Sovremennye problemy distantsionnogo zondirovaniya Zemli iz kosmosa, 2024, Vol. 21, No. 6, pp. 159-167
Statistical scale relationships of relief characteristics (based on runoff model rasters)
1 Space Research Institute RAS, Moscow, Russia
Accepted: 30.09.2024
DOI: 10.21046/2070-7401-2024-21-6-159-167
The article describes in detail the method of obtaining multiscale relationships of relief characteristics from the frequency functions of those rasters that a standard runoff model creates in a GIS (geographic information system) using a DTM (digital terrain model). Experimental results are presented for five dissimilar territories, with different DEM resolutions, but with fluvial relief. The frequency of occurrence of certain values in the raster is measured. The frequency functions of the corresponding rasters are the same in all territories, both in shape and structure, and are close in absolute values. Accordingly, the approximations of the frequency functions of different territories are also close. The length of the drainage line was used as a scaling parameter. Formulas are presented for the dependence on the scale of the number and density of watercourses and their average catchment area. These relations generalize such well-known regularities as Horton’s relations and Hack’s law. The relationships are homogeneous power relations and are determined by their exponents, which we called Horton exponents. The total catchment area of all points with the same drainage line length is practically constant and does not depend on the length value. This area is close to 0.3 of the area of the territory with very small difference between territories. The three relationships we obtained are interconnected — two can be obtained from the third (any) and from the constancy of the total catchment area. The exact correspondence of the dimensions of the left and right sides of these relations suggests that statistical physical laws have been determined. We assume that one law is the dependence of the number of drainage lines on their length, and the second is the constancy of the total catchment area of all points with the same drainage line length. The other two relations are consequences of them. It is shown that the found relationships are not affected in any way by changing the DEM projection from equiangular to equal-area.
Keywords: DTM, pixel catchment area, length of drainage line in pixel, scale of streams, statistical characteristics of streams, Horton’s laws, Horton’s exponents, Hack’s law
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