Geomatics / surveying III course: Module 1 refraction, Module 2 heighting

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30 Lectures , some AssignmentsRecommended Texts:Electronic Surveying instruments :ReugerHydrography for the Surveyor and Engineer : Ingham (rev Abbot)IHO Manual of Hydrography+ many other texts

APG3017DSURVEYING IIIAssessmentAssignments 20%Refraction in Heighting - tut Precise Levelling - tut Hydrographic Survey - site visitTests 20%

Exam end of year 60%

Why Surveying III?Observations in the real world:Reduce to plane/ellipsoidNot in a vacuum refractionGravityDetail of instrumentation, errors, operations etc.OffshoreAdvanced instruments eg gyrotheodolites

Theory of Atmospheric RefractionHeightingMeasurement with Electronic TheodolitesElectronic Distance MeasurementIntroduction to Hydrographic SurveyingSpecialised Instrumentation and Techniques Outline of ModulesRetardation of signalsCurvature of light pathRefraction in GPS .... Surveying IIModule 1: Refraction5RETARDING affects distance measurementBENDING affects direction and distance measurement

Electromagnetic Spectrum:Visible and infrared: 0.5mm < l < 1.0mmMicrowave: 5mm < l < 100mm

Atmospheric RefractionElectromagnetic spectrum

from: of EM radiationVelocity of light in a vacuum: 299792.458km.s-1

If n = 1.000273 , then N = 273Fermat's principle:EM radiation follows the path that takes the least time -

Because n varies along the path, the optical path is not the geometrical path.n is a function of air density, which is in turn a function of pressure, temperature & humidityVisible & Infrared:Modulus of refraction is a function of wavelength for group velocity. For standard temperature (To=273.15K) and pressure (Po=1013.25mbar):

For other atmospheric conditions:

Humidity:Relationship to Relative Humidity (RH):

Microwave:No dispersive effect in troposphere, hence no frequency dependence:

This is only valid for a ray path that is entirely within the atmosphere. For measurements to bodies outside the atmosphere (GPS, VLBI, SLR) a different approach is followed.Magnitude of Effects:ParameterEffect on Visible/InfraredEffect on Microwave1mbar change in air pressure0.3ppm0.3ppm1 change in temperature1.0ppm1.6ppm1 change in td-tw0.05ppm8.0ppmMeasurements to extra-terrestrial bodies:GNSS, VLBI use microwaves (1.2GHz to 8GHz)SLR uses visible laserApproach is to model the Earth's atmospheric as spherical layers, and to compute the magnitude of the retarding effect in units of metres.Correction to range computed using c is given by:Microwave: -2.3secq metresLaser: -2.5secq metres(both a function of P, T, e and valid for troposphere only)

Curvature of Light Path:Affects:Trigonometrical heightingLevellingEDMCurvature in trigonometrical heighting

Curvature:Now:Hence:Curvature - 2:Substituting for ds and dz in the expression for curvature:Snell's Law: nsinz = constantDifferentiating: + sinz.dn = 0Hence:

Now we need to find an expression for the vertical gradient of the refractive index nCurvature - 3:Using No = 293 andRemembering that n = 1 + N.10-6, we can use the formula for N to get:

We then get:

S-sdzdLsSPQWTotal refraction angle:

But, from the definition of curvature:Hence:

and:Total refraction angle - 2:

If the total refraction angle is:we just need to substitute for P, Td and to get W

This is not practical, and we need to make further approximationsand: Total refraction angle - 3:

If the total refraction angle is:If we use the coefficient of refraction,

Then: we may further approximate this by setting the radius of curvature s to be a constant r. Then: (R = radius of Earth)Curvature in levelling:

bfebefProvided the temperature gradient is the same, eb = ef and the refraction error cancels with equal sight lengthsCurvature in levelling - 2: On a hill, the temperature gradient is steeper close to the ground:

ef > eb , hence (b f) is too small and hill appears to be too lowCurvature in levelling - 3: The correction can be modelled as a function of the vertical derivative of the temperature gradient:Here:

can be deduced from temperature measurements on the stavesCurvature in EDM: Major effect of refraction on EDM is retardationBut curvature of light path also causes a change in path length:desired path length is d1 , but measured path length is doddPQ01Curvature in EDM - 2: Approximating the ray path by an arc of a circle:Using a series expansion:rr2d12d1qqdo

Curvature in EDM - 3: Optical coefficient of refraction: k = 0.13Microwave coefficient of refraction: k = 0.25 The EDM curvature is flatter than that of the Earth, so the ray passes through different layers of the atmosphere:

Correction:Combined:Specifications for height networksDesigning level networksSelecting a heighting datum and systemPrecise levellingAccuracy estimates for levelling and adjustmentSpecial techniques of heighting

Module 2: Heighting28Heighting Why do we need height networks?Control for mappingControl for engineering projectsDeformation measurementsReduction of distances to the ellipsoidReduction of gravity observations to the geoidStudy of variations in Mean Sea Level

