Components of interferometric phase

2. Realistic error modelling for InSARTim J Wright, Gareth J Funning (COMET, Oxford Earth Sciences)Peter J Clarke (Geomatics, University of Newcastle)Charles W Wicks (USGS, Menlo Park, California)

3. Components of interferometric phase Dfint = Dfgeom +Dftopo +Dfatm + Dfnoise + Dfdef Dfint = Dfgeom +Dftopo +Dfatm+ Dfnoise + Dfdef A foggy morning, near ancient Mycenae, Greece

4. June to December July to December June to July Components of interferometric phase Dfint = Dfgeom +Dftopo +Dfatm + Dfnoise + Dfdef Dfint = Dfgeom +Dftopo +Dfatm+ Dfnoise + Dfdef Turbulent atmosphere Athens Earthquake – September 1999

5. Comparison with Rain Radar (The Netherlands) Figures courtesy Ramon Hanssen (Delft)

6. The “Dutch Alps” Figures courtesy Ramon Hanssen (Delft)

7. 17 August 1999 Izmit (Turkey) earthquake ERS2 – ERS2 13-Aug-1999 – 17 Sept-1999 ERS1 – ERS1 12-Aug-1999 – 16 Sept-1999

8. mm 84 0 -84 • ERS1 interferogram – ERS2 interferogram • Atmosphere + Orbital Errors.

9. Solutions to atmospheric water vapour problem? • Ignore (most common) • Quantify • Model based on other observations • (e.g. GPS, meteorology…?) • Increase SNR by stacking • For some it is the signal not the noise.

10. Solutions to atmospheric water vapour problem? • Ignore (most common) • Quantify • Model based on other observations • (e.g. GPS, meteorology…?) • Increase SNR by stacking • For some it is the signal not the noise.

11. Question. • How can we quantify the influence of spatially-correlated noise on our models?

12. Monte Carlo Simulation of Correlated Noise A. Determine an empirical Variance-Covariance Matrix (VCM) for the noise in the interferogram. B. Create a suite of pseudo-random realisations of the noise, and add to the original data. C. Invert each dataset – the range of values for each model parameter gives their error.

13. A: Create a VCM – a practical approach • Calculate the covariance function - chose part of image away from signal or remove a 1st pass model. - remove plane, set mean to zero. - covariance function is radially-averaged autocovariance. - this can be calculated spatially or in the frequency domain (e.g. Hanssen, 2001).

14. radians radians 10 km 10 km Covariance / radians2 20 km Distance / km A: Create a VCM – a practical approach

16. sub-image size variance length scale 2 10 km 0.24 rad 1.6 km 30 2 25 km 1.1 rad 4.8 km 2 50 km 5.2 rad 10 km 2 100 km 28 rad 24 km 2 160 km* 30 rad 33 km 20 *= whole image 10 0 -10 0 20 40 60 80 100 120 Distance / km A word of caution…

17. A: Create a VCM – a practical approach • Calculate the covariance function • Try functional fit: e.g. cov = s2 e-ar cov = s2 e-ar cos(br) [s2= variance, r = distance] 3. Use this function to construct VCM for the sampled points of the interferogram.

18. B. Create synthetic noise • Construct X[vector of Gaussian noise, mean 0, s 1] • Want Y[vector of correlated noise with known VCM Sy] • Let Y = LXSy = LSxLT[L is a matrix] • But Sx= I Sy = LLT • i.e. L is the Cholesky Decomposition of Sy

19. mm

20. C. Invert perturbed datasets • Chose your favourite inversion method!

21. Test – correlated vs independent noise • Grid denser than 5km: independent noise  error estimate too small • correlated noise  “correct” error estimate • Grid sparser than 5km: both methods overestimate error.

22. Slip (m) 4 3.5 80 100 Dip Example 1: Fault geometry

23. 25 Volume (10-3 km3) 15 3.5 5.5 Depth (km) Example 2: Mogi Models Three Sisters: ‘clean’ (8/96 to 10/00) Three Sisters: ‘noisy’ (8/97 to 9/00)

24. 20 km Example 3: Distributed Slip (Izmit Earthquake) Dip: 87s 88s 86s 88n 81n 61n Rake: 174 171 178 -178 -164 -168 6-segment model (Wright et al, 2001) r.m.s. misfit = 28 mm; M0 = 265 x 1018 Nm

25. 20 km Example 3: Distributed Slip (Izmit Earthquake) 6-segment model (Wright et al., 2001) r.m.s. misfit = 28 mm; M0 = 265 x 1018 Nm Geometry of Wright et al., but with variable slip (5km patches) r.m.s. misfit = 24 mm; M0 = 260 x 1018 Nm Vertical fault, pure strike slip (5km patches) r.m.s. misfit = 45 mm; M0 = 288 x 1018 Nm

26. 20 km Example 3: Distributed Slip (Izmit Earthquake) 25 km 6-segment model (Wright et al., 2001) r.m.s. misfit = 28 mm; M0 = 265 x 1018 Nm Geometry of Wright et al., but with variable slip (5km patches) r.m.s. misfit = 24 mm; M0 = 260 x 1018 Nm Difference between 6-segment model and variable slip model Vertical fault, pure strike slip (5km patches) r.m.s. misfit = 45 mm; M0 = 288 x 1018 Nm

27. Example 3: Distributed Slip (Izmit Earthquake) Slip: Error:

28. Model Covariance 2 km 6 km 10 km 14 km 18 km 22 km

29. Inversion Method Elastic structure Fault Geometry EARTHQUAKE SLIP MODEL Smoothness criteria Data sampling scheme Data Noise

30. Mitigation using additional data • InSAR alone Simulations using one possible configuration of a dedicated InSAR satellite

31. Mitigation using additional data • InSAR alone Black dots – 1 int (descending, right-looking) Red dots – 2 ints (asc + dsc, right-looking) Cyan dots – 3 ints (asc + dsc, right + left-looking)

32. Mitigation using additional data 2. GPS data slip error

33. InSAR Data Only Long Period Seismology Data Only InSAR and seismology Mitigation using additional data 3. Seismic Data

34. Conclusions • Spatially-correlated noise must be considered when determining errors. • The method presented is relatively simple to use, and can be applied to any model derived from InSAR data. • Earthquake slip distributions derived from geodetic data are highly non-unique. • Use of additional data can often dramatically reduce parameter errors and trade-offs

36. Components of interferometric phase Dfint = Dfgeom +Dftopo +Dfatm + Dfnoise + Dfdef Dfint = Dfgeom +Dftopo +Dfatm+ Dfnoise + Dfdef Layered atmosphere 29/8/1995 to 29/7/1997 30/8/1995 to 29/7/1997 Topography