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

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}); ${S}_{s}$ is the specific storage of porous media (L

^{−1}); h is the groundwater head (L); W is the volumetric flux per unit volume representing sources (recharge) and/or sinks (pumpage) of the groundwater system, $W>0$ for the flow into the system (T

^{−1}); t is the time (T).

## 2. Methodology

^{th}and (k − 1)

^{th}iteration for Hessian approximation.

#### 2.1. Proposing Optimization Model for Model Calibration in a Multi-Layer Groundwater System

#### 2.2. Modification of Traditional Algorithms in Nonlinear Programming for Proposing SR3 VLS Quasi-Newton Algorithm to Solve Hydrogeological Parameters

_{f}. This study uses the factor score ${\left(\tilde{{C}_{\xb7,\xb7,f}^{}}\right)}^{k}$ and the factor loading ${\left({F}_{\xb7,\xb7,f}^{\mathrm{T}}\right)}^{k}$ calculated from the simulated storage difference ${\vartheta}_{m}^{k}-{\vartheta}_{m}^{k-1}$ through SVD between iteration k and k − 1 to compute the elements of ${\left({\Psi}_{\mathrm{D},l}^{\mathrm{T}}\right)}^{k}$ and ${\left({\Gamma}_{\mathrm{D},l}^{\mathrm{T}}\right)}^{k}$, as expressed in Equations (13)–(16), thereby eliminating the need for additional model runs to approximate the Hessian. ${D}_{\xb7,f}^{k}$ is the diagonalized standard deviation matrix of ${\vartheta}_{m}^{k}-{\vartheta}_{m}^{k-1}$, where $m=1,2,\dots ,M$, and $M={T}_{\mathrm{s}}\times {S}_{f}\times F$. Before SVD, the simulated storage difference ${\vartheta}_{m}^{k}-{\vartheta}_{m}^{k-1}$ will first be normalized by the average ${a}^{k}$ and standard deviation ${\epsilon}_{l}^{k}$ for each zonal ${s}_{f}$ and each aquifer f across multiple simulated time steps ${t}_{\mathrm{s}}$, such that the average of the simulated storage difference of each zone equals to 0 and the variance equals to 1. Hence, the vectorized step size ${\mathsf{\alpha}}^{k}$ is a switchable and limited set between $-{\epsilon}_{l}^{k}$ and ${\epsilon}_{l}^{k}$, namely ${\mathsf{\alpha}}^{k}=\left[{\alpha}_{l}^{k}|l=1~L\right]\in \left(-{\epsilon}_{l}^{k},{\epsilon}_{l}^{k}\right)$.

#### 2.3. Modification from Traditional Scalar Line Search to Vectorized Multi-Order Derivative Exact Double False Position Bracketing Using SVD-Related Rank and Depth

^{th}unit vector; ${m}_{1}$ is the halving vector of the proposing 0 order derivative’s double false position method; ${m}_{2}$, ${m}_{3}$, and ${m}_{4}$ are the halving vectors of the vectorized Anderson−Björk algorithm, vectorized Illinois algorithm, and vectorized conventional false position method, respectively [22]; and ${m}_{5}$ is the halving vector of the proposing SVD depth rank component-based double false position method, which uses the j

^{th}component in d

^{th}depth as the high-rank loading score ${\Gamma}_{\mathrm{D}}^{k}$ for the Hessian correction.

#### 2.4. Procedures of the Methodology

_{y}) estimated from pumping tests, drilled hydrogeological structure, and observed/simulated groundwater head hydrographs.

#### 2.5. SVD of the Change in Storage between k^{th} and (k − 1)^{th} Iteration for Hessian Approximation

**X**by the sample standard deviation of the variable ${\epsilon}_{{s}_{f},f}^{}$ for an aquifer f, that is,

^{th}and (k − 1)

^{th}iteration at the s

_{f}-th observation well in aquifer f, respectively. Set $D=\mathrm{diag}\left({\epsilon}_{1,f},\dots ,{\epsilon}_{{S}_{f},f}\right)$.

^{th}and (k − 1)

^{th}iteration). Considering the SVD of the deviation matrix

**X**:

^{th}principal component coefficient, called factor loading, and let $F=\left[{\varphi}_{j,{s}_{f},f}\right]$ be the ${S}_{f}\times {S}_{f}$ order factor loading matrix. Because $\tilde{X}$ and $\tilde{C}$ contain the standardized data, substituting $\tilde{X}=X{D}^{-1},\tilde{C}=XV{\Lambda}^{-1/2},S=\frac{1}{{T}_{\mathrm{s}}-1}{X}^{\mathrm{T}}X=V\Lambda {V}^{\mathrm{T}},{\Lambda}^{1/2}=\frac{1}{\sqrt{{T}_{\mathrm{s}}-1}}\mathrm{\Sigma}$, we can derive:

^{th}and (k − 1)

^{th}iterations by using SVD. The deviation matrix

**X**can be expressed by the factor score $\tilde{C}$ and the factor loading

**F**as follows:

_{f}factors, as shown in Figure 1.

