# 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
##### References:
 [1] K. Addy and L. Green, Dissolved oxygen and temperature, PLA Notes, 22 (2004), 48-65. Google Scholar [2] V. Z. Antonopoulos and S. K. Gianniou, Simulation of water temperature and dissolved oxygen distribution in Lake Vegoritis, Greece, Ecol. Model., 160 (2003), 39-53. doi: 10.1016/S0304-3800(02)00286-7. Google Scholar [3] L. Arvola, K. Salonen, J. Keskitalo, T. Tulonen, M. Järvinen and J. Huotari, Plankton metabolism and sedimentation in a small boreal lake -a long-term perspective, Boreal Environ. Res., 19 (2014), 83-97. Google Scholar [4] J. Babin and E. E. Prepas, Modelling winter oxygen depletion rates in ice-covered temperate zone lakes in Canada, Can. J. Fish. Aquat. Sci., 42 (1985), 239-249. doi: 10.1139/f85-031. Google Scholar [5] Q. Bai, R. Li, Z. Li, M. Leppäranta, L. Arvola and M. Li, Time-series analyses of water temperature and dissolved oxygen concentration in Lake Valkea-Kotinen (Finland) during ice season, Ecol. Inform., 36 (2016), 181-189. doi: 10.1016/j.ecoinf.2015.06.009. Google Scholar [6] J. Barica and J. A. Mathias, Oxygen depletion and winterkill risk in small prairie lakes under extended ice cover, J. Fish. Res. Board Can., 36 (1979), 980-986. doi: 10.1139/f79-136. Google Scholar [7] F. Evrendilek and N. Karakaya, Monitoring diel dissolved oxygen dynamics through integrating wavelet denoising and temporal neural networks, Environ. Monit. Assess., 186 (2014), 1583-1591. doi: 10.1007/s10661-013-3476-9. Google Scholar [8] X. Fang and H. G. Stefan, Simulated climate change effects on dissolved oxygen characteristics in ice-covered lakes, Ecol. Model., 103 (1997), 209-229. doi: 10.1016/S0304-3800(97)00086-0. Google Scholar [9] B. Foley, I. D. Jones, S. C. Maberly and B. Rippey, Long-term changes in oxygen depletion in a small temperate lake: Effects of climate change and eutrophication, Freshwater Biol., 57 (2012), 278-289. doi: 10.1111/j.1365-2427.2011.02662.x. Google Scholar [10] S. Golosov, O. A. Maher, E. Schipunova, A. Terzhevik, G. Zdorovennova and G. Kirillin, Physical background of the development of oxygen depletion in ice-covered lakes, Oecologia, 151 (2007), 331-340. doi: 10.1007/s00442-006-0543-8. Google Scholar [11] K. Jylhä, M. Laapas, K. Ruosteenoja, L. Arvola, A. Drebs, J. Kersalo, S. Saku, H. Gregow, H. R. Hannula and P. Pirinen, Climate variability and trends in the Valkea-Kotinen region, southern Finland: Comparisons between the past, current and projected climates, Boreal Environ. Res., 19 (2014), 4-30. Google Scholar [12] U. T. Khan and C. Valeo, A new fuzzy linear regression approach for dissolved oxygen prediction, Hydrolog. Sci. J., 60 (2015), 1096-1119. doi: 10.1080/02626667.2014.900558. Google Scholar [13] G. Kirillin, M. Leppäranta, A. Terzhevik, N. Granin, J. Bernhardt, C. Engelhardt, T. Efremova, S. Golosov, N. Palshin, P. Sherstyankin, G. Zdorovennova and R. Zdorovennov, Physics of seasonally ice-covered lakes: A review, Aquat. Sci., 74 (2012), 659-682. doi: 10.1007/s00027-012-0279-y. Google Scholar [14] J. A. Mathias and J. Barica, Factors controlling oxygen depletion in ice-covered lakes, Can. J. Fish. Aquat. Sci., 37 (1980), 185-194. doi: 10.1139/f80-024. Google Scholar [15] M. E. Meding and L. J. Jackson, Biological implications of empirical models of winter oxygen depletion, Can. J. Fish. Aqua. Sci., 58 (2001), 1727-1736. doi: 10.1139/f01-109. Google Scholar [16] J. C. Patterson, B. R. Allanson and G. N. Ivey, A dissolved oxygen budget model for Lake Erie in summer, Freshwater Biol., 15 (1985), 683-694. doi: 10.1111/j.1365-2427.1985.tb00242.x. Google Scholar [17] V. Ranković, J. Radulović, I. Radojević, A. Ostojić and L. Čomić, Neural network modeling of dissolved oxygen in the Gruža reservoir, Serbia, Ecol. Model., 221 (2010), 1239-1244. Google Scholar [18] J. Ruuhijärvi, M. Rask, S. Vesala, A. Westermark, M. Olin, J. Keskitalo and A. Lehtovaara, Recovery of the fish community and changes in the lower trophic levels in a eutrophic lake after a winter kill of fish, Hydrobiologia, 646 (2010), 145-158. Google Scholar [19] H. G. Stefan, M. Hondzo, X. Fang, J. G. Eaton and J. H. Mccormick, Simulated long-term temperature and dissolved oxygen characteristics of lakes in the north-central United States and associated fish habitat limits, Limnol. Oceanogr., 41 (1996), 1124-1135. doi: 10.4319/lo.1996.41.5.1124. Google Scholar [20] H. G. Stefan and X. Fang, Dissolved oxygen model for regional lake analysis, Ecol. Model., 71 (1994), 37-68. doi: 10.1016/0304-3800(94)90075-2. Google Scholar [21] J. Vuorenmaa, K. Salonen, L. Arvola, J. Mannio, M. Rask and P. Horppila, Water quality of a small headwater lake reflects long-term variations in deposition, climate and in-lake processes, Boreal Environ. Res., 19 (2014), 47-65. Google Scholar [22] Y. Wang, L2 Theory of Partial Differential Equations, 1st edition, Peking University Press, Beijing, 1989.Google Scholar [23] Y. Zhang, Z. Wu, M. Liu, J. He, K. Shi, Y. Zhou, M. Wang and X. Liu, Dissolved oxygen stratification and response to thermal structure and long-term climate change in a large and deep subtropical reservoir (Lake Qiandaohu, China), Water Res., 75 (2015), 249-258. doi: 10.1016/j.watres.2015.02.052. Google Scholar

show all references

##### References:
 [1] K. Addy and L. Green, Dissolved oxygen and temperature, PLA Notes, 22 (2004), 48-65. Google Scholar [2] V. Z. Antonopoulos and S. K. Gianniou, Simulation of water temperature and dissolved oxygen distribution in Lake Vegoritis, Greece, Ecol. Model., 160 (2003), 39-53. doi: 10.1016/S0304-3800(02)00286-7. Google Scholar [3] L. Arvola, K. Salonen, J. Keskitalo, T. Tulonen, M. Järvinen and J. Huotari, Plankton metabolism and sedimentation in a small boreal lake -a long-term perspective, Boreal Environ. Res., 19 (2014), 83-97. Google Scholar [4] J. Babin and E. E. Prepas, Modelling winter oxygen depletion rates in ice-covered temperate zone lakes in Canada, Can. J. Fish. Aquat. Sci., 42 (1985), 239-249. doi: 10.1139/f85-031. Google Scholar [5] Q. Bai, R. Li, Z. Li, M. Leppäranta, L. Arvola and M. Li, Time-series analyses of water temperature and dissolved oxygen concentration in Lake Valkea-Kotinen (Finland) during ice season, Ecol. Inform., 36 (2016), 181-189. doi: 10.1016/j.ecoinf.2015.06.009. Google Scholar [6] J. Barica and J. A. Mathias, Oxygen depletion and winterkill risk in small prairie lakes under extended ice cover, J. Fish. Res. Board Can., 36 (1979), 980-986. doi: 10.1139/f79-136. Google Scholar [7] F. Evrendilek and N. Karakaya, Monitoring diel dissolved oxygen dynamics through integrating wavelet denoising and temporal neural networks, Environ. Monit. Assess., 186 (2014), 1583-1591. doi: 10.1007/s10661-013-3476-9. Google Scholar [8] X. Fang and H. G. Stefan, Simulated climate change effects on dissolved oxygen characteristics in ice-covered lakes, Ecol. Model., 103 (1997), 209-229. doi: 10.1016/S0304-3800(97)00086-0. Google Scholar [9] B. Foley, I. D. Jones, S. C. Maberly and B. Rippey, Long-term changes in oxygen depletion in a small temperate lake: Effects of climate change and eutrophication, Freshwater Biol., 57 (2012), 278-289. doi: 10.1111/j.1365-2427.2011.02662.x. Google Scholar [10] S. Golosov, O. A. Maher, E. Schipunova, A. Terzhevik, G. Zdorovennova and G. Kirillin, Physical background of the development of oxygen depletion in ice-covered lakes, Oecologia, 151 (2007), 331-340. doi: 10.1007/s00442-006-0543-8. Google Scholar [11] K. Jylhä, M. Laapas, K. Ruosteenoja, L. Arvola, A. Drebs, J. Kersalo, S. Saku, H. Gregow, H. R. Hannula and P. Pirinen, Climate variability and trends in the Valkea-Kotinen region, southern Finland: Comparisons between the past, current and projected climates, Boreal Environ. Res., 19 (2014), 4-30. Google Scholar [12] U. T. Khan and C. Valeo, A new fuzzy linear regression approach for dissolved oxygen prediction, Hydrolog. Sci. J., 60 (2015), 1096-1119. doi: 10.1080/02626667.2014.900558. Google Scholar [13] G. Kirillin, M. Leppäranta, A. Terzhevik, N. Granin, J. Bernhardt, C. Engelhardt, T. Efremova, S. Golosov, N. Palshin, P. Sherstyankin, G. Zdorovennova and R. Zdorovennov, Physics of seasonally ice-covered lakes: A review, Aquat. Sci., 74 (2012), 659-682. doi: 10.1007/s00027-012-0279-y. Google Scholar [14] J. A. Mathias and J. Barica, Factors controlling oxygen depletion in ice-covered lakes, Can. J. Fish. Aquat. Sci., 37 (1980), 185-194. doi: 10.1139/f80-024. Google Scholar [15] M. E. Meding and L. J. Jackson, Biological implications of empirical models of winter oxygen depletion, Can. J. Fish. Aqua. Sci., 58 (2001), 1727-1736. doi: 10.1139/f01-109. Google Scholar [16] J. C. Patterson, B. R. Allanson and G. N. Ivey, A dissolved oxygen budget model for Lake Erie in summer, Freshwater Biol., 15 (1985), 683-694. doi: 10.1111/j.1365-2427.1985.tb00242.x. Google Scholar [17] V. Ranković, J. Radulović, I. Radojević, A. Ostojić and L. Čomić, Neural network modeling of dissolved oxygen in the Gruža reservoir, Serbia, Ecol. Model., 221 (2010), 1239-1244. Google Scholar [18] J. Ruuhijärvi, M. Rask, S. Vesala, A. Westermark, M. Olin, J. Keskitalo and A. Lehtovaara, Recovery of the fish community and changes in the lower trophic levels in a eutrophic lake after a winter kill of fish, Hydrobiologia, 646 (2010), 145-158. Google Scholar [19] H. G. Stefan, M. Hondzo, X. Fang, J. G. Eaton and J. H. Mccormick, Simulated long-term temperature and dissolved oxygen characteristics of lakes in the north-central United States and associated fish habitat limits, Limnol. Oceanogr., 41 (1996), 1124-1135. doi: 10.4319/lo.1996.41.5.1124. Google Scholar [20] H. G. Stefan and X. Fang, Dissolved oxygen model for regional lake analysis, Ecol. Model., 71 (1994), 37-68. doi: 10.1016/0304-3800(94)90075-2. Google Scholar [21] J. Vuorenmaa, K. Salonen, L. Arvola, J. Mannio, M. Rask and P. Horppila, Water quality of a small headwater lake reflects long-term variations in deposition, climate and in-lake processes, Boreal Environ. Res., 19 (2014), 47-65. Google Scholar [22] Y. Wang, L2 Theory of Partial Differential Equations, 1st edition, Peking University Press, Beijing, 1989.Google Scholar [23] Y. Zhang, Z. Wu, M. Liu, J. He, K. Shi, Y. Zhou, M. Wang and X. Liu, Dissolved oxygen stratification and response to thermal structure and long-term climate change in a large and deep subtropical reservoir (Lake Qiandaohu, China), Water Res., 75 (2015), 249-258. doi: 10.1016/j.watres.2015.02.052. Google Scholar
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
 [1] Pingping Niu, Shuai Lu, Jin Cheng. On periodic parameter identification in stochastic differential equations. Inverse Problems & Imaging, 2019, 13 (3) : 513-543. doi: 10.3934/ipi.2019025 [2] Roberto Camassa, Pao-Hsiung Chiu, Long Lee, W.-H. Sheu. A particle method and numerical study of a quasilinear partial differential equation. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1503-1515. doi: 10.3934/cpaa.2011.10.1503 [3] Angelo Favini. A general approach to identification problems and applications to partial differential equations. Conference Publications, 2015, 2015 (special) : 428-435. doi: 10.3934/proc.2015.0428 [4] Heiko Enderling, Alexander R.A. Anderson, Mark A.J. Chaplain, Glenn W.A. Rowe. Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology. Mathematical Biosciences & Engineering, 2006, 3 (4) : 571-582. doi: 10.3934/mbe.2006.3.571 [5] Hedia Fgaier, Hermann J. Eberl. Parameter identification and quantitative comparison of differential equations that describe physiological adaptation of a bacterial population under iron limitation. Conference Publications, 2009, 2009 (Special) : 230-239. doi: 10.3934/proc.2009.2009.230 [6] Eleonora Messina. Numerical simulation of a SIS epidemic model based on a nonlinear Volterra integral equation. Conference Publications, 2015, 2015 (special) : 826-834. doi: 10.3934/proc.2015.0826 [7] Walid K. Abou Salem, Xiao Liu, Catherine Sulem. Numerical simulation of resonant tunneling of fast solitons for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1637-1649. doi: 10.3934/dcds.2011.29.1637 [8] Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061 [9] Avner Friedman, Harsh Vardhan Jain. A partial differential equation model of metastasized prostatic cancer. Mathematical Biosciences & Engineering, 2013, 10 (3) : 591-608. doi: 10.3934/mbe.2013.10.591 [10] Yuepeng Wang, Yue Cheng, I. Michael Navon, Yuanhong Guan. Parameter identification techniques applied to an environmental pollution model. Journal of Industrial & Management Optimization, 2018, 14 (2) : 817-831. doi: 10.3934/jimo.2017077 [11] François James, Nicolas Vauchelet. One-dimensional aggregation equation after blow up: Existence, uniqueness and numerical simulation. Networks & Heterogeneous Media, 2016, 11 (1) : 163-180. doi: 10.3934/nhm.2016.11.163 [12] Gong Chen, Peter J. Olver. Numerical simulation of nonlinear dispersive quantization. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 991-1008. doi: 10.3934/dcds.2014.34.991 [13] Nicolas Vauchelet. Numerical simulation of a kinetic model for chemotaxis. Kinetic & Related Models, 2010, 3 (3) : 501-528. doi: 10.3934/krm.2010.3.501 [14] Petr Bauer, Michal Beneš, Radek Fučík, Hung Hoang Dieu, Vladimír Klement, Radek Máca, Jan Mach, Tomáš Oberhuber, Pavel Strachota, Vítězslav Žabka, Vladimír Havlena. Numerical simulation of flow in fluidized beds. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 833-846. doi: 10.3934/dcdss.2015.8.833 [15] Tuhin Ghosh, Karthik Iyer. Cloaking for a quasi-linear elliptic partial differential equation. Inverse Problems & Imaging, 2018, 12 (2) : 461-491. doi: 10.3934/ipi.2018020 [16] Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9 [17] Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907 [18] Susanna V. Haziot. Study of an elliptic partial differential equation modelling the Antarctic Circumpolar Current. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4415-4427. doi: 10.3934/dcds.2019179 [19] Shihchung Chiang. Numerical optimal unbounded control with a singular integro-differential equation as a constraint. Conference Publications, 2013, 2013 (special) : 129-137. doi: 10.3934/proc.2013.2013.129 [20] Qinqin Chai, Ryan Loxton, Kok Lay Teo, Chunhua Yang. A unified parameter identification method for nonlinear time-delay systems. Journal of Industrial & Management Optimization, 2013, 9 (2) : 471-486. doi: 10.3934/jimo.2013.9.471

2018 Impact Factor: 1.025

## Metrics

• PDF downloads (56)
• HTML views (936)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]