Surrogate Model Application to the Identification of Optimal Groundwater Exploitation Scheme Based on Regression Kriging Method—A Case Study of Western Jilin Province
Abstract
:1 Introduction
2. Study Area and Methods
2.1. Study Area
2.2. Methods
2.2.1. Latin Hypercube Sampling Method
- (1)
- Determining the sampling scale of random variable (N).
- (2)
- Dividing each variable into N equiprobable intervals, , and the probability of each interval is 1/N.
- (3)
- Extracting a random sample from each interval of variable xi, κ refers to variables, and then there are κ×N samples.
- (4)
- The N samples extracted respectively from variable x1 and x2 are matched randomly without repetition. Then let the matching process go on until the samples extracted from all the variables xi are completely matched. The eventual matched form is as follows:
2.2.2. Regression Kriging Method
- (1)
- r, the correlation matrix between m samples and prediction points x, and R, the correlation matrix between m samples, are calculated by Equation (8).
- (2)
- f (X), referring to the known regression functions of p-order, is calculated through equation 5.
- (3)
- β* is the estimated value of β, which is obtained by the generalized least-squares method.
- (4)
- The estimated value of σ2 is obtained by the following equation.
- (5)
- The parameter θk is obtained when the following equation achieves its maximum value, and this method is named as the maximum likelihood estimation method. The basic idea of this method (Maximum Likelihood, ML) is that the most reasonable parameter estimator is determined when extracting an n group sample observation value from the sample population of the model randomly and making the n group sample observation value selected from the overall model have a maximum probability.
2.2.3. Numerical Simulation of Groundwater Flow
Partitions Parameters | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Hydraulic conductivity (m/d) | 14 | 135 | 27 | 17 | 20 | 26 | 9 | 11 | 12 | 13 | 28 | 44 | 15 |
Specific yield | 0.09 | 0.23 | 0.10 | 0.12 | 0.18 | 0.08 | 0.08 | 0.10 | 0.11 | 0.08 | 0.09 | 0.10 | 0.15 |
Specific storage (m−1) | 0.008 | 0.008 | 0.009 | 0.008 | 0.008 | 0.008 | 0.009 | 0.008 | 0.008 | 0.008 | 0.007 | 0.008 | 0.008 |
2.2.4. Genetic Algorithm
3. Results and Discussions
3.1. Numerical Simulation of Groundwater Flow
Exploitation Scheme | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Training samples | 1 | 3240 | 0.808 | 7070 | 0.925 | 1551 | 0.609 | 2383 | 0.625 | 9949 | 1.848 | 8780 | 1.797 | 7840 | 2.395 | 1762 | 2.633 |
2 | 5713 | 0.896 | 88 | 0.565 | 7889 | 0.967 | 1480 | 0.588 | 9378 | 1.814 | 6227 | 2.110 | 6331 | 2.451 | 6961 | 3.154 | |
3 | 1159 | 0.558 | 5852 | 0.799 | 1677 | 0.584 | 3829 | 0.743 | 9705 | 1.678 | 2826 | 1.380 | 6499 | 1.978 | 4905 | 1.969 | |
4 | 3786 | 0.844 | 3100 | 0.911 | 4463 | 1.016 | 6899 | 1.194 | 6296 | 1.324 | 2397 | 1.559 | 6756 | 1.963 | 7091 | 2.342 | |
5 | 9330 | 1.178 | 1742 | 0.823 | 367 | 0.732 | 4703 | 0.794 | 1990 | 0.956 | 5481 | 2.017 | 7740 | 2.162 | 7827 | 3.189 | |
