Tachometric Survey: Principles and Applications

Tachometry is an advanced surveying method that uses a tachometer, a specialized surveying instrument, to measure horizontal distances, vertical elevations, and directions simultaneously. It is particularly useful in challenging terrains where traditional methods like chaining are impractical or time-consuming. The core principle of tachometry relies on the relationship between the angle subtended by a known interval on a graduated staff (the staff intercept) and the distance to the staff. This relationship is typically derived from the optical principles of the tachometer and its internal constants.

Tachometer and its Components

A tachometer is essentially a theodolite fitted with a tacheometric diaphragm or a telescope with a specific optical design. The diaphragm has two horizontal stadia hairs (upper and lower) in addition to the central horizontal hair. The telescope's internal optics are calibrated to provide two constants: the stadia constant (K) and the additive constant (C). The stadia constant relates the staff intercept to the distance, while the additive constant accounts for the distance from the instrument's center of trunnion axis to the vertical crosshair.

Tacheometric Principle

Consider a tachometer with a stadia constant K and an additive constant C. Let the horizontal distance between the instrument and the staff be D, the vertical height of the instrument axis be HI, and the staff intercept be 's'. The staff is held vertically, and readings are taken using the upper and lower stadia hairs. The difference between the readings of the upper and lower stadia hairs on the staff is the staff intercept 's'.

The horizontal distance D can be calculated using the formula:

D = K * s + C

Where:

K = Stadia Constant (typically 100 for most modern instruments)

s = Staff Intercept (difference between upper and lower stadia readings)

C = Additive Constant (often 0 for modern instruments, but can be around 0.3m for older ones)

Determination of Constants

The constants K and C need to be determined for accurate measurements. This is usually done by setting up the tachometer at a known point and taking readings on a staff held at various known distances (e.g., 50m, 100m, 150m). By plotting these values or using algebraic methods, the constants can be calculated. For most modern instruments, K is 100 and C is 0, simplifying calculations significantly.

Methods of Tachometry

There are two primary methods of tachometry:

1. Stadia Method:

This is the most common method and uses the stadia hairs within the telescope. The principles described above apply here. Readings are taken on a graduated staff.

2. Subtense Bar Method:

In this method, a specialized bar with precisely fixed lengths (e.g., 2 meters) is used. The bar is set up at the distant station, and the angle subtended by the fixed length of the bar at the instrument is measured using a theodolite. The distance is then calculated using trigonometry. This method is more accurate for moderate distances but requires the subtense bar.

Vertical Angle Measurement and Elevation Determination

When the line of sight is inclined, the calculation of horizontal distance and elevation becomes slightly more complex. Let 'α' be the vertical angle measured from the horizontal. The staff intercept 's' is still measured. The horizontal distance D is calculated as:

D = K * s * cos²(α) + C * cos(α)

The vertical distance V (height difference between instrument axis and point on staff intersected by line of sight) is calculated as:

V = K * s * sin(α) * cos(α) + C * sin(α)

The reduced level (RL) of a point is then determined by:

RL of Point = RL of Instrument Station + HI + V - Reading on staff at the point

Where HI is the height of the instrument above the station's benchmark.

Advantages of Tachometry

Tachometry offers several advantages:

• Speed: It is significantly faster than chaining, especially over long distances and difficult terrain.
• Efficiency: It combines distance and elevation measurement in a single operation.
• Accuracy: For moderate distances, it provides good accuracy.
• Terrain Independence: It is well-suited for hilly, undulating, or obstructed areas.
• Reduced Manpower: It requires fewer personnel compared to chaining methods.

Disadvantages of Tachometry

Despite its advantages, tachometry has some limitations:

• Instrument Cost: Tachometers or specialized theodolites can be expensive.
• Atmospheric Effects: Refraction and temperature variations can affect accuracy over very long distances.
• Staff Reading: Reading the staff accurately, especially with inclined lines of sight, can be challenging.
• Staff Holder Position: The staff holder needs to be positioned carefully to ensure clear sightlines.

Applications of Tachometry

Tachometry finds wide application in various surveying tasks:

• Topographical surveys for mapping.
• Setting out curves (horizontal and vertical).
• Cross-sectioning for earthwork calculations.
• Reconnaissance surveys.
• Hydrographic surveys.

Curve Setting in Surveying

Curves are essential elements in route surveying for roads, railways, and canals, providing a smooth transition between two straight sections (tangents). They allow vehicles or trains to change direction gradually, ensuring safety and comfort. There are two main types of curves used: horizontal curves and vertical curves.

