1.10. Example 1.28: R.L. Calculation of the Staff Station

An instrument was set up at P and the angle of depression to a vane 2 m above the foot of the staff held at Q was 5° 36′. The horizontal distance between P and Q was known to be 3000 metres.

In this first part of the problem, we are given a scenario where a survey instrument is placed at point P, and it is used to measure an angle of depression to a point (vane) located at Q. The vane is positioned 2 meters above the foot of the staff that is held vertically. The angle of depression is the angle between the horizontal line from the instrument and the line of sight to the vane. Knowing the horizontal distance between the two points (P and Q) allows us to relate this angle and distance to the height difference between the instrument and the vane, which is critical for calculating the reduced level (R.L.) of the staff station at Q.

Imagine you are standing on a hill (point P) and looking down at a friend (the vane) standing at the bottom of the hill (point Q) holding a stick. The angle at which you are looking down is similar to the angle of depression. The distance you are from your friend helps you understand how high you are standing relative to them.

Determine the R.L. of the staff station Q given that staff reading on a B.M. of elevation 436.050 was 2.865 metres. The difference in elevation between the vane and the instrument axis = D tan α = 3000 tan 5° 36′ = 294.153 m.

To find the difference in elevation between the vane and the instrument, we apply the formula: difference = horizontal distance x tan(angle of depression). In this case, we calculate that the elevation difference amounts to 294.153 meters. This information is crucial as it informs us how much lower the staff is positioned compared to the height of the instrument itself. This calculation is foundational as it exemplifies how trigonometric functions can help resolve vertical height issues in surveying.

Consider a flat piece of ground where you have a tall tree (the instrument). If you are 3000 meters away from the tree and observe that you need to look down at an angle of 5° 36’ to see the top of an object below, this angle helps you calculate how much lower the object is than your eye level. Just like with measuring the tree's height, the angle and distance help you determine how high off the ground the top of the object must be.

Combined correction due to curvature and refraction C = 0.06735 D² metres when D is in km = 0.606 m. Since the observed angle is negative, the combined correction due to curvature and refraction is subtractive.

The correction for curvature and refraction accounts for the fact that the Earth is not flat, and light bends when passing through different atmospheric layers. Here, we calculate C using the formula where D is the distance in kilometers. Once we find C to be approximately 0.606 m, we apply this correction to ensure our final elevation account reflects the true height of the point being measured. It's essential to subtract this value, as it adjusts our previous positive difference in elevation.

Imagine looking at a distant mountain through a thick layer of air filled with heat from the ground; just as the air can distort your view, the Earth’s curve can affect how we observe heights over long distances. The correction is akin to taking off sunglasses that distort your view, so you can see the actual height of the mountain more clearly.

RL of instrument axis = 436.050 + 2.865 = 438.915, RL of the vane = RL of instrument axis – h = 438.915 – 293.547 = 145.368, RL of Q = 145.368 – 2 = 143.368 m.

In this final part, we calculate the Reduced Level (R.L.) of the instrument axis by adding the B.M. elevation to the staff reading. The R.L. of the vane is then calculated by subtracting the height difference (h) found earlier from the instrument’s R.L. Finally, we determine the R.L. for the staff station Q by subtracting the height of the vane from the R.L. of the vane. This step-by-step approach reveals the central process of deriving R.L.s in surveying.

Think of it like figuring out the total height of a multi-story building. First, you find how high you are above the ground (the instrument's R.L.), then you measure how far you need to go down to reach a balcony (height of the vane) before finally determining how high the ground level is compared to your starting point. This method helps you 'connect the dots' to see the full elevation picture.

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