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. 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.”
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 Station1 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 =lsinθ , ∆N =lcosθ 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
random2 or systematic3 errors or reveal their effects and random errors do not accumulate in direct proportion to distance
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.
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
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 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 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]T , and the variables x,y,z in a vector x = [ x
y z]T where [ ]T represents the vector (or matrix) transpose; then we may write: and the Law of Propagation of Variances is expressed as (Mikhail & Gracie 1981) where the variance matrices are Σ yy = σ p2
σ qp and the matrix of partial derivatives is J yx = ∂x ∂q ∂x
If y = f (x)
T Σ yy = J yxΣxx J yx σ pq σ q2 Σxx = σ yx σ zx σ x2 ∂p ∂p ∂y ∂q ∂y ∂p ∂z ∂q ∂z σ xy σ 2
y σ zy σ xz
σ yz σ z2 σ w2 ≈ ∂∂wx 2 σ x2 + ∂w 2 ∂w 2
∂y σ y2 + ∂∂wz 2 σ z2 + + ∂t σ t2 + 2 ∂∂wx ∂∂wyσ xy + 2 ∂∂wx ∂∂wzσ xz + + 2 ∂∂wx ∂∂wtσ xt + 2 ∂∂wy ∂∂wzσ xz + + 2 ∂∂wy ∂∂wtσ yt + + 2 ∂∂wz ∂∂wtσ zt σ w2 ≈ ∂∂wx 2 σ x2 + ∂w 2 ∂w 2 ∂y σ y2 + ∂∂wz 2 σ z2 + + ∂t σ t2 (1) (2) (3) (4) (5) (6) 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 s2 (variance) calculated from small samples of measurements; or just assumed. So equations (1) and (2) can be expressed with estimates sx2 , sy2 , 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 Qxx containing estimates of variances and covariances replacing variance matrices Σ yy and Σxx . 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 sinθ + Ek −1 =∆E + Ek −1
Nk =l cosθ + Nk −1 =∆N + Nk −1
y = f (x) where y = [Ek
Nk ]T and x = [l θ
Nk −1 ]T . Applying PoV to (6) gives (7) (8) (9) (10) Q yy =sE2ksEk2Nk , Qxx sEk Nk sNk =s00θ2sE200k−1 s00l2 0 0 sEk−1Nk−1 0 0 sEk−1Nk−1 2 sNk−1 Q yy and Qxx are cofactor matrices containing estimates of variances and covariances of the elements of y and x respectively.
J yx ∂Ek ∂Ek ∂Ek ∂l ∂θ ∂Ek −1 = ∂∂Nlk ∂∂Nθk ∂∂ENk k−1 ∂Ek
∂Nk −1
∂Nk ∂Nk −1
sinθ l cosθ 1 = cosθ −l sinθ 0 0 1 =∆N ∆lE 1 0 ∆lN −∆E 0 1 Carrying out the matrix multiplications of (7) gives the cofactor matrix of the computed coordinates ∆E 2 l sl2 + ( ∆N )2 sθ2 + sE2k−1 Q yy = ∆El ∆2N sl2 − ( ∆E∆N ) sθ2 + sEk−1Nk−1 ∆El ∆2N sl2 − ( ∆E∆N ) sθ2 + sEk−1Nk−1 ∆N 2 sl2 + ( ∆E )2 sθ2 + sN2k−1 l 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 Qxx 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
1. The Total Station is pointed to the back-sight and the horizontal circle (or direction) read, α Back .
circle read again, α For .
=α For −α Back .
β 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 ( sθ2 ) , equation (2) can be applied to the equation θ For =θ Back + β , assuming that the measured angle β and the bearing of the back-sight θ Back are independent.
2 sθFor =s2 θBack + sβ2 (11) N
α1 ° θ β
A ° B N − A
N θ1 (12) (13) (14) Instrument Point B In equation (11) an estimate of the variance of the measured angle ( s2 ) is required and this may be considered as consisting of β 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 sP2R and sCENT are estimates of the variances of pointing and reading error and centring error respectively and how these are 2 obtained is discussed in the following sections.
Estimating the Precision of Pointing and Reading Errors horizontal directions to A and C respectively and β is the horizontal angle between the two traverse lines.
Back-sight Target 2 Assuming the variances of the FL/FR directions are equal and independent of direction, then we may write sαFL =s2 αFR =sα2 and the variances of the FL/FR angles become sβ2FL =2sα2and sβ2FR
=2sα2 The angle β is the average of the FL/FR angles, i.e., β
=12(β FL + β FR ) sβ2 = ( 12 )2 s2 βFL
+ ( 12 )2 sβ2FR = sα2 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 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 sα2 is the variance of a single-face direction of the Total Station.
