ADJUSTMENT COMPUTATIONS STATISTICS AND LEAST SQUARES IN SURVEYING AND GIS PAUL WOLF

1. ADJUSTMENT COMPUTATIONS STATISTICS AND LEAST SQUARES IN SURVEYING AND GIS PAUL WOLF CHARLES D. GHILANI

2. TRAVERSE CLOSURE √ ΔX2 +ΔY2 =Distance Error Distance Error/ Total Distance = Error per foot Or Error Ratio Tan -1 (ΔY / ΔX) = Angular Error (Azimuth)

3. ACCURACY VS. PRECISION PRECISE BUT NOT ACCURATE PRECISE AND ACCURATE

4. ACCURACY VS. PRECISION • ACCURACY-the degree of conformity with a standard or measure of closeness to a true value. • An exact value, such as the sum of three angles of a triangle equals 180° • A value of a conventional unit by physical representation, such as U.S. Survey foot. • A survey or map deemed sufficiently near the ideal or true value to be held constant for the control of dependent operations.

5. ACCURACY VS. PRECISION • Precision – the degree of refinement in the performance of an operation (procedures and instrumentation) or in the statement of a result. • Applied to methods and instruments used to attain a high order of accuracy. • The more precise the survey method, the higher the probability that the results can be repeated.

6. ACCURACY VS. PRECISION • Survey observations can have a high precision, but still be inaccurate. • Poorly adjusted instrument • Poor methods and procedures • Instrument set up • Not checking work • Human error

7. STANDARDS • National Geodetic Control Networks are based on accuracy. • Consistent with the network not just a particular survey • Not the mathematical closure but the ability to duplicate established control values

8. TOTAL STATIONS • DIN NUMBER (DIN 18723) • DeutschesInstitutfϋr Normung • DIN accuracy is not inferred from the least count • Example of DIN use • Accuracy according to DIN of 5” in a face 1 and face 2 direction • Standard Deviation of a Face 1 and Face 2 reading is ±5” • Standard Deviation of an angle σ =√2 * 5” = 7”

9. What is a mgon? milligon 1 grad = 1,000 mgon = 54’ of arc 1 mgon = 3.24” of arc= 0.001 grad

10. TRAVERSE BY TOTAL STATION POSSIBLE SOURCES OF ERROR • READING ERRORS • SET UP ERRORS • INSTRUMENT AND REFLECTOR • POINTING ERRORS • INSTRUMENT LEVELING ERRORS • MEASUREMENT ERRORS BY EDM

11. TOTAL STATION • ESTIMATED POINTING AND READING ERROR σαpr = 2σDIN √n

12. Example: • An angle is read six times (3 direct and 3 reverse) using a total station having a published DIN 18723 value for pointing and reading of ± 5” . What is the estimated error in the angle due to pointing and reading? σαpr = 2 * 5” = ± 4.1” √6

13. TARGET CENTERING ERRORS • Setting a target over a point • Weather conditions • Optical plummet • Quality of optical plummet • Plumb bob centering • Personal abilities • Others? • Usually set up within 0.001’ to 0.01’

14. TARGET CENTERING ERRORS Possible variations in centering target Variation (d) maximum error

15. TARGET CENTERING ERRORS • Maximum error in an individual direction due to target decentering e = ± σd (RAD) D e = uncertainty σd=the amount of centering error at the time of pointing D= distance from the instrument center to the target.

16. TARGET CENTERING ERRORS • Two directions are required for an angular measurement σσt = σd1 + σd2 D1 D2 σσt = angular error due to target centering σd1 & σd2 = target center errors at sta. 1 & 2 2 2

17. TARGET CENTERING ERRORS • σσt= ± (D1)2 + (D2)2 σt ρ D1D2 ρ= 206,264.8”/radian Assumes ability to center the target is independent of the particular direction. This makes σ1 = σ2 = σt

18. TARGET CENTERING ERRORS

19. TARGET CENTERING ERRORS

20. INSTRUMENT CENTERING ERRORS • Set-up location vs. True Location • Dependent • on quality of instrument • State of adjustment of optical plummet • Skill of observer • Can be compensating • Error is maximized when the individual setup is on the angle bisector.

