Least Squares Fitting Method Based on Bivariate Nonuniform B-Spline and Its Applications in Surveying Engineering

摘要

As a well-known numerical approximation method, the least squares fitting method is widely applied in surveying engineering. The most important and difficult problem in this method is to determine the type of base function to be used, and polynomial is the most frequently used type. But if the degree of polynomial is too high (3≥7), the normal equations are usually ill-conditioned. When we choose the uniform B-spline as the basis, there are two problems: (1) if we require the known data points to be an equidistant grid, it can only be applied in limited field; and (2) if the known data points are scattered, it is very difficult to obtain satisfactory results by using the uniform B-spline method. The purpose of this paper is to present the least squares fitting method based on a bivariate nonuniform B-spline. The principles and algorithms of this least squares fitting method are developed. Based on this method, two numerical simulations and one application in regional gravity field approximation are discussed.