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# Implementation of the Analysis step for the LSEIK 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 as 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 several routine names for routines that operate on the local analysis domains (marked by `_l`

at then end of the routine name). In addition, there are names for routines that consider all available observations required to perform local analyses with LSEIK within some sub-domain of a domain-decomposed model (marked by `_full`

at then end of the routine name). In case of a serial execution of the assimilation program, this will be all globally available observations. However, if the program is executed with parallelization, this might be a smaller set of observations. To explain the difference it is assumed for simplicity that a local analysis domain consists of a single vertical column of the model grid. In addition, we assume that the domain decomposition splits the global model domain by vertical boundaries as is typical for ocean models and that the observations are spatially distributed observations of model fields that are part of the state vector. Under these assumptions, the situation is the following: When a model uses domain decomposition, the LSEIK filter is executed such that for each model sub-domain a loop over all local analysis domains (e.g. vertical columns) that belong to the model sub-domain is performed. For the update of each single vertical column observations from some larger domain surrounding the vertical column are considered. If the influence radius for the observations is sufficiently small there will be vertical columns for which all relevant observations reside inside the model sub-domain of the process. However, if a vertical column is considered that is located close to the boundary to the model sub-domain, there will be some observations that don't belong spatially to the local model sub-domain, but to a neighboring model sub-domain. These observations nonetheless are required on the local model sub-domain. Thus, a simple way to handle this situation is to initialize for each process all globally available observations, together with their coordinates. While this method is simple, it can also become inefficient if the assimilation program is executed using a large number of processes. As for an initial implementation, it is usually not the concern to obtain the highest parallel efficiency, the description below assumes that 'full' refers to the global model domain.

The interface when using the LSEIK filter is the following:

SUBROUTINE PDAF_put_state_lseik(U_collect_state, U_init_dim_obs_full, U_obs_op_full, U_init_obs_full, & 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 all user-supplied routines are described that are required in the call to `PDAF_put_state_lseik`

. 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 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_collect_state`

(collect_state.F90)

See here for the description of this routine.

`U_init_dim_obs_full`

(init_dim_obs_full.F90)

This routine is used by all local filter algorithms (LSEIK, LETKF).

The interface for this routine is:

SUBROUTINE init_dim_obs_f(step, dim_obs_f) INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(out) :: dim_obs_f ! Dimension of full observation vector

The routine is called at the beginning of each analysis step, before the loop over all local analysis domains is entered. It has to initialize the size `dim_obs_f`

of the full observation vector according to the current time step. For simplicity, `dim_obs_f`

can be the size for the global model domain.
When a domain-decomposed model is used, `dim_obs_f`

will be the size of the observation vector for the sub-domain of the calling process plus .

Some hints:

- It can be useful if not only the size of the observation vector is determined at this point. One can also already gather information about the location 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`

.

`U_obs_op`

(obs_op.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-decompostion one has to gather the information from the other model domains using MPI communication.

`U_init_obs`

(init_obs.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_prodRinvA`

(prodrinva.F90)

This routine is used by all filters whose algorithm uses the inverse of the observation error covariance matrix (SEEK, SEIK, and ETKF).

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 SEIK filter 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.

`U_init_obsvar`

(init_obsvar.F90)

This routine is used by the global filter algorithms SEIK and ETKF as well as the local filters LSEIK and LETKF. 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 rotine for the case that the observation error variance is relative to the value of the observations.

`U_prepoststep`

(prepoststep_seik.F90)

See here for the description of this routine.