246
7. Mathematical Models of Groundwater Quality
was assumed to be stable and only In K and head measurements were used
for parameter identification. In Sun and Yeh (1992), the SIP for transient flow
fields was solved, and an adjoint state method was developed for efficientiy
calculating the cross-covariance matrices between head observations at different times. The solution of a SIP includes the following steps:
1. Using kriging to estimate a set of initial values of Jly, (Ty, and Iy.
2. Calculating the covariance matrices C IIIIII and Cllly relevant to known observation locations and times, where cl> is the head.
3. Using MLE to improve the estimations of Jly, (Ty, and Iy iteratively with
the aid of both Y and cl> measurements.
4. Using the same set of measurements and the co-kriging estimator to
generate realization of the Y field. The so-obtained realization is regarded
as the inverse solution.
For a complete discussion on SIP, readers may refer to Sun (1994). The
solution of SIP can be easily extended to the case that both head and concentration observations are used for the identification of stochastic parameters.
Exercises
7.1. Assuming that there are two tracer components, oe and p, in a confined
aquifer, write the hydrodynamic system for this situation. Then, use a
flow chart to describe the solution procedure.
7.2. Assuming that only the longitudinal dispersivity is unknown in a
advection-dispersion model, formulate an inverse problem for this ca se
and translate it into an optimization problem.
7.3. Average Eq. (7.2.32) along the y direction to find the expressions of
"apparent velocity" and "apparent dispersion coefficient."
7.4. Derive the adjoint system for one-dimensional coupled flow and mass
transport problems and find the expressions of ohjoK and oC;oK.
7.5. Explain the differences between Fickian expressions Eqs. (7.4.1) and
(7.4.9).
7.6. What terms are neglected in the derivation of the perturbation equation
(7.4.1O)?
7.7. Derive Eqs. (7.4.15) and (7.4.16) based on the first-order approximation.
7.8. Draw a flow chart to describe the procedure of Monte-Carlo simulation
method for estimating the uncertainty of a water quality model.
7.9. Draw a flow chart to describe the procedure of conditional simulation
method for estimating the uncertainty of a water quality model.
7. Mathematical Models of Groundwater Quality
was assumed to be stable and only In K and head measurements were used
for parameter identification. In Sun and Yeh (1992), the SIP for transient flow
fields was solved, and an adjoint state method was developed for efficientiy
calculating the cross-covariance matrices between head observations at different times. The solution of a SIP includes the following steps:
1. Using kriging to estimate a set of initial values of Jly, (Ty, and Iy.
2. Calculating the covariance matrices C IIIIII and Cllly relevant to known observation locations and times, where cl> is the head.
3. Using MLE to improve the estimations of Jly, (Ty, and Iy iteratively with
the aid of both Y and cl> measurements.
4. Using the same set of measurements and the co-kriging estimator to
generate realization of the Y field. The so-obtained realization is regarded
as the inverse solution.
For a complete discussion on SIP, readers may refer to Sun (1994). The
solution of SIP can be easily extended to the case that both head and concentration observations are used for the identification of stochastic parameters.
Exercises
7.1. Assuming that there are two tracer components, oe and p, in a confined
aquifer, write the hydrodynamic system for this situation. Then, use a
flow chart to describe the solution procedure.
7.2. Assuming that only the longitudinal dispersivity is unknown in a
advection-dispersion model, formulate an inverse problem for this ca se
and translate it into an optimization problem.
7.3. Average Eq. (7.2.32) along the y direction to find the expressions of
"apparent velocity" and "apparent dispersion coefficient."
7.4. Derive the adjoint system for one-dimensional coupled flow and mass
transport problems and find the expressions of ohjoK and oC;oK.
7.5. Explain the differences between Fickian expressions Eqs. (7.4.1) and
(7.4.9).
7.6. What terms are neglected in the derivation of the perturbation equation
(7.4.1O)?
7.7. Derive Eqs. (7.4.15) and (7.4.16) based on the first-order approximation.
7.8. Draw a flow chart to describe the procedure of Monte-Carlo simulation
method for estimating the uncertainty of a water quality model.
7.9. Draw a flow chart to describe the procedure of conditional simulation
method for estimating the uncertainty of a water quality model.