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LECTURES 22 AND 23, MARCH 8 AND 17, 2004

TRIANGULATION The positions of a network of points (control points) are established to provide accurate control for survey work. Among the common methods for establishing the positions are triangulation, trilateration, and satellite positioning.

If the positions of the ends of a line (called baseline) and two angles forming a triangle are known, the position of the third vertex of the triangle can be determined. Positions of other control points can then be established using a series of interconnected and/or overlapped triangles forming the so-called triangulation figure or triangulation system (for examples see Figure 1). This method is called triangulation and their vertices are called triangulation stations. In trilateration, vertices of the triangles are established from distance measurements only.

Figure 1. Triangulation Figures

In a triangulation scheme, the error tends to accumulate with distance from the baseline. To account for this limitation subsidiary bases are introduced at various locations within the network. At certain locations astronomical observations are made. These stations are called Laplace Stations.

If a point is located from triangulation stations without occupying, it is called intersected point. It is also possible to locate a point by making direction measurements to three known stations. A point located in this manner is called resected point.

CLASSIFICATION In first-order triangulation, an area as large as a country is covered. A second-order triangulation network comprises a network of triangulation stations connected to and within the first-order triangulation network and covers a relatively smaller area. A third-order triangulation network is established within and connected to the second-order triangulation network in a similar manner to cover a smaller area. Typical particulars of these systems are listed in Table 1.

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Table 1. Characteristics of Triangulation Networks First Order Second

Order Third Order

Baseline length in km 8 - 12 2 - 5 0.1 - 0.5 Lengths of sides in km 16 - 150 10 - 25 2 - 10

Triangle closure error after correcting spherical excess: average / maximum

<1″ / 3″ 3″ / 8″ 6″ / 12″

Actual error of base 1 in 50000 1 in 25000 1 in 10000 Probable error of base 1 in 106 1 in 5×105 1 in 2.5×105

Discrepancy between two measures in mm (k is distance in km)

5√k 10√k 25√k

Probable error of computed distances 1 in 50000 to 1 in 250000

1 in 20000 to 1 in 50000

1 in 5000 to 1 in 20000

Probable error in astronomical azimuth 0.5″ 5″ 10″

PROCEDURAL DETAILS OF TRIANGULATION FIELD WORK a. Reconnaissance b. Erection of signals (Devices that define the exact position of the station so that it can

be observed from other stations: can be non-luminous or luminous) and towers (Structures erected atop a station to support the instrument and the observer: needed when signal and/or the station are to be elevated)

c. Baseline measurement: While using tape, corrections need to be applied for absolute length (Ca), temperature (Ct), pull (Cp)and sag (Cs) given by:

( )22111 24/

)/()(

/

PWlC

AELPPCTLC

lLcC

s

op

t

a

=

−===α

(1)

where L is the measured length, c is the correction per tape length, l is the designated tape length, α is the coefficient of thermal expansion, T is equal to ambient temperature minus that at which the tape was standardized, P is the pull applied during measurement, Po is the standard pull, A is the tape cross sectional area, E is the Young’s Modulus of the tape material, Cs1 is the sag correction per span, l1 is the length pf tape per span, and W1 is the weight of tape. In other methods of baseline measurement, such as that based on the EDM, appropriate corrections should be applied.

d. Measurement of horizontal angles. e. Measurement of vertical angles. f. Astronomical observations to establish the azimuth of the lines.

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DATA REDUCTION g. Spherical excess Correction: For geodetic triangles the spherical excess must be

accounted for before figure adjustment. Due to the convexity of earth surface, the sum of the internal angles of geodetic triangles exceeds 180° by 1 second for a triangle with an area of 197 km2. Thus, the total adjustment to be applied to the three angles is the sum of the three angle measurements minus 180° minus 1″ times the area of the triangle in km2 divided by 197.

h. Phase Correction: Opaque signals may be partly illuminated and the observer may (a) bisect the bright portion or (b) take angle reading on the bright line of the signal (Figure 2).

