M. Holschneider 1 A. Eicker 2 R. Schachtschneider 1 T. Mayer-Guerr 2 K. Ilk 2

Transcription

1 SPP project TREGMAT: Tailored Gravity Field Models for Mass Distributions and Mass Transport Phenomena in the Earth System by by M. 1 A. Eicker 2 R. Schachtschneider 1 T. Mayer-Guerr 2 K. Ilk 2 1 University of Potsdam, Institute of Mathematics, Applied and Industrial Mathematics Department 2 University of Bonn, Institute of Geodesy and Geoinformation

2 Other collaborators by Many more people have contributed A. Chambodut, B. Minchev, I. Panet, M. Mandea, I. Iglewska-Nowak

3 Content by

4 Mathematical I by Equivalent formulations are: Spline, Kriging, Bayesian We are looking for a field satisfying Laplace equation Φ = 0, in source free region. A priori knowledge: Φ is a Gaussian process with probability density P(Φ) e λγ(φ), Γ(Φ) is a generalized energy ( e.g. Kaula s rule in Gravity, Geomagnetic norms in Geomagnetism ). Realized through rotation invariant operator G Γ(Φ) =< Φ, GΦ >, Γ(Φ) = l=0 γ 2 l l ˆΦ l,m 2 m= l

5 Mathematical II by A priori knowledge is increased through measurements. Measurement are functionals of Φ plus noise ( i I ) α i =< L i, Φ > +ɛ i. Noise is modelled by Gaussian process e.g.δ xi, δ xi E(ɛ i ) = 0, E(ɛ i ɛ k ) = Σ i,k. This yields a posteriory probability density in model ( P(Φ) exp λγ(φ) + ) (L i Φ α i )Σ 1 i,j (L j Φ α j ) All the physical information is in Γ, L i and ɛ. The "best" model, given our knowledge is the uniquely determined most probable model Φ opt = argmin λγ(φ) + (L i Φ α i )Σ 1 i,j (L j Φ α j )

6 Direct inversion by Theoretically Φ opt can be computed directly ( e.g. Wahaba 1982 ) Let η i = G 1 L i, This is the optimal model given the single measurement L i. Then Φ opt = γ i η i, where γ i are solution of an explicitly known linear system. Problem: dimension of system is #{observations} 2 Possible solution: Compression of observational data ( Minchev et. al. ) Lower dimensional approximations

7 Linear Models by amounts to compute an approximation to the minimizer of the cost functional. Linear models: superposition of "a priori" basic F n s(x) = α n F n (x). where the sum is finite and α n R. Quality of approximation s s depends on choice of F n ( SH, mascons, wavelets, local splines ) Often F n are ONB but not necessary.

8 Requirements by Requirements for basic easy physical interpretation numerically easy to compute universal ( = span dense in L 2 ) simple transition between local and global models for the of potential fields F n = 0. Models are approximants of the form α n F n.

9 Normal equations by System matrix A i,j = F i (x j ). Minimization of cost function leads to normal equations for expansion coefficients x n from observations α with and (M + λλ) α = Σ 1 A α M i,j = (A Σ 1 A) i,j = k,l Λ i,j = Γ(F i, F j ). F i (x k )Σ 1 k,l F j(x l ) Optimal λ may be determined through cross validation / variance component estimation.

10 Covariance Matrix by The spherical harmonics form an ONB. Covariance matrix diagonal Λ i,j =< SH i, SH j >

11 Correlation observations by Observation in x l {Paris, Sidney}. M i,j = l SH i(x l )SH j (x l ). Matrix is not sparse observations in Sidney influence the local model in Europe.

12 Localizing : by No "natural" dilation operator. Consider Ω R the sphere of radius R, Ω R = {(x 1, x 2, x 3 ) : x x x 2 3 = R2 }. Definition of exterior wavelets Wx ext,d (y) = R y l=0 ( ) x l l d Q l (ˆx ŷ) x Int Ω R, y Ext Ω R y d is the called the degree and Q l = (2l + 1)P l is reproducing kernel of Σ l = span{y l,m : l m +l} f Σ l Q l f = δ l,l f Convolution defined as K s(x) = y =R K (ˆx ŷ) s(y) dσ(y). (1)

13 Wavelet as multipoles by evaluation of wavelets is based on the following Proposition Exterior and interior wavelet with d N 0 may be uniquely harmonically continued to R 3 R 3 \{x = y}. W ext,d x (y) = d+1 l=0 (2α d+1 l d+1 = ( 1) d+1 l=0 + α d l ) Rl! x l P l (ŷ x ˆx) y x l+1 (2β d+1 with explicit coefficients α k and β k. l βl d ) Rl! y l P l ( x y ŷ) y x l+1.

