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CHAPTER 8
ERROR PROPAGATION IN TRAVERSE SURVEYS

8.1 INTRODUCTION

Even though the specifications for a project may allow lower accuracies, the presence of blunders in observations is never acceptable. Thus, an important question for every surveyor is, “How can I tell when blunders are present in the data?” In this chapter, we begin to address that question, and in particular, stress traverse analysis. This topic is discussed further in Chapters 21 and 25.

In Chapter 6, it was shown that the estimated error in a function of observations is dependent on the individual errors in the observations. Generally, observations in horizontal surveys (e.g., traverses) are independent. That is, the measurement of a distance observation is independent of the azimuth observation. But the latitude and departure of a line, which are computed from the distance and azimuth observations, are not independent. Figure 8.1 shows the effects of the errors in distance and azimuth observations on the computed latitude and departure. In Figure 8.1, it can be seen that there is correlation between the latitude and departure; that is, if either distance or azimuth observation changes, it causes changes in both latitude and departure.

Illustration of Latitude and departure uncertainties due to (a) the distance standard error (σD) and (b) the azimuth standard error (σα). — **FIGURE 8.1** Latitude and departure uncertainties due to (a) the distance standard error (σ_D) and (b) the azimuth standard error (σ_α). Note that if either the distance or azimuth changes, both the latitude and departure are affected.

Because the observations from which latitudes and departures are computed are assumed to be independent with no correlation, the SLOPOV approach [Equation (6.16)] can be used to determine the estimated error in these computed values. However, for proper computation of estimated errors in functions that use these computed values (i.e., latitudes and departures), the effects of correlation must be considered, and thus the GLOPOV approach [Equation (6.13)] will be used.

8.2 DERIVATION OF ESTIMATED ERROR IN LATITUDE AND DEPARTURE

When computing the latitude and departure of a line, the following well-known equations are used:

(8.1) $images$

where Lat is the latitude, Dep the departure, Az the azimuth, and D the horizontal length of the line. To derive the estimated error in the line's latitude or departure, the following partial derivatives from Equation (8.1) are required in using Equation (6.13).

(8.2) $images$

EXAMPLE 8.1

A traverse course has a length of 456.87 ± 0.02 ft and an azimuth of 23°35′26″ ± 9″. What are the latitude and departure, and their estimated errors?

SOLUTION

Using Equation (8.1), the latitude and departure of the course are

$images$

The estimated errors in these values is solved using Equation (6.16) as

$images$

Substituting partial derivatives into the above yields

(8.3) $images$

Notice that the angular values must be converted to radians to prevent a mixing of units. A close look at the units of the matrix multiplication shows that the final units will be ft². Entering the appropriate numerical values into Equation (8.3), the covariance matrix is

$images$

from which

(8.4) $images$

In Equation (8.4), $images$ is the variance of the latitude, $images$ is the variance of the departure, and σ₁₂ and σ₂₁ are the covariances. Thus, the standard errors are: $images$ and $images$ . Note that the off-diagonal of Σ_{Lat, Dep} is not equal to zero, and thus, the computed values are correlated as illustrated in Figure 8.1.

8.3 DERIVATION OF ESTIMATED STANDARD ERRORS IN COURSE AZIMUTHS

Equation (8.1) is based on the azimuth of a course. However, in practice traverse azimuths are normally computed from observed angles rather than measured directly. Thus, another level of error propagation exists in calculating the azimuths from angular values. In the following analysis, consider that angles-to-the-right are observed and that azimuths are computed in a counterclockwise direction successively around the traverse using the formula

(8.5) $images$

where Az_C is the azimuth for the current course, Az_P the previous course azimuth, and θ_i the appropriate interior angle to use in computing the current course azimuth. By applying Equation (6.18), the error in the current azimuth, Az_C, is

(8.6) $images$

In Equation (8.6) $images$ is the estimated error in the appropriate interior angle used in computation of the current azimuth, $images$ is the estimated error in the azimuth of the previous course, and $images$ is the estimated error in the course being computed. This equation is also valid for azimuth computations going clockwise around the traverse. The proof of this is left as an exercise.

