SU3250

SURVEY MEASUREMENTS AND ADJUSTMENTS

Course Notes

Prepared by

Indrajith D. Wijayratne Associate Professor of surveying Michigan Technological University

Houghton, MI 49931

Copyright © 2002 by Indrajith Wijayratne

PROPAGATION OF RANDOM ERRORS

• Propagation of random errors • Errors in computed values (indirect measurements) • Pre-analysis of planned surveys

Understanding error propagation is important in the following: • Combining two or more components of error in the same

measurement e.g. pointing error and reading error in an angle measurement

• Finding error in a sum or difference of two or more measurements

e.g. adding several tape lengths • Finding error in a computed value from measurements using a

mathematical relationship e.g. coordinates computed from distances and bearings Unlike systematic errors, random errors cannot be simply added or subtracted because the exact magnitude or the sign of the error is not known Even though the sum of several random errors tends to be zero it is not quite zero, in general Random errors belong to statistical distributions, and hence, follow rules of propagation of variances The variance of a random variable formed by combining two random variables is given by

σ2

comb = σ21 + σ2

2 + 2. σ12 Where σ12 = covariance between variable 1 and variable 2 If the two original random variables are independent, then σ12 = 0, and therefore, σ2

comb = σ21 + σ2

2 For several independent variables σ2

comb = σ21 + σ2

2 + σ23 + σ2

4+ .. Etc. Statistical independence means that any change in one variable does not have any influence on the other variable(s) This means that the variables are not statistically correlated or the covariance between them is zero In surveying, when combining different random error components or in sums and differences of similar measurements, statistical independence of individual components is assumed This is also true for independent measurements such as distances and angles, which are used for computation of other values

e.g. coordinates Two values computed from the same measurements are correlated e.g. latitude and departure of a traverse line computed from the

same angle and distance When combining random errors, estimates of all errors must have the same probability and the probability of the resulting error is also the same General guidelines for random error propagation: • clearly understand the measurement process • determine basic measurement • analyze each basic measurement to find sources of random

errors • compute combined random error in basic measurement • propagate the errors using rules of propagation to the final

measurements used in computations • propagate errors to computed values Propagation in Distance Measurements:

• Tape • Stadia • Subtense • EDM or GPS

Two main random errors in s distance measurement by taping are marking of the end of the tape on ground and reading the tape graduations. Other random errors such as those due to fluctuations in teperature or tension are generally small. EDM measurements are subject to random errors introduced by centering of the instrument and prism, and those due to random variations in electronic center and frequency of transmission The last two are generally supplied by the manufacturer and given in the form ± (a + b ppm. ) in which 'a' is constant and 'b' is proportional to the distance measured. ppm. = parts per million Since both 'a' and 'b' are random components of the error, the error in a measured EDM distance can be given as σM = √ a2 + (b* 10-6 * distance)2

Now, the combined error in a distance measured by EDM can be given as σD = √ ( σ2

i + σ2r + σ2

M) where σi = instrument centering error σr = prism (reflector) centering error Propagation in Angle Measurements

• Single measurement • Multiple measurements • Electronic theodolites Random errors in angle measurements are • instrument and target centering • plate bubble centering (leveling the instrument) • pointing to the target • reading (verniers, micrometers, scales) The instrument/target centering errors and plate bubble centering errors remain the same for each angle measured from the same setup Instrument/target centering errors and plate bubble centering errors become random from setup to setup, and therefore, will have a random effect on the angular closure of a triangle or a traverse Above are discussed in sections 6.6, 6.7 and 6.8 An angle is made up of two directions, and therefore, each direction is affected by all the errors stated above Following discussion is limited to the effect of only the pointing and reading errors in a measured angle

Multiple Measurements Two basic methods based on the type of instrument

• Direction method • Repetition method

Only difference is the effect of reading error Direction method Two pointings and two readings for each measurement of angle Each angle measurement is independent Standard deviation of the mean value of angle due to reading and pointing is σα = √ (2.σ2

p + 2.σ2r)

