Implementation of the Analysis step for the LKNETF (Local Kalman-Nonlinear Ensemble Transform Filter) algorithm

This page describes the implementation of the analysis step without using PDAF-OMI. Please see the page on the analysis with OMI for the more modern and efficient implementation variant using PDAF-OMI.

The LKNETF algorithm was added with Version 2.1 of PDAF.

Overview

The LKNETF algorithm is a hybrid method that combines the nonlinear LNETF and linear LETKF analysis updates. The hybridization allows to choose the strength of both filter methods according to the effective ensemble size and the non-Gaussianity of the observed ensemble. The details of the LKNETF method are described in the article: Nerger, L. (2022) Data assimilation for nonlinear systems with a hybrid nonlinear-Kalman ensemble transform filter. Q. J. Meteorol. Soc., 148, 620-640 ​doi:10.1002/qj.4221.

For the analysis step of the LKNETF 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_lknetf in the fully-parallel implementation (or PDAF_put_state_lknetf 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_lknetf. Since the LKNETF method combined the LNETF and LETKF methods, one can use user routines that were implemented for these filters. Note, that there is no global variant of the LKNETF implemented in PDAF.

PDAF_put_state_lknetf

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. The interface for the routine PDAF_assimilate_lknetf 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 LKNETF within some sub-domain of a domain-decomposed model (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, this might be a smaller set of observations.

To explain the difference, it is assumed, for simplicity, that a local analysis domain consists of a single vertical column of the model grid. In addition, we assume that the domain decomposition splits the global model domain by vertical boundaries as is typical for ocean models and that the observations are spatially distributed observations of model fields that are part of the state vector. Under these assumptions, the situation is the following: When a model uses domain decomposition, the LKNETF algorithm is executed such that for each model sub-domain a loop over all local analysis domains (e.g. vertical columns) that belong to the model sub-domain is performed. As each model sub-domain is treated by a different process, all loops are executed in parallel to each other.

For the update of each single vertical column, observations from some larger domain surrounding the vertical column are required. If the influence radius for the observations is sufficiently small there will be vertical columns for which the relevant observations reside completely inside the model sub-domain of the process. However, if a vertical column is considered that is located close to the boundary of the model sub-domain, there will be some observations that don't belong spatially to the local model sub-domain, but to a neighboring model sub-domain. Nonetheless, these observations are required on the local model sub-domain. A simple way to handle this situation is to initialize for each process all globally available observations, together with their coordinates. While this method is simple, it can also become inefficient if the assimilation program is executed using a large number of processes. As for an initial implementation it is usually not the concern to obtain the highest parallel efficiency, the description below assumes that 'full' refers to the global model domain.

The interface when using the LKNETF algorithm is the following:

  SUBROUTINE PDAF_assimilate_lknetf(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_prodRinvA_hyb_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_likelihood_l, U_likelihood_hyb_l, U_next_observation, outflag)

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_prodRinvA_hyb_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 including the hybrid weight.
• 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_likelihood_l: The name of the user-supplied routine that computes the likelihood of the local observations for an ensemble member provided when the routine is called.
• U_likelihood_hyb_l: The name of the user-supplied routine that computes the likelihood of the local observations for an ensemble member provided when the routine is called and accounting for the hybrid weigt.
• 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_lknetf 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_put_state_lknetf

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

The interface when using the LKNETF algorithm is the following:

  SUBROUTINE PDAF_put_state_lknetf(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_prodRinvA_hyb_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_likelihood_l, U_likelihood_hyb_l, outflag)

User-supplied routines

Here, all user-supplied routines are described that are required in the call to PDAF_assimilate_lknetf. 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 from 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_f (init_dim_obs_f_pdaf.F90)

This routine is used by all filter algorithms with domain-localization 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 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 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 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 LKNETF 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)

This routine can generally be identical to that used for the global SEIK filter, which has already been described on the page on modifying the model code for the ensemble integration. For completeness, the description is repeated:

The interface of the routine is identical for all filters. However, the particular operations that are performed in the routine can be specific for each filter algorithm. Here, we exemplify the interface on the example of the ETKF.

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.
• 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 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.
• 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_l (prodrinva_l_pdaf.F90)

This routine is used by the local filters. There is a slight difference between LSEIK and other local filters 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_prodRinvA_hyb_l (prodrinva_hyb_l_pdaf.F90)

This routine is used by the local hybrid filter LKNETF. It is a variant of U_proRinvA_l accounting for hybridization.

The interface for this routine is:

SUBROUTINE prodRinvA_hyb_l(domain_p, step, dim_obs_l, dim_ens, obs_l, gamma, 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(in)    :: gamma                ! Hybrid weight
  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.

The routine also has to apply the hybrid weight gamma. This is a simple multiplication with the input value in the loop where C_l is initialized.

Hints:

• This routine is a simple extension of prodRinvA_l. One can implement the hybrid variant by copying this routine and adapting it. gamma is computed inside PDAF and provided to the routine.

U_init_n_domains (init_n_domains_pdaf.F90)

This routine is used by all filter algorithms with domain-localization 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 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 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 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 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 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 and local square-root filter algorithms. 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 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_likelihood_l (likelihood_l_pdaf.F90)

This routine is used by the LNETF and LKNETF filters.

