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Improving the space-domain methods of forward modelling for geodynamic studies by GRACE

Improving the space-domain methods of forward modelling for geodynamic studies by GRACE. by Gabriel Strykowski, Danish National Space Center. Contents Motivation Theory Results Conclusions. Motivation

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Improving the space-domain methods of forward modelling for geodynamic studies by GRACE

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  1. Improving the space-domain methods of forward modelling for geodynamic studies by GRACE by Gabriel Strykowski, Danish National Space Center

  2. Contents • Motivation • Theory • Results • Conclusions Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  3. Motivation Geodynamic studies from space (e.g. GRACE) call on revision of methods of forward modelling of the gravitational signal generated by known sources. Why? The geodynamic signals are often weak and masked by the effect of stronger sources. For example, the gravitational effect of the post glacial land uplift in Scandinavia (0-1 cm/year) is masked by much stronger gravitational effect of topography and other sources. Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  4. Motivation. Cons of space domain methods “Definite integral – problem” (exact formulas) Ex. Elementary body: homogenous, rectangular prism where and involves arctan and log Field point located far away from the prism: Inadequate approximation (in double precision) of . Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  5. Motivation. Cons of space domain methods (cont.) Power series expansion is numerically more stable then closed formulas (McMillan, 1958; Forsberg, 1984; Sigl, 1985; Strykowski, 2003) series Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  6. Motivation. Cons of space domain methods (cont.) • Other important problems: • ‘Memory-less’- system: Computations start from ´scratch´ • every time the field point is changed. • Difficult to increase the model complexity by including • varying mass density. • (few ‘closed expressions’ exist for some gravitational quantities • and/or the number of prism can be increased) Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  7. Theory Sigl (1985) - the ´reciprocal distance´-function expanded into power series. Strykowski (2005) - expansion both with respect to the source- and the field domain P and Q are fixed support points for the field- and the source domain. P* and Q* are variable points in, respectively, the source domain (blue rectangle) and the field domain (red rectangle). chosen convention: domains are cubes with a fix point in its centre Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  8. Theory (cont.) Strykowski, 2005: Observe: qPQ is scaledby rPQ. Subsequent studies of the power series approximation using qPQ can be conducted for all distances rPQ on a’normalised problem’ rPQ=1. Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  9. Theory (cont.) Binomial series convergence criterion: truncation degree: N Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  10. Theory (cont.) In matrix notation where - unit matrix Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  11. Theory (cont.) Differentiation w.r.t field coordinates where Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  12. Theory (cont.) Differentiation w.r.t field coordinates where Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  13. Theory (cont.) Differentiation w.r.t field coordinates In general General structure ’field strength’ ’appropriate truncation’ ’relative position to fixed’ ’type of derivative’ Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  14. Relative accuracy of representation of the reciprocal distance function by truncated power series expansion. ‘Normalized problem’ rPQ = 1 N=100,LQ=.2, LP=.8 <4.4510-11 <1.9910-8 <5.9510-5 <3.3810-3 N=100, LQ=.1, LP=.5 <4.4810-16 <1.4010-15 <1.0410-11 <6.2510-13 N=100,LQ=.006,LP=.5 <4.4110-16 <3.3410-15 <2.6810-13 <5.3110-13 N=50, LQ=.006,LP=.5 <4.4110-16 <3.3610-15 <3.1610-13 <4.7310-11 N=30, LQ=.006,LP=.5 <6.0110-12 <1.1210-9 <6.3810-7 <5.9910-5 N=25, LQ=.006, LP=.5 <2.8410-10 <4.4410-8 <2.2110-5 <1.7510-3 N=15, LQ=.006, LP=.5 <6.7810-7 <6.5410-5 <2.1510-2 <1.02 N – truncation degree of power series for the potential LQ – side length of the source point domain (a cube of size LQ LQ LQ) LP – side length of the source point domain (a cube of size LP LP LP) Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  15. Scandinavian DEM and GRACE Altitude of GRACE orbit: 300 km – 500 km DEM: Scandem.01 Area: 52°N - 72°N , 4°E - 32°E (approx. 2224 km × 1462 km) Spacing: 0.01° × 0.02° (approx. 1.1 km × 1.0 km) heights: 0 m - 2266 m Choice of domains N: 30 Min rPQ: 400000 m Vertical level of “source support points”: 1133 m Vertical level of “field support points”: 300000 m Side length of “source-domain cube”: 2400 m Side length of “field-domain cube”: 200000 m Spacing of “source support points”: 0.02° × 0.04° (i.e. 2 x 2 source grid points) Spacing of “field support points”: 1° × 2° Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  16. Theory (cont.) Ellipsoidal and/or spherical Earth model • Main idea: local Cartesian frames of reference used (i.e. global ellipsoidal-/spherical coordinates not needed!!!) • Important properties of the proposed method: • Once for all-,flexible-, local source integration (i.e. independent from the field point). • Source information weighted by the fixed point configuration is stored in the fixed field-points. • Different quantities of the gravitational field can be modelled locally in the field domain (associated with the closest fixed field point). Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  17. Theory (cont.) Ellipsoidal and/or spherical Earth model Needed: rotation of field domain w.r.t source domain Mathematical structure prior to rotation: Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  18. Theory (cont.) Ellipsoidal and/or spherical Earth model Transforming from ’Q Cartesian frame of reference’ to ’P Cartesian frame of reference’ where is a rotation matrix Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  19. Theory (cont.) Ellipsoidal and/or spherical Earth model Transforming from ’P Cartesian frame of reference’ to ’P* Cartesian frame of reference’ where is a rotation matrix Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  20. Theory (cont.) qPQ in a convenient matrix notation ’field point’: ’Fix points’: ’source point’: Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  21. Theory (cont.) qPQ in a convenient matrix notation multinomial formula highest degree: 2n-1 where Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  22. Theory (cont.) Source integration Ωj is source domain with index j, G the gravitational constant, ρ density. Contribution from Ωj to the gravitational potential in P* is: Defining for each n, n=0,1,…N: Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  23. Theory (cont.) Source integration where Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  24. Theory (cont.) Source integration Examples of mass density functions that can be handled Examples of layer boundaries that can be handled Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  25. Theory (cont.) Source integration DEM constructed from rectangular, homogenous prisms Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  26. Theory (cont.) Stacking info from all source domains where stacking Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  27. Theory (cont.) Mass density integration (cont.) • Results • All key elements of the method accounted for: • choice of the truncation degree and the size of domains • once-for-all source integration • weighting according to the configuration of the fix points • stacking of contributions from all source domains • To be investigated: • efficient summation (interrelation between the terms) • spatial derivatives • how many terms are significant for a particular derivative Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

  28. Conclusions Presented A new method of space-domain forward modelling based on power series approximation of the reciprocal distance function. Merits of the method Once-for-all and flexible source integration and storing. For any point in space within the field domain (in 3D), the stored information can be used to compute any fieldcomponent to a chosen degree of accuracy. Strykowski, G.:Improving the space-domain methods of forward modelling for geodynamic studies by GRACE.

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