1.20. Trigonometrical Levelling

It is an indirect method of levelling in which the elevation of the point is determined from the observed vertical angles and the measured distances. It is commonly used in topographical work to find out the elevations of the top of buildings, chimneys, churches etc., from a distance. This is a faster method to get the elevations of top of structures and objects. Elevation of a BM in the area must be known.

Trigonometrical levelling is a technique used in surveying that relies on indirect measurements. Instead of measuring the height of an object directly, we determine its elevation based on vertical angles observed from a distance and the horizontal distance to the object. This method is particularly effective for assessing tall structures, such as buildings or churches, as it saves time and effort compared to traditional measuring techniques. However, to use trigonometrical levelling, we must first know the elevation of a benchmark (BM), which serves as a reference point.

Imagine trying to find the height of a tall building without a ladder. Instead of climbing up, you stand at a distance and use a theodolite to measure the angle to the top of the building. By knowing how far away you are and using some math, you can calculate the height of the building without ever leaving the ground.

Let PP’ is a tower whose elevation of the top is to be determined. Set up the theodolite at a convenient ground point A so that the top of tower and a staff kept on the BM can be bisected. Measure vertical angle α of the top of tower as well as take the staff reading at the BM. Measure D, the horizontal distance between theodolite station and tower.

To find the height of a structure, we first set up our theodolite at a point where we can see both the top of the tower and a staff at the benchmark. We then measure the vertical angle from our position up to the top of the tower (denoted as α) and note the reading from the staff at the benchmark, which gives us the lower reference. Lastly, we measure the horizontal distance (D) from our position to the base of the tower. These measurements allow us to calculate the height of the tower using trigonometric functions.

Think about using a smartphone app to measure the height of a tall tree. You stand a certain distance away from the tree and point your phone at the top of it. The app calculates the angle to the top and, knowing your distance from the tree, it automatically tells you how tall the tree is based on those numbers.

To find the height of the object above a BM: Let H = height of the object above the BM h = height of the tower above the instrument axis h_s = height of the instrument axis above the BM s α = vertical angle of the top of tower at the instrument station D = horizontal distance from the instrument station to the base of the tower. Then, h = D tan α H = h + h_s = D tan α + h_s.

In the calculations, we consider the height of the object (H) to be the sum of the height above the instrument axis (h) and the height of the instrument above the benchmark (h_s). The height above the instrument axis can be calculated using the horizontal distance (D) and the tangent of the vertical angle (α). Thus, the formula connects the angle measured, distance away, and known height of the instrument to ultimately derive the object's height.

Imagine adjusting the height of your camera to capture the perfect photo of a mountain while standing a specific distance away. Using the angle you look up at and your distance from the mountain, you can calculate just how high it towers above you, much like measuring a tall building's height using specific angles and distances.

To find the height of the tower PP’ above a BM, select two stations A and B suitably on a fairly level ground so that these points lie in a vertical plane with the tower, and measure the distance AB. Set up the theodolite at station A to take a staff reading kept on the BM. Read the vertical angle α. Shift the theodolite at B point and take similar observations as taken at A point. Read the vertical angle β.

When the object cannot be accessed directly, we employ two separate observation stations. From each station (A and B), we measure both the vertical angles (α from A to the object and β from B to the object). The distance between the two stations is also measured. These angles and distances create a geometric relationship allowing us to determine the height of the object indirectly using trigonometric principles.

Suppose you're trying to find the height of a tree growing on the edge of a steep cliff. You can't get close without risking a fall, so you position yourself at two safe spots (A and B) further back. By measuring the angles to the top of the tree from both points and knowing how far apart you are from each other, you can still work out how tall that tree really is!

If the distance D is large, correction for curvature and refraction, i.e., 0.00673(D^2 / 1000), is to be applied.

When using trigonometrical levelling over large distances, it is important to account for errors caused by the Earth’s curvature and atmospheric refraction. These factors can slightly alter the angle measurements and distances. The formula given (0.00673 times D squared divided by 1000) provides a correction factor that ensures our height calculations remain accurate even over long distances.

Think of trying to measure the distance on a large sports field. If you were at one end but couldn’t see the other due to the curvature of the ground, you would want to adjust your measurements to account for that curve, much like how the formula helps tweak readings for tall buildings or towers over long distances.

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