Abstract
Statistics has proved that when you increase number of observations
the mean of accidental errors goes to zero, in other words with statistical terms
we can say the expected value of accidental errors is zero
n
?i
E{?}=lim =0 .
n
Increasing the number of observed quantities will cause another problem, we will
have more equations than it is necessary to solve for unknown parameters.
Here is the point that mathematical solution differs from computation in
experimental sciences. The later method is the least squares solution. In this
me thod all the observed quantities are used in such a way that if we find
differences between observed quantities and the adjusted values of these quantities
(residuals), the sum of squares of residuals be minimum. We can summarize
this method by it's two major characteristics:
1-The solution is unique and independent of the path, and the order
of equations.
2 - The sum of squares of residuals is minimum.
The classic method of least squares (as we call it) can be found in such
books like geodesy, mathematical statistics, and numerical analysis. With the
application of matrix algebra we present the method of least squares adjustment
in different way, however the principle is the same, but it does have some
advantages. Let us summarize weaknesses of the classical method as bellow:
1 - With classical method computation is slow and uneconomical.
2 - The simultaneous adjustment of large networks is very difficult and
actualy imposible, therefore the network have to be devided in blocks, and
adjust each block respect to the previous one. This will cause a kind of controversy
with the first principle, because the final results depend on the path
and the starting point of adjustment.
3 - It is difficult to analyse the accuracy of computation, and the accuracy
of the adjusted values.
4 - Enforcing the effect of correlated observations (the effect of covariances)
is difficult and actually imposible.
In the method of least squares adjustment with the use of matrix algebra
most of the computations are systematic kind wich can be done by electronic
computers very fast. There are only two parts which shoud have to be found
or estimated by us:
1-To estimate a weight matrix (P) or variance-covariance matrix of
observations ( Lb= . P ) by the informations we have about instrument,
operator, and outer circumstances.
2-To find and write down a suitable mathematical structure (mathematical
model).
MATHEMATICAL MODEL
Mathematical model is some functions that exist between observations
and parameters. The mathematical model for observation equations is a special
kind, and that is when each observed quantity can be written as a function of
some parameters.
La=f(Xa)
Where X, is a vector of adjusted parameters with u components (number of
parameters), and La is avector of adjusted observations with n (number of
observations) components. The other notations that we are going to use are:
Xo , the initial approximate values of parameters chosen arbitrarily before
adjustment.
The differences of Xa, and Xo , or the variation of parameters are
X=X -X
The measured quantities are denoted by vector L with n components
using the following subscripts:
Lb , the measured values,
La' the adjusted values,
Lo , the values computed from values X, through the mathematical model
Lo=f(xo).
The differences of these sets are denoted as :
L=LO-Lb
V=La-Lb residuals.
We can rewrite the mathematical model (I) interms of L, and X,
Lb+V=F(Xo-X)
After expanding it we get
Lb+V=F(XO)+AX
where A is a matrix with n rows and u columns, the elements of A are partial
derivatives of measured quantities with respect to the parameters.
A
X =XO
Since
Lo=F(Xo), and L=Lo-Lb
equation (2) can be wirtten as
V=AX+L
The principle of lea
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