=Paper=
{{Paper
|id=Vol-1328/GSR2_Deakin
|storemode=property
|title=Traverse Analysis
|pdfUrl=https://ceur-ws.org/Vol-1328/GSR2_Deakin.pdf
|volume=Vol-1328
|dblpUrl=https://dblp.org/rec/conf/gsr/Deakin12
}}
==Traverse Analysis==
https://ceur-ws.org/Vol-1328/GSR2_Deakin.pdf
TRAVERSE ANALYSIS
R.E. Deakin
School of Mathematical & Geospatial Sciences
RMIT University, Melbourne, Australia.
Email: rod.deakin@rmit.edu.au
Abstract
Traversing is a fundamental operation in surveying and the assessment of the quality of a traverse is a skill that every surveyor
develops. Acceptable traverses have angular and linear misclosures that fall within acceptable bounds; which would permit the
adjustment of the traverse measurements to remove mathematical inconsistencies. Unfortunately these misclosures tell very little
about the precision of the location of the traverse stations – although large misclosures are good indicators of gross errors – and
more sophisticated mathematical techniques are required for proper traverse analysis. This paper presents some relatively simple
techniques that can be employed to give reliable estimations of the precision of traverse stations that allows a simple assessment
of the quality of a traverse.
Keywords: Surveying, traverse, propagation of variances.
Biography of Author Rod Deakin started work in 1968 (age 17) as a surveyor's assistant with John Horne of Frankston,
Victoria and fell in the Kananook Creek on his first day. In 1976, he graduated from the RMIT and returned to surveyor Horne's
employ until 1980. In 1981, he was appointed as a tutor in surveying and then a lecturer (1983) at RMIT where he has remained.
He has lectured in all aspects of surveying and in 2004 was awarded the Francis Ormond medal (RMIT University medal in
honour of the founder), and in the Vice Chancellor's address at the presentation it was noted that:
“Indeed, Rod even holds the honour of being cited in student evaluations at another university as the ‘best lecturer I
have had throughout my course at Melbourne University’ for a series of guest lectures he provided.”
Rod regards this as his finest achievement.
Introduction
A traverse is the fundamental component of many surveys and consists of a series of sides or lines whose bearings and distances
have been determined from Total Station 1 measurements; which for the purposes of this paper will be assumed to be horizontal
directions α and horizontal distances l. Traverse bearings θ are horizontal angles measured clockwise from north (0° to 360°);
traverse angles β are differences between directions or bearings; and a traverse line has east and north components
=∆E l sin θ , =∆N l cos θ respectively. A closed traverse starts and finishes at the same point and an open traverse starts and
finishes at different points.
A closed traverse is a polygon of n sides whose internal angles sum to ( 2n − 4 ) × 90 and this rule may be used to determine the
angular misclose: the difference between the rule and the sum of the measured angles. A closed traverse has a linear misclose
which is the length of an assumed ‘misclose vector’ (or missing line) whose east and north components are the sums of the east
and north components of the n traverse legs. A closed traverse also has a misclose ratio 1: x where
x = traverse perimeter linear misclose ; e.g. if the traverse perimeter is 850 m and the linear misclose is 0.050 m then the misclose
ratio is 1:17,000. The misclose ratio is often called the traverse accuracy, but this is wrong, since it does not distinguish between
random 2 or systematic 3 errors or reveal their effects and random errors do not accumulate in direct proportion to distance
(Valentine 1984).
1
Electronic surveying instrument combining the operation of a theodolite with EDM (Electronic Distance Measurement).
2
Random errors are the small errors remaining in measurements after mistakes, constant errors and systematic errors have been eliminated.
They are due to the imperfection of the equipment; the fallibility of the observer and the changing environmental conditions.
To determine the angular and linear misclosures of an open traverse; the coordinate differences between the start and end points
must be known and the bearings of the first and last lines of the traverse must be able to be determined from observations to other
known points. If the terminal points of an open traverse do not have a known coordinate relationship then no linear or angular
misclosures can be obtained from the traverse observations.
In this paper we will be primarily concerned with estimating precisions of stations in closed traverses although the methods we
outline can be just as easily applied to stations in an open traverses; and precision can be taken to mean variances sE2 , sN2 of east
and north coordinates respectively; variances sα2 , sβ2 , sθ2 , sl2 of directions, angles, bearings and distances respectively; or their
standard deviations sE , sN , sα , sβ , etc. noting that standard deviation is defined as the positive square-root of the variance.