8.4 COMPUTING AND ANALYZING POLYGON TRAVERSE MISCLOSURE ERRORS

From elementary surveying, it is known that the following geometric constraints exist for any closed polygon-type traverse:

(8.7) $images$

(8.8) $images$

Deviations from these conditions, normally called misclosures, can be calculated from the observations in any traverse. Statistical analyses can then be performed to determine the acceptability of the misclosures and check for the presence of blunders in the observations. If blunders appear to be present, the measurements must be rejected and the observations repeated.

However, while the initial azimuth will correctly orient the traverse, its error does not affect the linear misclosure of the traverse. This can also be stated about the observed angle at the control station, since the azimuth for the first course is either given or assumed. Thus, the errors in these observations can be ignored when checking the linear misclosure of the traverse. The error in the initial angle should be used in checking the angular misclosure in the traverse. The following example illustrates methods of making these computations for any closed polygon traverse.

EXAMPLE 8.2

Compute the angular and linear misclosures for the traverse illustrated in Figure 8.2. The observations for the traverse are given in Table 8.1. Determine the estimated misclosure errors at the 95% confidence level, and comment on whether or not the observations may contain any blunders.

FIGURE 8.2 Link traverse example.

TABLE 8.1 Distance and Angle Observations for Figure 8.2

Occupied	Sighted	Distance (ft)	S (ft)	BS	Occ	FS	Angle^*	S(″)
A	B	1435.67	0.020	E	A	B	110°24′40″	±3.5
B	C	856.94	0.020	A	B	C	87°36′14″	±3.1
C	D	1125.66	0.020	B	C	D	125°47′27″	±3.6
D	E	1054.54	0.020	C	D	E	99°57′02″	±3.1
E	A	756.35	0.020	D	E	A	116°14′56″	±3.9
							Σ = 540°00′19″

^*Each angle was measured using four repetitions.

SOLUTION

Angular Check: First the angular misclosure is checked to see if it is within the tolerances specified. From Equation (6.18), and using the standard deviations given in Table 8.1, the angular sum should have an error within $images$ 68.3% of the time. Since the angles were measured four times, each computed mean has three degrees of freedom, and the appropriate t value from Table D.3 (the t distribution) is t_0.025,3, which equals 3.183. This is a two-tailed test since we are looking for the range that is statistically equal to zero at the selected level of confidence. If this range contains the actual misclosure, then there is no statistical reason to believe that the observations contain a blunder. In this case, the angular misclosure at a 95% confidence level is estimated as

(8.9) $images$

Using the summation of the angles in Table 8.1, the actual angular misclosure in this problem is

$images$

Thus, the actual angular misclosure for the traverse (19″) is within its estimated range of error, and there is no reason to believe that a blunder exists in the angles.

Azimuth Computations: In this problem, no azimuth is given for the first course. However, to solve the problem, the azimuth of the first course can be assumed as 0°00′00″ and to be free of error. Since only geometric closure on the traverse is being checked, not the orientation of the traverse, this can be done even when the initial course azimuth is observed. For the data of Table 8.1, and using Equations (8.5) and (8.6), the values for the course azimuths and their estimated errors are computed and listed in the Table 8.2.

images — **TABLE 8.2** Estimated Errors in the Computed Azimuths of Figure 8.2

Computation of Estimated Linear Misclosure: Equation (6.13) properly accounts for correlation in the latitude and departure when computing the linear misclosure of a traverse. Applying the partial derivatives of Equation (8.2) to the latitudes and departures, the Jacobian matrix, A, has the form

(8.10) $images$

Because the lengths and angles were observed independently, they are uncorrelated. Thus, the appropriate covariance matrix, Σ, for solving this problem using Equation (6.13) is

(8.11) $images$

Again, notice that the estimated errors for the computed azimuths are converted to their equivalent radian values to ensure that the results of the computations will yield estimated errors in linear units. Substituting numerical values for this problem into Equations (8.10) and (8.11), the covariance matrix, Σ_{Lat, Dep}, is computed for the latitudes and departures AΣA^T, or

(8.12) $images$

By taking the square roots of the diagonal elements in the Σ_{Lat, Dep} matrix [Equation (8.12)], the errors for the latitude and departure of each course are found. That is, the estimated error in the latitude for course BC is the square root of the “3,3” element in Equation (8.12), and the estimated error in the departure of BC is the square root of the “4,4” element. In a similar fashion, the estimated errors in latitude and departure can be computed for any other course.

