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Using non-diagonal R matrices with OMI
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This feature was introduced with PDAF V2.3.
The default mode of PDAF-OMI is to use a diagonal observation error covariance matrix R and specifying the observation error variances, i.e. the diagonalof R only. This is in line with the common choice in data assimilation to assume that observation errors are uncorrelated.
However, there are also observation types with significant observation error correlations, which should be represented by a non-diagonal observation error covariance matrix. With PDAF V2.3 support for such nondiagonal R matrices was added to OMI.
PDAF-OMI's support for nondiagonal R-matrices consists in given the user access to the routines that perform operations involve R. This differs with the filter type, e.g. in LESTKF and LETKF a produce of some matrix with the inverse of R ha sto be computed, while in the traditional, perturbed observations, EnKF the matrix R has to be added to some other matrix. For the particle filter and the NETF the computation of the likelihood involves R.
Routines to perform the analysis step
For using a nondiagonal R there is a modified variant of the PDAF-OMI routines PDAFomi_assimilate_*
or PDAFomi_put_state_*
. Compared to the case of a diagonal R, the routines are slightly less generic. Below we provide an overview of the routines including the links to the interface description of each. The last column gives the name of the additional routine(s) provided in the interface to implement the non-diagonal R-matrix.
Global Filter | diagonal R | non-diagonal R | additional routine(s) |
---|---|---|---|
EnKF | PDAFomi_assimilate_global PDAFomi_put_state_global | PDAFomi_assimilate_enkf_nondiagR PDAFomi_put_state_enkf_nondiagR | add_obs_err_pdafomi init_obscovar_pdafomi |
ESTKF | PDAFomi_assimilate_global PDAFomi_put_state_global | PDAFomi_assimilate_global_nondiagR PDAFomi_put_state_global_nondiagR | prodRinvA_pdafomi |
ETKF | PDAFomi_assimilate_global PDAFomi_put_state_global | PDAFomi_assimilate_global_nondiagR PDAFomi_put_state_global_nondiagR | prodRinvA_pdafomi |
PF | PDAFomi_assimilate_global PDAFomi_put_state_global | PDAFomi_assimilate_nonlin_nondiagR PDAFomi_put_state_nonlin_nondiagR | likelihood_pdafomi |
NETF | PDAFomi_assimilate_global PDAFomi_put_state_global | PDAFomi_assimilate_nonlin_nondiagR PDAFomi_put_state_nonlin_nondiagR | likelihood_pdafomi |
SEIK | PDAFomi_assimilate_global PDAFomi_put_state_global | PDAFomi_assimilate_global_nondiagR PDAFomi_put_state_global_nondiagR | prodRinvA_pdafomi |
Local Filter | diagonal R | non-diagonal R | additional routine(s) |
---|---|---|---|
LEnKF | PDAFomi_assimilate_lenkf PDAFomi_put_state_lenkf | PDAFomi_assimilate_lenkf_nondiagR PDAFomi_put_state_lenkf_nondiagR | add_obs_err_pdafomi init_obscovar_pdafomi |
LESTKF | PDAFomi_assimilate_local PDAFomi_put_state_local | PDAFomi_assimilate_local_nondiagR PDAFomi_put_state_local_nondiagR | prodRinvA_l_pdafomi |
LETKF | PDAFomi_assimilate_local PDAFomi_put_state_local | PDAFomi_assimilate_local_nondiagR PDAFomi_put_state_local_nondiagR | prodRinvA_l_pdafomi |
LSEIK | PDAFomi_assimilate_local PDAFomi_put_state_local | PDAFomi_assimilate_local_nondiagR PDAFomi_put_state_local_nondiagR | prodRinvA_l_pdafomi |
LNETF | PDAFomi_assimilate_local PDAFomi_put_state_local | PDAFomi_assimilate_lnetf_nondiagR PDAFomi_put_state_lnetf_nondiagR | likelihood_l_pdafomi |
LKNETF | PDAFomi_assimilate_local PDAFomi_put_state_local | PDAFomi_assimilate_lknetf_nondiagR PDAFomi_put_state_lknetf_nondiagR | likelihood_l_pdafomi likelihood_hyb_l_pdafomi prodRinvA_l_pdafomi prodRinvA_hyb_l_pdafomi |
3D-Var | diagonal R | non-diagonal R | additional routine(s) |
---|---|---|---|
3DVar | PDAFomi_assimilate_3dvar PDAFomi_assimilate_3dvar | PDAFomi_put_state_3dvar_nondiagR PDAFomi_put_state_3dvar_nondiagR | prodRinvA_pdafomi |
En3DVar ESTKF | PDAFomi_assimilate_en3dvar_estkf PDAFomi_assimilate_en3dvar_estkf | PDAFomi_put_state_en3dvar_estkf_nondiagR PDAFomi_put_state_en3dvar_estkf_nondiagR | prodRinvA_pdafomi |
En3DVar LESTKF | PDAFomi_assimilate_en3dvar_lestkf PDAFomi_assimilate_en3dvar_lestkf | PDAFomi_put_state_en3dvar_lestkf_nondiagR PDAFomi_put_state_en3dvar_lestkf_nondiagR | prodRinvA_l_pdafomi |
