Groundwater Inverse Problem Research Articles

Groundwater inverse modeling: Physics-informed neural network with disentangled constraints and errors

This study combines a physics-informed neural network (PINN) and Karhunen-Loeve expansion (KLE) (i.e., KLE-PINN) to solve the groundwater inverse problem. The hydraulic head (u) distribution is approximated by a deep neural network (DNN), while the hydraulic conductivity (K) field is constructed by KLE with given prior geostatistical information. KLE-PINN is applied to investigate the inversion using data from a single pumping test, natural gradient flow (NG), and hydraulic tomography (HT). The results from these cases demonstrate that our inverse method is robust. Our error analysis endeavors to quantify the sources of error by using two custom reference indicators, eforward and ecoupling. Moreover, the study finds that the inversion using data from multiple pumping tests (HT) yields more accurate estimates, leads to faster training convergence, and maintains higher stability. In addition, by investigating cases with different outer boundary conditions (BCs), we find that KLE-PINN is more flexible. Precisely, in scenarios with missing BCs, our network still fits well with the observed data, and the estimates capture the approximate spatial pattern in the region where the observation wells are distributed. Even with incorrect BCs, our network still performs well because the observational data constraints are strongly enforced during training.

Symmetrical Rank-Three Vectorized Loading Scores Quasi-Newton for Identification of Hydrogeological Parameters and Spatiotemporal Recharges

In a multi-layered groundwater model, achieving accurate spatiotemporal identification and solving the ill-posed problem is the vital topic for model calibration. This study proposes a symmetry rank three vectorized loading scores (SR3 VLS) quasi-Newton algorithm by modifying the Levenberg–Marquardt algorithm and importing a rank three structure from Broyden–Fletcher–Goldfarb–Shanno algorithm for identification of hydrogeological parameters and spatiotemporal recharge simultaneously. To accelerate directional convergence and approach a global optimum, this study uses a vectorized limited switchable step size in the transmissive groundwater inverse problem. The Hessian approximation rank three uses high and low-rank factor loading scores analyzed from simulated storage fluctuation between adjacent iterations for calculation and matrix correction. Two numerical experiments were designed to validate the proposing algorithm, showing the SR3 VLS quasi-Newton reduced the error percentages of the identified parameters by 1.63% and 9.65% compared to the Jacobian quasi-Newton. The proposing method is applied to the Chou-Shui River alluvial fan groundwater system in Taiwan. Results show that the simulated storage error decreased rapidly in six iterations, and has good head convergence as small as 0.11% with a root-mean-square-error (RMSE) of 0.134 m, indicating that the proposing algorithm reduces the computational cost to converge to the true solution.

Parallel Inverse Modeling and Uncertainty Quantification for Computationally Demanding Groundwater-Flow Models Using Covariance Matrix Adaptation

This study investigates the performance of the covariance matrix adaptation-evolution strategy (CMA-ES), a stochastic optimization method, in solving groundwater inverse problems. The objectives of the study are to evaluate the computational efficiency of the parallel CMA-ES and to investigate the use of the empirically estimated covariance matrix in quantifying model prediction uncertainty due to parameter estimation uncertainty. First, the parallel scaling with increasing number of processors up to a certain limit is discussed for synthetic and real-world groundwater inverse problems. Second, through the use of the empirically estimated covariance matrix of parameters from the CMA-ES, the study adopts the Monte Carlo simulation technique to quantify model prediction uncertainty. The study shows that the parallel CMA-ES is an efficient and powerful method for solving the groundwater inverse problem for computationally demanding groundwater flow models and for deriving covariances of estimated parameters for uncertainty analysis.