Horizontal Curves

Horizontal curves are circular arcs used to connect two tangents that are at an angle to each other. The design of a horizontal curve involves several key elements:

Elements of a Circular Curve:

• Tangent Points (TP): The points where the curve begins and ends, connecting to the straight tangents.
• Point of Intersection (PI): The point where the two tangents, if extended, would intersect.
• Angle of Intersection (I): The angle between the two tangents at the PI. This is equal to the deflection angle.
• Radius (R): The radius of the circular arc.
• Back Tangent: The tangent preceding the curve.
• Forward Tangent: The tangent following the curve.
• Curve Tangent Length (T): The distance from the PI to the tangent point (TP). T = R * tan(I/2).
• Long Chord (LC): The straight line distance between the beginning and end of the curve. LC = 2 * R * sin(I/2).
• Length of Curve (L): The length of the circular arc. L = (π * R * I) / 180 (where I is in degrees).
• Radius of Curve (R): The radius of the circular arc.
• External Distance (E): The distance from the PI to the midpoint of the curve along the line bisecting the angle I. E = R * (sec(I/2) - 1).
• Middle Ordinate (M): The distance from the midpoint of the curve to the midpoint of the long chord. M = R * (1 - cos(I/2)).
• Super-elevation (e): The difference in elevation between the outer and inner edges of the pavement on a curve, used to counteract centrifugal force.
• Superelevation Runoff: The length of pavement over which the full superelevation is developed.

Setting Out Circular Curves

Setting out a curve involves physically marking its position on the ground using surveying instruments. Common methods include:

1. Method of Chords:

This is a widely used method for setting out simple circular curves. It involves dividing the curve into a series of chords of equal or unequal lengths. The curve is typically set out from the beginning tangent point (TP1).

• Step 1: Calculate the tangent length (T) and the deflection angle (I).
• Step 2: Locate the PI and TP1 and TP2 by measuring T from the PI along the tangents.
• Step 3: Set up the theodolite at TP1. Align it to the PI (back tangent).
• Step 4: Set the theodolite to zero and deflect by the calculated angle to establish the direction of the first chord. The length of the first chord (c1) is usually a standard length (e.g., 30m or 50m, depending on the scale of the project).
• Step 5: Measure the chord length c1 along this line and mark the first station point.
• Step 6: Move the instrument to the first station point. Backsight to TP1.
• Step 7: Set the vernier to zero and deflect by the angle corresponding to the chord length. The angle for a chord of length 'c' is given by θ = 2 * arcsin(c / 2R).
• Step 8: Measure the chord length 'c' along this new line and mark the next station point.
• Step 9: Repeat this process until the end of the curve (TP2) is reached. The last chord should ideally end exactly at TP2.

Tachometric Shortcut for Curve Setting: While the method of chords is common, tachometry can also be used. By setting up the instrument at known points along the curve and using the tacheometric principles to measure distances and bearings to pre-determined points, the curve can be laid out. This is particularly useful when direct chaining is difficult.

2. Tacheometric Method for Curve Setting:

This method is efficient for setting out curves, especially in difficult terrain. The instrument is set up at the tangent point or at intermediate points on the curve. Distances and angles are measured using the tachometer to locate points on the curve. The process involves calculating the coordinates of points on the curve and then locating them on the ground using tacheometric measurements.

3. Offsets from Tangents:

This method involves measuring offsets perpendicular to the tangent lines at specific intervals. The offsets can be radial or perpendicular. Radial offsets are taken from the center of the curve, while perpendicular offsets are measured from the tangent line itself.

Vertical Curves

Vertical curves are used to provide a smooth transition between two different grades (slopes) in route design. They are typically parabolic in shape, offering a constant rate of change in slope.

Types of Vertical Curves:

• Camber or Sag Curve: Connects two downward grades or a downward grade and a level grade.
• Crest Curve: Connects two upward grades or an upward grade and a level grade.

Elements of a Vertical Curve:

• Point of Vertical Intersection (PVI): The intersection of the two tangent grades.
• Beginning of Vertical Curve (BVC): The starting point of the curve.
• End of Vertical Curve (EVC): The ending point of the curve.
• Length of Vertical Curve (L): The horizontal distance between the BVC and EVC.
• G1: The initial grade (slope) of the incoming tangent.
• G2: The final grade (slope) of the outgoing tangent.
• Rate of Change of Grade: (G2 - G1) / L. For a parabolic curve, this is constant.

Setting Out Vertical Curves

Vertical curves are set out based on their horizontal lengths. The elevation of any point on the curve can be calculated using the parabolic equation. For a parabolic curve where the vertical tangent is at the BVC (x=0), the equation is:

y = G1 * x + ((G2 - G1) / 2L) * x²

Where 'y' is the vertical offset from the tangent grades, 'x' is the horizontal distance from the BVC, G1 and G2 are the grades expressed as decimals, and L is the length of the vertical curve.

