Triangulation Survey part 4

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Distance AC i.e. the curved length AC = S + d (McCaw’s assumption)Distance CB i.e. the curved length CB = S - d (McCaw’s assumption) Let ξ (Zii; eq to X of English) be the Zenith Distance i.e. 90o – vertical angle under consideration).McCaw’s Formula states that: Height HC of LOS at the point C is:HC = ½ (HB + HA) + ½ (HB - HA).(d) – (S2 – d2).(1 – 2m) Cosec2ξ ….. (i)                                                      S                        2RAnd Value of Zenith Distance is given by the following relation:Cosec2ξ = 1 + (HB + HA)2 In most of the practical surveys, this distance                              4S2        (Zenith Distance) is assumed to be very close to unity  Cosec2ξ = 1, making (HB + HA)2 /4S2 = 0. Putting the values of m and R in (1 – 2m)/2R, this term becomes = 0.067 in the equation (i).Therefore, the height of obstructive point C, HC comes to be:HC = ½ (HB + HA) + ½ (HB - HA).(d/S) – (S2 – d2) x 0.067 x 1 ……. (ii)Above-given equation (ii) is a deciding equation to calculate the height of obstructive point C, HC, whether the LOS is clear of this obstructive point C or the (target at) point B is required to be raised.McCaw’s Method:  If H > HC, then the LOS will be clear of obstruction and if H < HC then the height i.e. elevation of the target (at point B) will be required to be raised. H is the height i.e. vertical distance of apparent (flat and horizontal) horizon (known as “Plane of Horizon”) from Earth’s curvature.
Accuracy of Triangulation:Errors are inevitable and, therefore, in-spite of all the possible precautions the errors happen to occur and get accumulated. It is, therefore, essential to know the accuracy of the triangulation network, so that no appreciable error should be introduced while plotting a network on the paper and finally recording the values in the Record Book. The following formula of Root Mean Square of Errors is used:   M = (ΣE2)/3n]Where M is the Root Mean Square Error of unadjusted horizontal angles (in seconds) of the arcs as obtained from the triangular errors.ΣE2 is the sum of the squares of all the triangular errors in the triangulation series, and n is the total number of triangles in a triangulation network.    Example: “Root Mean Square” of 2, 3 & 4: [(22 + 32 + 42)/3] = 3.109

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