## Final Report Summary - INCLIDA (Initialization of global decadal climate forecast: a new challenge for multi-scale data assimilation)

The project has contributed to the development of advanced data assimilation techniques with the ultimate goal of their use in the context of seasonal-to-decadal predictions. Attention has been given in the one hand to the study of fundamental problems such as the treatment of model error and bias correction, and the control of chaotic error growth. On the other hand, efforts have been devoted to the study of the initialization methods use for long-term environmental predictions.

The project has been structured along two tasks. The objectives, the work performed and the main results achieved are given in the following.

TASK 1 - Data Assimilation & Initialization: Theory and Developments

- A novel Data Assimilation (DA) method for simultaneous state and parameter estimation has been introduced. The algorithm, referred to as Short Time Extended Kalman Filter (STAEKF), belongs to the class of sequential filters, and has been successfully applied to a low-order soil model for the assimilation of screen-level observations, i.e. the 2-meter atmospheric temperature and humidity. Two important parameters for soil modeling, the Leaf Area Index and the Albedo, were correctly retrieved along with an accurate reconstruction of the soil temperature and humidity (see Carrassi et al., 2012).

- The theory of the deterministic model error treatment in DA has been further formalized and generalized. This has lead to two studies in which the different aspects and applications of this approach are described in details (Carrassi, 2012; Carrassi and Vannitsem, 2014).

- Based on a limited number of noisy observations, estimation algorithms provide a complete description of the state of a system at current time. Estimation algorithms that go under the name of assimilation in the unstable subspace (AUS) exploit the nonlinear stability properties of the forecasting model in their formulation. Errors that grow due to sensitivity to initial conditions are efficiently removed by confining the analysis solution in the unstable and neutral subspace of the system, the subspace spanned by Lyapunov vectors with positive and zero exponents, while the observational noise does not disturb the system along the stable directions. The formulation of the AUS approach in the context of four-dimensional variational assimilation (4DVar-AUS) and the extended Kalman filter (EKF-AUS) and its application to chaotic models has been studied. In both instances, the AUS algorithms are at least as efficient but simpler to implement and computationally less demanding than their original counterparts. As predicted by the theory when error dynamics is linear, the optimal subspace dimension for 4DVar-AUS is given by the number of positive and null Lyapunov exponents, while the EKF-AUS algorithm, using the same unstable and neutral subspace, recovers the solution of the full EKF algorithm, but dealing with error covariance matrices of a much smaller dimension and significantly reducing the computational burden. The applications to simplified models of the atmospheric circulation and to the optimal velocity model for traffic dynamics demonstrate the efficiency of the AUS approach. Results are given in Palatella et al. (2013).

- Two main strategies exist nowadays for the initialization of climate predictions, namely the full-field and the anomaly initialization. Despite their increased use in climate services, a number of issues are still unclear on which methods is the more adequate and how their compare in different forecasting applications. We use the notation and concepts of data assimilation theory to propose a unified formalism from which full and anomaly initialization can be easily derived and seen as a particular case. Anomaly and full initialization are compared using an idealized coupled model under different scenarios of observational as well as parametric model error. The low order climate model (Pena and Kalnay, 2004) comprises nine ODEs coupled in order to simulate ocean and the tropical/extratropical atmospheres. We also introduce and assess the performance of two advanced schemes. Least-Square-Initialization (LSI) propagates information from observed to unobserved model compartments based on an approximation of the background error covariance based on the statistics of model anomalies. Exploring the Parameter Uncertainty (EPU) is an online short-time drift correction approach in which the drift caused by parametric error is estimated using a short-time evolution law and is then removed during the forecast run.

Results suggest better performances using FFI when a good observational network is available, and reveals a direct relationship between its skill and the observational accuracy. The skill of AI appears mostly related to the model quality and clear increases of skill can only be expected in coincidence with model upgrades. When comparing initialization of single compartments, the best performance is obtained when the slowest component (ocean) is initialized. Further results identify a range of parametric error in which anomaly initialization performs more favourable compared to full initialization. These results suggest that anomaly initialization performs better under circumstances in which the model bias is not accompanied by higher order differences between the real climate and the model PDF. The implementation of LSI improves FFI in all situations in which only a portion of the system’s state is observed. These results agree with those of Smith and Murphy (2007) and extend them to the case of cross-covariance between different model compartments, i.e. the atmosphere and the ocean. The use of EPU has clearly improved the skill of full initialization within the first forecast year, with minor improvements for longer time horizons. Results have demonstrated robustness of EPU with respect to the length of the correction time interval and the accuracy of the identified parameter uncertainty ranges. Results of this study can be found in Carrassi et al. (2014); we are currently studying new formulation of the AI incorporating higher moments of the model/observed PDFs.

