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# Implementation of the Analysis step for the ESTKF

#### 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
- Implementation for ESTKF
- Implementation for LESTKF
- Memory and timing information

#### Contents of this page

## Overview

With Version 1.8 of PDAF, the ESTKF [Error Subspace Transform Kalman Filter] algorithm has been introduced. The user-supplied routines required for the ESTKF are identical to those required for the SEIK filter.

For the analysis step of the ESTKF different operations related to the observations are needed. These operations are requested by PDAF by call-back routines supplied by the user. 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_estkf`

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_put_state_estkf. 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 ESTKF and the ETKF (Ensemble Transform Kalman Filter) are very similar. For this reason, the interface to the user-supplied routines is almost identical. Depending on the implementation it can be possible to use identical routines for the ESTKF and the ETKF. Differences are marked in the text below.

`PDAF_put_state_estkf`

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 once more the full interface. Subsequently, the full set of user-supplied routines specified in the call to `PDAF_put_state_estkf`

is explained.

The interface when using the ESTKF is the following:

SUBROUTINE PDAF_put_state_estkf(U_collect_state, U_init_dim_obs, U_obs_op, & U_init_obs, U_prepoststep, U_prodRinvA, U_init_obsvar, 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_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 SEIK, ETKF, and ESTKF algorithms.
- U_init_obsvar: The name of the user-supplied routine that provides a mean observation error variance to PDAF (This routine will only be executed, if an adaptive forgetting factor is used)
`status`

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

is exited without errors.

## User-supplied routines

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

. 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 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, 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_init_obs`

(init_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_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_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.)

`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, ETKF, ESTKF).

The interface for this routine is:

SUBROUTINE prodRinvA(step, dim_obs_p, rank, 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) :: rank ! Rank of initial covariance matrix REAL, INTENT(in) :: obs_p(dim_obs_p) ! PE-local vector of observations REAL, INTENT(in) :: A_p(dim_obs_p,rank) ! Input matrix from analysis routine REAL, INTENT(out) :: C_p(dim_obs_p,rank) ! 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 ESTKF 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 ESTKF and ETKF: For ETKF the third argument is the ensemble size (
`dim_ens`

), while for the ESTKF it is the rank (`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 ESTKF. (Practically, one can usually ignore this difference as the fourth argument of the interface can be named arbitrarily in the routine.)

`U_init_obsvar`

(init_obsvar_pdaf.F90)

This routine is used by the global filter algorithms SEIK, ETKF, and ESTKF as well as the local filters LSEIK, LETKF, ad LESTKF. The routine is only called if the adaptive forgetting factor is used (`type_forget=1`

in the example impementation).

The interface for this routine is:

SUBROUTINE init_obsvar(step, dim_obs_p, obs_p, meanvar) INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: dim_obs_p ! PE-local dimension of observation vector REAL, INTENT(in) :: obs_p(dim_obs_p) ! PE-local observation vector REAL, INTENT(out) :: meanvar ! Mean observation error variance

The routine is called in the global filters during the analysis or by the routine that computes an adaptive forgetting factor (PDAF_set_forget). The routine has to initialize the mean observation error variance. For the global filters this should be the global mean.

Hints:

- For a model with domain-decomposition one might use the mean variance for the model sub-domain of the calling process. Alternatively one can compute a mean variance for the full model domain using MPI communication (e.g. the function
`MPI_allreduce`

). - The observation vector
`obs_p`

is provided to the routine for the case that the observation error variance is relative to the value of the observations.

## Execution order of user-supplied routines

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

. 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 (A single call to operate on the ensemble mean state)
- U_init_obs
- U_obs_op (
`dim_ens`

calls: one call for each ensemble member) - U_init_obsvar (Only executed, if the adaptive forgetting factor is used (
`type_forget=1`

in the example implemention)) - U_prodRinvA
- U_prepoststep (Call to act on the analysis ensemble, called with (positive) value of the time step)