Multi-Objective Optimization of Managed Aquifer Recharge

This study demonstrates the utilization of a multi-objective hybrid global/local optimization algorithm for solving managed aquifer recharge (MAR) design problems, in which the decision variables included spatial arrangement of water injection and abstraction wells and time-variant rates of pumping and injection. The objective of the optimization was to maximize the efficiency of the MAR scheme, which includes both quantitative and qualitative aspects. The case study used to demonstrate the capabilities of the proposed approach is based on a published report on designing a real MAR site with defined aquifer properties, chemical groundwater characteristics as well as quality and volumes of injected water. The demonstration problems include steady state and transient scenarios. The steady state scenario demonstrates optimization of spatial arrangement of multiple injection and recovery wells, whereas the transient scenario was developed with the purpose of finding optimal regimes of water injection and recovery at a single location. Both problems were defined as multi-objective problems. The scenarios were simulated by applying coupled numerical groundwater flow and solute transport models: MODFLOW-2005 and MT3D-USGS. The applied optimization method was a combination of global (the non-dominated sorting genetic algorithm [NSGA-2]) and local (the Nelder-Mead downhill simplex search algorithms). The analysis of the resulting Pareto optimal solutions led to the discovery of valuable patterns and dependencies between the decision variables, model properties, and problem objectives. Additionally, the performance of the traditional global and the hybrid optimization schemes were compared.     

基于多目标优化的有限角度图像重建算法及实现

本文综合考虑了图像局部和整体的平滑性,以及最大熵准则,重新建立新的多目标优化模型,并在求解过程中引入同伦参数加快收敛速度。通过仿真实验,验证了所提算法能较好地改善有限角度下图像重建的质量及提高重建速度。     

Shuffled Complex Evolution model calibrating algorithm: enhancing its robustness and efficiency

Shuffled Complex Evolution—University of Arizona (SCE‐UA) has been used extensively and proved to be a robust and efficient global optimization method for the calibration of conceptual models. In this paper, two enhancements to the SCE‐UA algorithm are proposed, one to improve its exploration and another to improve its exploitation of the search space. A strategically located initial population is used to improve the exploration capability and a modification to the downhill simplex search method enhances its exploitation capability. This enhanced version of SCE‐UA is tested, first on a suite of test functions and then on a conceptual rainfall‐runoff model using synthetically generated runoff values. It is observed that the strategically located initial population drastically reduces the number of failures and the modified simplex search also leads to a significant reduction in the number of function evaluations to reach the global optimum, when compared with the original SCE‐UA. Thus, the two enhancements significantly improve the robustness and efficiency of the SCE‐UA model calibrating algorithm. Copyright © 2008 John Wiley & Sons, Ltd.     

改进的模拟退火－单纯形综合反演方法

实际中的大量地球物理反演是一个多参数、非线性优化问题,所采用的目标函数,即度量由参数化的理论模型得出的预测值与观测值的吻合程度,往往具有多个局部极值．针对这类问题,本文综合全局反演方法具有的全域搜索能力强、局部方法收敛速度快和“均匀设计”布点效率高的特点,提出了模拟退火－单纯性综合反演方法,并通过一维声波非线性反演验证了这种综合方法的搜索能力和效率．     

改进的模拟退火－单纯形综合反演方法

实际中的大量地球物理反演是一个多参数、非线性优化问题,所采用的目标函数,即度量由参数化的理论模型得出的预测值与观测值的吻合程度,往往具有多个局部极值．针对这类问题,本文综合全局反演方法具有的全域搜索能力强、局部方法收敛速度快和“均匀设计”布点效率高的特点,提出了模拟退火－单纯性综合反演方法,并通过一维声波非线性反演验证了这种综合方法的搜索能力和效率．     

Comparison of derivative-free optimization methods for groundwater supply and hydraulic capture community problems

Management decisions involving groundwater supply and remediation often rely on optimization techniques to determine an effective strategy. We introduce several derivative-free sampling methods for solving constrained optimization problems that have not yet been considered in this field, and we include a genetic algorithm for completeness. Two well-documented community problems are used for illustration purposes: a groundwater supply problem and a hydraulic capture problem. The community problems were found to be challenging applications due to the objective functions being nonsmooth, nonlinear, and having many local minima. Because the results were found to be sensitive to initial iterates for some methods, guidance is provided in selecting initial iterates for these problems that improve the likelihood of achieving significant reductions in the objective function to be minimized. In addition, we suggest some potentially fruitful areas for future research.     