**F**(customized spatial direction), where ${\varphi}_{j,{s}_{f},f}$ determines the loading weight (namely, the correlation coefficient) from the l

^{th}parameter’s factor perturbed between the k

^{th}and (k − 1)

^{th}iterations $\tilde{{c}_{\u2022,l,f}}$ to the change in storage at ${\left({s}_{f}\right)}^{\mathrm{th}}$ observation well ${x}_{\u2022,{s}_{f},f}$. Then, L variables pass through the stretching process

**D**(standard deviation) to enlarge or reduce the range of correction values. Finally, shift

**a**again to generate the component of the change in storage between the k

^{th}and (k − 1)

^{th}iterations ${\left({q}_{{t}_{\mathrm{s}},\u2022,f}^{\mathrm{sim}}\right)}^{k}-{\left({q}_{{t}_{\mathrm{s}},\u2022,f}^{\mathrm{sim}}\right)}^{k-1}$.

## 3. Numerical Experimental Validation

#### 3.1. Setup of the Test Case

#### 3.2. Results and Discussion of the Experimental Verification

#### 3.2.1. Experiment 1—Identification of a Single Type of Parameter

#### 3.2.2. Experiment 2—Identification of Multiple Types of Parameters Simultaneously

## 4. Practical Application

#### 4.1. Study Area

^{2}. According to terrain, geology, and stratigraphy, the Cho-Shui River alluvial fan can be divided into the apex, middle, and distal fans, as shown in Figure 6a. The aquifer stratification in the alluvial fan is shown in Figure 6b. It shows that rainfall, river, and irrigation water recharge the groundwater system through the unconfined aquifer at the fan apex and that the groundwater then flows west to the coast. The groundwater system in the alluvial fan is approximately 330 m thick, which can be divided into four aquifers and three aquitards. The specific yield (S

_{y}) of the unconfined aquifer at the apex of the fan is 0.137 to 0.237, and the storage coefficient of the confined aquifer varies from 10

^{−4}to 10

^{−3}[16]. Accordingly, the constructed conceptual groundwater flow model of the Cho-Shui River alluvial fan using MODFLOW-2005 is shown in Figure 6c.

#### 4.2. Calibration Outcome and Discussion of the Mega-Parameters Variables of Groundwater Flow Modeling

#### 4.2.1. Calculated Results and Discussion of Spatiotemporal Pumpage During 2012–2014

^{8}, 15.67 × 10

^{8}, and 15.80 × 10

^{8}m

^{3}, in consecutive years, while that for non-irrigation was 7.72 × 10

^{8}, 7.99 × 10

^{8}, and 7.46 × 10

^{8}m

^{3}, respectively. The annual total pumpage was 23.73 × 10

^{8}m

^{3}, of which irrigation pumpage accounted for 67.4% of the total.

^{8}and 2.13 × 10

^{8}m

^{3}) was in March (at the end of the dry season) and February (starting the mixed cultivation period of paddy and dry farming). Because of large water demand and insufficient surface water supply, the water sources for each demand during this month relied mainly on groundwater pumping. Moreover, the minimum pumpage during 2012–2014 (1.33 × 10

^{8}, 1.54 × 10

^{8}, and 1.18 × 10

^{8}m

^{3}) were in November and December, which are the end of the wet season and the dry farming cultivation period, respectively. Moreover, in the unusually dry year in 2014, there was only one typhoon Fung-Wong, which occurred at the end of September to provide rainfall and surface water. Therefore, the maximum pumpage was in August (2.30 × 10

^{8}m

^{3}), with the pumpage being closest to the mean value in April (2.08 × 10

^{8}m

^{3}). Overall, high-intensity pumping is mainly concentrated in the distal alluvial fan. During the dry season and the month with high water demand, the high-intensity pumping areas extend to the interior of the middle alluvial fan.

