Quasi-linear Geostatistical Approach Research Articles

Geostatistical inverse modeling for the characterization of aquitard heterogeneity using long-term multi-extensometer data

The classical aquitard drainage model COMPAC was used to simulate the deformation of a one-dimensional vertically heterogeneous aquitard (i.e. Heter-COMPAC). By coupling Heter-COMPAC with the quasi-linear geostatistical inversion approach, the vertical heterogeneities of hydraulic conductivity (Kv), elastic specific storage (Sske) and inelastic specific storage (Sskp) fields could be characterized with the availability of long-term multi-extensometer data. The methodology was first tested through a group of ten synthetic cases with randomly generated Kv, Sske and Sskp fields. Results of the synthetic cases revealed that a geostatistical model could characterize the variability of aquitard hydraulic parameter fields, and yields improved fits of simulated versus observed zone deformation than a geological model that treats each zone to be homogeneous. Subsequently, the methodology was applied to a research site situated in Changzhou city, Jiangsu Province, China, where long-term deformation behaviors of the strata within the aquifer-aquitard system have been monitored. Vertical heterogeneities of Kv, Sske and Sskp fields of the investigated aquitard at the site were characterized using the quasi-linear geostatistical approach. The correlation length of the three fields that yielded the best-quality fit was assessed to be approximately 50 m, and the fitting quality was slightly improved in comparison with that produced from calibrating a geological or zonation model. Through Monte Carlo analysis of aquitard deformation utilizing the inversed parameters and corresponding uncertainties, the investigated aquitard is estimated to have compacted −914.8 ± (42.6) mm before the rise in groundwater levels in adjoining aquifers.

Principal Component Geostatistical Approach for large-dimensional inverse problems.

The quasi-linear geostatistical approach is for weakly nonlinear underdetermined inverse problems, such as Hydraulic Tomography and Electrical Resistivity Tomography. It provides best estimates as well as measures for uncertainty quantification. However, for its textbook implementation, the approach involves iterations, to reach an optimum, and requires the determination of the Jacobian matrix, i.e., the derivative of the observation function with respect to the unknown. Although there are elegant methods for the determination of the Jacobian, the cost is high when the number of unknowns, m, and the number of observations, n, is high. It is also wasteful to compute the Jacobian for points away from the optimum. Irrespective of the issue of computing derivatives, the computational cost of implementing the method is generally of the order of m2n, though there are methods to reduce the computational cost. In this work, we present an implementation that utilizes a matrix free in terms of the Jacobian matrix Gauss-Newton method and improves the scalability of the geostatistical inverse problem. For each iteration, it is required to perform K runs of the forward problem, where K is not just much smaller than m but can be smaller that n. The computational and storage cost of implementation of the inverse procedure scales roughly linearly with m instead of m2 as in the textbook approach. For problems of very large m, this implementation constitutes a dramatic reduction in computational cost compared to the textbook approach. Results illustrate the validity of the approach and provide insight in the conditions under which this method perform best.

Three-dimensional geostatistical inversion of synthetic tomographic pumping and heat-tracer tests in a nested-cell setup

A main purpose of groundwater inverse modeling lies in estimating the hydraulic conductivity field of an aquifer. Traditionally, hydraulic head measurements, possibly obtained in tomographic setups, are used as data. Because the groundwater flow equation is diffusive, many pumping and observation wells would be necessary to obtain a high resolution of hydraulic conductivity, which is typically not possible. We suggest performing heat tracer tests using the same already installed pumping wells and thermometers in observation planes to amend the hydraulic head data set by the arrival times of the heat signals. For each tomographic combinations of wells, we recommend installing an outer pair of pumping wells, generating artificial ambient flow, and an inner well pair in which the tests are performed. We jointly invert heads and thermal arrival times in 3-D by the quasi-linear geostatistical approach using an efficiently parallelized code running on a mid-range cluster. In the present study, we evaluate the value of heat tracer versus head data in a synthetic test case, where the estimated fields can be compared to the synthetic truth. Because the sensitivity patterns of the thermal arrival times differ from those of head measurements, the resolved variance in the estimated field is 6 to 10 times higher in the joint inversion in comparison to inverting head data only. Also, in contrast to head measurements, reversing the flow field and repeating the heat-tracer test improves the estimate in terms of reducing the estimation variance of the estimate. Based on the synthetic test case, we recommend performing the tests in four principal directions, requiring in total eight pumping wells and four intersecting observation planes for heads and temperature in each direction.