Calibration of subsurface batch and reactive-transport models involving complex biogeochemical processes

In this study, the calibration of subsurface batch and reactive-transport models involving complex biogeochemical processes was systematically evaluated. Two hypothetical nitrate biodegradation scenarios were developed and simulated in numerical experiments to evaluate the performance of three calibration search procedures: a multi-start non-linear regression algorithm (i.e. multi-start Levenberg–Marquardt), a global search heuristic (i.e. particle swarm optimization), and a hybrid algorithm that combines the particle swarm procedure with a regression-based “polishing” step. Graphical analysis of the selected calibration problems revealed heterogeneous regions of extreme parameter sensitivity and insensitivity along with abundant numbers of local minima. These characteristics hindered the performance of the multi-start non-linear regression technique, which was generally the least effective of the considered algorithms. In most cases, the global search and hybrid methods were capable of producing improved model fits at comparable computational expense. In other cases, the multi-start and hybrid calibration algorithms yielded comparable fitness values but markedly differing parameter estimates and associated uncertainty measures.     

电测深曲线的遗传算法反演

电测深曲线作为地下介质电阻率和深度的非线性函数，其解具有高度的非唯一性．常规的基于局部线性化的最优化反演方法易使解估计陷入局部极大值中，而且严重地依赖初始模型的选择．遗传算法作为一种全局最优化方法，对初始模型的依赖性大为减弱，且不易陷人局部极大值之中，从而能有效地解决这类非线性最优化问题．本文首次将遗传算法用于电测深解释并对实测曲线进行反演，效果很好，显示了遗传算法独特的优越性．     

SCE-UA方法在新安江模型参数优化中的应用

以前在使用新安江模型时人们遇到的最大困难可归因于缺乏有效的参数全局优化的数学方法,事实上对于一个缺乏经验的人来说,模型参数的人工试错计算的过程是一个相当不容易的过程,并且耗时颇多,为此,近些年来研究者们正在探索把概念性水文模型中的专家经验与自动优化计算相结合的方法或者数学优化中的全局优化方法,如SEC-UA方法,本文首先简述新安江模型,而后采用3个大小和气候条件各不相同的流域对SCE-UA算法就在新安江模型计算的参数优化进行了研究,研究结果表明,SCE-UA算法用来进行新安江模型的参数优化所取得的效果是好的,从率定和检验的结果来看,SCE-UA算法可以使得率定的新安江模型的参数达到全局最优并且从概念上也合理。     

实数编码遗传算法在大地电磁测深二维反演中的应用

作为全局非线性优化的新方法之一的遗传算法,近年来已从生物工程流行到大地电磁测深资料解释中.然而,大地电磁反演问题具有不适定性,解的非唯一性.通过结合求解不适定问题的Tikhonov正则化方法,本文采用实数编码遗传算法求解大地电磁二维反演问题.此算法在构建目标函数时引入正则化的思想,利用遗传算法求解最优化问题.常规的基于局部线性化的最优化反演方法易使解陷入局部极小值,而且严重的依赖初始模型的选择.与传统线性化的迭代反演方法相比,实数编码遗传算法能够克服传统方法的不足且能获得更好的反演结果.通过对大地电磁测深理论模型进行计算,结果表明:该算法具有收敛速度快、解的精度高和避免出现早熟等优点,可用于大地电磁资料解释.     