Assessment of seawater intrusion potential from sea-level rise and groundwater extraction in a coastal aquifer

Groundwater is an important water resource in many coastal areas around the world. Excessive pumping can change the flow pattern so that seawater may migrate into the freshwater aquifer. In addition, the rise of the sea level due to climate change could accelerate the landward intrusion of seawater. This study addresses the problem of variable-density groundwater flow and miscible salt transport to assess the potential of seawater intrusion in coastal aquifers. Our conceptual model considers a complete hydrogeologic system including the river system, the seasonal groundwater recharge, the groundwater pumping, and the interaction between seawater and groundwater. Model calibration is performed using the covariance matrix adaptation–evolution strategy as a robust derivative-free global optimization algorithm. The advantage of this algorithm is its ability to reach a near global solution in solving the groundwater inverse problem, which is typically nonlinear and ill-posed. Simulations for the Nam Dinh Province, Vietnam over the period of 90 years indicate that seawater intrusion will occur, but the extent of it will vary depending on scenarios. The extraction of groundwater is the key factor governing the intrusion of seawater. The magnitude of seawater intrusion caused by rising sea level sturned out to be relatively small. Sensitivity analysis reveals that uncertainties of transport parameters could significantly affect the assessment of seawater intrusion potential.

Solving Groundwater Flow Inverse Problem Using Spreadsheet Solver

A simple spreadsheet cell-oriented methodology is proposed for solving a groundwater flow inverse problem. Initially, the numerical simulation of the groundwater flow process is carried out by using an embedded optimization approach and spreadsheet solver. The result of the simulation is compared with the result of a GMS MODFLOW simulation. The comparison shows that the result of the spreadsheet simulation is similar to the result of the GMS MODFLOW simulation. Two groundwater inverse problems are then solved using the proposed methodology. The first problem estimates the unknown pumping rates of an aquifer, and the second problem estimates the aquifer transmissivity. Using the spreadsheet method, it is easy to define boundary conditions and to initialize starting values as well as their visualization. The methodology is simple in concept and can be used for solving graduate- and undergraduate-level groundwater simulation and inverse problems. The methodology may also be used by the practicing engineer fo...

Incorporating subjective and stochastic uncertainty in an interactive multi-objective groundwater calibration framework

The interactive multi-objective genetic algorithm (IMOGA) combines traditional optimization with an interactive framework that considers the subjective knowledge of hydro-geological experts in addition to quantitative calibration measures such as calibration errors and regularization to solve the groundwater inverse problem. The IMOGA is inherently a deterministic framework and identifies multiple large-scale parameter fields (typically head and transmissivity data are used to identify transmissivity fields). These large-scale parameter fields represent the optimal trade-offs between the different criteria (quantitative and qualitative) used in the IMOGA. This paper further extends the IMOGA to incorporate uncertainty both in the large-scale trends as well as the small-scale variability (which can not be resolved using the field data) in the parameter fields. The different parameter fields identified by the IMOGA represent the uncertainty in large-scale trends, and this uncertainty is modeled using a Bayesian approach where calibration error, regularization, and the expert’s subjective preference are combined to compute a likelihood metric for each parameter field. Small-scale (stochastic) variability is modeled using a geostatistical approach and added onto the large-scale trends identified by the IMOGA. This approach is applied to the Waste Isolation Pilot Plant (WIPP) case-study. Results, with and without expert interaction, are analyzed and the impact that expert judgment has on predictive uncertainty at the WIPP site is discussed. It is shown that for this case, expert interaction leads to more conservative solutions as the expert compensates for some of the lack of data and modeling approximations introduced in the formulation of the problem.

Parameter and State Model Reduction for Large-Scale Statistical Inverse Problems

A greedy algorithm for the construction of a reduced model with reduction in both parameter and state is developed for an efficient solution of statistical inverse problems governed by partial differential equations with distributed parameters. Large-scale models are too costly to evaluate repeatedly, as is required in the statistical setting. Furthermore, these models often have high-dimensional parametric input spaces, which compounds the difficulty of effectively exploring the uncertainty space. We simultaneously address both challenges by constructing a projection-based reduced model that accepts low-dimensional parameter inputs and whose model evaluations are inexpensive. The associated parameter and state bases are obtained through a greedy procedure that targets the governing equations, model outputs, and prior information. The methodology and results are presented for groundwater inverse problems in one and two dimensions.

