However, in this experiment the moisture added by the SEKF was quickly lost from the model surface due to the enhanced surface fluxes (particularly drainage) induced by the wetter soil moisture states. Consequently, by the end of each winter, during which frozen conditions prevent the ASCAT data from being assimilated, the model land surface had returned to its original (dry-biased) climate. This highlights that it would be more effective to address the precipitation bias directly, than to correct it by constraining the model soil moisture through data assimilation.

Abstract:
The performance of the joint assimilation in a land surface model of a Soil Wetness Index (SWI) product provided by an exponential filter together with Leaf Area Index (LAI) is investigated. The data assimilation is evaluated with different setups using the SURFEX modeling platform, for a period of seven years (2001–2007), at the SMOSREX grassland site in southwestern France. The results obtained with a Simplified Extended Kalman Filter demonstrate the effectiveness of a joint data assimilation scheme when both SWI and Leaf Area Index are merged into the ISBA-A-gs land surface model. The assimilation of a retrieved Soil Wetness Index product presents several challenges that are investigated in this study. A significant improvement of around 13% of the root-zone soil water content is obtained by assimilating dimensionless root-zone SWI data. For comparison, the assimilation of in situ surface soil moisture is considered as well. A lower impact on the root zone is noticed. Under specific conditions, the transfer of the information from the surface to the root zone was found not accurate. Also, our results indicate that the assimilation of in situ LAI data may correct a number of deficiencies in the model, such as low LAI values in the senescence phase by using a seasonal-dependent error definition for background and observations. In order to verify the specification of the errors for SWI and LAI products, a posteriori diagnostics are employed. This approach highlights the importance of the assimilation design on the quality of the analysis. The impact of data assimilation scheme on CO2 fluxes is also quantified by using measurements of net CO2 fluxes gathered at the SMOSREX site from 2005 to 2007. An improvement of about 5% in terms of rms error is obtained.

Abstract:
The impact of assimilating near-surface soil moisture into the SAFRAN-ISBA-MODCOU (SIM) hydrological model over France is examined. Specifically, the root-zone soil moisture in the ISBA land surface model is constrained over three and a half years, by assimilating the ASCAT-derived surface degree of saturation product, using a Simplified Extended Kalman Filter. In this experiment ISBA is forced with the near-real time SAFRAN analysis, which analyses the variables required to force ISBA from relevant observations available before the real time data cut-off. The assimilation results are tested against ISBA forecasts generated with a higher quality delayed cut-off SAFRAN analysis. Ideally, assimilating the ASCAT data will constrain the ISBA surface state to correct for errors in the near-real time SAFRAN forcing, the most significant of which was a substantial dry bias caused by a dry precipitation bias. The assimilation successfully reduced the mean root-zone soil moisture bias, relative to the delayed cut-off forecasts, by close to 50 % of the open-loop value. The improved soil moisture in the model then led to significant improvements in the forecast hydrological cycle, reducing the drainage, runoff, and evapotranspiration biases (by 17 %, 11 %, and 70 %, respectively). When coupled to the MODCOU hydrogeological model, the ASCAT assimilation also led to improved streamflow forecasts, increasing the mean discharge ratio, relative to the delayed cut off forecasts, from 0.68 to 0.76. These results demonstrate that assimilating near-surface soil moisture observations can effectively constrain the SIM model hydrology, while also confirming the accuracy of the ASCAT surface degree of saturation product. This latter point highlights how assimilation experiments can contribute towards the difficult issue of validating remotely sensed land surface observations over large spatial scales.