= Implementation of the Analysis Step for 3D-Var without using OMI = 


{{{
#!html
<div class="wiki-toc">
<h4>Implementation Guide</h4>
<ol><li><a href="ImplementationGuide">Main page</a></li>
<li><a href="AdaptParallelization">Adaptation of the parallelization</a></li>
<li><a href="InitPdaf">Initialization of PDAF</a></li>
<li><a href="ModifyModelforEnsembleIntegration">Modifications for ensemble integration</a></li>
<li><a href="ImplementationofAnalysisStep">Implementation of the analysis step</a></li>
<ol>
<li><a href="ImplementAnalysisestkf">Implementation for ESTKF</a></li>
<li><a href="ImplementAnalysislestkf">Implementation for LESTKF</a></li>
<li><a href="ImplementAnalysisetkf">Implementation for ETKF</a></li>
<li>Implementation for LETKF</li>
<li><a href="ImplementAnalysisseik">Implementation for SEIK</a></li>
<li><a href="ImplementAnalysislseik">Implementation for LSEIK</a></li>
<li><a href="ImplementAnalysisseek">Implementation for SEEK</a></li>
<li><a href="ImplementAnalysisenkf">Implementation for EnKF</a></li>
<li><a href="ImplementAnalysislenkf">Implementation for LEnKF</a></li>
<li><a href="ImplementAnalysisnetf">Implementation for NETF</a></li>
<li><a href="ImplementAnalysislnetf">Implementation for LNETF</a></li>
<li><a href="ImplementAnalysispf">Implementation for PF</a></li>
<li>Implementation for 3D-Var</li>
<li><a href="ImplementAnalysis_3DEnVar_classical">Implementation for 3D Ensemble Var</a></li>
<li><a href="ImplementAnalysis_Hyb3DVar_classical">Implementation for Hybrid 3D-Var</a></li>
</ol>
<li><a href="AddingMemoryandTimingInformation">Memory and timing information</a></li>
<li><a href="EnsembleGeneration">Ensemble Generation</a></li>
<li><a href="DataAssimilationDiagnostics">Diagnostics</a></li>
</ol>
</div>
}}}

[[PageOutline(2-3,Contents of this page)]]

|| This page describes the implementation of the analysis step without using PDAF-OMI. Please see the [wiki:ImplementationofAnalysisStep page on the analysis with OMI] for the more modern and efficient implementation variant using PDAF-OMI. ||

== 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 parameterized 3D-Var in the classical way (without 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 calling user-supplied routines. Intentionally, the operations are split into separate routines in order to keep the operations rather elementary as this procedure should simplify the implementation. The names of the required routines are specified in the call to the routine `PDAF_assimilate_3dvar` in the fully-parallel implementation (or `PDAF_put_state_3dvar` for the 'flexible' implementation) described below. With regard to the parallelization, all these routines (except `U_collect_state`) are executed by the filter processes (`filterpe=.true.`) only.

For completeness we discuss here all user-supplied routines that are specified in the interface to `PDAF_assimilate_3dvar`. 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.


== `PDAF_assimilate_3dvar` ==

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.

The interface for using the parameterized 3D-Var is:
{{{
  SUBROUTINE PDAF_assimilate_3dvar(U_collect_state, U_distribute_state, &
                                 U_init_dim_obs, U_obs_op, U_init_obs, U_prodRinvA, &
                                 U_cvt, U_cvt_adj, U_obs_op_lin, U_obs_op_adj, &
                                 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_obsinit_dim_obs_pdaf.F90 U_init_dim_obs]: The name of the user-supplied routine that provides the size of observation vector
 * [#U_obs_opobs_op_pdaf.F90 U_obs_op]: The name of the user-supplied routine that acts as the observation operator on some state vector
 * [#U_init_obsinit_obs_pdaf.F90 U_init_obs]: The name of the user-supplied routine that initializes the vector of observations
 * [#U_prodRinvAprodrinva_pdaf.F90 U_prodRinvA]: The name of the user-supplied routine that computes the product of the inverse of the observation error covariance matrix with some matrix provided to the routine by PDAF. This operation occurs during the analysis step of the ETKF.
 * [#U_cvtcvt_pdaf.F90 U_cvt]: 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_adjcvt_adj_pdaf.F90 U_cvt_adj]: 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_linobs_op_lin_pdaf.F90 U_obs_op_lin]: The name of the user-supplied routine that acts as the linearized observation operator on some state vector
 * [#U_obs_op_adjobs_op_adj_pdaf.F90 U_obs_op_adj]: 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 the routine is exited without errors.


