Taking Advantage of Propagation of Error Fractions

Nigeria Journal of Geodesy Vol 2 (1) 2018 Accepted October 15, 2017

Taking Advantage of Propagation of Error Fractions in Survey Systems. Akajiaku C. Chukwuocha Ph.D. Department of Surveying and Geoinformatics [email protected] Abstract All information derived from survey measurements undergo processing through a set of specific computations. The errors in the measurements are transferred to the derived quantities through the process of propagation. Surveyors have always been trained to be cautious of propagation of errors. One to one propagation systems, are computational systems that are wholly additional in nature for example that in computing reduced levels from staff readings in spirit levelling exercises. Trigonometrical derivations are largely fractional in nature and may lead to better refined quantities than the observations if handled carefully. An example of deriving refined quantities from propagation of fractional errors is presented in a traverse exercise. It shows that the local accuracies and linear accuracies of such systems can be improved by a factor of about 2. Key words: Propagation, Errors, Survey, Traverse, Reorientation Traversing, Resected Traversing, Two Point Resection, Free Stationing, Angles, Electronic Distance Measurement (EDM), Total Station

Introduction Propagation of errors is a science that surveyors have studied over decades now in relation to their survey measurements. Measured quantities are processed in some equations to determine required quantities. The equations become the vehicles through which the required quantities are derived from the observed quantities. It is important to note that the errors in the measurements equally undergo the exactly the same transformation. Equations therefore act on the errors in the measured quantities to produce the errors in the derived quantities, which is error propagation. If a triangle is solved with the sine rule and the same triangle is solved with the cosine rule, the propagated errors will not be the same even though the measured quantities and their errors are the same in both cases. The implication is that the formula by which a desired quantity is derived from observed quantities is important in determining the quality of the quantity derived. In the practice of surveying the propagation of errors can be classified in three basic forms. The one to one error propagation exemplified in the determination of reduced level of a station from staff readings on a benchmark and on the desired station. The error in a staff reading is transferred as it was in the staff reading to the reduced level of the desired station. This first case occurs when the function is linear and additive with a coefficient of one. There are nonlinear fractional functions. An example of this second case is in the computation of coordinates from observed angles and distances. The propagation is of the


fraction of error because the trigonometric functions fractionalize the errors so to be propagated. The third case is of the polynomial form such as in the computation of areas and volumes. The form of propagation is a multiplication that must always propagate errors larger than a single value of the error. It should be noted that for traverses, even when the error in a station is fractionalized by the propagating equations used in determining the coordinates from observed angles and distances, these errors accumulate from a leg of the traverse to the next and so may still build up to significant magnitudes with the number of legs. Reorientation traversing has introduced solution of traverses by the derivation of base angles by indirect methods for cases where the takeoff and closing reference lines are not available (Chukwuocha et al, 2017). The method has found robust application in eliminating the short reference lines for connecting traverses to Global Navigation Satellite System (GNSS) controls. Another area of application is in solving the two point resection, where it resolves a number of difficulties in the method. An area where the propagation of error fractions has been found useful is in controlling the number of legs and average length of legs to be used in the scheme of reorientation traversing to eliminate short reference lines in traversing from Global Navigation Satellite System (GNSS) control stations. Another area where this is applicable is in reducing the magnitudes of random errors in a system for monitoring deformation of structures using total station observations. In this case deformation is easily visible by observing the raised magnitude of errors identified in the system and further rigorous analysis by least squares method will only be to determine the exact extent of deformation. The aim of this research is to show that better accuracy traverses will be obtained if the angles of a controlled system are determined from the traverse figure. The reasons behind this are from two facts. For a 1” readout theodolite or total station the precision is said to be 2”. For example Kolida Total Station details the precision of determining distance for the KTS 442 total station as ±(2mm+2ppm·D) (where ppm.D is parts per million multiplied by the distance), while for angle the precision is 2″ (Kolida undated). Table 1 presents the consequent uncertainties as a result of angular and distance errors projected over a distance. Taking other error sources such as centering into account, a 3” angular precision and a 3mm distance precision were used to estimate the displacements due to the errors. So it is seen that at 500m distance the uncertainties due to angular error is about 2 times that of the distance and at 1km it is about 3 times. Table 1: Displacement of Positions due to Errors in Angular and Distance Measurement. Projected Distance 1km 500m 250m 100m

