An analysis of errors introduced by the use of transformation grids

Abstract

In this paper we analyze the errors introduced by the use of transformation grids. A transformation grid is an intermediate step during the transformation of coordinates of points attached to distinct geodetic reference frames. The use of transformation grids simplifies the transformation when compared to the reference transformation that the grid represents. Transformation grids have become a standard way of making transformation distortion models available for end-users. They are widely accepted by the GIS industry, being already supported by a host of commercial and free programs. They are adopted in countries like Canada and Australia, and are currently being considered for adoption in Brazil. The work described in this paper was conducted by addressing a number of questions in the following sequence: (i) "Is there an upper bound in the error introduced by a transformation grid?"; (ii) "What is the coarsest spacing between nodes for a transformation grid to introduce only negligible errors?"; and, (iii) "How does the error introduced by a transformation grid vary spatially?" To answer these questions we transformed a set of random test points twice, once using a transformation grid and once using the reference transformation that the grid represents. Then, we analyzed the difference between the two results. We show that: (i) yes, there is an upper bound in the error introduced by the grid; (ii) the coarsest spacing can be found by plotting error versus distance to nearest grid node; and, (iii) the error varies spatially partially in proportion to the norm of the second derivative of the reference transformation. © Springer-Verlag Berlin Heidelberg 2007.