≈ 30 Lectures , some Assignments

Recommended Texts:

Electronic Surveying instruments : Reuger

Hydrography for the Surveyor and Engineer : Ingham (rev Abbot)

IHO Manual of Hydrography

+ many other texts

APG3017D

SURVEYING III

Assessment

Assignments 20%

Refraction in Heighting - tut

Precise Levelling - tut

Hydrographic Survey - site

visit

Tests 20%

Exam end of year 60%

Why Surveying III?

Observations in the real world:

- Reduce to plane/ellipsoid

- Not in a vacuum – refraction

- Gravity

Detail of instrumentation, errors, operations etc.

Offshore

Advanced instruments – eg gyrotheodolites

1. Theory of Atmospheric Refraction

2. Heighting

3. Measurement with Electronic Theodolites

4. Electronic Distance Measurement

5. Introduction to Hydrographic Surveying

6. Specialised Instrumentation and

Techniques

Outline of Modules

Retardation of signals

Curvature of light path

Refraction in GPS .... Surveying II

Module 1: Refraction5

RETARDING – affects distance measurement

BENDING – affects direction and distance

measurement

Electromagnetic Spectrum:

Visible and infrared: 0.5mm < l < 1.0mm

Microwave: 5mm < l < 100mm

cf e.g. for 100mm, f 3GHz

Atmospheric Refraction

Electromagnetic spectrum

from: http://www.lbl.gov/MicroWorlds/ALSTool/EMSpec/EMSpec2.html

EDM

InfraredGNSS

Reflectorless

Retardation of EM radiation

Velocity of light in a vacuum: 299792.458km.s-

1 cRefractive Index: n

v

6Modulus of Refraction: N n 1 10

If n = 1.000273 , then N = 273

Fermat's principle:

EM radiation follows the path that takes the

least time -1 n 1

dt min , or: ds ds n ds minv c c

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 &

humidity

Visible & Infrared:

Modulus of refraction is a function of wavelength

for group velocity. For standard temperature

(To=273.15K) and pressure (Po=1013.25mbar):

0 2 4

4.8864 0.068N 287.604

For other atmospheric

conditions:

o

d d

P eN 0.269578N 11.27

T T

Humidity:

Relationship to Relative Humidity

(RH):

d w we E 0.00066P t t 1 0.00115t mbars

w

24509.73

TE 10 mbars

a

eRH 100

E

d

24509.73

T

aE 10 mbars

Microwave:

No dispersive effect in troposphere, hence no

frequency dependence:

6

2

d d d

P e eN 77.624 0.372 10 12.92

T T T

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:

Parameter Effect on

Visible/Infrared

Effect on

Microwave

1mbar change

in air pressure

0.3ppm 0.3ppm

1° change in

temperature

1.0ppm 1.6ppm

1° change in td-

tw

0.05ppm 8.0ppm

Measurements to extra-

terrestrial bodies: GNSS, VLBI use microwaves (1.2GHz to

8GHz)

SLR uses visible laser

Approach 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 metres

Laser: -2.5secq metres

(both a function of P, T, e and valid for troposphere

only)

Curvature of Light Path:

Affects:

Trigonometrical heighting

Levelling

EDM

Curvature in trigonometrical

heighting

P

Q

z dz

z ds dh

1 dz

ds

dh

sin 90 zds

dhds

cos z

Curvature:

Now:

Hence:

Curvature - 2:

Substituting for ds and dz in the expression for

curvature:

Snell's Law: nsinz = constant

Differentiating: ncosz.dz + sinz.dn =

0

Hence:dn sin z

dzn cos z

1 1 dnsin z

n dh

Now we need to find an expression for the vertical

gradient of the refractive index n

Curvature - 3:

Using No = 293

and

Remembering that n = 1 + N.10-6, we can use the formula for N

to get:

d

dP P0.0342

dh T

6 d0 2

d d

dTdn 1 dP P0.26943 N 10

dh T dh T dh

We then get:

5 d

2

d

dT1 P7.88 10 0.0342 sin z

T dh

S-s

dzdℓ

L

s

S

P

Q

W

Total refraction angle:

L

SW L d

d S s dz

dsdz

But, from the definition of

curvature:

Hence

:

dsd S s

1 S sds

S

W

and:

Total refraction angle - 2:

5 d

2

d

dT1 P7.88 10 0.0342 sin z

T dh

ddT

dh

If the total refraction angle is:

we just need to substitute for P, Td and to get W

1 S sds

S

W

This is not practical, and we need to make further

approximations

and:

Total refraction angle - 3:

Sk (k generally taken as 0.13)

2RW

S

2rW

Rk

r

If the total refraction angle is:

If we use the coefficient of refraction,

1 S sds

S

W

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:

5 d

2

d

dT1 P7.88 10 0.0342 sin z but z 90

T dh

b f

eb ef

Provided the temperature gradient is the same, eb = ef and the

refraction error cancels with equal sight lengths

Curvature 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 low

Curvature in levelling - 3:

The correction can be modelled as a function of the vertical derivative

of the temperature gradient:

Here:

22

2

d Ts dn de e e f bf b 6 dT dh

d

5

2

d d

dn P7.88 10

dT T

2

d

2

d Tand

dh

can be deduced from temperature measurements on the

staves

Curvature in EDM:

Major effect of refraction on EDM is retardation

But curvature of light path also causes a change

in path length:

desired path length is d1 , but measured path length

is do

d

d

P

Q0

1

Curvature in EDM - 2:

Approximating the ray path by an arc of a circle:

Using a series expansion:r

r

2d1 2

d1

q q

do

1d 2r sin q

oo

dd 2 r

2r q q

o1

d d 2r sin

2r

3

o1 o 2

dd d

24r

3 2

o1 o 2

d .kRSubstituting r , we get d d

k 24R

Curvature in EDM - 3:

Optical coefficient of refraction: k = 0.13

Microwave 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:

23

0 2

k kd ' d

12R

23

0 2

2k kd d

24R

Correction: Combined:

Specifications for height networks

Designing level networks

Selecting a heighting datum and system

Precise levelling

Accuracy estimates for levelling and adjustment

Special techniques of heighting

Module 2: Heighting28

Heighting

Why do we need height networks?

Control for mapping

Control for engineering projects

Deformation measurements

Reduction of distances to the ellipsoid

Reduction of gravity observations to the geoid

Study of variations in Mean Sea Level

Specifications

Order Relative Accuracy Procedure

First 0.5 - 1.0mm per √km geodetic level with parallel plate micrometer or

digital level; invar staves; double run levelling

Second 1.0 - 3.0mm per √km as above, sometimes with only single run

levelling

Third 3.0 - 8.0mm per √km ordinary, digital or geodetic level (no

micrometer); single run levelling

Fourth 8.0 - 40mm per √km ordinary level or trigonometrical height traverse

or GPS heighting

Level network design

nodal point

tide gauge

Closed loops

Connection to tide gauges

Route based upon convenience and economic

need

South African levelling

networks

Vertical datum

Ideally – the geoid. Practically – MSL, measured using tide

gauges. Use a single tide gauge – possible bias wrt geoid,

discrepancies at other gauges

Use 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, DBN

15cm to 20cm below current MSL

Height system

Orthometric heights:

Normal heights:

Spheroidal orthometric

heights:

Dynamic heights:

dnH g

g

dnH g

dnH

dnH g

G

NB: these formulae are NOT exact

Benchmarks

Fundamental 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

Precise levelling - levels

High 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 staves

Pairs of invar staves

Digital Levels

Use of barcoded staves

Image of portion of staff captured by CCD

inside the level

Image matching enables precise

determination of position of crosshair on staff

Can also determine distance to staff, to a few

cms

Quick, accurate (no reading error), range of

up to 100m

Needs good lighting conditions

Automatic Compensators

Sources of

Error Earth curvature

Collimation

Overcompensation

Symmetrical

refraction

Change in height of

collimation axis due

to change of focus

Mitigation

Equal length foresight &

backsight

Equal length backsight &

foresight

Provided instrument is level,

equal length backsight &

foresight

Equal length foresight &

backsight

Equal 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 staff

Use short sight lengths

Measure temperature gradient and apply correction

Earths Magnetic Field

Daily checking of collimation error

(adjust if greater than 10")

Weekly checking of circular bubble

Weekly checking of staff bubbles

Annual calibration of staves

Observation procedures: Checks

Observation procedures

Equal 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 staff

Same staff (A) on benchmarks (equal number of

setups)

Level up while pointing to the same staff (A) every time

Always observe the same staff (A) first

Observe staves in the sequence: ABBA

(check that agreement is better than 0.5mm)

Observation procedures - 2

Keep instrument shaded

Use support rods for staves (do not lean on them), and

place footplates securely

Level at the time of day when shimmer is least (early

morning)

Level each section in both directions, at different times

Corrections to levelling

Scale 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 - 2

Temperature 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.

22

2

d Ts dn de e e f bf b 6 dT dh

d

5

2

d d

dn Pwhere: 7.88 10

dT T

Corrections to levelling

- 3 Earth 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 leveling

Influence 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

levelling

Assuming random tilt error a and equal sight lengths s, then the error in

the backsight is b = s.a and the error in the foresight is f = s.a. The error in

the height difference is:

For a line of levelling of length L:

s s

b f

a

2 2 2 2 2

h b f 2s a

1 2 3 n

2 2 2 2 2

L h h h h....

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

and:

1 2 3 n

2 2 2 2

h h h h.....

2 2 2

L n.2s a

But if all sight lengths are equal to

s:

Ln

2s

Therefore:2 2 2 2

L

L.2s L.s.

2sa a

L s. . La Hence:

L 1 L

(error in a section of levelling is proportional to the square root of the length of the

section)

Error propagation in levelling -

2

1 k s a

Precision estimates

1 (standard deviation per root km) can be determined as follows:

Manufacturer's specifications: generally too

optimistic

From comparison of forward and back levelling over

sections:

From loop closures:

2

nf b2 i

1

i 1 i

n n1

4n L

2n2 i1

i 1 i

1 h

n L

Precision estimates - 2

1 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 √L

If L is larger, then s1 is larger

This further implies that: where a >

0.5 L 1 La

Precision estimates - 3

Alternative model (Vignal), assumes that errors

propagate randomly, but with different values for s1,

depending upon L:

L < 5km, 1 = h

5km < L < Z, 1 = tL

L > Z, 1 = t

In South Africa: h = 0.6mm, Z = 25km, t = 1.1mm (optimistic)

Level network adjustment

Least squares adjustment by parametric or

condition equations

Free network or minimum constraint or constrained

Weighting of observations:

weight inversely proportional

to distance:

weight inversely proportional

to variance:

variance based upon:

L

kp

L

L 2

L

kp

2

L L 1 L

L 1 La

or Vignal's approach

Valley crossing

Reciprocal Levelling:

At A: HA = (bA + eS + rS +cS) - (fA + eL+S + rL+S + cL+S)

At B: HB = (bB + eL+S + rL+S + cL+S) - (fB + eS + rS + cS)

Reciprocal levelling

Collimation error can be eliminated if the two

levels are swopped and measurements repeated

Collimation error can be eliminated if a single

level is used, provided refraction conditions do

not change between setups

Or, specialised pairs of levels can be used, with

autocollimation

from: F. Deumlich – Surveying Instruments

Reciprocal levelling

Need special targets, as staff graduations cannot

be resolved beyond about 100m range:

Or, with digital level, use enlarged version of

bar code

from: http://www.fig.net/pub/athens/papers/pdf/ts_03_5_takalo_rouhiainen_ppt.pdf

Reciprocal 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

2 2B AB A A A A B

H H 1 kAt A: H H 1 . s.cot z .s cosec z i m

2R 2R

2 2B AA B B B B A

H H 1 kAt B: H H 1 . s.cot z .s cosec z i m

2R 2R

2 2 2B AB A A B A B A B B A

Subtracting:

H H 1 k 2 H H 1 s cot z cot z s cosec z cosec z i m i m

2R 2R

Reciprocal zenith angles

2 2 2B AB A A B A B A B B A

H H 1 k2 H H 1 s cot z cot z s cosec z cosec z i m i m

2R 2R

2 2

A B A A B BNow: cosec z cosec z and m i m i

B AB A A B A B

1 H HHence: H H 1 . s. cot z cot z i i

2 2R

(basic equation of simultaneous reciprocal zenith angles)

Note that the distance s is the distance on the ellipsoid

Error propagation

2 42 2 2 2 2

H s k4 2

s scot z

sin z 4R a

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 s = 10mm, the effect is

2mm. The effect of errors in the coefficient of refraction is much larger.

Putting estimates of s = 10mm, a = 2" and k = 0.04, we get the

simplified expression (assuming z equal to 80°):

2 2 4

H

H

4 100s 9.9s

in units of mm, s in units of km

Error propagation - 2

2 2 4

H 4 94s 9.9s

Distance Effect of a Effect of k Combined

effect

1km 10mm 3mm 11mm

2km 20mm 12mm 24mm

3km 30mm 28mm 41mm

5km 50mm 79mm 93mm

Error propagation - 3

2 2 4

H 4 94s 9.9s

0

200

400

600

800

1000

1200

1400

0 2 4 6 8 10 12 14 16 18 20

Distance (kms)

Err

or

(mm

s)

Hydrostatic levelling

Use undisturbed water to define a level surface

Water contained in a flexible tube, with vertical

glass tubes with scales at each end:

Connected to benchmarks at each end

Corrections for temperature and pressure

Hydrostatic levelling - 2

Time 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 Holland

On a smaller scale (20m to 100m) used for

deformation monitoring in structures (see

APG4005F)

Hydrodynamic levelling

Uses the open sea as a "level"

Tide gauges at each end eliminate/reduce the

effects of waves, seiches and tides

Effects of currents, winds and differential air

pressure must be modelled and corrected

Used to connect Uk and French level networks

across the English Channel – estimated accuracy

of 1.5cm over 70km

Trigonometrical height traversing

An extension of the simultaneous reciprocal

vertical angle approach, using two electronic

tacheometers:

z

z

A

B

S

AB A B

SH cos z cos z

2

S is the slope distance, and DHAB is the height difference

between the collimation axes of the two telescopes

Trigonometrical height traversing

Connection to benchmark involves a single staff

and reading of zenith angles to a minimum of two

graduation lines:

2 1 1 2

1 2

cot z cot zh

cot z cot z

z1

z2

h

ℓ1

ℓ2

Trigonometrical height traversing

Residual refraction effects (k not the same at

both ends of the line) means that sight lengths

should be kept to less than 250m

Accuracy of 3 – 5mm per km can be achieved

Faster than conventional levelling: 2km per hour

vs 0.5km per hour

More flexible – can traverse rough, steep terrain

Motorised levelling

Can 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%