Limits of static processing in a dynamic environment Matt King, Newcastle University, UK

Static Processing Good for these examples

Static Processing But what about this? Detrended 5 min positions Whillans Ice Stream

Background Common GPS processing approaches in glaciology Kinematic approach Antenna assumed moving constantly Coordinates at each and every measurement epoch Kinematic solutions often difficult due to long between-site differences Quasi-static approach Antenna assumed stationary for certain periods (~0.5-24h) 24h common for solid earth <4h common for glaciology But is this always valid?

GPS Data Processing Approaches Quasi-static Kinematic Quasi-static assumption is that site motion during each session is “averaged out” ~0.5-24h White noise or random walk model

Motion and Least Squares Functional model Should fully describe the relationship between parameters X and observation l with normally distributed residuals v F(X)=l + v Stochastic model Can attempt to mitigate or account for functional model deficiencies Unmodelled (i.e., systematic) errors will propagate according to the geometry of the solution Station-satellite geometry Estimated parameters (e.g., undifferenced “Precise Point Positioning” solutions vs double-differenced; ambiguity fixed vs ambiguity float)

Systematic Error Propagation Estimated parameters Station coordinates (X,Y,Z) AND real-valued phase ambiguity (N) parameters Clock errors differenced out (in double difference solutions) Once ambiguities estimated, statistical tests applied to fix to integers Fixing not always possible Site motion could induce incorrect ambiguity fixing

Real vs Imaginary: Example on the Amery Ice Shelf GAMIT 1hr quasi- static solutions Track Kinematic solution King et al., J Geodesy, 2003

What’s happening? Presence of motion during ‘static’ sections Violates least-squares principle of normal residuals Leads to biased parameter estimates Simulation How does a ~1m/day signal and ~1m tidal signal in 1 hr ‘static’ solutions propagate into the parameters? Real broadcast GPS orbits Precise Point Positioning approach simulated Site ~70S

What’s happening? Latitude East (m) North (m) Height (m) Ambiguity (m) Ambiguity estimates mapped Ambiguities fixed Ambiguities not fixed Satellites East of site

Horizontal Motion Only GAMIT 1h solutions over modified “zero” baseline ~0°N ~90°S Period related to satellite pass time?

Horizontal Motion Only Simulation – grounded case How does a ~1m/day signal 1 hr ‘static’ solutions propagate into the parameters? Various flow directions (N, NE, E) 1hr solutions Various latitudes Site ~70S

What’s Happening? North (m) East (m) Height (m) Ambiguities not fixed Ambiguity estimates mapped Ambiguities fixed King et al., J Glac., 2004

Whillans Ice Stream Based on simulation would expect Agreement during ‘stick’ Biases during ‘slip’ But not in kinematic solutions 4hr quasi- static solutions 5min kinematic solutions

Solid Earth Issues Propagation of mis/un-modelled periodic signals (e.g., ocean tide loading displacements) in 24h solutions Well described by Penna & Stewart (GRL, 2003) and Penna et al., JGR, Admittances in float ambiguity PPP solutions >120% in worst case (S 2 north component into local up) Depends on coordinate component of mismodelled signal & frequency & “geometry” Output frequencies depend on input frequency Annual, semi-annual and fortnightly, amongst many others

Periodic Signals Penna et al., JGR, 2007 mm

Effect in real data King et al, GRL, 2008

Conclusions Biases may exist in positions on moving ice from GPS Up to 40-50% of unmodelled vertical signal Up to ~10% of unmodelled horizontal signal May be offsets, periodic signals or both in east, north and height components Height biases of concern when validating Lidar missions Periodic signals may result in wrong interpretation as tidal modulation (or contaminate real tidal modulation) To measure bias-free ice motion using GPS Fix ambiguities to correct integers (not always possible) Use kinematic solution (may require non-commercial software) For 24h solutions Periodic signals propagate Other sub-daily signals (e.g., multipath) need further study