Modification of the model code for the ensemble integration

Overview

Numerical models are typically implemented for normal integration of some initial state. For the data assimilation with filter algorithm, an ensemble of model states has to be integrated for limited time until observations are available and an analysis step of the filter is computed. Subsequently, the updated ensemble has to be integrated again. To allow for these alternating ensemble integrations and analysis steps the model code has to be extended. The recommended implementation strategy for PDAF is to add an additional loop outside of the regular time-stepping loop of the model. This strategy has the potential to reduce the required chances in the model code to the minimum. In addition, a routine that simulates model errors might be required to be inserted into the time stepping loop of the model. The required extensions are described below.

Some operations that are specific to the model and the observations that are assimilated are performed by routines that are supplied by the user and that are called through the defined interface of PDAF. Generally, these user-supplied routines have to provide quite elementary operations, like initializing a model state vector for PDAF from model fields or providing the vector of observations. PDAF provides examples for these routines and templates that can be used as the basis for the implementation. As only the interface of these routines is specified, the user can implement the routines like a routine of the model. Thus, the implementation of these routines should not be difficult.

External ensemble loop

The external loop for the ensemble integration has to enclose the time stepping loop of the model. Next to the external loop, a control structure for exiting the external loop as well as two calls to subroutines of PDAF have to be added. These are the calls to PDAF_get_state and a filter-specific routine like PDAF_put_state_seik for the SEIK filter. Both routines are described in sections below.

The extended model code can look like this for the SEIK filter:

 pdaf_modelloop: DO  

    CALL PDAF_get_state(nsteps, ..., doexit, ...)

    ! Check whether forecast has to be performed
    ifcontrol: IF (doexit /= 1) THEN
     
       IF (nsteps > 0) THEN

         ... Time stepping code of the model ...         

       END IF

       CALL PDAF_put_state_seik(...)

    ELSE ifcontrol

       ! No more assimilation work; exit loop
       EXIT pdaf_modelloop

    END IF ifcontrol

 END DO pdaf_modelloop

In this example, which is taken from the example implementation in testsuite/src/dummymodel_1D, we use an unconditional DO loop (while loop). The exit flag doexit for this loop is set within PDAF_get_state. In addition, the variable nsteps is initialized, which defines the number of time steps to be performed during the current forecast phase. Thus, we only execute the time stepping code if nsteps>0. (If this has to be implemented using an IF-clause as in the example should be checked for the particular code).

PDAF_get_state

The routine PDAF_get_state has the purpose to initialize the information, whether further model integrations have to be computed and how many time steps have to be performed. In addition, the model fields to be propagated are initialized from the array holding the ensemble states.

The interface of PDAF_get_state is the following:

  SUBROUTINE PDAF_get_state(nsteps, timenow, doexit, U_next_obs, U_distribute_state, &
                            U_prepoststep, status)

with the following arguments:

• nsteps: An integer specifying upon exit the number of time steps to be performed
• timenow: A real specifying upon exit the current model time.
• doexit: An integer variable defining whether the assimilation process is completed and the program should exit the while loop. For compatibility 1 should be used for exit, 0 for continuing in the loop.
• U_next_obs: The name of a user supplied routine that initializes the variables nsteps, timenow, and doexit
• U_distributed_state: The name of a user supplied routine that initializes the model fields from the array holding the ensemble of model state vectors
• U_prepoststep: The name of a user supplied routine that is called before and after the analysis step. Here the user has the possibility to access the state ensemble and can e.g. compute estimated variances or can write the ensemble states the state estimate into files.
• status: The integer status flag. It is zero, if PDAF_get_state is existed without errors.

PDAF_put_state_*

There is a separate routine PDAF_put_state_* for each of the filter algorithms. The name of the routine includes the name of the filter at its end. The purpose of the PDAF_put_state_* routines is to write back the forecast model fields into the array holding the ensemble of model state vectors. In addition, the routine checks if the current forecast phase is completed. If not, the routine is exited and the next cycle of the ensemble loop is performed. If the current forecast phase is completed, the routine executes the analysis step of the chosen filter algorithm. The interface to each put-state routine is specific for each filter algorithm, because the names of several user-supplied routines have to be specified, which are specific for each filter algorithm. For example, the interface when using the SEIK filter is the following:

  SUBROUTINE PDAF_put_state_seik(U_collect_state, U_init_dim_obs, U_obs_op, &
                                 U_init_obs, U_prepoststep, U_prodRinvA, U_init_obsvar, 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: The name of the user-supplied routine that provides the size of observation vector
• U_obs_op: The name of the user-supplied routine that acts as the observation operator on some state vector
• U_init_obs: The name of the user-supplied routine that initializes the vector of observations
• U_prepoststep: The name of the pre/poststep routine as in PDAF_get_state
• U_prodRinvA: 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_obsvar: The name of the user-supplied routine that provides a mean observation error variance to PDAF (This routine will only be executed, if an adaptive forgetting factor is used)
• status: The integer status flag. It is zero, if PDAF_get_state is existed without errors.

Simulating model errors

The implementation of the filter algorithms does not support the specification of a model error covariance matrix. This was left out, because in the SEEK and SEIK filter, the handling can be extremely costly, as the model error covariance matrix has to be projected onto the ensemble space. Instead PDAF support the simulation of model errors by disturbing fields during the model integration. For this, some routine will be required that is inserted into the time stepping loop of the model. As this procedure is specific to each model, the is no routine provided by PDAF for this.

Compilation and testing

To compile the extended model code with PDAF, one has to extend the Makefile for the model. The core part of PDAF can be compiled separately as a library. It can then simply be linked to the model code. This is the strategy followed in the PDAF-package. The user-supplied routines also need to exist and need to be compiled and linked. However, for testing at this stage, only the user-supplied routines used in PDAF_get_state as well as the routine U_collect_state need to be implemented with functionality. The other routine will only be executed, when an actual analysis is performed.

If one out-comments the analysis routines in the PDAF_*_update routine (e.g. PDAF_seik_update), the analysis is not performed.