A genetic algorithm-based procedure for 3D source identification at the Borden emplacement site

Finding the location and concentration of groundwater contaminant sources typically requires the solution of an inverse problem. A parallel hybrid optimization framework that uses genetic algorithms (GA) coupled with local search approaches (GA-LS) has been developed previously to solve groundwater inverse problems. In this study, the identification of an emplaced source at the Borden site is carried out as a test problem using this optimization framework by using a Real Genetic Algorithm (RGA) as the GA approach and a Nelder–Mead simplex as the LS approach. The RGA results showed that the minimum objective function did not always correspond to the minimum solution error, indicating a possible non-uniqueness issue. To address this problem, a procedure to identify maximally different starting points for LS is introduced. When measurement or model errors are non-existent or minimal it is shown that one of these starting points leads to the true solution. When these errors are significant, this procedure leads to multiple possible solutions that could be used as a basis for further investigation. Metrics of mean and standard deviation of objective function values was adopted to evaluate the possible solutions. A new selection criterion based on these metrics is suggested to find the best alternative. This suggests that this alternative generation procedure could be used to address the non-uniqueness of similar inverse problems. A potential limitation of this approach is the application to a wide class of problems, as verification has not been performed with a large number of test cases or other inverse problems. This remains a topic for future work.

An interactive multi-objective optimization framework for groundwater inverse modeling

The groundwater inverse problem of estimating heterogeneous groundwater model parameters (hydraulic conductivity in this case) given measurements of aquifer response (such as hydraulic heads) is known to be an ill-posed problem, with multiple parameter values giving similar fits to the aquifer response measurements. This problem is further exacerbated due to the lack of extensive data, typical of most real-world problems. In such cases, it is desirable to incorporate expert knowledge in the estimation process to generate more reasonable estimates. This work presents a novel interactive framework, called the ‘Interactive Multi-Objective Genetic Algorithm’ (IMOGA), to solve the groundwater inverse problem considering different sources of quantitative data as well as qualitative expert knowledge about the site. The IMOGA is unique in that it looks at groundwater model calibration as a multi-objective problem consisting of quantitative objectives – calibration error and regularization – and a ‘qualitative’ objective based on the preference of the geological expert for different spatial characteristics of the conductivity field. All these objectives are then included within a multi-objective genetic algorithm to find multiple solutions that represent the best combination of all quantitative and qualitative objectives. A hypothetical aquifer case-study (based on the test case presented by Freyberg [Freyberg DL. An exercise in ground-water model calibration and prediction. Ground Water 1988;26(3)], for which the ‘true’ parameter values are known, is used as a test case to demonstrate the applicability of this method. It is shown that using automated calibration techniques without using expert interaction leads to parameter values that are not consistent with site-knowledge. Adding expert interaction is shown to not only improve the plausibility of the estimated conductivity fields but also the predictive accuracy of the calibrated model.

Fusion of hydrologic and geophysical tomographic surveys

In this paper, we argue the need for high-resolution characterization of the subsurface and discuss difficulties of traditional characterization approaches to meet this need. Necessary and sufficient conditions are then presented for well-posedness of groundwater inverse problems associated with identifying spatially distributed parameters. Non-uniqueness and large uncertainty in model calibration are subsequently attributed to difficulties in collecting information to meet these conditions. Using an example, we show that a tomographic survey can make problems of identification of spatially distributed parameters better posed. We subsequently present some recent advances in hydrologic/geophysical characterization of the subsurface using information fusion based on tomographic survey concepts. This paper includes hydraulic and electrical resistivity tomographic surveys as well as fusion of hydraulic and resistivity tomography and fusion of hydraulic and tracer tomography.

Steady- and transient-state inversion in hydrogeology by successive flux estimation

A calibration method to solve the groundwater inverse problem under steady- and transient-state conditions is presented. The method compares kriged and numerical head field gradients to modify hydraulic conductivity without the use of non-linear optimization techniques. The process is repeated iteratively until a close match with piezometric data is reached. The approach includes a damping factor to avoid divergence and oscillation of the solution in areas of low hydraulic gradient and a weighting factor to account for temporal head variation in transient simulations. The efficiency of the method in terms of computing time and calibration results is demonstrated with a synthetic field. It is shown that the proposed method provides parameter fields that reproduce both hydraulic conductivity and piezometric data in few forward model solutions. Stochastic numerical experiments are conducted to evaluate the sensitivity of the method to the damping function and to the head field estimation errors.

