= Implementation of the Analysis step for the LKNETF (Local Kalman-Nonlinear Ensemble Transform Filter) using PDAF's full interface = {{{ #!html

Implementation Guide

  1. Implementing the analysis step

  2. Localized ensemble Kalman filters
    1. Implementation for LESTKF
    2. Implementation for LETKF
    3. Implementation for LSEIK
    4. Implementation for LEnKF
    5. Implementation for EnSRF/EAKF
  4. Global ensemble Kalman filters
    1. Implementation for ESTKF
    2. Implementation for ETKF
    3. Implementation for SEIK
    4. Implementation for EnKF
    5. Implementation for SEEK
  6. Nonlinear DA methods
    1. Implementation for NETF
    2. Implementation for LNETF
    3. Implementation for PF
    4. Implementation for LKNETF
  8. 3D-Var methods
    1. Implementation for 3D-Var
    2. Implementation for 3D Ensemble Var
    3. Implementation for Hybrid 3D-Var
}}} [[PageOutline(2-3,Contents of this page)]] || This page describes the implementation of the analysis step without using PDAF-OMI. Please see the [wiki:ImplementationofAnalysisStep page on the analysis with OMI] for the more modern and efficient implementation variant using PDAF-OMI. || The LKNETF algorithm was added with Version 2.1 of PDAF. == Overview == The LKNETF algorithm is a hybrid method that combines the nonlinear LNETF and linear LETKF analysis updates. The hybridization allows to choose the strength of both filter methods according to the effective ensemble size and the non-Gaussianity of the observed ensemble. The details of the LKNETF method are described in the article: Nerger, L. (2022) Data assimilation for nonlinear systems with a hybrid nonlinear-Kalman ensemble transform filter. Q. J. Meteorol. Soc., 148, 620-640 [https://doi.org/10.1002/qj.4221 doi:10.1002/qj.4221]. For the analysis step of the LKNETF algorithm, several 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 as this procedure should simplify the implementation. The names of the required routines are specified in the call to the routine `PDAF_assimilate_lknetf` in the fully-parallel implementation (or `PDAF_put_state_lknetf` for the 'flexible' implementation) described below. With regard to the parallelization, all these routines (except `U_collect_state`) 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_lknetf`. Since the LKNETF method combined the LNETF and LETKF methods, one can use user routines that were implemented for these filters. Note, that there is no global variant of the LKNETF implemented in PDAF. == `PDAF_put_state_lknetf` == This routine is used both in the ''fully-parallel'' and the ''flexible'' implementation variants of the data assimilation system. (See the page [wiki:OnlineModifyModelforEnsembleIntegration_PDAF3 Modification of the model code for the ensemble integration] for these variants) The interface for the routine `PDAF_assimilate_lestkf` contains several routine names for routines that operate on the local analysis domains (marked by `_l` at the end of the routine name). In addition, there are names for routines that consider all available observations required to perform local analyses with LKNETKF within some sub-domain of a domain-decomposed model (we refer to these as 'full' observations, marked by `_f` at the end of the routine name). In case of a serial execution of the assimilation program, these will be all globally available observations. However, if the program is executed with parallelization, one might choose a smaller set of observations. We will explain this is some detail below. 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_lknetf(U_collect_state, U_distribute_state, & U_init_dim_obs_f, U_obs_op_f, U_init_obs_f, & U_init_obs_l, U_prepoststep, U_prodRinvA_l, U_prodRinvA_hyb_l, & U_init_n_domains, U_init_dim_l, U_init_dim_obs_l, & U_g2l_state, U_l2g_state, U_g2l_obs, U_init_obsvar, U_init_obsvar_l, & U_likelihood_l, U_likelihood_hyb_l, U_next_observation, outflag) }}} 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 [ModifyModelforEnsembleIntegration#PDAF_get_state PDAF_get_state] * [#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_obs_finit_dim_obs_f_pdaf.F90 U_init_dim_obs_f]: The name of the user-supplied routine that provides the size of the full observation vector * [#U_obs_op_fobs_op_f_pdaf.F90 U_obs_op_f]: The name of the user-supplied routine that acts as the full observation operator on some state vector * [#U_init_obs_finit_obs_f_pdaf.F90 U_init_obs_f]: The name of the user-supplied routine that initializes the full vector of observations * [#U_init_obs_linit_obs_l_pdaf.F90 U_init_obs_l]: The name of the user-supplied routine that initializes the vector of observations for a local analysis domain * [#U_prepoststepprepoststep_ens_pdaf.F90 U_prepoststep]: The name of the pre/poststep routine as in `PDAF_get_state` * [#U_prodRinvA_lprodrinva_l_pdaf.F90 U_prodRinvA_l]: The name of the user-supplied routine that computes the product of the inverse of the observation error covariance matrix with some matrix provided to the routine by PDAF. * [#U_prodRinvA_hyb_lprodrinva_hyb_l_pdaf.F90 U_prodRinvA_hyb_l]: The name of the user-supplied routine that computes the product of the inverse of the observation error covariance matrix with some matrix provided to the routine by PDAF including the hybrid weight. * [#U_init_n_domainsinit_n_domains_pdaf.F90 U_init_n_domains]: The name of the routine that provides the number of local analysis domains * [#U_init_dim_linit_dim_l_pdaf.F90 U_init_dim_l]: The name of the routine that provides the state dimension for a local analysis domain * [#U_init_dim_obs_linit_dim_obs_l_pdaf.F90 U_init_dim_obs_l]: The name of the routine that initializes the size of the observation vector for a local analysis domain * [#U_g2l_stateg2l_state_pdaf.F90 U_g2l_state]: The name of the routine that initializes a local state vector from the global state vector * [#U_l2g_statel2g_state_pdaf.F90 U_l2g_state]: The name of the routine that initializes the corresponding part of the global state vector from the the provided local state vector * [#U_g2l_obsg2l_obs_pdaf.F90 U_g2l_obs]: The name of the routine that initializes a local observation vector from a full observation vector * [#U_init_obsvarinit_obsvar_pdaf.F90 U_init_obsvar]: The name of the user-supplied routine that provides a global mean observation error variance (This routine will only be executed, if an adaptive forgetting factor is used) * [#U_likelihood_llikelihood_l_pdaf.F90 U_likelihood_l]: The name of the user-supplied routine that computes the likelihood of the local observations for an ensemble member provided when the routine is called. * [#U_likelihood_hyb_llikelihood_hyb_l_pdaf.F90 U_likelihood_hyb_l]: The name of the user-supplied routine that computes the likelihood of the local observations for an ensemble member provided when the routine is called and accounting for the hybrid weigt. * [#U_init_obsvar_linit_obsvar_l_pdaf.F90 U_init_obsvar_l]: The name of the user-supplied routine that provides a mean observation error variance for the local analysis domain (This routine will only be executed, if a local adaptive forgetting factor is used) * [#U_next_observationnext_observation.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_lknetf` is exited without errors. Note: * The order of the routine names does not show the order in which these routines are executed. See the [#Executionorderofuser-suppliedroutines section on the order of the execution] at the bottom of this page. == `PDAF_assim_offline_lknetf ` == 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 [wiki:OfflineImplementationGuide_PDAF3 implementation of the offline mode]. The interface is : {{{ SUBROUTINE PDAF_assim_offline_lknetf( & U_init_dim_obs_f, U_obs_op_f, U_init_obs_f, & U_init_obs_l, U_prepoststep, U_prodRinvA_l, U_prodRinvA_hyb_l, & U_init_n_domains, U_init_dim_l, U_init_dim_obs_l, & U_g2l_state, U_l2g_state, U_g2l_obs, U_init_obsvar, U_init_obsvar_l, & U_likelihood_l, U_likelihood_hyb_l, outflag) }}} == `PDAF_put_state_lknetf` == 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 [wiki:OnlineFlexible_PDAF3 'flexible' parallelization variant] and for the [wiki:OfflineImplementationGuide_PDAF3 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_lknetf(U_collect_state, & U_init_dim_obs_f, U_obs_op_f, U_init_obs_f, & U_init_obs_l, U_prepoststep, U_prodRinvA_l, U_prodRinvA_hyb_l, & U_init_n_domains, U_init_dim_l, U_init_dim_obs_l, & U_g2l_state, U_l2g_state, U_g2l_obs, U_init_obsvar, U_init_obsvar_l, & U_likelihood_l, U_likelihood_hyb_l, outflag) }}} == Explanation of 'full observations' == Above we mention the concept of 'full' observations. We distinguish them from the globally available observations for efficiency. Note: For an initial implementation, one might not needd to worry about high efficiency, so that 'full' can refer to all available observations. To explain why 'full' observations can be different from globally available observations, we assume, for simplicity, that we have a 2-dimensional domain and that a local analysis domain consists of a single grid point of the model grid. In addition, we assume that the domain decomposition splits the global model domain in compact sub-domains and that the observations are spatially distributed observations of model fields that are part of the state vector. The LESTKF performs a loop over all local analysis domains, i.e. grid points. When a model uses domain decomposition, the loop is over all grid points that belong to a process sub-domain. As each model sub-domain is treated by a different process, all loops are executed parallel to each other. For the update of each local analysis domain (grid points), observations within the localization