Volume 12, no. 3Pages 63 - 73

# Mathematical Terrain Modelling with the Help of Modified Gaussian Functions

V.A. Rodin, S.V. Sinegubov
Based on a fundamentally new approach, we present a complete mathematical model for estimating the mass of water in the flooded coastal relief, taking into account the water in the basin of the reservoir in a given region. Taking into account stochastic studies, we construct an approximate model of the relief of the reservoir basin bottom, as well as the relief of a possible section of the flooding of this basin coastline. The modelling is based on the empirical data of measurements of the reservoir depths, as well as on the study on the architecture of the lines of the coastal maps of the possible flooding zone. Based on the measurements of the depths and bumps of the bottom surface, we verify the hypothesis that the use of the two-dimensional Gauss distribution is adequate. Numerous confirmation of this hypothesis on the basis of empirical measurements allows to use localized elliptic Gauss surfaces as a model function in order to construct an approximate model of hillocks and valleys. At the same time, the coordinates of local extremes of the depths, as well as the values of these extremes are constant. In order to simulate the surfaces of the underwater slopes, we construct planes according to depth measurements. This simulation is not a real copy, but is stochastic in nature and allows to take into account the main goal of the model, i.e. a full adequate estimation of the water mass of the flooded coastal relief included the water in the basin of the reservoir in the region. The equation of the model of the entire flooded region includes all local functions constructed for the mounds and troughs of the reservoir, as well as the functions of the planes of the slope models. For an approximate construction of the surface equations of the coastal zone, we use maps with detailed level lines as empirical data.
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Keywords
mathematical terrain modelling; numerical methods; computer modelling; statistical hypothesis verification.
References
1. Bizov L.M., Sankov V.A. [Mathematical Modelling of the Evolution of the Relief of the Waste Ledge on the Example of the Svyatonossky Uplift (Baikal Basin)]. Izvestiya Irkutskogo gosudarstvennogo universiteta. Series: Nauki o Zemle, 2015, vol. 12, pp. 12-22. (in Russian)
2. Dobretsov N.L., Kirdyashkin A.G. Glubinnayi geodinamika [Deep Geodynamics]. Novosibirsk, Geo, 2001. (in Russian)
3. Yampolsky S.M., Naumov A.I., Kichigin E.K., Rubinov V.I., Mokh Ahmed Medani Ahmed Elamin. [Statistical Model of the Predicted Profile of the Terrain in the Task of Performing Low-Altitude Flight of an Aircraft Using a Digital Elevation Map]. Trudy MAI, 2013, no. 76, pp. 392-393. (in Russian)
4. Naumov A.I., Kichigin E.K., Mokh Ahmed Medani Ahmed Elamin. [Techniques Calculating the Statistical Characteristics of the Elevation Estimate Using a Digital Elevation Map of the Terrain with a Random Character Coordinates Calculation]. Akademicheskie Zhukovskie chteniya, Voronezh, 2013, pp. 38-42. (in Russian)
5. Sizdikova G.D. [The Distribution Function of the Morphometric Trait, Taking into Account the Geometrical Parameters of the Dismemberment of the Relief]. Vestnik KazNRTU, 2015, vol. 110, no. 4, pp. 70-74. (in Russian)
6. Semenova O.M. Analysis and Modelling of the Formation of Runoff in Little-Studied Basins (For Example, the Lena River Basin). PhD Thesis, 2008, St. Petersburg. (in Russian)
7. Arifulin E.Z. [Evaluation of the Effectiveness of the Method of Restoring the Terrain. Based on Map Data]. Bulletin of the Voronezh State Technical University, 2014, vol. 10, no. 3, pp. 10-12. (in Russian)
8. Kulazhskiy A.V. Digital and Mathematical Modelling of the Terrain in Computer-Aided Design of Railways. PhD Thesis, St. Petersburg, 2011. (in Russian)
9. Korobeynikov S.N., Reverdatto V.V., Polyanskiy O.P., Sverdlova V.G., Babichev A.V. [Formation of the Surface Relief in the Area of Plate Collisions: Mathematical Modelling]. Applied Mechanics and Technical Physics, 2012, vol. 53, no. 4, pp. 124-137. (in Russian) DOI: 10.1134/S0021894412040128
10. Krivtsova I.E., Lebedev I.S., Nasteka A.V. Osnovi diskretnoi matematiki. Chast 1 [Basics of Discrete Mathematics. Part 1]. St. Petersburg, St. Petersburg State University of Information Technologies, Mechanics and Optics, 2016. (in Russian)
11. Khromykh V.V., Khromykh O.V. Cifrovie modeli relefa [Digital Elevation Models]. Tomsk, TML-Press, 2007. (in Russian)