Efficient Parallel Implementation of Hybrid Optimization Approaches for Solving Groundwater Inverse Problems

Inverse problems that are constrained by large-scale partial differential equation (PDE) systems demand very large computational resources. Solutions to these problems generally require the solution of a large number of complex PDE systems. Three-dimensional groundwater inverse problems fall under this category. In this paper, we describe the implementation of a parallel simulation-optimization framework for solving PDE-based inverse problems and demonstrate it for the solution of groundwater contaminant source release history reconstruction problems that are of practical importance. The optimization component employs several optimization algorithms, including genetic algorithms (GAs) and several local search (LS) approaches that can be used in a hybrid mode. This hybrid GA-LS optimizer is used to drive a parallel finite-element (FEM) groundwater forward transport simulator. Parallelism is exploited within the transport simulator (fine grained parallelism) as well as the optimizer (coarse grained parallelism) through the exclusive use of the Message Passing Interface (MPI) communication library. Algorithmic and parallel performance results are presented for an IBM SP3 supercomputer. Simulation and performance results presented in this paper illustrate that an effective combination of efficient optimization algorithms and parallel computing can enable solution to three-dimensional groundwater inverse problems of a size and complexity not attempted before.

Inverse problem in hydrogeology

The state of the groundwater inverse problem is synthesized. Emphasis is placed on aquifer characterization, where modelers have to deal with conceptual model uncertainty (notably spatial and temporal variability), scale dependence, many types of unknown parameters (transmissivity, recharge, boundary conditions, etc.), nonlinearity, and often low sensitivity of state variables (typically heads and concentrations) to aquifer properties. Because of these difficulties, calibration cannot be separated from the modeling process, as it is sometimes done in other fields. Instead, it should be viewed as one step in the process of understanding aquifer behavior. In fact, it is shown that actual parameter estimation methods do not differ from each other in the essence, though they may differ in the computational details. It is argued that there is ample room for improvement in groundwater inversion: development of user-friendly codes, accommodation of variability through geostatistics, incorporation of geological information and different types of data (temperature, occurrence and concentration of isotopes, age, etc.), proper accounting of uncertainty, etc. Despite this, even with existing codes, automatic calibration facilitates enormously the task of modeling. Therefore, it is contended that its use should become standard practice.

A representer‐based inverse method for groundwater flow and transport applications

Groundwater inverse problems are concerned with the estimation of uncertain model parameters, such as hydraulic conductivity, from field or laboratory measurements. In practice, model and measurement errors compromise the ability of inverse procedures to provide accurate results. It is important to account for such errors in order to determine the proper weight to give to each source of information. Probabilistic descriptions of model and measurement errors can be incorporated into classical variational inverse procedures, but the computational demands are excessive if the model errors vary over time. An alternative approach based on representer expansions is able to efficiently accommodate time‐dependent errors for large problems. In the representer approach, unknown variables are expanded in finite series which depend on unknown functions called representers. Each representer quantifies the influence of a given measurement on the estimate of a particular variable. This procedure replaces the original inverse problem by an equivalent problem where the number of independent unknowns is proportional to the number of measurements. The representer approach is especially advantageous in groundwater problems, where the total number of measurements is often small. This approach is illustrated with a synthetic flow and transport example that includes time‐dependent model errors. The representer algorithm is able to provide good estimates of a spatially variable hydraulic conductivity field and good predictions of solute concentration. Its computational demands are reasonable, and it is relatively easy to implement. The example reveals that it is beneficial to account for model errors even when they are difficult to estimate.

A multipopulation genetic algorithm to solve the inverse problem in hydrogeology

The inverse problem of groundwater flow is treated with an automatic method that can produce several alternative solutions at once. During their joint optimization, these solutions can exchange information in order to maintain some diversity and thus avoid a systematic premature convergence toward a single local minimum. Although genetic algorithms are capable of doing this, they have not often been used in groundwater inverse optimization. First, a specific multipopulation genetic algorithm is developed. It is then tested on two synthetic cases of steady state flow with transmissivity values extending over 4 orders of magnitude. The first test is nonparametric and optimizes as many parameters as those used to define the reference case. The second test uses a sort of “pilot point” parametrization. The optimization is carried out on a limited number of perturbations that are interpolated and superimposed on an initial transmissivity field. In view of the good quality of the results, these initial attempts provide incentives to further develop genetic algorithms in groundwater inverse problems.

