Trigonometrical Levelling and Hydrographic Surveying

Contents

**Define Trigonometric Levelling** 1

**Recall the concept of Terrestrial Refraction** 8

**Describe the Correction of Refraction and Curvature** 8

**Derive an expression for the axis signal correction** 9

** ****Define Trigonometric Levelling**

Trigonometric Levelling is a method of determining the height of an object or the difference in height between two points. This method makes use of the principles of trigonometry to calculate the height of an object or the difference in height between two points.

The basic principle behind Trigonometric Levelling is to measure the angle between the line of sight from a levelling instrument (such as a level or theodolite) and the horizontal plane. This angle, known as the vertical angle, is used to calculate the height of the object being measured.

The process of Trigonometric Levelling involves setting up the levelling instrument at one point, sighting an object or point at a known height (such as the top of a building or a benchmark), and measuring the vertical angle between the line of sight and the horizontal plane. The height of the object or the difference in height between two points can then be calculated using trigonometry.

Trigonometric Levelling is used in many applications, including land surveying, civil engineering, construction, and cartography. It is a precise method of determining height and is particularly useful when a direct line of sight is not possible, such as when measuring the height of a tall building or a mountain.

In summary, Trigonometric Levelling is a method of determining the height of an object or the difference in height between two points using the principles of trigonometry. It is a precise method of height determination that is widely used in many fields, including surveying, engineering, construction, and cartography.

**Derive an expression for determination of Elevation of the given object When the base of the object is Accessible**

The determination of elevation can be calculated using trigonometry when the base of the object is accessible. In this case, the vertical angle, the horizontal distance between the levelling instrument and the object, and the height of the levelling instrument are used to calculate the elevation of the object.

Here is the expression used to determine the elevation of an object when the base of the object is accessible:

Elevation = (H_{1} + h) + d * tan(θ), where

H_{1} = height of the levelling instrument

h = height of the object’s base

d = horizontal distance between the levelling instrument and the object

θ = vertical angle between the line of sight from the levelling instrument and the horizontal plane

The expression uses the tangent of the vertical angle (θ) and the horizontal distance (d) to calculate the height of the object above the base (h). The height of the levelling instrument (H_{1}) and the height of the object’s base (h) are then added to the calculation to determine the elevation of the object.

In summary, the determination of elevation when the base of the object is accessible can be calculated using the expression Elevation = (H_{1} + h) + d * tan(θ), where H1 is the height of the levelling instrument, h is the height of the object’s base, d is the horizontal distance between the levelling instrument and the object, and θ is the vertical angle between the line of sight from the levelling instrument and the horizontal plane.

** ****Recall the determination of R.L of a point when the base of the object is Inaccessible under the following conditions: i. Instrument Axes at the same level ii. Instrument Axes at Different level iii. Instrument Axes at very Different level**

- Instrument Axes at the Same Level: When the instrument axes are at the same level, the R.L of a point can be determined by measuring the vertical angle between the line of sight from the levelling instrument and the horizontal plane, and the horizontal distance between the levelling instrument and the point. The R.L is then calculated using the following formula:

R.L = H_{1} + d * tan(θ), where

H_{1} = height of the levelling instrument

d = horizontal distance between the levelling instrument and the point

θ = vertical angle between the line of sight from the levelling instrument and the horizontal plane

- Instrument Axes at Different Levels: When the instrument axes are at different levels, the R.L of a point can be determined by measuring the vertical angle between the line of sight from the levelling instrument and the horizontal plane, the horizontal distance between the levelling instrument and the point, and the difference in height between the two instrument axes. The R.L is then calculated using the following formula:

R.L = H_{1} + (H_{2} – H_{1}) + d * tan(θ), where

H_{1} = height of the levelling instrument at the first point

H2_{ }= height of the levelling instrument at the second point

d = horizontal distance between the levelling instrument and the point

θ = vertical angle between the line of sight from the levelling instrument and the horizontal plane

- Instrument Axes at Very Different Levels: When the instrument axes are at very different levels, the R.L of a point can be determined by measuring the vertical angle between the line of sight from the levelling instrument and the horizontal plane, the horizontal distance between the levelling instrument and the point, and the difference in height between the two instrument axes. The R.L is then calculated using the following formula:

R.L = H_{1} + (H_{2} – H_{1}) + d * tan(θ), where

H_{1} = height of the levelling instrument at the first point

H_{2} = height of the levelling instrument at the second point

d = horizontal distance between the levelling instrument and the point

θ = vertical angle between the line of sight from the levelling instrument and the horizontal plane

In summary, the determination of R.L of a point when the base of the object is inaccessible depends on the level of the instrument axes. When the instrument axes are at the same level, the R.L is calculated using the formula R.L = H_{1} + d * tan(θ). When the instrument axes are at different levels, the R.L is calculated using the formula R.L = H_{1 }+ (H_{2} – H_{1}) + d * tan(θ). When the instrument axes are at very different levels, the R.L is calculated using the same formula.

