Implementation of the Analysis step for the LSEIK filter

Overview

For the analysis step of the LSEIK filter 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_put_state_lseik described below. With regard to the parallelization, all these routines are executed by the filter processes (filterpe=1) only.

For completeness we discuss here all user-supplied routines that are specified in the interface to PDAF_put_state_lseik.

PDAF_put_state_lseik

The general espects of the filter specific routines PDAF_put_state_* have been described on the page Modification of the model core for the ensemble integration. The interface for the routine PDAF_put_state_lseik contains several routine names for routines that operate on the local analysis domains (marked by _l at then end of the routine name). In addition, there are names for routines that consider all available observations required to perform local analyses with LSEIK within some sub-domain of a domain-decomposed model (marked by _full at then end of the routine name). In case of a serial execution of the assimilation program, this 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 LSEIK filter 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. For the update of each single vertical column observations from some larger domain surrounding the vertical column are considered. If the influence radius for the observations is sufficiently small there will be vertical columns for which all relevant observations reside inside the model sub-domain of the process. However, if a vertical column is considered that is located close to the boundary to 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. These observations nonetheless are required on the local model sub-domain. Thus, 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 LSEIK filter is the following:

  SUBROUTINE PDAF_put_state_lseik(U_collect_state, U_init_dim_obs_full, U_obs_op_full, U_init_obs_full, &
                                  U_init_obs_local, U_prepoststep, U_prodRinvA_local, U_init_n_domains, &
                                  U_init_dim_local, U_init_dim_obs_local, &
                                  U_global2local_state, U_local2glocal_state, U_glocal2local_obs, &
                                  U_init_obsvar, U_init_obsvar_local, 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 ensembel of model states from the model fields. This is basically the inverse operation to U_distribute_state used in PDAF_get_state
• U_init_dim_obs_full: The name of the user-supplied routine that provides the size of observation vector
• U_obs_op_full: The name of the user-supplied routine that acts as the observation operator on some state vector
• U_init_obs_full: The name of the user-supplied routine that initializes the vector of observations
• U_init_obs_local: 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_local: The name of the user-supplied routine that computes the product of the inverse of the observation error covariance matrix with some matrix provided to the routine by PDAF. This operation occurs during the analysis step of the SEIK filter.
• U_init_n_domains: The name of the routine that provides the number of local analysis domains
• U_init_dim_local: The name of the routine that provides the state domains for a local analysis domain
• U_init_dim_obs_local: The name of the routine that initializes the size of the observation vector for a local analysis domain
• U_global2local_state: The name of the routine that initializes a local state vector from the global state vector
• U_local2global_state: The name of the routine that initializes the part of the global state vector corresponding to the provided local state vector
• U_global2local_obs: The name of the routine that initialized 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 to PDAF (This routine will only be executed, if an adaptive forgetting factor is used)
• U_init_obsvar_local: The name of the user-supplied routine that provides a mean observation error variance for the local analysis domain to PDAF (This routine will only be executed, if a local adaptive forgetting factor is used)
• status: The integer status flag. It is zero, if PDAF_get_state is exited without errors.

Note:

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

User-supplied routines

Here all user-supplied routines are described that are required in the call to PDAF_put_state_lseik. 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/ these routines are provided in files with the routines name without this prefix. In the example implementation in testsuite/src/dummymodel_1D the routines exist without the prefix, but with the extension _dummy_D.F90. In the section titles below we provide the name of the template file in parentheses.

U_collect_state (collect_state.F90)

This routine is independent from the filter algorithm used. See here for the description of this routine.

U_init_dim_obs_full (init_dim_obs_full.F90)

This routine is used by all local filter algorithms (LSEIK, LETKF).

The interface for this routine is:

SUBROUTINE init_dim_obs_full(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 will 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. Arrays for the locations 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. Anyway, one can base on an implemented routine init_dim_obs to implement init_dim_obs_full.

U_obs_op_full (obs_op_full.F90)

This routine is used by all local filter algorithms (LSEIK, LETKF).

The interface for this routine is:

SUBROUTINE obs_op_full(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 that 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:

• If the observation operator involves a global operation, e.g. some global integration, while using domain-decompostion one has to gather the information from the other model domains using MPI communication.
• Analogously to the situation with init_dim_obs_full, the routine is similar to init_dim_obs used for the global filters. However, with a domain-decompoared model also here 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_init_obs_full (init_obs_full.F90)

This routine is used by all local filter algorithms (LSEIK, LETKF). 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_full(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. The caller is the routine that computes an adaptive forgetting factor (PDAF_set_forget). It has to provide the full vector of observations in observation_f for the current time step.

Hints:

• As for the other 'full' routines: While the global counterpart of this routine (init_obs) has to initialize the observation vector 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.

U_init_obs_local (init_obs_local.F90)

This routine is used by all local filter algorithms (LSEIK, LETKF).

The interface for this routine is:

SUBROUTINE init_obs_local(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 analysis of the local analysis domain of index domain_p in observation_l for the current time step.

