Manual Page for Command >>>> DEVIATION
>>>> DEVIATION: sigma
all third-level commands in block > PARAMETER
This command specifies the standard deviation sigma of the initial parameter
guess. Prior information about model parameter will be weighted by 1/sigma,
i.e., the difference between the prior information value p* and the estimate
p contributes to the objective function.
Commands for specifying the standard deviation are:
>>>> DEVIATION: sigma
>>>> VARIANCE: sigma2
>>>> WEIGHT: 1/sigma
By default, prior information is not weighted, i.e. sigma=infinity.
The standard deviation reflects the uncertainty associated with the initial
guess. If the initial guess is to be weighted, prior information should
originate from an independent source. For example, if porosity will be
estimated based on transient pressure data, the prior information value
should be taken from a "direct" porosity measurement, e.g. using
mercury-porosimetry or oven-drying methods. In these cases, the measured
parameter values p* are considered to be additional data points which serve
as a physical plausibility criterion for the estimate p. The p* values,
along with the observations of the system state z*, are then weighted
according to their uncertainties (see >>>> DEVIATION (o)).
Note that the relative weighting between prior information and the
observations z* depends on the number of calibration points selected.
If many transient data points are available, a smaller standard deviation
sigma may be specified to increase the relative weight of prior information.
In many cases, appropriately weighting the initial guess makes an ill-posed
inverse problem unique. Furthermore, the solution becomes more stable if
a parameter is not very sensitive. However, using 1/sigma as a regularization
parameter to improve the ability to obtain a unique solution with a poorly
conceptualized inverse problem inverse problem is not recommended.
Erratic behavior of a parameter during the inversion should be taken as an
indication that the data do not contain sufficient information for the
determination of the parameter. Differences between parameter values that
are independently determined from laboratory experiments and inverse modeling
suggest the presence of a systematic error or scaling problem. These
inconsistencies should be resolved rather than averaged out.
The standard deviation sigma is also used to scale the columns of the
Jacobian matrix. While the solution of the inverse problem is not affected
by the choice of the scaling factor, all the qualitative sensitivity measures
are directly proportional to sigma. If prior information is not weighted,
the scaling factor is taken to be 10 % of the respective parameter value.
Command >>>> VARIATION should be used to change the default scaling factor
without concurrently assigning a weight to prior information.
When performing uncertainty propagation analyses, sigma designates the
parameter uncertainty affecting the model prediction. It is used to generate
a set of random parameter values for Monte Carlo simulations, and it represents
the standard deviation of a normal distribution if performing linear
uncertainty propagation analysis (for more details see commands
>>> MONTE CARLO and >>> FOSM, respectively).
>>> MATERIAL: TUFFn
>>>> PRIOR information : 0.38 (laboratory measurement)
>>>> standard DEVIATION: 0.04 (measurement error)
>>> MATERIAL: ALLUV
>>>> PRIOR information : 0.30 (from experience)
>>>> VARIANCE : 0.01 (uncertainty of guess)
>>> MATERIAL: FAULT
>>>> initial GUESS : 0.25 (no measurements available)
>>>> WEIGHT : 0.00 (default)
>>>> VARIATION : 0.10 (for scaling of Jacobian)
>> GUESS |
>>> FOSM |
>>> MONTE CARLO |
>>>> DEVIATION (o) |
>>>> PRIOR |
>>>> VARIANCE |
>>>> VARIATION |
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Page updated: July 29, 1997