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