Least Squares Weighted Coordinate Transformation Formulas and Their Applications

Coordinate transformation is the process of determining and applying the relationship between two sets of coordinates. It answers the following question: “I know the coordinates of a point on one data source, what are the coordinate values for this point on another data source?” In most cases, points do not have homogeneous accuracies; thus, they must be weighted according to their relative values. An important objective of this procedure is to keep it as simple as possible so that it can be easily adopted by professional land surveyors or geographic information system (GIS) technical staff. Another important objective of the coordinate transformation process is to make it statistically sound. A weighted least squares is probably one of the better solutions for the transformation problem. Weighted least squares-based coordinate transformation has many applications in surveying practice. One application is to compute NAD83 coordinate values for points with known coordinate values in another plane coordinate system such as NAD27, UTM or NAD83 (19xx), where “xx” is the year in which the adjustment was computed. Another application is to combine maps from different sources or to register a map to an orthophoto when building a GIS. A third example of an application is to sort out evidence in boundary surveys.