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

1. A quick GPS Primer (assumed knowledge on the course!) • Observables • Error sources • Analysis approaches • Ambiguities If only it were this easy…

2. Review of GPS positioning • Dealing with errors • Clock errors (review) • Ionosphere (review) • Troposphere (part review) • Earth body deformations (new) • Orbit errors (new) Orbit Error Clock Error Epsilon (SA) Dither (SA) Ionospheric refraction Tropospheric refraction Receiver Noise Multipath Clock Error

3. GPS Undifferenced observable A somewhat simplified view, but all these need to be dealt with (at least) for precise GPS geodesy Carrier phase ambiguity Tropospheric Delay True range Satellite j Includes Multipath Observed range Receiver and Satellite clock errors (multiplied by speed of light) Ionospheric Delay Station A

4. 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 Satellite j Satellite k Station B Station A

5. 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)

6. 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)

7. 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

8. 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

9. 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

10. Matrix Form • Static case – solving for parameters x A x = b + V Obs1 1-4hrs Obsn

11. 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

12. 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

13. 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

14. 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