It is common practice to assess the quality of traverses by comparing angular/linear misclosures and or misclose ratios with
‘practical standards’. For example the Victorian Surveying (Cadastral Surveys) Regulations 2005 states in part (Regulation 7)
(1) A licensed surveyor must ensure that—
(a) the internal closure of any cadastral survey is such that the
length of the misclose vector does not exceed—
(i) 15 millimetres + 100 parts per million of the perimeter
for boundaries crossing level or undulating land; and
(ii) ....
These practical standards are not a ‘modern’ way of assessing traverse quality and are often based on historical survey practices
and equipment that may not reflect modern techniques. Instead, a method is proposed that is based on simple assumptions of
survey practice; knowledge of precisions of Total Station measurements and Propagation of Variances (PoV). This method
assesses the quality of a traverse by comparing linear and angular misclosures with statistical estimates that are functions of the
actual traverse measurements and the geometry of the traverse. In addition, this method: (i) provides estimates of the precision of
individual traverse stations; (ii) is easily programmed on calculators (and computer spread sheets) and (iii) can be enhanced with
error ellipse displays.
As a first step the rule for PoV is introduced – with a special case when variables are independent – and then it is shown how this
rule can be extended by using matrix algebra and applied to the computation of coordinates of traverse stations. Then, it is shown
how the determination of traverse bearings can be broken down into a sequence of simple Total Station measurements with errors
than can be plausibly explained and modelled; which in turn enables reasonable estimates of variances of traverse bearings (using
PoV). Finally, using the traverse observations (bearings and distances) with estimates of their variances it is shown how they are
combined in a sequential application of PoV to give precision estimates of the coordinates of the traverse stations. A rule for
assessing the quality of a traverse follows logically from these precision estimates.
Propagation of Variances (PoV)
In surveying, propagation involves obtaining information about a function (or process, or computation) involving variables
(measurements or functions of measurements) that are subject to systematic or random variation. Systematic errors are most often
due to poor measurement technique or perhaps equipment that is not properly calibrated or used incorrectly. In this paper, there is
an assumption that the surveyor properly understands her traversing equipment (Total Station + tribrachs, tripods, prisms, etc.); it
is calibrated; her field technique is adequate and measurements have been corrected for the effects of systematic errors. That
leaves the effects of random errors to be dealt with and Propagation of Variances (PoV) is also known as propagation of random
errors.
Consider a function w of variables x, y, z , , t that are affected by small random errors ε x , ε y , ε z , , ε t that are assumed to be
random variables of infinite populations having means µ x , µ y , µ z , , µt and variances σ x2 , σ y2 , σ z2 , , σ t2 then the Law of
Propagation of Variances allows us to say:
For a function w = w ( x, y, z , , t ) the variance σ w2 is
3
Systematic errors follow some fixed law (possibly unknown) dependent on local conditions and/or the equipment being used. Propagation of
systematic errors can be modelled by using the Total Increment Theorem (or Total Differential) of mathematics
2
∂w 2 ∂w 2 ∂w 2 ∂w 2
2 2 2
σ w2 ≈ σx + σy + σ z + + σt
∂x ∂y ∂ z ∂t
∂w ∂w ∂w ∂w ∂w ∂w
+2 σ xy + 2 σ xz + + 2 σ xt
∂x ∂y ∂x ∂z ∂x ∂t
∂w ∂w ∂w ∂w ∂w ∂w
+2 σ xz + + 2 σ yt + + 2 σ zt (1)
∂y ∂z ∂y ∂t ∂z ∂t
where σ xy , σ xz , σ xt , etc. are covariances that measure of how much two variables change together.
