An integrated optimization algorithm for parameter structure identification in groundwater modeling

The inverse problem of parameter structure identification in a distributed parameter system remains challenging. Identifying a more complex parameter structure requires more data. There is also the problem of over-parameterization. In this study, we propose a modified Tabu search for parameter structure identification. We embed an adjoint state procedure in the search process to improve the efficiency of the Tabu search. We use Voronoi tessellation for automatic parameterization to reduce the dimension of the distributed parameter. Additionally, a coarse-fine grid technique is applied to further improve the effectiveness and efficiency of the proposed methodology. To avoid over-parameterization, at each level of parameter complexity we calculate the residual error for parameter fitting, the parameter uncertainty error and a modified Akaike Information Criterion. To demonstrate the proposed methodology, we conduct numerical experiments with synthetic data that simulate both discrete hydraulic conductivity zones and a continuous hydraulic conductivity distribution. Our results indicate that the Tabu search allied with the adjoint state method significantly improves computational efficiency and effectiveness in solving the inverse problem of parameter structure identification.     

Parameter structure identification using tabu search and simulated annealing

In groundwater modeling the identification of an optimal flow or transport parameter that varies spatially should include both the values and structure of the parameter. However, most existing techniques for parameter identification only consider the parameter values. In this study, the problem of identifying optimal parameter structure is treated as a large combinatorial optimization problem. Two recently developed heuristic search techniques, simulated annealing and tabu search, are used to solve the large combinatorial optimization problem. The effectiveness and flexibility of these two techniques are evaluated and compared with simple grid search and descent search, using preliminary results from one-dimensional examples. Among the techniques examined in this paper, tabu search performs extremely well in terms of the total number of function evaluations required.     

Simultaneous determination of aquifer parameters and zone structures with fuzzy c-means clustering and meta-heuristic harmony search algorithm

This study proposes an inverse solution algorithm through which both the aquifer parameters and the zone structure of these parameters can be determined based on a given set of observations on piezometric heads. In the zone structure identification problem fuzzy c-means (FCM) clustering method is used. The association of the zone structure with the transmissivity distribution is accomplished through an optimization model. The meta-heuristic harmony search (HS) algorithm, which is conceptualized using the musical process of searching for a perfect state of harmony, is used as an optimization technique. The optimum parameter zone structure is identified based on three criteria which are the residual error, parameter uncertainty, and structure discrimination. A numerical example given in the literature is solved to demonstrate the performance of the proposed algorithm. Also, a sensitivity analysis is performed to test the performance of the HS algorithm for different sets of solution parameters. Results indicate that the proposed solution algorithm is an effective way in the simultaneous identification of aquifer parameters and their corresponding zone structures.     

Role of Geological and Geophysical Data in Modeling a Southwestern Alluvial Basin

A parameter-estimation technique based on existing hydrological, geophysical, and geological data was developed to approximate transmissivity values for use in a ground-water flow model of the Animas Valley, southwest New Mexico. Complete Bouguer gravity anomaly maps together with seismic-refraction profiles, geologic maps, geologic, geophysical, and drillers' logs, water levels, and pumping-test data provide insight into the transmissivity of bolson deposits throughout the basin. The transmissivity distribution was primarily based on reported pumping and specific-capacity tests in conjunction with complete Bouguer gravity anomaly maps and well log data. Reported transmissivity values were characterized by gravity values and well log data. In grid blocks lacking pumping and specific-capacity tests, transmissivity values were assigned based on the relationship of gravity values and well log data within the grid block to gravity values and well log data within other grid blocks for which transmissivity values are available. A two-dimensional, finite-difference, ground-water flow computer code was used to evaluate the effectiveness of the parameter-estimation technique. Although the trial-and-error method of calibration was employed, the actual computer time necessary for model calibration was minimal. The conceptually straightforward approach for parameter estimation utilizing existing hydrological, geophysical, and geological data provides realistic parameter estimates.     

