ensemble data assimilation

After this step, all involved processes of the program are active (for the parallelization aspects see Sect. 3.3). MPI allows one to compute a program using several processes with distributed memory. The resolution of the observations is 0.1∘ and hence higher than the resolution of the model in most regions. Data Assimilation comprehensively covers data assimilation and inverse methods, including both traditional state estimation and parameter estimation. If the DA would be performed in a separate program coupled to AWI-CM through files, Preliminary programs, registration, hotel, and general information will be posted on the AMS Web site (www.ametsoc.org). For offline-coupled DA, one could use the same variable names and the same names for the modules. For high-dimensional models, a localized analysis is computed following Nerger et al. Meteor. The call-back routines are called by PDAF to perform operations that are specific to the model or the observations. Res., 113, C05015, https://doi.org/10.1029/2007JC004482, 2008. a, Yu, L., Fennel, K., Bertino, L., Gharamti, M. E., and Thompson, K. R.: Insights Here, only the online coupling for DA is discussed. Finally, Sect. 3.4 explains the aspect of the call-back functions. To this end, separating the processes for the analysis step would mainly be a choice if the available memory on the first model task is not sufficient to execute the analysis step. For the ensemble DA, the parallelization of AWI-CM is modified. For coupled ocean–biogeochemical models, Yu et al. R., Savcenko, R., Bosch, W., Skachko, S., and Danilov, S.: On the Only if this is true will the analysis step be executed; otherwise, the time stepping is continued. Pycnocline Ocean Model, J. Thus, the observation operator has to be implemented by taking into account the specific character of the model grid such as the unstructured structure of FESOM's grid. 2019. , Rackow, T., Sein, D. V., Semmler, T., Danilov, S., Koldunov, N. V., Sidorenko, D., Wang, Q., and Jung, T.: Sensitivit, Sakov, P. and Oke, P. R.: A deterministic formulation of hte ensemlbe Kalman To assess the parallel performance of the assimilation system described above, AWI-CM is run here in the same global configuration as described by Sidorenko et al. Assimilation of Volcanic Ash Clouds from Satellite Observations: Application filter: an alternative to ensemble square root filters, Tellus, 60A, Weather Rev., 126, 1719–1724, 1998. a, Chang, Y.-S., Zhang, S., Rosati, A., Delworth, T. L., and Stern, W. F.: An The setup builds on the strategy introduced by Nerger and Hiller (2013). If only one compartment is observed, e.g., having only ocean observations yO, then we also have HXCf=(HXf)O, and the weights are only computed from these observations. To validate the assimilation with independent observations, temperature and salinity profiles from the EN4 data set (EN4.2.1) of the UK Met Office (Good et al., 2013) are used. The distribution of the processes is exemplified in Fig. 3a for the case of six processes in MPI_COMM_WORLD. Compared to the default setup in PDAF for a single-compartment model, we have adapted the routine to account for the existence of two model compartments. The atmosphere uses a horizontal spectral resolution (T63, about 180 km) with 47 layers. problems, Tellus A, 70, 1445364. Dynam., 44, 757–780, 2015. , Sluka, T. C., Penny, S. G., Kalnay, E., and Miyoshi, T.: Assimilating In the coupled model, this routine is executed before the parallelization of the coupler is initialized. World Scientific, 63–83, 2005. a, b, c, Nerger, L., Danilov, S., Hiller, W., and Schröter, J.: Using sea level data http://www.openmp.org/ (last access: 14 September 2020), 2008. a, Pardini, F., Corradini, S., Costa, A., Ongari, T. E., Merucci, L., Neri, A., While this discussion shows that technically it is straightforward to apply strongly coupled DA with these filter methods, one has to account for the model parallelization, which is discussed in Section 3.3. Nerger, L., Tang, Q., and Mu, L.: Efficient ensemble data assimilation for For FESOM, as a single-compartment model, the adaption of the parallelization was described by Nerger et al. It combines the information from the model state and the observations by taking into account the estimated error of the two information sources and computes an updated model state ensemble, which represents the analysis state estimate and its uncertainty. Improving the ocean and atmosphere in a coupled ocean-atmosphere model by In particular, for Ne=40 the execution time is almost identical to that of Ne=2. Implications of the chosen strategy for the coupled model and data assimilation are discussed in Sect. 6. Softw., 68, 122–128, 2015. a, b, Browne, P. A., de Rosnay, P., Zuo, H., Bennett, A., and Dawson, A.: Weakly While the RMSE of the salinity