If the variables in the function w are independent of each other then their covariance is zero and the Special Law of Propagation of
Variances follows as
2
∂w 2 ∂w 2 ∂w 2 ∂w 2
2 2 2
σ w2 ≈ σx + σy + σ z + + σt (2)
∂x ∂y ∂z ∂t
Suppose that p and q are both functions of variables x, y and z that are affected by small random errors ε x , ε y , ε z assumed to be
drawn from populations having variances σ x2 , σ y2 , σ z2 . The functions p and q may be combined in a vector y = [ p q ] , and the
T
variables x,y,z in a vector x = [ x z ] where [ ] represents the vector (or matrix) transpose; then we may write:
T T
y
If y = f ( x ) (3)
and the Law of Propagation of Variances is expressed as (Mikhail & Gracie 1981)
Σ yy = J yx Σ xx J Tyx (4)
σ x2 σ xy σ xz
σ 2 σ pq
where the variance matrices are Σ yy = p 2
Σ xx = σ yx σ y2 σ yz
σ qp σ q
σ zx σ zy σ z2
∂p ∂p ∂p
∂x ∂y ∂z
and the matrix of partial derivatives is J yx =
∂q ∂q ∂q
∂x ∂y ∂z
Note: (i) covariances σ ab = σ ba ; (ii) variance matrices are square and symmetric; and (iii) if variables are independent then the
off-diagonal elements of variance matrices are zero.
In measurement sciences, random errors are assumed to be members of infinite populations having means µ and variances σ 2
but in practice these quantities are unknown and instead we use estimates x (mean) and s 2 (variance) calculated from small
samples of measurements; or just assumed. So equations (1) and (2) can be expressed with estimates sx2 , s y2 , sz2 , and sxy , sxz ,
replacing the population quantities σ x2 , σ y2 , σ z2 , and σ xy , σ xz , . And equation (4) can be expressed with cofactor matrices
Q yy and Q xx containing estimates of variances and covariances replacing variance matrices Σ yy and Σ xx .
The matrix approach to PoV [equations (3) and (4)] can be demonstrated by the example of a traverse line having a bearing θ and
length l connecting points k − 1 and k of a traverse. The east and north coordinates of the kth traverse station are
Ek =l sin θ + Ek −1 =∆E + Ek −1
(5)
N k =l cos θ + N k −1 =∆N + N k −1
Ek , N k are functions of l , θ , Ek −1 , N k −1 which can be expressed symbolically in the matrix equation
y = f (x) (6)
where y = [ Ek N k ] and x = [l θ N k −1 ] . Applying PoV to (6) gives
T T
Ek −1
Q yy = J yx Q xx J Tyx (7)
Q yy and Q xx are cofactor matrices containing estimates of variances and covariances of the elements of y and x respectively.
sl2 0 0 0
sE2k sEk Nk 0 sθ 2
0 0
=Q yy = , Q 0 (8)
sEk Nk
2
sNk
xx
0 s 2
sEk −1 Nk −1
Ek −1
0 0 sEk −1 Nk −1 sN2 k −1
J yx is the matrix of partial derivatives
∂Ek ∂Ek ∂Ek ∂Ek ∆E
∂l ∂θ ∂Ek −1 ∂N k −1 sin θ l cos θ 1 0 l ∆N 1 0
=J yx = =
(9)
∂N k ∂N k ∂N k ∂N k cos θ −l sin θ 0 1 ∆N
−∆E 0 1
∂l ∂θ ∂Ek −1 ∂N k −1 l
Carrying out the matrix multiplications of (7) gives the cofactor matrix of the computed coordinates
∆E 2 2 ∆E ∆N 2
sl + ( ∆N ) sθ + sEk −1 sl − ( ∆E ∆N ) sθ + sEk −1 Nk −1
2 2
2
2
l l
2
Q yy = (10)
∆E ∆N 2 ∆N 2 2
sl − ( ∆E ∆N ) sθ + sEk −1 Nk −1 sl + ( ∆E ) sθ + sNk −1
2 2 2 2
l l
2
The elements of Q yy are the estimates of variances and covariances of the computed coordinates of point k. These elements
replace the lower-right block in Q xx in the computation of the precision estimates of the next point in the traverse. The matrix
multiplications of equation (4) can easily be done with spreadsheet computer programs.
To carry out this PoV we require estimates of the precisions of traverse bearings and distances. The determination of these is the
subject of the following sections.
Estimating the Precision of Traverse Bearings
Consider the operation of determining a traverse bearing.
1. The Total Station is pointed to the back-sight and the horizontal circle (or direction) read,
α Back .
2. The Total Station is turned clockwise to the forward-sight (or for-sight) and the horizontal
circle read again, α For .