6 | 3701 | 0.994 | 7724 | 1.151 | 3603 | 0.921 | 4171 | 0.958 | 1025 | 0.463 | 2156 | 0.830 | 4760 | 1.037 | 3034 | 1.304 | |
7 | 9759 | 1.732 | 7895 | 1.600 | 4834 | 1.413 | 6581 | 1.405 | 9128 | 1.722 | 9877 | 2.187 | 3480 | 2.242 | 4409 | 3.289 | |
8 | 204 | 0.395 | 838 | 0.535 | 6831 | 0.902 | 5248 | 0.960 | 1736 | 0.775 | 6610 | 1.449 | 7379 | 1.727 | 2606 | 2.280 | |
9 | 9174 | 1.262 | 2532 | 0.793 | 4706 | 0.854 | 393 | 0.451 | 4662 | 1.186 | 5797 | 2.007 | 4820 | 2.025 | 7566 | 3.110 | |
10 | 4479 | 0.933 | 3469 | 0.931 | 5121 | 1.035 | 5637 | 1.084 | 5798 | 1.213 | 9607 | 1.912 | 1892 | 1.755 | 3396 | 2.930 | |
11 | 6170 | 1.108 | 4151 | 0.851 | 7308 | 0.975 | 22 | 0.514 | 6172 | 1.354 | 9075 | 1.831 | 5945 | 2.073 | 2578 | 2.785 | |
12 | 2540 | 0.798 | 6649 | 1.051 | 1914 | 0.815 | 6084 | 1.080 | 3254 | 0.667 | 5078 | 0.852 | 2505 | 0.960 | 349 | 1.285 | |
13 | 7677 | 1.306 | 4740 | 1.200 | 3494 | 1.136 | 6789 | 1.250 | 3793 | 0.893 | 7353 | 1.603 | 1648 | 1.427 | 3701 | 2.483 | |
14 | 4990 | 0.896 | 4252 | 0.836 | 2507 | 0.731 | 3374 | 0.726 | 2265 | 0.548 | 703 | 1.059 | 606 | 0.890 | 6730 | 1.647 | |
15 | 4106 | 0.919 | 7451 | 1.081 | 71 | 0.680 | 4881 | 0.903 | 2559 | 0.701 | 4057 | 1.474 | 1147 | 1.236 | 6304 | 2.307 | |
16 | 477 | 0.497 | 5428 | 0.664 | 4359 | 0.597 | 964 | 0.469 | 7196 | 1.354 | 7816 | 1.443 | 5053 | 1.776 | 981 | 2.135 | |
17 | 1275 | 0.404 | 1098 | 0.399 | 5253 | 0.630 | 1961 | 0.508 | 8612 | 1.219 | 162 | 0.777 | 253 | 0.963 | 4337 | 1.032 | |
18 | 1887 | 0.562 | 2336 | 0.583 | 5738 | 0.773 | 2653 | 0.661 | 5325 | 1.135 | 9330 | 1.704 | 2898 | 1.679 | 2037 | 2.607 | |
19 | 7164 | 1.123 | 3718 | 1.053 | 710 | 0.907 | 7674 | 1.209 | 8122 | 1.711 | 8556 | 2.165 | 7467 | 2.529 | 5020 | 3.273 | |
20 | 6440 | 1.314 | 7240 | 1.238 | 6091 | 1.143 | 2830 | 0.916 | 7733 | 1.605 | 6778 | 1.956 | 7181 | 2.342 | 5364 | 2.946 | |
21 | 2993 | 0.683 | 2633 | 0.643 | 5531 | 0.796 | 2437 | 0.652 | 5137 | 0.935 | 345 | 0.861 | 4008 | 1.186 | 4615 | 1.250 | |
22 | 4699 | 0.701 | 1290 | 0.568 | 2016 | 0.610 | 3710 | 0.653 | 2144 | 0.621 | 3799 | 0.787 | 6022 | 1.171 | 696 | 1.205 | |
23 | 1570 | 0.510 | 2824 | 0.632 | 3314 | 0.682 | 4471 | 0.802 | 4863 | 1.104 | 4859 | 1.607 | 4583 | 1.738 | 5564 | 2.459 | |
24 | 8262 | 1.050 | 277 | 0.794 | 518 | 0.826 | 7581 | 1.100 | 561 | 0.398 | 3118 | 1.033 | 2647 | 0.970 | 4012 | 1.646 | |
25 | 6686 | 1.173 | 3845 | 1.103 | 3868 | 1.118 | 7098 | 1.261 | 7271 | 1.449 | 3308 | 1.575 | 6944 | 2.037 | 6132 | 2.344 | |
26 | 2492 | 0.802 | 5630 | 0.867 | 6446 | 0.896 | 1652 | 0.675 | 6907 | 1.316 | 5608 | 1.403 | 5268 | 1.753 | 2898 | 2.077 | |
27 | 2240 | 0.556 | 3392 | 0.684 | 1352 | 0.596 | 5016 | 0.819 | 7944 | 1.104 | 1088 | 0.486 | 1516 | 0.841 | 1122 | 0.578 | |
28 | 8070 | 1.443 | 6986 | 1.430 | 2211 | 1.175 | 7823 | 1.403 | 196 | 0.537 | 4293 | 1.689 | 3336 | 1.465 | 7617 | 2.712 | |
29 | 8956 | 1.439 | 4502 | 1.037 | 7600 | 1.144 | 401 | 0.625 | 2839 | 0.786 | 8151 | 1.703 | 986 | 1.393 | 3895 | 2.671 | |