first increases during the first month, it is reduced from day 60, but until day 140 it is sometimes larger than at the initial time. Meteor. In the case of AWI-CM when the two executables for ECHAM and FESOM are jointly started, they share the same MPI_COMM_WORLD so that parallel communication between the processes running ECHAM and those running FESOM is possible. Model. A state vector is the collection of all model fields at all model grid points in the form of a vector. The output files containing the timing information, the outputs from the 1-year experiments, and plotting scripts are available at Zenodo (https://doi.org/10.5281/zenodo.3823816, Nerger et al., 2019b). With this configuration, the assimilation can be performed independently for both compartments. The ensemble-based data assimilation techniques have shown great promise to initialize TC forecast. The weights are computed using Eqs. (4) to (6). If there are only observations in one of the compartments, one can also compute the weights in that compartment and provide them to the other compartment. studies, Mon. The maximum ensemble size was here limited by the batch job size of the used computer. The setup of a DA system is described in Sect. 3. (2016). in a simple coupled climate model: The role of ocean-atmopshere interaction, where x‾kf is the forecast ensemble mean state and 1Ne is a vector of size Ne holding the value one in all elements. Pycnocline Ocean Model, J. J. Roy. The assimilation in DA-SST strongly reduces the RMSE during the first 2 months. For the coupled model, there are different routines for FESOM and ECHAM. We must assign probabilities not only to the parameters of the models but also to the models themselves. (2016) have combined PDAF with the coupled terrestrial model system TerrSysMP. OASIS3-MCT computes the fluxes between the ocean and the atmosphere and performs the interpolation between both model grids. Soc., 136, 1991–1999, 2010. , van Leeuwen, P. J., Künsch, H. R., Nerger, L., Potthast, R., and Reich, S.: Here, one typically computes the ensemble mean and writes it into a file. Data Assimilation comprehensively covers data assimilation and inverse methods, including both traditional state estimation and parameter estimation. To use this updated scheme, one has to execute the coupled model with enough processors so that all ensemble members can be run at the same time. Climate, 26, 10218–10231, 2013. , Harlim, J. and Hunt, B. R.: Four-dimensional local ensemble transform Kalmn For the second issue regarding disk operations, one has to take into account that the direct outputs written by each coupled ensemble task are usually not relevant, because the assimilation focuses on the ensemble mean state. For a single grid point, this would be the number of variables stored at this grid point. Weather Rev., 143, 1347–1467, 2015. L=XfT, The modification of the model parallelization for ensemble DA is a core element of the DA online coupling. Weather Rev., 136, 4629–4640, 2008. , Stevens, B., Giorgetta, M., Esch, M., Mauritsen, T., Crueger, T., Rast, S., The DA is started on 1 January 2016, and satellite observations of the sea surface temperature obtained from the European Copernicus initiative (data set SST_GLO_SST_L3S_NRT_OBSERVATIONS_010_010 available at https://marine.copernicus.eu, last access: 14 September 2020), interpolated to the model grid, are assimilated daily. Since the model grid is unstructured with varying resolution, super-observations are generated by averaging onto the model grid. While there are typical observational data sets for the different Earth system compartments, the observation operator links the observations with the model fields on the model grid. The code modifications for online coupling are described in Sect. 3.2, and the modifications of the parallelization are described in Sect. 3.3. Here, the parallelization of AWI-CM and the required changes for the extension for the DA are described. localization scheme for ensemble-based Kalman filters, Q. J. Roy. Soc., https://doi.org/10.1002/qj.3885, in press, 2020. a, b, c, Tödter, J. and Ahrens, B.: A second-order exact ensemble square root filter World Scientific, 63–83, 2005. , Nerger, L., Danilov, S., Hiller, W., and Schröter, J.: Using sea level data The possibility to implement most parts of a filter algorithm in a generic model-agnostic way has motivated the implementation of software frameworks for ensemble DA. Further, the abstraction in the analysis step, which uses only state and observation vectors without accounting for the physical fields, allows one to separate the development of advanced DA algorithms from the development of the model. In this way the coupler will also be initialized for an ensemble of model states. 