= Implementation of the Analysis Step for 3D Ensemble Var with OMI = {{{ #!html

Implementation Guide

  1. Main page
  2. Adaptation of the parallelization
  3. Initialization of PDAF
  4. Modifications for ensemble integration
  5. Implementation of the analysis step with OMI
    1. Implementation for Global Filters
    2. Implementation for Local Filters
    3. Implementation for LEnKF
    4. Implementation for 3D-Var
    5. Implementation for 3D Ensemble Var
    6. Implementation for hybrid 3D-Var
    7. PDAF-OMI Overview
  6. Memory and timing information
  7. Ensemble Generation
  8. Diagnostics
}}} [[PageOutline(2-3,Contents of this page)]] == Overview == With Version 2.0 with introduced 3D variational assimilation methods to PDAF. There are genenerally three different variants: parameterized 3D-Var, 3D Ensemble Var, and hybrid (parameterized + ensemble) 3D-Var. This page describes the implementation of the analysis step for the 3D Ensemble Var using PDAF-OMI. For the analysis step of 3D-Var we need different operations related to the observations. These operations are requested by PDAF by call-back routines supplied by the user and provided in the OMI structure. The names of the routines that are provided by the user are specified in the call to the routine `PDAFomi_assimilate_en3dvar` in the fully-parallel implementation (or `PDAFomi_put_state_en3dvar` for the 'flexible' implementation) that was discussed before. With regard to the parallelization, all these routines (except `U_collect_state`, `U_distribute_state`, and `U_next_observation`) are executed by the filter processes (`filterpe=.true.`) only. For 3D Ensemble Var the ensemble is used to represent the background covariance matrix '''B'''. This ensemble perturbations need to be transformed by means of an ensemble Kalman filter. PDAF uses for this the error-subspace transform filter ESTKF. There are two variants: The first uses the localized filter LESTKF, while the second uses the global filter ESTKF. For completeness we discuss here all user-supplied routines that are specified in the interface to `PDAFomi_assimilate_en3dvar`. Thus, some of the user-supplied routines that are explained on the page describing the modification of the model code for the ensemble integration are repeated here. == Analysis Routines == The general aspects of the filter (or solver) specific routines `PDAF_assimilate_*` have been described on the page [ModifyModelforEnsembleIntegration Modification of the model code for the ensemble integration] and its sub-page on [InsertAnalysisStep inserting the analysis step]. The routine is used in the fully-parallel implementation variant of the data assimilation system. When the 'flexible' implementation variant is used, the routines `PDAF_put_state_*` is used as described further below. Here, we list the full interface of the routine. Subsequently, the user-supplied routines specified in the call is explained. There are two variants that either compute the transformataion of the ensemble transformation using the local LESTKF method, or the global ESTKF. === `PDAFomi_assimilate_en3dvar_lestkf` === This routine is called for the case of transforming the ensemble perturbations using the local LESTKF. The interface is: {{{ SUBROUTINE PDAFomi_assimilate_en3dvar_lestkf(U_collect_state, U_distribute_state, & U_init_dim_obs_pdafomi, U_obs_op_pdafomi, & U_cvt_ens, U_cvt_adj_ens, U_obs_op_lin_pdafomi, U_obs_op_adj_pdafomi, & U_init_n_domains_p, U_init_dim_l, U_init_dim_obs_l_pdafomi, & U_g2l_state, U_l2g_state, U_prepoststep, U_next_observation, outflag) }}} with the following arguments: * [#U_collect_statecollect_state_pdaf.F90 U_collect_state]: The name of the user-supplied routine that initializes a state vector from the array holding the ensemble of model states from the model fields. This is basically the inverse operation to `U_distribute_state` used in `PDAF_get_state` as well as here. * [#U_distribute_statedistribute_state_pdaf.F90 U_distribute_state]: The name of a user supplied routine that initializes the model fields from the array holding