The formula for determining the linear misclosure of a closed polygon traverse is

(8.13) $images$

where LC is the linear misclosure. To determine the estimated error in the linear misclosure, Equation (6.13) is applied to the linear misclosure formula in Equation (8.13). The necessary partial derivatives from Equation (8.13) for substitution into Equation (6.13) must first be determined. The partial derivatives with respect to the latitude and departure of course AB are

(8.14) $images$

Notice that these partial derivatives are independent of the course. Also, the other courses have the same partial derivatives as given by Equation (8.14), and thus, the Jacobian matrix for Equation (6.13) has the form

(8.15) $images$

As shown in Table 8.3, the sum of the latitudes is −0.083, the sum of the departures is 0.022, and LC = 0.086 ft. Substituting these values into Equation (8.15), which, in turn, is substituted into Equation (6.13), yields

(8.16) $images$

In Equation (8.16), Σ_LC is a single element covariance matrix; that is, the variance of the linear closure and can be called $images$ . Also, $images$ is the matrix given by Equation (8.12). To compute the 95% confidence interval, a t value from Table D.3 (the t distribution) must be used with $images$ and 3 degrees of freedom.¹ The estimated misclosure for a traverse at a (1 − α) level of confidence is t_α/2,3σ_LC. Again we are checking to see if the traverse misclosure falls within a range of errors that are statistically equal to zero. This requires placing α/2 into the upper and lower tails of the distribution. Thus, the estimated error in the traverse closure at a 95% level of confidence is

$images$

This value is well above the actual traverse linear misclosure of 0.086 ft, and thus there is no reason to believe that the traverse contains any blunders.

In Example 8.2, we failed to reject the null hypothesis; that is, there was no statistical reason to believe there were errors in the data. However, it is important to remember that this does necessarily imply that the observations are free from error. There is always the possibility of a Type II error. For example, if the computations were supposed to be performed on a map projection grid,² but the observations were not reduced, the traverse would still close within acceptable tolerances. However, the computed results would be incorrect since all the distances would be either too long or too short. Another example of an undetectable systematic error is an incorrectly entered EDM-reflector constant (see Problem 2.23). Again, all the observed distances would be either too long or too short, but the traverse misclosure would still be within acceptable tolerances.

Surveyors must always be aware of systematic instrumental errors and follow proper field and office procedures to remove these errors. As discussed, simply passing a statistical test does not imply directly that the observations are error or mistake free. However, when the test rejects the null hypothesis, only a Type I error can occur at an α level of confidence. Dependent on the value of α, rejection of the null hypothesis is a strong indicator of problems within the data.

8.5 COMPUTING AND ANALYZING LINK TRAVERSE MISCLOSURE ERRORS

As illustrated in Figure 8.3, a link traverse begins at one station and ends on a different one. Normally, they are used to establish the positions of intermediate stations, as in A through D of the figure. The coordinates at the endpoints, Stations 1 and 2 of the figure, are known. Angular and linear misclosures are also computed for these types of traverse, and the resulting values used as the basis for accepting or rejecting the observations. In a link traverse, the error in the initial azimuth and angle are important in checking both the angular and linear misclosure of the traverse. Furthermore, the error in the closing azimuth is considered when checking the angular misclosure. It does not affect the linear misclosure of the traverse. Example 8.3 illustrates the computational methods.

Geometrical depiction of Closed-link traverse. — **FIGURE 8.3** Closed-link traverse.