Estimating the Precision of Instrument and Target Centring Errors
β = θ 2 −θ1 = arctan
EC − EB − arctan NC − NB
EA − EB
NN − NB
T
Q yy = J yxQxxJ yx so β = β ( EA , N A , EB , NB , EC , NC ) or in symbolic matrix form y = f (x) and applying Propagation of Variances gives ∂β where Q yy = sβ2 ; J yx = ∂EA ∂β ∂N A ∂β ∂EB ∂β ∂NB ∂β ∂EC ∂β
and ∂NC sE2A sEANA sN2A Qxx = sEAEB sNAEB sE2B sEANB sNANB sEBNB sN2B sEAEC sNAEC sEBEC sNBEC sE2C sEANC sNANB
sEBNC
sNBNC sEC NC sN2C variables. expressed as
estimating the effects of centring errors, it is assumed that all covariances terms in Qxx equal zero, i.e.,
This means that equation (17) can be 2 sCENT = l22s12 + l12s32 + (l12 + l22 − 2l1l2 cos β ) s2
2 l12l22 2 2 2
2 sβ2 = ∂∂EβA sE2A + ∂∂NβA sN2A + ∂∂EβB sE2B + ∂∂NβB sN2B + ∂∂EβC sE2C + ∂∂NβC sN2C 2 2 Furthermore let sE2A =sN2A =s12 ; sE2B =sN2B =s22 ; sE2C =sN2C centring errors, i.e., sβ2 = sC2ENT then equation (18) becomes an expression for the variance of the centring errors written as =s32 and denote the variance of the angle as the variance of the sC2ENT = ∂∂EβA 2 + ∂∂NβA 2 s12 + ∂∂EβB 2 + ∂∂NβB 2 s22 + ∂∂EβC 2 + ∂∂NβC 2 s32
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.,
s12 = s32 and l1 = l2 = l then equation (20) becomes sCENT = {2s12 + 2 (1− cos β ) s22} l 2 which is a maximum when β = 180 and
2
cos β = −1, in which case sCENT (max ) ={2s12 + 4s22} l 2 . The conclusion from this equation is that Total Station centring errors
2
have twice the effect as target centring errors. This makes it clear that greater care should be taken in centring the Total Station
In traversing operations it is plausible to consider that Total Station and target centring errors will be similar and s12 =s22 =s32 =sc2 and equation (20) can be modified to give an expression of standard deviation
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 β2 β3 βn is obtained which will have a sample standard deviation sβ = sC′ENT 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 sC′ENT 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) ; 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 sC′ENT = 32.32′′ from 10000 simulations and sCENT = 45.87′′ from equation (21). The number in parentheses is sCENT sC′ENT
=45.87 32.32 =1.42that is the ratio between the computed and simulated standard deviation and is approximately equal to
the values in Table 1 shows that the standard deviation sCENT computed from equation (21) is consistently larger, by a factor of
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. sCENT =sc l112 + l122 − cos β l1l2 2r π 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). average angular error due to imperfect centring = ±
probable error due to imperfect centring = ± p 2π
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 sθBack :
(i)
(ii)
(iii)
(iv)
compute the estimate of the precision of the centring error sCENT [equation (22)]
set the estimate of precision of pointing and reading sPR = sα [equation (15)]
compute the estimate of the precision of a traverse angle sβ [equation (12)]
compute the precision of the forward bearing of the traverse line sθFor [equation (11)]
Estimating the Precision of Traverse Distances
Most manufacturers of Total Stations state the accuracy of their instrument’s EDM in the following fo
sE2i sEiNi =sEiNisN2i sEiEk sNiEk sEiNk sNiNk sEiEk sNiEk sE2k sEk Nk J yx =∂∂∂∂θNNsikii∂∂∂∂θEEsikkk ∂∂∂∂θEEsikii ∂∂∂∂NθNsikkk =−−csoliinksθθiikk− csoinlsiθkθiikk cosθ ik
lik sinθ ik − sinθ ik lik cosθ ik
−bik −aik bik aik = −dik −cik dik cik (29) 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 In this limited analysis: (i) the covariances between points [the upper-right and lower-left blocks of Qxx 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
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 sE2 , sN2 and the covariance sEN and defines the pedal curve of the standard error ellipse 290° 42′ 25″} 30″
35″ 110° 42′ 45″
55″} 50″
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.
1. Tabulate the traverse bearings and distances (last line 4-1 not included) then calculate standard deviation of centring errors and sC2ENT = l112 + l122 − col1sl2β sc2 sθ2For
=sθ2Back + sβ2
4.
line 1-2; read bearing line 1-4
along line 2-1; read bearing line 2-3 Set up at 3; set bearing 285° 22′ 20″ along line 3-2; read bearing line 3-4 Set up at 4; set bearing 10° 16′ 15″ along line 4-3; read bearing line 4-1
Line sBack 1-2 2-3 3-4 0′′ 10′′ sFor 4. Calculate estimates of standard deviations of the bearing and distance of the last line 4-1 using equations (32) and (33)
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) 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.