Adjustment Computations Adjustment Computations: Spatial Data Analysis

CHAPTER 15

Adjustment of Horizontal Surveys: Triangulation


Fundamental functions distance, azimuth, and angle computations

Note: The following routines are based on the station information being stored in a matrix called crds with column 0 containing the identifiers, 1 the x coordinate, and 2 the y coordinate.

The observation matrices for the observed distances, angles, and azimuths have indices to the crds matrix instead the station identifiers.

Azimuth Observation Equation Coefficients - Equation 15.9

dxi

dyi

dxj

dyj

Angle Observation Equation Coefficients - Equation 15.13

dxb

dyb

dxi

dyi

dxf

dxf

Example 15.1 -- Adjustment of Intersections

Using the method of least squares, compute the most probable coordinates of station U in the figure to thebelow by the least squares intersection procedure. The following equally-weighted horizontal angles were observed from control stations R, S, and T


θ1 = 50°06'50" θ2 = 101°30'47"

θ3 = 98°41'17" θ4 = 59°17'01"


The coordinates for the control stations R, S, and T are

xr = 865.40 xs = 2432.55 xt = 2865.22

yr = 4527.15 ys = 2047.25 yt = 27.15

Given:

Control stations

Unknown station

Angle Observations

equal-weight case

Routine to build matrices

Offset row to avoid angle entries

Note that all angular units are in radians and there is no need to convert the angles to arc seconds and suggested in the book.

Solve problem using least squares method

Note that the following routines were generically created so that they could be used repeatedly.

Solution

Compute adjusted angles

Compute statistics

SOLUTION

Note: Solution slightly different in book due to rounding in book.


Adjustment of Resections - Example 15.2

The following data are obtained for Figure .4. Control stations P, Q, R, and S have the following (x, y) coordinates: P (1303.599, 1458.615), Q (1636.436, 1310.468), R (1503.395, 888.362), and S (1506.262, 785.061). The observed values for angles 1, 2, and 3 with standard deviations are


BIF angle S

PUQ 30°29'33" ± 5"

QUR 38°30'31" ± 6"

RUS 10°29'57" ± 6"

What are the most probable coordinates of station U?

Given:

Unknown station

Control stations

Angle observations

SOLUTION

Set previously used matrices to 0.

Compute adjusted angles

Compute statistics

SOLUTION


Example 15.3 - Adjustment of Triangulated Quadrilaterals

This example will demonstrate using a file reading routine.

Parse file for coordinates and observations.

Set previously used matrix to 0

findsta is used to match the station identifier to its storage location in the sta input table.

Search through station table until "s" is found

j(i) represents the offset in the data file from the first line.

Angles and standard deviations are converted to radians

to make units consistent in linearized equations.

SOLUTION

Set previously used matrices to 0.

Compute adjusted angles

Compute statistics

SOLUTION

 

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