EXAMPLE 8.3

Compute the angular and linear misclosures for the traverse illustrated in Figure 8.3. The observed data for the traverse are given in Table 8.4. Determine the estimated misclosures at the 95% confidence level, and comment on whether or not the observations contain any blunders.

SOLUTION

Angular Misclosure: In a link traverse, angular misclosure is found by computing initial azimuths for each course, and then subtracting the final computed azimuth from its given counterpart. The initial azimuths and their estimated errors are computed using Equations (8.5) and (8.6) and are shown in Table 8.5.

The difference between the azimuth computed for course D2 (84°19′22″), and its actual value (264°19′13″ − 180°) is +9″. Using Equation (6.18), the estimated error in the difference is $images$ , and thus there is no reason to assume that the angles contain blunders.

TABLE 8.4 Data for Link Traverse in Example 8.3

Distance Observations				Control Stations
From	To	Distance (ft)	S (ft)	Station	X (ft)	Y (ft)
1	A	1069.16	±0.021	1	1248.00	3979.00
A	B	933.26	±0.020	2	4873.00	3677.00
B	C	819.98	±0.020
C	D	1223.33	±0.021
D	2	1273.22	±0.021

Angle Observations					Azimuth Observations
BS	Occ	FS	Angle	S (″)	From	To	Azimuth	S (″)
1	A	B	66°16′35″	±4.9	1	A	197°04′47″	±4.3
A	B	C	205°16′46″	±5.5	D	2	264°19′13″	±4.1
B	C	D	123°40′19″	±5.1
C	D	2	212°00′55″	±4.6

TABLE 8.5 Computed Azimuths and Their Uncertainties

Course	Azimuth	σ (″)
1A	197°04′47″	± 4.3
AB	83°21′22″	± 6.5
BC	108°38′08″	± 8.5
CD	52°18′27″	± 9.9
D2	84°19′22″	±11.0

Linear Misclosure: First, the actual traverse misclosure is computed using Equation (8.1). From Table 8.6, the total change in latitude for the traverse is −302.128 ft and the total change in departure is 3624.968 ft. From the control coordinates, the cumulative change in X and Y coordinate values is

$images$

TABLE 8.6 Computed Latitudes and Departures

Course	Latitude (ft)	Departure (ft)
1A	−1022.007	−314.014
AB	107.976	926.993
BC	−262.022	776.989
CD	747.973	968.025
D2	125.952	1266.975
	∑ = −302.128	∑ = 3624.968

The actual misclosures in departure and latitude are computed as

(8.17) $images$

In Equation (8.17), ΔDep represents the misclosure in departure and ΔLat represents the misclosure in latitude. The linear misclosure for the traverse is

(8.18) $images$

Estimated Misclosure for the Traverse: Following procedures similar to those described earlier for polygon traverses, the expected misclosure in this link traverse can be estimated. The Jacobian matrix of the partial derivative for the latitude and departure with respect to distance and angle observations is

(8.19) $images$

Similarly, the corresponding covariance matrix Σ in Equation (6.16) has the form

(8.20) $images$

Substituting the appropriate numerical values into Equations (8.19) and (8.20) and applying Equation (6.16), the covariance matrix is

$images$

To estimate the error in the traverse misclosure, Equation (6.16) must be applied to Equation (8.18). As was the case for closed polygon traverse, the terms of the Jacobian matrix are independent of the course for which they are determined, and thus, the Jacobian matrix has the form

(8.21) $images$

Following procedures similar to those used in Example 8.2, the estimated error in the misclosure of the link traverse is

$images$

From these results and using a t value from Table D.3 for 3 degrees of freedom, the estimated linear misclosure error for a 95% level of confidence is

$images$

Because the actual misclosure of 0.13 ft is within the range of values that are statistically equal to zero at the 95% level (±0.29 ft), there is no reason to believe that the traverse observations contain any blunders. Again, this test does not remove the possibility of a Type II error occurring.

