Adjustment of Triangulation. Introduction Triangulation was the preferred method for horizontal...
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Transcript of Adjustment of Triangulation. Introduction Triangulation was the preferred method for horizontal...
Adjustment of Triangulation
Introduction
• Triangulation was the preferred method for horizontal control surveys until the EDM was developed
• Angles could be measured to a high level of accuracy
• Measured baseline distances were included every so often to strengthen the network
Azimuth Observation Equation
Arctangent Function for Azimuth
xi yi xj yj xj-xi yj-yi atan2(dy,dx) atan(dx/dy)
0 0 0.500 0.866 0.500 0.866 30 30
0 0 0.866 0.500 0.866 0.500 60 60
0 0 1.000 0.000 1.000 0.000 90 #DIV/0!
0 0 0.866 -0.500 0.866 -0.500 120 -60
0 0 0.500 -0.866 0.500 -0.866 150 -30
0 0 0.000 -1.000 0.000 -1.000 180 0
0 0 -0.500 -0.866 -0.500 -0.866 -150 30
0 0 -0.866 -0.500 -0.866 -0.500 -120 60
0 0 -1.000 0.000 -1.000 0.000 -90 #DIV/0!
0 0 -0.866 0.500 -0.866 0.500 -60 -60
0 0 -0.500 0.866 -0.500 0.866 -30 -30
0 0 0.000 1.000 0.000 1.000 0 0
Azimuth Examples
0°30°
60°
90°
120°
150°
180°
-150°
-120°
-90°
-60°
-30°
X
Y
Correction Term
• Even if we use a full-circle arc tangent function we may still need a correction term
• This can happen where the azimuth is near ±180°
• Check the K-matrix term (measured minus computed)
• If it is closer to ±360° than it is to 0°, correction is needed
Linearizing the Azimuth Equation
Other Partials
Linearized Azimuth Observation Equation
Angle Observation Equation
Angle Observation Equation
Linearized Form
Example 14.1
First – Initial Approximations
Approximations - Continued
Approximations - Continued
Determine Computed Values for Angles and Distances
Computed Values - Continued
Set Up MatricesFirst, we need to define the Backsight, Instrument, and Foresight stations for the observed angles.
angle B I F
θ1 U R S
θ2 R S U
θ3 U S T
θ4 S T U
J Matrix
Note: Rho (ρ) is the conversion factor from radians to seconds. This complication can be avoided by keeping all angles in radian units (for example, in the K matrix).
K Matrix
If this was in radians, we wouldn’t need Rho. Also, the second value should be zero. (why?)
Compute Solution and Update Coords
Note: Further iterations produce negligible corrections.
Compute StatisticsResiduals: V = J X - K
S0
Coordinate Standard Errors
Other Angle Networks
• Resection – more than 3 points is redundant
• Triangulated quadrilaterals
• Other geometric shapes
Resection
Triangulated QuadrilateralA
B
CD