Simulação de Monte Carlo em redes triangulares irregulares / Monte Carlo simulation in triangular irregular networks

Abstract

Triangular Irregular Networks (TINs) are one of the most used ways to represent surface topology when working with Digital Terrain Models (DTM) or Geographic Information Systems (GIS). Given this form of representation, this article investigates one probabilistic demonstration to quantify how each point's accuracy σ (sigma) can be considered to have ambiguity, from the topological point of view, in any new 2D Delaunay Triangulation. To achieve it, this research designed an initial demonstration that, there is a maximum precision for which the network topology remains constant in a new Delaunay Triangulation, at each point and in the TIN as a whole. The methodological approach was experimental, with various mathematical experiments carried out using the Monte Carlo Simulation method. First, for each point of the network, and then for all network points for varied σ. The experiments culminate in helping to solve the problem of the existence of maximum σ for which the probability of occurrence in constant TIN topology is 100%. The mathematical results originated the following statement: Considering a TIN generated by Delaunay Triangulation, if any point of coordinates (x i , y i ) in a Triangular Irregular Network is disrupted (i.e., has its place altered), according to a Normal distribution N(μ, σ 2 ), then, exists a value σ max (sigma maximum) for which the topology of the network remains constant. For example, it was found that σ max.1 of one point exists and is obtained by σ max.1 = 0.30866, and at another point, σ max.2 = 0.2. The results also indicate the following for TIN: Every two-dimensional Triangular Irregular Network generated by the Delaunay Triangulation has a value σ * (sigma asterisk) to which the network topology remains constant. In this work, simulating the worst case of a Triangular Irregular Network: σ * = 0.2. Finally, it is concluded that the σ maximum for each point exists, as well as for the network as a whole. However, the results need to be tested in more extensive networks to prove (or not) if it always