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

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 LETKF (Local Ensemble Transform Kalman Filter, Hunt et al., 2007) is a widely used ensemble square-root Kalman filter. For the analysis step of the LETKF algorithm, 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_assimilate_letkf in the fully-parallel implementation (or PDAF_put_state_letkf for the 'flexible' implementation) described below. With regard to the parallelization, all these routines (except U_collect_state) 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_letkf. Many of the routines are localized versions of those that are needed for the global ETKF method. Hence, if the user-supplied routines for the global ETKF method have been already implemented, one can base on these routines to speed up the implementation. Due to this, it can also be reasonable to first fully implement a global filter version and subsequently implement the corresponding localized filter by modifying and extending the global routines.

The LESTKF and the LETKF 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 LESTKF filter and the LETKF. Differences are marked in the text below.

PDAF_put_state_letkf

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)

The interface for the routine PDAF_assimilate_letkf contains several routine names for routines that operate on the local analysis domains (marked by _l at the end of the routine name). In addition, there are names for routines that consider all available observations required to perform local analyses with LESTKF within some sub-domain of a domain-decomposed model (we refer to these as 'full' observations, marked by _f at the end of the routine name). In case of a serial execution of the assimilation program, these will be all globally available observations. However, if the program is executed with parallelization, one might choose a smaller set of observations. We will explain this is some detail below.

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_letkf(U_collect_state, U_distribute_state, &
                                  U_init_dim_obs_f, U_obs_op_f, U_init_obs_f, &
                                  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, 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
• 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_f: The name of the user-supplied routine that provides the size of the full observation vector
• U_obs_op_f: The name of the user-supplied routine that acts as the full observation operator on some state vector
• U_init_obs_f: The name of the user-supplied routine that initializes the full 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.
• 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 dimension 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 corresponding part of the global state vector from the the provided local state vector
• U_g2l_obs: The name of the routine that initializes 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 (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 (This routine will only be executed, if a local 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_letkf is exited without errors.

Note:

• The order of the routine names does not show the order in which these routines are executed. See the section on the order of the execution at the bottom of this page.

PDAF_assim_offline_letkf

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_letkf(&
                                  U_init_dim_obs_f, U_obs_op_f, U_init_obs_f, &
                                  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)

PDAF_put_state_letkf

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. 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 when using the LETKF algorithm is the following:

  SUBROUTINE PDAF_put_state_letkf(U_collect_state, &
                                  U_init_dim_obs_f, U_obs_op_f, U_init_obs_f, &
                                  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)

Explanation of 'full observations'

Above we mention the concept of 'full' observations. We distinguish them from the globally available observations for efficiency.

Note: For an initial implementation, one might not needd to worry about high efficiency, so that 'full' can refer to all available observations.

To explain why 'full' observations can be different from globally available observations, we assume, for simplicity, that we have a 2-dimensional domain and that a local analysis domain consists of a single grid point of the model grid. In addition, we assume that the domain decomposition splits the global model domain in compact sub-domains and that the observations are spatially distributed observations of model fields that are part of the state vector.

The LESTKF performs a loop over all local analysis domains, i.e. grid points. When a model uses domain decomposition, the loop is over all grid points that belong to a process sub-domain. As each model sub-domain is treated by a different process, all loops are executed parallel to each other.

For the update of each local analysis domain (grid points), observations within the localization radius around its location are required. If the influence radius for the observations is sufficiently small, there will be grid points a for which the relevant observations reside completely inside the model sub-domain of the process. However, if a grid point is located close to the boundary of the model sub-domain, there will be some observations that reside on a neighboring process sub-domain, but are within the localization radius. One needs to assimilate these observations as otherwise, there could be unrealistic steps in the analysis field. However, there will also be observations that reside far away from the process sub-domain and will never influence the analysis result in this domain.

A simple way to handle this situation is to initialize for each process all globally available observations, together with their coordinates. The observation operator would be applied on each sub-domain and then the observed ensemble would be collect using parallel communication with MPI. While this method is simple, it can also become inefficient if the assimilation program is executed using a large number of processes. In particular, all observation would need to be checked even if they are far away from the process sub-domain.

More efficient is hence to select as 'full' observations only those observations that can have an effect on the local analyses of a process sub-domain. These are the observations that reside within the sub-domain, plus observations in neighboring sub-domains that reside within the localization radius. Setting up 'full' observations in this way leads to a smaller number of observations whose distance need to be checked for each local analysis domain. Howeever, one would need to find an implementation that provides the 'full' observations.

