Sovremennye problemy distantsionnogo zondirovaniya Zemli iz kosmosa, 2025, V. 22, No. 6, pp. 182-190
Scaling of relief characteristic equations
1 Space Research Institute RAS, Moscow, Russia
Accepted: 02.10.2025
DOI: 10.21046/2070-7401-2025-22-6-182-190
The article studies the demonstration of scale invariance of relief in equations describing its hydrological characteristics. Equation is scale-invariant if all distances and time intervals change by the same number of times. We experimentally found relationships between relief characteristics that generalize such fundamental dependencies as Hack’s law, several Horton’s relations and Tokunaga matrix. These relationships turned out to be scale-invariant. Moreover, if we assume that the dependencies that connect the number, length of watercourses and the area of their catchment are scale-invariant, then we can formally find these relationships with an accuracy of a factor. The factors we determine experimentally, focusing on the scales range that yields a formula close to the one found formally. Thus, these dependencies follow from the principle of scale invariance of relief. Based on this principle, we can try to find other patterns for relief characteristics before empirical verification. With a different, geometric method of relief analysis, characteristics with close values and also with scale-invariant relationships are obtained. Not all dependencies of relief characteristics are scale-invariant. Thus, orientation characteristics are tied to a certain scale.
Keywords: scale invariance of relief, DTM, runoff model, valley axes, statistical characteristics of streams and valleys
Full textReferences:
- Zakharov V. S., Simonov D. A., Bryantseva G. V., Kosevich N. I., Self-similarity parameters of the Kerch Peninsula water streams system and their comparison with the results of structural and geomorphological analysis, Geofizicheskie protsessy i biosfera, 2019, V. 18, No. 1, pp. 50–60 (in Russian), DOI: 10.21455/gpb2019.1-5.
- Zlatopolsky A. A., Multiscale digital terrain map analysis. Experimental regularities, Sovremennye problemy distantsionnogo zondirovaniya Zemli iz kosmosa, 2015, V. 12, No. 3, pp. 27–35 (in Russian).
- Zlatopolsky A. A., Using LESSA technology to obtain territory orientation characteristics. Methodology and testing with digital elevation model for the pre-Baikal region, Sovremennye problemy distantsionnogo zondirovaniya Zemli iz kosmosa, 2020, V. 17, No. 4, pp. 98–110 (in Russian), DOI: 10.21046/2070-7401-2020-17-4-98-110.
- Zlatopolsky A. A., Scale terrain statistics — orders, ranges, tributary distribution, orientation, age, scaling, Sovremennye problemy distantsionnogo zondirovaniya Zemli iz kosmosa, 2024, V. 21, No. 2, pp. 103–121 (in Russian), DOI: 10.21046/2070-7401-2024-21-2-103-121.
- Zlatopolsky A. A. (2025a), Statistical regularities of the length of runoff lines (the basis for calculating the hydrological characteristics of the relief), Sovremennye problemy distantsionnogo zondirovaniya Zemli iz kosmosa, 2025, V. 22, No. 3, pp. 109–118 (in Russian), DOI: 10.21046/2070-7401-2025-22-3-109-118.
- Zlatopolsky A. A. (2025b), Multi-scale statistical analysis of hydrological characteristics of the relief (based on runoff model rasters), Sovremennye problemy distantsionnogo zondirovaniya Zemli iz kosmosa, 2025, V. 22, No. 4, pp. 164–172 (in Russian), DOI: 10.21046/2070-7401-2025-22-4-164-172.
- Zlatopolsky A. A., Zaitsev V. A., Relationship between order and width of valleys automatically found using a digital terrain model, Sovremennye problemy distantsionnogo zondirovaniya Zemli iz kosmosa, 2021, V. 18, No. 6, pp. 141–151 (in Russian), DOI: 10.21046/2070-7401-2021-18-6-141-151.
- Mandelbrot B., The fractal geometry of nature, New York: W. H. Freeman and Co., 1982. 461 p.
- Fizicheskaya ehntsiklopediya, Prokhorov A. M. (ed.), V. 3, Moscow: Great Russian Encyclopedia, 1992, 672 p.
- Sobol’ I. S., Krasil’nikov V. M., The Sura River basin water bodies fractal parameters, Vodnoe khozyaistvo Rossii: problemy, tekhnologii, upravlenie, 2018, No. 6, pp. 4–15 (in Russian), DOI: 10.35567/1999-4508-2018-6-1.
- Chernova I. Yu., Nugmanov I. I., Dautov A. N., Application of GIS analytic functions for improvement and development of the structural morphological methods of the neotectonics studies, Geoinformatica, 2010, No. 4, pp. 9–23 (in Russian).
- Li Y., Yue Z. Q., Lee C. F. et al., Hack’s law of debris-flow basins, Intern. J. Sediment Research, 2009, V. 24, No. 1, pp. 74–87.
- Pelletier J. D., Self-organization and scaling relationships of evolving river networks, J. Geophysical Research: Solid Earth, 1999, V. 104, No. B4, pp. 7359–7375.