30 | 8525 | 1.266 | 2056 | 0.832 | 7166 | 1.065 | 1217 | 0.629 | 3511 | 0.862 | 7065 | 1.152 | 5628 | 1.452 | 7 | 1.759 | |
31 | 7758 | 1.252 | 760 | 0.997 | 7616 | 1.387 | 7268 | 1.355 | 348 | 0.447 | 2552 | 1.385 | 2193 | 1.163 | 7287 | 2.218 | |
32 | 5984 | 1.025 | 5310 | 0.979 | 984 | 0.734 | 4325 | 0.832 | 1283 | 0.548 | 8428 | 1.450 | 1298 | 1.173 | 1916 | 2.301 | |
33 | 979 | 0.588 | 6563 | 0.915 | 1058 | 0.641 | 5425 | 0.937 | 8359 | 1.330 | 7649 | 1.371 | 18 | 1.340 | 1324 | 1.995 | |
34 | 532 | 0.537 | 4979 | 0.850 | 2987 | 0.780 | 6210 | 1.058 | 3206 | 0.929 | 4557 | 1.581 | 5536 | 1.725 | 5745 | 2.458 | |
35 | 5259 | 0.848 | 1951 | 0.674 | 4033 | 0.778 | 3028 | 0.678 | 4433 | 0.988 | 6495 | 1.512 | 3125 | 1.524 | 3543 | 2.319 | |
36 | 6849 | 1.162 | 6251 | 0.994 | 2756 | 0.758 | 1174 | 0.550 | 4163 | 0.888 | 884 | 1.174 | 3674 | 1.327 | 6595 | 1.780 | |
37 | 5164 | 0.838 | 554 | 0.627 | 5950 | 0.919 | 3516 | 0.772 | 5687 | 0.916 | 1485 | 0.479 | 4378 | 0.981 | 501 | 0.621 | |
38 | 7293 | 1.311 | 6169 | 1.084 | 6217 | 1.034 | 755 | 0.641 | 6593 | 1.150 | 3712 | 1.427 | 561 | 1.343 | 5830 | 2.129 | |
39 | 3372 | 0.700 | 1530 | 0.582 | 6783 | 0.861 | 2087 | 0.630 | 851 | 0.294 | 1719 | 0.513 | 2310 | 0.596 | 1572 | 0.801 | |
40 | 9738 | 1.513 | 5192 | 1.275 | 3026 | 1.139 | 5848 | 1.157 | 8827 | 1.363 | 1888 | 0.820 | 3998 | 1.331 | 2248 | 1.091 | |
Validation samples | 1 | 2032 | 0.512 | 952 | 0.573 | 4167 | 0.783 | 5642 | 0.932 | 2861 | 0.626 | 1031 | 0.842 | 2757 | 0.956 | 4332 | 1.279 |
2 | 5050 | 1.079 | 6829 | 1.137 | 3100 | 0.926 | 4497 | 0.971 | 1064 | 0.647 | 2940 | 1.280 | 7119 | 1.556 | 5172 | 2.026 | |
3 | 553 | 0.346 | 4431 | 0.487 | 1193 | 0.304 | 866 | 0.300 | 6786 | 1.623 | 9843 | 2.472 | 7295 | 2.662 | 6212 | 3.804 | |
4 | 6490 | 1.368 | 7557 | 1.311 | 6734 | 1.237 | 3334 | 1.015 | 9399 | 1.520 | 5333 | 1.162 | 4142 | 1.606 | 1232 | 1.628 | |
5 | 4363 | 1.079 | 6000 | 1.156 | 6113 | 1.167 | 5259 | 1.153 | 8178 | 1.242 | 4296 | 0.886 | 1879 | 1.160 | 791 | 1.217 | |
6 | 1630 | 0.475 | 3426 | 0.561 | 2283 | 0.503 | 2514 | 0.535 | 5859 | 0.997 | 3302 | 1.054 | 1484 | 1.132 | 3431 | 1.547 | |
7 | 9206 | 1.509 | 5357 | 1.134 | 7365 | 1.181 | 759 | 0.690 | 738 | 0.766 | 7649 | 2.126 | 5331 | 1.966 | 7055 | 3.400 | |
8 | 3615 | 0.786 | 2909 | 0.869 | 3417 | 0.932 | 7145 | 1.175 | 3935 | 0.823 | 6468 | 1.390 | 448 | 1.182 | 3200 | 2.137 | |
9 | 8124 | 1.097 | 1635 | 0.904 | 100 | 0.839 | 7783 | 1.148 | 7514 | 1.203 | 134 | 0.591 | 5865 | 1.275 | 2166 | 0.753 | |
10 | 7193 | 0.970 | 118 | 0.571 | 5252 | 0.822 | 1685 | 0.531 | 4985 | 1.237 | 8026 | 2.241 | 3244 | 2.062 | 7281 | 3.481 |
3.2. Surrogate Model of Numerical Simulation Model of Groundwater Flow
Parameter | ||||||||
---|---|---|---|---|---|---|---|---|
Value | 0.7922 | 0.9961 | 0.5000 | 1.6490 | 0.6476 | 0.4952 | 0.5737 | 1.0513 |
Scheme | Mean Relative Error | Root Mean Square Error | ||
---|---|---|---|---|
1 | 1.21 | 1.87 | 1.48 | 2.27 |
2 | 0.99 | 1.06 | ||
3 | 1.90 | 2.37 | ||
4 | 2.60 | 2.93 | ||
