7.4. Statistic Theory and Uncertainty Analysis
245
2. Solve the inverse problem with the method suggested by Neuman (1980),
and Clifton and Neuman (1982) to identify the logarithm transmissivity.
That is, using the initial estimate of the parameter in combination with the
generalized least squares method to obtain the logarithmic distribution
of transmissivity (denoted as Y = In T(x)) and covariance matrix of the
estimation errors (denoted as [V]) simultaneously;
3. Carry out many simulations on the computer along the lines of the
Monte-Carlo method. From the calculated Y and [V], a lot of possible
values of transmissivity may be genera ted by calculating
y= Y + [M]~,
(7.4.41)
where Y is a randomly genera ted "realization" of the logarithm transmissivity; [M] the upper triangular matrix of the LU decomposition of the
covariance matrix [V], i.e., [M] [MY = [V]; and ~ is a sampling value of
multidimensional normal distribution with zero mean and unit variance.
Nelson et al. (1985) obtained 600 "realizations" in their case study;
4. With the 600 "realizations" of transmissivity, 600 "realizations" of head
distribution can be obtained by solving the water flow model. Then,
using Darcy's law, 600 "realizations" ofvelocity distribution can be calculated, that is,
TVij = - b~ grad(hij), i = 1,2, ... ,600; j = 1,2, ... , M (7.4.42)
where Vij ' Tij' hij are the ith "realizations" of velocity, transmissivity and
hydraulic head, respectively, at node j; M the total number of nodes; b the
thickness of the aquifer; and n the porosity;
5. Substituting the 600 "realizations" of velocity distribution successively in
to a pure advection quality model to calculate the migration ofthe pollutant and its arrival time at the out-flow boundary, or at a fixed location, we
can obtain 600 "realizations" of the solution.
6. With all the aforementioned data available, we can calculate the mean
arrival time and statistical characteristics of the concentration distribution, including their variances and confidence intervals. As a result, a
quantitative estimate of the uncertainties of the model has been obtained.
The stochastic inverse problem (SIP) involves the identification of mean
and covariance functions of unknown parameters with the aid of head and
concentration measurements. The SIP was first considered by Kitanidis and
Vomvoris (1983). The mean value I1r of the Y = InK field and covariance
parameters Ur and Ir in Eq. (7.4.4) were estimated by the maximum likelihood estimator (MLE). This SIP was further considered by Hoeksema and
Kitanidis (1984, 1985), Dagan (1985), Rubin and Dagan (1987a, b) and Dagan
and Rubin (1988). Some field applications were given in these papers. However, there are limitations associated with the previous research: the flow field
245
2. Solve the inverse problem with the method suggested by Neuman (1980),
and Clifton and Neuman (1982) to identify the logarithm transmissivity.
That is, using the initial estimate of the parameter in combination with the
generalized least squares method to obtain the logarithmic distribution
of transmissivity (denoted as Y = In T(x)) and covariance matrix of the
estimation errors (denoted as [V]) simultaneously;
3. Carry out many simulations on the computer along the lines of the
Monte-Carlo method. From the calculated Y and [V], a lot of possible
values of transmissivity may be genera ted by calculating
y= Y + [M]~,
(7.4.41)
where Y is a randomly genera ted "realization" of the logarithm transmissivity; [M] the upper triangular matrix of the LU decomposition of the
covariance matrix [V], i.e., [M] [MY = [V]; and ~ is a sampling value of
multidimensional normal distribution with zero mean and unit variance.
Nelson et al. (1985) obtained 600 "realizations" in their case study;
4. With the 600 "realizations" of transmissivity, 600 "realizations" of head
distribution can be obtained by solving the water flow model. Then,
using Darcy's law, 600 "realizations" ofvelocity distribution can be calculated, that is,
TVij = - b~ grad(hij), i = 1,2, ... ,600; j = 1,2, ... , M (7.4.42)
where Vij ' Tij' hij are the ith "realizations" of velocity, transmissivity and
hydraulic head, respectively, at node j; M the total number of nodes; b the
thickness of the aquifer; and n the porosity;
5. Substituting the 600 "realizations" of velocity distribution successively in
to a pure advection quality model to calculate the migration ofthe pollutant and its arrival time at the out-flow boundary, or at a fixed location, we
can obtain 600 "realizations" of the solution.
6. With all the aforementioned data available, we can calculate the mean
arrival time and statistical characteristics of the concentration distribution, including their variances and confidence intervals. As a result, a
quantitative estimate of the uncertainties of the model has been obtained.
The stochastic inverse problem (SIP) involves the identification of mean
and covariance functions of unknown parameters with the aid of head and
concentration measurements. The SIP was first considered by Kitanidis and
Vomvoris (1983). The mean value I1r of the Y = InK field and covariance
parameters Ur and Ir in Eq. (7.4.4) were estimated by the maximum likelihood estimator (MLE). This SIP was further considered by Hoeksema and
Kitanidis (1984, 1985), Dagan (1985), Rubin and Dagan (1987a, b) and Dagan
and Rubin (1988). Some field applications were given in these papers. However, there are limitations associated with the previous research: the flow field