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# Implementation of the Analysis step for the EnKF (Ensemble Kalman Filter)

#### Implementation Guide

- Main page
- Adaptation of the parallelization
- Initialization of PDAF
- Modifications for ensemble integration
- Implementation of the analysis step
- Implementation for SEIK
- Implementation for LSEIK
- Implementation for ETKF
- Implementation for LETKF
- Implementation for SEEK
- Implementation for EnKF
- Memory and timing information

#### Contents of this page

## Overview

For the analysis step of the EnKF different operations related to the observations are needed. 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. This procedure should simplify the implementation. The names of the required routines are specified in the call to the routine `PDAF_put_state_enkf`

. With regard to the parallelization, all these routines are executed by the filter processes (`filterpe=1`

) only.

For completeness we discuss here all user-supplied routines that are specified in the interface to PDAF_put_state_enkf. Thus, some of the user-supplied routines that are explained on the page explaining the modification of the model code for the ensemble integration are repeated here.

The EnKF implemented in PDAF follows the original EnKF by Evensen (1994) including the correction for perturbed observations (Burgers et al. 1998). The EnKF implemented in PDAF is reviewed by Nerger et al (2005) and described in more detail by Nerger (2004). (See the page on publications and presentations for publications and presenations involving and about PDAF)

There is no localized variant of the EnKF in PDAF. In our studies (Nerger et al. 2005, Neger et al. 2007), the EnKF showed performance deficiencies compared to the SEIK filter. Due to this, we focused more on the SEIK filter and the ETKF after these comparison studies. For real applications, we generally recommend using SEIK or ETKF, or their local variants LSEIK or LETKF.

`PDAF_put_state_enkf`

The general espects of the filter specific routines `PDAF_put_state_*`

have been described on the page Modification of the model code for the ensemble integration. Here, we list the full interface specifically for the EnKF. Subsequently, the full set of user-supplied routines specified in the call to `PDAF_put_state_enkf`

is explained.

The interface when using the EnKF is the following:

SUBROUTINE PDAF_put_state_enkf(U_collect_state, U_init_dim_obs, U_obs_op, & U_init_obs, U_prepoststep, U_add_obs_err, U_init_obscovar, 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`

- 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_init_obs: The name of the user-supplied routine that initializes the vector of observations
- U_prepoststep: The name of the pre/poststep routine as in
`PDAF_get_state`

- U_add_obs_err: The name of the user-supplied routine that adds the observation error covariance matrix to the ensemble covariance matrix projected onto the observation space.
- U_init_obscovar: The name of the user-supplied routine that initializes the observation error covariance matrix.
`status`

: The integer status flag. It is zero, if`PDAF_put_state_enkf`

is exited without errors.

## User-supplied routines

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

. 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/`

these routines are provided in files with the routine's name without this prefix. In the example implementation in `testsuite/src/dummymodel_1D`

, the 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 modifying the model code for the ensemble integration 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).

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).

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_init_obs`

(init_obs_pdaf.F90)

This routine is used by all global filter algorithms (SEEK, SEIK, EnKF, ETKF).

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_prepoststep`

(prepoststep_enkf_pdaf.F90)

The general aspects of this routines have already been described on the page on modifying the model code for the ensemble integration for the SEIK filter. For completeness, the description is repeated specifically for the EnKF:

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

The interface for this routine is for the EnKF

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. ! It can be used freely in this routine. REAL, INTENT(inout) :: Uinv(1, 1) ! Not used not EnKF 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 array
`Uinv`

is not used in the EnKF. Internally to PDAF, it is allocated to be of size (1,1). - Apart from the size of the array
`Uinv`

, the interface is identical for all ensemble filters (SEIK/ETKF/EnKF/LSEIK/LETKF). In general it should be possible to use an identical pre/poststep routine for all these filters.

`U_add_obs_err`

(add_obs_err_pdaf.F90)

This routine is only used for the EnKF.

The interface for this routine is:

SUBROUTINE add_obs_err(step, dim_obs, C) INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: dim_obs ! Dimension of obs. vector REAL, INTENT(inout) :: C(dim_obs, dim_obs) ! Matrix to that the observation ! error covariance matrix is added

The routine is called during the analysis step. During the analysis step of the EnKF, the projection of the ensemble covariance onto the observation space is computed. This matrix is provided to the routine as `C_p`

. The routine has to add the observation error covariance matrix to `C_p`

.

The operation is for the global observation space. Thus, it is independent of whether the filter is executed with or without parallelization.

Hints:

- The routine does not require that the observation error covariance matrix is added as a full matrix. If the matrix is diagonal, only the diagonal elements have to be added.

`U_init_obscovar`

(init_obscovar_pdaf.F90)

This routine is only used for the EnKF.

The interface for this routine is:

SUBROUTINE init_obscovar(step, dim_obs, dim_obs_p, covar, m_state_p, & isdiag) INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: dim_obs ! Dimension of observation vector INTEGER, INTENT(in) :: dim_obs_p ! PE-local dimension of observation vector REAL, INTENT(out) :: covar(dim_obs, dim_obs) ! Observation error covariance matrix REAL, INTENT(in) :: m_state_p(dim_obs_p) ! PE-local observation vector LOGICAL, INTENT(out) :: isdiag ! Whether the observation error covar. matrix is diagonal

The routine is called during the analysis step and is required for the generation of an ensemble of observations. It has to initialize the global observation error covariance matrix `covar`

. In addition, the flag `isdiag`

has to be initialized to provide the information, whether the observation error covariance matrix is diagonal.

The operation is for the global observation space. Thus, it is independent of whether the filter is executed with or without parallelization.

Hints:

- The local observation vector
`m_state_p`

is provided to the routine for the case that the observation errors are relative to the value of the observation.

## Execution order of user-supplied routines

For the EnKF, the user-supplied routines are essentially executed in the order they are listed in the interface to `PDAF_put_state_enkf`

. The order can be important as some routines can perform preparatory work for later routines. For example, `U_init_dim_obs`

can prepare 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:

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 (
`dim_ens`

calls: one call for each ensemble member) - U_add_obs_err
- U_init_obs
- U_init_obscovar
- U_obs_op (
`dim_ens`

calls: one call for each ensemble member, repeated to reduce storage) - U_prepoststep (Call to act on the analysis ensemble, called with (positive) value of the time step)