# The Flux Recovering at the Ecosystem-Atmosphere Boundary by Inverse Modelling

E.I. Safonov, S.G. PyatkovWe consider the heat and mass transfer models in the quasistationary case, i. e., all coefficients and the data of the problem depends on time while the time derivative in the equation is absent. Under consideration is the inverse problem of recovering the surface flux through the values of a solution at some collection of points lying inside the domain. The flux is sought in the form of a finite segment of the Fourier series with unknown Fourier coefficients depending on time. The problem of determining the Fourier coefficient is reduced to a system of algebraic equations with the use of special solutions to the adjoint problem. The equation is considered in a cylidrical space domain. We prove the existence and uniqueness theorems for solutions of the corresponding direct problem. The results are employed in the proof of the corresponding results for the inverse problem. The corresponding numerical algorithm in the three-dimensional case is constructed and the results of the numerical experiments are exhibited. It is demonstrated that the algorithm is stable under random perturbations of the data. The finite element method is used. The results can be used in the problem of the determination of the fluxes of green house gases from soils from the concentration measurements.Full text

- Keywords
- inverse problem; flux; parabolic equation heat and mass transfer.
- References
- 1. Glagolev M.V., Sabrekov A.F., Determination of Gas Exchange on the Border between Ecosystem and Atmosphere: Inverse Modeling. Mathematical Biology and Bioinformatics, 2012, vol. 7, no. 1, pp. 81-101. DOI: 10.17537/2012.7.81

2. Sabrekov A.F., Glagolev M.V., Terent'eva I.E. Determination of Specific Methane Flux from the Soil using Inverse Modeling based on Adjoint Equations. Proceedings of the International Conference "Mathematical Biology and Bioinformatics", Puschino, 2018, vol. 7, article ID: e94. DOI: 10.17537/icmbb18.85 (in Russian)

3. Belolipetskiy V.M., Belolipetskiy P.V. Estimate of Carbon Flows between Atmosphere and Soil Ecossystem from Vertical Distribution of CO_2 Concentrations Measured at a Tower. Bulletin of the Novosibirsk State University. Series: Information Technologies, 2011, vol. 9, no. 1, pp. 75-81. (in Russian)

4. Yagovkina S.V., Karol' I.L., Zubov V.A., at al. Estimates of Methane Flows in Atmosphere from Methane Flows into the Atmosphere from Gas Fields of the North of Western Siberia Using Three-Dimensional Regional Transport Models. Meteorology and Hydrology, 2003, no. 4, pp. 49-62. (in Russian)

5. Borodulin A.I., Desyatkov B.D., Makhov G.A., Sarmanaev S.R. Determination of the Wetland Methane Emission from the Measured Values of its Concentration in the Surface Layer of the Atmosphere. Meteorology and Hydrology, 1997, no. 1, pp. 66-74. (in Russian)

6. Borodulin A.I., Makhov G.A., Desyatkov B.D., Sarmanaev S.R. Statistical Characteristics of the Methane Flow Released Swampy Underlying Surface. Doklady RAN, 1996, vol. 349, no. 2, pp. 256-258. (in Russian)

7. Borodulin A.I., Makhov G.A., Sarmanaev S.R., Desyatkov B.D. On Methane Distribution over a Wetlands. Meteorology and Hydrology, 1995, no. 11, pp. 72-79. (in Russian)

8. Berlyand M.Е. Pollution Forecast and Regulation of the Atmosphere, Leningrad, Gidrometeizdat, 1985. (in Russian)

9. Onyango T.M., Ingham D.B., Lesnic D. Restoring Boundary Conditions in Heat Conduction. Journal of Engineering Mathematics, 2007, vol. 62, pp. 85-101.

10. Hussein M.S., Lesnic D., Ivanchov M.I. Simultaneous Determination of Time-Dependent Coefficients in the Heat Equation. Computers and Mathematics with Applications, 2014. vol. 67, no. 5, pp. 1065-1091. DOI: 10.1016/j.camwa.2014.01.004

11. Kostin A.B., Prilepko A.I. Some Problems of Restoring the Roundary Condition for a Parabolic Equation, II. Differential Equations. 1996, vol. 32, no. 11, pp. 1515-1525.

12. Kostin A.B., Prilepko A.I. On Some Problems of Restoration of a Boundary Condition for a Parabolic Equation, I. Differential Equations, 1996, vol. 32, no. 1, pp. 113-122.

13. Pyatkov S.G., Baranchuk V.A. Determination of the Heat Transfer Coefficient in Mathematical Models of Heat and Mass Transfer. Mathematical Notes, 2023, vol. 113, no. 1, pp. 93-108. DOI: 10.1134/S0001434623010108

14. Pyatkov S., Shilenkov D. Existence and Uniqueness Theorems in the Inverse Problem of Recovering Surface Fluxes from Pointwise Measurements. Mathematics, 2022, vol. 10, no. 9, article ID: 1549, 23 p. DOI: 10.3390/math10091549

15. Wang Shoubin, Zhang Li, Sun Xiaogang, Jia Huangchao. Solution to Two-Dimensional Steady Inverse Heat Transfer Problems with Interior Heat Source Based on the Conjugate Gradient Method. Mathematical Problems in Engineering, 2017, vol. 8, article ID: 2861342, 9 p. DOI: 10.1155/2017/2861342

16. Knupp D.C., Abreu L.A.S. Explicit Boundary Heat Flux Reconstruction Employing Temperature Measurements Regularized via Truncated Eigenfunction Expansions. International Communications in Heat and Mass Transfer, 2016, vol. 78, pp. 241-252. DOI: 10.1016/j.icheatmasstransfer.2016.09.012

17. Glagolev M.V. Inverse Modelling Method for the Determination of the Gas Flux from the Soil. Environmental Dynamics and Global Climate Change, 2010, vol. 1, no. 1, pp. 17-36. DOI: 10.17816/edgcc1117-36

18. Glagolev M.V., Kotsyurbenko O.R., Sabrekov A.F., Litti Y.V., Terentieva I.E. Methodologies for Measuring Microbial Methane Production and Emission from Soils. A Review. Microbiology, 2021, vol. 90, no. 1, pp. 1-19. DOI: 10.1134/S0026261721010057

19. Desyatkov B.M., Borodulin A.I., Kotlyarova S.S. Determination of Aerosol Particle Flow Emitting from an Underlying Surface by Solving the Inverse Problem of their Propagation in the Atmosphere. Optics of Atmosphere and Ocean, 1997, vol. 10, no. 6, pp. 639-643. (in Russian)

20. Glagolev M.V., Sabrekov A.F. About a Features of Flux Measurements at Ecosystem-Atmosphere Boundary by Inverse Modelling. Conference: Seventh All-Russian Scientific School of Young Scientists with International Participation ``SWAMPS AND BIOSPHERE'', Tomsk, 2010, p. 31-35. DOI: 10.13140/2.1.5107.6800 (in Russian)

21. Marchuk G.I. Mathematical Models in Environmental Problems. Studies in Mathematics and its Applications, vol. 16, Amsterdam, Elsevier Science Publishers, 1986.

22. Triebel H. Interpolation Theory. Function Spaces. Differential Operators. Berlin, VEB Deutscher Verlag der Wissenschaften, 1978.

23. Amann H. Compact Embeddings of Vector-Valued Sobolev and Besov Spaces. Glasnik Matematicki, 2000, vol. 35 (55), pp. 161-177.

24. Denk R., Hieber M., Pruss J. R-Boundedness, Fourier Multipliers, and Problems of Elliptic and Parabolic Type. Memoirs of the American Mathematical Society, 2003, vol. 166, no. 788, pp. 1-114.

25. Grisvard P. Equations Differentielles Abstraites. Annales Scientifiques de l'Ecole Normale Superieure, 1969, vol. 2, no. 3, pp. 311-395. (in French)

26. Amann H. Nonautonomous Parabolic Equations Involving Measures. Journal of Mathematical Sciences, 2005, vol. 130, no. 4, pp. 4780-4802. DOI: 10.1007/s10958-005-0376-8

27. Ciarlet P.G. The Finite Element Method for Elliptic Problems. Amsterdam, Noth-Holland Publishing Company, 1978.