This example leads to an interesting discussion. When using traditional methods of adjusting link traverse data, such as the compass rule, the control is assumed to be perfect. However, since control coordinates are themselves derived from observations, they contain errors that are not accounted for in these computations. This fact is apparent in Equation (8.21) where the coordinate values are assumed to have no error and thus are not represented. These equations can easily be modified to consider the control errors, but this is left as an exercise for the student.

One of the principal advantages of the least squares adjustment method is that it allows application of varying weights to the observations, and control can be included in the adjustment with appropriate weights. A full discussion of this subject is presented in Section 21.7.

8.6 SOFTWARE

The computations in this chapter can be challenging or tedious at best to perform using a calculator. Thus, it is recommended that these computations be programmed in the software. On the companion website for this book, an Excel® spreadsheet (Chapter 8.xls) for Examples 8.2 and 8.3 is available for study. Programming allows for an easier and more reliable method of developing the required matrices. Once the initial matrices are computed, they can be manipulated in software such as MATRIX, which is on the companion website, or in the spreadsheet itself.

Illustration of Traverse computations option dialog box. — **FIGURE 8.4** Traverse computations option dialog box.

A spreadsheet often provides routines to manipulate the matrices. In Excel®, matrices can be multiplied using the MMULT() function. Similarly, matrices can be transposed using the TRANSPOSE() function. These functions are demonstrated in the spreadsheet CHAPTER 8.xls. To simplify the selection of the matrices, the block of cells for each matrix can be “named.” The help file that accompanies Excel® discusses how to name cells or groups of cells and use the matrix functions.

For those who wish to develop a more robust program, a higher-level programming language or Mathcad can be used. ADJUST demonstrates this in its traverse computations option. The dialog box for this option is shown in Figure 8.4, where a polygon traverse with angles is selected. At the bottom of the box is the option to compute the estimated errors in the traverse. Selecting this option informs the software to not adjust the angles in the traverse, but rather, to use the entered distance and angle uncertainties to compute the misclosures in both the angles and the traverse. Figure 8.5 depicts the file used in computing the estimated errors for Example 8.2. Notice that with a polygon traverse there is no error given for the initial azimuth of 0°00′00″, and that the estimated errors in the distances and angles follow the observations for each course of the traverse. The help file that accompanies ADJUST describes the format of this file.

Illustration of ADJUST data file. — **FIGURE 8.5** ADJUST data file for Example 8.2.

8.7 CONCLUSIONS

This chapter discussed the propagation of observational errors through traverse computations. Error propagation is a powerful tool for the surveyor, enabling an answer to be obtained for the question, “What is an acceptable traverse misclosure?” This is an example of surveying engineering. Surveyors are constantly designing measurement systems and checking their results against personal or legal standards. The subjects of error propagation and detection of observational blunders are discussed further in later chapters.

PROBLEMS

Note: Partial answers to problems marked with an asterisk can be found in Appendix H.

8.1 Which of the following are direct (independent) or indirect (dependent) observations?
1. *(a) Angles
2. (b) Latitudes and departures
3. (c) Coordinates
8.2 Discuss why latitudes and departures are considered correlated observations.
*8.3 Given a course with an azimuth of 78°16′08″ with an estimated error of ±5″ and a distance of 485.32 ft with an estimated error of ±0.018 ft, what are:
1. (a) the latitude and departure?
2. (b) the estimated errors in the latitude and departure?
8.4 Same as Problem 8.3, except the azimuth is 328°54′16″ ± 11.4″ and the distance is 402.153 ± 0.005 m.
8.5 Same as Problem 8.3, except the azimuth is 40°03′57″ ± 3.3″ and the distance is 1254.98 ± 0.013 ft.
8.6 Same as Problem 8.3, except the azimuth is 88°33′44″ ± 15.4″ and the distance is 202.408 m ± 3.7 mm.
8.7 Same as Problem 8.3, except the azimuth is 44°06′12″ ± 9.8″ and the distance is 156.022 m ± 3.2 mm.
*8.8 A polygon traverse has the following angle measurements and related standard deviations. Each angle was observed twice (1D and 1R). Do the angles meet acceptable closure limits at a 95% level of confidence?

