ADJUSTMENT COMPUTATIONSSTATISTICS AND LEAST SQUARES

IN SURVEYING AND GIS

PAUL WOLFCHARLES D. GHILANI

TRAVERSE CLOSURE

√ ΔX2 +ΔY2 =Distance ErrorDistance Error/ Total Distance = Error per foot

Or Error Ratio Tan -1 (ΔY / ΔX) = Angular Error (Azimuth)

ACCURACY VS. PRECISION

PRECISE BUT NOT ACCURATE PRECISE AND ACCURATE

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 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.

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

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

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 angleσ =√2 * 5” = 7”

What is a mgon? milligon1 grad = 1,000 mgon = 54’ of arc1 mgon = 3.24” of arc= 0.001 grad

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

TOTAL STATION

•ESTIMATED POINTING AND READING ERROR

σαpr = 2σDIN

√n

• 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

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’

TARGET CENTERING ERRORSPossible variations in centering targetVariation (d) maximum error

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 pointingD= distance from the instrument center to the target.

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

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

TARGET CENTERING ERRORS

TARGET CENTERING ERRORS

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.

INSTRUMENT CENTERING ERRORS

INSTRUMENT CENTERING ERRORS

INSTRUMENT CENTERING ERRORS

• σαi2 = ± D3 σi ρ

D1D2 √2

ρ = 206,264.8”/radian

INSTRUMENT CENTERING ERRORS

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

EFFECTS OF LEVELING ERROR

σαl = ± fdμ tan (vb) 2 + fdμ tan (vf) 2

√ n

EFFECTS OF LEVELING ERROR

σαl = ± fdμ tan (vb) 2 + fdμ tan (vf) 2

√ n

Fd = the fractional division the instrument is off levelVb and vf = vertical angles to the BS and FS respectivelyn = the number of repetitions

EFFECTS OF LEVELING ERROR

TRAVERSING BY GPSPOSSIBLE SOURCES OF ERRORS

• REFERENCE POSITION ERRORS• ANTENNA POSITION ERRORS• TIMING ERRORS• SIGNAL PATH ERRORS• HUMAN ERRORS• COMPUTING ERRORS• SATELLITE CONSTELLATION ERRORS• NOISE CAUSING ERRORS

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.

COLLIMATION

INSTRUMENTEYE PIECE

MAIN MIRROR

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.

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.

HUMAN ERRORS

• Check the optical plummet after the instrument is set up and just before moving to another point.

• Recheck the instrument level

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

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.

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

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.