Implementation of the Analysis step for the NETF (Nonlinear Ensemble Transform Filter) using PDAF's full interface

This page describes the implementation of the analysis step using PDAF's full interface, i.e. without using PDAF-OMI. This approach is supported by all versions of PDAF. However, this approach is mainly used in older implementations of PDAF and can be seen as a expert-mode. Please see the page on the analysis step in PDAF3 for the current implementation recommendation using the PDAF3 interface. The page also provides links to some other variants that were introduced in verisons of PDAF2.

The NETF algorithm was added with Version 1.12 of PDAF.

Overview

The NETF (nonlinear ensemble transform filter, Toedter and Ahrens, 2015) is a second-order exact particle filter with ensemble transofrmation (see, e.g. Vetra-Carvalho et al., 2018). There are different options to set perturbation noise or a stabilizing factor (type_winf, limit_winf) based on the effective sample size, see the Page on available options.

For the analysis step of the NETF, different 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. This procedure should simplify the implementation. The names of the required routines are specified in the call to the routine PDAF_assimilate_netf in the fully-parallel implementation (or PDAF_put_state_netf for the 'flexible' implementation) that was discussed before. With regard to the parallelization, all these routines 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_netf. Thus, some of the user-supplied routines that are explained on the page explaining the modification of the model code for the ensemble integration are repeated here.

The analysis step of the NETF is is wide parts similar to that of the ETKF, ESTKF, and SEIK filter. For this reason, the interface to the user-supplied routines is almost identical. Depending on the implementation it can be possible to use identical routines for the NETF and the ETKF with the exception of the routine U_likelihood. Differences are marked in the text below.

PDAF_assimilate_netf

This routine is used both in the fully-parallel and the flexible implementation variants of the data assimilation system. (See the page Modification of the model code for the ensemble integration for these variants)

Here, we list the full interface of the routine. Subsequently, the user-supplied routines specified in the call are explained.

The interface is:

  SUBROUTINE PDAF_assimilate_netf(U_collect_state, U_distribute_state, &
                                 U_init_dim_obs, U_obs_op, U_init_obs, U_prepoststep, &
                                 U_likelihood, U_next_observation, 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 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_prepoststep: The name of the pre/poststep routine as in PDAF_get_state
• U_likelihood: The name of the user-supplied routine that computes the likelihood of the observations for an ensemble member provide when the routine is called.
• 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 PDAF_assimilate_netf is exited without errors.

PDAF_assim_offline_netf

This routine is used to perform the analysis step for the offline mode of PDAF. The interface of the routine is identical with that of the 'assimilate'-routine, except that the user-supplied routines U_distribute_state, U_collect_state and U_next_observation are missing.

The 'assim_offline' routines were introduced with PDAF V3.0 to simplify the implementation of the offline mode.

The interface is:

  SUBROUTINE PDAF_assim_offline_netf( &
                                 U_init_dim_obs, U_obs_op, U_init_obs, U_prepoststep, &
                                 U_likelihood, status)

PDAF_put_state_netf

This routine exists for backward-compatibility. In implementations that were done for PDAF V2.3.1 and before, a 'put_state' routine was used for the 'flexible' parallelization variant and for the offline mode. This routine allows to continue using the previous implementation structure. The interface of the routine is identical with that of the 'assimilate'-routine, except that the user-supplied routines U_distribute_state and U_next_observation are missing.

The interface is:

  SUBROUTINE PDAF_put_state_netf(U_collect_state, &
                                 U_init_dim_obs, U_obs_op, U_init_obs, U_prepoststep, &
                                 U_likelihood, status)

User-supplied routines

Here all user-supplied routines are described that are required in the call to the analysis routine. 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 tutorials in tutorial/ and in the template directory templates/ these routines exist without the prefix, but with the extension _pdaf. The files are named correspondingly. 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 modifying the model code for the ensemble integration 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 modifying the model code for the ensemble integration 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, NETF).

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, NETF).

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, NETF).

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_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_likelihood (likelihood_pdaf.F90)

This routine is used only by the NETF and PF algorithms.

The interface for this routine is:

SUBROUTINE U_likelihood(step, dim_obs_p, obs_p, residual, likely)

  INTEGER, INTENT(in) :: step                ! Current time step
  INTEGER, INTENT(in) :: dim_obs_p           ! PE-local dimension of obs. vector
  REAL, INTENT(in)    :: obs_p(dim_obs_p)    ! PE-local vector of observations
  REAL, INTENT(in)    :: residual(dim_obs_p) ! Input vector holding the residual y-Hx
  REAL, INTENT(out)   :: likely              ! Output value of the likelihood

The routine is called during the analysis step. In the NETF, as in other particle filters, the likelihood of the observations has to be computed for each ensemble member. The likelihood is computed from the observation-state residual according to the assumed observation error distribution. Commonly, the observation errors are assumed to be Gaussian distributed. In this case, the likelihood is exp(-0.5*(y-Hx)^T*R^-1*(y-Hx)).

For a model with domain decomposition, resid contains the part of the matrix that resides on the model sub-domain of the calling process. The likelihood has to be computed for the global state vector. Thus some parallel communication might be required to complete the computation.

Hints:

• The routine is very similar to the routine U_prodRinvA. The main addition is the computation of the likelihood after computing R^-1*(y-Hx), which corresponds to R^-1*A_p in U_prodRinvA.
• The information about the inverse observation error covariance matrix has to be provided by the user. Possibilities are to read this information from a file, or to use a Fortran module that holds this information, which one could already prepare in init_pdaf.
• The routine does not require that the product is implemented as a real matrix-vector 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 inverse diagonal with the vector resid 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.

U_next_observation (next_observation_pdaf.F90)

This routine is independent of the filter algorithm used. See the page on modifying the model code for the ensemble integration for the description of this routine.

Execution order of user-supplied routines

For the NETF, the user-supplied routines are essentially executed in the order they are listed in the interface to PDAF_assimilate_netf. The order can be important as some routines can perform preparatory work for later routines. For example, U_init_dim_obs can prepare an index array that provides the information for executing the observation operator in U_obs_op.

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
3. U_init_obs
4. U_obs_op (dim_ens calls; one call for each ensemble member)
5. U_likelihood (dim_ens calls; one call for each ensemble member)
6. U_prepoststep (call to act on the analysis ensemble, called with (positive) value of the time step)

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

1. U_distribute_state
2. U_next_observation