Changes between Initial Version and Version 1 of Implement3DVarAnalysisPDAF3Universal

Timestamp:

May 26, 2025, 2:36:02 PM (7 days ago)

Author:

lnerger

Comment:

--

Implement3DVarAnalysisPDAF3Universal

@@ +1 @@
+= Implementation of the Analysis Step of 3D-Var using the universal PDAF3 interface = 
+
+{{{
+#!html
+<div class="wiki-toc">
+<h4>Implementation Guide - Analysis Step</h4>
+<ol><li>Implementing the analysis step</li>
+<ol>
+<li><b>Ensemble filters</b></li>
+<ol>
+<li> <a href="ImplementFilterAnalysisOverviewPDAF3"> General overview for ensemble filters</a></li>
+<li><a href="ImplementAnalysisPDAF3Universal">Universal interface </a></li>
+<li><a href="ImplementAnalysisPDAF3UniversalLocal">Universal interface using g2l/l2g_state</a></li>
+<li><a href="ImplementanalysisPDAF3Gloval">Interface specific for global filters</a></li>
+</ol>
+<li><b>3D-Var methods</b></li>
+<ol>
+<li> <a href="Implement3DVarAnalysisOverviewPDAF3"> General overview for 3D-Var methods</a></li>
+<li><a href="Implement3DVarAnalysisPDAF3Universal">Universal interface for 3D-Var</a></li>
+<li><a href="Implement3DVarAnalysisPDAF3_3DVar">Implementation for 3D-Var</a></li>
+<li><a href="Implement3DVarAnalysisPDAF3_3DEnVar">Implementation for 3D Ensemble Var</a></li>
+<li><a href="Implement3DVarAnalysisPDAF3_Hyb3DVar">Implementation for Hybrid 3D-Var</a></li>
+</ol>
+
+<li><a href="nondiagonal_observation_error_covariance_matrices_PDAF3">Using nondiagonal R-matrices</a></li>
+<li><a href="PDAF_OMI_Overview">PDAF-OMI Overview</a></li>
+</ol>
+</div>
+}}}
+
+
+[[PageOutline(2-3,Contents of this page)]]
+
+== Overview ==
+
+This page describes the recommended implementation of the analysis step for the different 3D-Var schemes using the universal interface of PDAF3.
+
+There are genenerally three different variants of 3D-Var provided by PDAF: parameterized 3D-Var, 3D Ensemble Var, and hybrid (parameterized + ensemble) 3D-Var. All can be called using the universal interface routines described here.
+
+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 `PDAF3_assimilate_3dvar_all` in the online mode of PDAF or `PDAF3_assim_offline_3dvar_all` for the offline mode. 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.
+
+The universal interface has more arguments than the specific interfaces for the parameterized 3D-Var or the 3D ensemble Var methods. It is useful if one implements both the 3D-Var with parameterized covariances and rhe 3D ensemble Var. The hybrid 3D-Var using the LESTKF is always called using this unversal interface.
+
+The different 3D-Var methods in PDAF were explained on the [wiki:Implement3DVarAnalysisOverview page providing the verview of the Analysis Step for 3D-Var Methods]. Depending the type of 3D-Var, the background covariance matrix '''B''' is represented either in a parameterized form, by an ensemble, or by a combination of both. The 3D-Var methods that use an ensemble need to transform the ensemble perturbations using 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_hyb3dvar_X`. 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.
+
+=== `PDAF3_assimilate_3dvar_all` ===
+
+This routine is used both in the ''fully-parallel'' and the ''flexible'' implementation variants of the data assimilation system. (See the page [ModifyModelforEnsembleIntegration Modification of the model code for the ensemble integration] for these variants)
+
+The interface is:
+{{{
+SUBROUTINE PDAF3_assimilate_3dvar_all(collect_state_pdaf, distribute_state_pdaf, &
+                                 init_dim_obs_pdafomi, obs_op_pdafomi, &
+                                 cvt_ens_pdaf, cvt_adj_ens_pdaf, cvt_pdaf, cvt_adj_pdaf, &
+                                 obs_op_lin_pdafomi, obs_op_adj_pdafomi, &
+                                 init_n_domains_p_pdaf, init_dim_l_pdaf, init_dim_obs_l_pdafomi, &
+                                 prepoststep_pdaf, next_observation_pdaf, outflag)
+}}}
+where all arguments, except the last one, are the names of call-back routines:
