Monte-Carlo-based uncertainty propagation in the context of Gauss–Markov model: a case study in coordinate transformation

Resumo

One of the main tasks of professionals in the Earth sciences is to convert coordinates from one reference frame into another. The coordinates transformation is widely needed and applied in all branches of modern geospatial activities. To convert from one reference frame to another, it is necessary initially to determine the parameters of a coordinate transformation model. A common way to estimate the transformation parameters is using the least-squares theory within a linearized Gauss-Markov Model (GMM). Another approach arises from a numerical method. Here, a Monte Carlo Method (MCM) is used to infer the uncertainty of estimators. This method is based on: (i) the assignment of probability distributions to the coordinates in the two reference frames, (ii) the determination of a discrete representation of the probability distribution for the transformation parameters, and (iii) the determination of the associated uncertainties from this discrete representation of the estimates of the transformation parameters. In this contribution, we compare the weighted least-squares within the GMM (WLSE-GM) and the proposed method regarding the transformation problem of computing 2D similarity transformation parameters. The results show that, the transformation parameters uncertainties are higher for the LS-MC than the WLSE-GM. This is due to the fact that the WLSE-GM solution does not take into account the uncertainties associated with the system matrix. In future studies the Monte Carlo method should be applied to the nonlinear least-squares solution.