5 | 1.55 | 1.71 | ||
6 | 2.31 | 2.60 | ||
7 | 2.25 | 2.58 | ||
8 | 2.26 | 2.53 | ||
9 | 2.16 | 2.60 | ||
10 | 1.52 | 2.15 |
3.3. Optimization Model
Well | ||||||||
---|---|---|---|---|---|---|---|---|
Cost coefficient | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 |
Exploitation Well | ||||||||
---|---|---|---|---|---|---|---|---|
Exploitation quantity () | 7.597 | 7.585 | 7.737 | 7.592 | 7.593 | 7.724 | 7.585 | 7.596 |
Groundwater table drawdown () | 0.400 | 0.411 | 0.422 | 0.426 | 0.427 | 0.607 | 0.675 | 0.929 |
Water cost () | 15.194 | 15.170 | 15.474 | 15.184 | 22.779 | 23.172 | 22.755 | 22.788 |
4. Conclusions
- (1)
- The groundwater table values calculated by the numerical simulation model of groundwater flow are very close to the actual measured values both at the stage of model calibration and model verification, which demonstrates that the selected hydrogeological conceptual model generalization, partial differential equations and algorithm are reasonable and feasible in the study area, and the established numerical simulation model of groundwater flow can objectively and accurately describe the groundwater flow characteristics of the study area. These research results can provide a good foundation for establishing a surrogate model.
- (2)
- Due to the regression kriging method with accurate approximation ability, the surrogate model results are much closer to that of the numerical simulation model of groundwater flow, and could effectively substitute the numerical simulation model of groundwater flow.
- (3)
- The huge computational burden of coupled operations during simulation and optimization hinders the success of the simulation optimization model in groundwater exploitation. According to this study, replacing the simulation models with surrogate models could reduce the huge computational burden effectively and maintain considerably high accuracy so as to obtain an optimal exploitation scheme.
Acknowledgments
Author Contributions
Conflicts of Interest
References
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An, Y.; Lu, W.; Cheng, W. Surrogate Model Application to the Identification of Optimal Groundwater Exploitation Scheme Based on Regression Kriging Method—A Case Study of Western Jilin Province. Int. J. Environ. Res. Public Health 2015, 12, 8897-8918. https://doi.org/10.3390/ijerph120808897
An Y, Lu W, Cheng W. Surrogate Model Application to the Identification of Optimal Groundwater Exploitation Scheme Based on Regression Kriging Method—A Case Study of Western Jilin Province. International Journal of Environmental Research and Public Health. 2015; 12(8):8897-8918. https://doi.org/10.3390/ijerph120808897
Chicago/Turabian StyleAn, Yongkai, Wenxi Lu, and Weiguo Cheng. 2015. "Surrogate Model Application to the Identification of Optimal Groundwater Exploitation Scheme Based on Regression Kriging Method—A Case Study of Western Jilin Province" International Journal of Environmental Research and Public Health 12, no. 8: 8897-8918. https://doi.org/10.3390/ijerph120808897