SUG533 kuliah 3a - LSA principle

8/7/2019 SUG533 kuliah 3a - LSA principle

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Least Square Adjustmentis a method of finding the most probablevalues for unknown quantities based on observations equations or

functionsthat can be expressed in terms of observed quantities, itsresiduals and the unknown quantities, that fulfilled the condition where

the sum of the residuals squared is minimum

min.....22

3

2

2

2

1

2

n

The minimum value of a function can be obtained from the first

derivative of the functions with respect to the unknowns.

0

2

x


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Observation equations are formulated by relating the observed

quantities and its residuals as well as the unknown quantities inmathematical expression.

One equation is formed for one observation.

A unique solution for the unknown would be obtained if the numberof observation equations is equal to the number of unknowns.

If the observations were redundant, i.e the number of observations

exceeds the number of unknown, then the observations need to beadjusted using the least squares adjustment method in order to reacha unique solution for the unknown


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Example: Linear equation

Different results in x and y indicate the presence of errorsin the observed values.

To get a unique solution, the observations need to be adjusted


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Formation of observation equations

10.3 yx

25.12 yx

32.0 yx

Which are the observed quantities, its residuals and theunknown?

Find the value of the sum of residuals using least squares

principle/condition


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Formation of observation equations

10.3 yx

25.12 yx

32.0 yx

Which are the observed quantities, its residuals and theunknown?

Find the value of the sum of residuals using least squares

principle/condition


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Residuals

LAx

?2

Sum of residuals squared


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Adjustment of elevation differences for the

determination of height above MSL


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Find elevations of A, B and C


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The distance between points A, B, C and D were measured using two (2)different Total Station that have constant error. The mean values of themeasured distance are given above. Using the observations of instrument 1,compute the (a) instrument constant error and its standard deviation, and (b)most probable value for distance AB, BC, CD and AD, and it standarddeviations based on Least-Square method.


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Unknown quantities (u = 4)


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Observed quantities (n = 12)


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SUG533 kuliah 3a - LSA principle