529–538, 2018. . coupled models with the Parallel Data Assimilation Framework: Example of The model is based on the three‐dimensional Richards equation for variably saturated porous media and a diffusion wave approximation for overland and channel flow. The EDA is an ensemble of 4D‑Var data assimilations that reflects uncertainties in observations; atmospheric boundary conditions, such as sea-surface temperature; and the model physics. Further, it provides functionality to adapt a model parallelization for parallel ensemble forecasts as well as routines for parallel communication linking the model and filters. into an ensemble data assimilation system using MPI, Environ. Commonly, DA is applied to separate models simulating, for example, the atmospheric dynamics or the ocean circulation. The first half of the course will be interactive tutorials that combine lectures with complementary online exercises. The fluctuation in the time is caused by parallel communication and file operations. The operations performed in each routine are rather elementary to keep the complexity of the routines low. Any operations specific to the model fields, the model grid, or to the assimilated observations are performed in program routines provided by the user based on existing template routines. Meteor. strongly coupled ocean-atmosphere data assimilation with an interface solver, In the implementation of AWI-CM-PDAF 1.0, the analysis is only performed in FESOM. Res.-Oceans, 124, 470–490, In general this is a template routine, which can be adapted by the user according to particular needs. The initial conditions for COSMO-1E and COSMO-2E forecasts, also known as analyses, are based on ensemble data assimilation that provides a multitude of slightly differing initial conditions. Weather Rev., 135, 3541–3564, 2007. , Geoscientific Instrumentation, Methods and Data Systems, Natural Hazards and Earth System Sciences, Parallel performance of the coupled data assimilation system, http://www.mpimet.mpg.de/en/science/models/mpi-esm/echam/, https://mpimet.mpg.de/en/science/models/availability-licenses, https://doi.org/10.1080/16000870.2017.1327766, https://doi.org/10.1080/16000870.2018.1445364. Meteor. In DA experiments on ,a regional scale, a "Big Data Assimilation" system has been ,developed to capture rapidly changing convective weather by ,combining a 100,,m mesh limited,,area simulation with advan,ced ,weather radar [2,7,]. strongly coupled ocean-atmosphere data assimilation with an interface solver, Nonetheless, the ensemble DA is computationally demanding; for larger applications, one might need to obtain a compute allocation at larger computing sites, such as national compute centers. This ensures optimal compatibility with these models, while it is still usable with models coded, for example, in the programming language C. The filter methods are model agnostic and only operate on abstract state vectors as described for the ESTKF in Sect. 2. Ensemble data assimilation methods such as the ensemble Kalman filter (EnKF) are a key component of probabilistic weather forecasting. The forecast ensemble represents an error subspace of dimension Ne−1, and the ESTKF computes the ensemble transformation matrix and vector in this subspace. Brasseur, P., Kirchgessner, P., and Beckers, J.-M.: State-of-the-art Both QT and LM worked on optimizing the compute performance of the implementation of PDAF with AWI-CM. Thus, a separation of concerns is ensured, which is mandated for efficient development of complex model codes and their adaptions to modern computers (Lawrence et al., 2018). First studies (Mu et al., 2020; Tang et al., 2020) based on this implementation have been published. In contrast, the forecast time without DA coupling only increases by about 3.5 % (black dashed line). Here, we discuss a strategy to build an online-coupled DA system for coupled models with the example of the coupled atmosphere–ocean model AWI-CM. In addition, the assimilation program would also need to read these restart files and write new restart files after the analysis step. Attendees will leave with a basic understanding of ensemble filter data assimilation that will allow them to better interpret ensemble forecasts. Soc., 145, 2335–2365, 2019. a, Vetra-Carvalho, S., van Leeuwen, P. J., Nerger, L., Barth, A., Altaf, M. U., ensemble square root Kalman filters, Mon. Q. J. Roy. Thus, their atmosphere had a higher resolution than used here, while the ocean resolution was comparable to the coarse FESOM resolution in the open ocean, which was then regionally refined. temperature and salinity profiles and monthly objective analyses with on multivariate updates of physical and biogeochemical ocean variables using This smaller effect on the salinity is expected, because there are no strong correlations between the SST and the salinity at different depths. The required modifications to the model source codes consist essentially of adding four subroutine calls in each of the two compartment models. Geosci., 55, 110–118, 2013. , Nerger, L., Hiller, W., and Schröter, J.: PDAF - The Parallel uncertainty estimates, J. Geophys. Nerger et al. Softw., 68, 122–128, 2015. . (2016) used a 3D variational DA in the ocean but 4D variational DA in the atmosphere. Karspeck et al. This routine of PDAF also checks whether all time steps of a forecast phase have been computed. 