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

Overview

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. This page described the implementation of the analysis step for the local EnKF (LEnKF).

For the analysis step of the global filters we need different operations related to the observations. These operations are requested by PDAF by call-back routines supplied by the user and provided in the OMI structure. 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 (except U_collect_state, U_distribute_state, and U_next_observation) 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, the ETKF and the 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 the full interface of the routine. Subsequently, the user-supplied routines specified in the call 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_pdafomi, U_obs_op_pdafomi, &
                                 U_prepoststep, U_localize_covar_pdafomi, &
                                 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_pdafomi: The name of the user-supplied routine that initializes the observation information and provides the size of observation vector
• U_obs_op_pdafomi: 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_pdafomi, U_obs_op_pdafomi, &
                                 U_prepoststep, U_localize_covar_pdafomi, &
                                 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. The user-routines relating to OMI are collected in the file callback_obs_pdafomi.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_pdafomi (callback_obs_pdafomi.F90)

This is a call-back routine for PDAF-OMI initializing the observation information. The routine just calls a routine from the observation module for each observation type.

See the documentation on callback_obs_pdafomi.F90 for more information.

U_obs_op_pdafomi (callback_obs_pdafomi.F90)

This is a call-back routine for PDAF-OMI applying the observation operator to the state vector. The routine calls a routine from the observation module for each observation type.

See the documentation on callback_obs_pdafomi.F90 for more information.

U_prepoststep (prepoststep_ens_pdaf.F90)

The routine has already been described for modifying the model for the ensemble integration and for inserting the analysis step.

See the page on inserting the analysis step for the description of this routine.

U_localize_covar_pdafomi (callback_obs_pdafomi.F90)

This is a call-back routine for PDAF-OMI. It applies covariance localization in the LEnKF to the matrices PH^T and HPH^T, which are intermediate results of the EnKF. The routine calls a routine from the observation module for each observation type.

See the documentation on callback_obs_pdafomi.F90 for more information.

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:

1. U_collect_state

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:

1. U_prepoststep (Call to act on the forecast ensemble, called with negative value of the time step)
2. U_init_dim_obs_pdafomi
3. U_obs_op_pdafomi (Called dim_ens times; once for each ensemble member)
4. U_localize_covar_pdafomi
5. U_obs_op (dim_ens calls: one call for each ensemble member, repeated to reduce storage)
6. 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:

1. U_distribute_state
2. U_next_observation