October 2017
Intermediate to advanced
720 pages
18h 37m
English
Content preview from Adjustment Computations, 6th Edition
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CHAPTER 22THE GENERAL LEAST SQUARES METHOD AND ITS APPLICATION TO CURVE FITTING AND COORDINATE TRANSFORMATIONS
22.1 INTRODUCTION TO GENERAL LEAST SQUARES
When fitting points to a straight line it must be recognized that both the x and y coordinates contain errors. Yet in the mathematical model presented in Section 11.11.1, the residuals, as illustrated in Figure 11.2 are only applied to the y coordinate. Because both coordinates contain errors, this mathematical model fails to account for the x coordinate being an observation. In this chapter, the general least squares method is presented, and its use in performing adjustments where the observation equations involve more than a single observation is demonstrated.
22.2 GENERAL LEAST SQUARES EQUATIONS FOR FITTING A STRAIGHT LINE
Consider the data illustrated in Figure 11.2. To account properly for both the x and y coordinates being observations, the observation equation must contain residuals for all observations. That is, Equation (11.40) must be rewritten as
In Equation (22.1), x and y are a point's coordinate pair with residuals vx and vy, respectively, m is the slope of the line, and b is the y intercept. Equation (22.1) contains vx, vy, m, and b as unknowns and is nonlinear. Thus, its solution is obtained by using ...
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