the ensemble of model state vectors. * [#U_init_dim_obs_pdafomicallback_obs_pdafomi.F90 U_init_dim_obs_pdafomi]: The name of the user-supplied routine that initializes the observation information and provides the size of observation vector * [#U_obs_op_pdafomicallback_obs_pdafomi.F90 U_obs_op_pdafomi]: The name of the user-supplied routine that acts as the observation operator on some state vector * [#U_cvt_enscvt_ens_pdaf.F90 U_cvt_ens]: The name of the user-supplied routine that applies the ensemble control-vector transformation (square-root of the B-matrix) on some control vector to obtain a state vector. * [#U_cvt_adj_enscvt_adj_ens_pdaf.F90 U_cvt_adj_ens]: The name of the user-supplied routine that applies the adjoint ensemble control-vector transformation (with square-root of the B-matrix) on some state vector to obtain the control vector. * [#U_obs_op_pdafomicallback_obs_pdafomi.F90 U_obs_op_lin_pdafomi]: The name of the user-supplied routine that acts as the linearized observation operator on some state vector * [#U_obs_op_pdafomicallback_obs_pdafomi.F90 U_obs_op_lin_pdafomi]: The name of the user-supplied routine that acts as the adjoint observation operator on some state vector * [#U_init_n_domainsinit_n_domains_pdaf.F90 U_init_n_domains]: The name of the routine that provides the number of local analysis domains * [#U_init_dim_linit_dim_l_pdaf.F90 U_init_dim_l]: The name of the routine that provides the state dimension for a local analysis domain * [#U_init_dim_obs_l_pdafomicallback_obs_pdafomi.F90 U_init_dim_obs_l_pdafomi]: The name of the routine that initializes the size of the observation vector for a local analysis domain * [#U_g2l_stateg2l_state_pdaf.F90 U_g2l_state]: The name of the routine that initializes a local state vector from the global state vector * [#U_l2g_statel2g_state_pdaf.F90 U_l2g_state]: The name of the routine that initializes the corresponding part of the global state vector from the provided local state vector * [#U_prepoststepprepoststep_ens_pdaf.F90 U_prepoststep]: The name of the pre/poststep routine as in `PDAF_get_state` * [#U_next_observationnext_observation.F90 U_next_observation]: The name of a user supplied routine that initializes the variables `nsteps`, `timenow`, and `doexit`. The same routine is also used in `PDAF_get_state`. * `status`: The integer status flag. It is zero, if `PDAFomi_assimilate_global` is exited without errors. === `PDAFomi_assimilate_en3dvar_estkf` === This routine is called for the case of transforming the ensemble perturbations using the global ESTKF. The interface is: {{{ SUBROUTINE PDAFomi_assimilate_en3dvar_estkf(U_collect_state, U_distribute_state, & U_init_dim_obs_pdafomi, U_obs_op_pdafomi, & U_cvt_ens, U_cvt_adj_ens, U_obs_op_lin_pdafomi, U_obs_op_adj_pdafomi, & U_prepoststep, U_next_observation, outflag) }}} with the following arguments: * [#U_collect_statecollect_state_pdaf.F90 U_collect_state]: The name of the user-supplied routine that initializes a state vector from the array holding the ensemble of model states from the model fields. This is basically the inverse operation to `U_distribute_state` used in `PDAF_get_state` as well as here. * [#U_distribute_statedistribute_state_pdaf.F90 U_distribute_state]: The name of a user supplied routine that initializes the model fields from the array holding the ensemble of model state vectors. * [#U_init_dim_obs_pdafomicallback_obs_pdafomi.F90 U_init_dim_obs_pdafomi]: The name of the user-supplied routine that initializes the observation information and provides the size of observation vector * [#U_obs_op_pdafomicallback_obs_pdafomi.F90 U_obs_op_pdafomi]: The name of the user-supplied routine that acts as the observation operator on some state vector * [#U_cvt_enscvt_ens_pdaf.F90 U_cvt_ens]: The name of the user-supplied routine that applies the ensemble control-vector transformation (square-root of the B-matrix) on some control vector to obtain a state vector. * [#U_cvt_adj_enscvt_adj_ens_pdaf.F90 U_cvt_adj_ens]: The name of the user-supplied routine that applies the adjoint ensemble control-vector