The final elevation of a point on the curve is calculated as:

Elevation = Elevation of BVC + G1 * x + ((G2 - G1) / 2L) * x²

Earthwork Calculations

Earthwork refers to the process of excavating or filling material to create a desired contour or level, commonly required in civil engineering projects like road construction, canal excavation, dam building, and building foundations. Accurate calculation of the volume of earthwork is crucial for estimating costs, planning material transport, and ensuring project feasibility.

Methods for Calculating Earthwork Volumes

Several methods are used to calculate earthwork volumes, depending on the complexity of the terrain and the required accuracy. These methods generally involve determining the average area of cross-sections or using formulas based on spot levels or contours.

1. Mid-Section Formula:

This is one of the simplest methods. It assumes that the area of a cross-section is approximately the average of the areas of the two end cross-sections. It is suitable for relatively uniform terrain.

Volume (V) = A * ((A1 + A2) / 2)

Where:

A = Average cross-sectional area

A1 = Area of the first cross-section

A2 = Area of the second cross-section

The length between the two cross-sections is assumed to be unit length (e.g., 1 meter or 1 chain) if the areas are calculated per unit length.

2. Mean Section Formula:

This method is similar to the mid-section formula but calculates the volume as the product of the length and the average of the areas of the two end cross-sections. It is also known as the Average End Area method.

Volume (V) = L * ((A1 + A2) / 2)

Where:

L = Length between the two cross-sections

A1 = Area of the first cross-section

A2 = Area of the second cross-section

This is the most commonly used method for preliminary estimates and is generally considered sufficiently accurate for many projects.

3. Prismoidal Formula:

This is the most accurate formula for calculating earthwork volumes, especially when the side slopes and formation levels change significantly between cross-sections. It requires calculating the area of the mid-section (Am) in addition to the end sections (A1 and A2).

Volume (V) = (L / 6) * (A1 + 4Am + A2)

Where:

L = Length between the two end cross-sections

A1 = Area of the first cross-section

Am = Area of the mid-section (obtained by averaging the dimensions of the end sections)

A2 = Area of the second cross-section

The prismoidal formula is derived from the prismoidal solid, which has a polygonal base and top and trapezoidal sides. It provides a more precise volume by accounting for the curvature or change in shape between the end sections.

4. Trapezoidal Formula:

This formula is used when calculating volumes from contour plans or when dealing with irregular areas. It's essentially an application of Simpson's rule or related principles for areas.

Calculating Cross-Sectional Areas

The calculation of cross-sectional areas is fundamental to earthwork volumes. The shape of a cross-section depends on the ground profile and the proposed formation level.

Case 1: Level Ground

If the ground is level and the proposed formation is also level, the cross-section is a simple rectangle. The area is calculated as:

Area = Width of formation * Depth of cutting/filling

Case 2: Ground with Uniform Slopes (Side Slopes)**

When the ground has a uniform side slope (e.g., 1 horizontal to n vertical, or 1:n) and the formation is level, the cross-section is a trapezoid. Let 'w' be the width of the formation, 'd' be the depth of cutting/filling at the center, and 's' be the side slope (horizontal distance for 1 vertical unit).

The width of the cross-section at the surface is w + 2sd.

Area = (Width of formation + Width at surface) / 2 * Depth

Area = ((w + (w + 2sd)) / 2) * d = (2w + 2sd) / 2 * d = (w + sd) * d

Case 3: Ground with One Side Sloping and One Side Vertical

If one side is vertical and the other has a side slope 's', and the depth at the vertical side is 'd', the area is calculated by dividing the cross-section into a rectangle and a triangle.

Area = (w * d) + (s * d * d)

Case 4: Ground with Two Different Side Slopes

If the ground has different side slopes on the left (sL) and right (sR) and the center depth is 'd', the area is calculated as:

Area = (w + sL*d + sR*d) * d = (w + (sL + sR)d) * d

Earthwork Volume Calculation with Contours

When dealing with large areas like dams or reservoirs, earthwork volumes are calculated from contour plans. Methods include:

• Grid Method: The area is divided into a grid of squares. Spot levels are taken at the grid points. The volume is calculated by considering the grid squares as prisms or prismoids.
• Contour Method: This method calculates the volume between two contours. It's often visualized as a series of frustums. The volume between two contours (e.g., contour C1 and C2) is calculated using:

  V = (h/3) * (A1 + A2 + sqrt(A1*A2)) (for a frustum of a pyramid)

  Where h is the contour interval, A1 and A2 are the areas enclosed by the contours.

