Implementation of the Analysis step for the LEnKF (Localized Ensemble Kalman Filter)
Implementation Guide
 Main page
 Adaptation of the parallelization
 Initialization of PDAF
 Modifications for ensemble integration
 Implementation of the analysis step
 Implementation for ESTKF
 Implementation for LESTKF
 Implementation for ETKF
 Implementation for LETKF
 Implementation for SEIK
 Implementation for LSEIK
 Implementation for SEEK
 Implementation for EnKF
 Implementation for LEnKF
 Implementation for NETF
 Implementation for LNETF
 Implementation for PF
 Memory and timing information
 Ensemble Generation
 Diagnostics
Contents of this page
 Overview

PDAF_assimilate_lenkf

PDAF_put_state_lenkf

Usersupplied routines

U_collect_state
(collect_state_pdaf.F90) 
U_distribute_state
(distribute_state_pdaf.F90) 
U_init_dim_obs
(init_dim_obs_pdaf.F90) 
U_obs_op
(obs_op_pdaf.F90) 
U_init_obs
(init_obs_pdaf.F90) 
U_prepoststep
(prepoststep_ens_pdaf.F90) 
U_localize
(localize_covar_pdaf.F90) 
U_add_obs_err
(add_obs_err_pdaf.F90) 
U_init_obscovar
(init_obscovar_pdaf.F90) 
U_next_observation
(next_observation_pdaf.F90)

