SUR-2250 Error Theory

Preliminary Every measurement includes errors. The real (true) value can never be known. The best we can do is to provide the best estimate of the measurement.

Accuracy vs Precision

Types of Error

Three Main Types of Error Gross Error (Mistakes or Blunders) Systematic Errors Random Errors

Three Main Types of Error

Gross Error These are not really errors under error theory, but they are one of the main reasons why surveyors purchase errors and omissions insurance. Some examples are: Transposing digits i.e. writing 144.03 instead of 114.03 Recording the wrong point number for a traverse point. Not recording a change in prism pole height. Blunders that cause large errors in the final survey are usually detected by normal survey checks, however small blunders may remain undetected.

Systematic Errors Systematic errors, or biases errors, result from the physical properties of the measuring system. Systematic errors are constant under constant measuring conditions and change as conditions change. A classical example is the change in length of a tape as the temperature changes.

Correcting Systematic Errors Because systematic errors are caused by the physics of the measurement system, they can be mathematically modeled and the corrections computed to offset these errors. For example temperature correction for a steel tape: 𝐶=𝑘 𝑇 𝑚 − 𝑇 𝑠 𝐿

Total Station EDM Corrections for Systematic Errors The EDM (Electronic Distance Measurement) part of a total station measures distances using light waves. The velocity of light in air varies according to the air density. If the operator enters air temperature and pressure, the systematic error caused by this variation is corrected by most total stations. The atmospheric pressure reported by The National Weather Service and used in weather reports is not an actual pressure. It is corrected to sea level pressure, so that a stated pressure means the same thing, as far as weather trends go, in areas of different elevations. If the survey is not at sea level, using sea level pressure introduces a systematic error. This error, of course, increases with elevation. Ignoring the correction at an elevation of 900 feet causes a distance error of about 10 ppm (parts per million).

Random Errors Random Errors remain after eliminating gross and systematic errors. Impossible to compute or eliminate. They follow the probability laws, so they can be adjusted. Their signs are not constant (±). Present in all surveying measurements. More observations result in a better estimate, and reduction of, random errors. There are limiting factors, time and cost. Random errors are as likely to be positive as negative. Small random errors are much more likely than large ones.

Probability Error analysis usually involves random errors only. However some systematic errors are governed by probability and can be used to predict accuracy. Instrument accuracies for example Random errors occurrence is governed by the probability laws (normal distribution), as any random phenomena. The most probable value of a single quantity observed many times under the same condition is the mean

Notation and Equations The lower case Greek letter σ, sigma denotes the standard deviation of a series. A lower case 𝑥 denotes a series of measurements. An x subscript i 𝑥 𝑖 denotes an individual measurement. Upper case sigma, ∑ 𝑥 denotes the sum of a series. A Lower case 𝑥 with an over-score denotes the mean or average of a series.

Mean of a Series The Mean is the average, or most probable value. 𝑥 = 𝑥 𝑛 Where: 𝑥 is the Mean of the series 𝑥 represents all the measurements in a series n is the number of data points in the series

Variance Variance (Residual) is the difference between any observation and it’s most probable value. 𝑣 𝑖 = 𝑥 − 𝑥 𝑖 = 𝜎 2 Where: 𝑣 𝑖 is the residual between a measurement and the mean. 𝑥 is the mean. 𝑥 𝑖 is an individual measurement. 𝜎 is the Standard Deviation.

Error Distribution Random errors are randomly distributed, a bell shape distribution that is approximated by the probability curve. Some General Laws of Probability: Small errors occur more often than large ones Positive and negative errors of the same size happen with equal frequency. They are equally probable. Therefore the mean is the most probable value.

Normal (Gaussian) Distribution In probability theory, the normal (or Gaussian) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known.

Normal (Gaussian) Distribution A normal distribution can be used to model surveying error. The following formula is used to generate a probability density graph. 𝑓 𝑥 = 1 2𝜋 𝜎 2 ∙ 𝑒 − 𝑥−𝜇 2 2 𝜎 2 Where: 𝜎 2 is the variance. 𝜎 is the standard deviation 𝜇 is the mean.

Standard Deviation Standard deviation is a measurement that is used to quantify the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the data points tend to be close to the mean high standard deviation shows that the data points are spread out over a wide range of values

Standard Deviation 𝜎= 𝑖=1 𝑛 𝑥 −𝑥 2 𝑛−1 Where: 𝜎 is the standard deviation of the data set. 𝑥 −𝑥 is the variance of each data point. n is the number of data points in the set.

Standard Error of the Mean (SEM) The standard error of the mean (SEM) is the standard deviation of the error in the data-set mean with respect to the true mean. SEM is usually estimated by the sample estimate of the population standard deviation (sample standard deviation) divided by the square root of the sample size.

Standard Error of the Mean (SEM) 𝑆𝐸𝑀= 𝜎 𝑛 Where: 𝜎 is the standard deviation of the series. 𝑛 is the number of data points in the series.

Probability Density Graphs The probability of having a measurement fall within one standard deviation of the mean is 68%.

Probability Density Graphs The probability of having a measurement fall within two standard deviations of the mean is 95%.

Probability Density Graphs The probability of having a measurement fall within three standard deviations of the mean is 99.7%.

Reducing Random Errors Make repeated measurements – Making 4 times as many measurements will reduce the random error by half. 𝑆𝐸𝑀= 𝜎 𝑛 Where: 𝜎 is the standard deviation of the series. 𝑛 is the number of data points in the series.