Bias and uncertainty in regression-calibrated models of groundwater flow in heterogeneous media

Groundwater models need to account for detailed but generally unknown spatial variability (heterogeneity) of the hydrogeologic model inputs. To address this problem we replace the large, m-dimensional stochastic vector β that reflects both small and large scales of heterogeneity in the inputs by a lumped or smoothed m-dimensional approximation γθ_∗, where γ is an interpolation matrix and θ_∗ is a stochastic vector of parameters. Vector θ_∗ has small enough dimension to allow its estimation with the available data. The consequence of the replacement is that model function f(γθ_∗) written in terms of the approximate inputs is in error with respect to the same model function written in terms of β, f(β), which is assumed to be nearly exact. The difference f(β) − f(γθ_∗), termed model error, is spatially correlated, generates prediction biases, and causes standard confidence and prediction intervals to be too small. Model error is accounted for in the weighted nonlinear regression methodology developed to estimate θ_∗ and assess model uncertainties by incorporating the second-moment matrix of the model errors into the weight matrix. Techniques developed by statisticians to analyze classical nonlinear regression methods are extended to analyze the revised method. The analysis develops analytical expressions for bias terms reflecting the interaction of model nonlinearity and model error, for correction factors needed to adjust the sizes of confidence and prediction intervals for this interaction, and for correction factors needed to adjust the sizes of confidence and prediction intervals for possible use of a diagonal weight matrix in place of the correct one. If terms expressing the degree of intrinsic nonlinearity for f(β) and f(γθ_∗) are small, then most of the biases are small and the correction factors are reduced in magnitude. Biases, correction factors, and confidence and prediction intervals were obtained for a test problem for which model error is large to test robustness of the methodology. Numerical results conform with the theoretical analysis.     

Practical Postcalibration Uncertainty Analysis: Yucca Mountain, Nevada

The values of parameters in a groundwater flow model govern the precision of predictions of future system behavior. Predictive precision, thus, typically depends on an ability to infer values of system properties from historical measurements through calibration. When such data are scarce, or when their information content with respect to parameters that are most relevant to predictions of interest is weak, predictive uncertainty may be high, even if the model is "calibrated." Recent advances help recognize this condition, quantitatively evaluate predictive uncertainty, and suggest a path toward improved predictive accuracy by identifying sources of predictive uncertainty and by determining what observations will most effectively reduce this uncertainty. We demonstrate linear and nonlinear predictive error/uncertainty analyses as applied to a groundwater flow model of Yucca Mountain, Nevada, the United States' proposed site for disposal of high-level radioactive waste. Linear and nonlinear uncertainty analyses are readily implemented as an adjunct to model calibration with medium to high parameterization density. Linear analysis yields contributions made by each parameter to a prediction's uncertainty and the worth of different observations, both existing and yet-to-be-gathered, toward reducing this uncertainty. Nonlinear analysis provides more accurate characterization of the uncertainty of model predictions while yielding their (approximate) probability distribution functions. This article applies the above methods to a prediction of specific discharge and confirms the uncertainty bounds on specific discharge supplied in the Yucca Mountain Project License Application.     

Mapping groundwater quality in the Netherlands

Maps of 25 groundwater quality variables were obtained by estimating 4 km × 4 km block median concentrations. Estimates were presented as approximate 95% confidence intervals related to four concentration levels mostly obtained from critical levels for human consumption. These maps were based on measurements from 425 monitoring sites of national and provincial groundwater quality monitoring networks. The estimation procedure was based on a stratification by soil type and land use. Within each soil-land use category, measurements were interpolated. Spatial dependence between measurements and regional differences in mean level were taken into account. Stratification turned out to be essential: no or partial stratification (using either soil type or land use) results in essentially different maps. The effect of monitoring network density was studied by leaving out the 173 monitoring sites of the provincial monitoring networks. Important changes in resulting maps were assigned to loss of information on short-distance variation, as well as loss of location-specific information. For 12 variables, maps of changes in groundwater quality were made by spatial interpolation of short-term predictions calculated for each well screen from time series of yearly measurements over 5–7 years, using a simple regression model for variation over time and taking location-specific time-prediction uncertainties into account.

