Surveying I. Lecture 7. Distance observations Distance observations Different ways to measure the distance between points: • tapes; • optical methods, e.g. using the stadia lines in the surveyor’s telescope; • physical methods: electrooptical distance measurements (EDM) Measuring distances with tapes How long is a 30-meter-long tape? Error sources: • graduation error (1m on the tape is not equal to the standard 1 m); • expansion caused by the pulling force; • thermal expansion. The effects must be taken into account to measure the distances accurately! The standardization of tapes The tape should be compared to a standard length. Therefore standard baselines have been created. 20 10 0 -10 -20 -20 -10 0 10 20 30 00 0 00 Tape standard baseline length = 29.998m The standardization of tapes Nominal length of the tape = 30.000m. 20 10 0 -10 -20 -20 -10 0 10 20 30 00 0 00 Tape standard baseline length = 29.998m The position of the two ends of the tape should be read on the scales, thus the true length of the tape can be computed. LL=+4.5 mm LR=+3.2 mm True length = LL+LR+Lbaseline=30,006 mm The standardization of tapes The standardization must be carried out using a standard pulling force, and the temperature should also be recorded to compute the effect of thermal expansion later. The true length is valid for the standardization temperature only! True length = LL+LR+Lbaseline=30,006 mm (T=18 °C) The effect of pulling force and temperature The effect of pulling force: • when the same pulling force is used during the measurements as during the standardization, then the effect of this error source is eliminated. The effect of temperature change: • the temperature during the measurements must be recorded; • the thermal expansion of the tape can be computed using the equation of thermal expansion of linear objects: L L T T0 where is the expansion coefficient of the material of the tape (steel: =1.1x10-5 1/°C). Physical Methods of Distance Observations Indirect way of distance observation: the distance is measured by the observation of other physical variables. Usually these methods are based on electromagnetic signals: • carrier signal – carries the measuring signal between the points; measuring signal – the carrier is modulated with this signal, and this is used for the distance observations Two parts of the electromagnetic spectrum are used: • microwave (wavelengths are in the order of centimeters) • electrooptical (wavelengths are in the order of micrometers – visible light or infrared rays Electrooptical Distance Measurements (EDM) Two different realization: • time pulse measurements • the carrier is modulated with a pulse (short peak), and the travel time of the signal between the station and the target is measured. • the speed of the carrier must be known • phase angle measurements • the phase angle must be measured at the two endpoints; • the distance is linked with the phase difference in the signals; • usually the phase difference of less than a full cycle can be measured; • the speed of the carrier must also be known; The realization of EDM Receiver Transmitter A transmitter is place on the station, and a receiver on the target. The carrier is modulated with the measurement signal, and transmitted by the instrument to the target. The receiver receives the signal, demodulates it from the carrier, and uses the measurement signal to carry out the measurement. The realization of EDM The distance can be computed using the phase difference or the travel time. However time synchronization is necessary to be able to measure these quantities. We use a reflector at the target, and the receiver is placed in the instrument together with the transmitter: Reflector Transmitter Receiver Time pulse measurement The instrument emits a pulse, which propagates to the target, where the reflector reflects it back to the instrument, where the receiver receives the signal. The time-of-flight (t) is measured. Thus the covered distance can be computed: 2 D t D t 2 Where u is the velocity of light in the atmosphere. Phase angle measurement Reflector Transmitter Receiver The signal travels the length of 2D. Let’s suppose that u and the frequency ( f ) of the measurement signal are known. In this case the signal travels during a full cycle: u f Where is the wavelength of the signal. Phase angle measurement d Receiver Transmitter N full wa ves = N 2D The observed phase angle can be used to compute the residual distance (d): 2 d Thus the distance: 2D N d D N 2 d 2 The equation of phase measurements D N 2 d 2 D N 2 2 2 The effective wavelength of EDMs: eff 2 The realization of phase measurements D N 2 2 2 Method of constant measurement frequency: is measured and the residual distance (d) is computed. Method of variable measurement frequency: The measurement frequency is changed so, that equals zero. In this case the residual distance is zero as well. The determination of N D N 2 2 2 When the effective wavelength is shorter than the observed distance, the observation is ambiguous (N is not known), but accurate. When the effective wavelength is larger than the observed distance, the observation is unambiguous (N=0), but inaccurate due to the long wavelength. The determination of N N2 full waves = N2 2D Two different frequencies are used for the measurements: • 1 is larger than the range of the EDM • 2 provides the higher accuracy d2 Receiver Transmitter N1= 0 d1 The determination of N N2 full waves = N2 d2 Receiver Transmitter N1= 0 d1 2D The accuracy of phase angle measurement is usually 2-3/10000. Let’s see the first observation (1eff=2000m): N=0, accuracy is 0.6 m d1 = 843.5 m The determination of N Let’s see the second observation (2eff= 10m): N 2 0, d N 2 int 1 2 eff 2 2 . 1765 d 2 2 2 2 84 eff 3 , 464 m D N 2 2 eff d 2 840 m 3 . 464 m 843 . 464 m accuracy 2 3 mm Error sources of EDMs Instrumental and reflector constants The transmitter is not aligned with the standing axis of the instrument, or the point of reflection is not aligned with the vertical of the target. Instrumental offsets are automatically included in the results, but it is not the case with the prism offset. Determining the reflector constant (c): Set out three collinear points (A,B and C), and measure the distances between AC in one set, and in two sets (AB and BC)! The computation of the reflector constant A C B True distances: DAB, DAC, DBC Observed distances: (D)AB, (D)AC, (D)BC D AC D AC c D AB D AB c since : D AC D AB D BC , D AC c D AB c D BC c D AC D AB D BC c, D BC D BC c Frequency error Frequency error: the oscillators, which create the measurement signal are stable, but a drift can occur over a longer operational time. A change in the frequency causes an error in the wavelength, thus in the observed distance as well. D N 2 2 2 Instruments must be calibrated frequently, corrections can be applied to the observed distances (k). k 1 f0 f f0 D k D raw Atmospheric correction The wavelength depends on the velocity of the signal in the atmosphere and the frequency. u f The velocity of the signal depends on the refractive index of the atmosphere: u c Where c is the velocity of light in vacuo, is refractive index. The refractive index depends mostly on the air pressure and the temperature. Atmospheric correction The atmospheric correction: atm . corr . mm / km 1, 0 T0 T 0 , 4 p 0 p The multiplicator of atmospheric corr: m 1 atm . corr .mm / km 10 6 1 T 0 T 0 , 4 p 0 p 10 6 The corrected distance observation (reflector constant, frequency error, atmospheric correction): D c k m D raw The reduction of distance observations Definition: the distance between two points is defined by the distance between the two points projected to a reference surface (e.g. the Mean Sea Level). Thus the measured distance should be reduced to this reference surface. Note: Later we’ll see that the distances should also be reduced to the applied projection plane. Reduction to the local horizon The distance observations are usually measured on a slope. Therefore they should be converted to horizontal distances. Slope distances can be: • straight slopes, • sections with different slopes. Straight slope (e.g. EDM) Sections with diff. slopes (e.g. tape) Reduction to the local horizon Sections with constant slopes can be converted to horizontal distances: B s A h d If is known: d=s cos, if h is known: d=s-h2/2s. Reduction to the Mean Sea Level d AB A’ H d ’AB R B’ MSL dAB is the horizontal distance at the mean elevation ( H ) of the two points d AB d AB O R RH , d AB d AB d AB H R Thank You for Your Attention!