Note: The handling of 'full' observations is one of the aspects that motivated the development of PDAF-OMI and the relaed avanced interface (now the PDAF3 interface). Here, PDAF-OMI does take case of the 'full' observations. See the Implementation Guide for the Analysis Step for the advanced interface using PDAF-OMI.

User-supplied routines

Here, all user-supplied routines are described that are required in the calls to the analysis routines. 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 (short for 'process'). 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. In addition, there will be variables with the suffix _f (for 'full') and with the suffix _l (for 'local').

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_f (init_dim_obs_f_pdaf.F90)

This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) and is independent of the particular algorithm.

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.

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 location of the observations, which can be used later, e.g. to implement the observation operator. In addition, one can already prepare an array that holds the full observation vector. This can be used later by U_init_obs_l to initialize a local vector of observations by selecting the relevant parts of the full observation vector. The required arrays can be defined in a module like mod_assimilation.
• The routine is similar to init_dim_obs used in the global filters. However, if the global filter is used with a domain-decomposed model, it only initializes the size of the observation vector for the local model sub-domain. This is different for the local filters, as the local analysis also requires observational data from neighboring model sub-domains. Nonetheless, one can base on an implemented routine init_dim_obs to implement init_dim_obs_f.

U_obs_op_f (obs_op_f_pdaf.F90)

This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) and is independent of the particular algorithm.

The interface for this routine is:

SUBROUTINE obs_op_f(step, dim_p, dim_obs_f, state_p, m_state_f)

  INTEGER, INTENT(in) :: step               ! Current time step
  INTEGER, INTENT(in) :: dim_p              ! PE-local dimension of state
  INTEGER, INTENT(in) :: dim_obs_f          ! Dimension of the full observed state
  REAL, INTENT(in)    :: state_p(dim_p)     ! PE-local model state
  REAL, INTENT(out) :: m_state_f(dim_obs_f) ! Full observed state

The routine is called during the analysis step, before the loop over the local analysis domain is entered. It has to perform the operation of the observation operator acting on a state vector, which is provided as state_p. The observed state has to be returned in m_state_f. It is the observed state corresponding to the 'full' observation vector.

Hint:

• The routine is similar to init_dim_obs used for the global filters. However, with a domain-decomposed model m_state_f will need to contain parts of the state vector from neighboring model sub-domains. Thus, one needs to collect this information which resides in the memory of other processes. PDAF provides the routine PDAF_gather_obs_f for this task. The example implementation in tutorial/classical/online_2D_parallelmodel shows the use of PDAF_gather_obs_f.

U_init_obs_f (init_obs_f_pdaf.F90)

This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) and is independent of the particular algorithm. The routine is only called if the globally adaptive forgetting factor is used (type_forget=1 in the example implementation). For the local filters there is also the alternative to use locally adaptive forgetting factors (type_forget=2 in the example implementation)

The interface for this routine is:

SUBROUTINE init_obs_f(step, dim_obs_f, observation_f)

  INTEGER, INTENT(in) :: step                     ! Current time step
  INTEGER, INTENT(in) :: dim_obs_f                ! Dimension of full observation vector
  REAL, INTENT(out)   :: observation_f(dim_obs_f) ! Full observation vector

The routine is called during the analysis step before the loop over the local analysis domains is entered. It has to provide the full vector of observations in observation_f for the current time step. The caller is the routine that computes an adaptive forgetting factor (PDAF_set_forget).

Hints:

• As for the other 'full' routines: While the global counterpart of this routine (init_obs) has to initialize the observation vector only for the local model sub-domain, the 'full' routine has to include observations that spatially belong to neighboring model sub-domains. As an easy choice one can simply initialize a vector of all globally available observations.
• If the adaptive forgetting factor is not used, this routine only has to exist. However, no functionality is required.

U_init_obs_l (init_obs_l_pdaf.F90)

This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) and is independent of the particular algorithm.

The interface for this routine is:

SUBROUTINE init_obs_l(domain_p, step, dim_obs_l, observation_l)

  INTEGER, INTENT(in) :: domain_p                 ! Current local analysis domain
  INTEGER, INTENT(in) :: step                     ! Current time step
  INTEGER, INTENT(in) :: dim_obs_l                ! Local dimension of observation vector
  REAL, INTENT(out)   :: observation_l(dim_obs_l) ! Local observation vector

The routine is called during the analysis step during the loop over the local analysis domain. It has to provide the vector of observations for the analysis in the local analysis domain with index domain_p in observation_l for the current time step.