21. INSTRUMENT CENTERING ERRORS

22. INSTRUMENT CENTERING ERRORS

23. INSTRUMENT CENTERING ERRORS • σαi2 = ± D3 σiρ D1D2√2 ρ = 206,264.8”/radian

24. INSTRUMENT CENTERING ERRORS

25. EFFECTS OF LEVELING ERROR • If instrument is not level, then its vertical axis is not vertical and the horizontal circle is not horizontal • Errors are most severe when backsight and or foresight is steeply inclined. • Error tends to be random

26. EFFECTS OF LEVELING ERROR σαl = ± fdμtan (vb)2 + fdμtan (vf) 2 √ n

27. EFFECTS OF LEVELING ERROR σαl = ± fdμtan (vb)2 + fdμtan (vf) 2 √ n Fd= the fractional division the instrument is off level Vband vf= vertical angles to the BS and FS respectively n = the number of repetitions

28. EFFECTS OF LEVELING ERROR

36. TRAVERSING BY GPS POSSIBLE SOURCES OF ERRORS • REFERENCE POSITION ERRORS • ANTENNA POSITION ERRORS • TIMING ERRORS • SIGNAL PATH ERRORS • HUMAN ERRORS • COMPUTING ERRORS • SATELLITE CONSTELLATION ERRORS • NOISE CAUSING ERRORS

37. TOTAL STATION • POSSIBLE SOURCES OF ERROR • COLLIMATION-TO ADJUST THE LINE OF SIGHT OR LENS AXIS OF AN OPTICAL INTRUMENT SO THAT IT IS IN ITS PROPER POSITION RELATIVE TO OTHER PARTS OF THE INSTRUMENT.

38. COLLIMATION MAIN MIRROR INSTRUMENT EYE PIECE

40. PARALLAX A change in the apparent position of an object with respect to the reference marks of an instrument which is due to imperfect adjustment of the instrument, to a change in the position of the observer, or both.

41. HUMAN ERRORS • Measuring the height of the instrument and reflector. • Setting up the instrument and reflector • Push the tripod shoes firmly into the ground • Place the legs in positions that will require minimum walking around the setup. • Ensure the instrument is set properly over the point.

42. HUMAN ERRORS • Check the optical plummet after the instrument is set up and just before moving to another point. • Recheck the instrument level

43. ACCURACY OF A GPS SURVEY ACCURACY DEPENDENT UPON MANY COMPLEX, INTERACTIVE FACTORS, INCLUDING • OBSERVATION TECHNIQUE USED, e.g., static vs. kinematic, code vs. phase, etc. • Amount and quality of data acquired • GPS signal strength and continuity • Ionosphere and troposphere conditions • Station site stability, obstructions, and multipath

44. ACCURACY OF A GPS SURVEY • Satellite orbit used, e.g., predicted vs. precise orbits • Satellite geometry, described by the dilution of precision (DOP) • Network design, e.g., baseline length and orientation • Processing methods used, e.g., double vs. triple differencing, etc.

45. OPERATIONAL PROCEDURES IDENTIFY AND MINIMIZE ALL ERRORS BY REDUNDANCY, ANALYSIS, AND CAREFUL OPERATIONAL PROCEDURES, INCLUDING: • REPETITION OF MEASUREMENTS UNDER INDEPENDENT CONDITIONS • REDUNDANT TIES TO MULTIPLE, HIGH-ACCURACY CONTROL STATIONS • GEODETIC GRADE INSTRUMENTATION, FIELD AND OFFICE PROCEDURES

46. OPERATIONAL PROCEDURES • ENSURE PROCESSING WITH THE MOST ACCURATE STATION COORDINATES, SATELLITE EPHEMERIDES, AND ATMOSPHERIC AND ANTENNA MODELS AVAILABLE. CAUTION: BE AWARE THAT THESE PROCEDURES CANNOT DISCLOSE ALL PROBLEMS.