电测深曲线的遗传算法反演

电测深曲线作为地下介质电阻率和深度的非线性函数，其解具有高度的非唯一性．常规的基于局部线性化的最优化反演方法易使解估计陷入局部极大值中，而且严重地依赖初始模型的选择．遗传算法作为一种全局最优化方法，对初始模型的依赖性大为减弱，且不易陷人局部极大值之中，从而能有效地解决这类非线性最优化问题．本文首次将遗传算法用于电测深解释并对实测曲线进行反演，效果很好，显示了遗传算法独特的优越性．     

A multiscale method for subsurface inverse modeling: Single-phase transient flow

High-resolution geologic models that incorporate observed state data are expected to effectively enhance the reliability of reservoir performance prediction. One of the major challenges faced is how to solve the large-scale inverse modeling problem, i.e., to infer high-resolution models from the given observations of state variables that are related to the model parameters according to some known physical rules, e.g., the flow and transport partial differential equations. There are typically two difficulties, one is the high-dimensional problem and the other is the inverse problem. A multiscale inverse method is presented in this work to attack these problems with the aid of a gradient-based optimization algorithm. In this method, the model responses (i.e., the simulated state data) can be efficiently computed from the high-resolution model using the multiscale finite-volume method. The mismatch between the observations and the multiscale solutions is then used to define a proper objective function, and the fine-scale sensitivity coefficients (i.e., the derivatives of the objective function with respect to each node’s attribute) are computed by a multiscale adjoint method for subsequent optimization. The difficult high-dimensional optimization problem is reduced to a one-dimensional one using the gradient-based gradual deformation method. A synthetic single-phase transient flow example problem is employed to illustrate the proposed method. Results demonstrate that the multiscale framework presented is not only computationally efficient but also can generate geologically consistent models. By preserving spatial structure for inverse modeling, the method presented overcomes the artifacts introduced by the multiscale simulation and may enhance the prediction ability of the inverse-conditional realizations generated.     

波动方程反演的全局优化方法研究

复杂介质波动方程反演是地球物理研究中的重要问题,通常表述为特定目标函数最优化,难点是多参数、非线性和不适定性.局部和全局优化方法都不能实现快速全局优化.本文概述了地震波勘探反演问题的理论基础和研究进展,阐述了反演中优化问题的解决方法和面临的困难,并提出了一种确定性全局优化的新方法.通过在优化参数空间识别并划分局部优化解及其附近区域,只需有限次参数空间划分过程就能发现所有局部解（集合）;基于复杂目标函数多尺度结构分析,提出多尺度参数空间分区优化方法的研究方向.该方法收敛速度快,优化结果不依赖初始解的选取,是对非线性全局优化问题的一个新探索.     

Sensitivity Kernel for the Weighted Norm of the Frequency-Dependent Phase Correlation

Full-3D waveform tomography (F3DT) is often formulated as an optimization problem, in which an objective function defined in terms of the misfit between observed and model-predicted (i.e., synthetic) waveforms is minimized by varying the earth structure model from which the synthetic waveforms are calculated. Because of the large dimension of the model space and the computational cost for solving the 3D seismic wave equation, it is often mandatory to use Newton-type local optimization algorithms; in which case, spurious local optima in the objective function can prevent the global convergence of the descent algorithm if the initial estimate of the structure model is not close enough to the global optimum. By appropriate design of the objective function, it is possible to enlarge the attraction domain of the global optimum so that Newton-type local optimization algorithms can achieve global convergence. In this article, an objective function based on a weighted L ₂ norm of the frequency-dependent phase correlation between observed and synthetic waveforms is proposed and studied, and its full-3D Fréchet kernel is constructed using the adjoint state method. The relation between the proposed objective function and the conventional frequency-dependent group-delay is analyzed and illustrated using numerical examples. The methodology has been successfully applied on a set of ambient-noise Green’s function observations collected in northern California to derive a full-3D crustal structure model.     