3. The first direction is subtracted from the second direction to obtain the clockwise angle
β α For − α Back .
=
4. β is added to the back-sight bearing to give the bearing of the for-sight θ=
For θ Back + β
This might not be the simplest or most common field technique but it suffices for separating the operation into certain parts.
To estimate the variance of an observed traverse bearing ( s ) , equation (2) can be applied to the equation θ= θ
2
θ For Back +β ,
assuming that the measured angle β and the bearing of the back-sight θ Back are independent.
This gives
s=
2
θ For sθ2Back + sβ2 (11)
In equation (11) an estimate of the variance of the measured angle ( sβ2 ) is required and this may be considered as consisting of
two parts; (i) the precision of pointing and reading and (ii) the precision of centring at the observing and target stations. Now
since any errors in pointing and reading are independent of direction we may apply equation (2) and write
s=
2
β
2
sPR + sCENT
2
(12)
2 2
sPR and sCENT are estimates of the variances of pointing and reading error and centring error respectively and how these are
obtained is discussed in the following sections.
Estimating the Precision of Pointing and Reading Errors
Figure 1 shows two lines, AB and BC, of a traverse. The Total Station is at B, the back-sight target is at A and the for-sight target
is at C. θ1 , l1 are the bearing and distance respectively of leg 1 and θ 2 , l2 are the bearing and distance of leg 2. α1 , α 2 are the
horizontal directions to A and C respectively and β is the horizontal angle between the two traverse lines.
A Back-sight Target
°
1
θ
l1
N α1
NA − NB
θ
EA − EB
Instrument Point B β
° EC − EB
NC − NB
α2
l2 θ
2 ° C Forward-sight Target
Figure 1
Good survey practice dictates that directions are read on Face Left (FL) and Face Right (FR) of the Total Station and averaged to
eliminate the effects of collimation and so we may write
β FL =
α 2 − α1 and β FR =
FL FL
α 2 − α1 FR FR
(13)
β FL , β FR are FL/FR angles, α1 , α1
FL FR
are FL/FR directions to the back-sight target and α 2FL , α 2FR are FL/FR directions to the for-
sight station. Applying the Special Law of Propagation of Variances to equation (13) gives the variances of the FL/FR angles as
sβ2FL =
sα2 2FL + sα21FL and sβ2FR =
sα2 2FR + sα21FR
Assuming the variances of the FL/FR directions are equal and independent of direction, then we may write s=
2
α FL s=
2
α FR sα2 and
the variances of the FL/FR angles become
=sβ2FL 2=
sα2 and sβ2FR 2 sα2
The angle β is the average of the FL/FR angles, i.e.,
=β 1
2 ( β FL + β FR ) (14)
and applying the Special Law of Propagation of Variances to equation (14) and using results above gives the variance of a mean
angle from a single pair of FR/FL pointings as
sβ2 = ( 12 ) sβ2FL + ( 12 ) sβ2FR = sα2
2 2
Denoting the variance of the angle as the variance due to pointing and reading errors we say for a single pair of FR/FL pointings
2
sPR = sα2 (15)
sα2 is the variance of a single-face direction of the Total Station.