TASK 2 - Data Assimilation: Initialization of realistic coupled systems for seasonal-to-decadal prediction

A nudging DA scheme in the Earth System Simulator (ESS), EC-Earth, has been implemented. EC-Earth is the state-of-the art ESS in use at the IC3, and has a global atmosphere, ocean, soil and sea-ice components (http://ecearth.knmi.nl). We use here the version 2.3 of EC-Earth (Hazeleger et al., 2012) coupled atmosphere-ocean general circulation model that comprises:

- the IFS atmospheric component with 62 vertical levels and a TL159 horizontal resolution (http://www.ecmwf.int/research/ifsdocs/CY36r1/index.html)

- the NEMO (Nucleus for European Modelling of the Ocean) ocean component in the ORCA1 configuration with 42 levels (Madec, 2008; Ethe et al., 2006)

- the Louvain-la-Neuve (LIM2) sea-ice model version 2 embedded into NEMO (Fichefet and Maqueda, 1997; Goosse and Fichefet, 1999). The atmospheric and ocean components are coupled through OASIS3 (Valcke, 2006).

The initial conditions of our simulations on 1 January 1960 are provided by the 5-member NEMOVAR-ORAS4 ocean reanalysis (Mogensen et al., 2012). The sea ice initial conditions come from an ocean-sea ice model forced with Drakkar Forcing Set V4.3 surface fluxes (Brodeau et al., 2009). The atmosphere and land surface initial conditions were taken from the ERA-40 reanalysis (Uppala et al., 2005) with different members provided by singular vectors.

From 1 January 1960, we perform 4 simulations lasting until 31 December 2012 and comprising 5 members distinct by their initial conditions:

1. A free experiment (without any ocean nudging) - Unforced Run

2. An experiment with nudging of the ocean temperature and salinity below the ocean mixed layer down to the ocean floor toward the monthly temperature and salinity from NEMOVAR- ORAS4 with restoring timescales of 10 days above 800m and 2 years below. - Global Ocean Nudging Run.

3. An experiment with nudging of the ocean temperature and salinity only above 2000m which corresponds to the depth down to which the ocean is observed through profiles. - Upper Ocean Nudging Run.

4. An experiment with nudging of the ocean temperature and salinity outside the tropical (30S-30N) and polar regions (90S-60S, 65N-90N). - Mid-Latitudes Ocean Nudging Run.

Results show the sensitivity of the climate system to different geographical areas and suggest which region has to be constraint to improve skill at different time horizon and/or different variables or climate indexes. Two papers are currently in preparations that contain results of this research.

The project has been structured along two tasks. The objectives, the work performed and the main results achieved are given in the following.

TASK 1 - Data Assimilation & Initialization: Theory and Developments

- A novel Data Assimilation (DA) method for simultaneous state and parameter estimation has been introduced. The algorithm, referred to as Short Time Extended Kalman Filter (STAEKF), belongs to the class of sequential filters, and has been successfully applied to a low-order soil model for the assimilation of screen-level observations, i.e. the 2-meter atmospheric temperature and humidity. Two important parameters for soil modeling, the Leaf Area Index and the Albedo, were correctly retrieved along with an accurate reconstruction of the soil temperature and humidity (see Carrassi et al., 2012).

- The theory of the deterministic model error treatment in DA has been further formalized and generalized. This has lead to two studies in which the different aspects and applications of this approach are described in details (Carrassi, 2012; Carrassi and Vannitsem, 2014).

- Based on a limited number of noisy observations, estimation algorithms provide a complete description of the state of a system at current time. Estimation algorithms that go under the name of assimilation in the unstable subspace (AUS) exploit the nonlinear stability properties of the forecasting model in their formulation. Errors that grow due to sensitivity to initial conditions are efficiently removed by confining the analysis solution in the unstable and neutral subspace of the system, the subspace spanned by Lyapunov vectors with positive and zero exponents, while the observational noise does not disturb the system along the stable directions. The formulation of the AUS approach in the context of four-dimensional variational assimilation (4DVar-AUS) and the extended Kalman filter (EKF-AUS) and its application to chaotic models has been studied. In both instances, the AUS algorithms are at least as efficient but simpler to implement and computationally less demanding than their original counterparts. As predicted by the theory when error dynamics is linear, the optimal subspace dimension for 4DVar-AUS is given by the number of positive and null Lyapunov exponents, while the EKF-AUS algorithm, using the same unstable and neutral subspace, recovers the solution of the full EKF algorithm, but dealing with error covariance matrices of a much smaller dimension and significantly reducing the computational burden. The applications to simplified models of the atmospheric circulation and to the optimal velocity model for traffic dynamics demonstrate the efficiency of the AUS approach. Results are given in Palatella et al. (2013).

