GIAA Mailbag: On Errors in Trigonometric Leveling
Q: I use my total station for trigonometric leveling. Most of the time I have excellent results over fairly long distances, but sometimes I get errors that I can’t explain. What is going on with my instrument? I use the “curvature and refraction” correction available in my software.A: With modern electronic total stations, rapid determination of vertical heights is a procedure that can be done effortlessly. It is also easy to do this at long distances, seemingly only limited by the EDM range of the instrument. However, as with all surveying, the entire system must be understood and considered to ensure that accurate measurements are made. Depending on the circumstances, it is possible to determine a theoretical maximum possible accuracy to be considered when the work is done. Unfortunately, it is not possible to easily determine the minimum possible accuracy for these same measurements. This is because the refraction of the line of sight becomes more and more uncertain as the distance between the instrument and the target increases. Refraction is impossible to predict with 100 percent certainty because it depends on the current density of the atmosphere. This is more complicated than it sounds when measuring over long distances because the average needs to be computed over the length of the entire line as it traverses air of different densities.
At any range the significant parts of the measurement system include the sub-system used to measure the height of the instrument above the ground point and the target over the ground point. Any errors introduced here will be in the final result determined for the elevation of the point being measured. The method the surveyor uses to measure the vertical angle is also important. Is it measured once in the F1 (direct) position? Is it measured several times in the F1 position? Is it measured in the F1 and F2 positions? How many times? For example, if the instrument is rated at ±5 arc seconds, it means that the likelihood of achieving a vertical angle reliable to within ±5" after measuring the angle once in F1 and once in F2 is approximately 68 percent. To achieve a likelihood of 99.9 percent, the reliability with that same pair of measurements is ±15". At 1,000 ft, this means a vertical distance accuracy of ±0.07 ft. If only measured in a single face, any errors in the collimation will not be compensated for, even if your instrument has software that corrects for collimation unless you have determined the collimation error immediately prior to your trigonometric leveling.
The intrinsic angle measuring accuracy of the instrument is another component in achieving the total accuracy of the vertical distance measurement. This is the ±5" used in the example above. This is best determined by the surveyor through experimentation, a subject too comprehensive to cover here. But a good estimate can be the vertical angle accuracy published by the manufacturer. Remember that accuracy is not least count. Knowledge of how to apply error propagation rules (another subject too comprehensive to cover here) must be used to determine the maximum error component from the combination of the instrument’s basic accuracy and the method used to determine the angle. While intrinsic angle accuracy pertains to random errors, remember that there may be systematic errors due to vertical circle indexing, or a bias in the mechanical or electronic vertical axis tilt sensor that should be considered as well and adjusted, removed through procedures or removed through calculations of the error.
Trigonometric leveling using curvature and refraction correction only ensures that errors introduced by curvature (which is relatively constant) and refraction (which changes with air density) will be correctly compensated for when measuring under average conditions. A good surveying textbook will cover the procedures (and underlying theory) for good practices in trigonometric leveling. The only way to be sure that natural errors are properly compensated for is with reciprocal observations, especially if they can be done simultaneously. The accuracy desired in trigonometric leveling should determine the rigor with which the procedures are followed.
These and many other types of errors obviously imply a well-adjusted instrument, good observation, evaluation of instrument accuracy, accessories in good condition and a thorough understanding of the natural errors involved in trigonometric leveling.