Backsight Occupied Foresight Angle S (″)

A B C 98°06′59″ ±2.8

B C D 85°56′57″ ±2.6

C D A 92°33′50″ ±3.0

D A B 83°22′19″ ±3.2
8.9 Same as Problem 8.8, except the angles were measured four times (2D and 2R) with the following observations.

Backsight Occupied Foresight Angle S (″)

A B C 82°42′10″ ±5.6

B C D 68°43′20″ ±5.3

C D A 93°03′30″ ±4.4

D A B 115°31′15″ ±4.8
*8.10 Given an initial azimuth for course AB of 345°16′29″ with an estimated error of ±4.5″, what are the azimuths and their estimated standard errors for the remaining three courses of Problem 8.8?
8.11 Same as Problem 8.10, except for Problem 8.9.
8.12 Using the distances listed in the following table and the data from Problems 8.8 and 8.10, compute:
1. (a) the misclosure of the traverse.
2. *(b) the estimated misclosure error.
3. (c) the 95% error in the traverse misclosure.
From To Distance (ft) S (ft)

A B 247.86 ±0.012

B C 302.49 ±0.012

C D 254.32 ±0.012

D A 319.60 ±0.012
8.13 Given the traverse misclosures in Problem 8.12, does the traverse meet acceptable closure limits at a 95% level of confidence? Justify your answer statistically.
8.14 Using the distances listed in the following table and the data from Problems 8.9 and 8.11, compute:
1. (a) the misclosure of the traverse.
2. *(b) the estimated misclosure error.
3. (c) the 95% error in the traverse misclosure.
4. (d) Given the traverse misclosures, does the traverse meet acceptable closure limits at a 95% level of confidence? Justify your answer statistically.
From To Distance (ft) S (ft)

A B 256.69 ±0.012

B C 303.45 ±0.012

C D 326.99 ±0.012

D A 160.17 ±0.012

Backsight	Occupied	Foresight	Angle	S (″)
A	B	C	98°06′59″	±2.8
B	C	D	85°56′57″	±2.6
C	D	A	92°33′50″	±3.0
D	A	B	83°22′19″	±3.2

Backsight	Occupied	Foresight	Angle	S (″)
A	B	C	82°42′10″	±5.6
B	C	D	68°43′20″	±5.3
C	D	A	93°03′30″	±4.4
D	A	B	115°31′15″	±4.8

From	To	Distance (ft)	S (ft)
A	B	247.86	±0.012
B	C	302.49	±0.012
C	D	254.32	±0.012
D	A	319.60	±0.012

From	To	Distance (ft)	S (ft)
A	B	256.69	±0.012
B	C	303.45	±0.012
C	D	326.99	±0.012
D	A	160.17	±0.012

8.15 Using the data for the link traverse listed below, compute:

(a) the angular misclosure and its estimated error.
(b) the misclosure of the traverse.
(c) the estimated misclosure error.
(d) the 95% error in the traverse misclosure.

Distance Observations				Angle Observations
From	To	Distance (m)	S (m)	BS	Occ	FS	Angle	S (″)
W	X	180.395	±0.005	Mk1	W	X	157°39′42″	±3.6
X	Y	146.622	±0.005	W	X	Y	96°27′11″	±3.8
Y	Z	251.526	±0.006	X	Y	Z	172°52′34″	±3.7
				Y	Z	Mk2	185°53′49″	±3.1

Control Azimuths				Control Stations
From	To	Azimuth	S (″)	Station	Northing (m)	Easting (m)
Mk1	W	345°57′50″	±9″	W	5,000.000	10,000.000
Z	Mk2	58°51′09″	±8″	Z	5079.412	10,434.819

8.16 Does the link traverse of Problem 8.15 have acceptable traverse misclosure at a 95% level of confidence? Justify your answer statistically.