From a policy point of view, the resulting maps can be used either for quantifying diffuse groundwater contamination and location-specific background concentrations (in order to assist local contamination assessment) or for input and validation of policy supporting regional or national groundwater quality models. The maps can be considered as a translation of point information obtained from the monitoring networks into information on spatial units, the size of which is used in regional groundwater models. The maps enable location-specific network optimization. In general, the maps give little reason for reducing the monitoring network density (wide confidence intervals).     

Estimating streambed parameters for a disconnected river

Evaluation of stream–aquifer interaction and water balance for a catchment often requires specific information on streambed parameters, such as streambed hydraulic conductivity, seepage flux across the streambed and so on. This paper describes a simple, inexpensive instrument that is used to measure these streambed parameters under the condition of a stream disconnected from groundwater. Our method includes a seepage cylinder for simulation of river water depth. The proposed method was applied to estimate the vertical hydraulic conductivity of a streambed and the changes in vertical seepage rate from stream to groundwater with varied stream water depth in the Manasi River of Xinjiang Uygur Autonomous Region, China. The vertical hydraulic conductivities of the streambed determined from 12 sites along the Manasi River vary from 1.01 to 29.m/day where the stream disconnects from the groundwater. The experimental results suggest that there are two kinds of relations between the vertical seepage rate and the simulated stream water depth. One is a linear relation between the two variables with low Reynolds numbers (less than 10); the other is a nonlinear relation (exponential relation) between the two variables with larger Reynolds numbers (greater than 10). This second relationship is quite different from the traditional model that usually calculates the vertical seepage rate from stream to groundwater under the condition of disconnection using a linear relation (Darcy's Law). Our results suggest that a linear relation can only be used for a limited range of river water depth. This method gives a convenient tool for rapidly estimating the streambed hydraulic conductivity and the changes in the vertical seepage rate across streambed with varied stream water depths for the case of a stream disconnected from groundwater. Copyright © 2013 John Wiley & Sons, Ltd.     

SOME IMPROVEMENTS IN THE INTERPRETATION OF VERTICAL ELECTRIC SOUNDING CURVES*

The process of VES interpretation is discussed, including the following points. (a) Preliminary interpretation by means of master curves. It is shown that the positions of the auxiliary points K and Q depend on the resistivity of the substratum. The interpretation is improved if the auxiliary points are determined separately for each master curve. (b) The individual parts of the measured curves are shifted in overlapping MN electrode positions so that the total sum of squares of the shifts is minimal. (c) The ambiguity may be reduced by means of supplementary information or assumptions on the resistivities. Fixing the resistivities is not always possible because discrepancies may arise between the ground measurement and the well-logging data. The simultaneous interpretation of several VES curves is recommended assuming constant resistivities. This assumption may be subsequently verified by means of the F-test. (d) A nonlinear algorithm is proposed for the determination of confidence intervals. As the multi-dimensional confidence intervals are often very complicated, it is recommended to construct only one-dimensional confidence intervals for the estimable parametric functions. (e) A ‘double-least-squares’ optimization technique is presented. The optimization is performed on the estimable parametric functions, and the individual parameters are determined so that the solution remains near the initial guess. This technique is faster than the singular value decomposition.     

On estimating the hydraulic properties of soil,Part 1. Comparison between forms to estimate the soil-water characteristic function

The hydraulic properties of soil include the soil-water characteristic function [h(θ), in which θ is water content and h is pressure head (suction)], and the hydraulic conductivity function [K(θ) or K(h)]. These functions are essential to the solution of unsaturated groundwater flow problems. A number of empirical and semiempirical forms have been proposed in the literature to estimate these functions. The present paper employs a nonlinear least-square analysis for comparison between some of the available forms, using a large number of experimental measurements of h(θ) for different classes of soil. Suitability of the forms for predicting the hydraulic conductivity function is examined. In the absence of accurate measurements, the paper facilitates modeling by providing estimates for the parameters of the soil-water characteristic function.     

