Adding an Assimilation Method to PDAF

PDAF provides an internal interface that makes it easy to add another assimilation method. Here we describe the implementation strategy and internal structure of PDAF valid for version 2.0 and later. In this text, we assume that the reader is already familiar with PDAF to the extend that it is known how PDAF is connected to a model as is described in the Implementation Guide.

The internal structure of PDAF is organized into a generic part providing the infrastructure to perform ensemble forecasts and filter analysis steps. This generic part is independent of the particular filter algorithm and only distinguishes between ensemble based filters (all filters except SEEK) and mode based filters (currently only SEEK, which integrates r modes plus one central model state). The specific routines for a DA method are called through an internal interface.

In PDAF, each DA algorithm consists of 3 mandatory routines plus 2 optional routines. All routines are described below. They are called through the internal interface of PDAF, except for the "PDAF_assimilate" or "PDAF_put_state" routine ('PDAF_assimilate_X' or PDAF_put_state_X where X is the name of the selected DA method), which is directly called in the model code.

PDAF's Internal Interface

Before explaining the method-specific routines and the calling interface of each routine, we provide an overview of the internal interface routines of PDAF. The structure of the internal interface of PDAF is depicted in Figure 1 (For the method-specific routines, 'X' is the name of the DA method). Shown are only the routines that are relevant for the implementation of a new filter method grouped by type. To add a filter algorithm, new method-specific routines (right column of Fig. 1) need to be implemented. These routines are registered in PDAF by modifying the internal interface routines in the middle column of Fig. 1.

Figure 1: Structure of the internal interface of PDAF. There are 6 interface routines (middle column) that connect the generic part with filter-specific routines. For each filter there are 5 filter-specific routines (right column). The three routines marked in blue are called inside the model code, while the routines marked in yellow are internal routines of PDAF. For PDAF-OMI there are seprate routines for local filters (PDAFomi_assimilate_local) and global filters (PDAFomi_assimilate_global).

The separate routines are the following:

Calls by PDAF_init

The routine PDAF_init calls

PDAF_init_filters	Interface routine to PDAF_X_init. PDAF_X_init performs the filter-specific initialization of parameters and calls the user-supplied routine that initializes the initial ensemble of model states.
PDAF_alloc_filters	Interface routine to PDAF_X_alloc. PDAF_X_alloc allocates the filter-specific arrays.
PDAF_options_filters	interface routine to PDAF_X_options. PDAF_X_options is an optional routine. Its purpose is to display an overview of available options for the filter algorithm.

When PDAF_init is called, the DA method is chosen by its ID number. Internally to PDAF, each DA method is identified by a string that is defined in PDAF_init_filters. The interface routines have a very simple structure. In general, they select the method-specific routine based on the string identifying the filters. When a DA method is added, a line for the corresponding method-specific routine has to be inserted to each of the interface routines. One can also remove a DA method from PDAF by deleting the corresponding lines form the internal interface routines.

See the section on filter-specific routines for a detailed description.

Calls by PDAF_print_info

The routine PDAF_print_info only includes the interface to PDAF_X_memtime

• PDAF_X_memtime displays information on the run time of the different parts of the assimilation process as well as information on the amount of allocated memory. This functionality is optional. See the section on filter-specific routines for a detailed description.

Calls by PDAFomi_assimilate_local or PDAFomi_assimilate_global

The routine PDAFomi_assimilate_local, for local ensemble filters and smoothers, or analogously the other available PDAF_omi_assimilate routines, like for global ensemble filters and smoothers (PDAFomi_assimilate_global) or the different variants of 3D-Var (PDAFomi_assimilate_*3dvar*) are called directly from the model code. These are used for the fully-parallel implementation variant. For the flexible-parallelization variant the analogous routines PDAFomi_put_state_* are provided. These are generic routines which internally call the method-specific routine (PDAF_assimilate_X or PDAF_put_state_X) according to the chosen filter which then calls the actual update routine (PDAF_X_update). This call structure is explained in Figure 2.

