In surveying, accurate measurements and precise calculations are paramount in obtaining reliable results. However, simply obtaining results is only half the battle. It is equally important to estimate the accuracy of the results obtained. The accuracy depends on various factors such as errors in measurement techniques, round-off errors, and even mistakes that could occur during calculations. In this discussion, we will focus on calculation errors and how to minimize their effects.
Calculation errors can occur in many different ways, including inaccurate calculators or computers, round-off errors due to expressing intermediate values to a finite number of significant figures, errors in the calculation model, and mistakes. These errors can lead to incorrect answers and must be avoided by considering all sources of error and taking appropriate action to minimize their effects.
You must consider all sources of error in your calculations and take appropriate action to minimise
their effects. Calculation errors can occur in a number of ways:
- Calculators and computers have limits and they are not perfectly
accurate so sometimes the answers are wrong!
- Round-off errors because intermediate values are expressed to
a finite number of significant figures.
- Errors in the calculation model, e.g. using equations that are only
Even when no apparent errors occur, the finite number of significant figures that calculators and computers can handle can still result in incorrect answers. For instance, taking the square of the square root of 2 should result in 0, but some calculators may display 1 x 10 -9 as an answer. Another example is when the cosine of 1” is taken and then the inverse cosine function. Ideally, the answer should be 1”, but the actual result may differ. Also, round-off errors can accumulate over a series of calculations, leading to significant inaccuracies.
It is vital to note that while computing machines can handle very small numbers, they can only carry a specific number of significant figures. Thus, a calculator or computer may not distinguish between two seemingly similar numbers. As such, it is crucial to use techniques to avoid calculation-induced errors as much as possible.
One technique is to thoroughly check calculations and results for mistakes. Also, using modified methods, such as converting 23° 46' 53.1234" to decimal degrees, can save space in a calculator display, thus avoiding the machine precision problem. We will discuss the accuracy of trigonometric functions in machines later.
Checking Your Calculations
In order to ensure accuracy in surveying calculations, it is crucial to perform thorough checks. Merely verifying the original calculation step by step is not sufficient, as there is a risk of blindly accepting incorrect figures. There are several methods for carrying out effective checks, including obtaining an independent calculation from another person, performing a second calculation using a different mathematical process, reversing the calculation, and utilizing built-in checks. It is also important to check data input, estimate the likely magnitude of the answer prior to the calculation, and use test data from textbooks to ensure that equations and steps are being followed correctly. Common sources of errors include illegible figures, untidy arrangements, transcription errors, and typographical errors. Additionally, it is essential to ensure that the data is correct and to design measurement techniques to minimize or locate any mistakes. One useful method is to make an estimate of the likely magnitude of the answer before doing the calculations. If using a program, it is important to check the program itself and to compare the results with those from a program independently written by someone else. It is inevitable that mistakes will occur, but good surveyors find and correct them before reporting results to their employer or client.
Here are the steps for checking calculations:
- Get an independent calculation by another person and compare the results of the calculations. This involves having someone else perform the same calculation using the same data and comparing the results for any discrepancies. This helps to ensure that the original calculation was accurate and can identify any mistakes made by the first person.
- Do a second calculation but use a different mathematical process. This involves performing a second calculation using a different method or equation than the original calculation to ensure that the result is the same. This eliminates the possibility of repeating the same mistake from the original calculation.
- Reverse the calculation. This involves using the results of the calculation to calculate the input data. For example, if you converted data from feet to meters, you can reverse the calculation by converting the result back to feet to check that it matches the original input.
- Check for built-in checks. Some survey problems have built-in checks, such as the angles in a triangle adding up to 180 degrees. These can be used to verify the accuracy of the calculation.
- Check data entry. Verify that the data was correctly entered by having the computer "echo" or display what was inputted.
- Estimate the answer. Before performing the calculation, make an estimate of the likely magnitude of the answer. Then, after completing the calculation, compare the result to the original estimate to check for accuracy. Use test data.
- Use test data from textbooks or generate simple data to check the calculation.
- Identify potential sources of error. Check for potential sources of error, such as illegible figures, untidy arrangements, unsystematic work, trying to do too many steps at once, transcription errors, and typographical errors.
- Ensure correct data and measurement techniques.
- Ensure that the data is correct and the measurement techniques are designed to avoid, minimize or locate any mistakes.
By following these steps, you can verify the accuracy of your calculations and identify any errors before reporting your results.
At the end of the day, obtaining accurate surveying results requires more than just measurements and calculations. The accuracy of the results obtained must also be estimated, and all sources of error must be considered and minimized. By doing so and using appropriate techniques, surveyors can obtain reliable and accurate results.