prodRinvA_hyb_l_pdaf

The page document the user-supplied call-back routine prodRinvA_hyb_l_pdaf.

The routine prodRinvA_hyb_l_pdaf (Called U_prodRinvA_hyb_l inside the PDAF core routines) is a call-back routine that has to be provided by the user. In the simplified interface the predefined name of the routine is prodRinvA_hyb_l_pdaf, but in the full interface, the user can choose the name of the routine.

This routine is a variant of prodRinvA_l_pdaf including the hybrid weight used for the hybrid Kalman-nonlinear ensemble transform filter LKNETF. If you already implemented prodRinvA_l_pdaf you can implement the hybrid variant by adding the hybrid weight gamma to the interface and add a multiplication with this value in the loop where C_l is computed.

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. This matrix holds the observed part of the ensemble perturbations for the specified local analysis domain. The matrix is provided as input argument A_l and the product has to be given in the output array 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.

Difference to prodRinvA_l_pdaf: The routine also has to apply the hybrid weight gamma. This is a simple multiplication with the input value in the loop where C_l is initialized.

The interface is the following:

SUBROUTINE prodRinvA_hyb_l_pdaf(domain_p, step, dim_obs_l, rank, obs_l, gamma, A_l, C_l)

with

• domain_p : integer, intent(in)
  Index of current local analysis domain
• step : integer, intent(in)
  Current time step
• dim_obs_l : integer, intent(in)
  Number of local observations at current time step (i.e. the size of the local observation vector)
• rank : integer, intent(in)
  Number of the columns in the matrix processes here. This is usually the ensemble size minus one (or the rank of the initial covariance matrix)
• obs_l : real, intent(in), dimension(dim_obs_l)
  Local vector of observations
• gamma : real, intent(in)
  Hybrid weight provided by PDAF
• A_l : real, intent(in), dimension(dim_obs_l, rank)
  Input matrix provided by PDAF
• C_l : real, intent(out), dimension(dim_obs_l, rank)
  Output matrix

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_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.
• To perform observation localization (i.e. observation weighting by modifying the inverse observation error covariance matrix) one computes for each observations the distance of it from the local analysis domain and then computes a weight for each observation according to this distance. For the computation of the weight, the routine PDAF_local_weight can be used.