1. Sophie RICCI CALTECH/JPL Post-doc Advisor : Ichiro Fukumori The diabatic errors in the formulation of the data assimilation Kalman Filter/Smoother system at JPL : Extending the control space

2. The diabatic errors in the formulation of data assimilation system Why: Sea level variability dominated by wind-driven baroclinic changes in the tropics, BUT for mid-latitude near-surface steric effects are much more important Need to resolve the dominant response to diabatic errors in the forcing, especially heat flux. Need to spread the correction vertically and correct the vertical mixing coefficient Kz How: Simplified physics: 1D vertical diffusion model for each grid point Reduced space: Temperature along the vertical Control space : errors on heat flux Qnet and vertical mixing coefficient Kz

3. Steps to the doubling algorithm: (Anderson et Moore, 1979) Transition matrix A (simplified linear model) Projection matrix for the errors onto the reduced space Γ Error covariance matrix for the heat flux and vertical mixing coefficient Q Formulation of the Kalman filter under these assumptions Transition matrix A at 214°W, 29°N Identification of the state error covariance matrix P (constant in time)

4. T surface (C) Control Assim Truth Obs What we learn from a single point twin experiment Control Q+δQ Q SST obs. Assimilation Truth Tassim Simulation 1D experiment at 214°W, 29°N Assimilate daily SST observations for 1 year (obs-Tassim)² < (obs-Truth) ² The filter has skills at the surface

5. Negatif impact of the assimilation at sub- surface Kz profils at 214°W, 29°N Dynamic up-date for T Need to apply a filter with statistics coherent with the vertical mixing struture at the instant of the assimilation. Need to derive several filters translating the variability of the vertical mixing coefficient profile. What we learn from a single point twin experiment The assumption made on the error profil for Kz in Q is crucial. It can lead to an irrealist correction below the mixed layer depth

6. Building a set of filters based on the variability of kz profile First try was to build 12 filters based on monthly profiles for kz at a given location (214°W, 29°N) The fit between the instantaneaous and monthly profile is computed with a least square procedure. The skill of the assimilation at sub-surface improves when a criteria on the depth of the mixed layer is added and when additional profiles are taken into account. The bigger the set is, the better the skill gets at depth Matching the depth of the mixed layer is the major key to avoid irrealistic correction The monthly set at 214°W, 29°N is not representative of all latitudes Skill at sub-surface

9. Our new set is a function of mixed-layer depth and the maximum Kz value within the mixed-layer. This set of 411 profiles is computed binning and averaging profiles of Kz over a 8 years period for a simulation run. Building a set of filters based on the variability of kz profile 10 profils per Mixed Layer Depth (MLD) MLD varies from surface to level 41 1 level for extremely low mixing (<5.10-4) Set of profiles for MLD=7 and MLD=20 Depth Kz Depth

10. Fitting for a summer time at 214°W, 56.5°S : Instantaneous kz profil Fit with monthly profils set Fit with a larger set of profils Perspectives: Compute the associated filters to this large set Perform a global assimilation experiment with real SST observations Building a set of filters based on the variability of kz profile

11. Associated set of 411 filters P for the assimilation of SST Estimation of the model error from a control run : Average value is 0.2 °C² Purpose: The Model error matrices of our set must approximate this value. The transition matrice to consider for the doubling represents 30 days of integration of the model Estimation of the observation error from Reynolds product : Average value is 0.16 °C²

12. Diagonal terms of the 411 filters The P matrices are computing with the doubling algorithm described earlier The observation error is assumed constant in time and space (0.16 °C²) The heat flux error is assumed constant in space and time (5.10ˉ 7 ²) The control space is reduced to the heat flux error. The treatment of the vertical mixing coefficient error seems to require additional work. P matrice for MLD=41 Depth °C²°C² Associated set of 411 filters P for the assimilation of SST

13. Assimilation of SST The SST data to assimilated are extracted from Reynolds SST The assimilation occurs every 30 days The data are located on a coarse grid at the surface, covering the entire globe (963 points) The SST and SSS relaxation are turned on The assimilation aims at controlling the variability of the SST but not an eventual bias. The model mean is computed over 8 years of a control run (1993- 2000) The SST mean is computed over the same period using the Reynolds product.

14. Assimilation of SST SST control dynup measup Examples of time series of temperature at 2 different locations where data are assimilated over 1993: dynup: dynamic up-date measup: measurement up-date The assimilation is dragging the control towards the observations.

15. Skill of the assimilation of SST If the assimilation works correctly, the skill must be negative Skill at the surface for 1993 The measurement up-date and the dynamic up-date lead to a negative skill for more the 90 % of the location where data are assimilated for the 1993-1995 period.

16. Independent in-situ data set Skill of the assimilation versus independent in-situ data (mostly XBT): The equivalent of the control and assimilation are computed at the locations in space and time of the in-situ data. These in-situ data are NOT assimilated and are used as an independent set of data to validate the impact of the assimilation of SST at the surface AND at the sub-surface. If the assimilation works correctly, the skill must be negative

17. Independent in-situ data set : validation at the surface T anomalies for the North Atlantic in 1995 XBT Assimilation Control Skill versus XBT for 1993: The assimilation is closer to the observation then the control. The skill is almost every where negative (improvement).

18. Independent in-situ data set : validation at the sub-surface Skill versus XBT for 1993 at level 7 (75 m): degradation T anomalies for the North Atlantic 1995 : XBT Assimilation Control The assimilation is further to the observation then the control. The skill shows some degradation areas.

19. Why is there a degradation at the sub-surface T profiles of anomalies for the North Atlantic 1995 at level 7 : XBT Assimilation Control In this region, even though the skill is good at the surface, there is a degradation at sub-surface. T anomalie Depth level 7

20. The anomalies of the control at level 7 are always positive (same for 1993 and 1994), then change sign for 1996-1999. There seems to be a low frequency signal in this region that the assimilation is not enable to correct. T Depth XBT Assimilation Control If the control space is reduced to an error on heat forcing, the T increment from the assimilation will be one sign through out the water column. The assimilation won’t be able to improve the control in such situation. Why is there a degradation at the sub-surface Schematic behavior of the model for this region:

21. Conclusion and perspectives The assimilation scheme works fine and gives satisfying results at the sub-surface. At sub-surface, in some regions, the model presents a low frequency variability that the assimilation is enable to correct under the currents assumptions. Solutions: Increase the observation error in these regions so that the assimilation is inactive Use a reference for the model that includes this low-frequency signal, such as a 1993-1995 mean. Once the Filter gives satisfying results at every depth, the ultimate step of this study is to run the Smoother to get an new estimate of the heat forcing. This new estimate should lead to a better representation of the ocean.