Factors Affecting Earthwork Calculations

• Nature of Terrain: Flat, undulating, or steep terrain significantly impacts the calculation method and accuracy.
• Accuracy of Survey Data: Errors in spot levels or cross-section measurements directly affect volume calculations.
• Side Slopes: The angle of repose of the soil and the required stability of slopes are critical.
• Shrinkage and Swellage: Excavated soil often occupies a different volume when compacted (swellage) or dried (shrinkage). This factor must be accounted for in bulk earthwork calculations.
• Formation Width and Depth: The design parameters of the project.

Advanced Surveying Equipment

Modern surveying relies on sophisticated equipment that enhances accuracy, efficiency, and data processing capabilities. These tools go beyond traditional methods, enabling complex measurements and detailed mapping.

Global Navigation Satellite Systems (GNSS) / Global Positioning System (GPS)

GNSS receivers use signals from constellations of satellites (like GPS, GLONASS, Galileo, BeiDou) to determine precise positions on Earth. They have revolutionized surveying by enabling rapid data acquisition over large areas.

Key Components:

• Satellites: Transmit signals with timing and orbital data.
• Receivers: Capture signals from multiple satellites to calculate position.
• Control Stations: Monitor satellite health and performance.

Types of GNSS Surveys:

• Absolute Positioning: Uses a single receiver to determine its absolute coordinates. Accuracy varies (meters to sub-meter).
• Relative Positioning (Differential GNSS - DGNSS): Uses two or more receivers simultaneously. A base station at a known point transmits corrections to a mobile rover. This significantly improves accuracy (centimeter-level). Techniques include Real-Time Kinematic (RTK) and Post-Processed Kinematic (PPK).

Applications:

Land boundary surveys, topographic mapping, construction layout, asset management, geodetic control networks.

Total Stations

A total station is an electronic optical instrument used in modern surveying and building construction. It integrates an electronic theodolite (for measuring angles) with an electronic distance meter (EDM) and a microprocessor. It can measure angles, distances, and coordinates of points.

Features:

• Electronic Theodolite: Measures horizontal and vertical angles digitally.
• EDM: Emits a beam of light or laser to a reflector (or directly to a surface) and measures the time it takes to return, calculating the distance.
• Microprocessor: Calculates coordinates, stores data, and performs various surveying computations.
• Data Collector: Stores survey data, allowing for easy transfer to computer software.
• Reflectorless Measurement: Some advanced total stations can measure distances to non-cooperative targets (e.g., building facades) without a reflector.

Applications:

Land surveying, construction staking, monitoring deformation, setting out points for infrastructure projects.

Laser Scanners (Terrestrial Laser Scanners - TLS)

TLS systems capture highly dense and accurate 3D point clouds of objects and environments. They use lasers to measure millions of points on a surface rapidly.

How it Works:

The scanner emits laser pulses and measures the time-of-flight or phase difference to determine the distance to a point. Combined with angle measurements, this creates a precise 3D coordinate for each scanned point.

Types:

• Time-of-Flight (ToF): Measures the time for a laser pulse to travel to the target and back.
• Phase-Shift: Measures the phase difference of a continuous laser beam reflected from the target.

Applications:

As-built surveys, architectural documentation, heritage preservation, forensic analysis, BIM (Building Information Modeling), infrastructure monitoring, virtual reality environments.

Photogrammetry and Drones (UAVs)

Photogrammetry is the science of making measurements from photographs. Drones equipped with high-resolution cameras are now widely used to capture aerial imagery for creating detailed maps and 3D models.

Process:

Drones fly a pre-programmed path, capturing overlapping aerial images. Software then uses photogrammetric principles (e.g., Structure from Motion - SfM) to reconstruct the 3D geometry and texture of the surveyed area.

Advantages:

• Rapid data acquisition over large areas.
• Cost-effective compared to traditional aerial surveys.
• Access to difficult-to-reach locations.
• Generation of orthomosaics, Digital Surface Models (DSM), and 3D point clouds.

Applications:

Topographic mapping, agricultural monitoring, construction progress tracking, inspection of infrastructure (bridges, power lines), environmental monitoring.

Geographic Information Systems (GIS)

While not an instrument, GIS is a critical advanced surveying technology. It is a system designed to capture, store, manipulate, analyze, manage, and present spatial or geographic data.

Functionality:

GIS integrates various data layers (e.g., topography, property boundaries, utility networks, demographic data) to provide insights and support decision-making.

Applications:

Urban planning, environmental management, resource allocation, disaster response, public utility management.