Seismic Event Location: Nonlinear Inversion Using a Neighbourhood Algorithm

—?A recently developed direct search method for inversion, known as a neighbourhood algorithm (NA), is applied to the hypocentre location problem. Like some previous methods the algorithm uses randomised, or stochastic, sampling of a four-dimensional hypocentral parameter space, to search for solutions with acceptable data fit. Considerable flexibility is allowed in the choice of misfit measure.¶At each stage the hypocentral parameter space is partitioned into a series of convex polygons called Voronoi cells. Each cell surrounds a previously generated hypocentre for which the fit to the data has been determined. As the algorithm proceeds new hypocentres are randomly generated in the neighbourhood of those hypocentres with smaller data misfit. In this way all previous hypocentres guide the search, and the more promising regions of parameter space are preferentially sampled.¶The NA procedure makes use of just two tuning parameters. It is possible to choose their values so that the behaviour of the algorithm is similar to that of a contracting irregular grid in 4-D. This is the feature of the algorithm that we exploit for hypocentre location. In experiments with different events and data sources, the NA approach is able to achieve comparable or better levels of data fit than a range of alternative methods; linearised least-squares, genetic algorithms, simulated annealing and a contracting grid scheme. Moreover, convergence was achieved with a substantially reduced number of travel-time/slowness calculations compared with other nonlinear inversion techniques. Even when initial parameter bounds are very loose, the NA procedure produced robust convergence with acceptable levels of data fit.     

Bayesian model averaging assessment on groundwater management under model structure uncertainty

This study introduces Bayesian model averaging (BMA) to deal with model structure uncertainty in groundwater management decisions. A robust optimized policy should take into account model parameter uncertainty as well as uncertainty in imprecise model structure. Due to a limited amount of groundwater head data and hydraulic conductivity data, multiple simulation models are developed based on different head boundary condition values and semivariogram models of hydraulic conductivity. Instead of selecting the best simulation model, a variance-window-based BMA method is introduced to the management model to utilize all simulation models to predict chloride concentration. Given different semivariogram models, the spatially correlated hydraulic conductivity distributions are estimated by the generalized parameterization (GP) method that combines the Voronoi zones and the ordinary kriging (OK) estimates. The model weights of BMA are estimated by the Bayesian information criterion (BIC) and the variance window in the maximum likelihood estimation. The simulation models are then weighted to predict chloride concentrations within the constraints of the management model. The methodology is implemented to manage saltwater intrusion in the “1,500-foot” sand aquifer in the Baton Rouge area, Louisiana. The management model aims to obtain optimal joint operations of the hydraulic barrier system and the saltwater extraction system to mitigate saltwater intrusion. A genetic algorithm (GA) is used to obtain the optimal injection and extraction policies. Using the BMA predictions, higher injection rates and pumping rates are needed to cover more constraint violations, which do not occur if a single best model is used.     

Groundwater Recharge Rate and Zone Structure Estimation Using PSOLVER Algorithm

The quantification of groundwater recharge is an important but challenging task in groundwater flow modeling because recharge varies spatially and temporally. The goal of this study is to present an innovative methodology to estimate groundwater recharge rates and zone structures for regional groundwater flow models. Here, the unknown recharge field is partitioned into a number of zones using Voronoi Tessellation (VT). The identified zone structure with the recharge rates is associated through a simulation‐optimization model that couples MODFLOW‐2000 and the hybrid PSOLVER optimization algorithm. Applicability of this procedure is tested on a previously developed groundwater flow model of the Tahtal? Watershed. Successive zone structure solutions are obtained in an additive manner and penalty functions are used in the procedure to obtain realistic and plausible solutions. One of these functions constrains the optimization by forcing the sum of recharge rates for the grid cells that coincide with the Tahtal? Watershed area to be equal to the areal recharge rate determined in the previous modeling by a separate precipitation‐runoff model. As a result, a six‐zone structure is selected as the best zone structure that represents the areal recharge distribution. Comparison to results of a previous model for the same study area reveals that the proposed procedure significantly improves model performance with respect to calibration statistics. The proposed identification procedure can be thought of as an effective way to determine the recharge zone structure for groundwater flow models, in particular for situations where tangible information about groundwater recharge distribution does not exist.     

A two‐dimensional inverse model to identify transmissivity in an anisotropic aquifer

This study suggested a numerical model using the Tabu search algorithm along with the Adjoint State method to identify the hydrogeological characteristics of an anisotropic groundwater aquifer. The Tabu search algorithm was applied to identify the anisotropic transmissivity components to avoid a local optimum. Then, the Adjoint State method was used to calculate the sensitivity of the parameters in order to increase the efficiency of the optimization. For an anisotropic and homogeneous aquifer, results showed that the optimal procedure presented combining the Tabu search algorithm and the Adjoint State method might successfully identify the values of the transmissivity components. If the duration of the pumping test was long enough (12‐h pumping test), the value of the transmissivity components could be optimized with type‐curve, straight‐line, and Tabu search methods, along with the Adjoint State methods. If the duration of the pumping test was short (0·5‐h pumping test), the Tabu search method, along with the Adjoint State method proposed herein, might successfully optimize the transmissivity components. For an anisotropic but heterogeneous aquifer, results showed that the suggested optimal procedure still successfully identified the values of the transmissivity components. Copyright © 2008 John Wiley & Sons, Ltd.     