The interface for this routine is:

SUBROUTINE likelihood_l(domain_p, step, dim_obs_l, obs_l, resid_l, likely_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
  REAL, INTENT(in)    :: obs_l(dim_obs_l)     ! Local vector of observations
  REAL, INTENT(inout) :: resid_l(dim_obs_l)   ! Input vector holding the local residual y-Hx
  REAL, INTENT(out)   :: likely_l(dim_obs_l)  ! Output value of the likelihood

The routine is called during the loop over the local analysis domains. The likelihood of the local observations has to be computed for each ensemble member. The likelihood is computed from the observation-state residual according to the assumed observation error distribution. Commonly, the observation errors are assumed to be Gaussian distributed. In this case, the likelihood is exp(-0.5*(y-Hx)^T*R^-1*(y-Hx)).

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_likelihood. Thus if that routine has been implemented before, it can be adapted here for the local filter.
• The routine is very similar to the routine U_prodRinvA_l. The main addition is the computation of the likelihood after computing R^-1*(y-Hx), which corresponds to R^-1*A_p in U_prodRinvA_l.
• The information about the inverse observation error covariance matrix has to be provided by the user. Possibilities are to read this information from a file, or to use a Fortran module that holds this information, which one could already prepare in init_pdaf.
• The routine does not require that the product is implemented as a real matrix-vector 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 inverse diagonal with the vector resid 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.

U_likelihood_hyb_l (likelihood_hyb_l_pdaf.F90)

This routine is used by the local hybrid filter LKNETF. It is a variant of U_likelihood_l accounting for hybridization.

The interface for this routine is:

SUBROUTINE likelihood_hyb_l(domain_p, step, dim_obs_l, obs_l, resid_l, gamma, likely_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
  REAL, INTENT(in)    :: obs_l(dim_obs_l)     ! Local vector of observations
  REAL, INTENT(inout) :: resid_l(dim_obs_l)   ! Input vector holding the local residual y-Hx
  REAL, INTENT(in)    :: gamma                ! Hybrid weight
  REAL, INTENT(out)   :: likely_l(dim_obs_l)  ! Output value of the likelihood

This routine is a variant for U_likelihood_l. See the description of this routine for its functionality. The routine is called during the loop over the local analysis domains. The likelihood of the local observations has to be computed for each ensemble member. The likelihood is computed from the observation-state residual according to the assumed observation error distribution. Commonly, the observation errors are assumed to be Gaussian distributed. In this case, the likelihood is exp(-0.5*(y-Hx)^T*R^-1*(y-Hx)).

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.

The routine also has to apply the hybrid weight gamma. This is a simple multiplication with 1-gamma in the loop where Rinvresid_l is initialized.

Hints:

• This routine is a simple extension of U_likelihood_l. One can implement the hybrid variant by copying this routine and adapting it. gamma` is computed inside PDAF and provided to the routine.

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

The executation order and how ofter the user routines are called depends on the chosen hybrid filter variant. The two-step variants HNK (subtype=0) and HKN (subtype=1) perform two local analysis loops (one for LNETF and one for LETKF), while the synchronous variance (Hsync, subtype=4) perform only a single loop and computes the LETKF and NETF updates synchronously (which still requires multiple calls to user routines). 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 is often used to 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 in the same way for all three hybrid filter variants:

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))

For Hsync: 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 (Called dim_ens+1 times: Once for each ensemble member and once for the mean state estimate)
5. U_init_obs_l
6. U_init_obsvar_l (Only called, if the local adaptive forgetting factor is used (type_forget=2 in the example implementation))
7. U_prodRinvA_l
8. U_likelihood_l (Calls dim_ens times, once for each ensemble state)
9. U_l2g_state (Called dim_ens+1 times: Once for each ensemble member and once for the mean state estimate)

For HNK and HKN two local analysis loops are performed with additional initialization of observation information in between:

1. First local analysis loop:
   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 (Called dim_ens+1 times: Once for each ensemble member and once for the mean state estimate)
   5. U_init_obs_l
   6. U_likelihood_l (Called dim_ens times to determine hybrid weight gamma)
   7. Execute LNETF (for HNK) or LETKF (for HKN)
   8. U_l2g_state (Called dim_ens+1 times: Once for each ensemble member and once for the mean state estimate)
2. In between both local analysis loops:
   1. U_obs_op_f (Called dim_ens times; once for each ensemble member)
3. Second local analysis loop:
   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 (Called dim_ens+1 times: Once for each ensemble member and once for the mean state estimate)
   5. U_init_obs_l
   6. U_likelihood_l (Called dim_ens times to determine hybrid weight gamma)
   7. Execute LETKF (for HNK) or LNETF (for HKN)
   8. U_l2g_state (Called dim_ens+1 times: Once for each ensemble member and once for the mean state estimate)

In step 7: When the LNETF is executed the routine

1. U_likelihood_hyb_l (Called dim_ens times, once for each ensemble member)

is called. In contrast when the LETKF is computed the routines

1. U_init_obsvar_l (Only called, if the local adaptive forgetting factor is used (type_forget=2 in the example implementation))
2. U_prodRinvA_hyb_l

are called.

After the loop(s) 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_lknetf, the following routines are executed after the analysis step:

1. U_distribute_state
2. U_next_observation