Symmetrical Rank-Three Vectorized Loading Scores Quasi-Newton for Identification of Hydrogeological Parameters and Spatiotemporal Recharges

Chien-Lin Huang,Chun-Hao Yao,Nien-Sheng Hsu,Gene J.-Y You,Fu-Jian Hsu

doi:10.3390/w12040995

Abstract

In a multi-layered groundwater model, achieving accurate spatiotemporal identification and solving the ill-posed problem is the vital topic for model calibration. This study proposes a symmetry rank three vectorized loading scores (SR3 VLS) quasi-Newton algorithm by modifying the Levenberg–Marquardt algorithm and importing a rank three structure from Broyden–Fletcher–Goldfarb–Shanno algorithm for identification of hydrogeological parameters and spatiotemporal recharge simultaneously. To accelerate directional convergence and approach a global optimum, this study uses a vectorized limited switchable step size in the transmissive groundwater inverse problem. The Hessian approximation rank three uses high and low-rank factor loading scores analyzed from simulated storage fluctuation between adjacent iterations for calculation and matrix correction. Two numerical experiments were designed to validate the proposing algorithm, showing the SR3 VLS quasi-Newton reduced the error percentages of the identified parameters by 1.63% and 9.65% compared to the Jacobian quasi-Newton. The proposing method is applied to the Chou-Shui River alluvial fan groundwater system in Taiwan. Results show that the simulated storage error decreased rapidly in six iterations, and has good head convergence as small as 0.11% with a root-mean-square-error (RMSE) of 0.134 m, indicating that the proposing algorithm reduces the computational cost to converge to the true solution.

Highlights

In arid and semi-arid areas, if there are insufficient surface water facilities to store runoff, groundwater provides an important water supply source [1]
This study proposes a symmetry rank three vectorized loading scores (SR3 VLS) quasi-Newton algorithm by modifying the Levenberg–Marquardt algorithm and importing a rank three structure from the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm to solve the constrained hydrogeological parameter identification and recharge spatiotemporal patterns simultaneously in a large-scale multi-layered groundwater model
This study proposes a systematic strengthening approach of vectorized multi-order derivative exact double false position bracketing using singular value decomposition (SVD)-related rank and depth, which guarantee convergence for solving the vectorized step size corresponding to different kinds of hydrogeological parameters and recharges

Summary

Introduction

In arid and semi-arid areas, if there are insufficient surface water facilities to store runoff, groundwater provides an important water supply source [1]. Conjunctive use planning for surface water and groundwater plays an important role in total water resource management and can be optimized through a sound groundwater management model based on mathematical system analysis. To develop such a model, it is necessary to understand and to be able to estimate the spatiotemporal patterns of pumpage, net recharge, and hydrogeological parameters of the groundwater system, in which the associated simulation model and optimization model are mostly nonlinear. The governing equation of three-dimensional groundwater flow can be expressed as [2] ∂

Methods

Results

Conclusion