= Implementation of the Analysis step for the PF (Particle Filter) algorithm =
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The PF algorithm was added with Version 1.14 of PDAF.
== Overview ==
For the analysis step of the PF, 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_pf` in the fully-parallel implementation (or `PDAF_put_state_pf` 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_pf`. 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 PF is is wide parts similar to that of filters like ETKF, and ESTKF filters and strongly related to the NETF method. For this reason, the interface to the user-supplied routines is almost identical, and in fact identical to the NETF. Depending on the implementation it can be possible to use identical routines for the PF and the ETKF with the exception of the routine `U_likelihood`. Differences are marked in the text below. However, the NETF and PF have the same interface, but there are some different options that can be set when PDAF_init is called.
== `PDAF_assimilate_pf` ==
The general aspects of the filter-specific routines `PDAF_assimilate_*` have been described on the page [ModifyModelforEnsembleIntegration Modification of the model code for the ensemble integration] and its sub-page on [InsertAnalysisStep 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 once more the full interface. Subsequently, the full set of user-supplied routines specified in the call to `PDAF_assimilate_pf` is explained.
The interface when using the PF method is the following:
{{{
SUBROUTINE PDAF_assimilate_pf(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_statecollect_state_pdaf.F90 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_statedistribute_state_pdaf.F90 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_obsinit_dim_obs_pdaf.F90 U_init_dim_obs]: The name of the user-supplied routine that provides the size of observation vector
* [#U_obs_opobs_op_pdaf.F90 U_obs_op]: The name of the user-supplied routine that acts as the observation operator on some state vector
* [#U_init_obsinit_obs_pdaf.F90 U_init_obs]: The name of the user-supplied routine that initializes the vector of observations
* [#U_prepoststepprepoststep_ens_pdaf.F90 U_prepoststep]: The name of the pre/poststep routine as in `PDAF_get_state`
* [#U_likelihoodlikelihood_pdaf.F90 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_observationnext_observation_pdaf.F90 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_pf` is exited without errors.
== `PDAF_put_state_pf` ==
When the 'flexible' implementation variant is chosen for the assimilation system, the routine `PDAF_put_state_pf` has to be used instead of `PDAF_assimilate_pf`. The general aspects of the filter specific routines `PDAF_put_state_*` have been described on the page [ModifyModelforEnsembleIntegration Modification of the model code for the ensemble integration]. The interface of the routine is identical with that of `PDAF_assimilate_pf` with the exception of the specification of the user-supplied routines `U_distribute_state` and `U_next_observation` are missing.
The interface when using the PF method is the following:
{{{
SUBROUTINE PDAF_put_state_pf(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 `PDAF_assimilate_pf`. For some of the generic routines, we link to the page on [ModifyModelforEnsembleIntegration 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 example implementation in `testsuite/src/dummymodel_1D` these routines exist without the prefix, but with the extension `_pdaf.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 [InsertAnalysisStep#U_collect_statecollect_state_pdaf.F90 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 [InsertAnalysisStep#U_distribute_statedistribute_state_pdaf.F90 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, NETF, PF).
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, PF).
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, PF).
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 been described on the [ModifyModelforEnsembleIntegration#U_prepoststepprepoststep_ens_pdaf.F90 page on modifying the model code for the ensemble integration] for the SEIK filter. For the PF, the interface is generally identical. For completeness, we repeat the description here.
The interface for this routine is
{{{
SUBROUTINE prepoststep(step, dim_p, dim_ens, dim_ens_p, dim_obs_p, &
state_p, Uinv, ens_p, flag)
INTEGER, INTENT(in) :: step ! Current time step
! (When the routine is called before the analysis -step is provided.)
INTEGER, INTENT(in) :: dim_p ! PE-local state dimension
INTEGER, INTENT(in) :: dim_ens ! Size of state ensemble
INTEGER, INTENT(in) :: dim_ens_p ! PE-local size of ensemble
INTEGER, INTENT(in) :: dim_obs_p ! PE-local dimension of observation vector
REAL, INTENT(inout) :: state_p(dim_p) ! PE-local forecast/analysis state
! The array 'state_p' is not generally not initialized in the case of SEIK/EnKF/ETKF/NETF/PF.
! It can be used freely in this routine.
REAL, INTENT(inout) :: Uinv(dim_ens, dim_ens) ! Inverse of matrix U
REAL, INTENT(inout) :: ens_p(dim_p, dim_ens) ! PE-local state ensemble
INTEGER, INTENT(in) :: flag ! PDAF status flag
}}}
The routine `U_prepoststep` is called once at the beginning of the assimilation process. In addition, it is called during the assimilation cycles before the analysis step and after the ensemble transformation. The routine is called by all filter processes (that is `filterpe=1`).
The routine provides for the user the full access to the ensemble of model states. Thus, user-controlled pre- and post-step operations can be performed. For example the forecast and the analysis states and ensemble covariance matrix can be analyzed, e.g. by computing the estimated variances. In addition, the estimates can be written to disk.
Hint:
* If a user considers to perform adjustments to the estimates (e.g. for balances), this routine is the right place for it.
* The vector (`state_p`) is allocated but not initialized with the ensemble mean. It can be used freely during the execution of `U_prepoststep`, for example to compute the ensemble mean state.
* The interface has a difference for PF and ESTKF/ETKF: For the PF, the array `Uinv` has size `1` x `1`. In contrast it has size `dim_ens` x `dim_ens` for the ESTKF/ETKF filters. (For most cases, this will be irrelevant, because most usually the ensemble array `ens_p` is used for computations, rather than `Uinv`.)
* The interface through which `U_prepoststep` is called does not include the array of smoothed ensembles. In order to access the smoother ensemble array one has to set a pointer to it using a call to the routine `PDAF_get_smootherens` (see page on [AuxiliaryRoutines auxiliary routines])
=== `U_likelihood` (likelihood_pdaf.F90) ===
This routine is used by the PF and NETF algorithms only.
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 PF and 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 [wiki: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 [wiki: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 [InsertAnalysisStep#U_next_observationnext_observation_pdaf.F90 inserting the analysis step] for the description of this routine.
== Execution order of user-supplied routines ==
For the PF, the user-supplied routines are essentially executed in the order they are listed in the interface to `PDAF_assimilate_pf`. 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_statecollect_state_pdaf.F90 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_prepoststepprepoststep_ens_pdaf.F90 U_prepoststep] (Call to act on the forecast ensemble, called with negative value of the time step)
1. [#U_init_dim_obsinit_dim_obs_pdaf.F90 U_init_dim_obs]
1. [#U_init_obsinit_obs_pdaf.F90 U_init_obs]
1. [#U_obs_opobs_op_pdaf.F90 U_obs_op] (`dim_ens` calls; one call for each ensemble member)
1. [#U_likelihoodlikelihood_pdaf.F90 U_likelihood] (`dim_ens` calls; one call for each ensemble member)
1. [#U_prepoststepprepoststep_ens_pdaf.F90 U_prepoststep] (call to act on the analysis ensemble, called with (positive) value of the time step)
In case of the routine `PDAF_assimilate_pf`, the following routines are executed after the analysis step:
1. [#U_distribute_statedistribute_state_pdaf.F90 U_distribute_state]
1. [#U_next_observationnext_observation_pdaf.F90 U_next_observation]