Implementation of the Analysis step for the ESTKF

This page describes the implementation of the analysis step using PDAF's full interface, i.e. without using PDAF-OMI. This approach is supported by all versions of PDAF. However, this approach is mainly used in older implementations of PDAF and can be seen as a expert-mode. Please see the page on the analysis step in PDAF3 for the current implementation recommendation using the PDAF3 interface. The page also provides links to some other variants that were introduced in verisons of PDAF2.

Overview

The ESTKF (Error Subspace Transform Kalman Filter, Nerger et al., 2012) is a particularly efficient ensemble-based Kalman filter. It was introcued with PDAF V1.8. The user-supplied routines required for the ESTKF are identical to those required for the SEIK filter and amost identical to those required for the ETKF method.

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 and efficient. This procedure should simplify the implementation. The names of the required routines are specified in the call to the routine PDAF_assimilate_estkf in the fully-parallel and flexible implementation (or PDAF_put_state_estkf for the offlie mode and the flexible implementation in PDAF2) that was discussed before. For the offline mode in PDAF3, the routine PDAF_assim_offline_estkf is used. 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_assimilate_estkf. Thus, some of the user-supplied routines that are explained on the page describing 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_assimilate_estkf

This routine is used both in the fully-parallel and the flexible implementation variants of the data assimilation system. (See the page Modification of the model code for the ensemble integration for these variants)

Here, we list the full interface of the routine. Subsequently, the user-supplied routines specified in the call are explained.

The interface is:

  SUBROUTINE PDAF_assimilate_estkf(U_collect_state, U_distribute_state, U_init_dim_obs, &
                                 U_obs_op, U_init_obs, U_prepoststep, U_prodRinvA, &
                                 U_init_obsvar, 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_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)
• 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_estkf is exited without errors.

where all arguments, except the last one, are names of used-supplied routines.

PDAF_assim_offline_estkf

This routine is used to perform the analysis step for the offline mode of PDAF. The interface of the routine is identical with that of the 'assimilate'-routine, except that the user-supplied routines U_distribute_state, U_collect_state and U_next_observation are missing.

The 'assim_offline' routines were introduced with PDAF V3.0 to simplify the implementation of the offline mode.

The interface is:

  SUBROUTINE PDAF_assim_offline_estkf(U_init_dim_obs, &
                                 U_obs_op, U_init_obs, U_prepoststep, U_prodRinvA, &
                                 U_init_obsvar, status)

PDAF_put_state_estkf

This routine exists for backward-compatibility. In implementations that were done for PDAF V2.3.1 and before, a 'put_state' routine was used for the flexible parallelization variant and for the offline mode. When the 'flexible' implementation variant is chosen for the assimilation system, the routine. This routine allows to continue using the previous implementation structure. The interface of the routine is identical with that of the 'assimilate'-routine, except that the user-supplied routines U_distribute_state and U_next_observation are missing.

The interface is:

  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)

User-supplied routines

Here all user-supplied routines are described that are required in the call to PDAF_assimilate_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 tutorials in tutorial/ and in the template directory templates/ these routines exist without the prefix, but with the extension _pdaf. The files are named correspondingly. 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 modifying the model code for the ensemble integration 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 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 for modifying the model for the ensemble integration and for inserting the analysis step.

See the page on modifying the model code for the ensemble integration for the description of this routine.

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, but also 3DEnVar and hybrid 3D-Var).

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.

U_next_observation (next_observation_pdaf.F90)

This routine is independent of the filter algorithm used. See the page on modifying the model code for the ensemble integration for the description of this routine.

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_assimilate_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:

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 (A single call to operate on the ensemble mean state)
4. U_init_obs
5. U_obs_op (dim_ens calls: one call for each ensemble member)
6. U_init_obsvar (Only executed, if the adaptive forgetting factor is used (type_forget=1 in the example implemention))
7. U_prodRinvA
8. U_prepoststep (Call to act on the analysis ensemble, called with (positive) value of the time step)

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

1. U_distribute_state
2. U_next_observation