3D adaptive tomography using Delaunay triangles and Voronoi polygons

The solutions of traveltime inversion problems are often not unique because of the poor match between the raypath distribution and the tomographic grid. However, by adapting the local resolution iteratively, by means of a singular value analysis of the tomographic matrix, we can reduce or eliminate the null space influence on our earth image: in this way, we get a much more reliable estimate of the velocity field of seismic waves. We describe an algorithm for an automatic regridding, able to fit the local resolution to the available raypaths, which is based on Delaunay triangulation and Voronoi tessellation. It increases the local pixel density where the null space energy is low or the velocity gradient is large, and reduces it elsewhere. Consequently, the tomographic image can reveal the boundaries of complex objects, but is not affected by the ambiguities that occur when the grid resolution is not adequately supported by the available raypaths.     

Stochastic delineation of capture zones: classical versus Bayesian approach

A Bayesian approach to characterize the predictive uncertainty in the delineation of time-related well capture zones in heterogeneous formations is presented and compared with the classical or non-Bayesian approach. The transmissivity field is modelled as a random space function and conditioned on distributed measurements of the transmissivity. In conventional geostatistical methods the mean value of the log transmissivity and the functional form of the covariance and its parameters are estimated from the available measurements, and then entered into the prediction equations as if they are the true values. However, this classical approach accounts only for the uncertainty that stems from the lack of ability to exactly predict the transmissivity at unmeasured locations. In reality, the number of measurements used to infer the statistical properties of the transmissvity field is often limited, which introduces error in the estimation of the structural parameters. The method presented accounts for the uncertainty that originates from the imperfect knowledge of the parameters by treating them as random variables. In particular, we use Bayesian methods of inference so as to make proper allowance for the uncertainty associated with estimating the unknown values of the parameters. The classical and Bayesian approach to stochastic capture zone delineation are detailed and applied to a hypothetical flow field. Two different sampling densities on a regular grid are considered to evaluate the effect of data density in both methods. Results indicate that the predictions of the Bayesian approach are more conservative.     

A new method for the interpretation of pumping tests in leaky aquifers

A novel methodology for the interpretation of pumping tests in leaky aquifer systems, referred to as the double inflection point (DIP) method, is presented. The method is based on the analysis of the first and second derivatives of the drawdown with respect to log time for the estimation of the flow parameters. Like commonly used analysis procedures, such as the type-curve approach developed by Walton (1962) and the inflection point method developed by Hantush (1956), the mathematical development of the DIP method is based on the assumption of homogeneity of the leaky aquifer layers. However, contrary to the two methods developed by Hantush and Walton, the new method does not need any fitting process. In homogeneous media, the two classic methods and the one proposed here provide exact results for transmissivity, storativity, and leakage factor when aquifer storage is neglected and the recharging aquifer is unperturbed. The real advantage of the DIP method comes when applying all methods independently to a test in a heterogeneous aquifer, where each method yields parameter values that are weighted differently, and thus each method provides different information about the heterogeneity distribution. Therefore, the methods are complementary and not competitive. In particular, the combination of the DIP method and Hantush method is shown to lead to the identification of contrasts between the local transmissivity in the vicinity of the well and the equivalent transmissivity of the perturbed aquifer volume.     

三维频率域可控源电磁反演研究

对于三维可控源电磁,反演计算效率、张量测量、旁侧效应以及阴影效应是目前研究的热点.本文正演采用基于库仑规范条件的耦合势有限体积算法,反演采用有限内存BFGS(L-BFGS)算法.合成数据反演结果表明:(1)有限内存BFGS法比非线性共轭梯度法,在反演计算效率上具有一定的优势,更适合求解大规模三维可控源电磁反演问题.(2)张量可控源电磁法相对于标量可控源电磁法,前者在模型分辨率上优于后者.(3)在某个区域无法布置测网的情况下,我们可利用旁侧效应在异常体周围布置测网进行三维反演,从而获得真实异常体的信息.同时,为避免阴影效应,我们应在测网外增加可控源电磁控制点,使得三维反演的数据更加完备.     

