Implementation of the Analysis Step for 3D-Var with OMI

Overview

With Version 2.0 with introduced 3D variational assimilation methods to PDAF. There are genenerally three different variants: parameterized 3D-Var, 3D Ensemble Var, and hybrid (parameterized + ensemble) 3D-Var.

This page describes the implementation of the analysis step for the parameterized 3D-Var using PDAF-OMI.

For the analysis step of 3D-Var we need different operations related to the observations. These operations are requested by PDAF by call-back routines supplied by the user and provided in the OMI structure. The names of the routines that are provided by the user are specified in the call to the routine PDAFomi_assimilate_3dvar in the fully-parallel implementation (or PDAFomi_put_state_3dvar for the 'flexible' implementation) that was discussed before. 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 PDAFomi_assimilate_3dvar. Thus, some of the user-supplied routines that are explained on the page describing the modification of the model code for the ensemble integration are repeated here.

PDAFomi_assimilate_3dvar

The general aspects of the filter (or solver) 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. Here, we list the full interface of the routine. Subsequently, the user-supplied routines specified in the call is explained.

The interface for using the parameterized 3D-Var is:

  SUBROUTINE PDAFomi_assimilate_3dvar(U_collect_state, U_distribute_state, &
                                 U_init_dim_obs_pdafomi, U_obs_op_pdafomi, &
                                 U_cvt, U_cvt_adj, U_obs_op_lin_pdafomi, obs_op_adj_pdafomi, &
                                 U_prepoststep, 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 as well as 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_pdafomi: The name of the user-supplied routine that initializes the observation information and provides the size of observation vector
• U_cvt: The name of the user-supplied routine that applies the control-vector transformation (square-root of the B-matrix) on some control vector to obtain a state vector.
• U_cvt_adj: The name of the user-supplied routine that applies the adjoint control-vector transformation (with square-root of the B-matrix) on some state vector to obtain the control vector.
• U_obs_op_pdafomi: The name of the user-supplied routine that acts as the observation operator on some state vector
• U_obs_op_lin_pdafomi: The name of the user-supplied routine that acts as the linearized observation operator on some state vector
• U_obs_op_lin_pdafomi: The name of the user-supplied routine that acts as the adjoint observation operator on some state vector
• U_prepoststep: The name of the pre/poststep routine as in PDAF_get_state
• 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 PDAFomi_assimilate_global is exited without errors.

PDAFomi_put_state_3dvar

When the 'flexible' implementation variant is chosen for the assimilation system, the routine PDAFomi_put_state_global has to be used instead of PDAFomi_assimilate_global. 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_global 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 global filters is the following:

  SUBROUTINE PDAFomi_put_state_3dvar(U_collect_state, &
                                 U_init_dim_obs_pdafomi, U_obs_op_pdafomi, &
                                 U_cvt, U_cvt_adj, U_obs_op_lin_pdafomi, obs_op_adj_pdafomi, &
                                 U_prepoststep, outflag)

User-supplied routines

Here all user-supplied routines are described that are required in the call to PDAFomi_assimilate_3dvar. 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 tutorial implementations in tutorial/ these routines exist without the prefix, but with the extension _pdaf.F90. The user-routines relating to OMI are collected in the file callback_obs_pdafomi.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. 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.

U_collect_state (collect_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_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_pdafomi (callback_obs_pdafomi.F90)

This is a call-back routine for PDAF-OMI initializing the observation information. The routine just calls a routine from the observation module for each observation type.

See the documentation on callback_obs_pdafomi.F90 for more information.

U_obs_op_pdafomi (callback_obs_pdafomi.F90)

This is a call-back routine for PDAF-OMI applying the observation operator to the state vector. The routine calls a routine from the observation module for each observation type.

See the documentation on callback_obs_pdafomi.F90 for more information.