Mutual information analysis to approach nonlinearity in groundwater stochastic fields

In heterogeneous porous media, transmissivity can be regarded as a spatial stochastic variable. Transmissivity fluctuations induce stochasticity in the groundwater velocity field and transport features. In order to model subsurface phenomena, it is important to understand the relationships that exist between the variables that characterize flow and transport. Linear relationships are easier to deal with. Nevertheless, it is well known that flow and transport variables exhibit interdependences that become more and more nonlinear as the heterogeneity increases. The aim of this work is to draw attention to the information contained in nonlinear linkages, and to show that it can be of great relevance with respect to the linear information content. Information theory tools are proposed to detect the presence of nonlinear components. By comparing the cross-covariance function and mutual information, the amount of linear linkage is compared with nonlinear linkage. In order to avoid analytical approximations, data from Monte Carlo simulations of heterogeneous transmissivity fields have been considered in the analysis. The obtained results show that the presence of nonlinear components can be relevant, even when the cross-covariance values are nil.     

基于精确Zoeppritz方程三变量柯西分布先验约束的广义线性AVO反演

常规AVO三参数反演是通过Zoeppritz方程的近似公式来建立AVO正演模拟的过程,然而在P波入射角过临界角和弹性参数在纵向上变化剧烈的情况下,Zoeppritz方程近似公式精度有限.针对这种情况,可以使用精确的Zoeppritz方程来构建反演目标函数,由于精确Zoeppritz方程中P波反射系数和弹性参数之间是一种复杂的非线性关系,通常解决途径是利用非线性的优化算法来进行数值计算,但是非线性优化算法的缺点是计算量过大;另外一种途径是利用广义线性反演的方法,通过泰勒一阶展开式将P波反射振幅展开后,用线性关系近似表达非线性关系,经过几次迭代后,在理论上可以达到很高的精度,但是广义线性反演算法的核心部分--Jacobian矩阵由于矩阵条件数过大,往往会造成反演算法的不稳定,其应用范围得到了限制.贝叶斯反演方法是通过引入模型参数的先验分布结合噪声的似然函数,生成模型参数的后验分布,通过求取模型参数的最大后验概率分布来得到模型参数的反演解,由于引入模型参数的先验分布信息,可以有效的降低反演的不适定问题.本文将两种反演算法的思想相结合,利用广义线性反演算法的思想,构建AVO正演模拟的过程来提高大角度地震数据反演的精度,同时结合贝叶斯理论,通过引入模型参数的先验分布信息构建反演目标函数的正则化项,可以有效降低由于Jacob矩阵条件数过大带来的反演不适定问题,该算法假设模型参数服从三变量柯西分布.     

Rehabilitation and replacement of water distribution system components considering uncertainties

This paper presents the mathematical development of an integer — nonlinear programming chance — constrained optimization model for the minimum cost rehabilitation/replacement of water distribution system components. Particular attention is given to the handling of uncertainties in the roughness factors and the loading conditions including both the random demand and preassure head requirements.The advantages of the proposed model include the ability to: 1) handle the optimal timing of rehabilitation/replacement for water distribution system components; 2) link a mixed-integer linear program solver, a nonlinear program solver, and a hydraulic simulator into an optimization framework; 3) handle the uncertainties of some of the variables; 4) incorporate various kinds of cost functions; and 5) handle multiple loading conditions.     

Rehabilitation and replacement of water distribution system components considering uncertainties

This paper presents the mathematical development of an integer — nonlinear programming chance — constrained optimization model for the minimum cost rehabilitation/replacement of water distribution system components. Particular attention is given to the handling of uncertainties in the roughness factors and the loading conditions including both the random demand and preassure head requirements.The advantages of the proposed model include the ability to: 1) handle the optimal timing of rehabilitation/replacement for water distribution system components; 2) link a mixed-integer linear program solver, a nonlinear program solver, and a hydraulic simulator into an optimization framework; 3) handle the uncertainties of some of the variables; 4) incorporate various kinds of cost functions; and 5) handle multiple loading conditions.     

