Extract deep geophysical signals from GPS data analysis Signals and noises inside GPS solutions Network adjustment Time series analysis Search subtle signals.

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• 20200 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 and noises in GPS solutions• Signals

Propagation medium: ionosphere, atmosphereSurface process: mass loading from atmosphere, ocean (tidal and

non-tidal) and ground water, thermal expansionUnderground 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-gassingFrame motion: polar motion, Earth rotation, geocenter

• NoisesSystematic: 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 troubleCommon mode error (CME): unknownRandom noise

Signals in space domain• Global

polar motion, Earth rotation, geocenter, sea level

• Continentalpost-glacial rebound, plate motion, micro-plate motion

• Regionalregional deformation, mass loading from atmosphere, ocean and groundwater, tidal variations, fault dislocation and creeping, co-seismic and post-seismic deformation

• Localaquifer undulation, thermal expansion, transient fault motion

Signals in time domain• Secular

tectonic motion, orogenic process, sediment or ice sheet compacting

• Decadalpost-glacial rebound, sea level change, post-seismic deformation

• Seasonal mass loading, thermal expansion, polar motion, UT1

• Intra-seasonalmagma intrusion and migration

• Short termco-seismic, transient, tidal motion

Network adjustment

• Adjusted parametersstation positions, velocities, jumps, network rotation, translation and scale, orbits and others

• Prosrigorous, 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 filteringcovariance matrix or normal matrix approaches

• Square root information filteringsuperior numerical stability

Reduce the burden of network analysis

• Eliminating uncorrelated parametersambiguities, troposphere parameters, orbitsin velocity field estimation

• Implicitly solving piecewise constant parameterstroposphere zenith delays and gradients, ambiguities

• Helmert blockingfor 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 parametersstation 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 functionGeophysical 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?

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Back to network analysis: taking advantage of spatial correlation

• Spatially correlated Green’s functionsfault 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

• Keynotethe 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)• XTX = VT*S2*V (PCA decomposition)• Rescale XTX -> correlation matrix (Karhunen-Loeve

expansion KLE), slight different eigenvectors

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Geodetic interpretation of the PCA analysis

• PCA decomposition

ak: k-th principal component (time domain)

vk: k-th eigenvector (space domain)

ak: k-th common time function for the network

vk: network spatial responses for the k-th PC

• First few PC: common modes

Last few PC: local modes

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PCA application example: regional network filtering

• Stacking approachCommon time functionSpatial responses: uniform distribution

• PCA decompositionCommon time functionSpatial responses: determined by data themselves

• Potential causes: satellite orbits, reference frame

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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-t0): fault slip time function (fault frame)

di(x,t): surface displacement at x in i-th direction (surface frame)m(t): reference frame termb(x,t): benchmark instability(x,t): observation noise

• Only spatio-temporal function s(,t-t0) is unknown• Can we estimate s(,t-t0) directly? Yes and No

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Constraints in Network filter estimation

• Fault geometry should be expanded by orthogonal spatial basis functions

spatial smoothing: temporal smoothing: state perturbation noiseminimum norm constraintpositivity constraint

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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

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More challenges ahead

• Moving targetHow to dig out the information of a moving source with varying position, shape and amplitude?

• Statistics not discussed here

More challenge: Cocktail-party problem

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