Figure 2: Internal call structure for the analysis step. The generic routine (here PDAFomi_assimilate_local as example) calls the method-specific routine. Here PDAF_assimilate_X is the routine that controls the ensemble run in case of the fully-parallel implementation. The routine PDAF_put_state_X is used for both the fully-parallel and flexible parallelization variants. For the flexible parallelization one uses PDAFomi_put_state_local which directly calls PDAF_put_state_X, while for the fully parallel variant this routine is called by PDAF_assimilate_X. PDAF_put_state_X controls the ensemble for the case that multiple ensmeble states are propagated by a single model task and collects the ensemble from the ensemble tasks before the assimilate update and distributes the ensemble to the model tasks afterwards. The actual main routine for the DA method is PDAF_X_update which is called by PDAF_put_state_X. The routines on the right without 'omi' in their name are those with the full interface so that they are usable without OMI.

See the section on filter-specific routines for a detailed description for PDAF_assimilate_X and PDAF_put_state_X.

Internal dimensions

PDAF internally stores the dimensions of the assimilation system. The dimensions are declared in the Fortran module PDAF_mod_filter. Important are the following dimensions:

dim_p	The size of the state vector (with parallelization the size of the local state vector for the current process)
dim_ens	The overall size of the ensemble
dim_ens_l	If the ensemble integration is distributed over several ensemble tasks, this variable stores the size of the sub-ensemble handled by the current process. (dim_ens_l equals dim_ens if no parallelization or if only a single model task is used.) For the fully parallel implementation it is din_ens_l=1. Note that this variable is only used in the ensemble handling of PDAF, but not in the DA update.
rank	The maximum rank of the ensemble covariance matrix. In almost all cases, it is dim_ens-1. Used in error-subspace filters, (L)ESTKF and (L)SEIK

Internal arrays

Several internal arrays are allocated when PDAF is initialized. These arrays are declared in PDAF_mod_filter. They are allocated in PDAF_X_alloc (see below for details) and remain allocated throughout the assimilation process. For the processes that computes the analysis (those with filterpe=.true.) the following arrays are defined:

Array	Dimension	Comment
state	dim_p	State vector. Used in all filters. Inside the filter code, it's usually called state_p
eofV	dim_p x dim_ens	Ensemble array. Used in all filters. Inside some filters the name is ens_p
eofU	dim_ens-1 x dim_ens-1 (SEEK, SEIK, ESTKF) dim_ens x dim_ens (ETKF)	Eigenvalue matrix U from P=VUVT (SEEK, SEIK) or transform matrix A (ETKF, ESTKF). Not used in EnKF. In some routines the matrix is called Uinv to indicate the inverse
state_inc	dim_p	state increment vector. Only allocated if incremental analysis updates are used

For the processes that only compute model forecasts but are not involved in the analysis step (i.e. filterpe=.false.), only one array is defined:

Array	Dimension	Comment
eofV	dim_p x dim_ens_l	Ensemble array on non-filter processes. Used in all filters.

Filter-specific routines

When a filter algorithm is added, the following filter routines have to be implemented and inserted to each interface routines described above.

• PDAF_X_init
• PDAF_X_alloc
• PDAF_X_options (optional)
• PDAF_X_memtime (optional)

The routines are very similar for all DA methods.

In addition, one has to implement the routines

• PDAF_put_state_X and PDAF_assimilate_X

As an example we recommend to see e.g. the routines for the ETKF or LETKF.

We recommend to base on the routines of an existing filter, as most of the routines can be easily adapted to a new filter method.

PDAF_X_init

The routine PDAF_X_init performs the initialization of filter-specific parameters. In addition, it prints information about the configuration.

The interface is as follows:

  SUBROUTINE PDAF_X_init(subtype, param_int, dim_pint, param_real, dim_preal, &
                                 ensemblefilter, fixedbasis, verbose, outflag)

  INTEGER, INTENT(inout) :: subtype                ! Sub-type of filter
  INTEGER, INTENT(in)    :: dim_pint               ! Number of parameters in param_int
  INTEGER, INTENT(inout) :: param_int(dim_pint)    ! The array of integer parameters
  INTEGER, INTENT(in)    :: dim_preal              ! Number of parameters in param_real
  REAL, INTENT(inout)    :: param_real(dim_preal)  ! The array of real parameters 
  LOGICAL, INTENT(out)   :: ensemblefilter         ! Flag, whether the filter is an ensemble filter or a mode-based filter
  LOGICAL, INTENT(out)   :: fixedbasis             ! Flag, whether the chosen `subtype` uses a fixed ensemble (only the ensemble mean is integrated by the model)
  INTEGER, INTENT(in)    :: verbose                ! Control screen output
  INTEGER, INTENT(inout) :: outflag                ! Status flag

The routine has to perform the following operations:

• initialize the PDAF-internal parameter variables specific for the DA method from the provided values of subtype, param_int, and param_real.
• set the logical flags ensemblefilter and fixedbasis.