Hints:

• For parallel efficiency the LSEIK algorithm is implemented in a way that first the full vectors are initialized. Thus, if observation_f has been initialized before U_init_obs_local is executed (e.g. by U_init_dim_obs_full), 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.

U_prepoststep (prepoststep_seik.F90)

This routine can be identical to that used for the global SEIK filter. See here for the description of this routine.

U_prodRinvA_local (prodrinva_local.F90)

This routine is used by the local filters (LSEIK and LETKF).

The interface for this routine is:

SUBROUTINE prodRinvA_local(domain_p, step, dim_obs_p, rank, obs_p, A_p, C_p)
SUBROUTINE prodRinvA_local(domain_p, step, dim_obs_l, rank, 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) :: rank                 ! Rank of initial covariance matrix
  REAL, INTENT(in)    :: obs_l(dim_obs_l)     ! Local vector of observations
  REAL, INTENT(inout) :: A_l(dim_obs_l, rank) ! Input matrix from analysis routine
  REAL, INTENT(out)   :: C_l(dim_obs_l, rank) ! Output matrix

The routine is called during the loop over the local analysis domains in the analysis step. 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_weights can be called. The procedure is used in the example implementation and also demonstrated in the template routine.

Hints:

• the routine does not require that the product is implemented as a real matrix-matrix product. Rather, the product can be implemented in its most efficient form. For example, if the observation error covariance matrix is diagonal, only the multiplication of the diagonal with matrix A_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.

U_init_n_domains (init_n_domains.F90)

This routine is used by all local filter algorithms (LSEIK, LETKF).

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-dmain of the calling process has to be initialized.

Hints:

• As a simple case, if the localization is only performed horizontally, the local analysis domain 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_local (init_dim_local.F90)

This routine is used by all local filter algorithms (LSEIK, LETKF).

The interface for this routine is:

SUBROUTINE init_dim_local(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_local (init_dim_obs_local.F90)

This routine is used by all local filter algorithms (LSEIK, LETKF).

The interface for this routine is:

SUBROUTINE init_dim_obs_local(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 observation 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 init_obs_local is executed. Thus, as 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 init_obs_local can be sped up.

U_global2local_state (global2local_state.F90)

This routine is used by all local filter algorithms (LSEIK, LETKF).

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_local2global_state (local2global_state.F90)

This routine is used by all local filter algorithms (LSEIK, LETKF).

The interface for this routine is:

SUBROUTINE local2global_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 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.

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_global2local_obs (global2local_obs.F90)

This routine is used by all local filter algorithms (LSEIK, LETKF).

The interface for this routine is:

SUBROUTINE global2local_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 of 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_full), this vector has to be restricted to the relevant observations for the current local analysis domain. For it, one can e.g. initialize an index array when U_init_dim_obs_local is executed. (Which happens before U_global2local_obs)

U_init_obsvar (init_obsvar.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 impementation).

The interface for this routine is:

SUBROUTINE init_obsvar(step, dim_obs_p, obs_p, meanvar)

  INTEGER, INTENT(in) :: step          ! Current time step
  INTEGER, INTENT(in) :: dim_obs_p     ! PE-local dimension of observation vector
  REAL, INTENT(in) :: obs_p(dim_obs_p) ! PE-local observation vector
  REAL, INTENT(out)   :: meanvar       ! Mean observation error variance

The routine is called in the global filters during the analysis or by the routine that computes an adaptive forgetting factor (PDAF_set_forget). The routine has to initialize the mean observation error variance. For the global filters this should be the global mean.

Hints:

• For a model with domain-decomposition one might use the mean variance for the model sub-domain of the calling process. Alternatively one can compute a mean variance for the full model domain using MPI communication (e.g. the function MPI_allreduce).
• The observation vector obs_p is provided to the rotine for the case that the observation error variance is relative to the value of the observations.

U_init_obsvar_local (init_obsvar_local.F90)

This routine is used by the local filters LSEIK and LETKF. The routine is only called if the local adaptive forgetting factor is used (type_forget=2 in the example impementation).

The interface for this routine is:

SUBROUTINE init_obsvar_local(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 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.

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 later routines. 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 this are given with the descriptions of the routine interfaces above.

Before the analysis step is called the following is executed:

1. U_collect_state

In the analysis step, before the loop over all local analysis domains, the following routines are executed:

1. U_prepoststep
2. U_init_n_domains
3. U_init_dim_obs_full
4. U_obs_op_full
5. U_init_obs_full (Only executed, if the global adaptive forgetting factor is used (type_forget=1 in the example implemention))
6. U_init_obsvar (Only execute, if the global adaptive forgetting factor is used (type_forget=1 in the example implemention))

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

1. U_init_dim_local
2. U_init_dim_obs_local
3. U_global2local_state
4. U_global2local_obs
5. U_init_obs_local
6. U_global2local_obs
7. U_init_obsvar_local (Only, if the local adaptive forgetting factor is used (type_forget=2 in the example implementation))
8. U_prodRinvA_local
9. U_local2global_state

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

1. U_prepoststep