Groundwater recharge estimation in arid hardrock‐alluvium aquifers using combined water‐table fluctuation and groundwater balance approaches

This paper proposes an approach to estimate groundwater recharge using an optimization‐based water‐table fluctuation method combined with a groundwater balance model in an arid hardrock‐alluvium region, located at the Oman–United Arab Emirates border. We introduce an “effective hardrock thickness” term to identify the percentage of the considered hardrock thickness in which effective groundwater flow takes place. The proposed method is based upon a Thiessen polygon zoning approach. The method includes subpolygons to represent specific geologic units and to enhance the confidence of the estimated groundwater recharge. Two linear and 1 nonlinear submodels were developed to evaluate the model components for the calibration (October 1996 to September 2008) and validation (October 2008 to September 2013) periods. Long‐term annual groundwater recharge from rainfall and return flow over the model domain are estimated as 24.62 and 5.71 Mm³, respectively, while the effective groundwater flow circulation is found to occur in the upper 7% of the known hardrock thickness (42 m), confirming conclusions of previous field studies. Considering a total difference in groundwater levels between eastern and western points of the study area of the order of 220 m and a 12‐year monthly calibration period, a weighted root mean squared error in predicted groundwater elevation of 2.75 m is considered quite reasonable for the study area characterized by remarkable geological and hydrogeological diversity. The proposed approach provides an efficient and robust method to estimate groundwater recharge in regions with a complex geological setting in which interaction between fractured and porous media cannot be easily assessed.     

Anisotropy and polymodal distribution of IMF fluctuations

The IMF statistical characteristics, depending on duration of the averaging intervals, are studied based on the ACE spacecraft measurements. The distributions of the induction (B) vector directions and the estimates of the variance and excess coefficient of the vector components are presented. The polymodal model of the density distribution function of the vector (B) component variations is proposed. The parameters of this model, which make it possible to approximate empirical distributions accurate within several hundredths of percent, are presented.     

Estimation of time dependent virus inactivation rates by geostatistical and resampling techniques: application to virus transport in porous media

A methodology is developed for estimating temporally variable virus inactivation rate coefficients from experimental virus inactivation data. The methodology consists of a technique for slope estimation of normalized virus inactivation data in conjunction with a resampling parameter estimation procedure. The slope estimation technique is based on a relatively flexible geostatistical method known as universal kriging. Drift coefficients are obtained by nonlinear fitting of bootstrap samples and the corresponding confidence intervals are obtained by bootstrap percentiles. The proposed methodology yields more accurate time dependent virus inactivation rate coefficients than those estimated by fitting virus inactivation data to a first-order inactivation model. The methodology is successfully applied to a set of poliovirus batch inactivation data. Furthermore, the importance of accurate inactivation rate coefficient determination on virus transport in water saturated porous media is demonstrated with model simulations.     

Using informative priors in facies inversion: The case of C-ISR method

Inverse problems involving the characterization of hydraulic properties of groundwater flow systems by conditioning on observations of the state variables are mathematically ill-posed because they have multiple solutions and are sensitive to small changes in the data. In the framework of McMC methods for nonlinear optimization and under an iterative spatial resampling transition kernel, we present an algorithm for narrowing the prior and thus producing improved proposal realizations. To achieve this goal, we cosimulate the facies distribution conditionally to facies observations and normal scores transformed hydrologic response measurements, assuming a linear coregionalization model. The approach works by creating an importance sampling effect that steers the process to selected areas of the prior. The effectiveness of our approach is demonstrated by an example application on a synthetic underdetermined inverse problem in aquifer characterization.     

A stepwise-cluster forecasting approach for monthly streamflows based on climate teleconnections

In this study, a stepwise cluster forecasting (SCF) framework is proposed for monthly streamflow prediction in Xiangxi River, China. The developed SCF method can capture discrete and nonlinear relationships between explanatory and response variables. Cluster trees are generated through the SCF method to reflect complex relationships between independent (i.e. explanatory) and dependent (i.e. response) variables in the hydrologic system without determining specific linear/nonlinear functions. The developed SCF method is applied for monthly streamflow prediction in Xiangxi River based on the local meteorological records as well as some climate index. Comparison among SCF, multiple linear regression, generalized regression neural network, and least square support vector machine methods would be conducted. The results indicate that the SCF method would produce good predictions in both training and testing periods. Besides, the inherent probabilistic characteristics of the SCF predictions are further analyzed. The results obtained by SCF can presented as intervals, formulated by the minimum and maximum predictions as well as the 5 and 95 % percentile values of the predictions, which can reflect the variations in streamflow forecasts. Therefore, the developed SCF method can be applied for monthly streamflow prediction in various watersheds with complicated hydrologic processes.     

