On the numerical computation of fields and field gradients from polyhedral gravitational and magnetic sources

Événement passé
20 février 2018
5, rue René Descartes, Salle du Conseil 2ème étage

Séminaire externe, équipe Dylbas, le 20 février à 13h45 dans la salle du conseil.

Intervenant : Horst Holstein, Aberystwyth University, UK

Titre : On the numerical computation of fields and field gradients from polyhedral gravitational and magnetic sources

Abstract : In this talk I shall survey the development of closed form solutions to the gravitational potential, and of its gradients, for a polyhedral target.  By Poisson's relation, this will include magnetic cases as well.   Since the 1960's, many non-identical looking solutions have been proposed, and there is much duplication of effort found in the related disciplines of geophysics, celestial mechanics and geodesy. I shall argue that desirable attributes in an anomaly solution for a polyhedral target include
1. coordinate-system independence (based on vectors and tensors);
2. similarity (links between solutions for the potential, field and field gradient);
3. low arithmetic complexity (leading to efficient code);
4. absence or flagging of singularities;
5. a priori numerical stability estimates (indication of the number of significant digits in the result).

I shall demonstrate that these goals can be realised for polyhedral targets of constant density. Recently, there have been important advances in the gravity field formulae for polyhedra with the polynomial density contrast. I shall argue that the above 5 attributes also serve as guidelines for good practice in this development. Finally, I shall illustrate specialisations of the general polyhedral theory, to two target cases, namely (1) thin polygonal sheets, and (2) 2D faults.  The latter case has recently been treated, in a setting of great generality, by le Maire and Munschy.

References (to be cited in the seminar)

2002 Holstein, H.  Gravimagnetic similarity in anomaly formulas for uniform polyhedra.   Geophysics, 67, 1126?1133.  

2002 Holstein, H.  Invariance in gravimagnetic formulas for uniform polyhedra. Geophysics, 67, 1134?1137.      

2003 Holstein, H.  Gravimagnetic anomaly formulas for polyhedra of spatially linear media".  Geophysics, 68, 157-167.

2009 Hamayun, I. Prutkin, and R. Tenzer.  The optimum expression for the gravitational potential of polyhedral bodies having a linearly varying density distribution. Journal of Geodesy, 83, no. 12,1163.          

2009 Holstein H., D. FitzGerald, C. Anastasiades.  Gravi-magnetic anomalies of uniform thin polygonal sheet. 11th SAGA Biennial Technical Meeting and Exhibition Session:  Numerical Methods. DOI10.3997/2214-4609.201400719           .

2014 D'Urso, M. G.  Gravity effects of polyhedral bodies with linearly varying density.  Celestial Mechanics and Dynamical Astronomy, 120, no. 4, 349-372.       

2014 Fitzgerald, D., and H Holstein. Structural foliations from gravity gradient data for a 2D fault. SEG Annual Meeting Post-convention Gravity and Magnetics Workshop, Denver. DOI: 10.13140/2.1.3440.0006.      

2014 H Holstein H., D. Hillan and D. Fitzgerald.  Gravity anomalies of polygonal sheets with linear density variation.  76th EAGE International Conference and Exhibition, 16-19 June, Amsterdam.  DOI: 10.3997/2214-4609.20140892.

2017 D'Urso, M. G., and S. Trotta. Gravity anomaly of polyhedral bodies having a polynomial density contrast. Surveys in Geophysics, 38, no. 4,781-832.

2018 Ren Z., Y. Zhong, C. Chen, J. Tang, and K. Pan (2018).  Gravity anomalies of arbitrary 3D polyhedral bodies with horizontal and vertical mass contrasts up to cubic order.  Geophysics, 83(1), G1-G13.

2018 Ren, Z., Y. Zhong, C. Chen, J. Tang, T. Kalscheuer and H. Maurer.  Gravity gradient tensor of arbitrary 3D polyhedral bodies with up to third-order polynomial horizontal and vertical mass contrasts. Submitted to Surveys in Geophysics.

2018 Le Maire, P., and M. Munschy.  2D Potential theory using complex algebra: new equations and visualisation for the interpretation of potential field data. Accepted for: Geophysics.