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

The LEnKF was added with verson 1.12 of PDAF.

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 PDAF_put_state_lenkf for the fully-parallel implementation (or PDAF_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 PDAF_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.

PDAF_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 PDAF_assimilate_lenkf` is explained.

The interface when using the LEnKF is the following:

  SUBROUTINE PDAF_assimilate_lenkf(U_collect_state, U_distribute_state, U_init_dim_obs, &
                                 U_obs_op, U_init_obs, U_prepoststep, U_localize, &
                                 U_add_obs_err, U_init_obscovar, 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_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_localize: Apply covariance localization to the matrices HP and HPH^T
• U_add_obs_err: The name of the user-supplied routine that adds the observation error covariance matrix to the ensemble covariance matrix projected onto the observation space.
• U_init_obscovar: The name of the user-supplied routine that initializes the observation error covariance matrix.
• 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 PDAF_assimilate_lenkf is exited without errors.

PDAF_put_statel_enkf

When the 'flexible' implementation variant is chosen for the assimilation system, the routine PDAF_put_state_lenkf has to be used instead of PDAF_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 PDAF_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 PDAF_put_state_lenkf(U_collect_state, U_init_dim_obs, U_obs_op, &
                                 U_init_obs, U_prepoststep, U_localize, &
                                 U_add_obs_err, U_init_obscovar, status)

User-supplied routines

Here all user-supplied routines are described that are required in the call to PDAF_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 (init_dim_obs_pdaf.F90)

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

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) and the LEnKF.

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) and the LEnKF.

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 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 (localize_covar_pdaf.F90)

This routine is only used for the LEnKF.

The interface for this routine is:

SUBROUTINE U_localize(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 matricesHP and HPH^T, which are provided as input/output arguments.

Notes:

• In case of a parallelization with domain decomposition, HP contains only the columns of the matrix that resides on the model sub-domain of the calling process. The number of rows is that of the global number of observations

Hints:

• To compute the localization one can use the routine PDAF_local_weight after computing the distance between two elements in the matrix HP or HPH^T.

U_add_obs_err (add_obs_err_pdaf.F90)

This routine is only used for the EnKF and LEnKF.

The interface for this routine is:

SUBROUTINE add_obs_err(step, dim_obs, C)

  INTEGER, INTENT(in) :: step                ! Current time step
  INTEGER, INTENT(in) :: dim_obs             ! Dimension of obs. vector
  REAL, INTENT(inout) :: C(dim_obs, dim_obs) ! Matrix to that the observation 
                                             !    error covariance matrix is added

The routine is called during the analysis step. During the analysis step of the LEnKF, the projection of the ensemble covariance onto the observation space is computed. This matrix is provided to the routine as C_p. The routine has to add the observation error covariance matrix to C_p.

The operation is for the global observation space. Thus, it is independent of whether the filter is executed with or without parallelization.

Hints:

• The routine does not require that the observation error covariance matrix is added as a full matrix. If the matrix is diagonal, only the diagonal elements have to be added.

U_init_obscovar (init_obscovar_pdaf.F90)

This routine is only used for the EnKF and LEnKF.

The interface for this routine is:

SUBROUTINE init_obscovar(step, dim_obs, dim_obs_p, covar, m_state_p, &
     isdiag)

  INTEGER, INTENT(in) :: step                ! Current time step
  INTEGER, INTENT(in) :: dim_obs             ! Dimension of observation vector
  INTEGER, INTENT(in) :: dim_obs_p           ! PE-local dimension of observation vector
  REAL, INTENT(out) :: covar(dim_obs, dim_obs) ! Observation error covariance matrix 
  REAL, INTENT(in)  :: m_state_p(dim_obs_p)  ! PE-local observation vector 
  LOGICAL, INTENT(out) :: isdiag             ! Whether the observation error covar. matrix is diagonal

The routine is called during the analysis step and is required for the generation of an ensemble of observations. It has to initialize the global observation error covariance matrix covar. In addition, the flag isdiag has to be initialized to provide the information, whether the observation error covariance matrix is diagonal.

The operation is for the global observation space. Thus, it is independent of whether the filter is executed with or without parallelization.

Hints:

• The local observation vector m_state_p is provided to the routine for the case that the observation errors are relative to the value of the observation.

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 PDAF_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
3. U_obs_op (dim_ens calls: one call for each ensemble member)
4. U_localize
5. U_add_obs_err
6. U_init_obs
7. U_init_obscovar
8. U_obs_op (dim_ens calls: one call for each ensemble member, repeated to reduce storage)
9. U_prepoststep (Call to act on the analysis ensemble, called with (positive) value of the time step)

In case of the routine PDAF_assimilate_enkf, the following routines are executed after the analysis step:

1. U_distribute_state
2. U_next_observation