Implementation of the Analysis step for the ETKF (Ensemble Transform Kalman Filter) algorithm

Overview

For the analysis step of the ETKF, 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_assimilate_etkf (or PDAF_put_state_etkf) that was discussed before. 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_etkf. 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 SEIK filter 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 SEIK filter and the ETKF. Differences are marked in the text below.

PDAF_assimilate_etkf

The general espects 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. Subsequently, the full set of user-supplied routines specified in the call to PDAF_assimilate_etkf` is explained.

The interface when using the ETKF method is the following:

  SUBROUTINE PDAF_assimilate_etkf(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 ETKF.
• 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_etkf is exited without errors.

PDAF_put_state_etkf

When the 'flexible' implementation variant is chosen for the assimilation system, the routine PDAF_put_state_etkf has to be used instead of PDAF_assimilate_etkf. 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_etkf with the exception the specification of the user-supplied routines U_distribute_state and U_next_observation are missing.

The interface when using the ETKF method is the following:

  SUBROUTINE PDAF_put_state_etkf(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_etkf. 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).

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).

The interface for this routine is:

SUBROUTINE obs_op(step, dim_p, dim_obs_p, state_p, m_state_p)

  INTEGER, INTENT(in) :: step               ! Current 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).

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 been described on the page on modifying the model code for the ensemble integration for the SEIK filter. For the ETKF, the interface is generally identical. For completeness, we repeat the description here.

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/EnKF/ETKF.
                                     ! It can be used freely in this routine.
  REAL, INTENT(inout) :: Uinv(dim_ens, dim_ens) ! 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.
• The vector (state_p) is allocated but not initialized with the ensemble mean. It can be used freely during the execution of U_prepoststep, for example to compute the ensemble mean state.
• The interface has a difference for ETKF and SEIK: For the ETKF, the array Uinv has size dim_ens x dim_ens. In contrast it has size dim_ens-1 x dim_ens-1 for the SEIK filter. (For most cases, this will be irrelevant, because most usually the ensemble array ens_p is used for computations, rather than Uinv. However, for the SEIK filter with fixed covariance matrix, Uinv is required to compute the estimate analysis error. The fixed covariance matrix mode is not available for the ETKF)
• 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_prodRinvA (prodrinva_pdaf.F90)

This routine is used by all filter algorithms that use the inverse of the observation error covariance matrix (SEEK, SEIK, and ETKF).

The interface for this routine is:

SUBROUTINE prodRinvA(step, dim_obs_p, dim_ens, 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) :: dim_ens             ! Ensemble size
  REAL, INTENT(in)    :: obs_p(dim_obs_p)    ! PE-local vector of observations
  REAL, INTENT(in)    :: A_p(dim_obs_p, dim_ens) ! Input matrix from analysis routine
  REAL, INTENT(out)   :: C_p(dim_obs_p, dim_ens) ! 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 ETKF, 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 SEIK and ETKF: For ETKF the third argument is the ensemble size (dim_ens), while for SEIK it is the 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 SEIK filter. (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 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 implementation).

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 inserting the analysis step for the description of this routine.

Execution order of user-supplied routines

For the ETKF, the user-supplied routines are essentially executed in the order they are listed in the interface to PDAF_assimilate_etkf. 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