Estimating the Precision of Instrument and Target Centring Errors
We may express the traverse angle β as
EC − EB E − EB
β = θ 2 − θ1 = arctan − arctan A (16)
NC − N B NN − NB
so β = β ( E A , N A , EB , N B , EC , N C ) or in symbolic matrix form y = f ( x ) and applying Propagation of Variances gives
Q yy = J yx Q xx J Tyx (17)
∂β ∂β ∂β ∂β ∂β ∂β
where Q yy = sβ2 ; J yx = and
∂E A ∂N A ∂EB ∂N B ∂EC ∂N C
sE2 A sE A N A s E A EB sE A N B sEA EC sE A NC
2
s NA s N A EB sN A N B sN A EC sN A N B
For the purposes of developing a working formula for
2
s s EB N B sEB EC s EB N C
Q xx =
EB estimating the effects of centring errors, it is assumed
sN2 B sN B EC s N B NC that all covariances terms in Q xx equal zero, i.e.,
sE2C sEC NC E A , N A , EB , N B , EC , N C are independent random
variables. This means that equation (17) can be
sN2 C expressed as
2 2 2 2 2 2
∂β 2 ∂β 2 ∂β 2 ∂β 2 ∂β 2 ∂β 2
sβ2 = sE A + sN A + s EB + sN B + sEC + s NC (18)
∂E A ∂N A ∂EB ∂N B ∂EC ∂N C
Furthermore let s=
2
EA s=
2
NA s12 ; s=
2
EB s=
2
NB s22 ; s=
2
EC s=
2
NC s32 and denote the variance of the angle as the variance of the
centring errors, i.e., sβ2 = sCENT
2
then equation (18) becomes an expression for the variance of the centring errors written as
∂β ∂β 2 ∂β ∂β 2 ∂β ∂β 2
2 2 2 2 2 2
2
sCENT = + 1 s + + s + + s3 (19)
∂E A ∂N A ∂EB ∂N B ∂EC ∂N C
2
Substituting the partial derivatives of equation (16) into equation (19) and simplifying gives
l22 s12 + l12 s32 + ( l12 + l22 − 2l1l2 cos β ) s22
2
sCENT = (20)
l12 l22
Equation (20) is the same as equation (16.34) given in Richardus (1966, p. 290). The estimates of variances s12 , s22 , s32 at the
traverse stations A, B, C respectively, are the same in any direction at those points, and can be considered as estimates of the
precision of centring. If the back-sight and for-sight target centring errors are considered equal and the traverse lengths equal; i.e.,
2 2 2 2
{ }
s12 = s32 and l1= l2= l then equation (20) becomes sCENT = 2 s1 + 2 (1 − cos β ) s2 l which is a maximum when β = 180 and
cos β = −1 , in which case sCENT
2
( max
= ) {2s + 4s } l . The conclusion from this equation is that Total Station centring errors
2
1
2
2
2
have twice the effect as target centring errors. This makes it clear that greater care should be taken in centring the Total Station
(Richardus, 1966, p. 291).
In traversing operations it is plausible to consider that Total Station and target centring errors will be similar and s=
2
1 s=
2
2 s=
2
3 sc2
and equation (20) can be modified to give an expression of standard deviation
1 1 cos β
=
sCENT sc 2 + − (21)
l12 l22 l1l2
But this simplification does not take into account the ‘fact’ established above that Total Station centring errors have more effect
than target centring errors. We may test equation (21) using Matlab and a Monte Carlo simulation 4.
Consider Figure 1 and imagine that the instrument point B moves a distance c in a random direction and that c is a random
variable drawn from a normal distribution having a mean of zero and standard deviation sc and that the back-sight and for-sight
targets also move in a similar random manner. The angle at the instrument point is β1 . If this randomized location of the
instrument point and target points is repeated then a sample of angles β1 β 2 β 3 β n is obtained which will have a sample standard
′
deviation sβ = sCENT that we call the simulated standard deviation. If n is large then the sample standard deviation will approach
the population standard deviation σ β = σ CENT .
′
A Matlab function angletest.m was used to test the value of sCENT from equation (21) against the simulated value sCENT for a range
of back-sight and for-sight distances and angles and Table 1 below shows the output from this function.