U_cvt (cvt_pdaf.F90)

The interface for this routine is:

SUBROUTINE cvt_pdaf(iter, dim_p, dim_cvec, cv_p, Vv_p)

  INTEGER, INTENT(in) :: iter           ! Iteration of optimization
  INTEGER, INTENT(in) :: dim_p          ! PE-local observation dimension
  INTEGER, INTENT(in) :: dim_cvec       ! Dimension of control vector
  REAL, INTENT(in)    :: cv_p(dim_cvec) ! PE-local control vector
  REAL, INTENT(inout) :: Vv_p(dim_p)    ! PE-local result vector (state vector increment)

The routine is called during the analysis step during the iterative minimization of the cost function. It has to apply the control vector transformation to the control vector and return the transformed result vector. Usually this transformation is the multiplication with the square-root of the background error covariance matrix B.

If the control vector is decomposed in case of parallelization it first needs to the gathered on each processor and afterwards the transformation is computed on the potentially domain-decomposed state vector.

U_cvt_adj (cvt_adj_pdaf.F90)

The interface for this routine is:

SUBROUTINE cvt_adj_pdaf(iter, dim_p, dim_cvec, Vv_p, cv_p)

  INTEGER, INTENT(in) :: iter           ! Iteration of optimization
  INTEGER, INTENT(in) :: dim_p          ! PE-local observation dimension
  INTEGER, INTENT(in) :: dim_cvec       ! Dimension of control vector
  REAL, INTENT(in)    :: Vv_p(dim_p)    ! PE-local result vector (state vector increment)
  REAL, INTENT(inout) :: cv_p(dim_cvec) ! PE-local control vector

The routine is called during the analysis step during the iterative minimization of the cost function. It has to apply the adjoint control vector transformation to a state vector and return the control vector. Usually this transformation is the multiplication with transposed of the square-root of the background error covariance matrix B.

If the state vector is decomposed in case of parallelization one needs to take care that the application of the trasformation is complete. This usually requries a comminucation with MPI_Allreduce to obtain a global sun.

U_obs_op_lin_pdafomi (callback_obs_pdafomi.F90)

This is a call-back routine for PDAF-OMI applying the linearized observation operator to the state vector. The routine calls a routine from the observation module for each observation type. If the full observation operator is lineaer the same operator can be used here.

See the documentation on callback_obs_pdafomi.F90 for more information.

U_obs_op_adj_pdafomi (callback_obs_pdafomi.F90)

This is a call-back routine for PDAF-OMI applying the adjoint observation operator to some vector inthe observation space. The routine calls a routine from the observation module for each observation type.

See the documentation on callback_obs_pdafomi.F90 for more information.

U_prepoststep (prepoststep_ens_pdaf.F90)

The routine has already been described for modifying the model for the ensemble integration and for inserting the analysis step.

See the page on inserting the analysis step for the description of this 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 user-supplied routines are essentially executed in the order they are listed in the interface to PDAFomi_assimilate_3dvar. The order can be important as some routines can perform preparatory work for later routines. For example, U_init_dim_obs_pdafomi prepares an index array that provides the information for executing the observation operator in U_obs_op_pdafomi. How this information is initialized is described in the documentation of OMI.

Before the analysis step is called the following routine is executed:

1. U_collect_state

The analysis step is executed when the ensemble integration of the forecast is completed. During the analysis step the following routines are executed in the given order:

1. U_prepoststep (Call to act on the forecast ensemble, called with negative value of the time step)
2. U_init_dim_obs_pdafomi
3. U_obs_op_pdafomi (multiple calls, one for each ensemble member)

Inside the analysis step the interative optimization is computed. This involves the repeated call of the routines:

1. U_cvt
2. U_obs_op_lin_pdafomi
3. U_obs_op_adj_pdafomi
4. U_cvt_adj

After the iterative optimization the following routines are executes to complte the analysis step:

1. U_cvt (Call to the control vector transform to compute the final state vector increment
2. U_prepoststep (Call to act on the analysis ensemble, called with (positive) value of the time step)

In case of the routine PDAFomi_assimilate_3dvar, the following routines are executed after the analysis step:

1. U_distribute_state
2. U_next_observation