混合式優化理論應用於耦合地下水流與溶質傳輸數值模式參數率定

Hybrid Optimization Scheme for the Parameter Identification of Coupled Groundwater Flow and Solute Transport Modeling

邱永嘉(Yung-Chia Chiu)

《土壤及地下水污染整治》 3卷4期 (2016/10) Pp. 291-304

https://doi.org/10.6499/JSGR.2016.0304.004

摘要

在土壤與地下水污染調查與整治上，受限於人、物力及經費上之限制，藉由地下水數值模式進行初步評估及整治策略的可行性分析已扮演舉足輕重的角色，對於高準確性、高效率的數值模式仰賴程度也與日遽增。欲建置準確且適用於現地環境中之數值模式，模式參數率定的過程及方法為模式建置過程中的關鍵一環。由於地下水流與溶質傳輸數值模式的非線性特性，利用傳統梯度法 (gradient search) 進行模式參數率定時，在求解過程中最佳解容易受困於局部最佳解 (local optimum)；利用非傳統的啟發式演算法 (heuristic algorithm) 時，又容易遭遇到率定效率過低，耗時太長的問題。因此，本研究提出了近年來廣受重視的混合式優化概念 (hybrid optimization scheme)，進行耦合地下水流與溶質傳輸數值模式參數率定。混合式優化方法結合了啟發式演算法與梯度法的優點，可大幅改善運算求解收斂的速度，並找尋到全域最佳解 (global optimum)。本研究以一個二維耦合地下水流與溶質傳輸數值模式為假想案例，利用混合式優化方法進行模式參數率定，並針對率定結果進行分析與比較。率定結果顯示，混合式優化方法不僅可準確的推估水文地質與溶質傳輸參數分布，且求解運算效率高，優於單獨使用梯度法或啟發式演算法的結果。藉由本研究之成果，未來可將混合式優化方法應用於實際的土壤與地下水污染整治場址，提升現地水文地質概況與污染物分布與傳輸情形的掌握，以作為後續整治策略評估上之參考依據。

關鍵字

地下水數值模擬 ； 參數率定 ； 梯度法 ； 啟發式演算法 ； 混合式優化方法

並列摘要

1. Introduction The use of a numerical model of coupled groundwater flow and solute transport is important when investigating and remedying a site with contaminated soil and groundwater. The development of an accurate and robust numerical model relies on an effective and efficient process of parameter identification (or model calibration). The problem of parameter identification, also called the inverse problem, in distributed parameter systems has been studied extensively over the last five decades. The inverse problem of parameter identification concerns the optimal determination of the parameters by observing the dependent variables collected in the spatial and time domains. However, the inverse problem is often illposed, and generally characterized by non-uniqueness and instability. Traditionally, the determination of aquifer parameter values is based on gradient-based search algorithms, although the resulting solution is easily trapped in a local optimum because of its strong nonconvexity. In order to overcome the difficulties related to gradient-based algorithms, a variety of heuristic algorithms have recently been considered. However, when the number of parameters is large, such heuristic algorithms become very ineffective. More recently an innovative approach has been developed to find the optimal solution of the inverse problem in groundwater modeling, called a hybrid optimization scheme. 2. Theory of Inverse Problem In parameter identification, the objective function to be minimized is a weighted least squares error (LSE), i.e., specifically, a weighted sum of the squared deviations between the model output and observations over the spatial and temporal domains where observations have been made. The sensitivity and parameter uncertainty analyses are conducted to avoid over-parameterization. In general, an increase in the parameter dimension (i.e., the number of parameters) will decrease the LSE, but this is accompanied by an increase in parameter uncertainty which is equivalent to a decrease in the reliability of the identified parameters. A hybrid optimization scheme, which couples a heuristic and a gradient-based search algorithm, is developed to solve the LSE in this study. The differential evolution (DE) and Levenberg-Marquardt (LM) algorithms are selected as the heuristic and gradient-based search algorithms, respectively, in the hybrid optimization scheme. 3. Numerical Example To verify the proposed hybrid optimization scheme, a numerical experiment for a synthetic 2-D heterogeneous confined aquifer with continuous hydraulic conductivity and zonal dispersivity distributions was conducted. Observations corrupted with Gaussian noise were generated using the model for identification. The pilot point method is implemented for parameterization during the parameter identification. The efficiency and effectiveness of the proposed optimization scheme for the purpose of identifying hydraulic conductivity and dispersivity distributions have been demonstrated. 4. Conclusion The hybrid optimization scheme of combining the DE algorithm and LM optimization scheme has been successfully applied to parameter identification in coupled groundwater flow and mass transport modeling. With the increasing complexity of real world optimization problems, the proposed hybrid optimization scheme with the pilot point method exhibits remarkable performance in terms of robustness, speed, and accuracy for parameter identification in groundwater modeling. The proposed scheme can thus serve as a reference tool for the investigation and remediation assessment of contaminated sites.

並列關鍵字

groundwater modeling ； parameter identification ； gradient search ； heuristic algorithm ； hybrid optimization scheme