>> angletest(0.005,10000,20)
d1 d2 beta sx rule1 (rule1/sx) rule1a (rule1a-sx)
---------------------------------------------------------------
123 34 104 32.32 45.87 (1.42) 32.43 ( 0.12)
59 119 267 19.57 27.88 (1.42) 19.71 ( 0.14)
138 71 132 18.25 26.06 (1.43) 18.43 ( 0.17)
126 166 175 12.41 17.68 (1.42) 12.50 ( 0.09)
128 164 260 10.69 15.05 (1.41) 10.64 (-0.04)
87 102 211 18.71 26.29 (1.41) 18.59 (-0.12)
115 103 75 12.63 17.74 (1.41) 12.55 (-0.08)
93 170 72 11.68 16.67 (1.43) 11.79 ( 0.10)
97 21 65 48.25 67.89 (1.41) 48.00 (-0.25)
81 48 187 30.13 42.32 (1.40) 29.92 (-0.21)
74 198 303 13.53 19.07 (1.41) 13.48 (-0.04)
95 200 337 9.64 13.63 (1.41) 9.64 (-0.00)
100 88 183 19.15 27.00 (1.41) 19.09 (-0.06)
85 180 163 15.69 22.20 (1.41) 15.70 ( 0.01)
69 196 106 16.59 23.36 (1.41) 16.52 (-0.07)
82 178 221 15.73 22.21 (1.41) 15.71 (-0.02)
124 186 126 11.22 15.94 (1.42) 11.27 ( 0.05)
172 43 12 21.87 30.68 (1.40) 21.69 (-0.18)
120 169 345 7.83 10.99 (1.40) 7.77 (-0.06)
92 46 305 21.97 31.12 (1.42) 22.00 ( 0.03)
Table 1 Output from Matlab function angletest.m
′
In Table 1 d1 and d2 are the back-sight and for-sight distances l1 , l2 ; sx is the simulated standard deviation that we denote sCENT
; and rule1 is sCENT computed from equation (21). rule1a = rule1/sqrt(2). angletest.m has the input parameters
sc = 0.005 m , n = 10000 simulations and 20 combinations of traverse distances and angles. The distances d1 and d2 are drawn
from a uniform distribution of random integers between 20 and 200 metres. The angle beta is drawn from a uniform distribution
of random integers between 10 and 350 degrees. The first line of Table 1 has traverse lines l1 = 124 m , l2 = 34 m and traverse
′
angle β = 104 ; then sCENT = 32.32′′ from 10000 simulations and sCENT = 45.87′′ from equation (21). The number in parentheses
is = ′ =
sCENT sCENT 45.87 32.32 1.42 that is the ratio between the computed and simulated standard deviation and is approximately
equal to 2 . The next number is sCENT 2 and the last number in parentheses is the difference sCENT ′ . Inspection of
2 − sCENT
the values in Table 1 shows that the standard deviation sCENT computed from equation (21) is consistently larger, by a factor of
′ .
2 , than the simulated value sCENT
This leads to a better rule for estimating the standard deviation of a centring error as
4
A method of repeated sampling to determine the properties of a particular function or phenomenon. The method employs a pseudo-random
number generator to simulate small random changes in function variables that can be used to assess their combined effect on the function.
1 1 cos β
= sc
sCENT + − (22)
l12 l22 l1l2
The rule (22) – developed from Richardus (1966, eq. 16.34, p. 290) – has some similarity with two other rules developed by
Briggs (1912, eq. 64, p.80) and Miller (1936, eq. 5, p.29).
2r 1 1 2 cos β
average angular error due to imperfect centring = ± + − Briggs (1912) (23)
π l12 l22 l1l2
1 1 cos β
probable error due to imperfect centring = ± p 2π + + Miller (1936) (24)
l12 l22 2l1l2
Rainsford (1957, pp.30-31) defines the probable error γ as that error that has a probability of occurrence of 0.5; and the average
error η as the mean of all errors taken with the same sign and gives the relationships γ = 0.6745 σ and η = 0.7979 σ where σ
is the standard deviation of a Normal distribution with mean zero.
In Briggs’ formula r is the average centring displacement of the instrument (targets considered error free) and in Miller’s formula
p is the probable error of plumbing over a station (instrument and targets). Miller’s derivation was motivated by Briggs only
considering the centring of the instrument. Both formula; modified by relationships connecting average and probable error with
standard deviation, were compared with the rule (22) but showed no real consistency.
Finally, the precision of a traverse bearing can be obtained from the following sequence given the precisions of a single face
pointing of a Total Station sα ; centring error sc and back-sight bearing sθ Back :
(i) compute the estimate of the precision of the centring error sCENT [equation (22)]
(ii) set the estimate of precision of pointing and reading sPR = sα [equation (15)]
(iii) compute the estimate of the precision of a traverse angle sβ [equation (12)]
(iv) compute the precision of the forward bearing of the traverse line sθ For [equation (11)]
Estimating the Precision of Traverse Distances
Most manufacturers of Total Stations state the accuracy of their instrument’s EDM in the following form (Rüeger 1990)
d
s=
± A+ B (25)
1000
where A is in mm; B in ppm 5; d is distance in metres. For example, if A = 5 mm, B = 3 ppm and the measured distance was
1355.310 m then s = ± 9.1 mm.