利用基于围陷波波形相关的网格搜索法确定昆仑山断裂带结构参数

提出了确定断层带结构的一种方法,它利用基于断层带围陷波波形相关的网格搜索法,以定量地确定断层带参数.该方法通过建立一系列由断层带的相关参数构建的断层带结构模型,利用有限差分方法数值模拟每组参数对应的断层带模型所产生的围陷波,并与实际观测围陷波波形进行相关分析,计算由以断层参数为多维网格坐标的相关系数变化趋势,网格搜索其峰值,最终获得最佳的一组参数,并确定断层带的结构.本文对昆仑山断裂带人工激发的围陷波进行分析,利用该方法确定了断裂带宽度和Q值参数.     

Reverse‐time migration from rugged topography using irregular,unstructured mesh

We developed a reverse‐time migration scheme that can image regions with rugged topography without requiring any approximations by adopting an irregular, unstructured‐grid modelling scheme. This grid, which can accurately describe surface topography and interfaces between high‐velocity‐contrast regions, is generated by Delaunay triangulation combined with the centroidal Voronoi tessellation method. The grid sizes vary according to the migration velocities, resulting in significant reduction of the number of discretized nodes compared with the number of nodes in the conventional regular‐grid scheme, particularly in the case wherein high near‐surface velocities exist. Moreover, the time sampling rate can be reduced substantially. The grid method, together with the irregular perfectly matched layer absorbing boundary condition, enables the proposed scheme to image regions of interest using curved artificial boundaries with fewer discretized nodes. We tested the proposed scheme using the 2D SEG Foothill synthetic dataset.     

Development and application of a coupled-process parameter inversion model based on the maximum likelihood estimation method

The coupled flow-mass transport inverse problem is formulated using the maximum likelihood estimation concept. An evolutionary computational algorithm, the genetic algorithm, is applied to search for a global or near-global solution. The resulting inverse model allows for flow and transport parameter estimation, based on inversion of spatial and temporal distributions of head and concentration measurements. Numerical experiments using a subset of the three-dimensional tracer tests conducted at the Columbus, Mississippi site are presented to test the model's ability to identify a wide range of parameters and parametrization schemes. The results indicate that the model can be applied to identify zoned parameters of hydraulic conductivity, geostatistical parameters of the hydraulic conductivity field, angle of hydraulic conductivity anisotropy, solute hydrodynamic dispersivity, and sorption parameters. The identification criterion, or objective function residual, is shown to decrease significantly as the complexity of the hydraulic conductivity parametrization is increased. Predictive modeling using the estimated parameters indicated that the geostatistical hydraulic conductivity distribution scheme produced good agreement between simulated and observed heads and concentrations. The genetic algorithm, while providing apparently robust solutions, is found to be considerably less efficient computationally than a quasi-Newton algorithm.     

Development and Applications of Double-difference Seismic Tomography

Double-difference (DD) tomography is a generalization of DD location; it simultaneously solves for the three-dimensional velocity structure and seismic event locations. DD tomography uses a combination of absolute and more accurate differential arrival times and hierarchically determines the velocity structure from larger scale to smaller scale. This method is able to produce more accurate event locations and velocity structure near the source region than standard tomography, which uses only absolute arrival times. We conduct a stability and uncertainty analysis of DD tomography based on a synthetic data set. Currently three versions of the DD tomography algorithms exist: tomoDD, tomoFDD and tomoADD. TomoDD assumes a flat earth model and uses a pseudo-bending ray-tracing algorithm to find rays between events and stations while tomoFDD uses a finite-difference travel-time algorithm and the curvature of the Earth is considered. Both codes are based on a regularly distributed inversion grid, with the former for a local scale and the latter for a regional scale. In contrast, tomoADD adapts the inversion mesh to match with the data distribution based on tetrahedral and Voronoi diagrams. We discuss examples of applying DD tomography to characterize fault zone structure, image high-resolution structure of subduction zones, and determine the velocity structure of volcanoes.     

