1.21. Traverse Computations

One the field observations are completed for a traverse; the next task is to compute the coordinates of traverse stations. These coordinates are required to be plotted to carry out the detailed mapping of the area.

After collecting measurements in the field during the traverse, the next essential step is to calculate the coordinates—specifically, the x, y, and z coordinates—of each traverse station. This computational process is crucial because it enables the creation of detailed maps, representing the geographical features of the surveyed area accurately.

Imagine you're a treasure hunter with a map. After marking the locations you visited, you need to compute the exact coordinates of those spots so that you can accurately navigate to them later. Similarly, surveyors use coordinate computations to ensure they can represent their measurements on maps effectively.

For the computation of coordinates (x, y and z), following observations are to be taken in the field: (i) Magnetic bearing of at least one traverse line (ii) Length of at least one traverse line (iii) Elevation of at least one traverse station (iv) Included angles between traverse lines (v) Vertical angles of traverse lines (vi) Real-world coordinates of one traverse station.

To accurately compute the coordinates of the traverse stations, several critical observations must be made during the fieldwork. These include capturing the magnetic bearing (the angle between the magnetic north and the line), measuring the length of at least one traverse line, recording the elevation of one station, and noting the angles between lines. Each piece of data serves as a key input for accurately determining the position of each point in the coordinate system.

Think of planning a hiking trip. You need to know the path you will follow (the traverse lines), the distance you will travel (length), how high you will climb (elevation), and your direction (magnetic bearing) to ensure that you reach your destination safely. Each measurement connects to form a clear pathway to follow.

If the length and bearing of a line are known, its projections on the y-axis and x-axis may be done, called latitude and departure of the line, respectively. Latitude is measured northward, and is also known as northing, and departure, if measured eastward is known as easting. The latitude of a line is determined by multiplying the length of the line with the cosine of its reduced bearing; and departure is computed by multiplying the length with the sine of its reduced bearing.

In surveying, when a line's length and direction (bearing) are known, you can determine its horizontal components or projections on a coordinate system. These components are referred to as latitude (northward component) and departure (eastward component). Latitude is calculated by taking the length of the line and multiplying it by the cosine of the line's bearing, while departure uses the sine function for its calculation. This helps in defining the exact position of the line in the 2D plane.

Consider a car driving from a parking lot. The distance your car travels directly north (latitude) and the distance it travels directly east (departure) portray your car's movements on a street grid. Just as you would navigate using these movements, surveyors use latitude and departure to describe positions on their maps.

Due to errors present in the observations, the coordinates of a closed traverse stations when plotted may not close itself, but will have a small difference. The errors in the linear and angular observations therefore are to be adjusted before using them for computational purpose. It is also called Balancing a Traverse.

Mistakes in measurements often occur when performing a traverse, which could lead to discrepancies in plotted coordinates, preventing them from forming a closed loop. To address this issue, errors due to both linear and angular observations need to be systematically adjusted. This process is commonly referred to as balancing the traverse and is essential to ensure that the plotted stations accurately represent their real-world positions.

Imagine you're assembling a jigsaw puzzle, and some pieces don't fit perfectly because of slight manufacturing errors. To ensure that everything fits together, you may need to adjust or trim some pieces. Similarly, surveyors adjust their measurements to correct for any errors and ensure that their plots are accurate.

(a) Adjustment of angular errors (b) Adjustment of bearings. (c) Adjustment of closing error of traverse.

Error adjustments can be addressed in several ways: (a) Angular errors are corrected by ensuring that the sum of measured angles equals the expected total based on the geometric properties of the traverse. (b) The bearings are checked to ensure they correspond to expected values, as the forward and backward bearings should differ by 180 degrees. (c) Closing errors occur when the traverse does not return to its starting point; these errors are resolved by analyzing the overall latitude and departure to ensure they balance to zero.

Consider a group of friends attempting to follow a map, where they might go off course due to misreads (angular errors), not ensuring they return to the starting point after their adventure (closing errors), or misinterpreting directions (bearings). Adjustments help get everyone back on the right path to their intended destination.

The magnitudes of two components of this error (A A and AA ) perpendicular to each other may be determined by finding the algebraic sum of the latitude (ΔL), as well as departures (ΔD).

When correcting errors in the traverse, we focus on the two primary components of the closing error, which are defined by their latitude and departure. By calculating the algebraic sums of both latitude and departure, we can quantify how much adjustment is needed to bring the traverse back to a closed position. This step is critical in ensuring that the sum of latitudes and departures equals zero.

Think of a student who miscalculated the distance back to their home after school. They may find they ended up left of where they started (negative latitude) and further down (negative departure). By adding their steps back synergistically, they can offset their misdirection and arrive back home. Similarly, surveyors adjust their calculations to complete the loop.

Any one of the two rules (Bowditch Rules and Transit Rules) may be used for finding the corrections to balance the survey.

Surveyors have proposed specific rules to distribute the errors effectively: the Bowditch Rule is utilized when all measurements are considered equally precise, allowing error to be distributed based on each line's length in relation to the total traverse perimeter. The Transit Rule is more appropriate when angular measurements are more accurate than linear ones, applying corrections based on individual latitudes and departures. These rules help ensure that errors are adjusted in a way that reflects the confidence in the individual measurements.

Consider a teacher grading essays from students of varying strengths. The teacher may opt for a fair average grading scale (Bowditch) for all, or give more weight to the essays with clearer arguments from the strongest students (Transit). This approach mirrors how surveyors apply corrections, ensuring the most accurate results.

The computations for a closed traverse may be made in the following steps and entered in a tabular form known as Gale’s Traverse, as shown in Table 1.8.

Gale's Traverse Table is a systematic method of organizing calculations for a traverse, ensuring that all necessary adjustments are made clearly. It facilitates the process of summarizing the adjustments to angles, calculating the resulting coordinates, and ensuring that the final adjusted latitudes and departures balance to zero, making it an essential tool in traverse computations.

Think of a project manager tracking tasks using a spreadsheet. Each adjustment and completed task is recorded methodically, making it easier to spot where corrections are needed and ensuring everything aligns by the deadline. Similarly, surveyors use Gale's Traverse Table to keep their measurements organized and accurate.

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