Adjusting a traverse–tools for understanding the technique
Sprinsky, William H
Students must be able to interpret observational errors when they enter professional practice. Common error treatment methods include Compass Rule, Transit Rule, and Crandall’s Rule, all of which can be understood with basic algebra skills possessed by first-year students, and the more complex Least Squares (LS) method.1,2,3 The LS (Gauss-Markov stationary) solution is well proven to give adjusted coordinates closest to the Maximum Likelihood solution to problems faced by surveyors and civil engineers alike. However, a common problem faced by students in both bachelor and associate degree Civil Engineering Technology programs is understanding how a Least Squares (LS) solution works. Firstyear students simply may not have the mathematical tools needed for this technique.
The technical programs at the Pennsylvania College of Technology (Penn College) include a two-year Surveying Technology degree and two-year and four-year Civil Engineering Technology degrees with an emphasis in surveying. The Surveying Technology and Civil Engineering Technology associate degree programs are accredited by the Accreditation Board for Engineering and Technology (ABET), and the four-year curriculum is being considered for ABET accreditation. A recent National Science Foundation (NSF) Instrumentation and Laboratory Initiatives grant allowed Penn College to change the thrust of its technical programs by purchasing equipment, computers, and software that support modern approaches in civil engineering and surveying. In technology programs that emphasize surveying, faculty try to develop student understanding of data adjustment processes, rather than merely developing repetitive skills in applying software to provide solutions. It is generally believed that fostering understanding of concepts behind very basic tools extends the value of these degrees.
A challenge faced by course designers is how to give each student not only theory but also actual experience with projects and equipment that are common and essential in modern civil engineering practice. One way to meet this challenge is to use modern equipment and techniques to solve problems that illustrate basic concepts. The exercise described here is designed to allow students to experiment with data under controlled conditions. It is healthy for students to be skeptical of procedures that promise to solve problems at hand, and they need opportunities to experiment with available options to understand why one is better than another. Discussed is Tuttle’s method, an algebraically derived technique that allows first-year students to see and understand the effects of weighting observations based upon their accuracy.
Teaching The Tools of Adjusting Traverses
Penn College used the NSF grant to purchase Intergraph’s SelectCad and Modular GIS Environment (MGE) for a student mapping project.4 Students were required to construct a map, a by-product of a Digital Terrain Model (DTM) coordinated in Pennsylvania State Plane (NAD83) North Zone 3701 form, as the unifying project of their first two– semester experience in surveying.
During the 1999 academic year, students were to observe a traverse that included an area of approximately 3.5 acres with 12 stations and a position closure after angle balancing of better than 1/34,000. Students were taught the use of Compass Rule, Transit Rule, and Crandall’s Rule through class discussions and homework that could be solved with calculators. Once they had mastered the concepts underlying these three methods, they were introduced to Intergraph’s InRoads Survey adjustment software, part of the InRoads SelectCAD package.
InRoads SelectCAD offers comprehensive transportation design and adapts to any workflow, regardless of the CAD platform. This specific product was chosen for instruction because
It will accept data from a variety of sources including stereoplotters, aerial photos, ASCII files, total station data collectors, and field books
It links total station data collectors to workstations using InRoads Survey, the data reduction application that allows transfer of survey data directly into InRoads SelectCAD for plans, designs, and reports
Students can enter information as surveyed using a variety of coordinate systems
It facilitates conversion and merging of information from different sources using the coordinate transformation tools discussed in class
Special emphasis is given to the effects of various user choices in the survey project. The loop traverse is first adjusted using routines included in the SelectCAD package, including the approximate methods of Compass Rule, Transit Rule, and Crandall’s Rule as well as the more accurate LS method. As demonstrated in a previous study, Compass Rule weighs correcting distance too heavily, regardless of the angle accuracy, when the angles and distances are observed with modern instruments.5 Transit Rule apportions errors based in part on the orientation of the traverse, but students make no azimuth observations in the first semester. Crandall’s Rule, the LS approach of its day, was developed when angles were observed much more accurately than distances.6 It apportions all the error to distance, correcting the angles only so their geometric sum is correct, based upon the number of sides of the closed traverse. In a comment attached to the original Crandall article, George W. Tuttle proposed an apportioning method based upon the error estimate for each observation.7 It is from that suggestion that the author developed Tuttle’s method.