The existing implementations also include some screen output about the configuration.

Please note:

• The routine should check, whether the provided value of subtype is a valid choice. If this is not the case, the error flag should be set to 2.
• Only parameters from param_int and param_real up to the value dim_pint and dim_preal, respectively, should be considered in the initialization. The arrays may be bigger, but the user defines which parameters are to be used be setting the values of dim_pint and dim_preal.
• The error flag outflag is initially set to 0.
• The internal parameters are declared in the Fortran module PDAF_mod_filter. If a new filter algorithm requires additional parameters, their declaration should be added to the module. Alternatively, one can introduce a new module the holds the parameters specific to the new DA method.

PDAF_X_alloc

The routine PDAF_X_alloc allocates arrays for the data assimilation. These are the arrays that are used in the forecasting and hence need to be persistently allocated, like the ensemble array and a state vector. The success of the allocation is checked.

The interface is as follows:

  SUBROUTINE PDAF_X_alloc(subtype, outflag)

  INTEGER, INTENT(in) :: subtype        ! Sub-type of filter
  INTEGER, INTENT(out):: outflag        ! Status flag

All arrays that need to be allocated are declared in the Fortran module PDAF_mod_filter. Here, also the shapes of the arrays are declared. For the allocation of arrays, one has to distinguish between processes that compute the analysis step (filterpe=.true.) and those that only participate in the ensemble forecast (filterpe=.false.).

For the processes that compute the analysis (those with filterpe=.true.) it is mandatory to allocate the following two arrays:

• state: The state vector of size dim_p.
• eofV: This is the ensemble matrix in all ensemble-based filters. For SEEK it is the matrix holding eigenvectors. eofV has size (dim_p, dim_ens).

Depending on the filter algorithm some of the following arrays also need to be allocated:

• eofU: This is the eigenvalue matrix U used in the SEIK and SEEK filters and the transform matrix of ESTKF (here, its size is (rank,rank)). For ETKF, it is the transform matrix A of size (dim_ens,dim_ens). The array should always be allocated. For method that don't use such transform matrix, for example the EnKF, it can be allocated with size (1,1).
• state_inc: The increment to the state vector computed in the analysis step. It only needs to be allocated in this routine, if incremental analysis updating is implemented. Otherwise, it is sufficient to allocate and deallocate state_inc in the routine performing the analysis step. The size of state_inc is dim_p.
• bias: If the filter algorithm is implemented with bias correction, the vector bias with size dim_bias_p is allocated.

Processes that only participate in the computation of the ensemble forecast, but are not involved in computing the analysis step (those with filterpe=.false.), operate only on a sub-ensemble. Accordingly, an ensemble array for this sub-ensemble has to be allocated. This is:

• eofV: This is the ensemble matrix in all ensemble-based filters. For SEEK it is the matrix holding eigenvectors. For the processes with filterpe=.false., eofV has size (dim_p, dim_ens_l).

PDAF_X_options

The optional routine PDAF_X_options displays information on the available options for the filter algorithm.

The routine has no arguments. Thus the interface is as follows:

  SUBROUTINE PDAF_X_options()

The following display is recommended:

• Available subtypes (At least '0' for standard implementation)
• Parameters used from the parameter arrays param_int and param_real.

PDAF_X_memtime

The optional routine PDAF_X_memtime displays information about allocated memory and the execution time of different parts of the filter algorithm.

The interface is as follows:

  SUBROUTINE PDAF_X_memtime(printtype)

  INTEGER, INTENT(in) :: printtype    ! Type of screen output:  
                                      ! (1) general timings
                                      ! (3) timing information for call-back routines and PDAF-internal operations
                                      ! (4) second-level timing information
                                      ! (5) very detailed third-level timing information
                                      ! (10) process-local allocated memory
                                      ! (11) globally allocated memory

The timing operations are implemented using the module PDAF_timer, which provides the function PDAF_timeit. Memory allocation is computed using PDAF_memcount, which is provided by the module PDAF_memcounting.