Assessment of NMR Logging for Estimating Hydraulic Conductivity in Glacial Aquifers

Glacial aquifers are an important source of groundwater in the United States and require accurate characterization to make informed management decisions. One parameter that is crucial for understanding the movement of groundwater is hydraulic conductivity, K. Nuclear magnetic resonance (NMR) logging measures the NMR response associated with the water in geological materials. By utilizing an external magnetic field to manipulate the nuclear spins associated with ¹H, the time-varying decay of the nuclear magnetization is measured. This logging method could provide an effective way to estimate K at submeter vertical resolution, but the models that relate NMR measurements to K require calibration. At two field sites in a glacial aquifer in central Wisconsin, we collected a total of four NMR logs and obtained measurements of K in their immediate vicinity with a direct-push permeameter (DPP). Using a bootstrap algorithm to calibrate the Schlumberger-Doll Research (SDR) NMR-K model, we estimated K to within a factor of 5 of the DPP measurements. The lowest levels of accuracy occurred in the lower-K (K < 10⁻⁴ m/s) intervals. We also evaluated the applicability of prior SDR model calibrations. We found the NMR calibration parameters varied with K, suggesting the SDR model does not incorporate all the properties of the pore space that control K. Thus, the expected range of K in an aquifer may need to be considered during calibration of NMR-K models. This study is the first step toward establishing NMR logging as an effective method for estimating K in glacial aquifers.     

Solving nonlinear water management models using a combined genetic algorithm and linear programming approach

Gradient-based nonlinear programming (NLP) methods can solve problems with smooth nonlinear objectives and constraints. However, in large and highly nonlinear models, these algorithms can fail to find feasible solutions, or converge to local solutions which are not global. Evolutionary search procedures in general, and genetic algorithms (GAs) specifically, are less susceptible to the presence of local solutions. However, they often exhibit slow convergence, especially when there are many variables, and have problems finding feasible solutions in constrained problems with “narrow” feasible regions. In this paper, we describe strategies for solving large nonlinear water resources models management, which combine GAs with linear programming. The key idea is to identify a set of complicating variables in the model which, when fixed, render the problem linear in the remaining variables. The complicating variables are then varied by a GA. This GA&LP approach is applied to two nonlinear models: a reservoir operation model with nonlinear hydropower generation equations and nonlinear reservoir topologic equations, and a long-term dynamic river basin planning model with a large number of nonlinear relationships. For smaller instances of the reservoir model, the CONOPT2 nonlinear solver is more accurate and faster, but for larger instances, the GA&LP approach finds solutions with significantly better objective values. The multiperiod river basin model is much too large to be solved in its entirety. The complicating variables are chosen here so that, when they are fixed, each period's model is linear, and these models can be solved sequentially. This approach allows sufficient model detail to be retained so that long-term sustainability issues can be explored.     

Bivariate drought frequency curves and confidence intervals: a case study using monthly rainfall generation

Although water resources management practices recently use bivariate distribution functions to assess drought severity and its frequency, the lack of systematic measurements is the major hindrance in achieving quantitative results. This study aims to suggest a statistical scheme for the bivariate drought frequency analysis to provide comprehensive and consistent drought severities using observed rainfalls and their uncertainty using synthesized rainfalls. First, this study developed a multi-variate regression model to generate synthetic monthly rainfalls using climate variables as causative variables. The causative variables were generated to preserve their correlations using copula functions. This study then focused on constructing bivariate drought frequency curves using bivariate kernel functions and estimating their confidence intervals from 1,000 likely replica sets of drought frequency curves. The confidence intervals achieved in this study may be useful for making a comprehensive drought management plan through providing feasible ranges of drought severity.