GOCE-GDC: Towards a better understanding of the Earth s interior and geophysical exploration research Work Package 5 - Geophysics B

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1 GOCE-GDC: Towards a better understanding of the Earth s interior and geophysical exploration research Work Package 5 - Geophysics B Zdeněk Martinec Dublin Institute for Advanced Studies 5 Merrion Square, Dublin 2, Ireland (i) GOCE solid Earth workshop (October 16-17, 212) (ii) Joint science meeting GOCE+ GeoExplore and GOCE+ GDC (October 18) (iii) GOCE+ GDC MTR (October 19) University of Twente, Enschede

2 Contents 1 Static-geoid interpretation 2 Dynamic-geoid interpretation 3 Seismic tomography of the upper mantle 4 Congo-basin negative free-air gravity anomaly 5 Mass-density Green s functions for gravity

3 What can we learn from the geoid on the solid Earth? Newton s law of gravity: ϱ(q) V (P) = G dv Q B d PQ The inverse gravimetric problem (IGP): Find the volume-mass distribution ϱ of the Earth, provided that the potential V is known outside the Earth. An inverse problem is called well-posed if each of the following three criteria is satisfied (Hadamard): a solution exists the solution is unique the solution is stable, i.e. it continuously depends on the given data (here: V )

4 Inverse gravimetric problem is ill-posed The IGP is an ill-posed inverse problem since violates all three of Hadamard s criteria. Existence: V has to be harmonic outside the Earth which may be violated by measurement erros Uniqueness: Only the harmonic part of the density function can uniquely be reconstructed, whereas the (in the sense of the L2-space) orthogonal, so-called anharmonic, part has the external potential and, therefore, does not leave any trace in gravitational measurements. This is the most serious problem of IGP. Stability: The density does not continuously depend on the gravitational potential.

5 Non-uniqueness Let ϱ 1 be a solution of the IGP and let B have a continuous normal n. Then a class of all solutions of the IGP in L 2 (B) can be represented as where ϱ = ϱ h h B = h n = B In other words, 2 h generates a null external gravitational field. The Green s third identity proves this: for P in E 3 \B it holds 2 [ h(q) h(q) 1 V (P) = G dv Q = G h(q) ( )] 1 ds Q = B d PQ B n d PQ n d PQ i.e., V = in E 3 \B and hence on B.

6 Minimum-norm solution The minimum-norm solution: B [δϱ(r, Ω)] 2 dv = min δϱ under the integral constraints for potential coefficients V jm = δϱ(r, Ω)r j Y jm (Ω)dV B The variational form using the Lagrange-multiplier method j 1 max j ] δ [δϱ(r, Ω)] 2 dv + α jm [V jm δϱ(r, Ω)r j Y jm (Ω)dV 2 B B = j= m= j where δ is the variation with respect to δϱ. harmonic density anomaly δϱ(r, Ω) = j max j= m= j j α jm r j Y jm (Ω)

7 The decomposition of L 2 space L 2 = H H { } H := r j Y jm (Ω), j =, 1,..., m = j,..., j { H := 2 h, h B = h } n = B H the set of gravitationally equivalent bodies a solution H the harmonic solution the minimum L 2 -norm solution

8 Instability Let σ = on B and let ϱ := in E 3 \B. The Laplace-Poisson equation for potential V : 2 V = 4πGϱ in E 3 Using the 3-D Fourier transform of the form f (k) = (2π) 3/2 E 3 f (r) e ik.r d 3 r The Laplace-Poisson equation in the Fourier domain: k 2 Ṽ (k) = 4πG ϱ(k) k = k Two possible views: Forward GP Ṽ (k) = A(k) ϱ(k) Inverse GP ϱ(k) = Ṽ (k)/a(k) The transfer function mapping the density ϱ onto the potential V : A(k) = 4πG k 2 For k, 1/A(k), and a solution of the IGP is unstable.

9 Do gradiometric data bring new information on mass density? Answer: NO if the gradiometric data are considered as the Stokes potential coefficients Spherical harmonic representation of V V (r, Ω) = GM a ( a ) j+1 j V jm Y jm (Ω) r j= m= j Stokes potential coefficients V jm = 4π 1 Ma j 2j + 1 B Gradiometric model (e.g. GOCO2) grad grad V (r, Ω) = GM j= a j r j ϱ(r, Ω )Y jm(ω )dv j m= j [ ] Vjm GOCE grad grad r j 1 Y jm (Ω)

10 Newton s integral: Green s function of geoid for a static mantle δv (r, Ω) = G a r =b Using the addition theorem for r > r, 1 L(r, ψ, r ) = 4π r j j= m= j δϱ(r, Ω ) Ω L(r, ψ, r ) r 2 dr dω 1 2j + 1 ( r and writing the spherical harmonic expansions: r ) j Y jm(ω )Y jm (Ω), δv (r, Ω) = jm δv jm (r)y jm (Ω), δϱ(r, Ω) = jm δϱ jm (r)y jm (Ω) Then for r = a: δv jm (a) = a r =b G sta. j (a, r )δϱ jm (r )dr where Green s function of a static geoid is Gj sta. (a, r ) = 4πGa 2j + 1 ( r a ) j+2

11 Green s functions of static geoid (j = 2, 4, 1) The smaller spatial scales in gravitational potential, the higher sensitivity to shallower small-scale density variations, or in other words, the lower sensitivity to deeper small-scale density variations.

12 Contents 1 Static-geoid interpretation 2 Dynamic-geoid interpretation 3 Seismic tomography of the upper mantle 4 Congo-basin negative free-air gravity anomaly 5 Mass-density Green s functions for gravity

13 The Earth s interior is closed to hydrostatic equilibrium Flattening of the Earth: hydrostatic: f = 1/299. observed : f = 1/ The Earth s gravity interior is slighlty perturbed from the hydrostatic equilibrium: ϱ = ϱ }{{} hydrostatic + δϱ }{{} non-hydrostatic Additionally: Seismic tomographic model of the compressional (P) and transverse (S) seismic elastic waves Conversion of seismic anomalies to density anomalies: p f... proportionality factor δϱ ϱ = p δv f v

14 The geoid response to a static mantle Induced geoid undulations hundreds of meters Earth s mantle is not static but dynamic

15 Dynamic mantle The Earth s mantle is slighlty perturbed from the hydrostatic equilibrium: where ϱ = ϱ + δϱ g = g + δg δϱ ϱ and δg g The gravitational force acting on a material partical: ϱg ϱ g }{{} + δϱ g }{{} + ϱ δg }{{} hydrostatic force buoyancy force self-gravitating force Driving forces from equilibrium = buoyancy + self-gravitation

16 Fluid dynamic model of the geoid V δϱ V t t < t > geoid surface + δϱ > δϱ < cmb V δϱ : potential induced by mantle density anomalies δϱ V t : potential induced by deformations of boundary topographies t Courtesy of N.Tosi

17 Stokes problem for present-day mantle convection Mass conservation for an incompressible mantle: divv = Momentum conservation with Pr : divτ + f = Constitutive relation for a Newtonian viscous medium: τ = pi + 2ηε ε = ( v + T v)/2 Boundary conditions at B = B top B cmb v n = τ n (n τ n)n =

18 Weak formulation Consider a suitable functional space and create the functional F (v, p, l 1, l 2 ) = η(ε : ε) dv + f v dv + p div v dv B B B + l 1 (n v) ds + l 2 (n v) ds B top B cmb Compute the variation of F and use of Green s identity δf = (div τ + f ) δv dv + [τ n (n τ n)n] δv ds B B + (n v) δl 1 ds + (n v) δl 2 ds B top B cmb At a stationary point of F, δf =, the differential and the variational problem are equivalent.