Hints:

• For parallel efficiency, the LETKF algorithm is implemented in a way that first the full vectors are initialized. These are then restricted to the local analysis domain during the loop over all local analysis domains. Thus, if the full vector of observations has been initialized before U_init_obs_l is executed (e.g. by U_init_dim_obs_f), the operations performed in this routine will be to select the part of the full observation vector that is relevant for the current local analysis domain.
• The routine U_init_dim_obs_l is executed before this routine. Thus, if that routine already prepares the information which elements of observation_f are needed for observation_l, this information can be used efficiently here.

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_l (prodrinva_l_pdaf.F90)

This routine is used by the local filters (LSEIK and LETKF). There is a slight difference between LSEIK and LETKF for this routine, which is described below.

The interface for this routine is:

SUBROUTINE prodRinvA_l(domain_p, step, dim_obs_l, dim_ens, obs_l, A_l, C_l)

  INTEGER, INTENT(in) :: domain_p             ! Current local analysis domain
  INTEGER, INTENT(in) :: step                 ! Current time step
  INTEGER, INTENT(in) :: dim_obs_l            ! Dimension of local observation vector
  INTEGER, INTENT(in) :: dim_ens              ! Ensemble size
  REAL, INTENT(in)    :: obs_l(dim_obs_l)     ! Local vector of observations
  REAL, INTENT(inout) :: A_l(dim_obs_l, dim_ens) ! Input matrix from analysis routine
  REAL, INTENT(out)   :: C_l(dim_obs_l, dim_ens) ! Output matrix

The routine is called during the loop over the local analysis domains. In the algorithm, 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 for the local analysis domain of index domain_p. The matrix is provided as A_l. The product has to be given as C_l.

This routine is also the place to perform observation localization. To initialize a vector of weights, the routine PDAF_local_weight can be called. The procedure is used in the example implementation and also demonstrated in the template routine.

Hints:

• The routine is a local variant of the routine U_prodRinvA. Thus if that routine has been implemented before, it can be adapted here for the local filter.
• 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_l has to be implemented.
• The observation vector obs_l 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 (rank) of the covariance matrix (usually ensemble size minus one). In addition, the second dimension of A_l and C_l 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_n_domains (init_n_domains_pdaf.F90)

This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) and is independent of the particular algorithm.

The interface for this routine is:

SUBROUTINE init_n_domains(step, n_domains_p)

  INTEGER, INTENT(in)  :: step        ! Current time step
  INTEGER, INTENT(out) :: n_domains_p ! Number of analysis domains for local model sub-domain

The routine is called during the analysis step before the loop over the local analysis domains is entered. It has to provide the number of local analysis domains. In case of a domain-decomposed model the number of local analysis domain for the model sub-domain of the calling process has to be initialized.

Hints:

• As a simple case, if the localization is only performed horizontally, the local analysis domains can be single vertical columns of the model grid. In this case, n_domains_p is simply the number of vertical columns in the local model sub-domain.

U_init_dim_l (init_dim_l_pdaf.F90)

This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) and is independent of the particular algorithm.

The interface for this routine is:

SUBROUTINE init_dim_l(step, domain_p, dim_l)

  INTEGER, INTENT(in)  :: step        ! Current time step
  INTEGER, INTENT(in)  :: domain_p    ! Current local analysis domain
  INTEGER, INTENT(out) :: dim_l       ! Local state dimension

The routine is called during the loop over the local analysis domains in the analysis step. It has to provide in dim_l the dimension of the state vector for the local analysis domain with index domain_p.

Hints:

• If a local analysis domain is a single vertical column of the model grid, the size of the state in the local analysis domain will be just the number of vertical grid points at this location.

U_init_dim_obs_l (init_dim_obs_l_pdaf.F90)

This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) and is independent of the particular algorithm.

The interface for this routine is:

SUBROUTINE init_dim_obs_l(domain_p, step, dim_obs_f, dim_obs_l)

  INTEGER, INTENT(in)  :: domain_p   ! Current local analysis domain
  INTEGER, INTENT(in)  :: step       ! Current time step
  INTEGER, INTENT(in)  :: dim_obs_f  ! Full dimension of observation vector
  INTEGER, INTENT(out) :: dim_obs_l  ! Local dimension of observation vector

The routine is called during the loop over the local analysis domains in the analysis step. It has to initialize in dim_obs_l the size of the observation vector used for the local analysis domain with index domain_p.