PDAF_assimilate_X / PDAF_put_state_X

These routines are called by PDAFomi_assimilate_* or PDAFomi_put_state_*, which are directly inserted into the model code, if the online mode of PDAF is used together with PDAF-OMI. The description of the implementation of the analysis step in the Implementation Guide explains the interface for the algorithms that are included in the PDAF package. PDAFomi_assimilate_* are interface routines for PDAF-OMI and call the PDAF_assimilate_X routines as described above. (Likewise the routines PDAFomi_put_state_* call PDAF_put_state_X.)

As described before, PDAF_assimilate_X calls PDAF_put_state_X which then calls the actual update routine of the DA method. Here. PDAF_assimilate_X mainly passes its arguments over to PDAF_put_state_X, but it also controls the ensemble run by counting the number of time steps.

Apart from the usual integer status flag, the interface of the routines contains the names of the user-supplied call-back 2routines that are required for the analysis step. Usually, the minimum set of routines are:

• U_collect_state: The routine that writes model fields into the state vector, i.e. a single column of the ensemble state array
• U_init_dim_obs: The routine that determines the size of the observation vector
• U_obs_op: The routine that contains the implementation of the observation operator
• U_init_obs: The routine that provdes the vector of observations
• U_prepoststep: The pre- and post-step routine in which the forecast and analysis ensembles can be analyzed or modified.

PDAF_assimilate_X uses in addition

• U_distribute_state: The routine that fills the field arrays of a model from the state vector, i.e. a single column of the ensemble state array
• U_next_observation: The routine that specified the number of time steps until the next DA analysis update.

Further routines can be added and depend on the requirements of the filter algorithm. Common is a routine that involves the observation error covariance matrix. For example, the (L)ETKF and (L)ESTKF methods use the routine

• U_prodRinvA which has to multiply some temporary matrix of the filter method with the inverse observation error covariance matrix.

When one plans to implement a new filter, we recommend to check whether the filter is compatible with the existing local or global filters, in particular whether it uses the same call-back routines. In this case, the new filter can be added into the existing routines (either PDAFomi_assimilate_global/PDAFomi_put_state_global for a global DA method or likewise PDAFomi_assimilate_local/PDAFomi_put_state_local for a DA method using domain localization).

The routine PDAF_assimilate_X and PDAF_put_state_X provide the actual infrastructure to manage the ensemble forecasting. Here PDAF_assimilate_X is used for the fully parallel setup and counts the time steps and finally calls PDAF_put_state_X.

The routines PDAF_put_state_X prepare for the actual analysis step, that is called inside these routines as a separate routine PDAF_X_update. PDAF_put_state_X also performs the ensemble management for the flexible parallelization variant in which more than one ensemble state can be integrated by a model task. The operations implemented in PDAF_put_state_X are:

• Write model fields back into the ensemble array (by calling U_collect_state)
• Increment the counter for the integrated ensemble members (named counter and provided by the module PDAF_mod_filter.
• Check, if the ensemble integration is completed (in that case, it is member = local_dim_ens + 1). If not, exit PDAF_put_state_X.
• When the ensemble integration is completed, the following operations are required:
  • If more than one model task is used: Collect the sub-ensembles from all model tasks onto the processes that perform the analysis step. This operation is done by the subroutine PDAF_gather_ens.
  • Call the routine that computes the analysis step for the chosen filter algorithm (typically named PDAF_X_update).
  • Reset the control variables for the ensemble forecast (initevol=1, member=1, step=step_obs+1).

In general, the PDAF_put_state routines of all ensemble-based filters have the same structure. For the implementation of a new filter we recommend to base on an existing routine, e.g. that of for the ETKF. Thus, one copies the routine to a new name. Then, one adapts the interface for the required user-supplied routines of the new DA method. In addition, the call of the routine PDAF_X_update holding the DA analysis update step has to be revised (name of the routine, required user-supplied routines).

