Implementation of the Analysis Step for 3D-Var without using 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 in the classical way without 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 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_assimilate_3dvar in the fully-parallel implementation (or PDAF_put_state_3dvar for the 'flexible' implementation) described below. With regard to the parallelization, all these routines (except U_collect_state) 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_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.

PDAF_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 PDAF_assimilate_3dvar(U_collect_state, U_distribute_state, &
                                 U_init_dim_obs, U_obs_op, U_init_obs, U_prodRinvA, &
                                 U_cvt, U_cvt_adj, U_obs_op_lin, U_obs_op_adj, &
                                 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: 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_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 ETKF.
• 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_lin: The name of the user-supplied routine that acts as the linearized observation operator on some state vector
• U_obs_op_adj: 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 the routine is exited without errors.

PDAFomi_put_state_global

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_assimilate_3dvar(collect_state_pdaf, &
                                 U_init_dim_obs, U_obs_op, U_init_obs, U_prodRinvA, &
                                 U_cvt, U_cvt_adj, U_obs_op_lin, U_obs_op_adj, &
                                 prepoststep_pdaf, 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 (init_dim_obs_pdaf.F90)

This routine is used by all global filter algorithms (SEEK, SEIK, EnKF, ETKF) and the 3D-Var methods.

The interface for this routine is:

SUBROUTINE init_dim_obs(step, dim_obs_p)

  INTEGER, INTENT(in)  :: step       ! Current time step
  INTEGER, INTENT(out) :: dim_obs_p  ! Dimension of observation vector

The routine is called at the beginning of each analysis step. It has to initialize the size dim_obs_p of the observation vector according to the current time step. Without parallelization dim_obs_p will be the size for the full model domain. When a domain-decomposed model is used, dim_obs_p will be the size of the observation vector for the sub-domain of the calling process.

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 locations of the observations, which will be used later, e.g. to implement the observation operator. An array for the locations can be defined in a module like mod_assimilation of the example implementation.

U_obs_op (obs_op_pdaf.F90)

This routine is used by all global filter algorithms (SEEK, SEIK, EnKF, ETKF) and the 3D-Var methods.

The interface for this routine is:

SUBROUTINE obs_op(step, dim_p, dim_obs_p, state_p, m_state_p)

  INTEGER, INTENT(in) :: step               ! Current time step
  INTEGER, INTENT(in) :: dim_p              ! PE-local dimension of state
  INTEGER, INTENT(in) :: dim_obs_p          ! Dimension of observed state
  REAL, INTENT(in)    :: state_p(dim_p)     ! PE-local model state
  REAL, INTENT(out) :: m_state_p(dim_obs_p) ! PE-local observed state

The routine is called during the analysis step. 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_p.

For a model using domain decomposition, the operation is on the PE-local sub-domain of the model and has to provide the observed sub-state for the PE-local domain.

Hint:

• If the observation operator involves a global operation, e.g. some global integration, while using domain-decomposition one has to gather the information from the other model domains using MPI communication.

U_init_obs (init_obs_pdaf.F90)

This routine is used by all global filter algorithms (SEEK, SEIK, EnKF, ETKF) and the 3D-Var methods.

The interface for this routine is:

SUBROUTINE init_obs(step, dim_obs_p, observation_p)

  INTEGER, INTENT(in) :: step             ! Current time step
  INTEGER, INTENT(in) :: dim_obs_p        ! PE-local dimension of obs. vector
  REAL, INTENT(out)   :: observation_p(dim_obs_p) ! PE-local observation vector

The routine is called during the analysis step. It has to provide the vector of observations in observation_p for the current time step.

For a model using domain decomposition, the vector of observations that exist on the model sub-domain for the calling process has to be initialized.

U_prodRinvA (prodrinva_pdaf.F90)

This routine is used by all filter algorithms that use the inverse of the observation error covariance matrix (SEEK, SEIK, and ETKF) and the 3D-Var methods.

