Extract deep geophysical signals from GPS data analysis Signals and noises inside GPS solutions Network adjustment Time series analysis Search subtle signals without a priori information Search subtle signals with a priori information

Intrinsic relation among various measurements

GPS nominal constellation 24 satellites 6 orbital planes, 60 degrees apart km altitude 55 degrees inclination 12 hours period Repeat the same track and configuration every 24 hours (4 minutes earlier each day)

Signals and noises in GPS solutions

Signals Propagation medium: ionosphere, atmosphere Surface process: mass loading from atmosphere, ocean (tidal and non-tidal) and ground water, thermal expansion Underground process: plate motion, fault dislocation, creeping, co- seismic and post-seismic deformation, post-glacial rebound, aquifer undulation, magma intrusion, eruption and migration, gassing and de-gassing Frame motion: polar motion, Earth rotation, geocenter Noises Systematic: modeling errors from satellite orbit, antenna phase center, clock, ionosphere and atmosphere, solid Earth tide and ocean tide, solar radiation, etc. Local effects: multipath, benchmark instability, receiver trouble Common mode error (CME): unknown Random noise

Signals in space domain Global polar motion, Earth rotation, geocenter, sea level Continental post-glacial rebound, plate motion, micro-plate motion Regional regional deformation, mass loading from atmosphere, ocean and groundwater, tidal variations, fault dislocation and creeping, co-seismic and post-seismic deformation Local aquifer undulation, thermal expansion, transient fault motion

Signals in time domain Secular tectonic motion, orogenic process, sediment or ice sheet compacting Decadal post-glacial rebound, sea level change, post-seismic deformation Seasonal mass loading, thermal expansion, polar motion, UT1 Intra-seasonal magma intrusion and migration Short term co-seismic, transient, tidal motion

Network adjustment Adjusted parameters station positions, velocities, jumps, network rotation, translation and scale, orbits and others Pros rigorous, full covariance matrix, stable reference frame Cons cpu and storage consuming, hard to identify outlier, hard to model complex signals

Estimators Least squares Cholesky decomposition Household transformation Kalman filtering covariance matrix or normal matrix approaches Square root information filtering superior numerical stability

Reduce the burden of network analysis Eliminating uncorrelated parameters ambiguities, troposphere parameters, orbits in velocity field estimation Implicitly solving piecewise constant parameters troposphere zenith delays and gradients, ambiguities Helmert blocking for huge network adjustment, breaking up a huge single computation into many small computational tasks

Time series analysis Pros fast, save space, flexible, easy to identify outliers, better modeling more complex signals Cons neglect correlations between stations Adjusted parameters station positions, velocities, jumps, seasonal and any harmonic terms, non-linear decay terms, modulations

Example 1: LBC1 vertical Harmonic approach annual, semi-annual and 4- months harmonics modulation is modeled by 4 spline parts, each uses degree 5 polynomials Spline function approach periodic pattern is modeled by 6 spline segments, each uses cubic fit modulation is modeled by 4 spline parts, each uses degree 4 polynomials

Example 2: Miyakejima volcano eruption

Estimated parameters: new consideration Green’s function Station coordinates and velocities can be considered as the spatially uncorrelated Green’s function Geophysical source caused surface deformation Green’s function are generally spatially correlated Can we estimate the amplitude of spatially correlated Green’s function from GPS data?

Back to network analysis: taking advantage of spatial correlation Spatially correlated Green’s functions fault slip, dike intrusion, magma migration, surface and underground mass loading, post- seismic relaxation common mode error, spatially correlated systematic error Raise signal to noise ratio through network analysis separate spatially correlated signals from the data Keynote the spatial responses of individual stations are different, but their time functions are the same

Without apriori information: Principal Component Analysis (PCA) Data matrix n(epoch) x m(site) time series to construct X matrix: surface representation (usually n > m) X = U*S*V (SVD decomposition) X T X = V T *S 2 *V (PCA decomposition) Rescale X T X -> correlation matrix (Karhunen-Loeve expansion KLE), slight different eigenvectors

Geodetic interpretation of the PCA analysis PCA decomposition a k : k-th principal component (time domain) v k : k-th eigenvector (space domain) a k : k-th common time function for the network v k : network spatial responses for the k-th PC First few PC: common modes Last few PC: local modes

PCA application example: regional network filtering Stacking approach Common time function Spatial responses: uniform distribution PCA decomposition Common time function Spatial responses: determined by data themselves Potential causes: satellite orbits, reference frame

SCIGN cme

GEONET cme

With apriori information: Network filter Source position and geometry are known example: fault segments caused surface displacements q: force acting at  in q-th direction (fault frame) p: displacement at x in p-th direction (fault frame) n: normal vector to the fault area element G(x,  ): elastostatic Green’s function at x due to fault source at  s( ,t-t 0 ): fault slip time function (fault frame) d i (x,t): surface displacement at x in i-th direction (surface frame) m(t): reference frame term b(x,t): benchmark instability  (x,t): observation noise Only spatio-temporal function s( ,t-t 0 ) is unknown Can we estimate s( ,t-t 0 ) directly? Yes and No

Constraints in Network filter estimation Fault geometry should be expanded by orthogonal spatial basis functions spatial smoothing: temporal smoothing: state perturbation noise minimum norm constraint positivity constraint

Aseismic slip along the Hayward fault

Deal with highly correlated parameters Single satellite vs. all spherical harmonics of the gravitational field Ocean tide with all side bands Phase center variations with vertical coordinate and troposphere parameters Scaled sensitivity matrix approach Challenge for conventional statistics

More challenges ahead Moving target How to dig out the information of a moving source with varying position, shape and amplitude? Statistics not discussed here

More challenge: Cocktail-party problem