8.17 Same as Problem 8.15, using the following data.

Distance Observations				Angle Observations
From	To	Distance (ft)	S (ft)	BS	Occ	FS	Angle	S (″)
W	X	223.59	±0.012	Mk1	W	X	171°52′06″	±5.5
X	Y	854.40	±0.013	W	X	Y	137°07′03″	±5.9
Y	Z	460.95	±0.012	X	Y	Z	290°02′35″	±3.4
				Y	Z	Mk2	77°44′55″	±4.8

Control Azimuths				Control Stations
From	To	Azimuth	S (″)	Station		Northing (ft)	Easting (ft)
Mk1	W	251°33′54″	±5.0″	W	1000.00			1000.00
Z	Mk2	28°21′00″	±3.0″	Z	1850.00			1600.00

8.18 Does the link traverse of Problem 8.17 have acceptable traverse misclosure at a 95% level of confidence? Justify your answer statistically.
8.19 Repeat Problem 8.15 with data and results from Problems 7.13 and 7.20. Assume the azimuth of AB is 24°02′16″ ± 6″ and the xy coordinates at station A are (10,000.00, 5,000.00) in units of feet.

8.20 A survey produces the following set of data. The angles were obtained from the average of four measurements (two face left and two face right) made with a total station. The estimated uncertainties in the observations are:

$images$

The EDM instrument has a specified accuracy of ± (3 mm + 3 ppm).

Distance Observations			Angle Observations
From	To	Distance (ft)	BS	Occ	FS	Angle
1	2	999.99	5	1	2	191°40′12″
2	3	801.55	1	2	3	56°42′22″
3	4	1680.03	2	3	4	122°57′10″
4	5	1264.92	3	4	5	125°02′11″
5	1	1878.82	4	5	1	43°38′10″

Control Azimuths				Control Stations
From	To	Azimuth	σ (″)	Station	Easting (ft)	Northing (ft)
1	2	216°52′11″	±3	1	1000.00	1000.00

Compute:

(a) the estimated errors in angles and distances.
(b) the angular misclosure and its 95% probable error.
(c) the misclosure of the traverse.
*(d) the estimated misclosure error and its 95% value.
(e) Did the traverse meet acceptable closures? Justify your response statistically.

8.21 Develop new matrices for the link traverse of Example 8.3 that considers the errors in the control.
8.22 Show that Equation (8.6) is valid for azimuth computations going clockwise around a traverse.

PROGRAMMING PROBLEMS

8.23 Develop a computational package that will compute the course azimuths and their estimated errors, given an initial azimuth and measured angles. Use this package to answer Problem 8.9.
8.24 Develop a computational package will compute estimated traverse misclosure error, given course azimuths, distances, and their estimated errors. Use this package to answer Problem 8.10.

NOTES

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Course	Latitude	Departure
AB	1435.670	0.000
BC	−35.827	−856.191
CD	−939.812	−619.567
DE	−723.829	766.894
EA	263.715	708.886
	Σ = −0.083	Σ = 0.022
$images$

From	To	Azimuth	Estimated Error
A	B	0°00′00″	0″
B	C	267°36′14″	±3.1″
C	D	213°23′41″	$images$
D	E	133°20′43″	$images$
E	A	69°35′39″	$images$

Table of Contents for CHAPTER 8: ERROR PROPAGATION IN TRAVERSE SURVEYS

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8.1 INTRODUCTION

8.2 DERIVATION OF ESTIMATED ERROR IN LATITUDE AND DEPARTURE

8.3 DERIVATION OF ESTIMATED STANDARD ERRORS IN COURSE AZIMUTHS

8.4 COMPUTING AND ANALYZING POLYGON TRAVERSE MISCLOSURE ERRORS

8.5 COMPUTING AND ANALYZING LINK TRAVERSE MISCLOSURE ERRORS

8.6 SOFTWARE

8.7 CONCLUSIONS

PROBLEMS

PROGRAMMING PROBLEMS

NOTES

Table of Contents for
CHAPTER 8: ERROR PROPAGATION IN TRAVERSE SURVEYS