s (in mm) is considered to be an estimate of standard deviation. The term A includes the electronic resolution of the EDM,
compatibility of reflectors, accuracy of pre-set additive constants, maximum effects of short periodic errors and non-linear
distance dependent errors. The B term is a scale error determined by calibration over known distances. An estimate of the
variance (in mm2) is obtained from
2
d
s=
2
l A+ B (26)
1000
Estimating the Precision of the Last Line of a Traverse
Suppose that estimates of precisions of the traverse bearings and distances are obtained using the method and equations set out
above; then applying Propagation of Variances [equations (7), (8), (9) and (10)] we can obtain variances and covariances
5
ppm is parts-per-million and since there are 1 million mm in a km then ppm is also mm per km
sE2 , sN2 , sEN of the coordinates of the traverse stations. These quantities define the size, shape and orientation of error ellipses 6 at a
traverse station but in pairs (end points of traverse lines) they can be used to estimate the precision of the bearing and distance of a
traverse line. This, of course, is perfectly reasonable since these estimates have been used to compute sE2 , sN2 and sEN but general
equations can be developed that will be useful for traverse analysis.
The bearing and distance between points Pi and Pk are functions of the east and north coordinates of the points
θik tan −1 ( Ek − Ei ) ( N k − N=
= i )
f1 ( Ei , N i , Ek , N k ) and lik = ( Ek − Ei ) + ( N k − Ni ) = f 2 ( Ei , N i , Ek , N k ) or symbolically;
2 2
y = f ( x ) where y = [θik lik ] and x = [ Ei N k ] ; and PoV can be written as
T T
Ni Ek
Q yy = J yx Q xx J Tyx (27)
where
sE2i sEi Ni sEi Ek sEi Nk
sθ2ik sθ l sE N s 2
sNi Ek s Ni N k
=Q yy =
Ni
2
, Q xx i i (28)
sθ l slik sEi Ek sNi Ek sE2k sEk Nk
s s Ni N k sEk Nk sN2 k
Ei Nk
and the matrix of partial derivatives is
∂θik ∂θik ∂θ ik ∂θ ik
∂E − cos θik sin θik cos θik − sin θik
∂N i ∂Ek ∂N k l −b −aik aik
lik ik
= bik
=
i
(29)
cik
J yx lik lik
∂s ∂s ∂s ∂s ik −d −cik
cos θik ik
dik
− sin θik − cos θik sin θik
∂Ei ∂N i ∂Ek ∂N k
aik , bik are direction coefficients and cik , dik are distance coefficients. Performing the matrix multiplications of (27) gives the
variances of the bearing and distance as
sθ2=
ik
( ) (
bik2 sE2i + sE2k − 2 sEi Ek + aik2 sN2 i + sN2 k − 2 sNi Nk )
(30)
(
+ 2aik bik sEi Ni + sEk Nk − sEi Nk − sEk Ni )
s=2
lik ( ) (
dik2 sE2i + sE2k − 2 sEi Ek + cik2 sN2 i + sN2 k − 2 sNi Nk )
(31)
(
+ 2cik dik sEi Ni + sEk Nk − sEi Nk − sEk Ni )
In this limited analysis: (i) the covariances between points [the upper-right and lower-left blocks of Q xx in equation (28)] are
assumed to be zero; (ii) Pi is the start point of a traverse and assumed to be fixed and ‘error free’ (variances and covariances =
zero) and (iii) Pk is the last point of the traverse. This means that the equations for the estimating standard deviations are
simplified to
sθik = bik2 sE2k + aik2 sN2 k + 2aik bik sEk Nk (32)
slik = dik2 sE2k + cik2 sN2 k + 2cik dik sEk Nk (33)
Conclusion
Equations (32) and (33) are proposed as measures to assess the quality of a traverse. And a rule is proposed:
reject a traverse if the angular or linear misclose is greater than two standard deviations.
6
Variance in any direction about a point is a function of variances s , s 2
E
2
N
and the covariance sEN and defines the pedal curve of the standard
error ellipse
This is a better approach than using misclose ratios or other ‘practical standards’ and modern computer/calculator software could
make this seemingly complex analysis a simple field task. The example below may assist in understanding the analysis.