**Recall the determination of R.L of a point when the base of the object is Inaccessible and when the Instrument Station is not in the Same Vertical Plane as the Elevated object**

In this scenario, the determination of the R.L of a point involves measuring the vertical angles between the line of sight from the levelling instrument to the point and the horizontal plane at both the instrument station and the elevated object. The horizontal distance between the instrument station and the elevated object is also measured.

The R.L of the elevated object is then calculated using the following formula:

R.L = H_{1} + d * (tan(θ_{1}) – tan(θ_{2})), where

H_{1} = height of the levelling instrument

d = horizontal distance between the instrument station and the elevated object

θ_{1 }= vertical angle between the line of sight from the levelling instrument to the point and the horizontal plane at the instrument station

θ_{2} = vertical angle between the line of sight from the levelling instrument to the point and the horizontal plane at the elevated object

It is important to note that the above formula assumes that the levelling instrument is set up at the same height at both the instrument station and the elevated object. If the height of the levelling instrument at the instrument station and the elevated object is different, then the height difference must be taken into consideration when calculating the R.L of the elevated object.

In conclusion, the determination of R.L of a point when the base of the object is inaccessible and the instrument station is not in the same vertical plane as the elevated object requires measuring the vertical angles and horizontal distance between the instrument station and the elevated object, and using these values in the formula R.L = H_{1} + d * (tan(θ_{1}) – tan(θ_{2})).

**Recall determination of Height of an Elevated object above ground when it’s base and top is visible but not accessible under the following conditions: i. Base Line Horizontal and in Line with the Object ii. Base Line Horizontal but not in Line with the Object**

When the base line is horizontal and in line with the object, the height of the elevated object can be determined by measuring the horizontal distance between the base of the object and the levelling instrument and the vertical angle between the line of sight from the levelling instrument to the top of the object and the horizontal plane at the levelling instrument. The height of the object is then calculated using the formula:

H = H_{1} + d * tan(θ), where

H = height of the elevated object

H_{1} = height of the levelling instrument

d = horizontal distance between the base of the object and the levelling instrument

θ = vertical angle between the line of sight from the levelling instrument to the top of the object and the horizontal plane at the levelling instrument.

When the base line is horizontal but not in line with the object, the height of the elevated object can be determined by measuring the horizontal distance between the levelling instrument and the base of the object, the vertical angle between the line of sight from the levelling instrument to the top of the object and the horizontal plane at the levelling instrument, and the vertical angle between the line of sight from the levelling instrument to the base of the object and the horizontal plane at the levelling instrument. The height of the object is then calculated using the formula:

H = H_{1} + d * (tan(θ1) – tan(θ2)), where

H = height of the elevated object

H_{1} = height of the levelling instrument

d = horizontal distance between the levelling instrument and the base of the object

θ_{1} = vertical angle between the line of sight from the levelling instrument to the top of the object and the horizontal plane at the levelling instrument

θ_{2} = vertical angle between the line of sight from the levelling instrument to the base of the object and the horizontal plane at the levelling instrument.

In conclusion, the determination of height of an elevated object above the ground when its base and top are visible but not accessible requires measuring the horizontal distance between the levelling instrument and the base of the object and the vertical angles between the line of sight from the levelling instrument to the top and base of the object and the horizontal plane at the levelling instrument, and using these values in the appropriate formula.

**Recall the concept of Terrestrial Refraction**

Terrestrial refraction is a phenomenon that occurs when light travels through the Earth’s atmosphere and bends or changes direction due to differences in the refractive index of air. This bending causes objects to appear in a different position or with a different shape than they would if viewed in a vacuum. In surveying and geospatial measurement, terrestrial refraction can cause significant errors in the determination of elevations and distances, particularly over large distances. To account for this phenomenon, surveyors use mathematical models to calculate the expected amount of refraction at a given location and time, and adjust their measurements accordingly. This helps to ensure accurate and reliable results in a variety of applications, including topographical surveys, construction projects, and geospatial mapping.

