Manual Page for Command >>> EMPIRICAL ORTHOGONAL FUNCTIONS

Syntax
>>> EMPIRICAL ORTHOGONAL FUNCTIONS (MATRIX: ndim) (iTOUGH2) (CORRELATION)

Parent Command
>> ERROR

Subcommand
-

Description
This command must be used in combination with command >>> MONTE CARLO to invoke stochastic simulations of correlated parameters using Empirical Orthogonal Functions (EOF). EOF is a variant of Monte Carlo simulations to quantify the uncertainty of model predictions as a result of parameter uncertainty. Many parameter sets are generated, following the predefined covariance matrix C_pp. The i-th parameter set Y_i is obtained by linear combination of the eigenvectors u_k of C_pp and stochastic coefficients phi(x_i):

where xsi is a standard normal distributed random variable, and a_k is the k-th eigenvalue of C_pp. For more details, see Kitterød and Gottschalk, [1997].

The elements of matrix C_pp can be supplied using one of the following options:

(1) Provide indices and elements of C_pp. Example:

>>> EMIRICAL ORTHOGONAL FUNCTIONS 1 1 0.80643E-04 2 2 0.71921E-04 2 1 0.64412E-04

Use keyword CORRELATION if off-diagonal term is correlation coefficient instead of covariance. Example:

>>> EMPIRICAL ORTHOGONAL FUNCTIONS, CORRELATION 1 1 0.80643E-04 2 2 0.71921E-04 2 1 0.864

(2) Provide keyword MATRIX, followed by a colon and the dimension ndim of the square matrix C_pp. The lower triangle of the covariance matrix is then provided on exactly ndim additional lines. If keyword CORRELATION is present, the off- diagonal terms represent correlation coefficients rather than covariances. Example:

>>> EMPIRICAL ORTHOGONAL FUNCTIONS, dim. of CORRELATION MATRIX: 3

`.80643E-04`
`.864`	`.71921E-04`
`-.253`	`-.500`	`.53843E-05`

(3) If calculated during a previous iTOUGH2 inversion, the covariance matrix can be taken from the iTOUGH2 output file and directly copied after the command line. This option is invoked by keyword iTOUGH2. The matrix will be read by formatted input, so it is crucial that the correct format is maintained. If ndim is greater than 6, the matrix is split in multiple submatrices. All submatrices must be copied exactly as they were printed to the iTOUGH2 output file. Example:

>>> EOF error analysis, read MATRIX of dim.: 3 in iTOUGH2 format

	log(abs. perm.)	`POROSITY SAND`	`Gas entrapped`
`log(abs. perm.)`	`.80643E-04`	`.846`	`-.253`
`POROSITY SAND`	`.64412E-04`	`.71921E-04`	`-.500`
`Gas entrapped`	`-.52623E-05`	`-.98296E-05`	`.53843E-05`

Example

> COMPUTATION >> STOP >>> number of Monte Carlo SIMULATIONS: 250 <<<

>> ERROR propagation analysis >>> MONTE CARLO simulations, SEED number: 777 >>> EMPIRICAL ORTHOGONAL FUNCTIONS, read MATRIX of dim.: 3 in iTOUGH2 format

	`log(abs. perm.)`	`POROSITY SAND`	`Gas entrapped`
`log(abs. perm.)`	`.80643E-04`	`.846`	`-.253`
`POROSITY SAND`	`.64412E-04`	`.71921E-04`	`-.500`
`Gas entrapped`	`-.52623E-05`	`-.98296E-05`	`.53843E-05`

<<< << <

See Also
>>> FOSM | >>> MONTE CARLO