Use of GA to Determine Areas of Accretionto Semiconfined Aquifer

The goal of any groundwater inverse problem is to identify the distribution of an input function or certain other variables describing the unique flow dynamics of an aquifer system. A genetic algorithm (GA) combined with a numerical modeling technique is useful in determining both the spatial distribution and the flux represented by the accretion component of the groundwater flow equation. The GA technique was compared to a modified Gauss-Newton iterative technique. Binary and hexadecimal representations provided mapping of parameters from genetic operators to the numerical model. The technique used the patterns that developed in the string representations to determine probability regions. Two synthetic test cases were used to test the effectiveness of the technique. The stability of the technique was evaluated by introducing random error into the observation data. The technique was capable of locating the accretion area and tended to converge to a flux most representative of the flux entering the aquifer. However, the technique was susceptible to typical problems affecting the inverse problem, such as nonuniqueness.

A full‐Bayesian approach to the groundwater inverse problem for steady state flow

A full‐Bayesian approach to the estimation of transmissivity from hydraulic head and transmissivity measurements is developed for two‐dimensional steady state groundwater flow. The approach combines both Bayesian and maximum entropy viewpoints of probability. In the first phase, log transmissivity measurements are incorporated into Bayes' theorem, and the prior probability density function is updated, yielding posterior estimates of the mean value of the log transmissivity field and covariance. The two central moments are generated assuming that the prior mean, variance, and integral scales are “hyperparameters”; that is, they are treated as random variables in themselves which is contrary to classical statistical approaches. The probability density functions (pdfs) of these hyperparameters are, in turn, determined from maximum entropy considerations. In other words, pdfs are chosen for each of the hyperparameters that are maximally uncommitted with respect to unknown information. This methodology is quite general and provides an alternative to kriging for spatial interpolation. The final step consists of updating the conditioned natural logarithm transmissivity (ln(T)) field with hydraulic head measurements, utilizing a linearized aquifer equation. It is assumed that the statistical properties of the noise in the hydraulic head measurements are also uncertain. At each step, uncertainties in all pertinent hyperparameters are removed by marginalization. Finally, what is produced is a ln(T) field conditioned on measurements of both hydraulic heads and log transmissivity and covariances of the ln(T) field. In addition, we can also produce resolution matrices, confidence (credibility) limits, and the like for the ln(T) field. It is shown that the application of the methodology yields good estimates of transmissivities, even when hydraulic head measurements are noisy and little or no information is specified on mean values of ln(T), variance of ln(T), and integral scales.

Comment on “A reassessment of the groundwater inverse problem” by D. McLaughlin and L. R. Townley

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A Reassessment of the Groundwater Inverse Problem

This paper presents a functional formulation of the groundwater flow inverse problem that is sufficiently general to accommodate most commonly used inverse algorithms. Unknown hydrogeological properties are assumed to be spatial functions that can be represented in terms of a (possibly infinite) basis function expansion with random coefficients. The unknown parameter function is related to the measurements used for estimation by a “forward operator” which describes the measurement process. In the particular case considered here, the parameter of interest is the large‐scale log hydraulic conductivity, the measurements are point values of log conductivity and piezometric head, and the forward operator is derived from an upscaled groundwater flow equation. The inverse algorithm seeks the “most probable” or maximum a posteriori estimate of the unknown parameter function. When the measurement errors and parameter function are Gaussian and independent, the maximum a posteriori estimate may be obtained by minimizing a least squares performance index which can be partitioned into goodness‐of‐fit and prior terms. When the parameter is a stationary random function the prior portion of the performance index is equivalent to a regularization term which imposes a smoothness constraint on the estimate. This constraint tends to make the problem well‐posed by limiting the range of admissible solutions. The Gaussian maximum a posteriori problem may be solved with variational methods, using functional generalizations of Gauss‐Newton or gradient‐based search techniques. Several popular groundwater inverse algorithms are either special cases of, or variants on, the functional maximum a posteriori algorithm. These algorithms differ primarily with respect to the way they describe spatial variability and the type of search technique they use (linear versus nonlinear). The accuracy of estimates produced by both linear and nonlinear inverse algorithms may be measured in terms of a Bayesian extension of the Cramer‐Rao lower bound on the estimation error covariance. This bound suggests how parameter identifiability can be improved by modifying the problem structure and adding new measurements.