Mathematical Models the Spatial Distribution of Pollutants in Soil Using Methods of Polynomial Interlination of Functions on a System of Observation Lines Passing Through Regularly Spaced Soil Sampling Points
Year: 2026
Type: article
Abstract: Monitoring and mathematical modeling of the spatial distribution of heavy metal soil pollution is a critical environmental task, particularly under theconditions of increasing anthropogenic pressure, industrial emissions, and the consequences of warfare. The object of this study is a threedimensional concentration field of heavy metals within a soil mass. The complexity of its modeling lies in the fact that traditional core sampling is destructive anddiscrete, whereas modern non-contact methods provide information in the form of continuous linear structures – namely, vertical measurement profiles.This paper addresses the problem of reconstructing a scalar function of three variables that describes the volumetric concentration of a pollutantbased on the known traces (profile functions) of this function on a system of regular vertical observation lines. To solve this problem, the framework offunction approximation theory is applied, specifically the methods of polynomial interlination of functions on a system of lines. Classical Lagrangebasis polynomials are utilized as approximation operators for individual spatial coordinates. A theorem on the structure of the polynomial interlination operator of a spatial field on a system of regular vertical profiles is formulated and mathematically proven. It is demonstrated that the constructed operator accurately reproduces experimental data at the nodes of the monitoring grid, thereby satisfying the interlination conditions. Analytical expressions for the interlination error at each fixed depth value are obtained, enabling the evaluation of the quality of the field approximation. A numerical experiment was conducted using a test function that simulates the behavior of pollutants in soil, modified by a piecewise-smooth barrier to mimic a geochemical barrier within the soil column. The results show that the proposed method ensures high accuracy in reconstructing the spatial field. Thisapproach allows for the reconstruction of linear surface scanning data from remote sensing alongside vertical profiles from subsurface geophysicalscreening, significantly reducing the need for costly and destructive core drilling.
Language: English
Topic: Heavy metals in environment
Subfield: Pollution
Field: Environmental Science
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W7224430307