[1] Hydraulic tomography is a method that images the hydraulic heterogeneity of the subsurface through the inversion of multiple pumping or cross-hole hydraulic test data. Transient hydraulic tomography is different from steady state hydraulic tomography in that it utilizes transient hydraulic head records to yield the distribution of hydraulic conductivity (K) as well as specific storage (S s ) of an aquifer. In this paper we demonstrate the robustness of transient hydraulic tomography through the use of hydraulic head data obtained from multiple cross-hole pumping tests conducted in a laboratory sandbox with deterministic heterogeneity. We utilize the algorithm developed by Zhu and Yeh (2005) to conduct the transient inversions and validate the K and S s tomograms using a multimethod and multiscale validation approach previously proposed by Illman et al. (2006). Validation data consist of cross-hole tests not used in the inversion as well as other hydraulic tests that provided local (core, single-hole tests) as well as large-scale (unidirectional flow-through tests) estimates of hydraulic parameters. Results show that the algorithm is able to yield consistent estimates that agree with independently collected local as well as large-scale hydraulic parameter data. In addition, we find that the transient hydraulic tomography requires a fewer number of pumping tests to estimate a similar quality K tomogram when compared with steady state hydraulic tomography, as the former approach utilizes more data from each pumping test. Overall, we find that transient hydraulic tomography is a robust subsurface characterization technique that can delineate the subsurface heterogeneity in both K and S s from multiple pumping or crosshole hydraulic tests.

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[1] Groundwater modeling has become a vital component to water supply and contaminant transport investigations. An important component of groundwater modeling under steady state conditions is selecting a representative hydraulic conductivity (K) estimate or set of estimates which defines the K field of the studied region. Currently, there are a number of characterization approaches to obtain K at various scales and in varying degrees of detail, but there is a paucity of information in terms of which characterization approach best predicts flow through aquifers or drawdowns caused by some drawdown inducing events. The main objective of this paper is to assess K estimates obtained by various approaches by predicting drawdowns from independent cross-hole pumping tests and total flow rates through a synthetic heterogeneous aquifer from flow-through tests. Specifically, we (1) characterize a synthetic heterogeneous aquifer built in the sandbox through various techniques (permeameter analyses of core samples, single-hole, cross-hole, and flow-through testing), (2) obtain mean K fields through traditional analysis of test data by treating the medium to be homogeneous, (3) obtain heterogeneous K fields through kriging and steady state hydraulic tomography, and (4) conduct forward simulations of 16 independent pumping tests and six flowthrough tests using these homogeneous and heterogeneous K fields and comparing them to actual data. Results show that the mean K and heterogeneous K fields estimated through kriging of small-scale K data (core and single-hole tests) yield biased predictions of drawdowns and flow rates in this synthetic heterogeneous aquifer. In contrast, the heterogeneous K distribution or "K tomogram" estimated via steady state hydraulic tomography yields excellent predictions of drawdowns of pumping tests not used in the construction of the tomogram and very good estimates of total flow rates from the flowthrough tests. These results suggest that steady state groundwater model validation is possible in this laboratory sandbox aquifer if the heterogeneous K distribution and forcing functions (boundary conditions and source/sink terms) are characterized sufficiently.

Hydraulic tomography has been developed as an alternative to traditional geostatistical methods to delineate heterogeneity patterns in parameters such as hydraulic conductivity (K) and specific storage (S s ). During hydraulic tomography surveys, a large number of hydraulic head data are collected from a series of cross-hole tests in the subsurface. These head data are then used to interpret the spatial distribution of K and S s using inverse modeling. Here, we use the Sequential Successive Linear Estimator (SSLE) of Yeh and Liu (2000) to interpret synthetic pumping test data created through numerical simulations and real data generated in a laboratory sandbox aquifer to obtain the K tomograms. Here, we define ''K tomogram'' as an image of K distribution of the subsurface (or the inverse results) obtained via hydraulic tomography. We examine the influence of signal-to-noise ratio and biases on results using inverse modeling of synthetic and real cross-hole pumping test data. To accomplish this, we first show that the pumping rate, which affects the signal-to-noise ratio, and the order of data included into the SSLE algorithm both have large impacts on the quality of the K tomograms. We then examine the role of conditioning on the K tomogram and find that conditioning can improve the quality of the K tomogram, but can also impair it, if the data are of poor quality and conditioning data have a larger support volume than the numerical grid used to conduct the inversion. Overall, these results show that the quality of the K tomogram depends on the design of pumping tests, their conduct, the order in which they are included in the inverse code, and the quality as well as the support volume of additional data that are used in its computation.

Application of Pontryagin’s optimal principle to control system problems eventually requires that a two-point boundary-value problem be solved. For plants whose equations of motion are greater than second order this represents a formidable barrier in realizing a practical feedback control. When the criterion for optimization is minimum-fuel consumption, choice of time for solution may be used as a free parameter, and this plus consideration of the efficiency of application of control leads to an approximate method which avoids this difficulty. A heuristic analysis of the geometry of state-space trajectories, using true optimal solutions as a guide, provides laws for constructing a feedback control from the state variables. It further provides a knowledge of the bounds of performance of the mechanized suboptimal control and an estimate on the performance of a minimum-time control for comparison purposes. As an example, the method is applied to the problem of minimum-fuel attitude control of an earth-orbiting satellite, a fourth-order plant with two controls.