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
REPEATABILITY TO CONFORM TO THE STANDARD
The target on the left could be acceptable if proper steps are taken to correct for the presence of systematic errors. (This is equivalent to a marksman realigning the sights after the shots.)
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.
Accuracy relates to the quality of the result obtained when compared to the standard.
The standard used to determine accuracy compared to the true value. Can the true value be determined?
No. Error is always present in all measurements. How much is determined by several factors.

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. READING ERRORS Repetition reading instrument
Repetition Method
Circle is zeroed
Reading errors
σαr = √σo2 + σr2 n
σσr - Estimated Standard Error in the average angle due to reading
σo - estimated error in setting zero
σr - estimated error in the final reading
n – number of repetitions
Standard deviation- square root of the sample variance.
68% of the time measurement will fall within this standard.
Square the Sum of the difference between the measurement and mean
Divided by the number of repetitions minus one.

9. READING ERRORS Ability to set zero and read the circle equal

10. READING ERRORS σαr = ±1.5√2 = ±0.4” 6 Example: Repetition Method
Suppose an angle is read six times using the repetition method. An operator having a personal reading error of ±1.5”, what is the estimated error in the angle due to circle reading?
σαr = ±1.5√2 = ±0.4” 6

11. READING ERRORS Direct Method σαr = σr√2 √ n
Backsight and Foresight readings
Angle is difference between to readings
Multiple measurements
σαr = σr√2
√ n
σσr - Estimated Standard Error in the average angle due to reading
σr - estimated error in the final reading
n – number of repetitions

12. READING ERRORS σαr = ±1.5√2 = ±0.9” √6 Example: Direct Method
Suppose an angle is read six times using the direct method. An operator having a personal reading error of ±1.5”, what is the estimated error in the angle due to circle reading?
σαr = ±1.5√2 = ±0.9” √6

13. POINTING ERRORS SEVERAL FACTORS AFFECT ACCURACY
OPTIC QUALITIES
TARGET SIZE
OBSERVER’S PERSONAL ABILITY TO PLACE CROSSHAIRS ON THE TARGET
WEATHER CONDITIONS
POINTING ERRORS ARE RANDOM
THEY WILL OCCUR
POINTING ERRORS NOT DEPENDENT ON INSTRUMENT.
Pointing errors occur at each back sight and fore sight, the pointing error will be the mean of the number of repetitions

14. POINTING ERRORS
Assume for any given instrument and observer the pointing error can be the same for each repetition.
σαp = σp√2
√ n

15. POINTING ERRORS Example:
Suppose an angle is read six times by an operator whose ability to point on a well-defined target is estimated to be ±1.8”, what is the estimated error in the angle due to pointing?
σαp = ±1.8√2 = ±1.0” √6

16. TOTAL STATIONS DIN NUMBER (DIN 18723) Deutsches Institut fϋr 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”
DIN is associated with statements of accuracy of theodolites since the introduction of electronic theodolites. DIN loosely translated German Institute for standards.
Least count- with the advent of electronic instruments, reliance on the least count for anything but an estimate of precision achievable (not accuracy) is highly inadvisable.

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

18. 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
Errors are always introduced in all instruments measuring distances and angles. Small or large depending on the operator, instrument and conditions at the time of measurement. Each source produce a small amount of random error.

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

20. Example:
An angle is read six times (3 direct and 3 reverse) using a total station having a published DIN 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

21. 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’
Whenever a target is set over a station, there will be some error due to faulty centering.
This type of error will appear as a random in the adjustment of a network involving many stations.

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

23. 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.
e = uncertainty in the direction due to target decentering

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

25. TARGET CENTERING ERRORS
σσt = ± (D1) (D2) 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

26. TARGET CENTERING ERRORS

27. TARGET CENTERING ERRORS
If a hand-held range pole were used in this example with an estimated centering error of +/- 0.01’, the estimated error due to target centering would be almost 10”.
This is why a range pole makes for a poor control point measuring tool.

28. 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.

29. INSTRUMENT CENTERING ERRORS
Bisector angle error maximized at b and c.

30. INSTRUMENT CENTERING ERRORS

31. INSTRUMENT CENTERING ERRORS
σαi2 = ± D σi ρ
D1D2 √2
ρ = 206,264.8”/radian

32. INSTRUMENT CENTERING ERRORS

33. 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
Random error- If the bubble of a theodolite were to remain off center by the same amount during the entire angle-measuring process at a station, the resulting error would be systematic. However, because an operator normally carefully monitors the bubble and attempts to keep it centered while turning angles, the amount and direction by which the instrument is out of level become random, and hence the resulting errors tend to be random.

34. EFFECTS OF LEVELING ERROR
σαl = ± fdμ tan (vb) 2 + fdμ tan (vf) 2
√ n

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

36. EFFECTS OF LEVELING ERROR
This error is generally small for traditional surveying when normal care in leveling the instrument is taken. Thus is can be ignored except for precise work.

44. POSSIBLE SOURCES OF ERRORS
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
REFERENCE POSITION-COORDINATE, MONUMENT STABILITY, CRUSTAL MOTION
ANTENNA POSITION-EQUIPMENT SETUP, PHASE CENTER VARIATION AND OFFSETS
SATELLITE POSITION-ORBIT EPHEMERIS ERRORS
TIMING ERRORS-SATELLITE OR RECEIVER CLOCK ERRORS
SIGNAL PATH ERRORS- ATMOSPHERIC DELAY AND REFRACTION, MULTIPATH
HUMAN ERRORS-FIELD OR OFFICE BLUNDERS
COMPUTING ERRORS-PROCESSING STATISTICAL MODELING ERRORS
SATELLITE CONSTELLATION- VDOP & PDOP & # OF SATELLITES
NOISE CAUSING-CELL PHONES, RADIOS, VEHICLES, POWER TRANSMISSION LINES, MICROWAVE SIGNALS

45. 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.

46. COLLIMATION
MAIN
MIRROR
INSTRUMENT
EYE PIECE

48. 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.
Parallax occurs when the focal point of the eyepiece does not coincide with the plane of the cross hairs. Depending on the shape of each observer’s eyeball, the focal length may vary.
This is also a major concern in the optical plummet.
To check for parallax in the telescope, focus on some well-defined distant object. Slowly move the head back and forth, about an inch from the eyepiece, while watching the relationship of the object to the cross hairs. If the object appears to move, parallax exists.
The optical plummet can be checked by rotating the knurled eyepiece until the cross hairs are the thickest and blackest, refocus and check for parallax.

49. 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.
Do not stamp on the tripod feet. Pressure should be parallel to each leg.
Legs- In windy conditions, additional stability can be achieved if one leg is set downwind.
On hillsides- one leg uphill and two legs downhill
If the ground in soft or muddy, drive hubs in the ground to support the tripod legs.

50. HUMAN ERRORS
Check the optical plummet after the instrument is set up and just before moving to another point.
Recheck the instrument level
If the instrument has moved, check the angle just measured.
The bubble should remain within one graduation when the instrument is smoothly turned through one circle (if the instrument is shaded).

51. 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

52. 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.

53. 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

54. 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.