Some hints:

• Usually, the observations to be considered for a local analysis are those which reside within some distance from the local analysis domain. Thus, if the local analysis domain is a single vertical column of the model grid and if the model grid is a regular ij-grid, then one could use some range of i/j indices to select the observations and determine the local number of them. More generally, one can compute the physical distance of an observation from the local analysis domain and decide on this basis, if the observation has to be considered.
• In the loop over the local analysis domains, the routine is always called before U_init_obs_l is executed. Thus, as U_init_dim_obs_local has to check which observations should be used for the local analysis domain, one can already initialize an integer array that stores the index of observations to be considered. This index should be the position of the observation in the array observation_f. With this, the initialization of the local observation vector in U_init_obs_l can be sped up.
• For PDAF, we could not join the routines U_init_dim_obs_l and U_init_obs_l, because the array for the local observations is allocated internally to PDAF after its size has been determined in U_init_dim_obs_l.

U_g2l_state (g2l_state_pdaf.F90)

This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) and is independent of the particular algorithm.

The interface for this routine is:

SUBROUTINE global2local_state(step, domain_p, dim_p, state_p, dim_l, state_l)

  INTEGER, INTENT(in) :: step           ! Current time step
  INTEGER, INTENT(in) :: domain_p       ! Current local analysis domain
  INTEGER, INTENT(in) :: dim_p          ! State dimension for model sub-domain
  INTEGER, INTENT(in) :: dim_l          ! Local state dimension
  REAL, INTENT(in)    :: state_p(dim_p) ! State vector for model sub-domain 
  REAL, INTENT(out)   :: state_l(dim_l) ! State vector on local analysis domain

The routine is called during the loop over the local analysis domains in the analysis step. It has to provide the local state vector state_l that corresponds to the local analysis domain with index domain_p. Provided to the routine is the state vector state_p. With a domain decomposed model, this is the state for the local model sub-domain.

Hints:

• In the simple case that a local analysis domain is a single vertical column of the model grid, the operation in this routine would be to take out of state_p the data for the vertical column indexed by domain_p.

U_l2g_state (l2g_state_pdaf.F90)

This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) and is independent of the particular algorithm.

The interface for this routine is:

SUBROUTINE l2g_state(step, domain_p, dim_l, state_l, dim_p, state_p)

  INTEGER, INTENT(in) :: step           ! Current time step
  INTEGER, INTENT(in) :: domain_p       ! Current local analysis domain
  INTEGER, INTENT(in) :: dim_p          ! State dimension for model sub-domain
  INTEGER, INTENT(in) :: dim_l          ! Local state dimension
  REAL, INTENT(in)    :: state_p(dim_p) ! State vector for model sub-domain 
  REAL, INTENT(out)   :: state_l(dim_l) ! State vector on local analysis domain

The routine is called during the loop over the local analysis domains in the analysis step. It has to initialize the part of the global state vector state_p that corresponds to the local analysis domain with index domain_p. Provided to the routine is the state vector state_l for the local analysis domain.

Hints:

• In the simple case that a local analysis domain is a single vertical column of the model grid, the operation in this routine would be to write into state_p the data for the vertical column indexed by domain_p.

U_g2l_obs (g2l_obs_pdaf.F90)

This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) and is independent of the particular algorithm.

The interface for this routine is:

SUBROUTINE g2l_obs(domain_p, step, dim_obs_f, dim_obs_l, mstate_f, mstate_l)

  INTEGER, INTENT(in) :: domain_p              ! Current local analysis domain
  INTEGER, INTENT(in) :: step                  ! Current time step
  INTEGER, INTENT(in) :: dim_obs_f             ! Dimension of full observation vector for model sub-domain
  INTEGER, INTENT(in) :: dim_obs_l             ! Dimension of observation vector for local analysis domain
  REAL, INTENT(in)    :: mstate_f(dim_obs_f)   ! Full observation vector for model sub-domain
  REAL, INTENT(out)   :: mstate_l(dim_obs_l)   ! Observation vector for local analysis domain

The routine is called during the loop over the local analysis domains in the analysis step. It has to provide a local observation vector mstate_l for the observation domain that corresponds to the local analysis domain with index domain_p. Provided to the routine is the full observation vector mstate_f from which the local part has to be extracted.

