Estimating coordinates of a 2D-point from multiple observations

It is a common practice in Geosciences to estimate coordinates of a two-dimensional (2D) point from multiple observations. For example, the “three-point resection” method in Geodesy typically applies a simple vector mean model to estimate the coordinates of a 2D point from three-point observations. In GIS, topological errors, such as unsnapped points or unaligned digital maps, are cleaned to generate a 2D point or multiple 2D points by node snapping or map conflations. In these data processes, the point observations are often assumed to be statistically independent. However, digital points collected from surveying in Geodesy and digitizing in GIS are often dependent in statistics and varied in accuracy. The dependency among digital points and the differences in their accuracy may have a considerable impact on the estimates of the coordinates for the 2D point. To examine the impact, this paper first develops a generalized model, in which the dependency and the differing accuracy of the multiple observations are taken into account by using the least-squares technique to estimate the finalized coordinates of the 2D point. The generalized model is further simplified into eight specialized models according to the accuracy of the observations and the degree of their correlations. The specialized models can be used to analytically examine the propagation of the correlations and accuracy from the multiple observations to the finalized coordinates estimated for the 2D point. In particular, numerical simulations are conducted to observe the impact of ignoring dependency among observations on the finalized coordinates. Results show that either the dependency of the observations or their varying accuracy has a substantial impact on the estimated coordinates of the 2D point, highlighting a need for caution in selecting an appropriate model for estimating the coordinates of a 2D point with multiple observations.