Mathematical Modelling of the Black Sea Coastline Considering its Fractal Structure and Grid Generation

Source abstract

Introduction . The coastline of a reservoir is a complex natural object, which geometry has irregularity and signs of selfsimilarity, and the length depends on the scale of measurement. Despite the widespread study of its fractal properties, the transition to a digital representation of the coastline is a complex procedure and is a current topic of research. The purpose of the article is to develop a mathematical-algorithmic scheme for analyzing the coastline, including geometry validation, geodetic measurement, scale dependence analysis, fractal dimension assessment and selection of an adaptive quadrilateral grid using the example of the Black Sea. Materials and Methods . Normalization and validation procedures are applied to the coastline represented as a finite sequence of geographic points to obtain an intermediate, geometrically plausible contour. The contour length is calculated on a sphere using the haversine formula. The “box-counting” method is used to address the coastline՚s multi-scale nature and fractal characteristics. To transition from the line to a 2D representation of the water area, the Lambert azimuthal equal-area (LAEA) projection is used, followed by the construction of a grid with the required cell size. Subsequently, detailed and coarse full-quadrilateral grids are generated using the Delaunay, Frontal-Delaunay for Quads, and Packing of Parallelograms algorithms. A comparison of the grid generators is then conducted. Results . A computational experiment was conducted using Black Sea coastline data. A comparison of two sets of grids demonstrated that Delaunay is the superior generator based on the specified multi-criteria metric. The Delaunay method with a range of 125–250 m should be used to preserve significant coastal features across the entire water area. A highresolution Delaunay configuration with a range of 50–250 m should be selected when accurate reconstruction of the seabed topography is the priority. Discussion . The practical significance lies in the ability to prepare verified contours and grids for seabed topography modelling, geoinformation analysis, and subsequent hydrodynamic calculations. Conclusions . Future research prospects involve expanding the range of coastal systems analyzed, comparing various methods for estimating fractal dimension, investigating the impact of source geodata spatial resolution on the stability of calculated metrics, and adapting the approach for multi-scale monitoring of coastal dynamics.