A quick GPS Primer (assumed knowledge on the course!)

ObservablesError sourcesAnalysis approachesAmbiguities

If only it were this easy…

Review of GPS positioning

Dealing with errors• Clock errors (review)• Ionosphere (review)• Troposphere (part review)• Earth body deformations (new)• Orbit errors (new)

Orbit ErrorClock ErrorEpsilon (SA)Dither (SA)

Ionosphericrefraction

Troposphericrefraction

MultipathReceiver Noise

Clock Error

GPS Undifferenced observable

other errorsS

Rc dt dt N I T Orbit

Station A

Satellite j

Observed range

True range

Receiver and Satellite clock errors (multiplied by speed of light)

Carrier phase ambiguity

Ionospheric Delay

Tropospheric Delay

Includes Multipath

A somewhat simplified view, but all these need to be dealt with (at least) for precise GPS geodesy

Dealing with clock errors

Undifferenced observable• Estimate both receiver and satellite clocks• Precise Point Positioning – Fix prior satellite clocks and estimate only

receiver clocks• Parameter hungry

Double-differenced observable• Undifferenced observations to

two satellites at two stations• Form two between-station

differences and then double-difference:

• Common clock terms difference

Station A Station B

Satellite j Satellite k

other errors

N I T

Orbit

Dealing with orbit errors

These days somewhat easy• Use the IGS final orbits (precise to 2-5cm)• Use Rapid or Ultra-rapid if quick turnaround needed (precise to ~5cm)• Probably no reason to use the broadcast orbits (precise to ~0.5-2m)

In practice• Need orbits from adjacent days when processing against the day

boundary• Orbit errors are rarely an error source when using IGS products (main

exception is pre-IGS data – earlier than 1994)

Dealing with the Tropospheric Delay (I)

Total delay • ~2.3m at zenith, greater at horizon• Elevation angle dependency may be relatively well modelled with a

mapping function (M) for each of two tropospheric components

Two components• Hydrostatic – could be well modelled with accurate pressure• Wet – not well modelled and must be parameterised• Over very short (<<10km) and small elevation difference (<100-200m)

baselines, effect cancels in double-difference

General approach• Model hydrostatic with standard pressure or (more accurate) use

ECMWF or station met data• Parameterise zenith wet delay (Twet), which also absorbs any residual

Thydro , once per 1-2 h (static) or every epoch (kinematic)

. ( ) . ( )Slant hydro hydro wet wetT T M El T M El

Dealing with the Tropospheric Delay (II)

Troposphere is not azimuthally uniform• Horizontal gradients are common, particularly N-S

Highest precision static processing will further estimate horizontal gradient terms

• 1-2 for each E-W and N-S per day common

In kinematic analysis, steps in estimated tropospheric zenith delay suggest likely wrong ambiguity fixed and hence quality control

Dealing with Ionospheric Delay (I)

Different frequency signals (in L-band) delayed by different amounts through Ionosphere

• Dual frequency GPS receivers allow 99.9% for effect to be removed• Higher order terms may be important for most precise geodetic work

Use a linear combination of L1 and L2 measurements to form new measurement ionosphere free combination for carrier (LC or alternatively L3)

Where are frequency of the L1 and L2 carrier phase signals

2

1 1 11

2

2 2 22

1( ) other errors

1( ) other errors

L L LL

L L LL

fcycles f N I

c f c

fcycles f N I

c f c

1 2,L Lf f

Dealing with Ionospheric Delay (II)

Differencing and re-arranging cancels I term

Ionosphere-free phase Linear Combination LC is defined:

Note: • Ambiguity terms are no longer integers – ambiguity fixing is not an option with

LC• Noise (“other errors”) is scaled up

General approach• Adopt LC for baselines >~10km• Fix ambiguities, where possible, using a different linear combination (e.g.,

wide-lane) then final solution using LC, holding ambiguities fixed

2 22 1 2 1

1 2 1 1 22 2 2 21 1 2 1 1 2

( ) ( ) other errors ( )L L L LL L L L L

L L L L L L

f f f fcycles cycles f N N cycles

f f f c f f f

1 1 2

1

2

( ) ( ) ( ) ( ) ( ) ( )0

( )0

AB AB AB

AB

AB

AB

ijAB

ikAB

F x F x F x F x F x F x

X Y Z T N N

d X

d Y

d Z

dT

F x

T d N

d N

Matrix Form

Static case – solving for parameters x

1-4hrs

A x = b + V

Obs1

Obsn

Multipath

Generally dealt with through • Stochastic model by assumption of elevation-dependence and down-

weighting lower elevation observations (GAMIT examines the residuals and allows iterative reweighting on a station-by-station basis)

• Assuming to “average toward zero” over 24h sessions• Possibly a blind spot in GPS geodesy today

Ambiguity Fixing

Ambiguity for each satellite pass and all cycle slips thereafter• Dozens of ambiguity terms for a 24 h period

Ambiguity fixing process is essentially a series of statistical tests• Can each ambiguity be confidently (given it’s uncertainty) be fixed to

an integer?• Iteration required, since uncertainties will change (normally reduce) as

ambiguities are fixed and removed from the least squares parameters set

Essential for kinematic (or stabilisation of real-valued estimates in, e.g., Kalman Filter such as in Track)

• Not always possible to fix all ambiguities

Less impact for static• Largest effect (normally <10mm) in E, then N & U (see Blewitt, 1989)• Can change the way systematic errors propagate

Double Difference vs PPP

Similar precision possible in 24 h solutions

Software• Few software do geodetic PPP (GIPSY mainly)• GAMIT/Track are Double Difference

PPP is requires extra care• modelling geophysical phenomena (e.g., ocean tide loading

displacements) which may be (partially) differenced in relative analysis• orbit/clock errors (some periodic) map 1:1 into positioning

Kinematic PPP requires longer periods of data – ambiguity fixing is not possible without a double difference second step

DD is more precise when short-baseline relative motion is all that is required (e.g., glacier monitoring), but depends on base station

Further Reading

Reference Texts• Hofmann-Wellenhof, B., H. Lichtenegger, and J. Collins. 2001. Global

Positioning System: theory and practice, Springer, Wien, 382 pp.

• Leick, A. 2004. GPS Satellite Surveying, John Wiley & Sons, New York, 435 pp.

Review Paper• Segall, P., and J.L. Davis. 1997. GPS applications for geodynamics

and earthquake studies, Annual Review of Earth Planet Science, 25, 301-336