Spatial positioning of sidewall stations in a narrow tunnel environment: a safe alternative to traditional mine survey practice

3.5. The geometry of observations

In the underground environment it is not always possible to install survey stations at the optimal position for ideal observation geometry. The text references discussing the geometry of survey stations to be observed for an intersection differ substantially in the approach to the ideal geometry. The placement of survey stations will depend on local conditions including rock structure visibility and the presence of mine services such as ventilation pipes, water and power cables.
In an article by Shewmon, a number of references are made to the geometry of the ideal survey geometry “Relation between limit of error and distance between known stations for three different triangles, when the transit error is 1 minute and the tape error is 0.02 ft. Note that the 2½º, 2½º, 175º triangle offers greater accuracy than the 45-45-90º triangle…” [106] He concluded from his observations that “…the bearing error is much less for the 2½º, 2½º, 175º triangle than for the 60º -60º -60º triangle.” . Using a graph, Shewmon illustrated that for angular accuracy a triangle with two small angles and one large angle provides the most accurate position fix. Shewmon argued that this is the reason why Weisbach triangles are considered to be a popular choice for shaft plumbing [106]. He further stated that the common assumption of a well-constructed triangle such as a “60º -60º -60º triangle is stronger than any other shaped triangle… is the least accurate both in limit of error in the triangle and angular error” [106]. His overall conclusion was that “angular accuracy as seen in the graphs can be surprisingly low when using triangles in which the sum or difference between the sides adjacent to the random setup approaches the length of the third side.” [106].
Arthur observed in his paper on the surveying techniques used at Sunrise Dam gold mine in Western Australia that “…the best geometry to use when surveying a new station is within 2-3m of the 2nd reference station and < or > than 90 degree angle from that station.” [43]. Briggs discussed the geometry of triangles used for observations and the effect thereof on errors in surveying where the unknown variable is the angle as “…that shape which ensures that errors in the linear and angular measurements have least influence on the angle to be calculated.” and the ideal shape of a triangle from which a length or area is to be determined as “…that shape which ensures that errors in the linear and angular measurement have a minimum proportional influence on the calculated result.” [114]. This statement seems to contradict the findings made by Shewmon regarding the ideal shape of a triangle, stating that “the ideal (strongest) triangle is one having angles close to 60 degrees (equilateral), although angles as small as 20 degrees may be acceptable.” [114]. Briggs concluded his observations with a statement that “It is evident that the accuracy of the fix depends not only on the magnitude of the angles, .., but also on the angle of intersection of the tangents. ... It is essential that when observing fixed points in a resection survey these points must be chosen in such a way that both necessary conditions are satisfied.” [114]. Briggs investigated the optimal geometry for a triangulation and found that “Providing that the measured angle lies between 165º and 180º, and the known sides are roughly equal in length, the effect of angular error is inappreciable as compared with that of linear errors; hence the angle need not be measured with more care than is taken with any other triangulation angle” [114].

3.5.1. The effect of redundant measurements

Watt argued that “In all adjustments by the method of conditions the number of conditions to be satisfied is equal to the number of excess or redundant observations.” [102]. He stated that in order to satisfy the mathematical model, a minimum number of observations (n_o) need to be taken, so that if an error occurs in one of the measurements the error will not be detected. [102]. Measurements in excess of the minimum number of observations are called a redundant measurement (r). If a total number of “n” measurements are obtained with respect to a certain mathematical model whose minimum number of elements is (n_o) the redundancy is given by the equation:

r = n - n_o ( 9 )

It is important to note that Watt stated that the minimum number of observed rays for a fix using a resection is three and using a triangulation is two but that “…This minimum will not afford the necessary check on the accuracy of the surveys in either case” [102]. In order to ensure that there is an acceptable level of redundancy in any survey it is considered good surveying practice to take more observations than the minimum to ensure an accurate survey. Schofield remarked that “In practical survey networks, it is usual to observe more than the strict minimum of observations required to solve the co-ordinates of the unknown points. The extra observations are ‘redundant’ and can be used to provide an ‘independent check’ but all the observations are incorporated into the solution of the network if the solution is by least squares” [115]. McCormack alluded to adjustment by least squares adjustment when using wall station surveying in Australia “Most computer programs and onboard instrument programs use least squares to solve the resection.” [110].
Least squares adjustments can only be made when a number of redundant observations have been made. Unless the original join distance between the two observed known points is considered as a redundant observation, there have not been a sufficient number of observations made in the Australian wall station survey method to warrant a least squares adjustment. From the paper written by Jaroz and Shepard about this surveying technique [96] it seems that wall stations were only installed on one side of the excavation, and only two pegs were used for the orientation of each survey and the calculation of the position of the new point.

3.5.2. The minimum number of observations

Middelton, Chadwick and Bogle observed that “as a system, however, observations to three points are unsatisfactory and lazy, the method is not susceptible of minute accuracy, and there is no check, unless four points are observed, also large error may creep in from mistakes in record, or in mistaking stations..” [116]. According to the observations made on the work completed at the DESY linear collider, Schwartz stated that “the required accuracy can be achieved if from each station the measurements are carried out to three reference points” [62] In some cases it might not be possible to observe more than two sights, that would meet the ideal of using a minimum of two pegs. Venter and Suttie stated in an article in the Journal of the Institute of Mine Surveyors that “It is intended to introduce a system whereby the total station may be set up anywhere in the road and the position fixed by measuring distances to two pegs and the angle at the set up.” [117] . A system of using two known survey points could arguably impact on the accuracy as no redundant observations are included. It is interesting to note that the standard operating procedure of one mine surveying company makes a clear distinction between the methods of observation and the minimum required observations “If angles and slope distances are present then resection observations to a minimum of two known stations are required. If only angles are present then resection observations to a minimum of three known stations are required.” [118]. Fowler observed that “Advice from experienced tunnel surveyors shows that the typical number of sets undertaken in a set up as being between four and eight (Henneker, pers comm.), with one set being defined as one left face and one right face observation to each target.” [45].

The Land Survey Act,1997 (Act No 8. OF 1997) stipulates that in the case of observations taken to determine the position of a survey point the minimum requirements for the determination of the position of such a point are:

“(b) when its position is determined by resection, at least four favourably situated known points shall be used, and sufficient observations shall be made to ensure the required accuracy of determination of its position: Provided that at least one arc shall be observed; and

(c) when its position is determined by a single triangle only, observations shall be made at all three points and on at least two different parts of the circle;” [33].

It could be argued that due to the advances in survey technology and the pressure of completing a survey in as short a time as possible time there is a possibility that the surveyor will neglect the fundamental principles of surveying by not taking observations to sufficient reference stations to ensure the accuracy of the survey.

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