The interface for this routine is:

SUBROUTINE prodRinvA(step, dim_obs_p, dim_ens, obs_p, A_p, C_p)

  INTEGER, INTENT(in) :: step                ! Current time step
  INTEGER, INTENT(in) :: dim_obs_p           ! PE-local dimension of obs. vector
  INTEGER, INTENT(in) :: dim_ens             ! Ensemble size
  REAL, INTENT(in)    :: obs_p(dim_obs_p)    ! PE-local vector of observations
  REAL, INTENT(in)    :: A_p(dim_obs_p, dim_ens) ! Input matrix from analysis routine
  REAL, INTENT(out)   :: C_p(dim_obs_p, dim_ens) ! Output matrix

The routine is called during the analysis step. In the algorithms the product of the inverse of the observation error covariance matrix with some matrix has to be computed. For the ETKF, this matrix holds the observed part of the ensemble perturbations. The matrix is provided as A_p. The product has to be given as C_p.

For a model with domain decomposition, A_p contains the part of the matrix that resides on the model sub-domain of the calling process. The product has to be computed for this sub-domain, too.

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_p has to be implemented.
• The observation vector obs_p is provided through the interface for cases where the observation error variance is relative to the actual value of the observations.
• The interface has a difference for SEIK and ETKF: For ETKF the third argument is the ensemble size (dim_ens), while for SEIK it is the rank of the covariance matrix (usually ensemble size minus one). In addition, the second dimension of A_p and C_p has size dim_ens for ETKF, while it is rank for the SEIK filter. (Practically, one can usually ignore this difference as the fourth argument of the interface can be named arbitrarily in the routine.)

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 (obs_op_pdaf_lin.F90)

This routine is used by all 3D-Var methods.

The interface for this routine is:

SUBROUTINE obs_op_lin(step, dim_p, dim_obs_p, state_p, m_state_p)

  INTEGER, INTENT(in) :: step               ! Current time step
  INTEGER, INTENT(in) :: dim_p              ! PE-local dimension of state
  INTEGER, INTENT(in) :: dim_obs_p          ! Dimension of observed state
  REAL, INTENT(in)    :: state_p(dim_p)     ! PE-local model state
  REAL, INTENT(out) :: m_state_p(dim_obs_p) ! PE-local observed state

The routine is called during the analysis step. It has to perform the operation of the linearized observation operator acting on a state vector increment that is provided as state_p. The observed state has to be returned in m_state_p.

For a model using domain decomposition, the operation is on the PE-local sub-domain of the model and has to provide the observed sub-state for the PE-local domain.

Hint:

• If the observation operator involves a global operation, e.g. some global integration, while using domain-decomposition one has to gather the information from the other model domains using MPI communication.

U_obs_op_adj (obs_op_pdaf_adj.F90)

This routine is used by all 3D-Var methods.

The interface for this routine is:

SUBROUTINE obs_op_adj(step, dim_p, dim_obs_p, state_p, m_state_p)

  INTEGER, INTENT(in) :: step                 ! Current time step
  INTEGER, INTENT(in) :: dim_p                ! PE-local dimension of state
  INTEGER, INTENT(in) :: dim_obs_p            ! Dimension of observed state
  REAL, INTENT(in)    :: m_state_p(dim_obs_p) ! PE-local observed state
  REAL, INTENT(out)   :: state_p(dim_p)       ! PE-local model state

The routine is called during the analysis step. It has to perform the operation of the adjoint observation operator acting on a vector in observation space that is provided as m_state_p. The resulting state vector has to be returned in m_state_p.

For a model using domain decomposition, the operation is on the PE-local sub-domain of the model and has to provide the observed sub-state for the PE-local domain.

Hint:

• If the observation operator involves a global operation, e.g. some global integration, while using domain-decomposition one has to gather the information from the other model domains using MPI communication.

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