- Two main strategies exist nowadays for the initialization of climate predictions, namely the full-field and the anomaly initialization. Despite their increased use in climate services, a number of issues are still unclear on which methods is the more adequate and how their compare in different forecasting applications. We use the notation and concepts of data assimilation theory to propose a unified formalism from which full and anomaly initialization can be easily derived and seen as a particular case. Anomaly and full initialization are compared using an idealized coupled model under different scenarios of observational as well as parametric model error. The low order climate model (Pena and Kalnay, 2004) comprises nine ODEs coupled in order to simulate ocean and the tropical/extratropical atmospheres. We also introduce and assess the performance of two advanced schemes. Least-Square-Initialization (LSI) propagates information from observed to unobserved model compartments based on an approximation of the background error covariance based on the statistics of model anomalies. Exploring the Parameter Uncertainty (EPU) is an online short-time drift correction approach in which the drift caused by parametric error is estimated using a short-time evolution law and is then removed during the forecast run.

Results suggest better performances using FFI when a good observational network is available, and reveals a direct relationship between its skill and the observational accuracy. The skill of AI appears mostly related to the model quality and clear increases of skill can only be expected in coincidence with model upgrades. When comparing initialization of single compartments, the best performance is obtained when the slowest component (ocean) is initialized. Further results identify a range of parametric error in which anomaly initialization performs more favourable compared to full initialization. These results suggest that anomaly initialization performs better under circumstances in which the model bias is not accompanied by higher order differences between the real climate and the model PDF. The implementation of LSI improves FFI in all situations in which only a portion of the system’s state is observed. These results agree with those of Smith and Murphy (2007) and extend them to the case of cross-covariance between different model compartments, i.e. the atmosphere and the ocean. The use of EPU has clearly improved the skill of full initialization within the first forecast year, with minor improvements for longer time horizons. Results have demonstrated robustness of EPU with respect to the length of the correction time interval and the accuracy of the identified parameter uncertainty ranges. Results of this study can be found in Carrassi et al. (2014); we are currently studying new formulation of the AI incorporating higher moments of the model/observed PDFs.

TASK 2 - Data Assimilation: Initialization of realistic coupled systems for seasonal-to-decadal prediction

A nudging DA scheme in the Earth System Simulator (ESS), EC-Earth, has been implemented. EC-Earth is the state-of-the art ESS in use at the IC3, and has a global atmosphere, ocean, soil and sea-ice components (http://ecearth.knmi.nl). We use here the version 2.3 of EC-Earth (Hazeleger et al., 2012) coupled atmosphere-ocean general circulation model that comprises:

- the IFS atmospheric component with 62 vertical levels and a TL159 horizontal resolution (http://www.ecmwf.int/research/ifsdocs/CY36r1/index.html)

- the NEMO (Nucleus for European Modelling of the Ocean) ocean component in the ORCA1 configuration with 42 levels (Madec, 2008; Ethe et al., 2006)

- the Louvain-la-Neuve (LIM2) sea-ice model version 2 embedded into NEMO (Fichefet and Maqueda, 1997; Goosse and Fichefet, 1999). The atmospheric and ocean components are coupled through OASIS3 (Valcke, 2006).

The initial conditions of our simulations on 1 January 1960 are provided by the 5-member NEMOVAR-ORAS4 ocean reanalysis (Mogensen et al., 2012). The sea ice initial conditions come from an ocean-sea ice model forced with Drakkar Forcing Set V4.3 surface fluxes (Brodeau et al., 2009). The atmosphere and land surface initial conditions were taken from the ERA-40 reanalysis (Uppala et al., 2005) with different members provided by singular vectors.

From 1 January 1960, we perform 4 simulations lasting until 31 December 2012 and comprising 5 members distinct by their initial conditions:

1. A free experiment (without any ocean nudging) - Unforced Run

2. An experiment with nudging of the ocean temperature and salinity below the ocean mixed layer down to the ocean floor toward the monthly temperature and salinity from NEMOVAR- ORAS4 with restoring timescales of 10 days above 800m and 2 years below. - Global Ocean Nudging Run.

3. An experiment with nudging of the ocean temperature and salinity only above 2000m which corresponds to the depth down to which the ocean is observed through profiles. - Upper Ocean Nudging Run.

4. An experiment with nudging of the ocean temperature and salinity outside the tropical (30S-30N) and polar regions (90S-60S, 65N-90N). - Mid-Latitudes Ocean Nudging Run.

Results show the sensitivity of the climate system to different geographical areas and suggest which region has to be constraint to improve skill at different time horizon and/or different variables or climate indexes. Two papers are currently in preparations that contain results of this research.