+ * [#U_collect_state_pdafcollect_state_pdaf.F90 collect_state_pdaf]: 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 `distribute_state` used in `PDAF_init_forecast` as well as here. 
+ * [#U_distribute_state_pdafdistribute_state_pdaf.F90 distribute_state_pdaf]:  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 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 obs_op_pdafomi]: The name of the user-supplied routine that acts as the observation operator on some state vector
+ * [#U_cvt_ens_pdafcvt_ens_pdaf.F90 cvt_ens_pdaf]: 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_ens_pdafcvt_adj_ens_pdaf.F90 cvt_adj_ens_pdaf]: 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_cvt_pdafcvt_pdaf.F90 cvt_pdaf]: The name of the user-supplied routine that applies the control-vector transformation (square-root of the B-matrix) on some control vector to obtain a state vector.
+ * [#U_cvt_adj_pdafcvt_adj_pdaf.F90 cvt_adj_pdaf]: The name of the user-supplied routine that applies the adjoint 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 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 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_domains_pdafinit_n_domains_pdaf.F90 init_n_domains_pdaf]: The name of the routine that provides the number of local analysis domains
+ * [#U_init_dim_l_pdafinit_dim_l_pdaf.F90 init_dim_l_pdaf]: The name of the routine that provides the state dimension for a local analysis domain
+ * [#U_init_dim_obs_l_pdafomicallback_obs_pdafomi.F90 init_dim_obs_l_pdafomi]: The name of the routine that initializes the size of the observation vector for a local analysis domain
+ * [#U_prepoststep_pdafprepoststep_ens_pdaf.F90 prepoststep_pdaf]: The name of the pre/poststep routine as in `PDAF_init_forecast`
+ * [#U_next_observation_pdafnext_observation.F90 next_observation_pdaf]: The name of a user supplied routine that initializes the variables `nsteps`, `timenow`, and `doexit`. The same routine is also used in `PDAF_init_forecast`.
+ * `status`: The integer status flag. It is zero, if the routine is exited without errors.
+
+
+=== `PDAF3_assim_offline_3dvar_all` ===
+
+For the offline mode of PDAF, the routine `PDAF3_assim_offline_3dvar_all` is used to perform the analysis step.
+The interface of the routine is identical with that of `PDAF3_assimilate_3dvar_all`, except that the user-supplied routines `U_distribute_state`, `U_collect_state` and `U_next_observation` are missing. 
+
+The interface when using one of the global filters is the following:
+{{{
+  SUBROUTINE PDAF3_assim_offline_3dvar_all(&
+                                 U_init_dim_obs_pdafomi, U_obs_op_pdafomi, &
+                                 U_cvt_ens, U_cvt_adj_ens, U_cvt, U_cvt_adj, &
+                                 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_prepoststep, outflag)
+}}}
+
+
+
+=== `PDAF3_put_state_3dvar_all` ===
+
+This routine exists for backward-compatibility. In implementations that were done before the release of PDAF V3.0, a 'put_state' routine was used for the `flexible` parallelization variant and for the offline mode. 
+When the 'flexible' implementation variant is chosen for the assimilation system, the routine. The routine `PDAF3_put_state_3dvar_all` allows to port such implemnetations to the PDAF3 interface with minimal changes.
+The interface of the routine is identical with that of `PDAF3_assimilate_3dvar_all`, except that 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 PDAF3_put_state_3dvar_all(U_collect_state, &
+                                 U_init_dim_obs_pdafomi, U_obs_op_pdafomi, &
+                                 U_cvt_ens, U_cvt_adj_ens, U_cvt, U_cvt_adj, &
+                                 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_prepoststep, outflag)
+}}}
+
+
+
+
+== User-supplied routines ==
+
+Here all user-supplied routines are described that are required in the call to the assimilation routines for hybrid 3D-Var. For some of the generic routines, we link to the page on [wiki:OnlineModifyModelforEnsembleIntegration_PDAF3 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_cvt` (cvt_pdaf.F90) ===
+
+The interface for this routine is:
+{{{
+SUBROUTINE cvt_pdaf(iter, dim_p, dim_cvec, cv_p, Vv_p)
+
+  INTEGER, INTENT(in) :: iter           ! Iteration of optimization
+  INTEGER, INTENT(in) :: dim_p          ! PE-local observation dimension
+  INTEGER, INTENT(in) :: dim_cvec       ! Dimension of control vector
+  REAL, INTENT(in)    :: cv_p(dim_cvec) ! PE-local control vector
+  REAL, INTENT(inout) :: Vv_p(dim_p)    ! PE-local result vector (state vector 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'''.
+
+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_pdaf(iter, dim_p, dim_cvec, Vv_p, cv_p)
+
+  INTEGER, INTENT(in) :: iter           ! Iteration of optimization
+  INTEGER, INTENT(in) :: dim_p          ! PE-local observation dimension
+  INTEGER, INTENT(in) :: dim_cvec       ! Dimension of control vector
+  REAL, INTENT(in)    :: Vv_p(dim_p)    ! PE-local result vector (state vector increment)
+  REAL, INTENT(inout) :: cv_p(dim_cvec) ! PE-local control 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 transposed of the square-root of the background error covariance matrix '''B'''.
+
+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. 
+For PDAF it has to provide in `dim_l` the dimension of the state vector for the local analysis domain with index `domain_p`. 
+
+In addition, for PDAFlocal the routine has to provide the index array containing the indices of the elements of the local state vector in the global (or domain-decomposed) state vector to PDAFlocal by calling `PDAFlocal_set_indices`. (in the template files, this array is called `id_lstate_in_pstate`)
+
+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. 
+  * The coordinates in `coords_l` have the same units as those used for the observations
+  * For geographic distance computations, the unit of the coordinates needs to be radian, thus (0, 2*pi) or (-pi,pi) for longitude and (-pi/2, pi/2) for latitude.
+ * Any form of local domain is possible as long as it can be describe as a single location. 
+  * If the local domain is a single grid point, `dim_l` will be the number of model variables at this grid point.
+  * The local analysis domain can also be a single vertical column of the model grid if observations are only horizontally distributed (a common situation with satellite data in the ocean). 
+   * In this case, `dim_l` will be the number of vertical grid points at this location times the number of model fields that exist in the vertical, plus possible variables at e.g. the surface. 
+   * In this case only the horizontal coordinates are used in `coords_l`.
+
+The index array `id_lstate_in_pstate` is an integer array in form of a one-dimensional vector. One initializes this vector by determining the indices of the elements of the local state vector in the global, or domain decomposed, state vector. After initializing `id_lstate_in_pstate`, one has to provided it to PDAFlocal by calling `PDAFlocal_set_indices'. The interface interface is:
+
+{{{
+SUBROUTINE PDAFlocal_set_indices(dim_l, id_lstate_in_pstate)
+
+  INTEGER, INTENT(in) :: dim_l                          ! Dimension of local state vector
+  INTEGER, INTENT(in) :: id_lstate_in_pstate(dim_l)     ! Index array for mapping
+}}}
+
+Hint for `id_lstate_in_pstate`:
+ * The initialization of the index vector `id_lstate_to_pstate` is analogous to a loop that directly performs the initialization of a local state vector. However, here only the indices are stored.
+ * See the [wiki:PDAFlocal_overview PDAFlocal overview page] for more information on the functionality of PDAFlocal.
+
+
+=== `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_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_cvtcvt_pdaf.F90 U_cvt] 
+ 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_adjcvt_adj_pdaf.F90 U_cvt_adj] 
+ 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_pdaf.F90 U_cvt] (Call to the parameterized part of the control vector transform to compute the final state vector increment)
+ 1. [#U_cvt_enscvt_ens_pdaf.F90 U_cvt_ens] (Call to the eensemble-part of 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]