Global Positioning System Reference
In-Depth Information
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
C
HAPTER 4
[Fir
[92
L
EAST-SQUARES ADJUSTMENTS
Lin
—
*
2
——
Sho
*PgE
Least-squares adjustment is a device for carrying out objective quality control of mea-
surements by processing sets of redundant observations according to mathematically
well-defined rules. The objectivity of least-squares quality control is especially useful
when depositing or exchanging observations. Least-squares solutions require redun-
dant observations, i.e., more observations are required than are necessary to deter-
mine a set of unknowns exactly. Details will be given as to what constitutes optimal
redundancy. This chapter contains compact but complete derivations of least-squares
algorithms.
First, the statistical nature of measurements is analyzed, followed by a discussion
of stochastic and mathematical models and the law of variance-covariance propa-
gation of random variables. The mixed adjustment model is derived in detail, and
the observation equation and the condition equation models are deduced from the
mixed model through appropriate specification. The cases of observed and weighted
parameters are presented as well. A special section is devoted to minimal and inner
constraint solutions and to those quantities that remain invariant with respect to a
change in minimal constraints. Whenever the goal is to perform quality control on
the observations, minimal or inner constraint solutions are especially relevant. Sta-
tistical testing is important for judging the quality of observations or the outcome of
an adjustment. A separate section deals with statistics in least-squares adjustments.
The chapter ends with a presentation of additional quality measures, such as inter-
nal and external reliability and blunder detection and a brief exposition of Kalman
filtering.
[92
92
