! Copyright (c) 2014-2024 Paul Kirchgessner ! ! This file is part of PDAF. ! ! PDAF is free software: you can redistribute it and/or modify ! it under the terms of the GNU Lesser General Public License ! as published by the Free Software Foundation, either version ! 3 of the License, or (at your option) any later version. ! ! PDAF is distributed in the hope that it will be useful, ! but WITHOUT ANY WARRANTY; without even the implied warranty of ! MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ! GNU Lesser General Public License for more details. ! ! You should have received a copy of the GNU Lesser General Public ! License along with PDAF. If not, see <http://www.gnu.org/licenses/>. ! !$Id$ !BOP ! ! !ROUTINE: PDAF_netf_analysis --- NETF analysis cf. Toedter & Ahrens (2015) ! ! !INTERFACE: SUBROUTINE PDAF_netf_analysis(step, dim_p, dim_obs_p, dim_ens, & state_p, ens_p, rndmat, T, type_forget, forget, & type_winf, limit_winf, noise_type, noise_amp, & U_init_dim_obs, U_obs_op, U_init_obs, U_likelihood, & screen, flag) ! !DESCRIPTION: ! Analysis step of the NETF following Toedter and Ahrens (2015) ! A Second-order Exact Ensemble Square Root Filter for Nonlinear ! Data Assimlation. ! The update is slightly modified to avoid computing the forecast ! mean state. ! ! Variant for domain decomposed states. ! ! ! This is a core routine of PDAF and ! should not be changed by the user ! ! ! !REVISION HISTORY: ! 2014-03 - Paul Kirchgessner Changed original ETKF code to NETF ! Later revisions - see svn log ! ! !USES: ! Include definitions for real type of different precision ! (Defines BLAS/LAPACK routines and MPI_REALTYPE) #include "typedefs.h" USE PDAF_timer, & ONLY: PDAF_timeit USE PDAF_memcounting, & ONLY: PDAF_memcount USE PDAF_mod_filtermpi, & ONLY: mype USE PDAF_mod_filter, & ONLY: obs_member, debug USE PDAFomi, & ONLY: omi_n_obstypes => n_obstypes, omi_omit_obs => omit_obs USE PDAF_analysis_utils, & ONLY: PDAF_omit_obs_omi IMPLICIT NONE ! !ARGUMENTS: INTEGER, INTENT(in) :: step ! Current time step INTEGER, INTENT(in) :: dim_p ! PE-local dimension of model state INTEGER, INTENT(out) :: dim_obs_p ! PE-local dimension of observation vector INTEGER, INTENT(in) :: dim_ens ! Size of ensemble REAL, INTENT(out) :: state_p(dim_p) ! on exit: PE-local forecast state REAL, INTENT(inout) :: ens_p(dim_p, dim_ens) ! PE-local state ensemble REAL, INTENT(in) :: rndmat(dim_ens, dim_ens) ! Orthogonal random matrix REAL, INTENT(inout) :: T(dim_ens, dim_ens) ! Ensemble transform matrix INTEGER, INTENT(in) :: type_forget ! Type of forgetting factor REAL, INTENT(in) :: forget ! Forgetting factor INTEGER, INTENT(in) :: type_winf ! Type of weights inflation REAL, INTENT(in) :: limit_winf ! Limit for weights inflation INTEGER, INTENT(in) :: noise_type ! Type of pertubing noise REAL, INTENT(in) :: noise_amp ! Amplitude of noise INTEGER, INTENT(in) :: screen ! Verbosity flag INTEGER, INTENT(inout) :: flag ! Status flag ! ! External subroutines ! ! (PDAF-internal names, real names are defined in the call to PDAF) EXTERNAL :: U_init_dim_obs, & ! Initialize dimension of observation vector U_obs_op, & ! Observation operator U_init_obs, & ! Initialize observation vector U_likelihood ! Compute observation likelihood for an ensemble member ! !CALLING SEQUENCE: ! Called by: PDAF_netf_update ! Calls: U_init_dim_obs ! Calls: U_obs_op ! Calls: U_init_obs ! Calls: U_likelihood ! Calls: PDAF_timeit ! Calls: PDAF_memcount ! Calls: gemmTYPE (BLAS; dgemm or sgemm dependent on precision) ! Calls: syevTYPE (LAPACK; dsyev or ssyev dependent on precision) !EOP ! *** local variables *** INTEGER :: i, j, member, col, row ! counters INTEGER, SAVE :: allocflag = 0 ! Flag whether first time allocation is done INTEGER :: ldwork ! Size of work array for SYEV INTEGER :: maxblksize, blkupper, blklower ! Variables for blocked ensemble update INTEGER :: syev_info ! Status flag for Lapack REAL :: fac ! Multiplication factor REAL :: effN ! Efective sample size REAL :: weight ! Ensemble weight (likelihood) INTEGER :: n_small_svals ! Number of small eigenvalues REAL, ALLOCATABLE :: resid_i(:) ! PE-local observation residual REAL, ALLOCATABLE :: obs_p(:) ! PE-local observation vector REAL, ALLOCATABLE :: ens_blk(:,:) ! Temporary block of state ensemble REAL, ALLOCATABLE :: svals(:) ! Singular values of Uinv REAL, ALLOCATABLE :: work(:) ! Work array for SYEV REAL, ALLOCATABLE :: T_tmp(:,:) ! Square root of transform matrix REAL, ALLOCATABLE :: A(:,:) ! Full transform matrix REAL, ALLOCATABLE :: Rinvresid(:) ! R^-1 times residual REAL, ALLOCATABLE :: weights(:) ! Weight vector REAL :: total_weight ! Sum of weights ! ********************** ! *** INITIALIZATION *** ! ********************** CALL PDAF_timeit(51, 'new') IF (debug>0) & WRITE (*,*) '++ PDAF-debug: ', debug, 'PDAF_netf_analysis -- START' IF (mype == 0 .AND. screen > 0) THEN WRITE (*, '(a, 5x, a)') & 'PDAF', 'Compute NETF filter update' END IF ! ************************ ! *** Inflate ensemble *** ! ************************ IF (type_forget==0 ) THEN IF (debug>0) & WRITE (*,*) '++ PDAF-debug PDAF_netf_analysis', debug, & 'Inflate forecast ensemble' CALL PDAF_timeit(34, 'new') ! Apply forgetting factor CALL PDAF_inflate_ens(dim_p, dim_ens, state_p, ens_p, forget) CALL PDAF_timeit(34, 'old') ENDIF CALL PDAF_timeit(51, 'old') ! ********************************* ! *** Get observation dimension *** ! ********************************* IF (debug>0) THEN WRITE (*,*) '++ PDAF-debug PDAF_netf_analysis:', debug, ' dim_p', dim_p WRITE (*,*) '++ PDAF-debug: ', debug, 'PDAF_netf_analysis -- call init_dim_obs' END IF CALL PDAF_timeit(15, 'new') CALL U_init_dim_obs(step, dim_obs_p) CALL PDAF_timeit(15, 'old') IF (debug>0) & WRITE (*,*) '++ PDAF-debug PDAF_netf_analysis:', debug, ' dim_obs_p', dim_obs_p IF (screen > 2) THEN WRITE (*, '(a, 5x, a13, 1x, i3, 1x, a, i8)') & 'PDAF', '--- PE-domain', mype, 'dimension of observation vector', dim_obs_p END IF ! *********************************************** ! *** Compute particle weights (=likelihood) *** ! *********************************************** CALL PDAF_timeit(12, 'new') ! Allocate weights ALLOCATE(weights(dim_ens)) IF (allocflag == 0) CALL PDAF_memcount(3, 'r', dim_ens) haveobs: IF (dim_obs_p > 0) THEN ! *** The weights only exist for domains with observations *** ALLOCATE(obs_p(dim_obs_p)) IF (allocflag == 0) CALL PDAF_memcount(3, 'r', dim_obs_p) ! Allocate tempory arrays for obs-ens_i ALLOCATE(resid_i(dim_obs_p)) ALLOCATE(Rinvresid(dim_obs_p)) IF (allocflag == 0) CALL PDAF_memcount(3, 'r', 2*dim_obs_p) ! Omit observations if innovation is too large ! This step also initializes obs_p IF (omi_omit_obs) THEN CALL PDAF_omit_obs_omi(dim_p, dim_obs_p, dim_ens, state_p, ens_p, & obs_p, U_init_obs, U_obs_op, 1, screen) END IF ! Get residual as difference of observation and observed state for each ensemble member IF (debug>0) & WRITE (*,*) '++ PDAF-debug: ', debug, & 'PDAF_netf_analysis -- call obs_op and likelihood', dim_ens, 'times' CALC_w: DO member = 1, dim_ens ! Store member index obs_member = member CALL PDAF_timeit(44, 'new') CALL U_obs_op(step, dim_p, dim_obs_p, ens_p(:, member), resid_i) CALL PDAF_timeit(44, 'old') IF (member==1 .and. (.not. omi_omit_obs)) THEN IF (debug>0) & WRITE (*,*) '++ PDAF-debug: ', debug, 'PDAF_netf_analysis -- call init_obs' ! get observation vector (has to be after U_obs_op for OMI) CALL PDAF_timeit(50, 'new') CALL U_init_obs(step, dim_obs_p, obs_p) CALL PDAF_timeit(50, 'old') END IF CALL PDAF_timeit(51, 'new') resid_i = obs_p - resid_i CALL PDAF_timeit(51, 'old') IF (debug>0) THEN WRITE (*,*) '++ PDAF-debug: ', debug, 'PDAF_netf_analysis -- member', member WRITE (*,*) '++ PDAF-debug PDAF_netf_analysis:', debug, ' innovation d', resid_i WRITE (*,*) '++ PDAF-debug: ', debug, 'PDAF_netf_analysis -- call likelihood' end IF ! Compute likelihood CALL PDAF_timeit(47, 'new') CALL U_likelihood(step, dim_obs_p, obs_p, resid_i, weight) CALL PDAF_timeit(47, 'old') weights(member) = weight END DO CALC_w IF (debug>0) & WRITE (*,*) '++ PDAF-debug PDAF_netf_analysis:', debug, ' raw weights', weights ! Compute inflation of weights according to N_eff IF (type_winf == 1) THEN IF (debug>0) & WRITE (*,*) '++ PDAF-debug: ', debug, & 'PDAF_netf_analysis -- inflate weights ' CALL PDAF_inflate_weights(screen, dim_ens, limit_winf, weights) END IF CALL PDAF_timeit(51, 'new') ! Normalize weights total_weight = 0.0 DO i = 1, dim_ens total_weight = total_weight + weights(i) END DO IF (total_weight /= 0.0) THEN ! Normalize weights weights = weights / total_weight IF (debug>0) & WRITE (*,*) '++ PDAF-debug PDAF_netf_analysis:', debug, ' normalized weights', weights ELSE ! ERROR: weights are zero WRITE(*,'(/5x,a/)') 'WARNING: Zero weights - reset to 1/dim_ens' weights = 1.0 / REAL(dim_ens) END IF DEALLOCATE(obs_p, resid_i, Rinvresid) ! Diagnostic: Compute effective sample size CALL PDAF_diag_effsample(dim_ens, weights, effN) IF (mype == 0 .AND. screen > 0) & WRITE (*, '(a, 5x, a, f10.2)') 'PDAF', '--- Effective sample size ', effN CALL PDAF_timeit(51, 'old') ELSE ! Without observations, all ensemble members have the same weight CALL PDAF_timeit(51, 'new') weights = 1/dim_ens CALL PDAF_timeit(51, 'old') ! For OMI we need to call observation operator also for dim_obs_p=0 ! in order to initialize pointer to observation type IF (omi_n_obstypes>0) THEN IF (debug>0) & WRITE (*,*) '++ PDAF-debug: ', debug, & 'PDAF_netf_analysis -- call obs_op', dim_ens, 'times' ALLOCATE(resid_i(1)) obs_member = 1 ! [Hx_1 ... Hx_N] CALL U_obs_op(step, dim_p, dim_obs_p, ens_p(:, 1), resid_i(:)) DEALLOCATE(resid_i) END IF END IF haveobs CALL PDAF_timeit(12, 'old') CALL PDAF_timeit(51, 'new') ! **************************************** ! *** Calculate the transform matrix *** ! *** A= (diag(w)-w*w^t) *** ! *** with the weights w *** ! **************************************** CALL PDAF_timeit(10, 'new') ALLOCATE(A(dim_ens,dim_ens)) IF (allocflag == 0) CALL PDAF_memcount(3, 'r', dim_ens*dim_ens) DO i = 1, dim_ens DO j = 1, dim_ens A(i,j) = -weights(i) * weights(j) ENDDO ENDDO DO i = 1, dim_ens A(i,i) = A(i,i) + weights(i) END DO IF (debug>0) & WRITE (*,*) '++ PDAF-debug PDAF_netf_analysis:', debug, ' A', A CALL PDAF_timeit(10, 'old') ! ******************************************************************** ! *** Compute ensemble transformation matrix W as square-root of A *** ! ******************************************************************** CALL PDAF_timeit(13, 'new') ! Compute symmetric square-root of A by EVD ALLOCATE(svals(dim_ens)) ALLOCATE(work(3*dim_ens)) IF (allocflag == 0) CALL PDAF_memcount(3, 'r', 4*dim_ens + dim_ens*dim_ens) ldwork = 3*dim_ens IF (debug>0) & WRITE (*,*) '++ PDAF-debug PDAF_netf_analysis:', debug, & ' Compute eigenvalue decomposition of A' CALL syevTYPE('v', 'l', dim_ens, A, dim_ens, svals, work, ldwork, syev_info) IF (syev_info == 0) THEN IF (debug>0) & WRITE (*,*) '++ PDAF-debug PDAF_netf_analysis:', debug, ' eigenvalues', svals ELSE WRITE(*,'(/5x,a/)') 'PDAF-ERROR(2): Problem in computing the SVD of W-ww^T' flag = 2 END IF ! Check for too small eigenvalues n_small_svals = 0 DO i = 1, dim_ens IF (svals(i)<1.0E-15) THEN svals(i) = 0.0 n_small_svals = n_small_svals + 1 END IF END DO ! subtract one, because A is rank dim_ens-1 n_small_svals = n_small_svals - 1 IF (mype == 0 .AND. screen > 0) & WRITE (*, '(a, 5x, a, i5)') & 'PDAF', '--- number of small singular values ', n_small_svals DO j = 1,dim_ens DO i = 1, dim_ens T(j,i) = A(j,i) * SQRT(svals(i)) END DO END DO DEALLOCATE(svals, work) ALLOCATE(T_tmp(dim_ens,dim_ens)) IF (allocflag == 0) CALL PDAF_memcount(3, 'r', dim_ens*dim_ens) ! Calculate transform matrix T CALL gemmTYPE('n', 't', dim_ens, dim_ens, dim_ens, 1.0, & T, dim_ens, A, dim_ens, 0.0, T_tmp, dim_ens) ! Multiply T by m/(m-1) to get unbiased ensemble fac = SQRT(REAL(dim_ens)) IF (type_forget==2) fac = fac / SQRT(forget) !analysis inflation ! Multiply random matrix with quare root of A (T) CALL gemmTYPE('n', 'n', dim_ens, dim_ens, dim_ens, & fac, T_tmp, dim_ens, rndmat, dim_ens, & 0.0, T, dim_ens) ! Compute W = sqrt(U) + w for efficient ensemble update DO col = 1, dim_ens DO row = 1, dim_ens T(row, col) = T(row, col) + weights(row) END DO END DO IF (debug>0) & WRITE (*,*) '++ PDAF-debug PDAF_netf_analysis:', debug, ' transform', T DEALLOCATE(weights, A, T_tmp) CALL PDAF_timeit(13, 'old') ! ******************************************* ! *** Transform state ensemble *** ! *** a f *** ! *** X = X W *** ! *** The weight matrix W is stored in T. *** ! ******************************************* ! Use block formulation for transformation maxblksize = 200 IF (mype == 0 .AND. screen > 0) & WRITE (*, '(a, 5x, a, i5)') & 'PDAF', '--- use blocking with size ', maxblksize ALLOCATE(ens_blk(maxblksize, dim_ens)) IF (allocflag == 0) CALL PDAF_memcount(3, 'r', maxblksize * dim_ens) blocking: DO blklower = 1, dim_p, maxblksize blkupper = MIN(blklower + maxblksize - 1, dim_p) ! Store forecast ensemble CALL PDAF_timeit(21, 'new') DO col = 1, dim_ens ens_blk(1 : blkupper - blklower + 1, col) & = ens_p(blklower : blkupper, col) END DO CALL PDAF_timeit(21, 'old') ! a _f ! Transform ensemble: X = X W CALL PDAF_timeit(22, 'new') CALL gemmTYPE('n', 'n', blkupper - blklower + 1, dim_ens, dim_ens, & 1.0, ens_blk, maxblksize, T, dim_ens, & 0.0, ens_p(blklower:blkupper, 1), dim_p) CALL PDAF_timeit(22, 'old') END DO blocking DEALLOCATE(ens_blk) ! ***************************************** ! *** Perturb particles by adding noise *** ! ***************************************** CALL PDAF_timeit(23, 'new') IF (noise_type>0) THEN IF (debug>0) & WRITE (*,*) '++ PDAF-debug PDAF_netf_analysis:', debug, ' add noise to particles' CALL PDAF_pf_add_noise(dim_p, dim_ens, state_p, ens_p, noise_type, noise_amp, screen) END IF CALL PDAF_timeit(23, 'old') CALL PDAF_timeit(51, 'old') ! ******************** ! *** Finishing up *** ! ******************** IF (allocflag == 0) allocflag = 1 IF (debug>0) & WRITE (*,*) '++ PDAF-debug: ', debug, 'PDAF_netf_analysis -- END' END SUBROUTINE PDAF_netf_analysis