Specifications OrderRelative AccuracyProcedureFirst0.5 - 1.0mm per kmgeodetic level with parallel plate micrometer or digital level; invar staves; double run levellingSecond1.0 - 3.0mm per kmas above, sometimes with only single run levellingThird3.0 - 8.0mm per kmordinary, digital or geodetic level (no micrometer); single run levellingFourth8.0 - 40mm per kmordinary level or trigonometrical height traverse or GPS heightingLevel network design nodal pointtide gaugeClosed loopsConnection to tide gaugesRoute based upon convenience and economic needSouth African levelling networks

Vertical datumIdeally the geoid. Practically MSL, measured using tide gauges.Use a single tide gauge possible bias wrt geoid, discrepancies at other gaugesUse all tide gauges and force levelling network to fit distortions?Use all tide gauges, and allow levelling network to "float" to best fitSouth African levelling datum Land Levelling Datum (LLD):Forced fit to four tide gauges: CPT, PLZ, ELN, DBN15cm to 20cm below current MSLHeight systemOrthometric heights: Normal heights:

Spheroidal orthometric heights:

Dynamic heights:

NB: these formulae are NOT exactBenchmarksFundamental benchmarks: 50km to 100km spacing; on bedrock; below ground; with reference benchmarks

Main benchmarks: 8km to 12km spacing; on bedrock where possible; with reference benchmarks

Ordinary benchmarks: 1km to 3km spacing; on bedrock, in bridges, culverts or concrete foundations

Tide Gauge benchmarks (TGBM): adjacent to tide gauge; frequent re-levelling of connection.

Precise levelling - levelsHigh magnification (30x to 50x); high sensitivity ( < 0.5")High accuracy (under ideal conditions: 0.3mm per km;practically: 1mm per km)Optical levels with parallel plate micrometers (obsolete)Digital levels with bar coded stavesPairs of invar staves

Digital Levels

Use of barcoded stavesImage of portion of staff captured by CCD inside the levelImage matching enables precise determination of position of crosshair on staffCan also determine distance to staff, to a few cmsQuick, accurate (no reading error), range of up to 100mNeeds good lighting conditions

Automatic Compensators

Sources of Error

Earth curvatureCollimationOvercompensation

Symmetrical refractionChange in height of collimation axis due to change of focusMitigationEqual length foresight & backsightEqual length backsight & foresightProvided instrument is level, equal length backsight & foresightEqual length foresight & backsightEqual length foresight & backsight will eliminate the need to change focus

Sources of Error & their Mitigation

Refraction changing uniformly with time: Use two staves (A & B) and observe A, then B, then B, then A;Compute two height differences and take mean.Steadily sinking (or rising) instrument or staff (single setup):Use two staves (A & B) and observe A, then B, then B, then A;Compute two height differences and take mean.Maladjustment of circular bubble:Level up pointing in alternate directions i.e always pointing towards same staff (e.g. staff A). Only works for even number of setups and with each pair of setups having similar sightlengths.Non-verticality of staff: Use even number of setups, with each pair of setups having similar sightlengths.Sources of Error & their Mitigation

Heating of the instrument (causes changes in collimation):Shield with an umbrella.Index error of staves:Use same staff on every benchmark i.e. even number of setups.Scale error in staves:Calibrate staves (calibrate entire system for digital level).Non-symmetric refraction:Never observe below 0.5m on the staffUse short sight lengthsMeasure temperature gradient and apply correctionEarths Magnetic FieldDaily checking of collimation error(adjust if greater than 10")Weekly checking of circular bubbleWeekly checking of staff bubblesAnnual calibration of stavesObservation procedures: ChecksObservation proceduresEqual sight lengths for forward and back (within 1m)Short sight lengths ( < 60m, but < 30m on long shallow inclines )Observe higher than 0.5m on the staffSame staff (A) on benchmarks (equal number of setups)Level up while pointing to the same staff (A) every timeAlways observe the same staff (A) firstObserve staves in the sequence: ABBA(check that agreement is better than 0.5mm)Observation procedures - 2Keep instrument shadedUse support rods for staves (do not lean on them), and place footplates securelyLevel at the time of day when shimmer is least (early morning)Level each section in both directions, at different timesCorrections to levellingScale correction: Requires determination of scale factor for staves Conventional invar staves are calibrated using a laser interferometer (see later) and either an average scale factor or a scale factor per graduation are applied.

Bar-coded staves must be calibrated together with the digital level, and the process requires (in addition to a laser interferometer) a special carriage to move the staff vertically through the field of view of the level.Corrections to levelling - 2Temperature correction:Not normally applied, as invar is not temperature sensitive.Non-symmetric refraction:Requires measurement or modelling of temperature gradient and its derivative at every setup. Seldom done.

Corrections to levelling - 3Earth Tides:Attraction of sun and moon causes small changes in direction of gravity and hence in direction of collimation axis (effect is 0.02").Generally ignored (also indirect effect of ocean loading).Few cm over 1000km levelingInfluence of Gravity: Spatial variations in gravity cause changes in shape of equipotential surfaces. Derivation is covered in Geodesy course. Effect is of the order of cms over 100km.Error propagation in levellingAssuming random tilt error sa and equal sight lengths s, then the error in the backsight is sb = and the error in the foresight is sf = The error in the height difference is:For a line of levelling of length L:


If all sight lengths are the same ( = s) then:and:

But if all sight lengths are equal to s:



(error in a section of levelling is proportional to the square root of the length of the section)Error propagation in levelling - 2

Precision estimatess1 (standard deviation per root km) can be determined as follows:Manufacturer's specifications: generally too optimisticFrom comparison of forward and back levelling over sections:

From loop closures:

Precision estimates - 2s1 derived from loop closures is generally larger than that from comparison of forward and back levelling.This implies that errors do not propagate strictly according to LIf L is larger, then s1 is largerThis further implies that: where a > 0.5

Precision estimates - 3Alternative model (Vignal), assumes that errors propagate randomly, but with different values for s1, depending upon L:

L < 5km, s1 = h

5km < L < Z, s1 = tL

L > Z, s1 = tIn South Africa: h = 0.6mm, Z = 25km, t = 1.1mm (optimistic)Level network adjustmentLeast squares adjustment by parametric or condition equationsFree network or minimum constraint or constrainedWeighting of observations:weight inversely proportional to distance: weight inversely proportional to variance:variance based upon:

or Vignal's approachValley crossingReciprocal Levelling:

At A: DHA = (bA + eS + rS +cS) - (fA + eL+S + rL+S + cL+S)At B: DHB = (bB + eL+S + rL+S + cL+S) - (fB + eS + rS + cS)Reciprocal levellingCollimation error can be eliminated if the two levels are swopped and measurements repeatedCollimation error can be eliminated if a single level is used, provided refraction conditions do not change between setupsOr, specialised pairs of levels can be used, with autocollimation

from: F. Deumlich Surveying InstrumentsReciprocal levellingNeed special targets, as staff graduations cannot be resolved beyond about 100m range:

Or, with digital level, use enlarged version of bar code

from: zenith angles

Where the height differences are more than a few metres, or the range exceeds 2km, reciprocal levelling cannot be used.Reciprocal zenith angles, with simultaneous observations, provide an alternative method, albeit with a loss of accuracy:Reciprocal zenith angles

Reciprocal zenith angles

(basic equation of simultaneous reciprocal zenith angles)Note that the distance s is the distance on the ellipsoidError propagation

If we differentiate the trigonometrical heighting equation, we get the model for error propagation:Provided that z is close to 90, the effect of a distance error on the height difference is small. E.g., for z = 80 and ss = 10mm, the effect is 2mm. The effect of errors in the coefficient of refraction is much larger. Putting estimates of ss = 10mm, sa = 2" and sk = 0.04, we get the simplified expression (assuming z equal to 80):

Error propagation - 2

DistanceEffect of saEffect of skCombined effect1km10mm3mm11mm2km20mm12mm24mm3km30mm28mm41mm5km50mm79mm93mmError propagation - 3

Hydrostatic levellingUse undisturbed water to define a level surfaceWater contained in a flexible tube, with vertical glass tubes with scales at each end:Connected to benchmarks at each endCorrections for temperature and pressureHydrostatic levelling - 2Time comsuming, but can achieve high accuracy over long distances (a few mm over 10km)Used to cross water bodies (pipe laid under water) e.g. in Denmark and HollandOn a smaller scale (20m to 100m) used for deformation monitoring in structures (see APG4005F)Hydrodynamic levellingUses the open sea as a "level"Tide gauges at each end eliminate/reduce the effects of waves, seiches and tidesEffects of currents, winds and differential air pressure must be modelled and correctedUsed to connect Uk and French level networks across the English Channel estimated accuracy of 1.5cm over 70kmTrigonometrical height traversingAn extension of the simultaneous reciprocal vertical angle approach, using two electronic tacheometers:zzABS

S is the slope distance, and DHAB is the height difference between the collimation axes of the two telescopes Trigonometrical height traversingConnection to benchmark involves a single staff and reading of zenith angles to a minimum of two graduation lines:

z1z2h12Trigonometrical height traversingResidual refraction effects (k not the same at both ends of the line) means that sight lengths should be kept to less than 250mAccuracy of 3 5mm per km can be achievedFaster than conventional levelling: 2km per hour vs 0.5km per hourMore flexible can traverse rough, steep terrainMotorised levellingCan be used to speed up levelling (conventional, digital, or trigonometrical height traversing).Requires up to three vehicles, which must be specially modified.Can improve speed of conventional levelling by up to 50%









Distance (kms)Error (mms)



Distance (kms)Error (mms)