Worth of head data in well-capture zone design: deterministic and stochastic analysis

In this study, we examine the effects of conditioning spatially variable transmissivity fields using head and/or transmissivity measurements on well-capture zones. In order to address the challenge posed by conditioning a flow model with spatially varying parameters, an innovative inverse algorithm, the Representers method, is employed. The method explicitly considers this spatial variability.

A number of uniform measurement grids with different densities are used to condition transmissivity fields using the Representers method. Deterministic and stochastic analysis of well-capture zones are then examined. The deterministic study focuses on comparison between reference well-capture zones and their estimated mean conditioned on head data. It shows that model performance due to head conditioning on well-capture zone estimation is related to pumping rate. At moderate pumping rates transmissivity observations are more crucial to identify effects arising from small-scale variations in pore water velocity. However, with more aggressive pumping these effects are reduced, consequently model performance, through incorporating head observations, markedly improves. In the stochastic study, the effect of conditioning using head and/or transmissivity data on well-capture zone uncertainty is examined. The Representers method is coupled with the Monte Carlo method to propagate uncertainty in transmissivity fields to well-capture zones. For the scenario studied, the results showed that a combination of 48 head and transmissivity data could reduce the area of uncertainty (95% confidence interval) in well-capture zone location by over 50%, compared to a 40% reduction using either head or transmissivity data. This performance was comparable to that obtained through calibrating on three and a half times the number of head observations alone.     

Identification of heterogeneous aquifer transmissivity using an AE-based method

Determination of hydraulic head, H, as a function of spatial coordinates and time, in ground water flow is the basis for aquifer management and for prediction of contaminant transport. Several computer codes are available for this purpose. Spatial distribution of the transmissivity, T(x,y), is a required input to these codes. In most aquifers, T varies in an erratic manner, and it can be characterized statistically in terms of a few moments: the expected value, the variance, and the variogram. Knowledge of these moments, combined with a few measurements, permits one to estimate T at any point using geostatistical methods. In a review of transmissivity data from 19 unconsolidated aquifers, Hoeksema and Kitanidis (1985) identified two types of the logtransmissivity Y= ln(T) variations: correlated variations with variance sigma2Yc and correlation scale, I(Y), on the order of kilometers, and uncorrelated variations with variance sigma2Yn. Direct identification of the logtransmissivity variogram, Gamma(Y), from measurements is difficult because T data are generally scarce. However, many head measurements are commonly available. The aim of the paper is to introduce a methodology to identify the transmissivity variogram parameters (sigma2Yc, I(Y), and sigma2Yn) using head data in formations characterized by large logtransmissivity variance. The identification methodology uses a combination of precise numerical simulations (carried out using analytic element method) and a theoretical model. The main objective is to demonstrate the application of the methodology to a regional ground water flow in Eagle Valley basin in west-central Nevada for which abundant transmissivity and head measurements are available.     

Assessing radial transmissivity variation in heterogeneous aquifers using analytical techniques

Pumping tests are one of the most commonly used in situ testing techniques for assessing aquifer hydraulic properties. Numerous researches have been conducted to predict the effects of aquifer heterogeneity on the groundwater levels during pumping tests. The objectives of the present work were as follows: (1) to predict drawdown conditions and to estimate aquifer properties during pumping tests undertaken in radially symmetric heterogeneous aquifers, and (2) to identify a method for assessing the transmissivity field along the radial coordinate in radially symmetric and fully heterogeneous transmissivity fields. The first objective was achieved by expanding an existing analytical drawdown formulation that was valid for a radially symmetric confined aquifer with two concentric zones around the pumping well to an N concentric zone confined aquifer having a constant transmissivity value within each zone. The formulation was evaluated for aquifers with three and four concentric zones to assess the effects of the transmissivity field on the drawdown conditions. The specific conditions under which aquifer properties could be identified using traditional methods of analysis were also evaluated. The second objective was achieved by implementing the inverse solution algorithm (ISA), which was developed for petroleum reservoirs to groundwater aquifer settings. The results showed that the drawdown values are influenced by a volumetric integral of a weighting function and the transmissivity field within the cone of depression. The weighting function migrates in tandem with the expanding cone of depression. The ability of the ISA to predict radially symmetric and log‐normally distributed transmissivity fields was assessed against analytical and numerical benchmarks. The results of this investigation indicated that the ISA method is a viable technique for evaluating the radial transmissivity variations of heterogeneous aquifer settings. Copyright © 2013 John Wiley & Sons, Ltd.