Syntax
>>> FOSM (MATRIX: ndim) (iTOUGH2) (CORRELATION) (DIAGONAL)
Parent Command
>> ERROR
Subcommand

Description
This command performs FirstOrderSecondMoment (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, C_{pp}, is translated into the covariance of the simulated system response, C_{pp}:
C_{zz}=JC_{pp}J^{T}The diagonal elements of matrix C_{pp}, i.e., the variances of the parameters, can be supplied by command >>>> VARIANCE (or related commands) in block > PARAMETER. The elements of matrix C_{pp} can be supplied using one of the following options:
(1) Provide indices and elements of C_{pp}. Example:
>>> FOSM
1 1 0.80643E04
2 2 0.71921E04
2 1 0.64412E04
Use keyword CORRELATION if offdiagonal term is correlation coefficient instead of covariance. Example:
>>> FOSM, CORRELATION
1 1 0.80643E04
2 2 0.71921E04
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:
>>> FOSM, dim. of CORRELATION MATRIX: 3
.80643E04  
.864  .71921E04  
.253  .500  .53843E05 
(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.)  .80643E04  .846  .253 
POROSITY SAND  .64412E04  .71921E04  .500 
Gas entrapped  .52623E05  .98296E05  .53843E05 
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 nonlinear. It should be realized, however, that Monte Carlo is the preferred method if dealing with highly nonlinear flow systems.
Example
> COMPUTATION
>> ERROR propagation analysis
>>> perform FirstOrderSecondMoment (FOSM) analysis
>>> draw error bands on (1ALPHA)=: 95 % confidence level
<<<
>> JACOBIAN
>>> use CENTERED finite difference quotient
>>> PERTURBATION factor at least: 5.0 %
<<<
<<
See Also
>>> ALPHA  >>> EMPIRICAL ORTHOGONAL FUNCTIONS  >>> MONTE CARLO