transformation (with square-root of the B-matrix) on some state vector to obtain the control vector. * [#U_obs_op_pdafomicallback_obs_pdafomi.F90 U_obs_op_lin_pdafomi]: The name of the user-supplied routine that acts as the linearized observation operator on some state vector * [#U_obs_op_pdafomicallback_obs_pdafomi.F90 U_obs_op_lin_pdafomi]: The name of the user-supplied routine that acts as the adjoint observation operator on some state vector * [#U_prepoststepprepoststep_ens_pdaf.F90 U_prepoststep]: The name of the pre/poststep routine as in `PDAF_get_state` * [#U_next_observationnext_observation.F90 U_next_observation]: The name of a user supplied routine that initializes the variables `nsteps`, `timenow`, and `doexit`. The same routine is also used in `PDAF_get_state`. * `status`: The integer status flag. It is zero, if `PDAFomi_assimilate_global` is exited without errors. Note that the interface of `PDAFomi_assimilate_en3dvar_estkf` is identical to that of `PDAFomi_assimilate_3dvar` apart from using the routines `U_cvt_ens` and `U_cvt_adj_ens` in case of the ensemble variational method. === `PDAFomi_put_state_en3dvar_lestkf` === When the 'flexible' implementation variant is chosen for the assimilation system, the routine `PDAFomi_put_state_*` has to be used instead of `PDAFomi_assimilate_*`. The general aspects of the filter specific routines `PDAF_put_state_*` have been described on the page [ModifyModelforEnsembleIntegration Modification of the model code for the ensemble integration]. The interface of the routine is identical with that of `PDAF_assimilate_*` with the exception the specification of the user-supplied routines `U_distribute_state` and `U_next_observation` are missing. The interface when using one of the global filters is the following: {{{ SUBROUTINE PDAFomi_put_state_en3dvar_lestkf(U_collect_state, & U_init_dim_obs_pdafomi, U_obs_op_pdafomi, & U_cvt_ens, U_cvt_adj_ens, U_obs_op_lin_pdafomi, U_obs_op_adj_pdafomi, & U_init_n_domains_p, U_init_dim_l, U_init_dim_obs_l_pdafomi, & U_g2l_state, U_l2g_state, U_prepoststep, outflag) }}} === `PDAFomi_put_state_en3dvar_estkf` === The interface of this routine is analogous to that of `PDAFomi_assimilate_en3dvar_estkf'. Thus it is identical to this routine with the exception the specification of the user-supplied routines `U_distribute_state` and `U_next_observation` are missing. The interface when using one of the global filters is the following: {{{ SUBROUTINE PDAFomi_put_state_en3dvar_estkf(U_collect_state, & U_init_dim_obs_pdafomi, U_obs_op_pdafomi, & U_cvt_ens, U_cvt_adj_ens, U_obs_op_lin_pdafomi, U_obs_op_adj_pdafomi, & U_prepoststep, outflag) }}} == User-supplied routines == Here all user-supplied routines are described that are required in the call to `PDAFomi_assimilate_3dvar`. For some of the generic routines, we link to the page on [ModifyModelforEnsembleIntegration modifying the model code for the ensemble integration]. To indicate user-supplied routines we use the prefix `U_`. In the template directory `templates/` as well as in the tutorial implementations in `tutorial/` these routines exist without the prefix, but with the extension `_pdaf.F90`. The user-routines relating to OMI are collected in the file `callback_obs_pdafomi.F90`. In the section titles below we provide the name of the template file in parentheses. In the subroutine interfaces some variables appear with the suffix `_p`. This suffix indicates that the variable is particular to a model sub-domain, if a domain decomposed model is used. Thus, the value(s) in the variable will be different for different model sub-domains. === `U_collect_state` (collect_state_pdaf.F90) === This routine is independent of the filter algorithm used. See the page on [InsertAnalysisStep#U_collect_statecollect_state_pdaf.F90 inserting the analysis step] for the description of this routine. === `U_distribute_state` (distribute_state_pdaf.F90) === This routine is independent of the filter algorithm used. See the page on [InsertAnalysisStep#U_distribute_statedistribute_state_pdaf.F90 inserting the analysis step] for the