hyb3DVar ESTKF | PDAFomi_assimilate_hyb3dvar_estkf PDAFomi_assimilate_hyb3dvar_estkf | PDAFomi_put_state_hyb3dvar_estkf_nondiagR PDAFomi_put_state_hyb3dvar_estkf_nondiagR | prodRinvA_pdafomi |
hyb3DVar LESTKF | PDAFomi_assimilate_hyb3dvar_lestkf PDAFomi_assimilate_hyb3dvar_lestkf | PDAFomi_put_state_hyb3dvar_lestkf_nondiagR PDAFomi_put_state_hyb3dvar_lestkf_nondiagR | prodRinvA_l_pdafomi |
Call-back routines handling observations
The tables above show the additional routine(s) that need to be implemented the nondiagonal R matrix for the different filters and 3D-Var variants. Here we describe the interface of these routines. Generally the required routine(s) for the chosen DA method should be added to callback_obs_pdafomi.F90
and to each observation module, where thisobs
or thisobs_l
can be included.
Note that a mixed implementation is not possible. If one observation type has a non-diagonal R matrix, the additional routine(s) have to be implemented for all observations, even if an observation type uses a diagonal R. However, for observation wit a diagonal R, the corresponding PDAF-OMI routine to be called in the observation module can be called. Below we provide the information where this routine can be found.
prodRinvA_pdafomi
The interface for this routine is:
SUBROUTINE prodRinvA_pdafomi(step, dim_obs_p, rank, 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) :: rank ! Rank of initial covariance matrix REAL, INTENT(in) :: obs_p(dim_obs_p) ! PE-local vector of observations REAL, INTENT(in) :: A_p(dim_obs_p,rank) ! Input matrix from analysis routine REAL, INTENT(out) :: C_p(dim_obs_p,rank) ! Output matrix
The routine computes the product of the inverse of the observation error covariance matrix with matrix A_p
. For the ESTKF this matrix holds the observed part of the ensemble perturbations. 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 matrix
A_p
relates to the observations of all observation types. Thus, one needs to take the offset of an observation in the observation vector of all observation types into account. This offset is given bythisobs%off_obs_f
. - 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 ESTKF and ETKF: For ETKF the third argument is the ensemble size (
dim_ens
), while for the ESTKF it is the rank (rank
) of the covariance matrix (usually ensemble size minus one). In addition, the second dimension ofA_p
andC_p
has sizedim_ens
for ETKF, while it isrank
for the ESTKF. (Practically, one can usually ignore this difference as the fourth argument of the interface can be named arbitrarily in the routine.) - For diagonal R PDAF-OMI uses the routine
PDAFomi_prodRinvA
in/src/PDAFomi_obs_f.F90
as the routine that is called within the observation module. This routine can serve as a template for an implementation by the user.
likelihood_pdafomi
The interface for this routine is:
SUBROUTINE likelihood_pdafomi(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 computes the likelihood of the observations 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 matrix
residual
relates to the observations of all observation types. Thus, one needs to take the offset of an observation in the observation vector of all observation types into account. This offset is given bythisobs%off_obs_f
. - 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. - For diagonal R PDAF-OMI uses the routine
PDAFomi_likelihood
in/src/PDAFomi_obs_f.F90
as the routine that is called within the observation module. This routine can serve as a template for an implementation by the user.
add_obs_err_pdafomi
The interface for this routine is:
SUBROUTINE add_obs_err_pdafomi(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
During the analysis step of the EnKF, 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:
- Matrix C relates to the observations of all types. Thus, for a single obsevation type one has to take the offset of this observtion type in the observation vector into account. For this, PDAF-OMI initializes an array
obsdims
, which can also be used by a user-implemented routine. See the routinePDAFomi_add_obs_error
in/src/PDAFomi_obs_f.F90
for how this can be implemented. - 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.
