Changes between Version 1 and Version 2 of ImplementAnalysisletkf


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Timestamp:
May 17, 2011, 3:39:42 PM (13 years ago)
Author:
lnerger
Comment:

Created text for LETKF based on text for LSEIK

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  • ImplementAnalysisletkf

    v1 v2  
    1 Implementation of the Analysis step for the LETKF (Local Ensemble Transform Kalman Filter) algorithm
     1= Implementation of the Analysis step for the LETKF (Local Ensemble Transform Kalman Filter) algorithm =
     2
     3{{{
     4#!html
     5<div class="wiki-toc">
     6<h4>Implementation Guide</h4>
     7<ol><li><a href="ImplementationGuide">Main page</a></li>
     8<li><a href="AdaptParallelization">Adaptation of the parallelization</a></li>
     9<li><a href="InitPdaf">Initialization of PDAF</a></li>
     10<li><a href="ModifyModelforEnsembleIntegration">Modifications for ensemble integration</a></li>
     11<li><a href="ImplementationofAnalysisStep">Implementation of the analysis step</a></li>
     12<ol>
     13<li><a href="ImplementAnalysisseik">Implementation for SEIK</a></li>
     14<li><a href="ImplementAnalysislseik">Implementation for LSEIK</a></li>
     15<li><a href="ImplementAnalysisetkf">Implementation for ETKF</a></li>
     16<li>Implementation for LETKF</li>
     17</ol>
     18<li><a href="AddingMemoryandTimingInformation">Memory and timing information</a></li>
     19</ol>
     20</div>
     21}}}
     22
     23
     24[[PageOutline(2-3,Contents of this page)]]
     25
     26== Overview ==
     27
     28For the analysis step of the LETKF 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_put_state_letkf` described below. With regard to the parallelization, all these routines (except `U_collect_state`) are executed by the filter processes (`filterpe=1`) only.
     29
     30For completeness we discuss here all user-supplied routines that are specified in the interface to `PDAF_put_state_letkf`. Many of the routines are localized versions of those that are needed for the global ETKF method. Hence, if the user-supplied routines for the global ETKF method have been already implemented, one can base on these routines to speed up the implementation. Due to this, it can also be reasonable to first fully implement a global filter version and subsequently implement the corresponding localized filter by modifying and extending the global routines.
     31
     32The LSEIK filter and the LETKF (Local Ensemble Transform Kalman Filter) are very similar. For this reason, the interface to the user-supplied routines is almost identical. Depending on the implementation it can be possible to use identical routines for the LSEIK filter and the LETKF. Differences are marked in the text below.
     33
     34== `PDAF_put_state_letkf` ==
     35
     36The general espects of the filter-specific routines `PDAF_put_state_*` have been described on the page [ModifyModelforEnsembleIntegration Modification of the model code for the ensemble integration].
     37The interface for the routine `PDAF_put_state_letkf` contains several routine names for routines that operate on the local analysis domains (marked by `_local` at then end of the routine name). In addition, there are names for routines that consider all available observations required to perform local analyses with LETKF within some sub-domain of a domain-decomposed model (marked by `_full` at then 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, this might be a smaller set of observations.
     38
     39To explain the  difference, it is assumed, for simplicity, that a local analysis domain consists of a single vertical column of the model grid. In addition, we assume that the domain decomposition splits the global model domain by vertical boundaries as is typical for ocean models and that the observations are spatially distributed observations of model fields that are part of the state vector.  Under these assumptions, the situation is the following: When a model uses domain decomposition, the LETKF algorithm is executed such that for each model sub-domain a loop over all local analysis domains (e.g. vertical columns) that belong to the model sub-domain is performed. As each model sub-domain is treated by a different process, all loops are executed parallel to each other.
     40
     41For the update of each single vertical column, observations from some larger domain surrounding the vertical column are required. If the influence radius for the observations is sufficiently small there will be vertical columns for which the relevant observations reside completely inside the model sub-domain of the process. However, if a vertical column is considered that is located close to the boundary of the model sub-domain, there will be some observations that don't belong spatially to the local model sub-domain, but to a neighboring model sub-domain. Nonetheless, these observations are required on the local model sub-domain. A simple way to handle this situation is to initialize for each process all globally available observations, together with their coordinates. While this method is simple, it can also become inefficient if the assimilation program is executed using a large number of processes. As for an initial implementation it is usually not the concern to obtain the highest parallel efficiency, the description below assumes that 'full' refers to the global model domain.
     42
     43The interface when using the LETKF algorithm is the following:
     44{{{
     45  SUBROUTINE PDAF_put_state_letkf(U_collect_state, U_init_dim_obs_full, U_obs_op_full, U_init_obs_full, &
     46                                  U_init_obs_local, U_prepoststep, U_prodRinvA_local, U_init_n_domains, &
     47                                  U_init_dim_local, U_init_dim_obs_local, &
     48                                  U_global2local_state, U_local2glocal_state, U_glocal2local_obs, &
     49                                  U_init_obsvar, U_init_obsvar_local, status)
     50}}}
     51with the following arguments:
     52 * [#U_collect_statecollect_state.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]
     53 * [#U_init_dim_obs_fullinit_dim_obs_full.F90 U_init_dim_obs_full]: The name of the user-supplied routine that provides the size of the full observation vector
     54 * [#U_obs_op_fullobs_op_full.F90 U_obs_op_full]: The name of the user-supplied routine that acts as the full observation operator on some state vector
     55 * [#U_init_obs_fullinit_obs_full.F90 U_init_obs_full]: The name of the user-supplied routine that initializes the full vector of observations
     56 * [#U_init_obs_localinit_obs_local.F90 U_init_obs_local]: The name of the user-supplied routine that initializes the vector of observations for a local analysis domain
     57 * [#U_prepoststepprepoststep_seik.F90 U_prepoststep]: The name of the pre/poststep routine as in `PDAF_get_state`
     58 * [#U_prodRinvA_localprodrinva_local.F90 U_prodRinvA_local]: 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.
     59 * [#U_init_n_domainsinit_n_domains.F90 U_init_n_domains]: The name of the routine that provides the number of local analysis domains
     60 * [#U_init_dim_localinit_dim_local.F90 U_init_dim_local]: The name of the routine that provides the state dimension for a local analysis domain
     61 * [#U_init_dim_obs_localinit_dim_obs_local.F90 U_init_dim_obs_local]: The name of the routine that initializes the size of the observation vector for a local analysis domain
     62 * [#U_global2local_stateglobal2local_state.F90 U_global2local_state]: The name of the routine that initializes a local state vector from the global state vector
     63 * [#U_local2global_statelocal2global_state.F90 U_local2global_state]: The name of the routine that initializes the corresponding part of the global state vector from the the provided local state vector
     64 * [#U_global2local_obsglobal2local_obs.F90 U_global2local_obs]: The name of the routine that initializes a local observation vector from a full observation vector
     65 * [#U_init_obsvarinit_obsvar.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)
     66 * [#U_init_obsvar_localinit_obsvar_local.F90 U_init_obsvar_local]: 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)
     67 * `status`: The integer status flag. It is zero, if `PDAF_put_state_letkf` is exited without errors.
     68
     69Note:
     70 * 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.
     71
     72== User-supplied routines ==
     73
     74Here, all user-supplied routines are described that are required in the call to `PDAF_put_state_letkf`. For some of the generic routines, we link to the page on [ModifyModelforEnsembleIntegration modifying the model code for the ensemble integration].
     75
     76To indicate user-supplied routines we use the prefix `U_`. In the template directory `templates/` these routines are provided in files with the routine's name without this prefix. In the example implementation in `testsuite/src/dummymodel_1D`, the routines exist without the prefix, but with the extension `_dummy_D.F90`. In the section titles below we provide the name of the template file in parentheses.
     77
     78In 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').
     79
     80=== `U_collect_state` (collect_state.F90) ===
     81
     82This routine is independent from the filter algorithm used.
     83See the mape on [ModifyModelforEnsembleIntegration#U_collect_statecollect_state.F90 modifying the model code for the ensemble integration] for the description of this routine.
     84
     85
     86=== `U_init_dim_obs_full` (init_dim_obs_full.F90) ===
     87
     88This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) and is independent of the particular algorithm.
     89
     90The interface for this routine is:
     91{{{
     92SUBROUTINE init_dim_obs_full(step, dim_obs_f)
     93
     94  INTEGER, INTENT(in)  :: step       ! Current time step
     95  INTEGER, INTENT(out) :: dim_obs_f  ! Dimension of full observation vector
     96}}}
     97
     98The 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.
     99
     100Some hints:
     101 * 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`.
     102 * 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_full`.
     103
     104=== `U_obs_op_full` (obs_op_full.F90) ===
     105
     106This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) and is independent of the particular algorithm.
     107
     108The interface for this routine is:
     109{{{
     110SUBROUTINE obs_op_full(step, dim_p, dim_obs_f, state_p, m_state_f)
     111
     112  INTEGER, INTENT(in) :: step               ! Currrent time step
     113  INTEGER, INTENT(in) :: dim_p              ! PE-local dimension of state
     114  INTEGER, INTENT(in) :: dim_obs_f          ! Dimension of the full observed state
     115  REAL, INTENT(in)    :: state_p(dim_p)     ! PE-local model state
     116  REAL, INTENT(out) :: m_state_f(dim_obs_f) ! Full observed state
     117}}}
     118
     119The 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.
     120
     121Hint:
     122 * The routine is similar to `init_dim_obs` used for the global filters. However, with a domain-decomposed model `m_state_f` will contain parts of the state vector from neighboring model sub-domains. To make these parts accessible, some parallel communication will be necessary (The state information for a neighboring model sub-domain, will be in the memory of the process that handles that sub-domain). The example implementation in `testsuite/dummymodel_1d` uses the function `MPI_AllGatherV` for this communication.
     123
     124=== `U_init_obs_full` (init_obs_full.F90) ===
     125
     126This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) and is independent of the particular algorithm.
     127The 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)
     128
     129The interface for this routine is:
     130{{{
     131SUBROUTINE init_obs_full(step, dim_obs_f, observation_f)
     132
     133  INTEGER, INTENT(in) :: step                     ! Current time step
     134  INTEGER, INTENT(in) :: dim_obs_f                ! Dimension of full observation vector
     135  REAL, INTENT(out)   :: observation_f(dim_obs_f) ! Full observation vector
     136}}}
     137
     138The 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).
     139
     140Hints:
     141 * 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.
     142 * If the adaptive forgetting factor is not used, this routine only has to exist. However, no functionality is required.
     143
     144
     145=== `U_init_obs_local` (init_obs_local.F90) ===
     146
     147This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) and is independent of the particular algorithm.
     148
     149The interface for this routine is:
     150{{{
     151SUBROUTINE init_obs_local(domain_p, step, dim_obs_l, observation_l)
     152
     153  INTEGER, INTENT(in) :: domain_p                 ! Current local analysis domain
     154  INTEGER, INTENT(in) :: step                     ! Current time step
     155  INTEGER, INTENT(in) :: dim_obs_l                ! Local dimension of observation vector
     156  REAL, INTENT(out)   :: observation_l(dim_obs_l) ! Local observation vector
     157}}}
     158
     159The routine is called during the analysis step during the loop over the local analysis domain.
     160It 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.
     161
     162Hints:
     163 * For parallel efficiency, the LETKF 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_local` is executed (e.g. by `U_init_dim_obs_full`), 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.
     164 * The routine `U_init_dim_obs_local` 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.
     165
     166
     167=== `U_prepoststep` (prepoststep_seik.F90) ===
     168
     169This routine can generally be identical to that used for the global SEIK filter, which has already been described on the [ModifyModelforEnsembleIntegration#U_prepoststepprepoststep_seik.F90 page on modifying the model code for the ensemble integration]. For completeness, the description is repeated:
     170
     171The interface of the routine is identical for all filters. However, the particular operations that are performed in the routine can be specific for each filter algorithm. Here, we exemplify the interface on the example of the ETKF.
     172
     173The interface for this routine is
     174{{{
     175SUBROUTINE prepoststep(step, dim_p, dim_ens, dim_ens_p, dim_obs_p, &
     176                       state_p, Uinv, ens_p, flag)
     177
     178  INTEGER, INTENT(in) :: step        ! Current time step
     179                         ! (When the routine is called before the analysis -step is provided.)
     180  INTEGER, INTENT(in) :: dim_p       ! PE-local state dimension
     181  INTEGER, INTENT(in) :: dim_ens     ! Size of state ensemble
     182  INTEGER, INTENT(in) :: dim_ens_p   ! PE-local size of ensemble
     183  INTEGER, INTENT(in) :: dim_obs_p   ! PE-local dimension of observation vector
     184  REAL, INTENT(inout) :: state_p(dim_p) ! PE-local forecast/analysis state
     185                                     ! The array 'state_p' is not generally not initialized in the case of SEIK/EnKF/ETKF.
     186                                     ! It can be used freely in this routine.
     187  REAL, INTENT(inout) :: Uinv(dim_ens, dim_ens)  ! Inverse of matrix U
     188  REAL, INTENT(inout) :: ens_p(dim_p, dim_ens)   ! PE-local state ensemble
     189  INTEGER, INTENT(in) :: flag        ! PDAF status flag
     190}}}
     191
     192The 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`).
     193
     194The 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.
     195
     196Hint:
     197 * If a user considers to perform adjustments to the estimates (e.g. for balances), this routine is the right place for it.
     198 * 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 of `U_prepoststep`.
     199 * The interface has a difference for ETKF and SEIK: For the ETKF, the array `Uinv` has size `dim_ens` x `dim_ens`. In contrast it has size `dim_ens-1` x `dim_ens-1` for the SEIK filter.
     200
     201
     202
     203=== `U_prodRinvA_local` (prodrinva_local.F90) ===
     204
     205This routine is used by the local filters (LSEIK and LETKF). There is a slight difference between LSEIK and LETKF for this routine, which is described below.
     206
     207The interface for this routine is:
     208{{{
     209SUBROUTINE prodRinvA_local(domain_p, step, dim_obs_l, dim_ens, obs_l, A_l, C_l)
     210
     211  INTEGER, INTENT(in) :: domain_p             ! Current local analysis domain
     212  INTEGER, INTENT(in) :: step                 ! Current time step
     213  INTEGER, INTENT(in) :: dim_obs_l            ! Dimension of local observation vector
     214  INTEGER, INTENT(in) :: dim_ens              ! Ensemble size
     215  REAL, INTENT(in)    :: obs_l(dim_obs_l)     ! Local vector of observations
     216  REAL, INTENT(inout) :: A_l(dim_obs_l, dim_ens) ! Input matrix from analysis routine
     217  REAL, INTENT(out)   :: C_l(dim_obs_l, dim_ens) ! Output matrix
     218}}}
     219
     220The 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`.
     221
     222This routine is also the place to perform observation localization. To initialize a vector of weights, the routine `PDAF_local_weights` can be called. The procedure is used in the example implementation and also demonstrated in the template routine.
     223
     224Hints:
     225 * 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.
     226 * 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.
     227 * 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.
     228 * 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.)
     229
     230
     231=== `U_init_n_domains` (init_n_domains.F90) ===
     232
     233This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) and is independent of the particular algorithm.
     234
     235The interface for this routine is:
     236{{{
     237SUBROUTINE init_n_domains(step, n_domains_p)
     238
     239  INTEGER, INTENT(in)  :: step        ! Current time step
     240  INTEGER, INTENT(out) :: n_domains_p ! Number of analysis domains for local model sub-domain
     241}}}
     242
     243The routine is called during the analysis step before the loop over the local analysis domains is entered.
     244It 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.
     245
     246Hints:
     247 * 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.
     248
     249
     250=== `U_init_dim_local` (init_dim_local.F90) ===
     251
     252This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) and is independent of the particular algorithm.
     253
     254The interface for this routine is:
     255{{{
     256SUBROUTINE init_dim_local(step, domain_p, dim_l)
     257
     258  INTEGER, INTENT(in)  :: step        ! Current time step
     259  INTEGER, INTENT(in)  :: domain_p    ! Current local analysis domain
     260  INTEGER, INTENT(out) :: dim_l       ! Local state dimension
     261}}}
     262
     263The routine is called during the loop over the local analysis domains in the analysis step.
     264It has to provide in `dim_l` the dimension of the state vector for the local analysis domain with index `domain_p`.
     265
     266Hints:
     267 * 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.
     268
     269
     270=== `U_init_dim_obs_local` (init_dim_obs_local.F90) ===
     271
     272This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) and is independent of the particular algorithm.
     273
     274The interface for this routine is:
     275{{{
     276SUBROUTINE init_dim_obs_local(domain_p, step, dim_obs_f, dim_obs_l)
     277
     278  INTEGER, INTENT(in)  :: domain_p   ! Current local analysis domain
     279  INTEGER, INTENT(in)  :: step       ! Current time step
     280  INTEGER, INTENT(in)  :: dim_obs_f  ! Full dimension of observation vector
     281  INTEGER, INTENT(out) :: dim_obs_l  ! Local dimension of observation vector
     282}}}
     283
     284The routine is called during the loop over the local analysis domains in the analysis step.
     285It has to initialize in `dim_obs_l` the size of the observation vector used for the local analysis domain with index `domain_p`.
     286
     287Some hints:
     288 * 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.
     289 * In the loop over the local analysis domains, the routine is always called before `U_init_obs_local` 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_local` can be sped up.
     290 * For PDAF, we could not join the routines `U_init_dim_obs_local` and `U_init_obs_local`, because the array for the local observations is allocated internally to PDAF after its size has been determined in `U_init_dim_obs_local`.
     291
     292
     293=== `U_global2local_state` (global2local_state.F90) ===
     294
     295This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) and is independent of the particular algorithm.
     296
     297The interface for this routine is:
     298{{{
     299SUBROUTINE global2local_state(step, domain_p, dim_p, state_p, dim_l, state_l)
     300
     301  INTEGER, INTENT(in) :: step           ! Current time step
     302  INTEGER, INTENT(in) :: domain_p       ! Current local analysis domain
     303  INTEGER, INTENT(in) :: dim_p          ! State dimension for model sub-domain
     304  INTEGER, INTENT(in) :: dim_l          ! Local state dimension
     305  REAL, INTENT(in)    :: state_p(dim_p) ! State vector for model sub-domain
     306  REAL, INTENT(out)   :: state_l(dim_l) ! State vector on local analysis domain
     307}}}
     308
     309The 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.
     310
     311Hints:
     312 * 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`.
     313
     314
     315=== `U_local2global_state` (local2global_state.F90) ===
     316
     317This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) and is independent of the particular algorithm.
     318
     319The interface for this routine is:
     320{{{
     321SUBROUTINE local2global_state(step, domain_p, dim_l, state_l, dim_p, state_p)
     322
     323  INTEGER, INTENT(in) :: step           ! Current time step
     324  INTEGER, INTENT(in) :: domain_p       ! Current local analysis domain
     325  INTEGER, INTENT(in) :: dim_p          ! State dimension for model sub-domain
     326  INTEGER, INTENT(in) :: dim_l          ! Local state dimension
     327  REAL, INTENT(in)    :: state_p(dim_p) ! State vector for model sub-domain
     328  REAL, INTENT(out)   :: state_l(dim_l) ! State vector on local analysis domain
     329}}}
     330
     331The 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.
     332
     333Hints:
     334 * 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`.
     335
     336
     337=== `U_global2local_obs` (global2local_obs.F90) ===
     338
     339This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) and is independent of the particular algorithm.
     340
     341The interface for this routine is:
     342{{{
     343SUBROUTINE global2local_obs(domain_p, step, dim_obs_f, dim_obs_l, mstate_f, mstate_l)
     344
     345  INTEGER, INTENT(in) :: domain_p              ! Current local analysis domain
     346  INTEGER, INTENT(in) :: step                  ! Current time step
     347  INTEGER, INTENT(in) :: dim_obs_f             ! Dimension of full observation vector for model sub-domain
     348  INTEGER, INTENT(in) :: dim_obs_l             ! Dimension of observation vector for local analysis domain
     349  REAL, INTENT(in)    :: mstate_f(dim_obs_f)   ! Full observation vector for model sub-domain
     350  REAL, INTENT(out)   :: mstate_l(dim_obs_l)   ! Observation vector for local analysis domain
     351}}}
     352
     353The 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.
     354
     355Hints:
     356 * 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`.
     357 * 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_full`), 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_local` is executed. (Which happens before `U_global2local_obs`)
     358
     359
     360=== `U_init_obsvar` (init_obsvar.F90) ===
     361
     362This routine is used by the global filter algorithms SEIK and  ETKF as well as the local filters LSEIK and LETKF. 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.
     363
     364The interface for this routine is:
     365{{{
     366SUBROUTINE init_obsvar(step, dim_obs_f, obs_f, meanvar_f)
     367
     368  INTEGER, INTENT(in) :: step             ! Current time step
     369  INTEGER, INTENT(in) :: dim_obs_f        ! Full dimension of observation vector
     370  REAL, INTENT(in)    :: obs_f(dim_obs_f) ! Full observation vector
     371  REAL, INTENT(out)   :: meanvar_f        ! Mean observation error variance
     372}}}
     373
     374The 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`).
     375The routine has to initialize an average full observation error variance, which should be consistent with the observation vector initialized in `U_init_ob_full`.
     376
     377
     378Hints:
     379 * 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`).
     380 * 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.
     381 * If the adaptive forgetting factor is not used, this routine has only to exist for the compilation, but it does not need functionality.
     382
     383
     384=== `U_init_obsvar_local` (init_obsvar_local.F90) ===
     385
     386This routine is used by all filter algorithms with domain-localization (LSEIK, LETKF) 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).
     387
     388The interface for this routine is:
     389{{{
     390SUBROUTINE init_obsvar_local(domain_p, step, dim_obs_l, obs_l, meanvar_l)
     391
     392  INTEGER, INTENT(in) :: domain_p         ! Current local analysis domain
     393  INTEGER, INTENT(in) :: step             ! Current time step
     394  INTEGER, INTENT(in) :: dim_obs_l        ! Local dimension of observation vector
     395  REAL, INTENT(in)    :: obs_l(dim_obs_l) ! Local observation vector
     396  REAL, INTENT(out)   :: meanvar_l        ! Mean local observation error variance
     397}}}
     398
     399The 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.
     400
     401Hints:
     402 * If the local adaptive forgetting factor is not used, this routine has only to exist for the compilation, but it does not need functionality.
     403
     404== Execution order of user-supplied routines ==
     405
     406The user-supplied routines are executed in the order listed below. 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_local` can 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.
     407
     408Before the analysis step is called the following is executed:
     409 1. [#U_collect_statecollect_state.F90 U_collect_state] (called once for each ensemble member)
     410
     411When 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:
     412 1. [#U_prepoststepprepoststep_seik.F90 U_prepoststep] (Call to act on the forecast ensemble, called with negative value of the time step)
     413 1. [#U_init_n_domainsinit_n_domains.F90 U_init_n_domains]
     414 1. [#U_init_dim_obs_fullinit_dim_obs_full.F90 U_init_dim_obs_full]
     415 1. [#U_obs_op_fullobs_op_full.F90 U_obs_op_full] (Called `dim_ens` times; once for each ensemble member)
     416 1. [#U_init_obs_fullinit_obs_full.F90 U_init_obs_full] (Only executed, if the global adaptive forgetting factor is used (`type_forget=1` in the example implemention))
     417 1. [#U_init_obsvarinit_obsvar.F90 U_init_obsvar] (Only executed, if the global adaptive forgetting factor is used (`type_forget=1` in the example implemention))
     418
     419In the loop over all local analysis domains, it is executed for each local analysis domain:
     420 1. [#U_init_dim_localinit_dim_local.F90 U_init_dim_local]
     421 1. [#U_init_dim_obs_localinit_dim_obs_local.F90 U_init_dim_obs_local]
     422 1. [#U_global2local_stateglobal2local_state.F90 U_global2local_state] (Called `dim_ens+1` times: Once for each ensemble member and once for the mean state estimate)
     423 1. [#U_global2local_obsglobal2local_obs.F90 U_global2local_obs] (A single call to localize the mean observed state)
     424 1. [#U_init_obs_localinit_obs_local.F90 U_init_obs_local]
     425 1. [#U_global2local_obsglobal2local_obs.F90 U_global2local_obs] (`dim_ens` calls: one call to localize the observed part of each ensemble member)
     426 1. [#U_init_obsvar_localinit_obsvar_local.F90 U_init_obsvar_local] (Only called, if the local adaptive forgetting factor is used (`type_forget=2` in the example implementation))
     427 1. [#U_prodRinvA_localprodrinva_local.F90 U_prodRinvA_local]
     428 1. [#U_local2global_statelocal2global_state.F90 U_local2global_state] (Called `dim_ens+1` times: Once for each ensemble member and once for the mean state estimate)
     429
     430After the loop over all local analysis domains, it is executed:
     431 1. [#U_prepoststepprepoststep_seik.F90 U_prepoststep] (Call to act on the analysis ensemble, called with (positive) value of the time step)
     432
     433
     434