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Adjustment of Triangulation. Introduction Triangulation was the preferred method for horizontal...
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![Page 1: Adjustment of Triangulation. Introduction Triangulation was the preferred method for horizontal control surveys until the EDM was developed Angles could.](https://reader030.fdocuments.us/reader030/viewer/2022033103/56649e4e5503460f94b44860/html5/thumbnails/1.jpg)
Adjustment of Triangulation
![Page 2: Adjustment of Triangulation. Introduction Triangulation was the preferred method for horizontal control surveys until the EDM was developed Angles could.](https://reader030.fdocuments.us/reader030/viewer/2022033103/56649e4e5503460f94b44860/html5/thumbnails/2.jpg)
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
![Page 3: Adjustment of Triangulation. Introduction Triangulation was the preferred method for horizontal control surveys until the EDM was developed Angles could.](https://reader030.fdocuments.us/reader030/viewer/2022033103/56649e4e5503460f94b44860/html5/thumbnails/3.jpg)
![Page 4: Adjustment of Triangulation. Introduction Triangulation was the preferred method for horizontal control surveys until the EDM was developed Angles could.](https://reader030.fdocuments.us/reader030/viewer/2022033103/56649e4e5503460f94b44860/html5/thumbnails/4.jpg)
Azimuth Observation Equation
![Page 5: Adjustment of Triangulation. Introduction Triangulation was the preferred method for horizontal control surveys until the EDM was developed Angles could.](https://reader030.fdocuments.us/reader030/viewer/2022033103/56649e4e5503460f94b44860/html5/thumbnails/5.jpg)
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
![Page 6: Adjustment of Triangulation. Introduction Triangulation was the preferred method for horizontal control surveys until the EDM was developed Angles could.](https://reader030.fdocuments.us/reader030/viewer/2022033103/56649e4e5503460f94b44860/html5/thumbnails/6.jpg)
Azimuth Examples
0°30°
60°
90°
120°
150°
180°
-150°
-120°
-90°
-60°
-30°
X
Y
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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
![Page 8: Adjustment of Triangulation. Introduction Triangulation was the preferred method for horizontal control surveys until the EDM was developed Angles could.](https://reader030.fdocuments.us/reader030/viewer/2022033103/56649e4e5503460f94b44860/html5/thumbnails/8.jpg)
Linearizing the Azimuth Equation
![Page 9: Adjustment of Triangulation. Introduction Triangulation was the preferred method for horizontal control surveys until the EDM was developed Angles could.](https://reader030.fdocuments.us/reader030/viewer/2022033103/56649e4e5503460f94b44860/html5/thumbnails/9.jpg)
Other Partials
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Linearized Azimuth Observation Equation
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Angle Observation Equation
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Angle Observation Equation
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Linearized Form
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Example 14.1
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First – Initial Approximations
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Approximations - Continued
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Approximations - Continued
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Determine Computed Values for Angles and Distances
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Computed Values - Continued
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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
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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).
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K Matrix
If this was in radians, we wouldn’t need Rho. Also, the second value should be zero. (why?)
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Compute Solution and Update Coords
Note: Further iterations produce negligible corrections.
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Compute StatisticsResiduals: V = J X - K
S0
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Coordinate Standard Errors
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Other Angle Networks
• Resection – more than 3 points is redundant
• Triangulated quadrilaterals
• Other geometric shapes
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Resection
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Triangulated QuadrilateralA
B
CD