2019. a, Rackow, T., Sein, D. V., Semmler, T., Danilov, S., Koldunov, N. V., Sidorenko, D., Wang, Q., and Jung, T.: Sensitivity of deep ocean biases to horizontal resolution in prototype CMIP6 simulations with AWI-CM1.0, Geosci. The left-hand side of Fig. 2 (Fig. 2a) shows the typical flow of a coupled compartment model. The method of hierarchical modelling allows us to calculate these probabilities. Meteor. Further, we discuss the particularities of the coupled model. During time stepping, the coupler exchanges the interface information between the different compartments. 323–340, 1998. a, b, Pradhan, H. K., Voelker, C., Losa, S. N., Bracher, A., and Nerger, L.: This large variation is due to the fact that here the communication happens in the communicators COMM_COUPLE, which are spread much wider over the computer than the communicators for each coupled model task (COMM_CPLMOD), as is visible in Fig. 3. Figure 5 shows the execution times per model day for different parts of the assimilation program. Data Assimilation comprehensively covers data assimilation and inverse methods, including both traditional state estimation and parameter estimation. For coupled models consisting of multiple executables, this call structure is used for each compartment model. To avoid these conflicts, it helped to distribute the execution of the different ensemble tasks to different directories, e.g.,mpirun -np  N_O 01/fesom.x : -np N_A \ 01/echam.x : -np  02/N_O fesom.x : \ -np N_A  02/echam.x …combined with a prior operation in the run script to generate the directories and distribute the model executables and input files. This is nowadays easier than in the past, because the number of processor cores is much larger in current high-performance computers compared to the past. For interfacing model fields and state vector (cyan), there are two routines called before and after the analysis step. This communication will only be a small part of the analysis time. Here, a configuration is used that computes the filter analysis step on the first coupled model task using the same domain decomposition as the coupled model. 137, 4089–4114, 2009. , van Leeuwen, P. J.: Nonlinear data assimilation in geosciences: An extremely Further, we expect a similar scalability in the case of strongly coupled DA. Etna Explosive Eruption, Atmosphere, 11, 359, https://doi.org/10.3390/atmos11040359, When repeating experiments with the same ensemble size, we found a variation of the execution time for the analysis step of up to 10 %. The experiments were conducted on the Cray XC40 system “Konrad” of the North-German Supercomputer Alliance (HLRN). Partly, the mentioned studies used twin experiments assimilating synthetic observations to assess the DA behavior. In the analysis step at time tk, the ESTKF transforms a forecast ensemble Xkf of Ne model states of size Nx stored in the columns of this matrix into a matrix of analysis states Xka as. 2017. a, b, Kunii, M., Ito, K., and Wada, A.: Preliminary Test of a Data Assimilation the Met Office coupled atmosphere-land-ocean-sea ice model, Mon. Assimilation of Volcanic Ash Clouds from Satellite Observations: Application coupled ocean–atmosphere data assimilation in the ECMWF NWP system, Q.: Towards multi-resolution global climate moeling with ECHAM6-FESOM. an ensemble Kalman filter and an idealized model of upwelling, Ocean Model., Nerger, L., Tang, Q., and Mu, L.: Efficient ensemble data assimilation for coupled models with the Parallel Data Assimilation Framework: example of AWI-CM (AWI-CM-PDAF 1.0), Geosci. For the temperature, the RMSE is reduced by 15.2 % in the upper 200 m but only 3.0 % below 200 m. Physica D, 230, 112–126, 2007. a, b, c, Karspeck, A. R., Danabasoglu, G., Anderson, J., Karol, S., Collins, N., system for seamless sea ice prediction based on the AWI climate model, J. assimilation, Ocean Dynam., 69, 1217–1237, 2019. a, b, Gropp, W., Lusk, E., and Skjellum, A.: Using MPI: Portable Parallel Programming This work discusses the technical aspects of how a coupled model that simulates the ocean and the atmosphere can be augmented by data assimilation functionality provided in generic form by the open-source software PDAF (Parallel Data Assimilation Framework). When the parallel region of a program is initialized, the communicator MPI_COMM_WORLD is initialized, which contains all processes of the program. For strongly