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# Implementation of the Analysis step for the SEIK filter

#### Implementation Guide

#### Contents of this page

## Overview

For the analysis step of the LSEIK filter several 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_lseik`

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

) only.

The following user-supplied routines for the SEIK filter are described on this page. (For completeness, we also repeat the generic routines that were described on the page Modification of the model core for the ensemble integration.

- U_init_dim_obs_full: The name of the user-supplied routine that provides the size of observation vector
- U_obs_op_full: The name of the user-supplied routine that acts as the observation operator on some state vector
- U_init_obs_full: The name of the user-supplied routine that initializes the vector of observations
- U_prodRinvA_local: 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 LSEIK filter.
- 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)

Below the names of the corresponding routines in the template directory are provided in parentheses. The the routines in the example implementation have the same name but include '`_dummy_D`

' in the name.

## PDAF_put_state_lseik

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

have been described on the page Modification of the model core for the ensemble integration.
The interface for the routine `PDAF_put_state_lseik`

contains routine names for routines that operate on the local analysis domains (marked by `_l`

at then end of the routine name), as well as routines that consider all available observations to be considered within some sub-domain of the model (marked by `_f`

('full') at then end of the routine name). In case of a serial execution of the assimilation program, this will be all available observation. However, if the program is execute with parallelization, this might be a limited number of observations.

The interface when using the LSEIK filter is the following:

SUBROUTINE PDAF_put_state_lseik(U_collect_state, U_init_dim_obs_f, U_obs_op_f, & U_init_obs_f, U_init_obs_l, U_prepoststep, U_prodRinvA_l, U_init_n_domains, & U_init_dim_l, U_init_dim_obs_l, U_g2l_state, U_l2g_state, U_g2l_obs, & U_init_obsvar, U_init_obsvar_l, 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 ensembel 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_f`

: The name of the user-supplied routine that provides the size of observation vector`U_obs_op_f`

: The name of the user-supplied routine that acts as the observation operator on some state vector`U_init_obs_f`

: The name of the user-supplied routine that initializes the vector of observations`U_init_obs_l`

: The name of the user-supplied routine that initializes the vector of observations for a local analysis domain`U_prepoststep`

: The name of the pre/poststep routine as in`PDAF_get_state`

`U_prodRinvA_l`

: 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 filter.`U_init_n_domains`

: The name of the routine that provides the number of local analysis domains`U_init_dim_l`

: The name of the routine that provides the state domains for a local analysis domain`U_init_dim_obs_l`

: The name of the routine that initializes the size of the observation vector for a local analysis domain`U_g2l_state`

: The name of the routine that initializes a local state vector from the global state vector`U_l2g_state`

: The name of the routine that initializes the part of the global state vector corresponding to the provided local state vector`U_g2l_obs`

: The name of the routine that initialized a local observation vector from a full observation vector`U_init_obsvar`

: The name of the user-supplied routine that provides a global mean observation error variance to PDAF (This routine will only be executed, if an adaptive forgetting factor is used)`U_init_obsvar_l`

: The name of the user-supplied routine that provides a mean observation error variance for the local analysis domain 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_get_state is exited without errors.

## User-supplied routines

Here, only the user-supplied routines are discussed that are required at this stage of the implementation (that is, the ensemble integration). The routines that are required to conduct the analysis step of some filter, are described in the main section about the implementation of the analysis step. For testing (see Compilation and testing), all routines need to exist, but only those described here in detail need to be implemented with functionality.

To indicate user-supplied routines we use the prefix `U_`

. In the template directory `templates/`

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

the routines exist without the prefix, but with the extension `_dummy_D.F90`

. In the section titles below we provide the name of the template file in parentheses.

`U_next_observation`

(next_observation.F90)

The interface for this routine is

SUBROUTINE U_next_obs(stepnow, nsteps, doexit, timenow) INTEGER, INTENT(in) :: stepnow ! Number of the current time step INTEGER, INTENT(out) :: nsteps ! Number of time steps until next obs INTEGER, INTENT(out) :: doexit ! Whether to exit forecasting (1 for exit) REAL, INTENT(out) :: timenow ! Current model (physical) time

The routine is called once at the beginning of each forecast phase. It is executed by all processes that participate in the model integrations.

Based on the information of the current time step, the routine has to define the number of time steps `nsteps`

for the next forecast phase. In addition, the flag `doexit`

has to be initialized to provide the information if the external ensemble loop can be exited. `timenow`

is the current model time. This variable should also be initialized. It is particularly important, if an ensemble task integrates more than one model state. In this case `timenow`

can be used to correctly jump back in time.

Some hints:

- If the time interval between successive observations is known,
`nsteps`

can be simply initialized by dividing the time interval by the size of the time step `doexit`

should be 0 to continue the assimilation process. In most cases`doexit`

is set to 1, when`PDAF_get_state`

is called after the last analysis for which observations are available.- At the first call to
`U_next_obs`

the variable`timenow`

should be initialized with the current model time. At the next call a forecast phase has been completed. Thus, the new value of`timenow`

follows from the timer interval for the previous forecast phase.

`U_distribute_state`

(distribute_state.F90)

The interface for this routine is

SUBROUTINE distribute_state(dim_p, state_p) INTEGER, INTENT(in) :: dim_p ! State dimension for PE-local model sub-domain REAL, INTENT(inout) :: state_p(dim_p) ! State vector for PE-local model sub-domain

This routine is called during the forecast phase as many times as there are states to be integrated by a model task. Again, the routine is executed by all processes that belong to model tasks.

When the routine is called a state vector `state_p`

and its size `dim_p`

are provided. As the user has defined how the model fields are stored in the state vector, one can initialize the model fields from this information. If the model is not parallelized, `state_p`

will contain a full state vector. If the model is parallelized using domain decomposition, `state_p`

will contain the part of the state vector that corresponds to the model sub-domain for the calling process.

Some hints:

- If the state vector does not include all model fields, it can be useful to keep a separate array to store those additional fields. This array has to be kept separate from PDAF, but can be defined using a module like
`mod_assimilation`

.

`U_prepoststep`

(prepoststep_seik.F90)

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. Here, we exemplify the interface on the example of the SEIK filter.

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. ! It can be used freely here. 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.

`U_collect_state`

(collect_state.F90)

The interface for this routine is

SUBROUTINE collect_state(dim_p, state_p) INTEGER, INTENT(in) :: dim_p ! State dimension for PE-local model sub-domain REAL, INTENT(inout) :: state_p(dim_p) ! State vector for PE-local model sub-domain

This routine is called during the forecast phase as many times as there are states to be integrated by a model task. It is called at the end of the integration of a member state of the ensemble. The routine is executed by all processes that belong to model tasks.

When the routine is called, a state vector `state_p`

and its size `dim_p`

are provided. The operation to be performed in this routine is inverse to that of the routine `U_distribute_state`

. That is, the state vector `state_p`

has to be initialized from the model fields. If the model is not parallelized, `state_p`

will contain a full state vector. If the model is parallelized using domain decomposition, `state_p`

will contain the part of the state vector that corresponds to the model sub-domain for the calling process.

Some hints:

- If the state vector does not include all model fields, it can be useful to keep a separate array to store those additional fields. This array has to be kept separate from PDAF, but can be defined using a module like
`mod_assimilation`

.