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# Implementation of the Analysis step for the LEnKF (Localized Ensemble Kalman Filter) with OMI

#### Implementation Guide

#### Contents of this page

With Version 1.16 of PDAF we introduced PDAF-OMI (observation module infrastructure). With OMI, a smaller number of routines needs to be supplied by the user than in the previous implementation approach.

## Overview

For the analysis step of the LEnKF 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 `PDAFomi_put_state_lenkf`

for the fully-parallel implementation (or `PDAFomi_put_state_lenkf`

for the 'flexible' implementation). 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 `PDAFomi_put_state_lenkf`

. 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 LEnKF implemented in PDAF follows the original LEnKF by Evensen (1994) including the correction for perturbed observations (Burgers et al. 1998). The LEnKF implemented in PDAF is reviewed by Nerger et al (2005) and described in more detail by Nerger (2004). The localization is covariance lozalization of PH^{T and HPH}T as described in Houtekamer & Mitchell (2001) (See the page on publications and presentations for publications and presenations involving and about PDAF)

In our studies (Nerger et al. 2005, Nerger 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 and ESTKF after these comparison studies. For real applications, we generally recommend using ESTKF or ETKF, or their local variants LESTKF or LETKF. However, the EnKF/LEnKF might have a good performance if very large ensemble can be used as this reduces the sampling errors.

`PDAFomi_assimilate_lenkf`

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 `

PDAFomi_assimilate_lenkf` is explained.

The interface when using the LEnKF is the following:

SUBROUTINE PDAFomi_assimilate_lenkf(U_collect_state, U_distribute_state, & U_init_dim_obs, U_obs_op, & U_prepoststep, U_localize_covar, & 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_localize_covar: Apply covariance localization to the matrices HP and HPH
^{T} - 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_lenkf`

is exited without errors.

`PDAFomi_put_state_lenkf`

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

has to be used instead of `PDAFomi_assimilate_lenkf`

. 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 `PDAFomi_assimilate_lenkf`

with the exception the specification of the user-supplied routines `U_distribute_state`

and `U_next_observation`

are missing.

The interface when using the LEnKF is the following:

SUBROUTINE PDAFomi_put_state_lenkf(U_collect_state, & U_init_dim_obs, U_obs_op, & U_prepoststep, U_localize, & status)

## User-supplied routines

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

. 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`

(callback_obs_pdafomi.F90)

This routine is used by all global filter algorithms (SEEK, SEIK, EnKF, ETKF, NETF, PF) and by the LEnKF.

The interface for this routine is:

SUBROUTINE init_dim_obs_pdafomi(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.

With PDAF-OMI, the routine just calls a routine from the observation module for each observation type.

`U_obs_op`

(calllback_obs_pdafomi.F90)

This routine is used by all global filter algorithms (SEEK, SEIK, EnKF, ETKF, NETF, PF) and the LEnKF.

The interface for this routine is:

SUBROUTINE obs_op_pdafomi(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.

With PDAF-OMI, the routine just calls a routine from the observation module for each observation type.

`U_prepoststep`

(prepoststep_ens_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 LEnKF

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 LEnKF 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/LEnKF). In general it should be possible to use an identical pre/poststep routine for all these filters. - 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_localize`

(callback_obs_pdafomi.F90)

This routine is only used for the LEnKF.

The interface for this routine is:

SUBROUTINE localize_covar_pdafomi(dim_p, dim_obs, HP, HPH) INTEGER, INTENT(in) :: dim_p ! PE-local state dimension INTEGER, INTENT(in) :: dim_obs ! Dimension of global observation vector REAL, INTENT(inout) :: HP(dim_obs, dim_p) ! Matrix HP REAL, INTENT(inout) :: HPH(dim_obs, dim_obs) ! Matrix HPH^T^

The routine is called during the analysis step and has to apply the element-wise Schur product for the covariance localization of the two matrices**HP** and **HPH ^{T}**, which are provided as input/output arguments.

With PDAF-OMI, the routine just calls a routine provided by PDAF-OMI.

`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

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

. 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_localize
- 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)

In case of the routine `PDAFomi_assimilate_enkf`

, the following routines are executed after the analysis step: