assimilation_routines Module Reference

Functions/Subroutines
subroutine	meanormedian_to_ensemble (nx, assimX, meanORmedian)
subroutine	initialize_assim_info (assimInfo)
subroutine	allocate_assim_ensemble_vectors (assimData, fid_0)
subroutine	allocate_assim_ensemble (assimVar, nens, nvar, varID, fileID, bindir, useBinFile, locID, coordID, xini, mini, maxi, allocateOutput, missing, assimilate, iniFromBin, transformation, lambda, epsilon, dxadd)
subroutine	initiate_fileid_assim_ensemble (assimVar, nens, varID, fileID, useBinFile, dxadd)
subroutine	allocate_assim_interface (assimVar, varName, varID, modID, nSubDim, subDimID)
subroutine	allocate_assim_generation (assimVar, nvar, ensgen, fixsigma, semimeta, restmeta, minsigma, lscale, gridsize, corrtype, xcoord, ycoord, tau)
subroutine	allocate_auxiliary_ensemble (assimVar, nens, nvar, varName, xini, varID, recID, locID, coordID, fileID, bindir, useBinFile, minimum, maximum, assimilate, missing, transformation, lambda, epsilon)
subroutine	allocate_0dim_state_ensemble (assimVar, nens, nvar, varName, xini, varID, locID, coordID, fileID, bindir, useBinFile, narray, limitsarray, assimilate, iniFromBin, missing, transformation, lambda, epsilon)
subroutine	allocate_1dim_state_ensemble (assimVar, nens, nvar, varName, xini, varID, locID, coordID, fileID, bindir, useBinFile, narray, limitsarray, n1, assimilate, iniFromBin, missing, transformation, lambda, epsilon)
subroutine	allocate_2dim_state_ensemble (assimVar, nens, nvar, varName, xini, varID, locID, coordID, fileID, bindir, useBinfile, narray, limitsarray, n1, n2, assimilate, iniFromBin, missing, transformation, lambda, epsilon)
subroutine	allocate_3dim_state_ensemble (assimVar, nens, nvar, varName, xini, varID, locID, coordID, fileID, bindir, useBinFile, narray, limitsarray, n1, n2, n3, assimilate, iniFromBin, missing, transformation, lambda, epsilon)
subroutine	initiate_fileid_0dim_state_ensemble (assimVar, nens, varID, fileID, useBinFile)
subroutine	initiate_fileid_1dim_state_ensemble (assimVar, nens, varID, fileID, useBinFile, n1)
subroutine	initiate_fileid_2dim_state_ensemble (assimVar, nens, varID, fileID, useBinfile, n1, n2)
subroutine	initiate_fileid_3dim_state_ensemble (assimVar, nens, varID, fileID, useBinFile, n1, n2, n3)
subroutine	initiate_fileid_forcing_ensemble (assimVar, nens, varID, fileID, useBinFile, usedx)
subroutine	allocate_assim_forcing_ensemble (assimVar, nens, nvar, varName, xini, varID, locID, coordID, fileID, minimum, maximum, ensgen, sigma, semimeta, restmeta, minsigma, lscale, gridsize, corrtype, xcoord, ycoord, bindir, useBinFile, missing, iniFromBin, transformation, lambda, epsilon, tau, usedx)
subroutine	allocate_assim_observation_ensemble (assimVar, nens, nvar, varName, xini, obsID, modID, coordID, fileID, minimum, maximum, ensgen, sigma, semimeta, restmeta, minsigma, lscale, gridsize, corrtype, xcoord, ycoord, missing, transformation, lambda, epsilon)
subroutine	assim_checkminmax (nx, ne, ensemble, minval, maxval, missing)
subroutine	generate_input_ensemble (n, assimVar, missing, usedx)
subroutine	get_spatially_correlated_random_data2 (n, nens, assimG, X, usedx)
subroutine	get_random_vector_gaussian (n, a, sigma, r)
subroutine	matrixmatrixmultiply (mat1, mat2, matout)
subroutine	choldc (a, p, n, failure)
subroutine	cholsl (a, p, b, x, n)
subroutine	choleskysolution (ny, ne, P, Y, M, failure)
subroutine	enkf_analysis_prepare (N, ND, D, HX, R, locCyy, M, Y, HA, status)
subroutine	enkf_analysis_apply (N, NX, ND, M, HA, locCxy, X)
subroutine	assim_ensemble_statistics (xin, DIM, NN, xmean, xmins, xmaxs, xsigma, xmedian, domedian, missing)
subroutine	assim_get_ensemble_data (NX, NE, ensData, x)
subroutine	assim_set_ensemble_data (NX, NE, ensData, x, checkMinMax, missing)
subroutine	assim_get_ensemble_data_dx (NX, NE, ensData, dx)
subroutine	assim_set_ensemble_data_dx (NX, NE, ensData, x, dx, checkMinMax, missing)
subroutine	updateensemblestatistics (assimData, total_time)
subroutine	assim_forward_transform (nx, ne, x, ensdata, missing)
subroutine	assim_backward_transform (nx, ne, x, ensdata, missing)
subroutine	enkf_analysis_main (assimData)

Detailed Description

Generic subroutines and functions for (Ensemble Kalman Filter) Data Assimilation. These functions are supposed to be general, and should not be changed Functions special for a specific model application are in assimilation_interface.f90.

Function/Subroutine Documentation

allocate_0dim_state_ensemble()

subroutine assimilation_routines::allocate_0dim_state_ensemble	(	type(assim_state_ensemble_type)	assimVar,
		integer	nens,
		integer	nvar,
		character(len=*)	varName,
		real, dimension(nvar)	xini,
		integer	varID,
		integer	locID,
		integer	coordID,
		integer	fileID,
		character(len=*), intent(in)	bindir,
		integer, intent(in)	useBinFile,
		integer, intent(in)	narray,
		real, dimension(narray,2), intent(in)	limitsarray,
		logical, intent(in)	assimilate,
		logical, intent(in)	iniFromBin,
		real	missing,
		integer, intent(in)	transformation,
		real, intent(in)	lambda,
		real, intent(in)	epsilon
	)

Allocate and initialize state ensemble variables; 0-dimensional.

Parameters

[in]	bindir	path to bin-file
[in]	usebinfile	flag for using bin-file
[in]	assimilate	flag for including variable in assimilation
[in]	inifrombin	flag for initializing ensemble from bin-files

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allocate_1dim_state_ensemble()

subroutine assimilation_routines::allocate_1dim_state_ensemble	(	type(assim_state_ensemble_type), dimension(:)	assimVar,
		integer	nens,
		integer	nvar,
		character(len=*)	varName,
		real, dimension(n1,nvar)	xini,
		integer	varID,
		integer	locID,
		integer	coordID,
		integer, dimension(:)	fileID,
		character(len=*), intent(in)	bindir,
		integer, intent(in)	useBinFile,
		integer, intent(in)	narray,
		real, dimension(narray,2), intent(in)	limitsarray,
		integer	n1,
		logical, intent(in)	assimilate,
		logical, intent(in)	iniFromBin,
		real	missing,
		integer, intent(in)	transformation,
		real, intent(in)	lambda,
		real, intent(in)	epsilon
	)

Allocate and initialize state ensemble variables; 1-dimensional.

Parameters

[in]	bindir	path to bin-file
[in]	usebinfile	flag for using bin-file
[in]	assimilate	flag for including variable in assimilation
[in]	inifrombin	flag for initializing ensemble from bin-files

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allocate_2dim_state_ensemble()

subroutine assimilation_routines::allocate_2dim_state_ensemble	(	type(assim_state_ensemble_type), dimension(:)	assimVar,
		integer	nens,
		integer	nvar,
		character(len=*)	varName,
		real, dimension(n2,n1,nvar)	xini,
		integer	varID,
		integer	locID,
		integer	coordID,
		integer, dimension(:)	fileID,
		character(len=*), intent(in)	bindir,
		integer, intent(in)	useBinfile,
		integer, intent(in)	narray,
		real, dimension(narray,2), intent(in)	limitsarray,
		integer	n1,
		integer	n2,
		logical, intent(in)	assimilate,
		logical, intent(in)	iniFromBin,
		real	missing,
		integer, intent(in)	transformation,
		real, intent(in)	lambda,
		real, intent(in)	epsilon
	)

Allocate and initialize state ensemble variables; 2-dimensional.

Parameters

[in]	bindir	path to bin-file
[in]	usebinfile	flag for using bin-file
[in]	assimilate	flag for including variable in assimilation
[in]	inifrombin	flag for initializing ensemble from bin-files

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allocate_3dim_state_ensemble()

subroutine assimilation_routines::allocate_3dim_state_ensemble	(	type(assim_state_ensemble_type), dimension(:)	assimVar,
		integer	nens,
		integer	nvar,
		character(len=*)	varName,
		real, dimension(n3,n2,n1,nvar)	xini,
		integer	varID,
		integer	locID,
		integer	coordID,
		integer, dimension(:)	fileID,
		character(len=*), intent(in)	bindir,
		integer, intent(in)	useBinFile,
		integer, intent(in)	narray,
		real, dimension(narray,2), intent(in)	limitsarray,
		integer	n1,
		integer	n2,
		integer	n3,
		logical, intent(in)	assimilate,
		logical, intent(in)	iniFromBin,
		real	missing,
		integer, intent(in)	transformation,
		real, intent(in)	lambda,
		real, intent(in)	epsilon
	)

Allocate and initialize state ensemble variables; 3-dimensional.

Parameters

[in]	bindir	path to bin-file
[in]	usebinfile	flag for using bin-files
[in]	narray	size of array
[in]	assimilate	flag for including variable in assimilation
[in]	inifrombin	flag for initializing ensemble from bin-files

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allocate_assim_ensemble()

subroutine assimilation_routines::allocate_assim_ensemble	(	type(assim_ensemble_type)	assimVar,
		integer	nens,
		integer	nvar,
		integer, intent(in)	varID,
		integer, intent(in)	fileID,
		character(len=*), intent(in)	bindir,
		integer, intent(in)	useBinFile,
		integer	locID,
		integer	coordID,
		real, dimension(nvar)	xini,
		real	mini,
		real	maxi,
		logical	allocateOutput,
		real	missing,
		logical, intent(in)	assimilate,
		logical, intent(in)	iniFromBin,
		integer, intent(in)	transformation,
		real, intent(in)	lambda,
		real, intent(in)	epsilon,
		logical, intent(in)	dxadd
	)

Allocate and initialize an assim_ensemble_type variable.

Parameters

	assimvar	ensemble data
	nens	number of ensemble members
	nvar	number of variables (model units, i.e. n:o subbasins in HYPE)
[in]	varid	to set record for binary file
[in]	fileid	file unit ID for direct access binary file I/O
	coordid	ID for coordinate system for this variable
	xini	initial values
	allocateoutput	flag for output variable allocation
	missing	initial output values
[in]	bindir	path to bin-file
[in]	usebinfile	flag for using bin-file
[in]	assimilate	flag for including variable in assimilation
[in]	inifrombin	flag for initializing ensemble from bin-files
[in]	dxadd	flag for allocate and initialize perturbation field (dx)

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allocate_assim_ensemble_vectors()

subroutine assimilation_routines::allocate_assim_ensemble_vectors	(	type(assim_data_type), intent(inout)	assimData,
		integer	fid_0
	)

Allocate ensemble vectors needed for the data assimilation application.

Parameters

[in,out]	assimdata	main assimilation variable containing all data
	fid_0	base fileunit for bin files to be used

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allocate_assim_forcing_ensemble()

subroutine assimilation_routines::allocate_assim_forcing_ensemble	(	type(assim_input_ensemble_type)	assimVar,
		integer	nens,
		integer	nvar,
		character(len=*)	varName,
		real, dimension(nvar)	xini,
		integer, intent(in)	varID,
		integer	locID,
		integer	coordID,
		integer	fileID,
		real	minimum,
		real	maximum,
		integer	ensgen,
		real	sigma,
		real	semimeta,
		real	restmeta,
		real	minsigma,
		real	lscale,
		real	gridsize,
		integer	corrtype,
		real, dimension(:)	xcoord,
		real, dimension(:)	ycoord,
		character(len=*), intent(in)	bindir,
		integer, intent(in)	useBinFile,
		real	missing,
		logical, intent(in)	iniFromBin,
		integer, intent(in)	transformation,
		real, intent(in)	lambda,
		real, intent(in)	epsilon,
		real, intent(in)	tau,
		logical, intent(in)	usedx
	)

Allocate and initialize forcing ensemble variables.

Parameters

[in]	varid	counter for variables in ensembles (is not all variables varID for F-variables but hardkoded)
[in]	bindir	path to bin-file
[in]	usebinfile	flag for using bin-file
[in]	inifrombin	flag for initializing ensemble from bin-files

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allocate_assim_generation()

subroutine assimilation_routines::allocate_assim_generation	(	type(assim_generation_type)	assimVar,
		integer	nvar,
		integer	ensgen,
		real	fixsigma,
		real	semimeta,
		real	restmeta,
		real	minsigma,
		real	lscale,
		real	gridsize,
		integer	corrtype,
		real, dimension(:)	xcoord,
		real, dimension(:)	ycoord,
		real	tau
	)

Allocate and initialize an enkf_generation_type variable.

Parameters

assimvar	generation_data variable
nvar	number of variables, ie n:o "model units" (for instance, number of sub-basins)
ensgen	type of ensemble generation (0 none, 1 unrestricted, 2 [min,+inf], 3 [-inf,max], 4 restricted [min,max])
fixsigma	fixed standard deviation (ensgen=1)
semimeta	relative sigma for semi-restricted (ensgen=2,3,4, following Turner et al 2008)
restmeta	relative sigma for restricted (ensgen=2,3,4)
minsigma	minimum sigma (ensgen=2,3,4)
lscale	correlation length for spatially correlated perturbation
gridsize	2D gridsize for spatially correlated perturbation generation
corrtype	spatial correlation FUNCTION option
xcoord	x coordinate, for spatially correlated perturbations
ycoord	y coordinate, for spatially correlated perturbations
tau	perturbation memory coefficient

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allocate_assim_interface()

subroutine assimilation_routines::allocate_assim_interface	(	type(assim_interface_type)	assimVar,
		character(len=*)	varName,
		integer	varID,
		integer	modID,
		integer	nSubDim,
		integer, dimension(:)	subDimID
	)

Allocate and initialize a assim_interface_type variable.

Parameters

assimvar	interface data
varname	character string for model variables (used for debugging, and possibly file names and outputs)
varid	variable ID (id number used by interface for linking to model variables) (in HYPE it can be an outvar index, or the order in which the state variables are considered by interface)
modid	model ID, link to the corresponding variables used for H(X) (in HYPE: outvar index)
nsubdim	number of sub-dimensions (if needed, for instance lateral sub-units, vertical layers, substances, etc, in HYPE for instance SLC, substances, landuse, or slc, etc)
subdimid	index in the sub-dimensions

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allocate_assim_observation_ensemble()

subroutine assimilation_routines::allocate_assim_observation_ensemble	(	type(assim_input_ensemble_type)	assimVar,
		integer	nens,
		integer	nvar,
		character(len=*)	varName,
		real, dimension(nvar)	xini,
		integer	obsID,
		integer	modID,
		integer	coordID,
		integer	fileID,
		real	minimum,
		real	maximum,
		integer	ensgen,
		real	sigma,
		real	semimeta,
		real	restmeta,
		real	minsigma,
		real	lscale,
		real	gridsize,
		integer	corrtype,
		real, dimension(:)	xcoord,
		real, dimension(:)	ycoord,
		real	missing,
		integer, intent(in)	transformation,
		real, intent(in)	lambda,
		real, intent(in)	epsilon
	)

Allocate and initialize observation ensemble variables.

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allocate_auxiliary_ensemble()

subroutine assimilation_routines::allocate_auxiliary_ensemble	(	type(assim_state_ensemble_type)	assimVar,
		integer	nens,
		integer	nvar,
		character(len=*)	varName,
		real, dimension(nvar)	xini,
		integer, intent(in)	varID,
		integer, intent(in)	recID,
		integer	locID,
		integer	coordID,
		integer	fileID,
		character(len=*), intent(in)	bindir,
		integer, intent(in)	useBinFile,
		real	minimum,
		real	maximum,
		logical, intent(in)	assimilate,
		real	missing,
		integer, intent(in)	transformation,
		real, intent(in)	lambda,
		real, intent(in)	epsilon
	)

Allocate and initialize auxiliary ensemble variables; 0-dimensional.

Parameters

[in]	varid	outvar-variable index
[in]	recid	record of variable in bin-file
[in]	bindir	path to bin-file
[in]	usebinfile	flag for using bin-file
[in]	assimilate	flag for including variable in assimilation

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assim_backward_transform()

subroutine assimilation_routines::assim_backward_transform	(	integer, intent(in)	nx,
		integer, intent(in)	ne,
		real, dimension(nx,ne), intent(inout)	x,
		type(assim_ensemble_type), intent(inout)	ensdata,
		real, intent(in)	missing
	)

assim_forward_transform

transformation (forward) of ensembles (observations, states, etc) to improve Gaussian nature of ensemble distributions

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assim_checkminmax()

subroutine assimilation_routines::assim_checkminmax	(	integer, intent(in)	nx,
		integer, intent(in)	ne,
		real, dimension(nx,ne), intent(inout)	ensemble,
		real, intent(in)	minval,
		real, intent(in)	maxval,
		real, intent(in)	missing
	)

Truncates an ensemble matrix to minimum and maximum allowed values.

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assim_ensemble_statistics()

subroutine assimilation_routines::assim_ensemble_statistics	(	real, dimension(dim,nn)	xin,
		integer	DIM,
		integer	NN,
		real, dimension(dim)	xmean,
		real, dimension(dim)	xmins,
		real, dimension(dim)	xmaxs,
		real, dimension(dim)	xsigma,
		real, dimension(dim)	xmedian,
		logical	domedian,
		real	missing
	)

Calculates summary statistics for an ensemble matrix.

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assim_forward_transform()

subroutine assimilation_routines::assim_forward_transform	(	integer, intent(in)	nx,
		integer, intent(in)	ne,
		real, dimension(nx,ne), intent(inout)	x,
		type(assim_ensemble_type), intent(inout)	ensdata,
		real, intent(in)	missing
	)

assim_forward_transform (Gaussian Anamorphosis)

transformation (forward) of ensembles (observations, states, etc) to improve Gaussian nature of ensemble distributions

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assim_get_ensemble_data()

subroutine assimilation_routines::assim_get_ensemble_data	(	integer, intent(in)	NX,
		integer, intent(in)	NE,
		type(assim_ensemble_type), intent(in)	ensData,
		real, dimension(nx,ne), intent(out)	x
	)

Routine that returns the ensemble data in a matrix. If needed, the data is read from binary file. For speed, records in binary files contain all states in one ensemble member (X transformated).

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assim_get_ensemble_data_dx()

subroutine assimilation_routines::assim_get_ensemble_data_dx	(	integer, intent(in)	NX,
		integer, intent(in)	NE,
		type(assim_ensemble_type), intent(in)	ensData,
		real, dimension(nx,ne), intent(out)	dx
	)

Routine that returns the perturbation field ensemble data in a matrix. If needed, the data is read from binary file. For speed, records in binary files contain all states in one ensemble member (X transformated).

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assim_set_ensemble_data()

subroutine assimilation_routines::assim_set_ensemble_data	(	integer, intent(in)	NX,
		integer, intent(in)	NE,
		type(assim_ensemble_type), intent(inout)	ensData,
		real, dimension(nx,ne), intent(inout)	x,
		logical, intent(in)	checkMinMax,
		real, intent(in)	missing
	)

Routine that writes a matrix into an ensemble data structure. If needed, the data is written to a binary file.

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assim_set_ensemble_data_dx()

subroutine assimilation_routines::assim_set_ensemble_data_dx	(	integer, intent(in)	NX,
		integer, intent(in)	NE,
		type(assim_ensemble_type), intent(inout)	ensData,
		real, dimension(nx,ne), intent(inout)	x,
		real, dimension(nx,ne), intent(inout)	dx,
		logical, intent(in)	checkMinMax,
		real, intent(in)	missing
	)

Routine that writes a matrix into an ensemble data structure. If wanted, the data is written to a binary file.

dx-version which also writes the dx field (perturbation)

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choldc()

subroutine assimilation_routines::choldc	(	real, dimension(:,:), intent(inout)	a,
		real, dimension(:), intent(out)	p,
		integer	n,
		logical, intent(inout)	failure
	)

Cholesky decomposition (from Numerical Recipes, adopted by David)

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choleskysolution()

subroutine assimilation_routines::choleskysolution	(	integer, intent(in)	ny,
		integer, intent(in)	ne,
		real, dimension(ny,ny), intent(inout)	P,
		real, dimension(ny,ne), intent(in)	Y,
		real, dimension(ny,ne), intent(out)	M,
		logical, intent(inout)	failure
	)

Cholesky solution.

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cholsl()

subroutine assimilation_routines::cholsl	(	real, dimension(:,:), intent(in)	a,
		real, dimension(:), intent(in)	p,
		real, dimension(:), intent(in)	b,
		real, dimension(:), intent(inout)	x,
		integer	n
	)

Cholesky solver (from Numerical Recipes, adopted by David)

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enkf_analysis_apply()

subroutine assimilation_routines::enkf_analysis_apply	(	integer, intent(in)	N,
		integer, intent(in)	NX,
		integer, intent(in)	ND,
		real, dimension(nd,n), intent(in)	M,
		real, dimension(nd,n), intent(in)	HA,
		real, dimension(nx,nd), intent(in)	locCxy,
		real, dimension(nx,n), intent(inout)	X
	)

Part 2 of ensemble Kalman filter analysis.

Apply ENKF filter on a model variable ensemble using the M matrix derived in the enkf_analysis_prepare function.

EnKF analysis, following the "basic implementation" in Jan Mandel report modified to include covariance localization:

           (K = Cxy .* loc_Cxy / (Cyy .* loc_Cyy + R))

The localization of Cyy is embedded in the M matrix which is prepared by the enkf_analysis_prepare routine.

The localization of Cxy is introduced in the final step of this "apply"-routine, which involves a change in the calculation of the AZ matrix compared to Mandel:

           Mandel:    AZ = A * (HA' * M)

           Here:      AZ = ((A*HA') .* loc_Cxy) * M

Matrix operations are made in an order defined by the parantheses (most inner operations first). This might have consequences on the number of operations compared to Mandel, but I have not checked that.

Parameters

[in]	n	number of ensemble members
[in]	nx	number of variables in ensemble (rows)
[in]	nd	number of observations
[in,out]	x	ensemble data
[in]	m	inverse of (R+CXY) multiplied with innovation Y
[in]	ha	HX deviation from mean(HX)
[in]	loccxy	localization matrix

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enkf_analysis_main()

subroutine assimilation_routines::enkf_analysis_main

(

type(assim_data_type), intent(inout)

assimData

)

Main routine in the ensemble Kalman filter analysis.

Ensemble Kalman filter equations (Evensen) following (a) Mandel, J. Efficient implementation of the ensemble kalman filter, and (b) DeChant, C. Examining the effectiveness and robustness of sequential data assimilation methods for quantification of uncertainty in hydrological forecasting. Localization following Magnusson, Gustafsson et al (2014)

Parameters

[in,out]

assimdata

main assimilation variable containing all data

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enkf_analysis_prepare()

subroutine assimilation_routines::enkf_analysis_prepare	(	integer, intent(in)	N,
		integer, intent(in)	ND,
		real, dimension(nd,n), intent(in)	D,
		real, dimension(nd,n), intent(in)	HX,
		real, dimension(nd), intent(in)	R,
		real, dimension(nd,nd), intent(in)	locCyy,
		real, dimension(nd,n), intent(out)	M,
		real, dimension(nd,n), intent(out)	Y,
		real, dimension(nd,n), intent(out)	HA,
		logical, intent(inout)	status
	)

Part 1 of ensemble Kalman filter analysis.

Innovations (Y=D-HX), and inversion of variances M=1/(var(HX) + R) which is later used in part 2; the update equation:

X_analysis = X_forecast + cov(X,HX)/(var(HX)+R) * (D-HX)

Parameters

[in]	n	number of ensemble members
[in]	nd	number of observations
[in]	d	observation ensemble
[in]	hx	predicted observation ensemble
[in]	r	observation error variance, uncorrelated
[in]	loccyy	localization matrix for cov(HX)
[out]	m	inverse of (R+CXY) multiplied with innovation Y
[out]	y	innovation ensemble
[out]	ha	HX deviation from mean(HX)
[in,out]	status	error status, failure of cholesky

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generate_input_ensemble()

subroutine assimilation_routines::generate_input_ensemble	(	integer	n,
		type(assim_input_ensemble_type), dimension(n)	assimVar,
		real	missing,
		logical	usedx
	)

General routine for Ensemble generation (forcing and observation data).

The ensemble generation is made by adding random numbers to the input data.

a) The basic assumption is that the random perturbations are gaussian with zero mean and standard deviation sigma.

b) Based on Turner et al (2008), sigma is a constant value for unrestricted variables, and a relative value for variables restricted by a minimum or a maximum or both: pertubation_unrestricted = G(0,sigma) pertubation_restricted,min = G(0,semi.meta * (X-minX)) pertubation_restricted,max = G(0,semi.meta * (maxX-X)) pertubation_restricted,minmax = G(0,rest.meta * (maxX-X)/(maxX-midX)) if X > midX
pertubation_restricted,minmax = G(0,rest.meta * (X-minX)/(midX-minX)) if X <= midX

(Turner et al(2008) also suggested that in order to get unbiased input ensembles, we need to consider two types of perturbations - systematic and random: Thus, the input x at time k for ensemble member j, x_kj = x_k0+eata_kj+chi_j, where eata_kj is regenerated every time step and chi_j is generated only once. Then, eata and chi can be generated with three different types of restrictions on variables.

However, we never implemented this part of Turner - we assume systematic error chi_j = 0.)

c) We now also take into account spatial correlation in the data, by generating spatially correlated random data using a FFT-based method (usually only on forcing)

d) Instead of the systematic error suggested by Turner et al (2008); see section in brackets under b) we introduce a memory in the perturbation (only for forcing):

pert(t) = pert(t-1) * TAU + pert * (1-TAU), where pert is generated with existing methods

for restricted and semi-restricted variables, we probably need to propagate relative perturbations instead of absolute perturbations.

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get_random_vector_gaussian()

subroutine assimilation_routines::get_random_vector_gaussian	(	integer, intent(in)	n,
		real, intent(in)	a,
		real, intent(in)	sigma,
		real, dimension(:), intent(out)	r
	)

Get an array of random gaussian values with specified mean and sigma.

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get_spatially_correlated_random_data2()

subroutine assimilation_routines::get_spatially_correlated_random_data2	(	integer, intent(in)	n,
		integer, intent(in)	nens,
		type(assim_generation_type), intent(inout)	assimG,
		real, dimension(n,nens), intent(out)	X,
		logical	usedx
	)

Function for spatially correlated random variable generation.

Parameters

[in]	n	number of element (nsub usually)
[in]	nens	number of ensembles
[in,out]	assimg	information on generation of input data
[out]	x	generated random matrix

Algorithm Loop over ensemble members:

• generate random numbers (with mean 0, standard deviation 1 and correlation length L) for all subbasins' location

• scale perturbations with the required sigma (standard deviation) and apply to the mean (if not perturbation is requested by USEDX

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initialize_assim_info()

subroutine assimilation_routines::initialize_assim_info

(

type(assim_info_type), intent(inout)

assimInfo

)

Initializes the information variable for assimilation.

Parameters

[in,out]

assiminfo

information on assimilation simulation

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initiate_fileid_0dim_state_ensemble()

subroutine assimilation_routines::initiate_fileid_0dim_state_ensemble	(	type(assim_state_ensemble_type)	assimVar,
		integer, intent(in)	nens,
		integer, intent(inout)	varID,
		integer, intent(in)	fileID,
		integer, intent(in)	useBinFile
	)

Initialize state ensemble variables regaring files; 0-dimensional.

Parameters

[in]

usebinfile

flag for using bin-file

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initiate_fileid_1dim_state_ensemble()

subroutine assimilation_routines::initiate_fileid_1dim_state_ensemble	(	type(assim_state_ensemble_type), dimension(:)	assimVar,
		integer	nens,
		integer	varID,
		integer, dimension(:)	fileID,
		integer, intent(in)	useBinFile,
		integer	n1
	)

Initialize state ensemble variables regaring files; 1-dimensional.

Parameters

[in]

usebinfile

flag for using bin-file

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initiate_fileid_2dim_state_ensemble()

subroutine assimilation_routines::initiate_fileid_2dim_state_ensemble	(	type(assim_state_ensemble_type), dimension(:)	assimVar,
		integer	nens,
		integer	varID,
		integer, dimension(:)	fileID,
		integer, intent(in)	useBinfile,
		integer	n1,
		integer	n2
	)

Initialize state ensemble variables regaring files; 2-dimensional.

Parameters

[in]

usebinfile

flag for using bin-file

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initiate_fileid_3dim_state_ensemble()

subroutine assimilation_routines::initiate_fileid_3dim_state_ensemble	(	type(assim_state_ensemble_type), dimension(:)	assimVar,
		integer	nens,
		integer	varID,
		integer, dimension(:)	fileID,
		integer, intent(in)	useBinFile,
		integer	n1,
		integer	n2,
		integer	n3
	)

Initialize state ensemble variables regaring files; 3-dimensional.

Parameters

[in]

usebinfile

flag for using bin-files

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initiate_fileid_assim_ensemble()

subroutine assimilation_routines::initiate_fileid_assim_ensemble	(	type(assim_ensemble_type)	assimVar,
		integer, intent(in)	nens,
		integer, intent(in)	varID,
		integer, intent(in)	fileID,
		integer, intent(in)	useBinFile,
		logical, intent(in)	dxadd
	)

Initialize file related variables of assim_ensemble_type variable.

Parameters

	assimvar	ensemble data
[in]	nens	number of ensemble members
[in]	varid	to set record for binary file
[in]	fileid	file unit ID for direct access binary file I/O
[in]	usebinfile	flag for using bin-file
[in]	dxadd	flag for allocate and initialize perturbation field (dx)

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initiate_fileid_forcing_ensemble()

subroutine assimilation_routines::initiate_fileid_forcing_ensemble	(	type(assim_input_ensemble_type), intent(inout)	assimVar,
		integer, intent(in)	nens,
		integer, intent(in)	varID,
		integer, intent(in)	fileID,
		integer, intent(in)	useBinFile,
		logical, intent(in)	usedx
	)

Initialize forcing ensemble variables regaring files.

Parameters

[in]	varid	counter for variables in forcing ensembles (is not all variables varID for F-variables but special hardkoded)
[in]	usebinfile	flag for using bin-file

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matrixmatrixmultiply()

subroutine assimilation_routines::matrixmatrixmultiply	(	real, dimension(:,:), intent(in)	mat1,
		real, dimension(:,:), intent(in)	mat2,
		real, dimension(:,:), intent(out)	matout
	)

Subroutine for matrix multiplication.

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meanormedian_to_ensemble()

subroutine assimilation_routines::meanormedian_to_ensemble	(	integer, intent(in)	nx,
		type(assim_state_ensemble_type), dimension(:)	assimX,
		logical, intent(in)	meanORmedian
	)

Parameters

	assimx	ensemble data
[in]	nx	number of variables
[in]	meanormedian	flag for setting mean (T) or median (F)

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updateensemblestatistics()

subroutine assimilation_routines::updateensemblestatistics	(	type(assim_data_type)	assimData,
		real, dimension(4), intent(inout), optional	total_time
	)

Calculates ensemble statistics for state, forcing and auxiliary ensembles.

Parameters

	assimdata	main assimilation variable containing all data
[in,out]	total_time	optional timing of the subroutine

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