In this article, we will explore the principles, methods, and applications of prismatic surveys, covering Magnetic North, True North, Azimuth, Bearings, Local Attraction, Magnetic Declination/Variation, and more.
Prismatic Survey
The Magnetic North (MN)
The Earth has a magnetic axis inclined to the line of longitude, dividing it into two equal parts. This magnetic axis influences the compass needle. When allowed to swing freely, a compass needle points to the northern pole of this axis, indicating the Magnetic North.
The Magnetic North is the direction of the Earth’s magnetic axis pole from any point on the Earth’s surface as shown by a freely suspended compass needle. It is important for all angular measurements with surveying instruments. Without it, prismatic surveys with the theodolite and compass would not be possible.
True North (TN)
True North points to the geographic North Pole of the Earth’s axis in the Northern Hemisphere. All bearings referenced to this direction are known as true north bearings. The figure below illustrates Magnetic North (MN), True North (TN), True South (TS), and Magnetic South
The Azimuth
Azimuth is the smallest angle measured from a reference north to a point, either eastward or westward. Azimuths can be based on Magnetic North or True North, referred to as Magnetic North Azimuths and True North Azimuths, respectively. The azimuth starts at 0° or 360° for North, and continues through 90° for East, 180° for South, 270° for West, and back to 360° for North.
Bearings
The mathematical analysis of survey data starts by converting the obtained bearings to quadrantal bearings.
Next, the measured ground distances are broken down into their horizontal and vertical components. This is achieved by calculating the sine and cosine of each quadrantal bearing and then multiplying these by the measured ground distance.
The sine of the bearing, when multiplied by the distance, provides the horizontal X-axis value, also known as the Easting Component or DEPARTURE. The cosine of the bearing, when multiplied by the distance, gives the vertical Y-axis value, known as the Northing Component or LATITUDE.
Quadrantal bearings are characterized by being measured relative to the North (N) and South (S) points, rather than the East (E) and West (W) points. They always include a preceding N or S followed by E or W, depending on the quadrant in which the bearing lies.
The primary purpose of converting whole circle bearings to quadrantal bearings is to simplify them to values between 0 and 90 degrees. This makes it easier to calculate the sines and cosines, which, when multiplied by the measured distances, yield the DEPARTURES and LATITUDES.
Angles of the First Quadrant
Angles that are 90 degrees or less keep their values and are positioned in the first quadrant. A 90-degree angle is labeled as Due East. For instance, a bearing of 75 degrees is expressed as N 75° E in quadrantal terms. A bearing of 90° is simply referred to as DUE EAST.
Angles of the Second Quadrant
Bearings exceeding 90° but remaining below 180° are typically subtracted from 180°. The resultant angle is then measured from the South cardinal point.
South is situated in the second quadrant. Bearings of 180 degrees are labeled as Due South. For instance, if a bearing is 170° and is reduced to quadrantal bearings, it becomes (180° – 170°) = S 10° E. A bearing of 180° is called DUE SOUTH.
Angles of the Third Quadrant
Bearings exceeding 180° have 180° subtracted from them to generate quadrantal bearings in the third quadrant. These bearings are measured from the South cardinal point and expressed as S “bearing” W. For instance, a full compass bearing of 196° would be (196° – 180°) = S 16° W in quadrantal terms. A bearing of 270° is termed as DUE WEST.
Angles of the Fourth Quadrant
Angles greater than 270° fall into the fourth quadrant. To obtain quadrantal bearings of the fourth quadrant, these bearings are subtracted from 360°. For example, a whole compass bearing of 288° would be (360° – 288°) = N 72° W. A bearing of 0° or 360° is referred to as DUE NORTH.
Local Attraction
Local attraction describes any nearby influence that causes the magnetic needle to deviate from the magnetic meridian for that area, leading to inaccurate measurements. Taking both the forward and back bearings aids in identifying local attraction.
Factors contributing to local attraction encompass fixed objects made of iron, steel, and magnetite in the ground. It also encompasses iron and steel items carried by individuals. High-tension lines can affect the compass needle and should be avoided when feasible. Typically, the disparity between the forward bearing (FB) and the back bearing (BB) is 180°.
The Forward Bearing and Back Bearing
The forward bearing is the direction from an original point to a terminal point. The back bearing is the opposite direction of the forward bearing.
If the difference between the forward and back bearings is exactly 180°, it indicates no local attraction at the two stations. For example, consider a survey line along stations A, B, and C. If the forward bearing from A to B is 95° and the back bearing to A from B is 275°, the difference is exactly 180°, indicating no local attraction at stations A and B.
If the forward bearing from B to C is 240° and the back bearing from C to B is 61°, the difference is 179°. Since stations A and B are free from local attraction, this suggests possible local attraction at station C. To confirm this, take a forward bearing from station C to A and a back bearing from A to C. If the difference is not exactly 180°, then station C has local attraction.
Magnetic Declination/Variation (MD or MV)
Magnetic Declination or Variation is the angle between the magnetic axis and the Earth’s geographic axis. This value changes from one location to another. The Isogonic chart of Ghana from 1958 shows this variability. The magnetic axis oscillates between West and East over an angle of about 22°, with a maximum deviation from True North of 11°.
In Ghana, the magnetic axis is currently inclined west of True North and decreasing at a rate of 6.5 minutes of arc per year (6.5′).
The magnetic declination of an area is subtracted from all magnetic bearings recorded in the field before plotting to adjust them to True North. It is crucial to check the forest reserve boundary schedule to determine if the bearings refer to True North or Magnetic North.
If the bearings refer to True North, add the magnetic declination of that year to each bearing before using the schedule in the field. If the schedule refers to Magnetic North, use an Isogonic Chart.
Understanding these principles and techniques in prismatic surveys ensures accurate and reliable surveying practices essential for various civil engineering projects.