coupled DA, the observation operator routine would also contain parallel communication that acts across the compartments. The number of excluded observations shrinks during the course of the assimilation and after 1 month less than 5 % of the days observations are excluded. coupled models with the Parallel Data Assimilation Framework: Example of This study discusses an approach to augment a coupled model with data assimilation functionality provided by the Parallel Data Assimilation Framework (PDAF). These routines are executed by all processes that participate in the model integrations, and each routine acts on its process subdomain. Here, an updated coupling strategy is discussed that requires less changes to the model code. There is a natural linkage between data assimilation and ensemble forecasting: ensemble forecasts are designed to estimate the flow-dependent uncertainty of the forecast; data assimilation techniques require accurate estimates of forecast uncertainty in order to optimally When the time for the DA coupling is subtracted from the forecast time, the variability is much reduced as the black dashed line shows. Note that compared to the single-compartment case discussed in Since then the WRF core, data assimilation scheme, ensemble perturbation approaches, and ensemble output post-processing have been continuously improved. System with a Regional High-Resolution Atmosphere-Ocean Coupled Model Based Arellano, A.: The Data Assimilation Research Testbed: A Community Facility, Soc., 144, 2404–2430, https://doi.org/10.1002/qj.3308, 2018. a, b, c, d, e, f, Kirchgessner, P., Toedter, J., Ahrens, B., and Nerger, L.: The smoother Data assimilation integrates information from observational measurements with numerical models. Meteor. This variation is mainly due to the operations for writing the ensemble mean state into a file. Meteorol. This study explains the required modifications to the programs with the example of the coupled atmosphere–sea-ice–ocean model AWI-CM (AWI Climate Model). Model Dev., 6, 373–388, https://doi.org/10.5194/gmd-6-373-2013, 2013. a, van Leeuwen, P. J.: Particle Filtering in Geophysical Systems, Mon. This section describes the assimilation framework and the setup of the DA program. Shown are the relative RMSEs for temperature (blue) and salinity (green). These methods are far more costly to compute than a single coupled model because of the required integration of the ensemble. Finally, the communicator COMM_FILTER (row 5 of Fig. 3b) is defined, which contains all processes of the first model task. SMOS sea ice thickness data simultaneously, Q. J. Roy. Adv. Use of High Performance Computing in Meteorology – Proceedings of the 11. The model-agnostic structure of the assimilation software ensures a separation of concerns in which the development of data assimilation methods can be separated from the model application. Nonetheless, repeated experiments showed that the timings in Fig. 5 are representative. In the free run, the RMSE increases first to about 1.4 ∘C and reaches nearly 1.6 ∘C a the end of the year. To discuss strongly coupled filtering, let us assume a two-compartment system (perhaps the atmosphere and the ocean). Another class of EnDA methods are particle filters (e.g., van Leeuwen, 2009). Here, DA can either be performed separately in the different compartment domains, commonly called weakly coupled DA, or it can be performed in a joint update, called strongly coupled DA. Accordingly, each ocean model will be placed distant from the atmospheric model to which it is coupled. These additions are calls to subroutines that interface between the model code and the DA framework. Thus, after this point, the coupler can distinguish the different model compartments. 126, 13–28, 2018. a, Zhang, S., Harrison, M. J., Rosati, A., and Wittenberg, A.: System design and Model Dev., 6, 373–388, van Leeuwen, P. J.: Particle Filtering in Geophysical Systems, Mon. coupled data assimilation, Clim. Counillon, F., Drper, C., Frolov, S., Fujii, Y., Kumar, A., Laloyaux, P., Further, wk is a vector of size Ne, which transforms the ensemble mean and W̃ is a matrix of size Ne×Ne, which transforms the ensemble perturbations. (AWI-CM-PDAF version 1.0), Zenodo. Here, at the very beginning of the program, the parallelization is initialized (“init. AWI-CM – output files and plot scripts, Zenodo, For the online coupling of PDAF with the coupled model TerrSysMP, the setup of the parallelization was described by Kurtz et al. After time stepping, some postprocessing can be performed, e.g., writing time averages or restart files to disk. In this configuration, the performance results of Sect. 4.1 were obtained. Data assimilation combines observations with numerical models to get an improved estimate of the model state. Q. J. Roy. filter: numerical experiments with