The routine PDAF_assimilate_X is mainy an interface routine to PDAF_put_state_X. It counts the time steps and calls PDAF_put_state_X when the forecast phase is complete. Thus, to implement a new filter, one can copy a routine and apart form adapting the name (X) and the interface specifying the call-back routines, there should be no need for changes.

Analysis update step PDAF_X_update

The actual DA analysis update is computed in the routine PDAF_X_update. The routine is provided with the names of the call-back routines, the arrays as well of the relevant dimensions as describe above.

The structure of the operations in PDAF_X_update can be designed freely when implementing a new DA method. The existing methods follow the recommended structure in which PDAF_X_update only performes preparations for the actual analysis update and then calls PDAF_X_analysis for the computation of the actual update. Here, the operations are different for global and local methods.

Global Methods

For global methods, e.g. ESTKF, ETKF, EnKF, NETF, the general structure of PDAF_X_update is:

   IF (fixed ensemble, i.e. subtype==2 or subtype==3) THEN
      Add central state and ensemble perturbations
   ENDIF

   CALL U_prepoststep    # Perform pre/poststep before analysis update

   CALL PDAF_X_analysis  # Compute global analysis update of ensemble

   CALL PDAF_smoother    # Compute smoother update (Only for filters for which a smoother is implemented)
  
   CALL U_prepoststep    # Perform pre/poststep after analysis update

You will find this structure e.g. in PDAF_ESTKF_update.F90. This structure is also valid for the LEnKF. The actual implementation, e.g. in PDAF_ESTKF_update.F90 contains additional functionality e.g. calls for timers and normal screen output or debug outputs. Also there can be several different calls PDAF_X_analysis depending on the subtype of a filter.

This general structure should be widely usable. Thus, when implementing a new global ensemble method one could copy an existing file and adapt it for the new method. The main functionality of the new DA method would then be implemented in PDAF_X_analysis.

Local Methods

For domain-localized Methods, e.g. LESTKF, LETKF or LNETF the general structure of PDAF_X_update is different from the global methods because the local analysis loop is contained in PDAF_X_update and only a local analysis routine is called. The general structure is:

   IF (fixed ensemble, i.e. subtype==2 or subtype==3) THEN
      Add central state and ensemble perturbations
   ENDIF

   CALL U_prepoststep          # Perform pre/poststep before analysis update

   CALL U_init_n_domains       # Determine number of local analysis domains (n_domains_p)

   # --- Perform global operations on observations and ensemble
   CALL U_init_dim_obs         # Determine number of observations (PDAF-OMI routine init_dim_obs_pdafomi)
   DO i = 1, dim_ens
      CALL U_obs_op            # Apply observation operator to all ensemble state (PDAF-OMI routine obs_op_pdafomi)
   ENDDO
   Compute mean forecast state (state_p)
   Compute mean observed forecast state (HXbar_f)
   # --- End of global operations

   # --- Perform local analysis loop
   DO i = 1, n_domains_p  
     CALL U_init_dim_l         # Set size of local state vector 
     CALL U_init_dim_obs_l     # Set number of observations within radius (PDAF-OMI routine init_dim_obs_l_pdafomi)
     DO i = 1, dim_ens
        CALL U_g2l_state       # Get local state vector for all ensemble states
     ENDDO

     CALL PDAF_X_analysis      # Compute local analysis update of ensemble

     DO i = 1, dim_ens
        CALL U_l2g_state       # Update global state vector for all ensemble states
     ENDDO
     CALL U_l2g_state          # Update global state vector for ensemble mean state

     CALL PDAF_smoother_local  # Only for filters for which a smoother is implemented
   ENDDO
   # --- End of local analysis loop
  
   CALL U_prepoststep          # Perform pre/poststep after analysis update

You will find this structure e.g. in PDAF_LESTKF_update.F90. This structure is not used for the LEnKF because this filter doe snot use a local analysis loop. The actual implementation, e.g. in PDAF_LESTKF_update.F90 contains additional functionality e.g. calls for timers and normal screen output or debug outputs. Also there can be different calls PDAF_X_analysis depending on the subtype of a filter.

This general structure should be widely usable for DA methods that use domain localization. Accordingly, when implementing a new local ensemble method one could copy an existing file and adapt it for the new method. The main functionality of the new DA method would then be implemented in PDAF_X_analysis.