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

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

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ACCURACY VS. PRECISION PRECISE BUT NOT ACCURATEPRECISE AND ACCURATE

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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.

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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.

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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

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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

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READING ERRORS Repetition reading instrument Repetition Method – Circle is zeroed – Reading errors σ αr = √σ o 2 + σ r 2 n σ σr - Estimated Standard Error in the average angle due to reading σ o - estimated error in setting zero σ r - estimated error in the final reading n – number of repetitions

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READING ERRORS Ability to set zero and read the circle equal σ αr = σ r √2 n

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READING ERRORS 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? σ αr = ±1.5√2 = ±0.4” 6

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

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READING ERRORS 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? σ αr = ±1.5√2 = ±0.9” √6

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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

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POINTING ERRORS Assume for any given instrument and observer the pointing error can be the same for each repetition. σ αp = σ p √2 √ n

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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? σ αp = ±1.8√2 = ±1.0” √6

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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”

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

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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

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TOTAL STATION ESTIMATED POINTING AND READING ERROR σ αpr = 2σ DIN √n

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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? σ αpr = 2 * 5” = ± 4.1” √6

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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’

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

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

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TARGET CENTERING ERRORS Two directions are required for an angular measurement σ σt = σ d1 + σ d2 D 1 D 2 σ σt = angular error due to target centering σ d1 & σ d2 = target center errors at sta. 1 & 2 22

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TARGET CENTERING ERRORS σ σt = ± (D 1 ) 2 + (D 2 ) 2 σ t ρ D 1 D 2 ρ= 206,264.8”/radian Assumes ability to center the target is independent of the particular direction. This makes σ 1 = σ 2 = σ t

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TARGET CENTERING ERRORS

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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.

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INSTRUMENT CENTERING ERRORS

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σ αi 2 = ± D 3 σ i ρ D 1 D 2 √2 ρ = 206,264.8”/radian

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INSTRUMENT CENTERING ERRORS

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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

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EFFECTS OF LEVELING ERROR σ αl = ± f d μ tan (v b) 2 + f d μ tan (v f ) 2 √ n

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EFFECTS OF LEVELING ERROR σ αl = ± f d μ tan (v b) 2 + f d μ tan (v f ) 2 √ n F d = the fractional division the instrument is off level V b and v f = vertical angles to the BS and FS respectively n = the number of repetitions

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EFFECTS OF LEVELING ERROR

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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

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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.

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COLLIMATION INSTRUMENT EYE PIECE MAIN MIRROR

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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.

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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.

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

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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

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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.

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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

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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.

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