Combination of FEM and CMA‐ES algorithm for transmissivity identification in aquifer systems

In this paper, we propose a coupling of a finite element model with a metaheuristic optimization algorithm for solving the inverse problem in groundwater flow (Darcy's equations). This coupling performed in 2 phases is based on the combination of 2 codes: This is the HySubF‐FEM code (hydrodynamic of subsurface flow by finite element method) used for the first phase allowing the calculation of the flow and the CMA‐ES code (covariance matrix adaptation evolution strategy) adopted in the second phase for the optimization process. The combination of these 2 codes was implemented to identify the transmissivity field of groundwater by knowing the hydraulic head in some point of the studied domain. The integrated optimization algorithm HySubF‐FEM/CMA‐ES has been validated successfully on a schematic case offering an analytical solution. As realistic application, the integrated optimization algorithm HySubF‐FEM/CMA‐ES was applied to a complex groundwater in the north of France to identify the transmissivity field. This application does not use zonation techniques but solves an optimization problem at each internal node of the mesh. The obtained results are considered excellent with high accuracy and fully consistent with the hydrogeological characteristics of the studied aquifer.However, the various numerical simulations performed in this paper have shown that the CMA‐ES algorithm is time‐consuming. Finally, the paper concludes that the proposed algorithm can be considered as an efficient tool for solving inverse problems in groundwater flow.     

一种新的地球物理反演方法——模拟原子跃迁反演法

详细研究了一般地球物理反问题的迭代优化求解过程与物理学中原子跃迁过程的对应关系,建立了反演问题中模型空间、初始模型、局部极值模型、最优化模型等与原子的态空间、定态、激发态、基态等的对应关系. 在此基础上,模拟了物理学中原子从激发态向基态跃迁的物理过程,建立了一种与原子跃迁过程相对应的非线性随机跃迁数学模型和模型解跃迁搜索准则,导出了适用于一般地球物理资料的模拟原子跃迁的非线性反演算法. 用理论测试函数对这种新的反演方法进行了数值试验,结果表明该方法具有解不依赖于初始模型、收敛速度快等优点.     

Parameter Estimation for Groundwater Models under Uncertain Irrigation Data

The success of modeling groundwater is strongly influenced by the accuracy of the model parameters that are used to characterize the subsurface system. However, the presence of uncertainty and possibly bias in groundwater model source/sink terms may lead to biased estimates of model parameters and model predictions when the standard regression‐based inverse modeling techniques are used. This study first quantifies the levels of bias in groundwater model parameters and predictions due to the presence of errors in irrigation data. Then, a new inverse modeling technique called input uncertainty weighted least‐squares (IUWLS) is presented for unbiased estimation of the parameters when pumping and other source/sink data are uncertain. The approach uses the concept of generalized least‐squares method with the weight of the objective function depending on the level of pumping uncertainty and iteratively adjusted during the parameter optimization process. We have conducted both analytical and numerical experiments, using irrigation pumping data from the Republican River Basin in Nebraska, to evaluate the performance of ordinary least‐squares (OLS) and IUWLS calibration methods under different levels of uncertainty of irrigation data and calibration conditions. The result from the OLS method shows the presence of statistically significant (p < 0.05) bias in estimated parameters and model predictions that persist despite calibrating the models to different calibration data and sample sizes. However, by directly accounting for the irrigation pumping uncertainties during the calibration procedures, the proposed IUWLS is able to minimize the bias effectively without adding significant computational burden to the calibration processes.     

An efficiency-improved genetic algorithm and its application on multimodal functions and a 2D common reflection surface stacking problem

Although Genetic Algorithms have found many successful applications in the field of exploration geophysics, the convergence speed remains a big challenge as Genetic Algorithms usually require a huge amount of fitness function evaluations. In this paper, we propose an efficiency-improved Genetic Algorithm, which has both a good global search capability and a good local search capability, and is also capable of robustly handling the premature convergence challenge commonly seen in linear and directed non-linear optimization methods. In our new genetic algorithm, the global search capability is performed via a modified island model, while the local search capability is provided by a novel self-adaptive differential evolution fine tuning scheme. Premature convergence is dealt with via a local exhaustive search method. We first demonstrate the much improved convergence speed of this efficiency-improved Genetic Algorithm over that of our previously proposed advanced Genetic Algorithm on several multimodal functions. We further demonstrate the effectiveness of our efficiency-improved Genetic Algorithm by applying it to a two-dimensional common reflection surface stacking problem, which is a highly nonlinear geophysical optimization problem, to obtain very encouraging results.