# Manual Page for Command >>> FOSM

Syntax
>>> FOSM (MATRIX: ndim) (iTOUGH2) (CORRELATION) (DIAGONAL)

Parent Command
>> ERROR

Subcommand
-

Description
This command performs First-Order-Second-Moment (FOSM) uncertainty propagation analysis. FOSM quantifies the uncertainty of model predictions as result of parameter uncertainty. FOSM is the analysis of the mean and covariance of a random function based on its first order Taylor series expansion. FOSM analysis presumes that the mean and covariance are sufficient to characterize the distribution of the dependent variables, i.e., the model results are assumed to be normally distributed, and perturbations about the mean can be approximated by linear functions J. The covariance of the uncertain parameters, Cpp, is translated into the covariance of the simulated system response, Cpp:

Czz=JCppJTThe diagonal elements of matrix Cpp, i.e., the variances of the parameters, can be supplied by command >>>> VARIANCE (or related commands) in block > PARAMETER. The elements of matrix Cpp can be supplied using one of the following options:

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

>>> FOSM
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:

>>> FOSM, 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:

>>> FOSM, 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:

>>> FOSM 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

If the full matrix is provided, but only the diagonal terms (variances) shall be used in the uncertainty analysis, use keyword DIAGONAL on the command line. This option makes it easy to study the impact of correlations on the uncertainty propagation analysis. The uncertainty of the model prediction as a result of parameter uncertainty is given as a standard deviation in the residual analysis. Furthermore, the plotfile contains the system response for the mean parameter set as well as error band on the specified confidence level (see command >>> ALPHA). It is suggested to also increase the perturbation factor for calculating the Jacobian matrix, and to use a centered finite difference quotient. This yields in an more realistic the error band if the model is highly non-linear. It should be realized, however, that Monte Carlo is the preferred method if dealing with highly non-linear flow systems.

Example
> COMPUTATION
>> ERROR propagation analysis
>>> perform First-Order-Second-Moment (FOSM) analysis
>>> draw error bands on (1-ALPHA)=: 95 % confidence level
<<<
>> JACOBIAN
>>> use CENTERED finite difference quotient
>>> PERTURBATION factor at least: 5.0 %
<<<
<<