Figure 2. Phase Correction

i. Refraction Correction: Horizontal angles measured using optical signal, are affected

by refraction in the presence of temperature gradient along the line of sight. The error due to refraction in seconds, ε, is estimated from:

dxdT

TSP

28=ε (2)

where S is the length of line of sight in meter, P is the barometric pressure in mb, and T is the temperature in Kelvin.

THE STRENGTH OF TRIANGULATION FIGURE The accuracy attained in a triangulation system depends on (a) the conditioning of the triangles and (b) the arrangement of the triangles. A well conditioned triangle is one in which error in angle measurements translates into small errors in lengths. Consider a

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triangle ABC comprised of angles A, B and C and sides a, b and c, with angle A faces a, etc., AB being the baseline. Error propagation is minimized when A = B. By sine rule

CAca sinsin×= , from which

AACAaa

CCCa

aC

CCAca

AAAa

aC

AAca

CC

AA

2cotcotcotcot

cotcotsin

cossin

cotcotsin

cos

2222

2

+=+=

⇒

×−=×−=⇒×××

−=

×=×=⇒××

=

ββδ

βδδδδ

βδδδδ

δδ

δδ

where δa and δA (=β) are the errors in side and angle measurements, respectively, and

Aa

δδ and

Ca

δδ are the components of δa due to errors in A and C, respectively. It can be

shown that aaδ is minimized when A ≈ 56-14 (how?).

Angles less than 30° are not preferred in triangulation. If use of such angles is unavoidable, then such angles should not be opposite the side whose length has to be computed to carry forward the triangulation system.

The US Coast and Geodetic Survey gives the following expression for the square of the probable error in the sixth place of logarithm of any side, L, for computations carried from a known side through a single chain of triangles after the system has been adjusted for side and angle conditions:

RdL 22

34

= (3)

where d is the probable error of an observed angle in seconds. The strength of figure depends on factor R. The smaller the value of R, greater is the strength of figure. This factor is calculated using:

{ }∑ +×+−

= 22BBAAD

CDR δδδδ (4)

where D is the number of directions observed (forward and back) excluding the known side of the figure, subscripted δ denotes the difference per second in the sixth place of logarithm of the sine of an angle that affect the length of the line of interest. C depends on the layout of the figure as follows:

( ) ( )321 +−++′−′= SnSnC (5)

where n is the total number of lines including the baseline in a figure, n′ is the number of lines observed in both directions including the baseline, S is the total number of stations and S′ is the number of stations from where at least one measurement was carried out (called occupied stations).

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Example 1: What would be the maximum value of R for first order triangulation based on single chain triangles?

Solution: From Table 1,

( ) ( )[ ]( ) ( )[ ] 74.11050000011log50000011log

69.81010000011log10000011log5.0

6min

6max

=×+−+=

=×+−+=

=

L

Ld

using symbols used in Equation 3. Substituting these values into Equation 3

55.22675.0 22maxmax =×= dLR

Example 2: Compute the strength of figure for a triangulation system shown on margin sketch for each of four routes for establishing BD from baseline AC.

Solution: D = 10, n = n′ = 6, S = S′ = 4. Hence, from Equation (14), C = 4.

Alt. 1: Uses ∆ACD and ∆ADB. Distance angles for ∆ACD are 55° and 58°. Hence

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32.110/3600)0.558sin(log -/3600)0.585sin(log

47.110/3600)0.555sin(log -/3600)0.555sin(log

22

61010

61010

=++⇒

=

×°+°°+°=

=

×°+°°+°=

BABA

B

A

δδδδ

δ

δ

Similarly for ∆ADB 1222 =++ BABA δδδδ and 1822 =++∑ BABA δδδδ . Hence, R for this alternative is

0.6×18 = 11.

Alt. 2: Uses ∆ACD and ∆DCB. For ∆ACD, distance angles are 55° and 67°. For ∆DCB, distance angles are 36° and 112°. Hence, R for this alternative is 7.

Alt. 3: Uses ∆ACB and ∆ABD. R for this alternative is 14.

Alt. 4: Uses ∆ACB and ∆BCD. R for this alternative is 8.

Alt. 2 is thus the best alternative.