Position Displacement due to angular error of 3" 14.5mm 7.3mm 3.6mm 1.4mm

Position Displacement due to Distance = 3mm +2pm.D 5.0mm 4.0mm 3.5mm 3.2mm

It is now commonly recognized that the old practice of distributing angular misclosures in the traverses equally to all angles at all traverse stations are not based on accurate facts. Recent literature refer to these


practice of non-rigorous distribution of errors as balancing (Ghilani, 2018, Subramanian 2010, Punmia et al 2006). The term balancing also describes the “arbitrary” distribution of positional misclosures in the traverse system that show in the computed coordinates of the closing control station as a function of the cumulative distances or as a function of the cumulative latitudes and departures of the traverse legs over the total latitudes and departure, respectively (Ghilani, 2018). The application of the propagation of error fractions has been found to resolve the angular errors in triangles and quadrilaterals completely. The procedure is simply to extract the angles from the figure of the triangle. Since the manual extraction by graphical figures would not suffice, the traverse figure is set up mathematically using the measurements derived from the field observations, grid distances and horizontal angles, without any form of adjustment. The theory of the propagation of error fractions is shown in the derivation of the propagated errors in the angles derived from the traverse figure using Figure 1.

1 δ1

δ2

n

2

n-1 Figure 1.

Sketch for the General Indirect Determination of Angles

From Figure 1, 𝛿1 = 𝐴𝑧1,2 − 𝐴𝑧1,𝑛

.

.

.

(1)

𝛿2 = 𝐴𝑧1,𝑛 − 𝐴𝑧𝑛,𝑛−1

.

.

.

(2)

.

.

.

(3)`

.

.

.

(4)`

Where, 𝑋2 − 𝑋1 ) 𝑌2 − 𝑌1

𝐴𝑧1,2 = 𝑎𝑟𝑐𝑡𝑎𝑛(

𝑋 −𝑋

𝐴𝑧1,𝑛 = 𝑎𝑟𝑐𝑡𝑎𝑛( 𝑌𝑛 − 𝑌1 ) 𝑛

1

Nigeria Journal of Geodesy Vol 2 (1) 2018 Accepted October 15, 2017 𝑋

−𝑋

𝐴𝑧𝑛,𝑛−1 = 𝑎𝑟𝑐𝑡𝑎𝑛( 𝑌𝑛−1 − 𝑌𝑛 ) 𝑛−1

.

𝑛

.

.

(5)`

The error in the azimuths will generally be determined from the partial derivatives where from equation 6, 𝛥𝑋

𝑋 −𝑋

𝐴𝑧1,2 = 𝑎𝑟𝑐𝑡𝑎𝑛 ( 𝑌2 − 𝑌1 ) = 𝑎𝑟𝑐𝑡𝑎𝑛 ( 𝛥𝑌1,2 ) 2

𝑑𝐴𝑧1,2 = 𝑙𝑒𝑡 𝑢 = (

𝜕𝐴𝑧1,2

𝛥𝑌1,2

1,2

𝜕𝐴𝑧1,2

𝑑𝛥𝑥 +

𝜕𝛥𝑥

𝛥𝑋1,2

1

𝜕𝛥𝑦

𝑑𝛥𝑦 .

.

) , 𝑠𝑜 𝑡ℎ𝑎𝑡 𝐴𝑧1,2 = arctan 𝑢

.

.

.

(6)

.

.

.

.

.

(8)

.

.

.

(9)

(7)

then, 𝑑𝐴𝑧1,2 =

𝑑𝐴𝑧1,2

( .

𝑑𝑢

𝑑𝐴𝑧1,2 𝑑𝑢

𝜕𝑢

(𝜕𝛥𝑥 𝑑𝛥𝑥 +

𝑑 arctan 𝑢 ) 𝑑𝑢 𝑑𝑢

) 𝑑𝑢 = (

𝜕𝑢 𝑑𝛥𝑦) 𝑑𝑢 𝜕𝛥𝑦

1 ) 𝑑𝑢 1+ 𝑢2

= (

= (

1 2 𝛥𝑋 1+ ( 1,2 ) 𝛥𝑌1,2

) 𝑑𝑢 = (

(𝛥𝑌1,2 )2 (𝛥𝑌1,2 )2 + (𝛥𝑋1,2 )2

) 𝑑𝑢

. .