radius around its location are required. If the influence radius for the observations is sufficiently small, there will be grid points a for which the relevant observations reside completely inside the model sub-domain of the process. However, if a grid point is located close to the boundary of the model sub-domain, there will be some observations that reside on a neighboring process sub-domain, but are within the localization radius. One needs to assimilate these observations as otherwise, there could be unrealistic steps in the analysis field. However, there will also be observations that reside far away from the process sub-domain and will never influence the analysis result in this domain. A simple way to handle this situation is to initialize for each process all globally available observations, together with their coordinates. The observation operator would be applied on each sub-domain and then the observed ensemble would be collect using parallel communication with MPI. While this method is simple, it can also become inefficient if the assimilation program is executed using a large number of processes. In particular, all observation would need to be checked even if they are far away from the process sub-domain. More efficient is hence to select as 'full' observations only those observations that can have an effect on the local analyses of a process sub-domain. These are the observations that reside within the sub-domain, plus observations in neighboring sub-domains that reside within the localization radius. Setting up 'full' observations in this way leads to a smaller number of observations whose distance need to be checked for each local analysis domain. Howeever, one would need to find an implementation that provides the 'full' observations. || Note: The handling of 'full' observations is one of the aspects that motivated the development of PDAF-OMI and the relaed avanced interface (now the PDAF3 interface). Here, PDAF-OMI does take case of the 'full' observations. See the [wiki:ImplementationofAnalysisStep_PDAF3 Implementation Guide for the Analysis Step for the advanced interface using PDAF-OMI]. || == User-supplied routines == Here, all user-supplied routines are described that are required in the calls to the analysis routines. For some of the generic routines, we link to the page on [wiki:OnlineModifyModelforEnsembleIntegration_PDAF3 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` (short for 'process'). 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. In addition, there will be variables with the suffix `_f` (for 'full') and with the suffix `_l` (for 'local'). === `U_collect_state` (collect_state_pdaf.F90) === This routine is independent of the filter algorithm used. See the page on [wiki:OnlineModifyModelforEnsembleIntegration_PDAF3#collect_state_pdafcollect_state_pdaf.F90 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 [wiki:OnlineModifyModelforEnsembleIntegration_PDAF3#distribute_state_pdafdistribute_state_pdaf.F90 modifying the model code for the ensemble integration] for the description of this routine. === `U_init_dim_obs_f` (init_dim_obs_f_pdaf.F90) === This routine is used by all filter algorithms with domain-localization and is independent of the particular algorithm. The interface for this routine is: {{{ SUBROUTINE init_dim_obs_f(step, dim_obs_f) INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(out) :: dim_obs_f ! Dimension of full observation vector }}} The routine is called at the beginning of each analysis step, before the loop over all local analysis domains is entered. It has to initialize the size `dim_obs_f` of the full observation vector according to the current time step. For simplicity, `dim_obs_f` can be the size for the global model domain. 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 location of the observations, which can be used later, e.g. to implement the observation operator. In addition, one can already prepare an array that holds the full observation vector. This can be used later by `U_init_obs_l` to initialize a local vector of observations by selecting the relevant parts of the full observation vector. The required arrays can be defined in a module like `mod_assimilation`. * The routine is similar to `init_dim_obs` used in the global filters. However, if the global filter is used with a domain-decomposed model, it only initializes the size of the observation vector for the local model sub-domain. This is different for the local filters, as the local analysis also requires observational data from neighboring model sub-domains. Nonetheless, one can base on an implemented routine `init_dim_obs` to implement `init_dim_obs_f`. === `U_obs_op_f` (obs_op_f_pdaf.F90) === This routine is used by all filter algorithms with domain-localization and is independent of the particular algorithm. The interface for this routine is: {{{ SUBROUTINE obs_op_f(step, dim_p, dim_obs_f, state_p, m_state_f) INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: dim_p ! PE-local dimension of state INTEGER, INTENT(in) :: dim_obs_f ! Dimension of the full observed state REAL, INTENT(in) :: state_p(dim_p) ! PE-local model state REAL, INTENT(out) :: m_state_f(dim_obs_f) ! Full observed state }}} The routine is called during the analysis step, before the loop over the local analysis domain is entered. It has to perform the operation of the observation operator acting on a state vector, which is provided as `state_p`. The observed state has to be returned in `m_state_f`. It is the observed state corresponding to the 'full' observation vector. Hint: * The routine is similar to `init_dim_obs` used for the global filters. However, with a domain-decomposed model `m_state_f` will need to contain parts of the state vector from neighboring model sub-domains. Thus, one needs to collect this information which resides in the memory of other processes. PDAF provides the routine [wiki:PDAF_gather_obs_f PDAF_gather_obs_f] for this task. The example implementation in `tutorial/classical/online_2D_parallelmodel` shows the use of `PDAF_gather_obs_f`. === `U_init_obs_f` (init_obs_f_pdaf.F90) === This routine is used by all filter algorithms with domain-localization and is independent of the particular algorithm. The routine is only called if the globally adaptive forgetting factor is used (`type_forget=1` in the example implementation). For the local filters there is also the alternative to use locally adaptive forgetting factors (`type_forget=2` in the example implementation) The interface for this routine is: {{{ SUBROUTINE init_obs_f(step, dim_obs_f, observation_f) INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: dim_obs_f ! Dimension of full observation vector REAL, INTENT(out) :: observation_f(dim_obs_f) ! Full observation vector }}} The routine is called during the analysis step before the loop over the local analysis domains is entered. It has to provide the full vector of observations in `observation_f` for the current time step. The caller is the routine that computes an adaptive forgetting factor (PDAF_set_forget). Hints: * As for the other 'full' routines: While the global counterpart of this routine (`init_obs`) has to initialize the observation vector only for the local model sub-domain, the 'full' routine has to include observations that spatially belong to neighboring model sub-domains. As an easy choice one can simply initialize a vector of all globally available observations. * If the adaptive forgetting factor is not used, this routine only has to exist. However, no functionality is required. === `U_init_obs_l` (init_obs_l_pdaf.F90) === This routine is used by all filter algorithms with domain-localization and is independent of the particular algorithm. The interface for this routine is: {{{ SUBROUTINE init_obs_l(domain_p, step, dim_obs_l, observation_l) INTEGER, INTENT(in) :: domain_p ! Current local analysis domain INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: dim_obs_l ! Local dimension of observation vector REAL, INTENT(out) :: observation_l(dim_obs_l) ! Local observation vector }}} The routine is called during the analysis step during the loop over the local analysis domain. It has to provide the vector of observations for the analysis in the local analysis domain with index `domain_p` in `observation_l` for the current time step. Hints: * For parallel efficiency, the LKNETF algorithm is implemented in a way that first the full vectors are initialized. These are then restricted to the local analysis domain during the loop over all local analysis domains. Thus, if the full vector of observations has been initialized before `U_init_obs_l` is executed (e.g. by `U_init_dim_obs_f`), the operations performed in this routine will be to select the part of the full observation vector that is relevant for the current local analysis domain. * The routine `U_init_dim_obs_l` is executed before this routine. Thus, if that routine already prepares the information which elements of `observation_f` are needed for `observation_l`, this information can be used efficiently here. === `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 [wiki:OnlineModifyModelforEnsembleIntegration_PDAF3#prepoststep_pdafprepoststep_ens_pdaf.F90 inserting the analysis step] for the description of this routine. === `U_prodRinvA_l` (prodrinva_l_pdaf.F90) === This routine is used by the local filters. There is a slight difference between LSEIK and other local filters for this routine, which is described below. The interface for this routine is: {{{ SUBROUTINE prodRinvA_l(domain_p, step, dim_obs_l, dim_ens, 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) :: dim_ens ! Ensemble size REAL, INTENT(in) :: obs_l(dim_obs_l) ! Local vector of observations REAL, INTENT(inout) :: A_l(dim_obs_l, dim_ens) ! Input matrix from analysis routine REAL, INTENT(out) :: C_l(dim_obs_l, dim_ens) ! 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 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 SEIK and ETKF: For ETKF the third argument is the ensemble size (`dim_ens`), while for SEIK it is the rank (`rank`) of the covariance matrix (usually ensemble size minus one). In addition, the second dimension of `A_l` and `C_l` has size `dim_ens` for ETKF, while it is `rank` for the SEIK filter. (Practically, one can usually ignore this difference as the fourth argument of the interface can be named arbitrarily in the routine.) === `U_prodRinvA_hyb_l` (prodrinva_hyb_l_pdaf.F90) === This routine is used by the local hybrid filter LKNETF. It is a variant of `U_proRinvA_l` accounting for hybridization. The interface for this routine is: {{{ SUBROUTINE prodRinvA_hyb_l(domain_p, step, dim_obs_l, dim_ens, obs_l, gamma, 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) :: dim_ens ! Ensemble size REAL, INTENT(in) :: obs_l(dim_obs_l) ! Local vector of observations REAL, INTENT(in) :: gamma ! Hybrid weight REAL, INTENT(inout) :: A_l(dim_obs_l, dim_ens) ! Input matrix from analysis routine REAL, INTENT(out) :: C_l(dim_obs_l, dim_ens) ! 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. 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. Hints: * This routine is a simple extension of `prodRinvA_l`. One can implement the hybrid variant by copying this routine and adapting it. `gamma` is computed inside PDAF and provided to the routine. === `U_init_n_domains` (init_n_domains_pdaf.F90) === This routine is used by all filter algorithms with domain-localization and is independent of the particular algorithm. The interface for this routine is: {{{ SUBROUTINE init_n_domains(step, n_domains_p) INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(out) :: n_domains_p ! Number of analysis domains for local model sub-domain }}} The routine is called during the analysis step before the loop over the local analysis domains is entered. It has to provide the number of local analysis domains. In case of a domain-decomposed model the number of local analysis domain for the model sub-domain of the calling process has to be initialized. Hints: * As a simple case, if the localization is only performed horizontally, the local analysis domains can be single vertical columns of the model grid. In this case, `n_domains_p` is simply the number of vertical columns in the local model sub-domain. === `U_init_dim_l` (init_dim_l_pdaf.F90) === This routine is used by all filter algorithms with domain-localization and is independent of the particular algorithm. The interface for this routine is: {{{ SUBROUTINE init_dim_l(step, domain_p, dim_l) INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: domain_p ! Current local analysis domain INTEGER, INTENT(out) :: dim_l ! Local state dimension }}} The routine is called during the loop over the local analysis domains in the analysis step. It has to provide in `dim_l` the dimension of the state vector for the local analysis domain with index `domain_p`. Hints: * If a local analysis domain is a single vertical column of the model grid, the size of the state in the local analysis domain will be just the number of vertical grid points at this location. === `U_init_dim_obs_l` (init_dim_obs_l_pdaf.F90) === This routine is used by all filter algorithms with domain-localization and is independent of the particular algorithm. The interface for this routine is: {{{ SUBROUTINE init_dim_obs_l(domain_p, step, dim_obs_f, dim_obs_l) INTEGER, INTENT(in) :: domain_p ! Current local analysis domain INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: dim_obs_f ! Full dimension of observation vector INTEGER, INTENT(out) :: dim_obs_l ! Local dimension of observation vector }}} The routine is called during the loop over the local analysis domains in the analysis step. It has to initialize in `dim_obs_l` the size of the observation vector used for the local analysis domain with index `domain_p`. Some hints: * Usually, the observations to be considered for a local analysis are those which reside within some distance from the local analysis domain. Thus, if the local analysis domain is a single vertical column of the model grid and if the model grid is a regular ij-grid, then one could use some range of i/j indices to select the observations and determine the local number of them. More generally, one can compute the physical distance of an observation from the local analysis domain and decide on this basis, if the observation has to be considered. * In the loop over the local analysis domains, the routine is always called before `U_init_obs_l` is executed. Thus, as `U_init_dim_obs_local` has to check which observations should be used for the local analysis domain, one can already initialize an integer array that stores the index of observations to be considered. This index should be the position of the observation in the array `observation_f`. With this, the initialization of the local observation vector in `U_init_obs_l` can be sped up. * For PDAF, we could not join the routines `U_init_dim_obs_l` and `U_init_obs_l`, because the array for the local observations is allocated internally to PDAF after its size has been determined in `U_init_dim_obs_l`. === `U_g2l_state` (g2l_state_pdaf.F90) === This routine is used by all filter algorithms with domain-localization and is independent of the particular algorithm. The interface for this routine is: {{{ SUBROUTINE global2local_state(step, domain_p, dim_p, state_p, dim_l, state_l) INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: domain_p ! Current local analysis domain INTEGER, INTENT(in) :: dim_p ! State dimension for model sub-domain INTEGER, INTENT(in) :: dim_l ! Local state dimension REAL, INTENT(in) :: state_p(dim_p) ! State vector for model sub-domain REAL, INTENT(out) :: state_l(dim_l) ! State vector on local analysis domain }}} The routine is called during the loop over the local analysis domains in the analysis step. It has to provide the local state vector `state_l` that corresponds to the local analysis domain with index `domain_p`. Provided to the routine is the state vector `state_p`. With a domain decomposed model, this is the state for the local model sub-domain. Hints: * In the simple case that a local analysis domain is a single vertical column of the model grid, the operation in this routine would be to take out of `state_p` the data for the vertical column indexed by `domain_p`. === `U_l2g_state` (l2g_state_pdaf.F90) === This routine is used by all filter algorithms with domain-localization and is independent of the particular algorithm. The interface for this routine is: {{{ SUBROUTINE l2g_state(step, domain_p, dim_l, state_l, dim_p, state_p) INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: domain_p ! Current local analysis domain INTEGER, INTENT(in) :: dim_p ! State dimension for model sub-domain INTEGER, INTENT(in) :: dim_l ! Local state dimension REAL, INTENT(in) :: state_p(dim_p) ! State vector for model sub-domain REAL, INTENT(out) :: state_l(dim_l) ! State vector on local analysis domain }}} The routine is called during the loop over the local analysis domains in the analysis step. It has to initialize the part of the global state vector `state_p` that corresponds to the local analysis domain with index `domain_p`. Provided to the routine is the state vector `state_l` for the local analysis domain. Hints: * In the simple case that a local analysis domain is a single vertical column of the model grid, the operation in this routine would be to write into `state_p` the data for the vertical column indexed by `domain_p`. === `U_g2l_obs` (g2l_obs_pdaf.F90) === This routine is used by all filter algorithms with domain-localization and is independent of the particular algorithm. The interface for this routine is: {{{ SUBROUTINE g2l_obs(domain_p, step, dim_obs_f, dim_obs_l, mstate_f, mstate_l) INTEGER, INTENT(in) :: domain_p ! Current local analysis domain INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: dim_obs_f ! Dimension of full observation vector for model sub-domain INTEGER, INTENT(in) :: dim_obs_l ! Dimension of observation vector for local analysis domain REAL, INTENT(in) :: mstate_f(dim_obs_f) ! Full observation vector for model sub-domain REAL, INTENT(out) :: mstate_l(dim_obs_l) ! Observation vector for local analysis domain }}} The routine is called during the loop over the local analysis domains in the analysis step. It has to provide a local observation vector `mstate_l` for the observation domain that corresponds to the local analysis domain with index `domain_p`. Provided to the routine is the full observation vector `mstate_f` from which the local part has to be extracted. Hints: * The vector `mstate_f` that is provided to the routine is one of the observed state vectors that are produced by `U_obs_op_full`. * Some operations performed here are analogous to those required to initialize a local vector of observations in `U_init_obs_l`. If that routine reads first a full vector of observations (e.g. in `U_init_dim_obs_f`), this vector has to be restricted to the relevant observations for the current local analysis domain. For this operation, one can for example initialize an index array when `U_init_dim_obs_l` is executed. (Which happens before `U_global2local_obs`) === `U_init_obsvar` (init_obsvar_pdaf.F90) === This routine is used by the global and local square-root filter algorithms. The routine is only called if the adaptive forgetting factor is used (`type_forget=1` in the example implementation). The difference in this routine between global and local filters is that the global filters use 'global' while the local filters use 'full' quantities. The interface for this routine is: {{{ SUBROUTINE init_obsvar(step, dim_obs_f, obs_f, meanvar_f) INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: dim_obs_f ! Full dimension of observation vector REAL, INTENT(in) :: obs_f(dim_obs_f) ! Full observation vector REAL, INTENT(out) :: meanvar_f ! Mean observation error variance }}} The routine is called in the local filters before the loop over all local analysis domains is entered. The call is by the routine that computes an adaptive forgetting factor (`PDAF_set_forget`). The routine has to initialize an average full observation error variance, which should be consistent with the observation vector initialized in `U_init_ob_f`. Hints: * For a model with domain-decomposition one might use the mean variance for the model sub-domain of the calling process. Alternatively one can compute a mean variance for the full model domain using MPI communication (e.g. the function `MPI_allreduce`). * The observation vector `obs_p` is provided to the routine for the case that the observation error variance is relative to the value of the observations. * If the adaptive forgetting factor is not used, this routine has only to exist for the compilation, but it does not need functionality. === `U_init_obsvar_l` (init_obsvar_l_pdaf.F90) === This routine is used by all filter algorithms with domain-localization and is independent of the particular algorithm. The routine is only called if the local adaptive forgetting factor is used (`type_forget=2` in the example implementation). The interface for this routine is: {{{ SUBROUTINE init_obsvar_l(domain_p, step, dim_obs_l, obs_l, meanvar_l) INTEGER, INTENT(in) :: domain_p ! Current local analysis domain INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: dim_obs_l ! Local dimension of observation vector REAL, INTENT(in) :: obs_l(dim_obs_l) ! Local observation vector REAL, INTENT(out) :: meanvar_l ! Mean local observation error variance }}} The routine is called in the local filters during the loop over all local analysis domains by the routine that computes a local adaptive forgetting factor (`PDAF_set_forget_local`). The routine has to initialize a local mean observation error variance for all observations used for the analysis in the current local analysis domain. Hints: * If the local adaptive forgetting factor is not used, this routine has only to exist for the compilation, but it does not need functionality. === `U_likelihood_l` (likelihood_l_pdaf.F90) === This routine is used by the LNETF and LKNETF filters. The interface for this routine is: {{{ SUBROUTINE likelihood_l(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. 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 routine is very similar to the routine [wiki: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 [wiki: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. === `U_likelihood_hyb_l` (likelihood_hyb_l_pdaf.F90) === This routine is used by the local hybrid filter LKNETF. It is a variant of `U_likelihood_l` accounting for hybridization. The interface for this routine is: {{{ SUBROUTINE likelihood_hyb_l(domain_p, step, dim_obs_l, obs_l, resid_l, gamma, 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(in) :: gamma ! Hybrid weight REAL, INTENT(out) :: likely_l(dim_obs_l) ! Output value of the likelihood }}} This routine is a variant for `U_likelihood_l`. See the description of this routine for its functionality. The routine is called during the loop over the local analysis domains. 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. The routine also has to apply the hybrid weight `gamma`. This is a simple multiplication with `1-gamma` in the loop where `Rinvresid_l` is initialized. Hints: * This routine is a simple extension of `U_likelihood_l. One can implement the hybrid variant by copying this routine and adapting it. `gamma` is computed inside PDAF and provided to the routine. === `U_next_observation` (next_observation_pdaf.F90) === This routine is independent of the filter algorithm used. See the page on [wiki:OnlineModifyModelforEnsembleIntegration_PDAF3#next_observation_pdafnext_observation_pdaf.F90 modifying the model code for the ensemble integration] for the description of this routine. == Execution order of user-supplied routines == The executation order and how ofter the user routines are called depends on the chosen hybrid filter variant. The two-step variants HNK (subtype=0) and HKN (subtype=1) perform two local analysis loops (one for LNETF and one for LETKF), while the synchronous variance (Hsync, subtype=4) perform only a single loop and computes the LETKF and NETF updates synchronously (which still requires multiple calls to user routines). The order can be important as some routines can perform preparatory work for routines executed later on during the analysis. For example, `U_init_dim_obs_l` is often used to prepare an index array that provides the information how to localize a 'full' vector of observations. Some hints one the efficient implementation strategy are given with the descriptions of the routine interfaces above. Before the analysis step is called the following is executed: 1. [#U_collect_statecollect_state_pdaf.F90 U_collect_state] (called once for each ensemble member) When the ensemble integration of the forecast is completed, the analysis step is executed. Before the loop over all local analysis domains, the following routines are executed in the same way for all three hybrid filter variants: 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_n_domainsinit_n_domains_pdaf.F90 U_init_n_domains] 1. [#U_init_dim_obs_finit_dim_obs_f_pdaf.F90 U_init_dim_obs_f] 1. [#U_obs_op_fobs_op_f_pdaf.F90 U_obs_op_f] (Called `dim_ens` times; once for each ensemble member) 1. [#U_init_obs_finit_obs_f_pdaf.F90 U_init_obs_f] (Only executed, if the global adaptive forgetting factor is used (`type_forget=1` in the example implementation)) 1. [#U_init_obsvarinit_obsvar_pdaf.F90 U_init_obsvar] (Only executed, if the global adaptive forgetting factor is used (`type_forget=1` in the example implementation)) For `Hsync`: In the loop over all local analysis domains, it is executed for each local analysis domain: 1. [#U_init_dim_linit_dim_l_pdaf.F90 U_init_dim_l] 1. [#U_init_dim_obs_linit_dim_obs_l_pdaf.F90 U_init_dim_obs_l] 1. [#U_g2l_stateg2l_state_pdaf.F90 U_g2l_state] (Called `dim_ens+1` times: Once for each ensemble member and once for the mean state estimate) 1. [#U_g2l_obsg2l_obs_pdaf.F90 U_g2l_obs] (Called `dim_ens+1` times: Once for each ensemble member and once for the mean state estimate) 1. [#U_init_obs_linit_obs_l_pdaf.F90 U_init_obs_l] 1. [#U_init_obsvar_linit_obsvar_l_pdaf.F90 U_init_obsvar_l] (Only called, if the local adaptive forgetting factor is used (`type_forget=2` in the example implementation)) 1. [#U_prodRinvA_lprodrinva_l_pdaf.F90 U_prodRinvA_l] 1. [#U_likelihood_llikelihood_l_pdaf.F90 U_likelihood_l] (Calls `dim_ens` times, once for each ensemble state) 1. [#U_l2g_statel2g_state_pdaf.F90 U_l2g_state] (Called `dim_ens+1` times: Once for each ensemble member and once for the mean state estimate) For `HNK` and `HKN` two local analysis loops are performed with additional initialization of observation information in between: 1. First local analysis loop: 1. [#U_init_dim_linit_dim_l_pdaf.F90 U_init_dim_l] 1. [#U_init_dim_obs_linit_dim_obs_l_pdaf.F90 U_init_dim_obs_l] 1. [#U_g2l_stateg2l_state_pdaf.F90 U_g2l_state] (Called `dim_ens+1` times: Once for each ensemble member and once for the mean state estimate) 1. [#U_g2l_obsg2l_obs_pdaf.F90 U_g2l_obs] (Called `dim_ens+1` times: Once for each ensemble member and once for the mean state estimate) 1. [#U_init_obs_linit_obs_l_pdaf.F90 U_init_obs_l] 1. [#U_likelihood_llikelihood_l_pdaf.F90 U_likelihood_l] (Called `dim_ens` times to determine hybrid weight `gamma`) 1. Execute LNETF (for `HNK`) or LETKF (for `HKN`) 1. [#U_l2g_statel2g_state_pdaf.F90 U_l2g_state] (Called `dim_ens+1` times: Once for each ensemble member and once for the mean state estimate) 1. In between both local analysis loops: 1. [#U_obs_op_fobs_op_f_pdaf.F90 U_obs_op_f] (Called `dim_ens` times; once for each ensemble member) 1. Second local analysis loop: 1. [#U_init_dim_linit_dim_l_pdaf.F90 U_init_dim_l] 1. [#U_init_dim_obs_linit_dim_obs_l_pdaf.F90 U_init_dim_obs_l] 1. [#U_g2l_stateg2l_state_pdaf.F90 U_g2l_state] (Called `dim_ens+1` times: Once for each ensemble member and once for the mean state estimate) 1. [#U_g2l_obsg2l_obs_pdaf.F90 U_g2l_obs] (Called `dim_ens+1` times: Once for each ensemble member and once for the mean state estimate) 1. [#U_init_obs_linit_obs_l_pdaf.F90 U_init_obs_l] 1. [#U_likelihood_llikelihood_l_pdaf.F90 U_likelihood_l] (Called `dim_ens` times to determine hybrid weight `gamma`) 1. Execute LETKF (for `HNK`) or LNETF (for `HKN`) 1. [#U_l2g_statel2g_state_pdaf.F90 U_l2g_state] (Called `dim_ens+1` times: Once for each ensemble member and once for the mean state estimate) In step 7: When the LNETF is executed the routine 1. [#U_likelihood_hyb_llikelihood_hyb_l_pdaf.F90 U_likelihood_hyb_l] (Called `dim_ens` times, once for each ensemble member) is called. In contrast when the LETKF is computed the routines 1. [#U_init_obsvar_linit_obsvar_l_pdaf.F90 U_init_obsvar_l] (Only called, if the local adaptive forgetting factor is used (`type_forget=2` in the example implementation)) 1. [#U_prodRinvA_hyb_lprodrinva_hyb_l_pdaf.F90 U_prodRinvA_hyb_l] are called. After the loop(s) over all local analysis domains, it is executed: 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_lknetf`, 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]