Hints:

• The vector mstate_f that is provided to the routine is one of the observed state vectors that are produced by U_obs_op_full.
• Some operations performed here are analogous to those required to initialize a local vector of observations in U_init_obs_l. If that routine reads first a full vector of observations (e.g. in U_init_dim_obs_f), this vector has to be restricted to the relevant observations for the current local analysis domain. For this operation, one can for example initialize an index array when U_init_dim_obs_l is executed. (Which happens before U_global2local_obs)

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 difference in this routine between global and local filters is that the global filters use 'global' while the local filters use 'full' quantities.

The interface for this routine is:

SUBROUTINE init_obsvar(step, dim_obs_f, obs_f, meanvar_f)

  INTEGER, INTENT(in) :: step             ! Current time step
  INTEGER, INTENT(in) :: dim_obs_f        ! Full dimension of observation vector
  REAL, INTENT(in)    :: obs_f(dim_obs_f) ! Full observation vector
  REAL, INTENT(out)   :: meanvar_f        ! Mean observation error variance

The routine is called in the local filters before the loop over all local analysis domains is entered. The call is by the routine that computes an adaptive forgetting factor (PDAF_set_forget). The routine has to initialize an average full observation error variance, which should be consistent with the observation vector initialized in U_init_ob_f.

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.
• If the adaptive forgetting factor is not used, this routine has only to exist for the compilation, but it does not need functionality.

U_init_obsvar_l (init_obsvar_l_pdaf.F90)

This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) and is independent of the particular algorithm. The routine is only called if the local adaptive forgetting factor is used (type_forget=2 in the example implementation).

The interface for this routine is:

SUBROUTINE init_obsvar_l(domain_p, step, dim_obs_l, obs_l, meanvar_l)

  INTEGER, INTENT(in) :: domain_p         ! Current local analysis domain
  INTEGER, INTENT(in) :: step             ! Current time step
  INTEGER, INTENT(in) :: dim_obs_l        ! Local dimension of observation vector
  REAL, INTENT(in)    :: obs_l(dim_obs_l) ! Local observation vector
  REAL, INTENT(out)   :: meanvar_l        ! Mean local observation error variance

The routine is called in the local filters during the loop over all local analysis domains by the routine that computes a local adaptive forgetting factor (PDAF_set_forget_local). The routine has to initialize a local mean observation error variance for all observations used for the analysis in the current local analysis domain.

Hints:

• If the local adaptive forgetting factor is not used, this routine has only to exist for the compilation, but it does not need functionality.

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

The user-supplied routines are executed in the order listed below. The order can be important as some routines can perform preparatory work for routines executed later on during the analysis. For example, U_init_dim_obs_l can prepare an index array that provides the information how to localize a 'full' vector of observations. Some hints one the efficient implementation strategy are given with the descriptions of the routine interfaces above.

Before the analysis step is called the following is executed:

1. U_collect_state (called once for each ensemble member)

When the ensemble integration of the forecast is completed, the analysis step is executed. Before the loop over all local analysis domains, the following routines are executed:

1. U_prepoststep (Call to act on the forecast ensemble, called with negative value of the time step)
2. U_init_n_domains
3. U_init_dim_obs_f
4. U_obs_op_f (Called dim_ens times; once for each ensemble member)
5. U_init_obs_f (Only executed, if the global adaptive forgetting factor is used (type_forget=1 in the example implementation))
6. U_init_obsvar (Only executed, if the global adaptive forgetting factor is used (type_forget=1 in the example implementation))

In the loop over all local analysis domains, it is executed for each local analysis domain:

1. U_init_dim_l
2. U_init_dim_obs_l
3. U_g2l_state (Called dim_ens+1 times: Once for each ensemble member and once for the mean state estimate)
4. U_g2l_obs (A single call to localize the mean observed state)
5. U_init_obs_l
6. U_g2l_obs (dim_ens calls: one call to localize the observed part of each ensemble member)
7. U_init_obsvar_l (Only called, if the local adaptive forgetting factor is used (type_forget=2 in the example implementation))
8. U_prodRinvA_l
9. U_l2g_state (Called dim_ens+1 times: Once for each ensemble member and once for the mean state estimate)

After the loop over all local analysis domains, it is executed:

1. U_prepoststep (Call to act on the analysis ensemble, called with (positive) value of the time step)

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

1. U_distribute_state
2. U_next_observation