(10) 𝛥𝑋

𝑆𝑖𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 (8) 𝑢 = ( 𝛥𝑌1,2 ) 1,2

𝛥𝑌1,2 .𝑑𝛥𝑋1,2 − 𝛥𝑋1,2 .𝑑𝛥𝑌1,2

𝑡ℎ𝑒𝑛, 𝑑𝑢 = (

(𝛥𝑌 2 1,2 )

𝑖𝑛 𝑙𝑖𝑛𝑒 𝑤𝑖𝑡ℎ (9), 𝑑𝐴𝑧1,2 = ((𝛥𝑌

)

.

.

(𝛥𝑌1,2 )2

𝛥𝑌1,2 .𝑑𝛥𝑋1,2 − 𝛥𝑋1,2 .𝑑𝛥𝑌1,2 (𝛥𝑌1,2 )2 + (𝛥𝑋1,2 )2

)(

From equations (2)

(𝛥𝑌1,2 )2

)

𝛥𝑌𝑎,𝑏 .𝑑𝛥𝑋𝑎,𝑏 − 𝛥𝑋𝑎,𝑏 .𝑑𝛥𝑌𝑎,𝑏

𝑔𝑒𝑛𝑒𝑟𝑎𝑙𝑙𝑦, 𝑑𝐴𝑧𝑎,𝑏 = (

(11)

𝛥𝑌1,2 .𝑑𝛥𝑋1,2 − 𝛥𝑋1,2 .𝑑𝛥𝑌1,2

2 2 1,2 ) + (𝛥𝑋1,2 )

ℎ𝑒𝑛𝑐𝑒, 𝑑𝐴𝑧1,2 = (

.

(𝛥𝑌𝑎,𝑏 )2 + (𝛥𝑋𝑎,𝑏 )2

)

) .

.

.

(12)

.

.

.

(13)

.

.

.

(14)

.

.

.

(15)

𝛿1 = 𝐴𝑧1,2 − 𝐴𝑧1,𝑛 𝑑𝛿1 = (𝑑𝐴𝑧1,2 − 𝑑𝐴𝑧1,𝑛)


Methodology Fig 2 illustrates the indirect determination of angles δ1 and δ3 in a traverse scheme based on the figure of the traverse with the angle δ2 and traverse legs, d1, d2, d3.The angles are first measured properly in the field. The deriving of base angles from traverse figures can be done by graphically plotting the traverse using only one of the observed angles. However the manual scaling of the angles will degrade the efforts in the derivation. So using point P1 as origin with coordinates 0,0 and adopting 1800 as the azimuth the line from the first point to the second point P2 any of the two angles may be determined first and then the third angle with a check are determined in the second iteration.

δ2

δ3

δ1

Fig 2. Scheme for Deriving Base Angles from the Figure of the Traverse. The procedure of extracting an angle at a point from the traverse figure is given in the fact that an angle at a point between two lines is equal to the azimuth of the succeeding line minus the azimuth of the preceding line, moving in a clockwise direction. The traverse figure is computed in the steps given that follow. Step 1: Coordinates of P1(x,y) = A(0,0) Step 2: Azimuth of line P1P2 = 1800 Step 3: Coordinates of P2(x,y) = P2(d1*sin180, d1*cos1800) = P2(0,-d3) Step 4: Azimuth of line P2P3 = δ2 Step 5: Coordinates of P3(x,y) = P3(d3*SIN1800 + d1*SIN δ2, d3*COS1800 + d1*COSδ2) = P3(d1*SIN δ2, -d3 + d1*COS δ2) Step 6: Azimuth P3P1 = arctan ((XP1 – X P3)/(YP1 – YP3)) = arctan((0 – d2*SIN δ2)/(0 – (-d3+ d1*COS δ2))) = arctan ((– d2*SIN δ2)/ (d3 – d1*COS δ2))) Step 7: By the direction of line P2P1: = arctan ((-d2*SIN δ2)/( d3 – d1*COS δ2)) + 1800 Step 8: Az. P1P2 = 00 (3600)


Step 9: Az. P2P3 = δ2 Angle δ1, (P2P1P3) = Az.P1P2 – Az.P1P3 = (arctan((0 - 0)/( -d3 - 0)) + 1800 – (arctan((d1*SIN δ2 - 0)/(-d3 + d1*COS δ2 - 0)) = 1800 – (arctan((d1*SIN δ2)/(d1*COS δ2 - d3)) . . . (1)

Angle δ2, (P1P2P3) = Az.P2P3 – Az.P2P1 = (arctan((d1*SIN δ2 - 0)/( (-d3 + d1*COS δ2) – (-d3))) – arctan ((0 – 0)/(0 – (-d3)) = (arctan((d1*SIN δ2)/(d1*COS δ2)) – arctan (0) = arctan(TAN δ2) = δ2 .