#### 4.2.2. Initially Estimated Spatiotemporal Pattern of Groundwater Recharge

#### 4.2.3. Calibration Process and Discussion

#### 4.2.4. Results and Discussions of Identified Parameters

## 5. Conclusions and Suggestion

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Schematic diagram of singular value decomposition of changes in storage between the k

^{th}and (k − 1)

^{th}iterations with once simulation perturbation: (

**a**) observed change in storage ${\left({q}_{{t}_{\mathrm{s}},\u2022,f}^{\mathrm{sim}}\right)}^{k}-{({q}_{{t}_{\mathrm{s}},\u2022,f}^{\mathrm{sim}})}^{k-1}$; (

**b**) decomposed components of change in storage perturbed by the correction in the parameter ${\left({\mathsf{\eta}}_{{t}_{\mathrm{s}},\u2022,f}^{}\right)}^{k}-{({\mathsf{\eta}}_{{t}_{\mathrm{s}},\u2022,f}^{})}^{k-1}$.

**Figure 3.**Comparison of descent footprint of objective function: (

**a**) Experiment 1; (

**b**) Experiment 2.

**Figure 4.**Error percentage of identified hydraulic conductivity using Jacobian quasi-Newton, LMA, and SR1 VLS quasi-Newton algorithms during iterations across multiple zones in Experiment 1.

**Figure 5.**Error percentage of identified hydraulic conductivity (

**a**), surface recharge (

**b**) and boundary recharge (

**c**) during iterations across multiple zones in Experiment 2.

**Figure 6.**Study area: (

**a**) distribution of groundwater monitoring wells, pumping wells, and sub-alluvial fan; (

**b**) aquifer/aquitard stratification profile across all drilling wells; (

**c**) constructed hydrogeological gridding structure of groundwater modeling.

**Figure 8.**The overlaid factor loadings for an initial spatial estimate of surface/boundary recharge in Cho-Shui River alluvial fan: (

**a**) Aquifer 1; (

**b**) Aquifer 2; (

**c**) Aquifer 3; (

**d**) Aquifer 4.

**Figure 9.**Simulated storage error percentages vs. number of iterations for the Chou-Shui River alluvial fan groundwater system.

**Figure 10.**The calculated $\underset{{s}_{f},k}{\mathrm{E}}\left[{\beta}_{{t}_{\mathrm{s}},{s}_{f},f}^{k}\right]$ hyetograph across four aquifers.

**Figure 11.**The identified spatial pattern of surface recharge ${R}_{{t}_{\mathrm{s}},{s}_{1}}^{\mathrm{surf}}$ and aquitard leakance conductivities ${K}_{{s}_{f},f}^{\mathrm{V}}$: (

**a**) Aquifer 1; (

**b**) Aquitard 1; (

**c**) Aquitard 2; and (

**d**) Aquitard 3.

**Figure 12.**The identified spatial pattern of aquifer’s hydraulic conductivity ${K}_{{s}_{f},f}^{\mathrm{aqu}}$: (

**a**) Aquifer 1; (

**b**) Aquifer 2; (

**c**) Aquifer 3; and (

**d**) Aquifer 4.

**Figure 13.**The spatial pattern of root mean square error (RMSE) between simulated and observed groundwater heads: (

**a**) Aquifer 1; (

**b**) Aquifer 2; (

**c**) Aquifer 3; and (

**d**) Aquifer 4.

**Figure 14.**Comparison among simulated and observed groundwater head hydrographs: (

**a**) MT station; (

**b**) TK station; (

**c**) TH station; (-1) Aquifer 1; (-2) Aquifer 2; (-3) Aquifer 3; and (-4) quartile analysis of simulated error.

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## Share and Cite

**MDPI and ACS Style**

Huang, C.-L.; Hsu, N.-S.; Hsu, F.-J.; You, G.J.-Y.; Yao, C.-H.
Symmetrical Rank-Three Vectorized Loading Scores Quasi-Newton for Identification of Hydrogeological Parameters and Spatiotemporal Recharges. *Water* **2020**, *12*, 995.
https://doi.org/10.3390/w12040995

**AMA Style**

Huang C-L, Hsu N-S, Hsu F-J, You GJ-Y, Yao C-H.
Symmetrical Rank-Three Vectorized Loading Scores Quasi-Newton for Identification of Hydrogeological Parameters and Spatiotemporal Recharges. *Water*. 2020; 12(4):995.
https://doi.org/10.3390/w12040995

**Chicago/Turabian Style**

Huang, Chien-Lin, Nien-Sheng Hsu, Fu-Jian Hsu, Gene J.-Y. You, and Chun-Hao Yao.
2020. "Symmetrical Rank-Three Vectorized Loading Scores Quasi-Newton for Identification of Hydrogeological Parameters and Spatiotemporal Recharges" *Water* 12, no. 4: 995.
https://doi.org/10.3390/w12040995