== `PDAF_put_state_3dvar` ==

When the 'flexible' implementation variant is chosen for the assimilation system, the routine `PDAF_put_state_3dvar` has to be used instead of `PDAF_assimilate_3dvar`. 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_global` with the exception the specification of the user-supplied routines `U_distribute_state` and `U_next_observation` are missing. 

The interface for using the parameterized 3D-Var is:
{{{
  SUBROUTINE PDAF_put_state_3dvar(collect_state_pdaf, &
                                 U_init_dim_obs, U_obs_op, U_init_obs, U_prodRinvA, &
                                 U_cvt, U_cvt_adj, U_obs_op_lin, U_obs_op_adj, &
                                 prepoststep_pdaf, outflag)
}}}

== User-supplied routines ==

Here all user-supplied routines are described that are required in the call to `PDAF_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 tutorials in `tutorial/` and in the template directory `templates/` these routines exist without the prefix, but with the extension `_pdaf`. The files are named correspondingly. 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` (init_dim_obs_pdaf.F90) ===

This routine is used by all global filter algorithms (SEEK, SEIK, EnKF, ETKF) and the 3D-Var methods.

The interface for this routine is:
{{{
SUBROUTINE init_dim_obs(step, dim_obs_p)

  INTEGER, INTENT(in)  :: step       ! Current time step
  INTEGER, INTENT(out) :: dim_obs_p  ! Dimension of observation vector
}}}

The routine is called at the beginning of each analysis step.  It has to initialize the size `dim_obs_p` of the observation vector according to the current time step. Without parallelization `dim_obs_p` will be the size for the full model domain. When a domain-decomposed model is used, `dim_obs_p` will be the size of the observation vector for the sub-domain of the calling process.

Some hints:
 * It can be useful to not only determine the size of the observation vector at this point. One can also already gather information about the locations of the observations, which will be used later, e.g. to implement the observation operator. An array for the locations can be defined in a module like `mod_assimilation` of the example implementation.


=== `U_obs_op` (obs_op_pdaf.F90) ===

This routine is used by all global filter algorithms (SEEK, SEIK, EnKF, ETKF) and the 3D-Var methods.

The interface for this routine is:
{{{
SUBROUTINE obs_op(step, dim_p, dim_obs_p, state_p, m_state_p)

  INTEGER, INTENT(in) :: step               ! Current time step
  INTEGER, INTENT(in) :: dim_p              ! PE-local dimension of state
  INTEGER, INTENT(in) :: dim_obs_p          ! Dimension of observed state
  REAL, INTENT(in)    :: state_p(dim_p)     ! PE-local model state
  REAL, INTENT(out) :: m_state_p(dim_obs_p) ! PE-local observed state
}}}

The routine is called during the analysis step. It has to perform the operation of the observation operator acting on a state vector that is provided as `state_p`. The observed state has to be returned in `m_state_p`.

For a model using domain decomposition, the operation is on the PE-local sub-domain of the model and has to provide the observed sub-state for the PE-local domain.

Hint:
 * If the observation operator involves a global operation, e.g. some global integration, while using domain-decomposition one has to gather the information from the other model domains using MPI communication. 


=== `U_init_obs` (init_obs_pdaf.F90) ===

This routine is used by all global filter algorithms (SEEK, SEIK, EnKF, ETKF) and the 3D-Var methods.

The interface for this routine is:
{{{
SUBROUTINE init_obs(step, dim_obs_p, observation_p)

  INTEGER, INTENT(in) :: step             ! Current time step
  INTEGER, INTENT(in) :: dim_obs_p        ! PE-local dimension of obs. vector
  REAL, INTENT(out)   :: observation_p(dim_obs_p) ! PE-local observation vector
}}}

The routine is called during the analysis step. 
It has to provide the vector of observations in `observation_p` for the current time step. 

For a model using domain decomposition, the vector of observations that exist on the model sub-domain for the calling process has to be initialized.




=== `U_prodRinvA` (prodrinva_pdaf.F90) ===

This routine is used by all filter algorithms that use the inverse of the observation error covariance matrix (SEEK, SEIK, and ETKF) and the 3D-Var methods.

The interface for this routine is:
{{{
SUBROUTINE prodRinvA(step, dim_obs_p, dim_ens, obs_p, A_p, C_p)

  INTEGER, INTENT(in) :: step                ! Current time step
  INTEGER, INTENT(in) :: dim_obs_p           ! PE-local dimension of obs. vector
  INTEGER, INTENT(in) :: dim_ens             ! Ensemble size
  REAL, INTENT(in)    :: obs_p(dim_obs_p)    ! PE-local vector of observations
  REAL, INTENT(in)    :: A_p(dim_obs_p, dim_ens) ! Input matrix from analysis routine
  REAL, INTENT(out)   :: C_p(dim_obs_p, dim_ens) ! Output matrix
}}}

The routine is called during the analysis step. In the algorithms the product of the inverse of the observation error covariance matrix with some matrix has to be computed. For the ETKF, this matrix holds the observed part of the ensemble perturbations. The matrix is provided as `A_p`. The product has to be given as `C_p`.