The Tuttle method is not proposed to take the place of LS solutions in practice. On the contrary, LS remains the gold standard to which all solutions should be compared. The Tuttle method is used for two reasons. First, it is algebraic in nature, as are Compass, Transit and Crandall’s Rules, and can be easily understood by first-year students. Secondly, the adjusted observations, which are derived from the resulting coordinates, can be studied and compared to other methods so that the mechanics behind the adjustments can be better understood.
For illustration, Section A describes a sample traverse, lists the observations of its interior angles and distances, and compares traverse adjustments from the Compass, LS, and Tuttle methods.
Understanding Traverse Adjustment
To understand the Tuttle approach, one should recognize that coordinates from any of these adjustments tell the viewer very little. One must look at the changes made to the original observations to judge the nature and size of the changes. The Compass, Crandall’s, and Tuttle methods adjust the latitude Li and departure Di of each observed course in the traverse with calculations based on changes in these observed quantities. Specifically,
Note that where i – 1 = 0, the observation is made at the initial station and i – 1 is either the last station in a closed traverse that starts and ends at different known stations (a nonloop traverse) with n observed courses, or the last new station in a traverse that starts and ends on the same station (a loop traverse) of-n observed courses.
The emphasis on examining adjusted observations provides an additional consideration. One of the problems with traverse adjustment is the case of a fixed initial station and azimuth, such as a point of beginning for a plat and a fixed centerline azimuth for an adjoining road. There is a tendency in this type of case to use Crandall’s Rule because the fixed azimuth is not changed, where in Compass and Transit Rules, it is. That is exactly the wrong choice in modern surveying. Teaching students to use adjusted observations reinforces the option of recomputing the Northing and Easting coordinates of stations with the adjusted values and the initial station’s coordinates and azimuth, retaining the fixed azimuth. This adjustment can also be done to the coordinates of new stations by mathematically applying a rotation so the initial azimuth is maintained as fixed.
Description of Tuttle’s Method
A summary of the technique suggested by Tuttle is presented. Copies of programs written to implement this technique are available by request from the author.5 Since all methods except the LS approach adjust latitudes and departures, observed interior angles and distances must be converted to azimuths (or bearings) and distances.
The first step in converting angles to azimuths involves the Propagation of Error equation for independent observations. Solving this equation may be a stretch for students with no knowledge of partial differentiation, but class discussions and examples that illustrate the numerical technique can mitigate any difficulties. Since the model converting interior angles to azimuths is linear with respect to the interior angles, rates of change of azimuth with respect to interior angle are easily understood. The solution, in the form of differential calculus, is presented to the class, which already understands generally how it is derived.
Starting from a survey’s initial azimuth Al, any subsequent forward azimuth is a function of that azimuth and the interior angles I (either observed or computed from directions) of the traverse. Once the angles have been balanced, this azimuth is of the form
Since p^sup 2^ is a part of the solution, the ratio of p influences the apportionment of the corrections. When p is unity, the result is the original Compass Rule with the errors in distance varying as the distance itself. When p is zero, the result is Crandall’s method with the same distance weighting.
In this particular application, the ratio of p^sup 2^ should be examined. It is assumed that the traverse will be observed as a “transit” traverse, either by angle or direction, starting with an initial azimuth from a known station. This initial azimuth can be treated as a known value by assigning a zero or a very small standard error to it. If actually observed, the standard error of the observation can be used. This error will affect the value of p for each course of the traverse.
Although it does not have all the attributes of a LS solution, the proposed Tuttle algorithm allows the modern surveyor to influence the results of a transit traverse adjustment through the accuracy of the instrumentation. The numerical application of this technique will be shown later, but it is informative to examine this technique’s capabilities before proceeding.
