UBC Theses and Dissertations
Simultaneous inversion of thermal and hydrogeologic data Woodbury, Allan David
The question that is addressed in this thesis is: can a simultaneous inverse scheme involving thermal and hydrologic data resolve hydrologic model parameters to a better degree than hydrologic data alone? The first chapter sets the framework for this question by first reviewing linear and non-linear inverse problems and then illustrating the advantages of a simultaneous inverse of two different data sets through the use of a simple example. It is the goal of Chapter 2 to examine current methodologies for stating and solving the inverse problem. A review of the maximum likelihood approach is presented, and a construction formalism is adopted by introducing a series of objective functionals (norms) which are minimized to yield a variety of possible models. The inverse is carried out using a modification of a constrained simplex procedure. The algorithm requires no derivative computations and can be used to minimize an arbitrarily complicated non-linear functional, subject to non-linear inequality constraints. The algorithm produces a wide variety of acceptable models clustered about a global minimum, each of which generates data that match observed values. The inverse technique is demonstrated on a series of one and two-dimensional synthetic data sets, and on a hydraulic head data set from Downie Slide, British Columbia, Canada. At this site, four parameters are determined; the free-surface position of the water table and three boundary conditions for the domain. Further simulations using a theoretical data set with assumed properties similar to that of Downie Slide show that with noise free data, and an adequate spacing between points it is possible to interpolate an unbiased estimate of hydraulic head data at all nodes in the equivalent discretized domain. When the inverse technique is applied, the domain's conductivity structure is correctly identified when enough prior log-conductivity information is available. The implications for Downie Slide are that in order to construct anything but a simple hydrogeologic model, accurate field measurements of hydraulic head are required, as well as well-defined estimates of hydraulic conductivity, a better spacing between measurements, and adequate knowledge of the boundary conditions. Chapter 3 is devoted to developing the idea of a joint inversion scheme involving both thermal and hydrologic data. One way of overcoming data limitations (sparse hydraulic head or few hydraulic conductivity estimates) in an inverse problem is to introduce an independently collected data set and apply simultaneous or joint inversion. The joint inversion method uses data from a number of different measurements to improve the resolution of parameters which are in common to one or more functional relationships. One such data set is subsurface temperature, which is sensitive to variations in hydraulic conductivity. In Chapter 3, the basic concepts of heat and fluid transfer in porous media with emphasis on forced convective effects are reviewed, followed by inversion of theoretical data and a re-investigation of the hydrogeology of Downie Slide, augmented with thermal data and a simultaneous inverse. Additional runs on a heterogenous medium presented in Chapter 2 are carried out. With a good temperature data base, thermal properties can be properly resolved. However, in this stochastic problem the addition of thermal data did not condition .the inverse to a greater degree than accomplished with the addition of prior information on log-conductivity. The benefits of including thermal data and applying a joint inversion can be substantial when considering the more realistic problem of uncertain boundary conditions. The simultaneous inverse is also applied to the Downie Slide data set examined in Chapter 2. Unfortunately, with a homogeneous hydraulic conductivity, all that can be determined from a hydraulic head inverse are ratios of flux to hydraulic conductivity. By including thermal data, the value of hydraulic conductivity can be determined at this site. Some of the model parameters (basal heat flux, thermal conductivity, specified head boundaries) are not resolved well by the joint scheme. None theless the constructed models do offer valuable insight into the hydrogeology of the field site. The constructed models persistently show a hydraulic conductivity value of about 1 x 10⁻⁷ m/sec, which is consistent with previous estimates of hydraulic conductivity at the site. A further comparison with the inverse results in Chapter 2 show good agreement between the two inverses for the hydraulic properties.
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