 Execution order of usersupplied routines
The LEnKF was added with verson 1.12 of PDAF.
Overview
For the analysis step of the LEnKF different operations related to the observations are needed. These operations are requested by PDAF by calling usersupplied 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_put_state_lenkf
for the fullyparallel implementation (or PDAF_put_state_lenkf
for the 'flexible' implementation). With regard to the parallelization, all these routines are executed by the filter processes (filterpe=.true.
) only.
For completeness we discuss here all usersupplied routines that are specified in the interface to PDAF_put_state_lenkf. Thus, some of the usersupplied routines that are explained on the page explaining the modification of the model code for the ensemble integration are repeated here.
The LEnKF implemented in PDAF follows the original LEnKF by Evensen (1994) including the correction for perturbed observations (Burgers et al. 1998). The LEnKF implemented in PDAF is reviewed by Nerger et al (2005) and described in more detail by Nerger (2004). The localization is covariance lozalization of PH^{T and HPH}T as described in Houtekamer & Mitchell (2001) (See the page on publications and presentations for publications and presenations involving and about PDAF)
In our studies (Nerger et al. 2005, Nerger et al. 2007), the EnKF showed performance deficiencies compared to the SEIK filter. Due to this, we focused more on the SEIK filter and the ETKF and ESTKF after these comparison studies. For real applications, we generally recommend using ESTKF or ETKF, or their local variants LESTKF or LETKF. However, the EnKF/LEnKF might have a good performance if very large ensemble can be used as this reduces the sampling errors.
PDAF_assimilate_lenkf
The general aspects of the filter specific routines PDAF_assimilate_*
have been described on the page Modification of the model code for the ensemble integration and its subpage on inserting the analysis step. The routine is used in the fullyparallel 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 of the routine. Subsequently, the full set of usersupplied routines specified in the call to
PDAF_assimilate_lenkf` is explained.
The interface when using the LEnKF is the following:
SUBROUTINE PDAF_assimilate_lenkf(U_collect_state, U_distribute_state, U_init_dim_obs, & U_obs_op, U_init_obs, U_prepoststep, U_localize, & U_add_obs_err, U_init_obscovar, U_next_observation, status)
with the following arguments:
 U_collect_state: The name of the usersupplied 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 inPDAF_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 usersupplied routine that provides the size of observation vector
 U_obs_op: The name of the usersupplied routine that acts as the observation operator on some state vector
 U_init_obs: The name of the usersupplied routine that initializes the vector of observations
 U_prepoststep: The name of the pre/poststep routine as in
PDAF_get_state
 U_localize: Apply covariance localization to the matrices HP and HPH^{T}
 U_add_obs_err: The name of the usersupplied routine that adds the observation error covariance matrix to the ensemble covariance matrix projected onto the observation space.
 U_init_obscovar: The name of the usersupplied routine that initializes the observation error covariance matrix.
 U_next_observation: The name of a user supplied routine that initializes the variables
nsteps
,timenow
, anddoexit
. The same routine is also used inPDAF_get_state
. status
: The integer status flag. It is zero, ifPDAF_assimilate_lenkf
is exited without errors.
PDAF_put_state_lenkf
When the 'flexible' implementation variant is chosen for the assimilation system, the routine PDAF_put_state_lenkf
has to be used instead of PDAF_assimilate_lenkf
. 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_lenkf
with the exception the specification of the usersupplied routines U_distribute_state
and U_next_observation
are missing.
The interface when using the LEnKF is the following:
SUBROUTINE PDAF_put_state_lenkf(U_collect_state, U_init_dim_obs, U_obs_op, & U_init_obs, U_prepoststep, U_localize, & U_add_obs_err, U_init_obscovar, status)
Usersupplied routines
Here all usersupplied routines are described that are required in the call to PDAF_assimilate_lenkf
. For some of the generic routines, we link to the page on modifying the model code for the ensemble integration.
To indicate usersupplied 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 subdomain, if a domain decomposed model is used. Thus, the value(s) in the variable will be different for different model subdomains.
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 by the LEnKF.
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 domaindecomposed model is used, dim_obs_p
will be the size of the observation vector for the subdomain 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 LEnKF.
The interface for this routine is:
SUBROUTINE obs_op(step, dim_p, dim_obs_p, state_p, m_state_p) INTEGER, INTENT(in) :: step ! Currrent time step INTEGER, INTENT(in) :: dim_p ! PElocal dimension of state INTEGER, INTENT(in) :: dim_obs_p ! Dimension of observed state REAL, INTENT(in) :: state_p(dim_p) ! PElocal model state REAL, INTENT(out) :: m_state_p(dim_obs_p) ! PElocal 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 PElocal subdomain of the model and has to provide the observed substate for the PElocal domain.
Hint:
 If the observation operator involves a global operation, e.g. some global integration, while using domaindecomposition 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 LEnKF.
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 ! PElocal dimension of obs. vector REAL, INTENT(out) :: observation_p(dim_obs_p) ! PElocal 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 subdomain for the calling process has to be initialized.
U_prepoststep