The Tuttle procedure closely approximates the adjusted observations from a LS technique without the computational complexity normally involved in the latter technique. Tuttle observations are adjusted to meet the same conditions in the LS approach, namely, the sum of the interior angles of a closed traverse is geometrically correct and the sums of course latitudes and departures are equal to the difference between the end and initial coordinates. Thus, the procedure does accomplish the task of imposing mathematical consistency.
While the procedure works well for closed traverses, those traverses that include multiple high quality azimuths are not accommodated. Where the angles and directions are not observed with equal quality, the Tuttle procedure less closely approximates the LS solution. Also, where a network of closed traverses with common stations is observed, this procedure, like Compass and Crandall’s, will not properly accomplish the adjustment. Despite these exceptions, the majority of traverse work done today remains well within the capabilities of the Tuttle procedure, which produces results comparable to those from more computationally complex techniques.
Application and Comparison of Results
Careful comparison of adjusted and original observations is essential to understanding how a particular adjustment scheme works. In the case of the Penn College traverse, the observations are angles and horizontal distances, as shown in figure 2. The initial starting station and azimuth are derived from GPS observations, a subject the students have not yet studied, but the values could also have been been in NAD27 coordinates or arbitrary settings to illustrate the concept. Table I lists interior angle and distance observations from this student exercise.
After angle balancing, the position closure of this traverse is about 1/34,000. Table 2 lists the corresponding coordinates and adjusted observations generated by the Compass Rule. These values will not change, regardless of whether the observations were made by handheld compass and pacing or by electronic angle and distance measuring devices, which can be a significant drawback to this and other approximate methods. From these coordinates, the adjusted angles and distances may be computed as shown in table 2 and compared to those in table 1. While none of the residuals to the observed quantities exceeds three times the standard error of the observations themselves, the Compass Rule solution may not be the best choice.
Table 3 contains these coordinates and adjusted observations from an Intergraph LS routine in which the angle accuracy (standard error) is 30″ and the distance accuracy is approximately 0.01 sf. These values are computed from equation (6), with a = 3 mm and b = 3 ppm, which is representative of modern distance measuring instrumentation. Comparing the observations in tables 2 and 3 reveals a marked difference in the amount of correction made to distances. Compass Rule overcorrects the distances because of the way it was originally derived by Bodwitch. This method may have been correct in his time, but it is certainly not correct in modern practice.
While the LS solution appears to give the most accurate adjusted observations, it can be difficult to demonstrate how this method works to students who are mastering college algebra and trigonometry. Table 4 shows the Tuttle solution of the same data, including observational accuracies. These Tuttle adjusted observations are remarkably similar to the LS adjusted observations in table 3. By comparing table 4 to the Compass Rule results in table 2, students can begin to see why the LS method is preferable with modern instruments, since observed distances are corrected much less radically.
To emphasize that the quality of the observations should influence apportionment of error, table 5 lists the same observations shown in table 1, except that the accuracies are now more typical of a vernier transit, steel tape survey. Again, the Compass Rule solution shown in table 2 and other approximate methods do not change.
Table 6 shows the Intergraph LS results of the data shown in table 2. Comparison of table 5 to tables 6 and 3 (the previous LS values) clearly reflects the difference in instrumentation, particularly when comparing a tape to an electronic distance measurer. Table 7 shows the Tuttle solution for table 2 data, again remarkably similar to table 6. These results indicate that the accuracies of observation do matter and they should influence the solution.
The traverse balancing exercise described in this paper is a second-semester project in both the associate and bachelor degree programs at Penn College. Because coordinates are the basis for many aspects of any civil engineering project, from layout to geographic information system, it is important for students and practitioners alike to understand how to interpret and analyze data in this form. Trying to judge acceptability from coordinates alone is, in the opinion of the author, not practicable if not possible. The procedures discussed here are an attempt to give students an insight not only into the process but also what lies behind the process so they can make an informed choice about the various solution methods available.
Copyright American Society for Engineering Education Spring 2001
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