description of this routine. === `U_init_dim_obs_pdafomi` (callback_obs_pdafomi.F90) === This is a call-back routine for PDAF-OMI initializing the observation information. The routine just calls a routine from the observation module for each observation type. See the [wiki:OMI_Callback_obs_pdafomi documentation on callback_obs_pdafomi.F90] for more information. === `U_obs_op_pdafomi` (callback_obs_pdafomi.F90) === This is a call-back routine for PDAF-OMI applying the observation operator to the state vector. The routine calls a routine from the observation module for each observation type. See the [wiki:OMI_Callback_obs_pdafomi documentation on callback_obs_pdafomi.F90] for more information. === `U_cvt_ens` (cvt_ens_pdaf.F90) === The interface for this routine is: {{{ SUBROUTINE cvt_ens_pdaf(iter, dim_p, dim_ens, dim_cv_ens_p, ens_p, cv_p, Vcv_p) INTEGER, INTENT(in) :: iter ! Iteration of optimization INTEGER, INTENT(in) :: dim_p ! PE-local observation dimension INTEGER, INTENT(in) :: dim_ens ! Ensemble size INTEGER, INTENT(in) :: dim_cv_ens_p ! Dimension of control vector REAL, INTENT(in) :: ens_p(dim_p, dim_ens) ! PE-local ensemble REAL, INTENT(in) :: cv_p(dim_cv_ens_p) ! PE-local control vector REAL, INTENT(inout) :: Vcv_p(dim_p) ! PE-local state increment }}} The routine is called during the analysis step during the iterative minimization of the cost function. It has to apply the control vector transformation to the control vector and return the transformed result vector. Usually this transformation is the multiplication with the square-root of the background error covariance matrix '''B'''. For the 3D Ensemble Var, this square root is usually expressed through the ensemble. If the control vector is decomposed in case of parallelization it first needs to the gathered on each processor and afterwards the transformation is computed on the potentially domain-decomposed state vector. === `U_cvt_adj` (cvt_adj_pdaf.F90) === The interface for this routine is: {{{ SUBROUTINE cvt_adj_ens_pdaf(iter, dim_p, dim_ens, dim_cv_ens_p, ens_p, Vcv_p, cv_p) INTEGER, INTENT(in) :: iter ! Iteration of optimization INTEGER, INTENT(in) :: dim_p ! PE-local observation dimension INTEGER, INTENT(in) :: dim_ens ! Ensemble size INTEGER, INTENT(in) :: dim_cv_ens_p ! PE-local dimension of control vector REAL, INTENT(in) :: ens_p(dim_p, dim_ens) ! PE-local ensemble REAL, INTENT(in) :: Vcv_p(dim_p) ! PE-local input vector REAL, INTENT(inout) :: cv_p(dim_cv_ens_p) ! PE-local result vector }}} The routine is called during the analysis step during the iterative minimization of the cost function. It has to apply the adjoint control vector transformation to a state vector and return the control vector. Usually this transformation is the multiplication with transpose of the square-root of the background error covariance matrix '''B'''. or the 3D Ensemble Var, this square root is usually expressed through the ensemble. If the state vector is decomposed in case of parallelization one needs to take care that the application of the trasformation is complete. This usually requries a comminucation with MPI_Allreduce to obtain a global sun. === `U_obs_op_lin_pdafomi` (callback_obs_pdafomi.F90) === This is a call-back routine for PDAF-OMI applying the linearized observation operator to the state vector. The routine calls a routine from the observation module for each observation type. If the full observation operator is lineaer the same operator can be used here. See the [wiki:OMI_Callback_obs_pdafomi documentation on callback_obs_pdafomi.F90] for more information. === `U_obs_op_adj_pdafomi` (callback_obs_pdafomi.F90) === This is a call-back routine for PDAF-OMI applying the adjoint observation operator to some vector inthe observation space. The routine calls a routine from the observation module for each observation type. See the [wiki:OMI_Callback_obs_pdafomi documentation on callback_obs_pdafomi.F90] for more information. === `U_init_n_domains` (init_n_domains_pdaf.F90) === The interface for this routine is: {{{ SUBROUTINE init_n_domains(step, n_domains_p) INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(out) :: n_domains_p ! Number of analysis domains for local model sub-domain }}} The routine is called during the analysis step before the loop over the local analysis domains is entered. It has to provide the number of local analysis domains. In case of a domain-decomposed model the number of local analysis domain for the model sub-domain of the calling process has to be initialized. Hints: * As a simple case, if the localization is only performed horizontally, the local analysis domains can be single vertical columns of the model grid. In this case, `n_domains_p` is simply the number of vertical columns in the local model sub-domain. === `U_init_dim_l` (init_dim_l_pdaf.F90) === The interface for this routine is: {{{ SUBROUTINE init_dim_l(step, domain_p, dim_l) INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: domain_p ! Current local analysis domain INTEGER, INTENT(out) :: dim_l ! Local state dimension }}} The routine is called during the loop over the local analysis domains in the analysis step. It has to provide in `dim_l` the dimension of the state vector for the local analysis domain with index `domain_p`. Hints: * For sharing through the module 'mod_assimilation', we further initialize an array 'coords_l' containing the coordinates that describe the local domain. These coordinates have to describe one location in space that is used in the OMI observation modules to compute the distance from observations. This requires that the coordinates in 'coords_l' have the same units as those used for the observations. * Any form of local domain is possible as long as it can be describe as a single location. If observations are only horizontally distributed (a common situation with satellite data in the ocean), the local analysis domain can be a single vertical column of the model grid. In this case, the size of the state in the local analysis domain will be just the number of vertical grid points at this location and the horizontal coordinates are used in 'coords_l' * Further, we recommend to initialize an array containing the indices of the elements of the local state vector in the global (or domain-decomposed) state vector (`id_lstate_in_pstate` in the template files). This array is also shared through 'mod_assimilation'. === `U_init_dim_obs_l_pdafomi` (callback_obs_pdafomi.F90) === This is a call-back routine for PDAF-OMI that initializes the local observation vector. The routine calls a routine from the observation module for each observation type. See the [wiki:OMI_Callback_obs_pdafomi documentation on callback_obs_pdafomi.F90] for more information. === `U_g2l_state` (g2l_state_pdaf.F90) === The interface for this routine is: {{{ SUBROUTINE g2l_state(step, domain_p, dim_p, state_p, dim_l, state_l) INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: domain_p ! Current local analysis domain INTEGER, INTENT(in) :: dim_p ! State dimension for model sub-domain INTEGER, INTENT(in) :: dim_l ! Local state dimension REAL, INTENT(in) :: state_p(dim_p) ! State vector for model sub-domain REAL, INTENT(out) :: state_l(dim_l) ! State vector on local analysis domain }}} The routine is called during the loop over the local analysis domains in the analysis step. It has to provide the local state vector `state_l` that corresponds to the local analysis domain with index `domain_p`. Provided to the routine is the state vector `state_p`. With a domain decomposed model, this is the state for the local model sub-domain. Hints: * In the simple case that a local analysis domain is a single vertical column of the model grid, the operation in this routine would be to take out of `state_p` the data for the vertical column indexed by `domain_p`. * Usually, one can initialize the indices of the local state vector elements in the global state vector in `U_init_dim_l` and just use these here. === `U_l2g_state` (l2g_state_pdaf.F90) === The interface for this routine is: {{{ SUBROUTINE l2g_state(step, domain_p, dim_l, state_l, dim_p, state_p) INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: domain_p ! Current local analysis domain INTEGER, INTENT(in) :: dim_p ! State dimension for model sub-domain INTEGER, INTENT(in) :: dim_l ! Local state dimension REAL, INTENT(in) :: state_p(dim_p) ! State vector for model sub-domain REAL, INTENT(out) :: state_l(dim_l) ! State vector on local analysis domain }}} The routine is called during the loop over the local analysis domains in the analysis step. It has to initialize the part of the global state vector `state_p` that corresponds to the local analysis domain with index `domain_p`. Provided to the routine is the state vector `state_l` for the local analysis domain. Hints: * In the simple case that a local analysis domain is a single vertical column of the model grid, the operation in this routine would be to write into `state_p` the data for the vertical column indexed by `domain_p`. * Usually, one can initialize the indices of the local state vector elements in the global state vector in `U_init_dim_l` and just use these here. === `U_prepoststep` (prepoststep_ens_pdaf.F90) === The routine has already been described for modifying the model for the ensemble integration and for inserting the analysis step. See the page on [InsertAnalysisStep#U_prepoststepprepoststep_ens_pdaf.F90 inserting the analysis step] for the description of this routine. === `U_next_observation` (next_observation_pdaf.F90) === This routine is independent of the filter algorithm used. See the page on [InsertAnalysisStep#U_next_observationnext_observation_pdaf.F90 inserting the analysis step] for the description of this routine. == Execution order of user-supplied routines == The user-supplied routines are essentially executed in the order they are listed in the interface to `PDAFomi_assimilate_3dvar`. The order can be important as some routines can perform preparatory work for later routines. For example, `U_init_dim_obs_pdafomi` prepares an index array that provides the information for executing the observation operator in `U_obs_op_pdafomi`. How this information is initialized is described in the documentation of OMI. Before the analysis step is called the following routine is executed: 1. [#U_collect_statecollect_state_pdaf.F90 U_collect_state] The analysis step is executed when the ensemble integration of the forecast is completed. During the analysis step the following routines are executed in the given order: 1. [#U_prepoststepprepoststep_ens_pdaf.F90 U_prepoststep] (Call to act on the forecast ensemble, called with negative value of the time step) 1. [#U_init_dim_obs_pdafomicallback_obs_pdafomi.F90 U_init_dim_obs_pdafomi] 1. [#U_obs_op_pdafomicallback_obs_pdafomi.F90 U_obs_op_pdafomi] (multiple calls, one for each ensemble member) Inside the analysis step the interative optimization is computed. This involves the repeated call of the routines: 1. [#U_cvt_enscvt_ens_pdaf.F90 U_cvt_ens] 1. [#U_obs_op_lin_pdafomicallback_obs_pdafomi.F90 U_obs_op_lin_pdafomi] 1. [#U_obs_op_adj_pdafomicallback_obs_pdafomi.F90 U_obs_op_adj_pdafomi] 1. [#U_cvt_adj_enscvt_adj_ens_pdaf.F90 U_cvt_adj_ens] After the iterative optimization the following routines are executes to complte the analysis step: 1. [#U_cvt_enscvt_ens_pdaf.F90 U_cvt_ens] (Call to the control vector transform to compute the final state vector increment 1. [#U_prepoststepprepoststep_ens_pdaf.F90 U_prepoststep] (Call to act on the analysis ensemble, called with (positive) value of the time step) The iterative optimization abovve computes an updated ensemble mean state. Subsequently, the ensemble perturbations are updated using the LESTKF or ESTKF. The execution of the routines for these filters is described for the LESTKF on the [wiki:ImplementAnalysisLocal page on implementing the local filter analysis step] and for the ESTKF on the [wiki:ImplementAnalysisGlobal page on implementing the global filter analysis step]. In case of the routine `PDAFomi_assimilate_*`, the following routines are executed after the analysis step: 1. [#U_distribute_statedistribute_state_pdaf.F90 U_distribute_state] 1. [#U_next_observationnext_observation_pdaf.F90 U_next_observation]