IUGG 2015 - Prague, June 22 July 3, 2015 A comparison of existing and new methods for the analysis of nonlinear variations in coordinate time series Miltiadis Chatzinikos Athanasios Dermanis Department of Geodesy and Surveying - Aristotle University of Thessaloniki
Comparison of four methods for the analysis of the GPS position time series: Existing: STD - Standard least-square fitting of periodic function with annual and semiannual frequencies SSA - Singular Spectrum Analysis New: MOD - Least-Square fitting of a amplitude and phase modulated annual signal using continuous piecewise-linear amplitude and phase models ECS - Least-Square fitting of equidistant cubic splines
Objectives: - Separate physically meaningful signals from noise in coordinate time series - Introduce easy-to-use models for future update of the ITRF to include also nonlinear coordinate variation terms
Coordinate Interpolation method: SSA Singular Spectral Analysis
SSA - Singular Spectrum Analysis Time series: 1 2 x, x,..., xn Lag-window size: M c c c c c c c c c c c c c c c c c c C c c c c c c c c c c c c c c c c c c 0 1 2 3 M2 M1 1 0 1 2 M3 M2 2 1 0 1 M4 M3 3 2 1 0 M5 M4 M 2 M 3 M 4 M 5 0 1 M 1 M 2 M 3 M 4 1 0 Ni 1 c x x, 0 i M 1 i k ki N i k1 a: principal components, E: eigenvectors M k k i i j j j1 a x E, 0 i N M Reconstruction of coordinates: k 1 i k k xi a, 1 1 j 1 i je j i M i k 1 M k k xi a, j 1 i je j M M i N M 1 x k 1 M k k i i j j, j i N M 1 N i N M 2
Coordinate Interpolation method: MOD Amplitude and phase modulation of annual carrier by piecewise linear functions
MOD - Modeling of coordinate time series by amplitude and phase modulation x( t) A( t)cos 2 t ( t) 0 0 = annual carrier frequency A(t) = amplitude (t) = phase t : k t tk 1 A A A( t) A ( t t ) k k1 k1 k1 tk tk 1 ( t) ( t t ) k k 1 k1 k1 tk tk 1 At () Ak 1 Ak........ t t t 0 1 t 2 t k 2 tk 1 t k tk 1 tn 2 t n1 t n
MOD - Modeling of coordinate time series by amplitude and phase modulation t : k t tk 1 A A x( t ) x( t ) e A ( t t ) cos 2 t ( t t ) e k k1 k k1 i i i k1 i k1 0 i k1 i k1 i tk tk 1 tk tk 1 Linearization f y f ( x) e b y f ( x0) ( x x0) e A x e x 0 x [ A A ] T 0 n 0 n T e Pe min leads to large variation of A k and k! Instead: Tikhonov regularization solution: T T e Pe x Wx min n n T 2 2 x Wx A( Ak Ak 1) ( k k 1) k1 k1 A, : regularization parameters
Coordinate Interpolation method: ECS Equidistant Cubic Splines
ECS - Modeling of coordinate time series by equidistant cubic splines t t t i1 i : Version 1: Spline in terms of second derivatives g g 2M M M M M x( t) S ( t) g h ( t ) ( t ) ( t ) h 6 2 6h i i1 i1 i i1 2 i i1 3 i i1 i1 i1 i1 N M ds dt ds dt i i1 i ( i ) ( i ) d S dt d S ( ) ( ) dt 2 2 i i1 i 2 i 2 i Constraints for equality of first derivatives 6 M i1 4 M i M i1 ( g 2 i1 2 gi gi 1) h xt () gi 2 S i () 1 t gi 1 Si () t () 1 t.... g i.... h S i t i2 i 1 i i 1
ECS - Modeling of coordinate time series by equidistant cubic splines t t t i1 i : Version 2: Spline in terms of first derivatives g g 2N N g g N N x( t) S ( t) g N ( t ) 3 ( t ) 2 ( t ) h h h h N i i1 i1 i 2 i i1 i1 i 3 i i1 i1 i1 2 i1 3 2 i1 ds dt ds dt i i1 i ( i ) ( i ) Constraints for equality of second derivatives M d S dt d S ( ) ( ) dt 2 2 i i1 i 2 i 2 i hn 4hN hn 3( g g ) i1 i i1 i1 i1 xt () gi 2 S i () 1 t gi 1 Si () t () 1 t.... g i.... h S i t i2 i 1 i i 1
ECS - Modeling of coordinate time series by equidistant cubic splines t t t i1 k i : Version 1: Spline in terms of second derivatives 1 2 1 1 2 1 3 ( ) ( ) ( ) g i g M i i M i 1 ( 1) M i i i k k k i k k i k i ( k i 1) M x t x t e S t e g h t t M ( tk i1) ek h 6 b Ax e 2 6h x [ g g M M ] T Constraints: 6 M i1 4 M i M i1 ( g 2 i1 2 gi gi 1) 0 h 0 n 0 Cx 0 n T e Pe min Subject to Cx 0 Least squares with constraints xt () gi 2 S i () 1 t gi 1 Si () t () 1 t.... g i.... h S i t i2 i 1 i i 1
ECS - Modeling of coordinate time series by equidistant cubic splines t t t i1 k i : Version 2: Spline in terms of first derivatives 2 ( ) ( ) ( ) 3 g g N N x t S t e g N t ( t ) 2 g g N N ( t ) e h h h h i i 1 i 1 i 2 i i 1 i 1 i 3 k i k k i1 i1 k i1 2 k i1 3 2 k i1 k b Ax e x [ g g N N ] T 0 n 0 n Constraints: hn 4hN hn 3( g g ) 0 i1 i i1 i1 i1 Cx 0 T e Pe min Subject to Cx 0 Least squares with constraints xt () gi 2 S i () 1 t gi 1 Si () t () 1 t.... g i.... h S i t i2 i 1 i i 1
Replacement of East, North coordinates with principal horizontal components E, N = East and North coordinate residuals after linear trend removal i i Sample dispersion matrix: S 2 2 se s 1 EN ( Ei me ) ( Ei me )( Ni mn ) 2 2 s N EN sn i ( Ei me )( Ni mn ) ( Ni mn ) 1 1 m E, m N N N E i N i i i Diagonalization: 2 2 se s EN smax 0 S ( ) ( ) 2 R 2 R sen sn 0 smin Principal horizontal components: max Hi Ei ( ) min R H N i i Identification of the horizontal direction with maximal coordinate variation!
Replacement of East, North coordinates with principal horizontal components
Hellas Network 17 GNSS stations Time span 2007-2014
Definition of the reference system Alignment to a global network (ITRF, IGS): - Best for global or large scale tectonic studies - Coordinate time series reflect station motion at global scale Regional stacking: - Best for regional scale tectonic studies - Removes from coordinate series contributions from motion of the region as whole with respect to global system systematic errors in GNSS data
Time-series of Transformation Parameters (Stacking procedure on 17 stations)
GPS station: RLSO (NW Peloponnese) Comparison of Coordinate Interpolation Methods
Coordinate Time Series Interpolated vs original nonlinear terms max horizontal (x 2) vertical STD SSA MOD ECS Time Time
Spectrum Analysis Covariance Functions GPS station: RLSO (Achaia, Northwestern Peloponnesus) max horizontal max horizontal ORIG STD SSA MOD ECS vertical vertical Cycles/Year Week
GPS station: KASI (Corfu, Ionian Sea) Comparison of Coordinate Interpolation Methods
Coordinate Time Series Interpolated vs original nonlinear terms max horizontal vertical STD SSA MOD ECS Time Time
Spectrum Analysis Covariance Functions GPS station: KASI (Corfu, Ionian Sea) max horizontal max horizontal ORIG STD SSA MOD ECS vertical vertical Cycles/Year Week
2 RMS of residuals after interpolation for all 17 stations (maximal horizontal component) 1.8 STD SSA MOD ECS 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 ANAV AUT1 DUTH KASI KLOK KORI LEMN MET0 NOA1 PAT0 PONT PRKV RLSO SPAN SPET TUC2 VLSM
4 RMS of residuals after interpolation for all 17 stations (vertical component) 3.5 3 2.5 2 1.5 1 0.5 0 ANAV AUT1 DUTH KASI KLOK KORI LEMN MET0 NOA1 PAT0 PONT PRKV RLSO SPAN SPET TUC2 VLSM
EUREF Network NYA1 NYA1 - Ny-Alesund, Norway QAQ1 - Qaqortoq/Julianehaab, Southern Greenland QAQ1
Coordinate Time Series Interpolated vs original nonlinear terms GPS station: NYA1 (Ny-Alesund, Norway) max horizontal ORIG STD SSA MOD ECS time
Coordinate Time Series Interpolated vs original nonlinear terms GPS station: NYA1 (Ny-Alesund, Norway) vertical ORIG STD SSA MOD ECS time
Spectral Analysis Interpolated vs original nonlinear terms GPS station: NYA1 (Ny-Alesund, Norway) max horizontal vertical ORIG STD SSA MOD ECS ORIG STD SSA MOD ECS Cycles/Year Cycles/Year
Covariance functions Interpolated vs original nonlinear terms GPS station: NYA1 (Ny-Alesund, Norway) C(0) = 3.52 mm 2 C(0) = 57.25 mm 2 max horizontal ORIG vertical ORIG STD STD SSA SSA MOD MOD ECS ECS week week
Coordinate Time Series Interpolated vs original nonlinear terms GPS station: QAQ1 (Qaqortoq/Julianehaab, Southern Greenland) max horizontal ORIG STD SSA MOD ECS time
Coordinate Time Series Interpolated vs original nonlinear terms GPS station: QAQ1 (Qaqortoq/Julianehaab, Southern Greenland) vertical ORIG STD SSA MOD ECS time
Spectral Analysis Interpolated vs original nonlinear terms GPS station: QAQ1 (Qaqortoq/Julianehaab, Southern Greenland) max horizontal vertical ORIG STD SSA MOD ECS ORIG STD SSA MOD ECS Cycles/Year Cycles/Year
Covariance functions Interpolated vs original nonlinear terms GPS station: QAQ1 (Qaqortoq/Julianehaab, Southern Greenland) C(0) = 6.19 mm 2 C(0) = 59.06 mm 2 max horizontal ORIG vertical ORIG STD STD SSA SSA MOD MOD ECS ECS week week
The adaptability issue for a coordinate interpolation-smoothing method A good method should be adaptive enough to recover the geophysical signal and at the same time not too adaptive to interpret noise as signal Optimal separation is not possible in an undisputable way because the noise characteristics are not fully known. This is mainly due to colored noise (effect of systematic errors) and biases, while only the white and zero mean part of the noise can be easily removed Within each method various degrees of adaptability are possible by tuning relevant parameters: STD: SSA: MOD: ECS: frequencies other than annual and semi-annual length of lag window length of interval of linear amplitude/phase regularization parameter values length of interval for splines
The suitability of a coordinate interpolation-smoothing method for geophysical interpretation STD: Standard use of annual, semiannual and other frequencies Limited interpretation: Amplitude of included frequencies MOD: Amplitude and phase modulation by piecewise linear functions Most challenging for geophysical interpretation. Variation in amplitude values reflects magnitude of meteorological & hydrological effects. Variation in phase values reflects different occurrence times of such effects. In plane words: Not all winters/summers are severe to the same degree each year! Severe phenomena do not occur at the same dates every year!
The suitability of the nodulation method for geophysical interpretation - Examples GPS station: KASI (Corfu island, Ionian Sea) - interval 3 months Amplitude max horizontal Phase max horizontal vertical vertical
The suitability of the nodulation method for geophysical interpretation - Examples GPS station: RLSO (Achaia, Northwestern Peloponnesus) - interval 3 months max horizontal Amplitude Phase max horizontal vertical vertical
The suitability of the nodulation method for geophysical interpretation - Examples GPS station: NYA1 (Ny-Alesund, Norway) - interval 3 months max horizontal Amplitude Phase max horizontal vertical vertical
The suitability of the nodulation method for geophysical interpretation - Examples GPS station: QAQ1 (Qaqortoq/Julianehaab, Southern Greenland) - interval 3 months max horizontal Amplitude Phase max horizontal vertical vertical
Conclusions: Standard annual and semiannual frequencies: ANALYTIC MODEL, INSUFFICIENT ADAPTIVITY GEOPHYSICAL INTERPRETATION? Singular Spectrum Analysis: HIGH ADAPTIVITY, NO ANALYTIC MODEL Amplitude & phase modulation: GOOD ADAPTIVITY ANALYTIC MODEL GEOPHYSICAL INTERPRETATION! NEED FOR REGULARIZATION Equidistant cubic splines: HIGH ADAPTIVITY ANALYTIC MODEL
Thanks for your attention!