A Flexible Krylov Solver for Shifted Systems with Application to Oscillatory Hydraulic Tomography

We discuss efficient solutions to systems of shifted linear systems arising in computations for oscillatory hydraulic tomography. The reconstruction of hydrogeological parameters such as hydraulic conductivity and specific storage using limited discrete measurements of pressure (head) obtained from sequential oscillatory pumping tests, leads to a nonlinear inverse problem. We tackle this using the quasi-linear geostatistical approach [Kitanidis, Water Resources Res., 31 (1995), pp. 2411--2419]. This method requires repeated solution of the forward (and adjoint) problem for multiple frequencies, for which we use flexible preconditioned Krylov subspace solvers specifically designed for shifted systems based on ideas in [Gu, Zhou, and Lin, J. Comput. Math., 25 (2007), pp. 522--530]. The solvers allow the preconditioner to change at each iteration. We analyze the convergence of the solver and perform an error analysis when an iterative solver is used for inverting the preconditioner matrices. Finally, we apply our algorithm to a challenging application taken from oscillatory hydraulic tomography to demonstrate the computational gains by using the resulting method.

Efficient parallelization of geostatistical inversion using the quasi-linear approach

Hydraulic conductivity is a key parameter for the simulation of groundwater flow and transport. Typically, it is highly variable in space and difficult to determine by direct methods. The most common approach is to infer hydraulic-conductivity values from measurements of dependent quantities, such as hydraulic head and concentration. In geostatistical inversion, the parameters are estimated as continuous, spatially auto-correlated fields, the most likely values of which are obtained by conditioning on the indirect data. In order to identify small-scaled features, a fine three-dimensional discretization of the domain is needed. This leads to high computational demands in the solution of the forward problem and the calculation of sensitivities. In realistic three-dimensional settings with many measurements parallel computing becomes mandatory.In the present study, we investigate how parallelization of the quasi-linear geostatistical approach of inversion can be made most efficient. We suggest a two-level approach of parallelization, in which the computational domain is subdivided and the evaluation of sensitivities is also parallelized. We analyze how these two levels of parallelization should be balanced to optimally exploit a given number of computing nodes.

A modified Levenberg–Marquardt algorithm for quasi-linear geostatistical inversing

The Quasi-Linear Geostatistical Approach is a method of inverse modeling to identify parameter fields, such as the hydraulic conductivity in heterogeneous aquifers, given observations of related quantities like hydraulic heads or arrival times of tracers. Derived in the Bayesian framework, it allows to rigorously quantify the uncertainty of the identified parameter field. Since inverse modeling in subsurface flow is in general a non-linear problem, the Quasi-Linear Geostatistical Approach employs an algorithm for non-linear optimization. Up to presence, this algorithm has been similar to the Gauss–Newton algorithm for least-squares fitting and fails in cases of strong non-linearity. In this study, we present and discuss a new modified Levenberg–Marquardt algorithm for the Quasi-Linear Geostatistical Approach. Compared to the original method, the new algorithm offers increased stability and is more robust, allowing for stronger non-linearity and higher variability of the parameter field to be identified. We demonstrate its efficiency and improved convergence compared to the original version in several test cases. The new algorithm is designed for the general case of an uncertain mean of the parameter field, which includes the cases of completely known and entirely unknown mean as special cases.

A modified Levenberg?Marquardt algorithm for quasi-linear geostatistical inversing

The Quasi-Linear Geostatistical Approach is a method of inverse modeling to identify parameter fields, such as the hydraulic conductivity in heterogeneous aquifers, given observations of related quantities like hydraulic heads or arrival times of tracers. Derived in the Bayesian framework, it allows to rigorously quantify the uncertainty of the identified parameter field. Since inverse modeling in subsurface flow is in general a non-linear problem, the Quasi-Linear Geostatistical Approach employs an algorithm for non-linear optimization. Up to presence, this algorithm has been similar to the Gauss–Newton algorithm for least-squares fitting and fails in cases of strong non-linearity. In this study, we present and discuss a new modified Levenberg–Marquardt algorithm for the Quasi-Linear Geostatistical Approach. Compared to the original method, the new algorithm offers increased stability and is more robust, allowing for stronger non-linearity and higher variability of the parameter field to be identified. We demonstrate its efficiency and improved convergence compared to the original version in several test cases. The new algorithm is designed for the general case of an uncertain mean of the parameter field, which includes the cases of completely known and entirely unknown mean as special cases.