應用遺傳演算法與試驗設計原則於地下水觀測井網設計

Abstract

本研究目的為應用遺傳演算法(Genetic Algorithm)結合試驗設計原則，發展地下水觀測井網設計模式。在本研究中，試驗設計部份採用D-Optimal準則，而D-Optimal之內涵，乃求取得估計參數變異數矩陣行列式值為最小。本研究除了根據試驗設計原則架構出井網設計的一般化表示式外，並將此一般化問題簡化成觀測井網設計、抽水井網設計、觀測頻率設計及參數分區型式設計等四類問題。再以遺傳演算法(Genetic Algorithm)進行近似全域搜尋，求得在一定成本限制下使估計參數可靠度達到最大之設計。本文之地下水流數值模式及參數檢定模式，乃採用美國地質調查局(U.S.G.S.)所發展之MODFLOW及UCODE程式。UCODE程式乃是應用非線性迴歸理論與地下水流模式結合，本文藉以優選由MODFLOW所建立之地下水流模式中的參數，並得到估計參數的敏感度矩陣。本研究針對四類問題，以各種簡化案例進一步探討，研究成果顯示計算參數對水位的敏感度--傑可比矩陣(Jacobian Matrix)，對最佳井網設計有顯著地影響，由於該矩陣乃由模擬模式求得，因此模式的最佳設計確能反應出系統的物理條件。由頻率設計結果顯示，系統誤差將會干擾敏感度資訊，使得在不同位置對應之條件下(邊界、外力、…)，應有不同觀測頻率的現象無法突顯。另外，分區設計在參數維度龐大時，不易收歛至最佳解(還原真實分區)，若能適度降低維度，可改善不易收歛至最佳解之情形。

This study presents a novel groundwater network design model by integrating the Genetic Algorithms (GA) and experimental design theory. The experimental design attempts to minimize the estimated parameters’ variance that can be represented by the determinant of covariance matrix (D-optimal Criteria). Based on the D-Optimal criteria, this study also develops a general formulation based on the optimal groundwater network design problem. The general problem is then classified into three classes of network design problems; a monitoring network designing problem, a pumping network design problem and a frequency design problem. The Genetic Algorithm seeks to obtain a network design or monitoring frequency that minimizes the determinant of the covariance matrix, i.e., maximizes the reliability of the estimated parameters. Additionally, the UCODE developed by U.S.G.S. is used to implement the parameter optimization and compute the sensitivity matrix. The sensitivity matrix is then used to calculate the covariance matrix. Focusing on the three classes of problems, several numerical examples are presented to investigate the mechanism affecting the optimal design system. Numerical results indicate that the sensitivity of the hydraulic head (the Jacobian matrix), which is computed by the simulation model, plays an important role in selecting the optimal network design. Its importance implies that the algorithm can accurately reflect the physical configuration of the problem when designing an optimal network. Regarding the frequency design problem, the system error can disturb the information associated with the Jacobian matrix and uniformized spatially the monitoring frequency.