a global corculation model, Tellus, 59A, where T is a projection matrix with j=Ne rows and i=Ne-1 columns defined by. During this initial transient phase, the RMSE is reduced to about 0.45 ∘C. Different shades of the same color mark the same communicator type (e.g., four orange communicators COMM_FESOM). Shown are the free run (blue) and the SST assimilation experiment (black). While for TerrSysMP, a different coupling strategy was used; the parallelization of the overall system is essentially the same as discussed here for AWI-CM. For AWI-CM without data assimilation extension, the parallelization is initialized by each program at the very beginning. (2019). These alternative parallelization strategies are, however, more complex to implement and hence not the default in PDAF. Given that both model compartments in AWI-CM scale to larger processor numbers than we used for the DA experiment, we expect that the DA in AWI-CM with ECHAM at a resolution of T127 (i.e., about 1∘) could be run at a similar execution time as for T63 given that a higher number of processors would be used. Res. Methods to make particle filters usable for high-dimension systems were reviewed by van Leeuwen et al. Then, a routine of OASIS3-MCT is called which splits MPI_COMM_WORLD into two communicators: one for ECHAM (COMM_ECHAM) and one for FESOM (COMM_FESOM). OASIS3-MCT is linked into each program as a library. with the Message-Passing Interface, The MIT Press, Cambridge, Massachusetts, For coupled ensemble DA in hydrology, Kurtz et al. The localization leads to individual transformation weights wk and W̃ for each local analysis domain. The authors declare that they have no conflict of interest. To obtain the scalability discussed above, important optimization steps have been performed. For the coupled model, the routine is called in both ECHAM and FESOM. Weather Rev., 135, 3541–3564, 2007. a, An interactive open-access journal of the European Geosciences Union, Efficient ensemble data assimilation for coupled models with the Parallel Data Assimilation Framework: example of AWI-CM (AWI-CM-PDAF 1.0). The time for the DA coupling (blue line) varies by a factor of 2.5. Generally, the introduction of the ensemble adds one additional level of parallelization to a model, which allows one to concurrently compute the ensemble of model integrations, i.e., several concurrent model tasks. The motivation for this call structure is that the call-back routines exist in the context of the model (i.e., the user space) and can be implemented like model routines. The PDAF code (version 1.14 was used here), as well as a full code documentation and a usage tutorial, is available at http://pdaf.awi.de (last access: 14 September 2020). Meteor. Good performance with small ensemble filters applied to models with many state variables may require ‘localizing’ the impact of an observation to state variables that are ‘close’ to the observation. and Thurlow, M.: Assessing a new coupled data assimilation system based on Modell. Anderson, J., Hoar, T., Raeder, K., Liu, H., Collins, N., Torn, R., and This routine is inserted directly after the parallelization is started. We discuss the strategy for both weakly and strongly coupled DA but assess the parallel performance only for weakly coupled DA into the ocean, which is supported in the code version AWI-CM-PDAF V1.0. Marine Syst., 16, uncertainty estimates, J. Geophys. Nerger, L., Tang, Q., and Mu, L.: The PDAF model binding for AWI-CM 142, 65–78, 2016. a, Lawrence, B. N., Rezny, M., Budich, R., Bauer, P., Behrens, J., Carter, M., Deconinck, W., Ford, R., Maynard, C., Mullerworth, S., Osuna, C., Porter, A., Serradell, K., Valcke, S., Wedi, N., and Wilson, S.: Crossing the chasm: how to develop weather and climate models for next generation computers?, Geosci. 126, 13–28, 2018. , Zhang, S., Harrison, M. J., Rosati, A., and Wittenberg, A.: System design and Res.-Oceans, 118, 6704–6716, 2013. , Goodliff, M., Bruening, T., Schwichtenberg, F., Li, X., Lindenthal, A., Three of these subroutine calls connect the models to the DA functionality provided by PDAF, while the fourth is optional and provides timing and memory information. Now, the model itself is initialized; e.g., the model grid for each compartment is initialized and the initial fields are read from files. Meteorol. AWI-CM – output files and plot scripts, Zenodo. In the forecast phase the ensemble of model states is integrated with the numerical model until the time when observations are available. To provide flexibility to adapt to such requirements, the routine init_parallel_pdaf is compiled with the model and is not part of the core routines of the PDAF library. on an Ensemble Kalman Filter, Mon. on multivariate updates of physical and biogeochemical ocean variables using

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