# 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 Cpp. The i-th parameter set Yi is obtained by linear combination of the eigenvectors uk of Cpp and stochastic coefficients phi(xi): where xsi is a standard normal distributed random variable, and ak is the k-th eigenvalue of Cpp. For more details, see Kitterød and Gottschalk, .

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

(1) Provide indices and elements of Cpp. 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 Cpp. 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

 8.0643e-05 0.864 .71921E-04 -0.253 -.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

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