**Describe the Correction of Refraction and Curvature**

Correction of refraction and curvature refers to the process of adjusting for the effects of terrestrial refraction and the Earth’s curvature on the measurement of elevations and distances in surveying and geospatial measurement.

Terrestrial refraction occurs when light travels through the Earth’s atmosphere and bends or changes direction due to differences in the refractive index of air. This bending causes objects to appear in a different position or with a different shape than they would if viewed in a vacuum, and can cause significant errors in the determination of elevations and distances, particularly over large distances. To correct for this phenomenon, surveyors use mathematical models to calculate the expected amount of refraction at a given location and time, and adjust their measurements accordingly.

The Earth’s curvature refers to the spherical shape of the planet and the way it affects the measurement of elevations and distances. Over large distances, the Earth’s curvature can cause significant errors in the determination of elevations and distances, as objects that are farther away appear lower in the sky than they would if the Earth were flat. To correct for this phenomenon, surveyors use mathematical models to calculate the expected amount of curvature at a given location and time, and adjust their measurements accordingly.

Both refraction correction and curvature correction are essential for ensuring accurate and reliable results in a variety of surveying and geospatial applications, including topographical surveys, construction projects, and geospatial mapping.

**Derive an expression for the axis signal correction**

Axis signal correction refers to the process of adjusting for the effects of terrestrial refraction on the measurement of elevations in surveying and geospatial measurement. The axis signal correction is applied when the instrument station (the location from which measurements are taken) is not in the same vertical plane as the elevated object, and it is used to correct for the bending of the light beam as it passes through the Earth’s atmosphere.

The correction is derived using the following expression:

Δh = R * (n – 1) * h_{0} / h_{0} + R * (n – 1) * d

where:

Δh = axis signal correction

R = radius of the Earth

n = refractive index of air (usually taken to be 1.000293)

h_{0} = height of the instrument station above the ground

h = height of the elevated object above the ground

d = horizontal distance between the instrument station and the elevated object

This expression calculates the amount of correction required to account for the bending of the light beam due to terrestrial refraction. The correction is then subtracted from or added to the raw measurement of the elevation, depending on the specific conditions, to obtain a corrected value that is more accurate and reliable.

It is important to note that this expression assumes a constant value of the refractive index of air and does not take into account changes in temperature, pressure, or other atmospheric conditions that can affect the amount of refraction. In practice, surveyors may use more sophisticated models or empirical measurements to determine the amount of correction required.

**Derive an expression for the determination of the difference in Elevation by: i. Single Observation ii. Reciprocal Observation**

In surveying and geospatial measurement, the difference in elevation between two points can be determined using either single observation or reciprocal observation. Both methods are used to correct for the effects of terrestrial refraction on the measurement of elevations.

Single Observation:

In single observation, an instrument station is set up at one point, and the elevation of a distant point is measured relative to the instrument station. The following expression can be used to determine the difference in elevation:

Δh = h_{2} – h_{1} + Δh_{1} – Δh_{2}

where:

Δh = difference in elevation

h_{1} = height of the first point above the ground

h_{2} = height of the second point above the ground

Δh_{1} = axis signal correction for the first point

Δh_{2} = axis signal correction for the second point

This expression calculates the difference in elevation by subtracting the height of the first point from the height of the second point, and then correcting for the effects of terrestrial refraction using the axis signal correction values for each point.

Reciprocal Observation:

In reciprocal observation, the instrument station is set up at two points and the elevations of a common point are measured relative to both instrument stations. The following expression can be used to determine the difference in elevation:

Δh = (h_{1} – h_{2}) / 2 + (Δh_{1} – Δh_{2}) / 2

where:

Δh = difference in elevation

h_{1} = height of the first point above the ground as measured from the first instrument station

h_{2 }= height of the second point above the ground as measured from the second instrument station

Δh_{1} = axis signal correction for the first instrument station

Δh_{2} = axis signal correction for the second instrument station

This expression calculates the difference in elevation by taking the average of the height of the common point as measured from each instrument station, and then correcting for the effects of terrestrial refraction using the axis signal correction values for each instrument station.

It is important to note that both single observation and reciprocal observation assume a constant value of the refractive index of air and do not take into account changes in temperature, pressure, or other atmospheric conditions that can affect the amount of refraction. In practice, surveyors may use more sophisticated models or empirical measurements to determine the amount of correction required.