.

.

(2)

Angle δ3, (P2P3P1) = Az.P3P1 – Az. P3P2 = (arctan((0 - d1*SIN δ2)/(0 – (-d3 + d1*COS δ2))) – arctan ((0 – d1*SIN δ2)/(-d3 + d1*COS δ2)) = arctan((-d1*SIN δ2)/(d3 - d1*COS δ2))) – arctan ((– d1*SIN δ2)/(d1*COS δ2 - d3)) . . . (3) In summary, Angle δ1, (P2P1P3) = 1800 – (arctan((d1*SIN δ2)/(d1*COS δ2 - d3))

. . . (4)

Angle δ2, (P1P2P3) is the angle used in mathematically constructing the traverse figure and will only get a computed value in the second iteration when another derived angle is used to construct the traverse. Angle δ3, (P2P3P1) = arctan((-d1*SIN δ2)/(d3 - d1*COS δ2))) – arctan((–d1*SIN δ2)/(d1*COS δ2 - d3)) . . . (5)

The coordinates of the control station are FUTOG 004: X = 502633.350m, Y = 154106.026m FUTOG 005: X = 503759.271m, Y = 153582.882m. Quantities measured in the field traverse is presented in Table 2. First the traverse was computed in the traditional method in Table 3. Next computed values were determined for angles at FUTOG 004 (δ1) and FUTOG 005 (δ2) were computed using equations (4) and (5) respectively. BACK SIGHT FUTOG 005 FUTOG 004 FUTOG 006

Table 2: Field Quantities Measured in the Triangle Case STN AT FORE SIGHT ANGLE FUTOG 004 FUTOG 006 0160 56’ 45.2” 0 FUTOG 006 FUTOG 005 136 20’ 29.8” FUTOG 005 FUTOG 004 0260 42’ 54.9”

DISTANCE 808.421m 524.156m 1241.522m

The result of the traverse computation in the traditional Bowditch method is presented in Table 3. Table 3. Results of Comptation of the Traditional and 2 Reorientation Traverse Cases Traverse Type Angular Linear Accuracy misclosure Traditional (all distances and all angles measured in field) 10" 1/83,973.75 Reorientation I 00" 1/328074.684 (all distances and 2 base angles indirectly determined from traverse figure)


Reorientation II (all distances and all 3 base angles indirectly determined from traverse figure )

00"

1/240561.931

Least Squares Adjustment Computation of the Traverses using Adjust Software Results of the Triangle Case are presented in tale 4 while those of the Quadrilateral case are resented in Table 5.

Table 4: Result of Least Squares Adjustment in the Triangle Cases Traditional Traverse Distance, residuals: 808.421m, 0.003m; 524.156m, 0.003m; Angles, residual 0160 56’ 45.2”, 3"; 1360 20’ 29.8” 3"; 0260 42’ 54.9”, 3" Station FUTOG006

X

Y

503,235.375

Sx

153,566.483

0.0025

Sy 0.0032

Su 0.0033

Sv 0.0023

t 23.84°

r(95%) 0.0071

Local accuracy: 0.0071 Reorientation Traverse Indirect Determination of Angles First Iteration Distance, residuals: 808.421m, 0.003m; 524.156m, 0.003m Angles, residuals: 160 56’ 43.1”, 3”; 1360 20’ 29.8”, 3”; 260 42’ 47.1”, 3” Station FUTOG006

X 503,235.376

Y

Sx

153,566.491

0.0022

Sy 0.0019

Su 0.0022

Sv 0.0019

t 79.60°

r(95%) 0.0051

Local accuracy: 0.0051 Second Iteration Distance Errors 3mm, 3mm, Angle Errors 1", 1", 1"

Angles: 160 56’ 42.9”; 1360 20’ 28.9”; 260 42’ 47.3”

Station

X

Y

Sx

Sy

Su

Sv

t

r(95%)

FUTOG006

503,235.377

153,566.491

0.0021

0.0013

0.0022

0.0013

100.50°

0.0045

Local accuracy: 0.0045


Table 5: Result of Least Squares Adjustment in the Quadrilateral Cases Traditional Traverse Distance errors 2mm, 2mm, 2mm, Angle errors 3", 3", 3", 3" Station

X

Y

Sx

Sy

Su

Sv

t

r(95%)

P1

502,859.641 153,882.976 0.0020

0.0024

0.0026

0.0017

31.96°

0.0055

FUTOG006

503,235.373 153,566.485 0.0018

0.0030

0.0031

0.0017

10.66°

0.0063

Local accuracy: 0.0059 Reorientation Traverse Indirect Determination of Angles First Iteration Distances, estimated residual 317.741m, 0.002m; 491.266m, 0.002m; 524.156m, 0.002m Angle Used, estimated residuals 19°39'54.3", 1.0"; 175°31'18.2", 1.0"; 138°05'59.7",1.0"; 26°42'48.0", 1.0" Station

X

Y

Sx

Sy

Su

Sv

t

r(95%)

P1

502,859.641 153,882.978 0.0014

0.0013

0.0016

0.0010

133.66°

0.0034

FUTOG006

503,235.374 153,566.490 0.0017

0.0013

0.0017

0.0013

106.95°

0.0037

Local accuracy: 0.0036 Second Iteration Distances, estimated residual 317.741m, 0.002m, 491.266m,0.002m, 524.156m, 0.002m Angle Used, estimated residuals 19°39'54.3", 1.0"; 175°31'20.2", 1.0"; 138°05'58.5",1.0"; 26°42'47.8", 1.0" Station

X

Y

Sx

Sy

Su

Sv

t

r(95%)

P1

502,859.642 153,882.979 0.0014

0.0013

0.0016

0.0010

133.66°

0.0034

FUTOG006

503,235.373 153,566.490 0.0017

0.0013

0.0017

0.0013

106.95°

0.0037

Local accuracy: 0.0036 Discussion of Result Table 3, Table 4 and Table 5 all show the improvements that the application of the propagation of error fractions bring to the results of the traverses. In Table 3 without using the old method of balancing of angles, the angular closure of the in traverse is accomplished in a scientific many. This is continually repeatable given equations (14) and (15). The same Table 3 presents the improved linear accuracy. Table 4 and Table 5 present the accuracy indicators in terms of the standard deviations in x and y coordinates. The local accuracy presents a more reliable accuracy indicator than the linear accuracy Conclusion and Recommendations This study has successfully demonstrated the advantages of applying the principles of propagation of error fractions. The local accuracy of the traverses were improved by an average of 1.6 factors in both the


triangle and quadrilateral cases. Accumulation of errors over more stations and legs of traverses may inhibit the application to traverses of not more than two unknown stations. This method recommends itself in the area of improving quality of traverses for critical control extension. It will find application in the areas where it is important to keep the residuals in the observations very low such as in the areas of deformation monitoring and other sensitive industrial uses where accuracy must be kept very high. Further uses of deriving values due to propagation of error fractions should be explored even beyond the field of Surveying and Geoinformatics.

References Chukwuocha, A. C., Moka, E. C., Uzodimma, V. N., Ono M. N. 2017. Solving Control Reference Azimuth Problems of Traversing Using Reorientation Traversing,. Surveying and Land Information Science May, 2017 Vol. 76 (1). Pp 23 – 37. http://www.ingentaconnect.com/content/aags/salis/2017/00000076/00000001/art00004. Accessed October 13, 2011 Davis, Raymond E., Francis S. Foote, M. Anderson, and Edward M. Mikhail. 1981. Surveying Theory and Practice. pp 319 – 333 Ghilani, C. D. 2010. Adjustment Computations – Spatial data Analysis 5th Ed. New Jersey, John Wiley and Sons pp. 74 – 76. –––––. 2018. Elementary Surveying: An Introduction to Geomatics. 15th ed. Prentice Hall, New York pp 239 – 269, 387 – 389. Kolida

(undated)

User’s

Operational

Manual.

https://www.geoappliedintl.com/wp-

content/uploads/2015/08/KTS-440-RCLC-Operation-Manual-2010-nov.pdf. Accessed October 13, 2011 Punmia, B. C., Jain, Ashkok. K., Jain, Arun. K. 2006. Surveying vol. 1. Laxmi Publications (P) Ltd., New Delhi. Subramanian, R. 2010. Surveying and Levelling. Oxford University Press, New Delhi. p. 243.
