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# Implementation of the analysis step for the global filters

#### Implementation Guide

- Main page
- Adaptation of the parallelization
- Initialization of PDAF
- Modifications for ensemble integration
- Implementation of the analysis step
- Implementation for global filters
- Implementation for local filters
- Implementation for ESTKF
- Implementation for LESTKF
- Implementation for ETKF
- Implementation for LETKF
- Implementation for SEIK
- Implementation for LSEIK
- Implementation for SEEK
- Implementation for EnKF
- Implementation for LEnKF
- Implementation for NETF
- Implementation for LNETF
- Implementation for PF
- Memory and timing information
- Ensemble Generation
- Diagnostics

#### Contents of this page

## Overview

With Version 1.16 of PDAF we introduced PDAF-OMI (observation module infrastructure) and with it generic routines for the analysis step, which only distinguish global and local filters. This page describes the implementation of the analysis step for global filters.

For the analysis step of the global filters different operations related to the observations are needed. 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 `PDAF_assimilate_global`

in the fully-parallel implementation (or `PDAF_put_state_global`

for the 'flexible' implementation) that was discussed before. With regard to the parallelization, all these routines 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_global. 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_global`

The general aspects of the filter specific routines `PDAF_assimilate_*`

have been described on the page Modification of the model code for the ensemble integration and its sub-page on 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 once more the full interface of the routine. Subsequently, the full set of user-supplied routines specified in the call to `PDAF_assimilate_global`

is explained.

The interface when using one of the global filters is the following:

SUBROUTINE PDAF_assimilate_global(U_collect_state, U_distribute_state, U_init_dim_obs, & U_obs_op, U_init_obs, U_prepoststep, U_next_observation, status)

with the following arguments:

- 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_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: The name of the user-supplied routine that provides the size of observation vector
- U_obs_op: The name of the user-supplied routine that acts as the observation operator on some state vector
- U_prepoststep: The name of the pre/poststep routine as in
`PDAF_get_state`

- 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`PDAF_assimilate_estkf`

is exited without errors.

`PDAF_put_state_global`

When the 'flexible' implementation variant is chosen for the assimilation system, the routine `PDAF_put_state_global`

has to be used instead of `PDAF_assimilate_global`

. The general aspects of the filter specific routines `PDAF_put_state_*`

have been described on the page 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 when using one of the global filters is the following:

SUBROUTINE PDAF_put_state_global(U_collect_state, U_init_dim_obs, & U_obs_op, U_init_obs, U_prepoststep, status)

## User-supplied routines

Here all user-supplied routines are described that are required in the call to `PDAF_assimilate_global`

. For some of the generic routines, we link to the page on 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 example implementation in `testsuite/src/dummymodel_1D`

these routines exist without the prefix, but with the extension `_pdaf.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 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 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, ESTKF).

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, ESTKF).

The interface for this routine is:

SUBROUTINE obs_op(step, dim_p, dim_obs_p, state_p, m_state_p) INTEGER, INTENT(in) :: step ! Currrent 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_prepoststep`

(prepoststep_ens_pdaf.F90)

The routine has already been described on the page on modifying the model code for the ensemble integration. For completeness, the description is repeated:

The interface of the routine is identical for all filters. However, the particular operations that are performed in the routine can be specific for each filter algorithm.

The interface for this routine is

SUBROUTINE prepoststep(step, dim_p, dim_ens, dim_ens_p, dim_obs_p, & state_p, Uinv, ens_p, flag) INTEGER, INTENT(in) :: step ! Current time step ! (When the routine is called before the analysis -step is provided.) INTEGER, INTENT(in) :: dim_p ! PE-local state dimension INTEGER, INTENT(in) :: dim_ens ! Size of state ensemble INTEGER, INTENT(in) :: dim_ens_p ! PE-local size of ensemble INTEGER, INTENT(in) :: dim_obs_p ! PE-local dimension of observation vector REAL, INTENT(inout) :: state_p(dim_p) ! PE-local forecast/analysis state ! The array 'state_p' is not generally not initialized in the case of SEIK/EnKF/ETKF/ESTKF. ! It can be used freely in this routine. REAL, INTENT(inout) :: Uinv(dim_ens-1, dim_ens-1) ! Inverse of matrix U REAL, INTENT(inout) :: ens_p(dim_p, dim_ens) ! PE-local state ensemble INTEGER, INTENT(in) :: flag ! PDAF status flag

The routine `U_prepoststep`

is called once at the beginning of the assimilation process. In addition, it is called during the assimilation cycles before the analysis step and after the ensemble transformation. The routine is called by all filter processes (that is `filterpe=1`

).

The routine provides for the user the full access to the ensemble of model states. Thus, user-controlled pre- and post-step operations can be performed. For example the forecast and the analysis states and ensemble covariance matrix can be analyzed, e.g. by computing the estimated variances. In addition, the estimates can be written to disk.

Hint:

- If a user considers to perform adjustments to the estimates (e.g. for balances), this routine is the right place for it.
- Only for the SEEK filter the state vector (
`state_p`

) is initialized. For all other filters, the array is allocated, but it can be used freely during the execution of`U_prepoststep`

. - The interface has a difference for ETKF and ESTKF: For the ETKF, the array
`Uinv`

has size`dim_ens`

x`dim_ens`

. In contrast it has size`dim_ens-1`

x`dim_ens-1`

for the ESTKF. (For most cases, this will be irrelevant, because most usually the ensemble array`ens_p`

is used for computations, rather than`Uinv`

. Only for the SEIK filter with fixed covariance matrix,`Uinv`

is required to compute the estimate analysis error. The fixed covariance matrix mode is not available for the ETKF or ESTKF.) - The interface through which
`U_prepoststep`

is called does not include the array of smoothed ensembles. In order to access the smoother ensemble array one has to set a pointer to it using a call to the routine`PDAF_get_smootherens`

(see page on auxiliary routines)

`U_next_observation`

(next_observation_pdaf.F90)

This routine is independent of the filter algorithm used. See the page on 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_global`

. 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`

. How this information is initialized is described in the documentation of OMI.

Before the analysis step is called the following routine is executed:

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:

- U_prepoststep (Call to act on the forecast ensemble, called with negative value of the time step)
- U_init_dim_obs
- U_obs_op (multiple calls for each ensemble members)
- U_prodRinvA
- U_prepoststep (Call to act on the analysis ensemble, called with (positive) value of the time step)

In case of the routine `PDAF_assimilate_global`

, the following routines are executed after the analysis step: