Solving planar intersection problem using Gauss quadrature rule exact for three-order monomials

Abstract

Intersection problem is a basic problem in geodesy and surveying. In planar intersection problem, if the observation noises of the intersection angles are assumed Gaussian, the calculation of the mean and covariance of the target’s coordinate is constructed as a multidimensional non-linear integral, i.e. the non-linear intersection functions’ Gauss weighted integral. From the perspective of numerical integration, conventional method based on the Jacobian matrix of a non-linear function is just the result of simplifying the integrated non-linear function by its one-order Taylor series truncation. The Gauss quadrature rule which is exact for all monomials of order not greater than 3, constructed by McNamee and Stenger, is adopted to numerically solve the integral, and a derivative free method (without the derivation of the Jacobian matrix of the non-linear intersection function) is proposed to calculate the mean and covariance of the target’s coordinate. The proposed method uses five elaborately sampled quadrature points and their corresponding weights to transform the mean and covariance of the intersection angles to the mean and covariance of the target’s coordinate. It is assured that both the mean and covariance of the coordinate calculated by the Gauss quadrature rule based method are exact for up to two-order terms of the Taylor series of the intersection function. Simulation is constructed in which the true mean and covariance of the target’s coordinate are obtained with Monte Carlo method using 100 000 randomly sampled intersection angle suits. The mean and covariance of the coordinate are estimated using both the conventional and the proposed method, both of which are compared to the Monte Carlo calculated true ones. Simulation results show that the Gauss quadrature rule based method can give higher precision of both the coordinate and its covariance than the conventional method. Hopefully, the work also has values for some other geodetic and surveying topics where the non-linear transformation of mean and/ or covariance is involved.