Example
57.9992
− 98 Assume:
.
sc = 0.002 m
↓
126. .305 centring errors
− 305
105° 2
22′ 1 5″
}
−
↓
↑
sα = 5′′
−
00
5″ 20″ st.dev. of direction
2
−
00 st.dev. of distance =sl 5 mm + 5 ppm
3 −↑
Use:
M
TU
s=
2 2
sPR + sCENT
2
DA
β
2
= sα2
00′
sPR
20″ 15″
25°
1 1 cos β 2
= 2+ 2−
}
2
sCENT sc
10″
l1 l2 l1l2
′
190° 16
=sθ2For sθ2Back + sβ2
Traverse operations:
−↑
−
00 1 1. Set up at 1; set bearing 25° 00′ along
290 line 1-2; read bearing line 1-4
° 42′ 35″
}
−
00
25″ 30
−
110°
↑
42′ 4 ″ 2. Set up at 2; set bearing 205° 00′
5″
}
55″ 50
″
4 − ↑ along line 2-1; read bearing line 2-3
133.5.543 3. Set up at 3; set bearing 285° 22′ 20″
37
along line 3-2; read bearing line 3-4
.3828
−
7
91.3↑
4. Set up at 4; set bearing 10° 16′ 15″
Figure 2 Traverse along line 4-3; read bearing line 4-1
Computation steps:
1. Tabulate the traverse bearings and distances (last line 4-1 not included) then calculate standard deviation of centring errors and
traverse angles.
Line Bearing Distance Point l1 l2 β sPR sCENT sβ
1-2 25 00′ 00′′ 126.305
2 126.305 57.995 260 22′ 20′′ 5′′ 8.07′′ 9.5′′
2-3 105 22′ 20′′ 57.995 3 57.995 133.545 264 53′ 55′′ 5′′ 7.88′′ 9.3′′
3-4 190 16′15′′ 133.545
Table 2 Traverse Bearings & Distances and standard deviations of traverse angles
2. Calculate the standard deviations of the traverse bearings and distances.
Line sBack sβ sFor Line sl
1-2 0′′ 1-2 0.006
2-3 0′′ 9.5′′ 10′′ 2-3 0.005
3-4 10′′ 9.3′′ 14′′ 3-4 0.006
Table 3 Standard deviations of traverse bearings and distances
3. Calculate estimates of precision at points 2, 3 and 4 using equation (7)
Point sE sN sE , N
1 0.0000 0.0000 0
2 0.0025 0.0054 1.3789 E − 04
3 0.0055 0.0062 9.4194 E − 06
4 0.0105 0.0087 1.3208E − 06
4. Calculate estimates of standard deviations of the bearing and distance of the last line 4-1 using equations (32) and (33)
sθ = 20.3′′
sl = 0.010 m
5. Determine if traverse is acceptable.
Angular misclose From the traverse shown in Figure 2, the angular misclose is 20″ (the difference between the two
observed bearings on the last line 4-1)
Linear misclose Using the mean bearing of line 4-1 (290° 42′ 40″) a closure of the traverse gives the linear misclose as
0.016 m.
The traverse is acceptable since the angular and linear misclosures are both less than two standard deviations of the
relevant estimates of the last line.
References
Briggs, H., 1912, The Effects of Errors in Surveying, Charles Griffen & Co., London, 1912.
http://www.archive.org/details/effectsoferrorsi00briguodt (accessed Sep 2012)
Mikhail, E.M. & Gracie, G., 1981, Analysis and Adjustment of Survey Measurements, Van Nostrand Reinhold Company, New
York.
Miller, A.W., 1936, ‘Analysis of the error in a traverse angle due to errors in plumbing over the station marks’, The Australian
Surveyor, Vol. 6, No. 1, pp. 28-31, March 1936.
Rainsford, H.F., 1957, Survey Adjustments and Least Squares, Constable & Co. Ltd, London.
Richardus, P., 1966, Project Surveying, North-Holland Publishing Company, Amsterdam.
Rüeger, J.M., 1990, Electronic Distance Measurement, 3rd edition, Springer-Verlag, Berlin.
Valentine, W., 1984, ‘Practical traverse analysis’, Journal of Surveying Engineering, Vol. 110, No. 1, pp. 58-65, March 1984.
Acknowledgements
The author wishes to thank the reviewers for their thoughtful comments.