19 Spherical-harmonic, finite-element parameterization Expansion in scalar and vector spherical harmonics with respect to angular coordinates: N j p = p jm (r)y jm (Ω) v = N j j= m= j j= m= j [U jm (r)s ( 1) jm (Ω) + V jm(r)s (1) jm (Ω) + W jm(r)s () jm (Ω)] Expansion into piecewise linear-finite elements with respect to radial coordinate: K +1 U jm (r) = Ujmψ k k (r) where: k=1 ψ k (r) = r k+1 r r k+1 r k and ψ k+1 (r) = r r k+1 r k+1 r k

20 Green s function of geoid for a dynamic mantle Assumption: 1-D viscosity model η = η(r). The gravitational effect of dynamic mantle + dynamic topographies: where a δϱ(r, Ω ) δv (r, Ω) = G r =b Ω L(r, ψ, r ) r 2 dr dω + G σ i (Ω) = ϱ i t i (Ω) n i=1 i = 1,..., n σ i (Ω ) Ω L(r, ψ, r i ) dω and t i (Ω) is the i-th dynamic topography. Easy manipulations results in δv jm (a) = a r =b G sta. j (a, r )δϱ jm (r )dr + n i=1 4πGa 2j + 1 where t jm are the coefficients of topography t(ω). Formally, δv jm (a) = a r =b G dyn. j (a, r )δϱ jm (r )dr ( ri ) j+2 ϱi (t i ) jm a Green s function of a dynamic geoid G dyn. j (a, r ) can only be modelled numerically.

21 Input data for forward dynamic-mantle modelling S-wave seismic tomographic model of Becker & Boschi (22). Conversion of seismic to density anomalies: proportionality factor p f =.2 3-layer viscosity model: δϱ ϱ = p δv f v η LT = Pa s r ( ) VM1 : η UM = Pa s r ( ) VM2 : η UM = Pa s r ( ) η LM = 4 η UM r ( )

22 Green s functions of dynamic geoid (j = 2, 8, 163) Depth Amplitude

23 Green s functions of surface dynamic topography (j = 2, 8, 16, 3) TBD

24 Geoid GOCO2 (j=2 12) SMEAN model LB model Geoid [m] Geoid [m]

25 Surface dynamic topography SMEAN model LB model Surface dynamic topography [km] Surface dynamic topography [km]

26 CMB dynamic topography SMEAN model LB model CMB dynamic topography [km] CMB dynamic topography [km]

27 SMEAN dynamic-mantle prediction SMEAN (j=2 3) 4 Free air gravity anomalies 4 Surface dynamic topography 4 CMB dynamic topography mgal km km

28 Observed vs. predicted g-fields 4 Geoid Anomalies 4 GOCO2 (j=2 3) Free Air Gravity Anomalies 4 d 2 V/dr meter mgal E Geoid Anomalies SMEAN dynamic mantle prediction Free Air Gravity Anomalies 4 4 d 2 V/dr

29 SMEAN dynamic-mantle prediction: VM1 vs. VM2 4 Geoid Anomalies 4 Viscosity model VM1 Free Air Gravity Anomalies 4 d 2 V/dr meter mgal E Geoid Anomalies 4 Viscosity model VM2 Free Air Gravity Anomalies 4 d 2 V/dr

30 Contents 1 Static-geoid interpretation 2 Dynamic-geoid interpretation 3 Seismic tomography of the upper mantle 4 Congo-basin negative free-air gravity anomaly 5 Mass-density Green s functions for gravity

31 Tomography of crust & shallow UM (Lebedev & Hilst, 28) 4 depth: 7 km 4 2 km 4 36 km km 4 8 km 4 11 km Shear velocity anomaly (%)

32 Tomography of deep UM (Lebedev & Hilst, 28) 4 depth: 15 km 4 2 km 4 26 km km 4 41 km km Shear velocity anomaly (%)

33 Lithospheric thickness Shear velocity anomaly (%) Fishwick & Bastow, 211 Lebedev & Hilst, 28

34 Contents 1 Static-geoid interpretation 2 Dynamic-geoid interpretation 3 Seismic tomography of the upper mantle 4 Congo-basin negative free-air gravity anomaly 5 Mass-density Green s functions for gravity

35 Free-air gravity anomalies GOCO2 (Free Air Gravity Anomalies) 4 Degree: mgal

36 Butterworth filter Spectral amplitude Spatial weight Cut-off frequency = 7.5 deg Angular degree Spherical distance (degree)

37 Masked GOCO2 free-air gravity anomalies j max = Free air gravity anomaly (mgal)

38 Possible contributors to the (static) gravity field Surface topography (Is ϱ = 267 kg/m 3 correct for sedimentary basin?) Lower-density sediments 3-D density variations within the crust (various geological units) Compensation at the Moho discontinuity 1-D (3-D) density variations in the lithosphere Compensation at the lithosphere boundary Pull-down or push-up by mantle flow at the lithosphere boundary 3-D density variations due to mantle convection

39 ETOPO1 topography ETOPO1 topography (m)

40 Total sediment thickness of Congo basin Sediment thickness (m) Kadima et al., 211

41 Degree amplitude spectra FA Gravity Sediments Degree amplitude spectrum (mgal) Degree j Degree amplitude spectrum (meter) Degree j

42 A density contrast The external gravitation of a body can be approximated by a material surface with either surface mass density ϱ t(ω) Airy s type σ(ω) = h ϱ(ω) Pratt s type The coefficients of the gravity induced by Airy s type mass surface: g ϱ jm = (j + 1) ϱt jm T jm = Let R jm be the residual gravity coefficients: R jm = g obs jm + g ϱ jm 3 a ϱ mean t jm 2j + 1 The density contrast ϱ will be searched by minimization of R jm : R jm Rjm = min ϱ It results in (Martinec, 1994): jm jm ϱ = (j + 1)T jm gjm obs jm (j + 1)2 T jm Tjm

43 Degree correlation and density contrast 1. 3 Degree correlation.5. Confidence level=.95 Density contrast (kg/m 3 ) kg/m Degree j Degree j Conclusions: Free-air gravity and gravity induced by sediments are correlated (in sense of statistical significance) up to degree j = 13 The density of sediments is in average of 21 kg/m 3 smaller than the density of surrounding crustal material

44 Gravity signals, j max = 13 6 FA Gravity 6 6 Sediments Free air gravity anomalies (mgal) Free air gravity anomalies (mgal)

45 Residual free-air gravity, j max = FA S gravity anomalies (mgal) Kadima et al. (211) hypothesis: Uplift of the Moho surface crustal thickening fill in by heavier mantle material

46 The effect of topography, j max = 13 Bouguer gravity corrections Geological units Bouguer gravity correction (mgal) Kadima et al., 211

47 The crustal thickness over Africa Crustal thickness (km) Crustal thickness (km) Pasyanos & Nyblade, 27

48 The Moho-induced gravity, j max = ϱ Moho = 2 kg/m Free air gravity anomalies (mgal)

49 A possible scenario The negative free-air gravity could be explained by lower-density sediments (it agrees with Kadima et al., 211), except the narrow, NW-SE oriented, positive gravity anomaly. The positive Bouguer gravity correction could be balanced by the negative Moho-induced gravity. However, the narrow, NW-SE oriented, positive gravity anomaly is amplified, not compensated (it contradicts to Kadima et al., 211 hypothesis). The Congo high-seismic velocity lithospheric craton may have a large density (due to thermal reason) and induced a higher gravity could be compensated by the negative gravity induced by mantle flow (not shown here, but also within the frame of Buiter at al., 211) Open question: How to interpret the narrow, NW-SE oriented, positive gravity anomaly?

50 Contents 1 Static-geoid interpretation 2 Dynamic-geoid interpretation 3 Seismic tomography of the upper mantle 4 Congo-basin negative free-air gravity anomaly 5 Mass-density Green s functions for gravity

51 Gravitational potential and its gradients Green s functions: G( r, r ) := V ( r) = κ B B ϱ( r ) G( r, r ) dv grad V ( r) = κ ϱ( r ) G( r, r ) dv grad grad V ( r) = κ ϱ( r ) G( r, r ) dv 1 L(r, ψ, r ) G( r, r ) := grad 1 L = r r L 3 G( r, r ) := grad grad 1 L = 1 [ I 3( r r ) ( r r ) ] L 3 L 2 B

52 An example in 2-D Cartesian geometry Observer x, x L ϱ(x, z ) Green s functions at (x =, z = ): z, z G xx = G xz = G zz = G x = G z = G = 1 L ( grad 1 ) L x ) ( grad 1 L ( grad grad 1 ) L ) ( grad grad 1 L ( grad grad 1 L ) z xx xz zz = = x L 3 = z L 3 1 x 2 + z 2 = 1 L 3 ( 1 = 3x z L 5 3x 2 L 2 ) = 1 L 3 ( 1 3z 2 L 2 )

53 Scalar, vector and tensor Green s functions for mass density 2 G zz z =const. Amplitude 1 G z G x G xz G x 1 G xx Conclusion: The higher the order of the derivative of the gravitational potential as a boundary datum, the smaller the contribution from far-zone mass-density anomalies.

54 Mass-density Green s functions for gradiometric data After lengthy derivation: in spherical geometry grad grad 1 L = 1 [K r 3 rr (t, x)e rr + K rω (t, x)(cos α e rϑ sin α e rϕ) ( ) +K ΩΩ (t, x) cos 2α (e ϑϑ e ϕϕ) 2 sin 2α e ϑϕ 1 ] 2 Krr (t, x) (e ϑϑ + e ϕϕ) Three isotropic kernels: ψ... spherical distance, x = cos ψ α... azimuth t = r /r K rr (t, x) = (j + 1)(j + 2)t j P j (x) j= K rω (t, x) = 2 1 x 2 (j + 2)t j dp j (x) dx j= K ΩΩ (t, x) = 1 2 (1 x 2 ) t j d 2 P j (x) dx 2 j=

55 Closed spatial forms K rr (t, x) = 1 3(1 tx)2 + g3 g 5 K rω (t, x) = 3t(1 tx) 1 x 2 g 5 K ΩΩ (t, x) = 1 2 (1 x 2 ) 3t 2 g 5 where g = 1 + t 2 2tx

56 Isotropic gradiometric Green s functions 6 1 Amplitude K rr K rω K ΩΩ h = 25 km Amplitude K rr K ΩΩ K rω h = 1 km ψ (degree) ψ (degree) Amplitude K rr K ΩΩ h = 5 km 2 K rω ψ (degree)

57 Integral equations for a surface-mass density For a fixed source position (r = const): Ω ϱ(ω )K α(ω, Ω )dω = d α(ω) α = 1,..., GOCE The Nyström method discretizes the integral equation by a quadrature rule: n w j ϱ(ω j )K α(ω k, Ω j ) = d α(ω k ) j=1 The Galerkin method: ϱ(ω) = n ϱ j φ j (Ω) j=1 k = 1,..., n φ j (Ω) a set of n linearly indep. functions (e.g. tapers on a spherical cap) n j=1 ϱ j dω dω Ω φ k (Ω)K α(ω, Ω )φ j (Ω ) = d α(ω)φ k (Ω)dΩ A Ω Ω A A Overdetermined s. of linear algebraic equations for a surface-mass density k = 1,..., n A α ϱ = d α α = 1,..., GOCE