Implementation of the Analysis Step for the Local Filters

Overview

With Version 1.16 of PDAF we introduced PDAF-OMI (observation module infrastructure). With OMI we provide generic routines for the analysis step, which only distinguish global and local filters. This page describes the implementation of the analysis step for domain-local filters (LESTKF, LETKF, LNETF, LSEIK).

For the analysis step of the local filters, several operations related to the observations are needed. These operations are requested by PDAF by calls to routines supplied by the user and provided in the OMI structure. The names of the required routines are specified in the call to the routine PDAF_assimilate_local in the fully-parallel implementation (or PDAF_put_state_local for the 'flexible' implementation) described below. With regard to the parallelization, all these routines (except U_collect_state, U_distribute_state, and U_next_observation) 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_local. Many of the routines are identical to those used for the global filters. Hence, when the user-supplied routines for the global filters 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.

PDAF_assimilate_local

The general aspects of the filter-specific routines PDAF_assimilate_* have been described on the page Modification of the model code for the ensemble integration and its sub-page on inserting the analysis step. The routine is used in the fully-parallel implementation variant of the data assimilation system. When the 'flexible' implementation variant is used, the routines `PDAF_put_state_*' is used as described further below. The interface for the routine PDAF_assimilate_local 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 (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 LESTKF 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 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 one of the local filters is the following:

  SUBROUTINE PDAF_assimilate_lestkf(U_collect_state, U_distribute_state, &
                                  U_init_dim_obs_f, U_obs_op_f, &
                                  U_prepoststep, U_init_n_domains, U_init_dim_l, &
                                  U_init_dim_obs_l, U_g2l_state, U_l2g_state, &
                                  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 and also 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_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_prepoststep: The name of the pre/poststep routine as in PDAF_get_state
• 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_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_lestkf 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_local

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

The interface when using one of the local filters is the following:

  SUBROUTINE PDAF_put_state_lestkf(U_collect_state, &
                                  U_init_dim_obs_f, U_obs_op_f, &
                                  U_prepoststep, U_init_n_domains, U_init_dim_l, &
                                  U_init_dim_obs_l, U_g2l_state, U_l2g_state, &
                                  status)

User-supplied routines

Here, all user-supplied routines are described that are required in the call to PDAF_assimilate_local or PDAF_put_state_local. 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 (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 mape 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 (LSEIK, LETKF, LESTKF) 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, LESTKF) 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               ! Currrent 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 contain parts of the state vector from neighboring model sub-domains. To make these parts accessible, some parallel communication will be necessary (The state information for a neighboring model sub-domain, will be in the memory of the process that handles that sub-domain). The example implementation in testsuite/dummymodel_1d uses the function MPI_AllGatherV for this communication.

U_prepoststep (prepoststep_ens_pdaf.F90)

This routine can be identical to that used for the global ESTKF algorithm, 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.

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/ESTKF.
                                     ! It can be used freely in this routine.
  REAL, INTENT(inout) :: Uinv(dim_ens-1, dim_ens-1) ! 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 LETKF and LESTKF: For the LETKF, the array Uinv has size dim_ens x dim_ens. In contrast it has size dim_ens-1 x dim_ens-1 for the LESTKF.
• 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_init_n_domains (init_n_domains_pdaf.F90)

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

The interface for this routine is:

SUBROUTINE g2l_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, LESTKF) 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_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 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_local 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)

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_prodRinvA_l
5. 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_local, the following routines are executed after the analysis step:

1. U_distribute_state
2. U_next_observation