(prepoststep_ens_pdaf.F90)
The general aspects of this routines have already been described on the page on modifying the model code for the ensemble integration for the SEIK filter. For completeness, the description is repeated specifically for the EnKF:
The interface of the routine is identical for all filters, but sizes can vary. Also, the particular operations that are performed in the routine can be specific for each filter algorithm.
The interface for this routine is for the LEnKF
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 ! PElocal state dimension INTEGER, INTENT(in) :: dim_ens ! Size of state ensemble INTEGER, INTENT(in) :: dim_ens_p ! PElocal size of ensemble INTEGER, INTENT(in) :: dim_obs_p ! PElocal dimension of observation vector REAL, INTENT(inout) :: state_p(dim_p) ! PElocal forecast/analysis state ! The array 'state_p' is not generally not initialized in the case of SEIK/EnKF/ETKF. ! It can be used freely in this routine. REAL, INTENT(inout) :: Uinv(1, 1) ! Not used not LEnKF REAL, INTENT(inout) :: ens_p(dim_p, dim_ens) ! PElocal 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, usercontrolled pre and poststep 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.
 Only for the SEEK filter the state vector (
state_p
) is initialized. For all other filters, the array is allocated, but it can be used freely during the execution ofU_prepoststep
.  The array
Uinv
is not used in the EnKF. Internally to PDAF, it is allocated to be of size (1,1).  Apart from the size of the array
Uinv
, the interface is identical for all ensemble filters (SEIK/ETKF/EnKF/LSEIK/LETKF/LEnKF). In general it should be possible to use an identical pre/poststep routine for all these filters.  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 routinePDAF_get_smootherens
(see page on auxiliary routines)
U_localize
(localize_covar_pdaf.F90)
This routine is only used for the LEnKF.
The interface for this routine is:
SUBROUTINE U_localize(dim_p, dim_obs, HP, HPH) INTEGER, INTENT(in) :: dim_p ! PElocal state dimension INTEGER, INTENT(in) :: dim_obs ! Dimension of global observation vector REAL, INTENT(inout) :: HP(dim_obs, dim_p) ! Matrix HP REAL, INTENT(inout) :: HPH(dim_obs, dim_obs) ! Matrix HPH^T^
The routine is called during the analysis step and has to apply the elementwise Schur product for the covariance localization of the two matricesHP and HPH^{T}, which are provided as input/output arguments.
Notes:
 In case of a parallelization with domain decomposition,
HP
contains only the columns of the matrix that resides on the model subdomain of the calling process. The number of rows is that of the global number of observations
Hints:
 To compute the localization one can use the routine
PDAF_local_weight
after computing the distance between two elements in the matrix HP or HPH^{T}.
U_add_obs_err
(add_obs_err_pdaf.F90)
This routine is only used for the EnKF and LEnKF.
The interface for this routine is:
SUBROUTINE add_obs_err(step, dim_obs, C) INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: dim_obs ! Dimension of obs. vector REAL, INTENT(inout) :: C(dim_obs, dim_obs) ! Matrix to that the observation ! error covariance matrix is added
The routine is called during the analysis step. During the analysis step of the LEnKF, the projection of the ensemble covariance onto the observation space is computed. This matrix is provided to the routine as C_p
. The routine has to add the observation error covariance matrix to C_p
.
The operation is for the global observation space. Thus, it is independent of whether the filter is executed with or without parallelization.
Hints:
 The routine does not require that the observation error covariance matrix is added as a full matrix. If the matrix is diagonal, only the diagonal elements have to be added.
U_init_obscovar
(init_obscovar_pdaf.F90)
This routine is only used for the EnKF and LEnKF.
The interface for this routine is:
SUBROUTINE init_obscovar(step, dim_obs, dim_obs_p, covar, m_state_p, & isdiag) INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: dim_obs ! Dimension of observation vector INTEGER, INTENT(in) :: dim_obs_p ! PElocal dimension of observation vector REAL, INTENT(out) :: covar(dim_obs, dim_obs) ! Observation error covariance matrix REAL, INTENT(in) :: m_state_p(dim_obs_p) ! PElocal observation vector LOGICAL, INTENT(out) :: isdiag ! Whether the observation error covar. matrix is diagonal
The routine is called during the analysis step and is required for the generation of an ensemble of observations. It has to initialize the global observation error covariance matrix covar
. In addition, the flag isdiag
has to be initialized to provide the information, whether the observation error covariance matrix is diagonal.
The operation is for the global observation space. Thus, it is independent of whether the filter is executed with or without parallelization.
Hints:
 The local observation vector
m_state_p
is provided to the routine for the case that the observation errors are relative to the value of the observation.
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 usersupplied routines
For the :EnKF, the usersupplied routines are essentially executed in the order they are listed in the interface to PDAF_assimilate_lenkf
. 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:
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:
 U_prepoststep (Call to act on the forecast ensemble, called with negative value of the time step)
 U_init_dim_obs
 U_obs_op (
dim_ens
calls: one call for each ensemble member)  U_localize
 U_add_obs_err
 U_init_obs
 U_init_obscovar
 U_obs_op (
dim_ens
calls: one call for each ensemble member, repeated to reduce storage)  U_prepoststep (Call to act on the analysis ensemble, called with (positive) value of the time step)
In case of the routine PDAF_assimilate_enkf
, the following routines are executed after the analysis step: