1. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Athanasios Dermanis & Christopher Kotsakis The Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges International Symposium on Geodetic Deformation Monitoring: From Geophysical to Engineering Roles 17 – 19 March 2005, Jaén (SPAIN)

2. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Athanasios Dermanis & Christopher Kotsakis The Aristotle University of Thessaloniki Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges International Symposium on Geodetic Deformation Monitoring: From Geophysical to Engineering Roles 17 – 19 March 2005, Jaén (SPAIN)

3. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges THE ISSUES:  What should be the end product of geodetic analysis? (Choice of parameters describing deformation)  Which is the role of the chosen reference system(s)?  How should the necessary spatial (and/or temporal) interpolation be performed? (Trend removal and/or minimum norm interpolation)  2-dimensional or 3-dimensional deformation? (Incorporating height variation information in a reasonable way)  Quality assessment (effect of data errors and interpolation errors on final results)  Data analysis strategy (Data  coordinates / displacements  deformation parameters)

4. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges geometric information (shape alteration) GEODESY gravity variation information GEOPHYSICS acting forces models for earth behavior (elasticity, viscocity,...) equations of motion for deforming earth - - constitutional equations density distribution hypotheses An interplay between Geodesy and Geophysics: Crustal Deformation as an Inverse Problem y = Ax y A x Geodetic product: Free of geophysical hypotheses!

5. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges geometric information (shape alteration) GEODESY gravity variation information GEOPHYSICS acting forces models for earth behavior (elasticity, viscocity,...) equations of motion for deforming earth - - constitutional equations density distribution hypotheses An interplay between Geodesy and Geophysics: Crustal Deformation as an Inverse Problem y = Ax y A x Geodetic product: Free of geophysical hypotheses!

6. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges The crustal deformation parameters to be produced by geodetic analysis

7. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges The deformation function f x (λ)x (λ)x0(λ)x0(λ) O0O0 O Shape S 0 Shape S f x = f ( x 0 )

8. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges x (λ)x (λ)x0(λ)x0(λ) O0O0 O Shape S 0 Shape S Two shapes of the same material curve x (λ) = f (x 0 (λ)) parametric curve descriptions with parameter λ The deformation gradient F

9. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges x (λ)x (λ)x0(λ)x0(λ) O0O0 O Shape S 0 Shape S tangent vectors to the curve shapes The deformation gradient F dx 0 dλdλ u0 =u0 = dx dλdλ u =u =

10. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges ds 0 d λd λ |u 0 | = ds d λd λ |u| = x (λ)x (λ)x0(λ)x0(λ) O0O0 O Shape S 0 Shape S tangent vector length = rate of length variation The deformation gradient F

11. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges The deformation gradient F dx 0 dλdλ u0 =u0 = dx dλdλ u =u = x (λ)x (λ)x0(λ)x0(λ) O0O0 O Shape S 0 Shape S F u = F(u 0 )

12. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges The deformation gradient F Representation by a matrix F in chosen coordinate systems d x0d x0 d λd λ u0 =u0 = d xd x d λd λ u =u = x (λ)x (λ)x0(λ)x0(λ) O0O0 O Shape S 0 Shape S F u = F u 0 d x  x du 0 d λ  x 0 d λ = xx x0x0 F =F =

13. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges time space (coordinates) Comparison of shapes at two epochs t 0 and t coordinate lines of material points

14. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges time space (coordinates) Comparison of shapes at two epochs t 0 and t t0t0 t x 0 = x(P,t 0 ) x = x(P,t ) Observation of coordinates of all material points at 2 epochs: t 0 and t Spatially continuous information

15. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges t0t0 t x0x0 x Comparison of shapes at two epochs t 0 and t

16. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges x0x0 x x0x0 Use initial coordinates as independent variables Comparison of shapes at two epochs t 0 and t

17. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges xx x0x0 Comparison of shapes at two epochs t 0 and t Use coordinates at epoch t as dependent variables

18. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges x x0x0 Comparison of shapes at two epochs t 0 and t Deformation function f : x = f(x 0 )

19. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges x x0x0 Deformation function f : x = f(x 0 ) Deformation gradient F at point P = Local slope of deformation function f : ff x0x0 F(P) = (P) Comparison of shapes at two epochs t 0 and t

20. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges x x0x0 Comparison of shapes at two epochs t 0 and t When only discrete spatial information is available In order to compute the deformation gradient F ff x0x0 F(P) = (P) We must perform spatial interpolation

21. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges x x0x0 Comparison of shapes at two epochs t 0 and t When only discrete spatial information is available In order to compute the deformation gradient F ff x0x0 F(P) = (P) I N T E R P O L A T I O N We must perform spatial interpolation

22. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Physical interpretation of the deformation gradient SVD Singular Value Decomposition F=QTQT LP From diagonalizations: C = U 2 = F T F = P T L 2 P B = V 2 = FF T = Q T L 2 Q Polar decomposition: F = Q T LP= (Q T P)(P T LP)= RU = (Q T LQ) Q T P= VR orthogonal diagonal λ 1, λ 2, λ 3 = singular values =L 0 λ 2 0 λ 1 0 0 0 0 λ 3

23. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges e1(t)e1(t) e2(t)e2(t) Physical interpretation of the deformation gradient SVD F=QTQT LP e1(t0)e1(t0) e2(t0)e2(t0) P RR Q

24. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges e1(t)e1(t) e2(t)e2(t) Physical interpretation of the deformation gradient SVD F=QTQT LP e1(t0)e1(t0) e2(t0)e2(t0) This is all we can observe at the two epochs No relation of coordinate systems possible due to deformation

25. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Physical interpretation of the deformation gradient e1(t0)e1(t0) e2(t0)e2(t0) P R e1(t)e1(t) e2(t)e2(t) Q R The reference systems and cannot be identified in geodesy! We live on the deforming body and not in a rigid laboratory! e(t0)e(t0)e(t)e(t) SVD F=QTQT LP

26. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges λ2λ2 Local deformation F = Q T L P consists of λ1λ1 1 elongations L (scaling by λ 1, λ 2, λ 3 ) along principal axes and a rotation R = Q T P inaccessible in geodesy due to lack of coordinate system identification R Under x 0 = S 0 x 0, x = Sx : R = SRS 0 T ~~ ~ principal axes S 0, S inaccessible but common for all points R(x 0 ) = SR(x 0 )S 0 T ~ ~

27. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Local deformation parameters (functions of F = Q T L P ) λ1λ1 λ2λ2 1 Singular values in L(λ 1, λ 2, λ 3 ) and functions ψ(λ 1,λ 2,λ 3 ) - Numerical invariants R principal axes Angles in P(θ 1, θ 2, θ 3 ) defining directions of principal axes (physical invariants) P Angles in R(ω 1, ω 2, ω 3 ) defining local rotation (not invariant) e1(t0)e1(t0) e2(t0)e2(t0)

28. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Local deformation parameters (functions of F = Q T L P ) λ1λ1 λ2λ2 1 Singular values in L(λ 1, λ 2, λ 3 ) and functions ψ(λ 1,λ 2,λ 3 ) - Numerical invariants R principal axes e1(t0)e1(t0) e2(t0)e2(t0) P 2D (areal) dilatation: Δ = λ 1 λ 2  1 3D3D (volume) dilatation: Δ = λ 1 λ 2 λ 3  1 shear: γ = (λ 1  λ 2 ) (λ 1  λ 2 )  1/2 shears within principal planes: γ ik = (λ i  λ k ) (λ i  λ k )  1/2 ik = 12, 23, 13

29. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges The role of the reference system

30. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Coordinates in an “preliminary” system, WGS 84, ITRF, or user defined: We need a reference system O(t), e 1 (t), e 2 (t), e 3 (t) for every epoch t ! (Dynamic or space-time reference system) epoch t 0 epoch t displacements too large !

31. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Definition of a network-intrinsic reference system: Center of mass preservation: Vanishing of relative angular momentum: Mean quadratic scale preservation: h R =  i [x i  ] v i = 0 L 2 =  i ||x i – m|| 2 = const. 1 n m =  i x i = const. (= 0) 1 n

32. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Advantages of a network-intrinsic reference system: Invariant deformation parameters the same –No advantage for continuous spatial information Dermination of motion of the network area as whole:translationand rotation

33. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Advantages of a network-intrinsic reference system: Invariant deformation parameters the same –No advantage for continuous spatial information Determination of motion of the network area as whole:translation& rotation Small displacements (trend removal):Essential for proper spatial interpolation of discrete spatial information Reference systems at 2 epochs identified

34. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges 2-dimensional or 3-dimensional deformation?

35. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Crustal deformation is a 3-dimensional physical process t0t0 t F

36. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Usually studied as 2-dimensional by projection of physical surface to a “horizontal” plane t0t0 t F

37. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Proper treatment: Deformation of the 2-dimensional physical surface as embedded in 3-dimensional space t0t0 t F

38. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Attempts for a 3-dimensional treatment Extension of the 2D finite element method (triangular elements) to 3D (quadrilateral elements)

39. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Attempts for a 3-dimensional treatment deformation of mountain deformation of air ! quadrilateral elements have much smaller vertical extension We can obtain good horizontal information by interpolation or virtual densification. Vertical information requires extrapolation (an insecure process) Derination of 3D crustal deformation from 2D deformation surface deformatiom (downward continuation) an improperly posed problem !

40. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Spatial (and/or temporal) interpolation

41. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges time space The “ideal” situation: Space continuous Time continuous To provide deformation parameters at any point for any 2 epochs No interpolation needed ! Geodetic information on crustal deformation - Coordinates x(P,t)

42. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges time space The “satisfactory” situation: Space continuous Time discrete To provide deformation parameters at any point for any 2 observation epochs No interpolation needed ! t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t7 Geodetic information on crustal deformation - Coordinates x(P,t k )

43. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges time space Geodetic information on crustal deformation - Coordinates x(P,t k ) The “satisfactory” situation: Space continuous Time discrete To provide deformation parameters at any point for any 2 epochs Temporal interpolation needed ! t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t7

44. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges time space The “realistic” situation: Space discrete Time discrete To provide deformation parameters at any point for any 2 observation epochs Spatial interpolation needed ! t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t7 P1P1 P2P2 P3P3 P4P4 P5P5 P6P6 Geodetic information on crustal deformation - Coordinates x(P i,t k )

45. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges time space The “realistic” situation: Space discrete Time discrete To provide deformation parameters at any point for any 2 epochs Spatial interpolation needed ! t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t7 P1P1 P2P2 P3P3 P4P4 P5P5 P6P6 Temporal interpolation also needed ! Geodetic information on crustal deformation - Coordinates x(P i,t k )

46. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges time space The “realistic” situation: Space discrete Time discrete Spatial interpolation needed ! t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t7 P1P1 P2P2 P3P3 P4P4 P5P5 P6P6 Not all points observed at each epoch Temporal interpolation needed ! Geodetic information on crustal deformation - Coordinates x(P i,t k ) To provide deformation parameters at any point for any 2 observation epochs

47. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges time space t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t7 P1P1 P2P2 P3P3 P4P4 P5P5 P6P6 Geodetic information on crustal deformation - Coordinates x(P i,t) GPS permanent stations Space discrete Time continuous To provide deformation parameters at any point for any 2 epochs Spatial interpolation needed !

48. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges time space t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t7 P1P1 P2P2 P3P3 P4P4 P5P5 P6P6 GPS permanent stations and SAR interferometry Space discrete Time discrete (SAR) time - continuous (GPS) To provide deformation parameters at any point for any 2 SAR observation epochs Geodetic information on crustal deformation - Coordinates x(P i,t), x(P i,t k ) No spatial interpolation needed !

49. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges time space t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t7 P1P1 P2P2 P3P3 P4P4 P5P5 P6P6 Geodetic information on crustal deformation - Coordinates x(P i,t), x(P i,t k ) GPS permanent stations and SAR interferometry Space discrete Time discrete (SAR) time - continuous (GPS) To provide deformation parameters at any point for any 2 epochs Spatial interpolation needed !

50. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Data analysis strategies

51. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Alternatives for spatial and/or temporal interpolation To be inerpolated: displacments u at discrete epochs t k and discrete points P i u(P i, t k ) = x(P i, t k )  x(P i, t 0 ) u(x 0i,t k ) = x i (t k )  x 0i expressed as Analytic least squares (smoothing) interpolation u(x, t)u(x, t) Minumum norm (exact) interpolation Sought: u(x0, t)u(x0, t) Given: simplified to for every x 0 and t 2 types of interpolation or Minimum norm (smoothing) interpolation equivalent to Minimum mean-square error linear prediction deterministic stochastic

52. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Temporal interpolation: Analytic least squares (smoothing) interpolation deformation evolves slowly with time Spatial interpolation: Analytic least squares (smoothing) interpolation Minumum norm (exact) interpolation or Minimum norm (smoothing) interpolation or combination of the two

53. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Alternatives for linear spatial interpolation Analytic least squares (smoothing) interpolation: u(x) = g(x, a) =  m f m (x) a m a = vector of free parameters (  observations) least square solution a of u(x i ) = g(x i,a) + v i (v T Pv = min ) Minumum norm (exact) interpolation: minimum-norm solution a of u(x i ) = g(x i,a)(a T Ra = min ) Minumum norm (smoothing) interpolation: (parameters a > observations, even infinite) hybrid solution a of u(x 0i ) = g(x 0i, a) + v i ( v T Pv+a T Ra = min) (parameters a > observations, even infinite)

54. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Deterministic and stochastic interpretation for linear interpolation Minumum norm (exact) interpolation: Minumum norm (smoothing) interpolation: a T Ra = min  u(x) = f T (x)R  1 F T (FR  1 F T )  1 b = = k(x) T K  1 b b = F a u(x i ) =  m f m (x i ) a m = f(x i ) T a a T Ra + v T Pv = min  u(P) = k T (P)(K+P  1 )  1 b Minumum mean square error prediction: b = F a + v b = s b = s + v e = s(x) - s(x), trE{ee T } = min  s(x) = C s(x)s C s  1 b ~ ~ “Collocation” in geodetic jargon s(x) = C s(x)s (C s + C v )  1 b ~ deterministicstochastic

55. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges s(x) = C s(x)s (C s + C v )  1 b ~ = = f(x) T C a F T (FC a F T )  1 b = = f(x) T C a F T (FC a F T )  1 b Deterministic and stochastic interpretation for linear interpolation Minumum norm (exact) interpolation: Minumum norm (smoothing) interpolation: a T Ra = min  u(x) = f T (x)R  1 F T (FR  1 F T )  1 b = = k(x) T K  1 b a T Ra + v T Pv = min  u(P) = k T (P)(K+P  1 )  1 b Minumum mean square error prediction: b = F a + v b = s = F a, s(x) = f(x) T a  C ss = FC a F T, C s(x)s = f(x) T C a F T Equivalence C a = R  1 C v = P  1 b = F a u(x i ) =  m f m (x i ) a m = f(x i ) T a s(x) = C s(x)s C s  1 b

56. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Temporal interpolation:least-squares analytic x(x 0,t) = x 0 + (t  t 0 ) v(x 0 ) u(x 0,t) = (t  t 0 ) v(x 0 ) F(x 0,t) = I + (t  t 0 ) L(x 0 ) L = vv x0x0 + Spatial interpolation:Combination of least-squares analytic and stochastic prediction v(x 0 ) = f(x 0 ) T a + z(x 0 ) s = s1s2s1s2 Csisk(x0,x0)Csisk(x0,x0) Spatial interpolation: Combination of least-squares analytic and stochastic prediction component covariance functions x(x 0 ) = f(x 0 ) T a + s(x 0 ) EXAMPLES OF INTERPOLATION MODELS Observation epochs t 0, t - No temporal interpolation Czizk(x0,x0)Czizk(x0,x0) component covariance functions

57. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges C s 2 s 2 (x 0,x 0 ) = = C s 2 s 2 (||x 0  x 0 ||) Invariance guaranteed by stationary and isotropic component covariance functions Invariant Interpolation = independent of used reference systems Analytic least squares (smoothing) interpolation: u(x) =  m f m (x) a m = f(x) T a Not invariant interpolation! Base functions f m (x) depend on coordinate system used Exception: Finite element method u T (x) = J T x + c T Different J T, for each triangular element, c T irrelevant ( F T = I + J T ) T 1 2 3 u(x 2 )-u(x 1 ) = J (x 2 -x 1 ) u(x 3 )-u(x 1 ) = J (x 3 -x 1 ) Solve for J Same equations for x = Rx+d  u = Ru Invariant interpolation ! ~~ Minumum mean square error prediction: C s 1 s 1 (x 0,x 0 ) = = C s 1 s 1 (||x 0  x 0 ||)C s 1 s 2 (x 0,x 0 ) = 0 s(x) = C s(x)s (C s + C v )  1 b

58. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Data analysis alternatives “raw” observations (e.g. GPS baselines) station coordinates displacement field deformation gradient adjustment interpolation “raw” observations (e.g. GPS baselines) displacement field deformation gradient combined adjustment and interpolation Potential problem: Incorrect separation of observation errors and displacements

59. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Quality assessment for deformation parameters

60. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges Problems in quality assessment for deformation parameters  Incorrect statistics for input data (coordinates)  Incorrect assessment of interpolation errors  Incorrect error propagation from deformation gradient to deformation parameters Typical for GPS coordinates Improper separation of signal (displacements) from noise Singular values highly nonlinear functions of deformation gradient Fit lines to time variation Estimate statistics from residuals Trial and error Use propagation with higher derivatives and moments Use Monte Carlo techniques

61. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges FUTURE OUTLOOK:  Permanent GPS stations provide initial coordinates and velocities with realistic variance-covariance matrices  Supplementary SAR Interferometry provides spatially interpolated velocities & identifies problematic points OPEN PROBLEMS:  Optimal merging of GPS with SAR Interferometry data  The missing third dimension in crustal deformation information (Introduce geophysical hypotheses ?)

62. Athanasios Dermanis & Christopher KotsakisThe Aristotle University of Thessaloniki, Department of Geodesy and Surveying Estimating crustal deformation parameters from geodetic data: Review of existing methodologies, open problems and new challenges This presentation will be available at http://der.topo.auth.gr/