7.4. Statistic Theory and Uncertainty Analysis
239
1. The hydraulic conductivity K, porosity n, flow velocity V, dispersivity IXL'
IX T and so forth in the governing equations are all obtained by measurements, data processing, or through solving inverse problems. Because of
their uncertainties, they should be regarded as stochastic variables with
certain statistical characteristics. Mathematical expectations, covariances,
or confidence intervals are often used to describe these random fields.
2. The initial and boundary conditions may contain uncertainties caused by
measurement errors, inaccurate geological inferences, and so forth. Sometimes, the boundary conditions themselves are stochastic processes. For
example, the water level of a river, which is regarded as a boundary
condition, may randomly vary.
3. The source/sink terms mayaiso change randomly. For instance, rainfall
infiltration, irregular pumping and recharge, and locations and times of
contaminants entering into the aquifer may all vary randomly.
Since there are random components in the models, their solutions (head
and concentration distributions) should be regarded as random functions
with certain probability distributions. Therefore, even in the framework of
the classical dispersion theory, we have to deal with stochastic partial differential equations (SPDE). To solve a SPDE means to find the probability
distribution of its solution based on the probability distributions of input
data. Generally, it is enough to obtain the first two moments, the mean and
variance functions, of the solution.
To solve a SPDE, we must know the probability distributions of input
parameters, at least their first two moments. If the input parameters are
obtained by solving inverse problems based on the observed head and
concentration, then we have to consider the uncertainties of the estimated
parameters caused by the insufficient quantity and quality of the observation data. When the observation errors of concentration C and head h are
normally distributed with zero means and independent of each other, the
covariance matrix of the estimated parameters can be approximately expressed by the following equation (Bard, 1974; Wagner and Gorelick, 1987):
(7.4.14)
where [W]c and [W]h have been defined in Eq. (7.3.15), and the sensitivity
matrices [J]c and [J]h have been defined by Eq. (7.3.17).
Let us now discuss how to obtain the estimate for the uncertainties of
solutions from the uncertainties of model parameters. If a linear relationship between the solution (head or concentration) and the model parameters is assumed based on the first-order approximation, then it is easy to
derive
E[(p)] ~ [E(p)],
(7.4.15)
where E[(p)] is the mathematical expectation of solution of the equation
and [E(p)] is the solution obtained by substituting the expectation of the
239
1. The hydraulic conductivity K, porosity n, flow velocity V, dispersivity IXL'
IX T and so forth in the governing equations are all obtained by measurements, data processing, or through solving inverse problems. Because of
their uncertainties, they should be regarded as stochastic variables with
certain statistical characteristics. Mathematical expectations, covariances,
or confidence intervals are often used to describe these random fields.
2. The initial and boundary conditions may contain uncertainties caused by
measurement errors, inaccurate geological inferences, and so forth. Sometimes, the boundary conditions themselves are stochastic processes. For
example, the water level of a river, which is regarded as a boundary
condition, may randomly vary.
3. The source/sink terms mayaiso change randomly. For instance, rainfall
infiltration, irregular pumping and recharge, and locations and times of
contaminants entering into the aquifer may all vary randomly.
Since there are random components in the models, their solutions (head
and concentration distributions) should be regarded as random functions
with certain probability distributions. Therefore, even in the framework of
the classical dispersion theory, we have to deal with stochastic partial differential equations (SPDE). To solve a SPDE means to find the probability
distribution of its solution based on the probability distributions of input
data. Generally, it is enough to obtain the first two moments, the mean and
variance functions, of the solution.
To solve a SPDE, we must know the probability distributions of input
parameters, at least their first two moments. If the input parameters are
obtained by solving inverse problems based on the observed head and
concentration, then we have to consider the uncertainties of the estimated
parameters caused by the insufficient quantity and quality of the observation data. When the observation errors of concentration C and head h are
normally distributed with zero means and independent of each other, the
covariance matrix of the estimated parameters can be approximately expressed by the following equation (Bard, 1974; Wagner and Gorelick, 1987):
(7.4.14)
where [W]c and [W]h have been defined in Eq. (7.3.15), and the sensitivity
matrices [J]c and [J]h have been defined by Eq. (7.3.17).
Let us now discuss how to obtain the estimate for the uncertainties of
solutions from the uncertainties of model parameters. If a linear relationship between the solution (head or concentration) and the model parameters is assumed based on the first-order approximation, then it is easy to
derive
E[(p)] ~ [E(p)],
(7.4.15)
where E[(p)] is the mathematical expectation of solution of the equation
and [E(p)] is the solution obtained by substituting the expectation of the