For a model with domain decomposition, `A_p` contains the part of the matrix that resides on the model sub-domain of the calling process. The product has to be computed for this sub-domain, too.

Hints:
 * The routine does not require that the product is implemented as a real matrix-matrix product. Rather, the product can be implemented in its most efficient form. For example, if the observation error covariance matrix is diagonal, only the multiplication of the diagonal with matrix `A_p` has to be implemented.
 * The observation vector `obs_p` is provided through the interface for cases where the observation error variance is relative to the actual value of the observations.
 * The interface has a difference for SEIK and ETKF: For ETKF the third argument is the ensemble size (`dim_ens`), while for SEIK it is the rank of the covariance matrix (usually ensemble size minus one). In addition, the second dimension of `A_p` and `C_p` has size `dim_ens` for ETKF, while it is `rank` for the SEIK filter.  (Practically, one can usually ignore this difference as the fourth argument of the interface can be named arbitrarily in the routine.)




=== `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` (obs_op_lin_pdaf.F90) ===

This routine is used by all 3D-Var methods.

The interface for this routine is:
{{{
SUBROUTINE obs_op_lin(step, dim_p, dim_obs_p, state_p, m_state_p)

  INTEGER, INTENT(in) :: step               ! Current time step
  INTEGER, INTENT(in) :: dim_p              ! PE-local dimension of state
  INTEGER, INTENT(in) :: dim_obs_p          ! Dimension of observed state
  REAL, INTENT(in)    :: state_p(dim_p)     ! PE-local model state
  REAL, INTENT(out) :: m_state_p(dim_obs_p) ! PE-local observed state
}}}

The routine is called during the analysis step. It has to perform the operation of the linearized observation operator acting on a state vector increment that is provided as `state_p`. The observed state has to be returned in `m_state_p`.

For a model using domain decomposition, the operation is on the PE-local sub-domain of the model and has to provide the observed sub-state for the PE-local domain.

Hint:
 * If the observation operator involves a global operation, e.g. some global integration, while using domain-decomposition one has to gather the information from the other model domains using MPI communication. 


=== `U_obs_op_adj` (obs_op_adj_pdaf.F90) ===

This routine is used by all 3D-Var methods.

The interface for this routine is:
{{{
SUBROUTINE obs_op_adj(step, dim_p, dim_obs_p, state_p, m_state_p)

  INTEGER, INTENT(in) :: step                 ! Current time step
  INTEGER, INTENT(in) :: dim_p                ! PE-local dimension of state
  INTEGER, INTENT(in) :: dim_obs_p            ! Dimension of observed state
  REAL, INTENT(in)    :: m_state_p(dim_obs_p) ! PE-local observed state
  REAL, INTENT(out)   :: state_p(dim_p)       ! PE-local model state
}}}

The routine is called during the analysis step. It has to perform the operation of the adjoint observation operator acting on a vector in observation space that is provided as m_state_p. The resulting state vector has to be returned in `m_state_p`.

For a model using domain decomposition, the operation is on the PE-local sub-domain of the model and has to provide the observed sub-state for the PE-local domain.

Hint:
 * If the observation operator involves a global operation, e.g. some global integration, while using domain-decomposition one has to gather the information from the other model domains using MPI communication. 



=== `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 `PDAF_assimilate_3dvar`. The order can be important as some routines can perform preparatory work for later routines. For example, `U_init_dim_obs` prepares an index array that provides the information for executing the observation operator in `U_obs_op`. 

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_obsinit_dim_obs_pdaf.F90 U_init_dim_obs]
 1. [#U_obs_opobs_op_pdaf.F90 U_obs_op] 
 1. [#U_init_obsinit_obs_pdaf.F90 U_init_obs] 

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_obs_op_linobs_op_lin_pdaf.F90 U_obs_op_lin]
 1. [#U_prodRinvAprodrinva_pdaf.F90 U_prodRinvA]
 1. [#U_obs_op_adjobs_op_adj_pdaf.F90 U_obs_op_adj]
 1. [#U_cvt_adjcvt_adj_pdaf.F90 U_cvt_adj] 

After the iterative optimization the following routines are executes to complte the analysis step:
 1. [#U_cvtcvt_pdaf.F90 U_cvt] (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)

In case of the routine `PDAF_assimilate_3dvar`, 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]