14 Interpretation as wavelet by Actually these are wavelets! Consider restriction to sphere Ω R. For x < R and y Ω R Wx ext,d (y) = R y l=1 ( ) x l l d Q l (ˆx ŷ) y = R > x. y This may be written as ( b = Rx/ x, x = e a b ) W ext,d x (y) = W (d) a d b,a (y) = R (al) d e al Q l (ŷ b) l

15 Scaling of wavelets by The distance of the position of the poles to the surface of the earth is related to the scale

16 Interpretation by The are wavelets on the sphere Ω R. Their harmonic extension into outer are W ext,d a = log(r/ x ) 1 x /R the scale b = Rˆx the position

17 Euclidean limit by Stereographic mapping Ψ : H Ω R at north-pole N. Theorem ( M.H. and I. Iglewska-Nowak) The following limit exists pointwise for d N and y H. V d (y) := lim a d+2 W ext,d (Ψ(ay)) (2) a 0 e a N = lim a d+2 W int,d a 0 e a N (Ψ(ay)) (3) = lim a 2 ga,n d (Ψ(ay)) (4) a 0 = 2(d + 1)! R d 1 P d+1(1/ 1 + y/r 2 ) (1 + y/r 2 ) (d+2)/2. (5) Extension to directional wavelets ( PhD M. Hayn )

18 Covariance matrix by The family of wavelets is NOT an ONB, but (generzalized) correlations (i.e. quadratic form of generalized energy) are low

19 Data correlation by The correlation matrix M i,j = l SH i(x l )SH j (x l ) with respect to the observations is sparse local is possible.

20 Local Spline by Idea: (quasi) diagonalizing the quadratic form Γ. Γ(Φ) = l γ 2 l l m= l ˆΦ l,m 2, with ˆΦ l,m the Fourier coefficients of Φ on the Earth. Then set with small regularizing η > 0 W = 2l + 1 γ l e ηl P l Therefore Γ(W ) = (1 e 2ηl ) 1 (2η) 1 and < W x, GW y > C η, x y We have < W x, GW y > δ x (y) ( η 0).

21 Local Spline by Grid of positions choose η to optimize compromise between diagonality of Γk,l =< Wxk, GWxl > and finiteness of generalized energy. Problem: spatial localization of W itself cannot be improved without bounds. But: smaller details than these kernels need exponential increase of precision for measurements anyway

22 Mathematical for regional by Mathematical analysis based on operator K : (Earth) (Satellite), Φ χ data-region Upward h Φ Extremely ill posed problem: recovering details δφ = Φ Φ 0 to a prior global model Φ 0 from noisy observations Φ ɛ = Φ 0 + K δφ + ɛ, δφ ɛ = Φ ɛ Φ 0.

23 SVD decomposition allows optimal regularization by The SVD of K allows to consider regularizing approximants of K 1. K = n λ n φ n >< ψ n, K 1 N = N n=1 λ 1 n ψ n >< φ n Trade-off between reconstruction N and amplification of noise ɛ( N n=1 λ 2 n ) 1/2. δφ K 1 N δφɛ δφ K 1 1 K δφ + K ɛ = "bias"+"variance" N N For local numerical approximation of ψ and φ.

24 Mathematical for regional by, Dependence of reconstruction error position in modelling region noise level sattellite hight size of modelling region degree of global a priori model Φ 0. First results see Poster R. Schachtschneider.

25 Local prior information by Sometimes the prior information may be dependent Γ(Φ) = < Φ, G 1 Φ > + < Φ, G 2 Φ > solidearth ocean No reference to in the physical formulation. However, in practice (reasonably good approximative) computation easy with, Φ = α x W x + β y W y, x solidearth upon neglecting cross terms. y ocean

26 refinement strategy by global reference field (spherical harmonics) EGM96 regional refinement: splines 1 month of data, resolution n = 140 comparison to global 3-year solutions

27 refinement by Spline solution (06/2005) RMS: 57.7 cm

28 Differences compared to the ITG-Grace02s (n=120) by Spline solution (06/2005) GFZ-RL04 (06/2005) RMS: 6.6 cm RMS: 13.9 cm

29 Differences compared to the EIGEN-GL04c (n=120) by Spline solution (06/2005) GFZ-RL04 (06/2005) RMS: 9.3 cm RMS: 16.5 cm

30 Patching by

31 Global Refinement by

32 Differences compared to ITG-Grace02s (n=140) by RMS: 9.98 cm

33 Temporal variations by Yearly period Sumatra-Andaman Earthquake

34 Conclusions by localized are useful tool in local theory of local needs more progress GRACE models can compete with global solutions regional temporal variations can be modelled