# American Institute of Mathematical Sciences

October  2018, 14(4): 1463-1478. doi: 10.3934/jimo.2018016

## Parameter identification and numerical simulation for the exchange coefficient of dissolved oxygen concentration under ice in a boreal lake

 †. School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, Liaoning Province, China ‡. State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, Liaoning Province, China

* Corresponding author: Zhijun Li

Received  January 2017 Revised  August 2017 Published  January 2018

Fund Project: The second author is supported by the National Natural Science Foundation of China (NNSFC) (Nos.51579028,41376186), the third author is supported by the NNSFC (No.11401073), and the fourth author is supported by the NNSFC (No.41306207).

Dissolved oxygen (DO) is one of the main parameters to assess the quality of lake water. This study is intended to construct a parabolic distributed parameter system to describe the variation of DO under the ice, and identify the vertical exchange coefficient K of DO with the field data. Based on the existence and uniqueness of the weak solution of this system, the fixed solution problem of the parabolic equation is transformed into a parameter identification model, which takes K as the identification parameter, and the deviation of the simulated and measured DO as the performance index. We prove the existence of the optimal parameter of the identification model, and derive the first order optimality conditions. Finally, we construct the optimization algorithm, and have carried out numerical simulation. According to the measured DO data in Lake Valkea-Kotinen (Finland), it can be found that the orders of magnitude of the coefficient K varying from 10-6 to 10-1 m2 s-1, the calculated and measured DO values are in good agreement. Within this range of K values, the overall trends are very similar. In order to get better fitting, the formula needs to be adjusted based on microbial and chemical consumption rates of DO.

Citation: Qinxi Bai, Zhijun Li, Lei Wang, Bing Tan, Enmin Feng. Parameter identification and numerical simulation for the exchange coefficient of dissolved oxygen concentration under ice in a boreal lake. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1463-1478. doi: 10.3934/jimo.2018016
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##### References:
Schematic diagram of model identification area and mesh generation
DO concentration curves at different depths at No. 2 station
Comparison curves of the measured and the calculated DO concentration at depth 0.70 m when $K$ varying from 10$^{ - 8}$ to 10 m$^{2}$ s$^{-1}$ with a total of ten orders of magnitude
Comparison curves of the measured and the calculated DO concentrations at different depths when the order of magnitude $K$ = 10$^{-4}$ m$^{2}$ s$^{-1}$
Relative errors (%) of the measured and the calculated DO concentrations at different depths when $K$ varying from 10$^{ - 8}$ to 10 m$^{2}$ s$^{-1}$ with ten orders of magnitude
 Relerr 10$^{-8}$ 10$^{-7}$ 10$^{-6}$ 10$^{-5}$ 10$^{-4}$ 10$^{-3}$ 10$^{-2}$ 10$^{-1}$ 10$^{0}$ 10$^{1}$ 0.45 m 0.6112 0.2582 0.3161 0.3270 0.3287 0.3294 0.3324 0.3419 0.3609 0.4018 0.70 m 0.6537 0.1574 0.1586 0.1675 0.1689 0.1694 0.1720 0.1627 0.1691 0.2215 0.95 m 1.6933 0.2649 0.1927 0.2279 0.234 0 0.2355 0.2370 0.2473 0.3164 0.9236 1.95 m 8.5907 4.0214 19.431 28.932 30.002 30.126 30.058 27.763 23.361 22.972 2.95 m 21.037 13.605 4.5830 3.7184 3.7672 3.7480 3.5150 3.4109 3.2001 6.6497
 Relerr 10$^{-8}$ 10$^{-7}$ 10$^{-6}$ 10$^{-5}$ 10$^{-4}$ 10$^{-3}$ 10$^{-2}$ 10$^{-1}$ 10$^{0}$ 10$^{1}$ 0.45 m 0.6112 0.2582 0.3161 0.3270 0.3287 0.3294 0.3324 0.3419 0.3609 0.4018 0.70 m 0.6537 0.1574 0.1586 0.1675 0.1689 0.1694 0.1720 0.1627 0.1691 0.2215 0.95 m 1.6933 0.2649 0.1927 0.2279 0.234 0 0.2355 0.2370 0.2473 0.3164 0.9236 1.95 m 8.5907 4.0214 19.431 28.932 30.002 30.126 30.058 27.763 23.361 22.972 2.95 m 21.037 13.605 4.5830 3.7184 3.7672 3.7480 3.5150 3.4109 3.2001 6.6497
Correlation coefficients of the measured and the calculated DO concentrations at different depths when $K$ varying from 10$^{ - 8}$ to 10 m$^{2}$ s$^{-1}$ with ten orders of magnitude
 Corcoef 10$^{-8}$ 10$^{-7}$ 10$^{-6}$ 10$^{-5}$ 10$^{-4}$ 10$^{-3}$ 10$^{-2}$ 10$^{-1}$ 10$^{0}$ 10$^{1}$ 0.45 m 0.9012 0.9247 0.9530 0.9552 0.9552 0.9550 0.9540 0.9506 0.9438 0.9310 0.70 m 0.9248 0.9475 0.9693 0.9702 0.9700 0.9698 0.9676 0.9609 0.9451 0.9257 0.95 m 0.9361 0.9429 0.9518 0.9458 0.9448 0.9444 0.9439 0.9417 0.9282 0.8409 1.95 m 0.9431 0.9276 0.9799 0.9728 0.9695 0.9687 0.9666 0.9637 0.9575 0.9383 2.95 m 0.8850 0.8883 0.9535 0.9706 0.9719 0.9723 0.9749 0.9684 0.9671 0.9279
 Corcoef 10$^{-8}$ 10$^{-7}$ 10$^{-6}$ 10$^{-5}$ 10$^{-4}$ 10$^{-3}$ 10$^{-2}$ 10$^{-1}$ 10$^{0}$ 10$^{1}$ 0.45 m 0.9012 0.9247 0.9530 0.9552 0.9552 0.9550 0.9540 0.9506 0.9438 0.9310 0.70 m 0.9248 0.9475 0.9693 0.9702 0.9700 0.9698 0.9676 0.9609 0.9451 0.9257 0.95 m 0.9361 0.9429 0.9518 0.9458 0.9448 0.9444 0.9439 0.9417 0.9282 0.8409 1.95 m 0.9431 0.9276 0.9799 0.9728 0.9695 0.9687 0.9666 0.9637 0.9575 0.9383 2.95 m 0.8850 0.8883 0.9535 0.9706 0.9719 0.9723 0.9749 0.9684 0.9671 0.9279
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