- For diagonal R, PDAF-OMI uses the routine
PDAFomi_add_obs_error
in/src/PDAFomi_obs_f.F90
as the routine that is called within the observation module. This routine can serve as a template for an implementation by the user.
init_obscovar_pdafomi
The interface for this routine is:
SUBROUTINE init_obscovar_pdafomi(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 ! PE-local 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) ! PE-local observation vector LOGICAL, INTENT(out) :: isdiag ! Whether the observation error covar. matrix is diagonal
The routine 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:
- Matrix
covar
relates to the observations of all types. Thus, for a single obsevation type one has to take the offset of this observtion type in the observation vector into account. For this, PDAF-OMI initializes an arrayobsdims
, which can also be used by a user-implemented routine. See the routinePDAFomi_init_obscovar
in/src/PDAFomi_obs_f.F90
for how this can be implemented. - 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. - For diagonal R, PDAF-OMI uses the routine
PDAFomi_init_obscovar
in/src/PDAFomi_obs_f.F90
as the routine that is called within the observation module. This routine can serve as a template for an implementation by the user.
prodRinvA_l_pdafomi
The interface for this routine is:
SUBROUTINE prodRinvA_l_pdafomi(domain_p, step, dim_obs_l, rank, obs_l, A_l, C_l) INTEGER, INTENT(in) :: domain_p ! Current local analysis domain INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: dim_obs_l ! Dimension of local observation vector INTEGER, INTENT(in) :: rank ! Rank of initial covariance matrix REAL, INTENT(in) :: obs_l(dim_obs_l) ! Local vector of observations REAL, INTENT(inout) :: A_l(dim_obs_l, rank) ! Input matrix from analysis routine REAL, INTENT(out) :: C_l(dim_obs_l, rank) ! Output matrix
The routine is called during the loop over the local analysis domains. In the algorithm, the product of the inverse of the observation error covariance matrix with some matrix has to be computed. For the SEIK filter this matrix holds the observed part of the ensemble perturbations for the local analysis domain of index domain_p
. The matrix is provided as A_l
. The product has to be given as C_l
.
This routine is also the place to perform observation localization. To initialize a vector of weights, the routine PDAF_local_weight
can be called. The procedure is used in the example implementation and also demonstrated in the template routine.
Hints:
- The routine is a local variant of the routine
U_prodRinvA
. Thus if that routine has been implemented before, it can be adapted here for the local filter. - The matrix
A_p
relates to the observations of all observation types. Thus, one needs to take the offset of an observation in the observation vector of all observation types into account. This offset is given bythisobs_l%off_obs_l
. - 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_l
has to be implemented. - The observation vector
obs_l
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 LESTKF and LETKF: For LETKF the third argument is the ensemble size (
dim_ens
), while for LESTKF it is the rank (rank
) of the covariance matrix (usually ensemble size minus one). In addition, the second dimension ofA_l
andC_l
has sizedim_ens
for LETKF, while it isrank
for LESTKF. (Practically, one can usually ignore this difference as the fourth argument of the interface can be named arbitrarily in the routine.) - For diagonal R PDAF-OMI uses the routine
PDAFomi_prodRinvA_l
in/src/PDAFomi_obs_k.F90
as the routine that is called within the observation module. This routine can serve as a template for an implementation by the user.
likelihood_l_pdafomi
The interface for this routine is:
SUBROUTINE likelihood_l_pdafomi(domain_p, step, dim_obs_l, obs_l, resid_l, likely_l) INTEGER, INTENT(in) :: domain_p ! Current local analysis domain INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: dim_obs_l ! Dimension of local observation vector REAL, INTENT(in) :: obs_l(dim_obs_l) ! Local vector of observations REAL, INTENT(inout) :: resid_l(dim_obs_l) ! Input vector holding the local residual y-Hx REAL, INTENT(out) :: likely_l(dim_obs_l) ! Output value of the likelihood
The routine is called during the loop over the local analysis domains. In the NETF, as in other particle filters, the likelihood of the local 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)).
This routine is also the place to perform observation localization. To initialize a vector of weights, the routine PDAF_local_weight
can be called. The procedure is used in the example implementation and also demonstrated in the template routine.
Hints:
- The routine is a local variant of the routine
U_likelihood
. Thus if that routine has been implemented before, it can be adapted here for the local filter.- The matrix
resid_l
relates to the observations of all observation types. Thus, one needs to take the offset of an observation in the observation vector of all observation types into account. This offset is given bythisobs_l%off_obs_l
. - The routine is very similar to the routine U_prodRinvA_l. 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_l.
- 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_l
is provided through the interface for cases where the observation error variance is relative to the actual value of the observations. - For diagonal R PDAF-OMI uses the routine
PDAFomi_likelihood_l
in/src/PDAFomi_obs_l.F90
as the routine that is called within the observation module. This routine can serve as a template for an implementation by the user.
- The matrix