Prof. Manoochehr Shirzaei.

Transcription

1 RaTlab.asu.edu

2 Introduction To Error Adjustment And Least Squares

3 Adjustment of Observations In practice, a unique solution is dangerous, as an error in a single observation can radically effect the final solution for the unknowns. Therefore, we typically collect redundant observations, i.e. having (taking) more observations than is necessary for a unique solution. Redundant observations lead to various solutions for the same parameters, which are slightly different. Any body knows why?

4 Example: Levelling Networks The example shows three different routes (A P, B P, C P) for estimating the elevation of point p. Due to the different uncertainty (errors) in the observations hi, the estimated elevation of point p is not unique.

5 Example: Area of Triangle Again, identical results may not (most probably) be obtained, due to the errors in the observation vector L

6 Least Squares and Observation Adjustments The above-mentioned problems associated with the over-determined math model can be overcome by adjusting the observations. The apparent inconsistency, due to measurement errors, model can be resolved through the replacement of the given observations by another set of the so-called best estimates of the observations such that the new set fits the model exactly. Where,,, Note: The estimated residuals are unknown and must be determined before the observations can be estimated.

7 How to choose? There are essentially an infinite number of possible sets of residuals that provide estimated observations that fits the math model. However, there is only one set of residuals that yield the optimal Least Squares solution: In addition to the fact that the adjusted observation must satisfy the mathematical model exactly, the corresponding residuals must satisfy the Least Squares criterion.

8 Geometric Interpretation of Given the values of, estimate the length X. The final value of (best estimate) can be obtained from the observation equations. Some estimates of ; Note that is the smallest, but is it the very minimum value when all possible combinations of corrections are considered?

9 Geometric Interpretation of The best estimate of observations is when, which forms a line with slope angle of 45 deg. can be represented by point (A) The projection of point (A) into the condition line have 3 possibilities The shortest is when the vector is perpendicular to the condition line, i.e.;

10 Note: least squares condition that is, the mean satisfies the Note: The condition of Least Squares of the residuals was found to satisfy the properties of the best estimate; 1) Maximum Likelihood (most probable) 2) Minimum variance (most precise) 3) Unbiased (most accurate)

11 Example The sketch shows a leveling network abcd, in which point (a) is assumed to be fixed with zero elevation. Estimate

12 The six observation equations Solution Adding six unknown observation residual and replacing and in matrix format; Exercise 1: use MATLB lsqlin command from optimization toolbox to find.

13 Exercise 2 The two-parameter transformation (rotation and scale), between two coordinate system (, ) and (, ) is represented by; Y bx ay, are the transformation parameters. In order to estimate and, table below gives the tree point of know coordinate in the both systems; Find best estimate of and.

14 References Mikhail, E. M. (1976), Observations and least squares 497 pp., IEP New York.