√n σα = σrp √2 √n Repetition Method

Only two readings regardless of number of repetitions Same number of pointings as before Standard deviation of the mean angle due to reading and pointing is σα = 2.σ2

p + 2.σ2r

√ n n2

Note that reading and pointing errors need to be known separately in the case of repetition method Angle Measurements Using Total Stations (Digital Theodolites) It is not possible to assess the reading error in digital theodolites as there is no human influence on reading A standard, called DIN18723, has been developed that gives the estimated error of the mean of two direction measurements, one with telescope direct and one telescope inverted Each direction measurement include both pointing and reading Assume two direction measurements are indicated by d1 and d2, and therefore, the mean direction is given by d = d1+ d2 = 1/2 d1+ 1/2 d2 2 By rules of random error propagation σ2

d = (1/2)2 σ2pr + (1/2)2σ2

pr

= 2. σ2pr = σ2

pr 4 2 σDIN = σpr/√2 DIN value has been estimated in the factory using a specific set of targets. Since the targets used in the field are different, one could get a realistic estimates of an angle measurements error by field testing. Self-study Problems 1. A distance of 334.56 feet was measured with a 100 ft. steel tape.

The uncertainty in one tape measurement due to random errors has been estimated to be ±0.03 feet. Find the unceratainty of the total distance measured.

2. If the above distance was measured with an EDM with a

manufacturer specification of ±(3 mm. + 3 ppm), what would be the uncertainty of the distance ?

3. If the above distance was added to another distance of 567.98

feet measured with the same EDM, what would be the uncertainty of the sum of two distances ?

4. What would be the expected uncertainty of an angle, with 95%

confidence, measured with a theodolite that has a pointing error of ±1.2" and a reading error of ±1.5". Disregard other errors.

Error propagation in Traverses

• Expected angular closure in a traverse • Estimated error in azimuths/bearings • Expected error in linear closure of traverses

Once the combined error in each angle has been estimated as discussed previously, the expected angular closure of a traverse can be computed (See sec. 6.9, page 110) Angular closure in a polygon traverse is computed by adding all interior angles, and then comparing this sum with a theoretical value The random error propagation is done assuming each angle was measured independently This is the error propagation in a sum as given by equation 5.18 on page 86 (See Example 6.8 on page 110) Expected angular closure is computed at a higher level of probability, e.g. 95%, and then compared with the actual angular

closure to ensure that measured angle do not contain blunders (see Example 6.9 on page 111) Expected error in a link traverse could be computed by first computing the estimated errors in azimuths/bearings Propagation of errors in computed azimuths is discussed in sec. 7.3 on page 125 and is given by equations 7.5/7.6 on page 126 (Also see example ) Expected Linear Closure in Traverse (Sec. 7.4, page 126) • Distance and angle measurements are independent • Computed latitudes and departures are correlated • Coordinates are correlated Latitude and departure of a traverse line computed from the same distance and azimuth/bearing are correlated even though the distance and azimuth/bearing are independent Northing and Easting of a point are correlated with each other as they are computed from latitude and departure that are correlated

It could be seen that, through the process of computing azimuths/bearings, each computed azimuth/bearing has an influence not only from the angle used in computing that azimuth but also from angles used for the computation of previous azimuths This in turn results in coordinates of all points in the traverse or network being correlated with each other The error propagation in horizontal surveys such as traverses is much complicated and an approach using matrices is often used Propagation in elevation measurements • Differential leveling • Trigonometric leveling Random Errors in differential leveling • Rod reading • Instrument leveling • Rod leveling • Residual errors in instrument collimation, refraction and

curvature, etc. due to unequal sight distances Random Errors in Trigonometric leveling

• Instrument and reflector heights • Vertical or zenith angle • Distance • Refraction and curvature Specifications for surveys • Analysis of errors in measurements is a useful tool when

writing specifications for surveys • Knowing the required accuracy/ tolerance in the final

coordinates/ elevations, detailed specifications for equipment/methods can be developed