Abstract
We develop the general mathematical basis for space magnetic gradiometry in spherical coordinates. The magnetic gradient tensor is a second rank tensor consisting of 3 × 3 = 9 spatial derivatives. Since the geomagnetic field vector B is always solenoidal \({(\nabla \cdot {\rm B}=0)}\) there are only eight independent tensor elements. Furthermore, in current free regions the magnetic gradient tensor becomes symmetric, further reducing the number of independent elements to five. In that case B is a Laplacian potential field and the gradient tensor can be expressed in series of spherical harmonics. We present properties of the magnetic gradient tensor and provide explicit expressions of its elements in terms of spherical harmonics. Finally we discuss the benefit of using gradient measurements for exploring the Earth’s magnetic field from space, in particular the advantage of the various tensor elements for a better determination of the small-scale structure of the Earth’s lithospheric field.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arfken G.B., Weber H.J.: Mathematical Methods for Physicists, 6th edn. Elsevier Academic Press, Amsterdam (2005)
Backus G.: Poloidal and toroidal fields in geomagnetic field modeling. Rev. Geophys. 24, 75–109 (1986)
Backus G., Parker R., Constable C.: Foundations of Geomagnetism. Cambridge University Press, Cambridge (1996)
Cain, J.C.: POGO (OGO-2, −4 and −6 spacecraft). In: Gubbins, D., Herrero-Bervera, E. (eds.) Encyclopedia of Geomagnetism and Paleomagnetism, pp. 828–829, Springer, Heidelberg (2007)
Christensen A., Rajagopalan S.: The magnetic vector and gradient tensor in mineral and oil exploration. Preview 84, 77 (2000)
Freeden W., Nutz H.: Satellite gravity gradiometry as tensorial inverse problem. GEM Int. J. Geomath. 2, 177–218 (2011)
Harrison C., Southam J.: Magnetic field gradients and their uses in the study of the Earth’s magnetic field. J. Geomagn. Geoelectr. 43, 585–599 (1991)
Langel, R.A.: The main field. In: Jacobs, J.A. (ed) Geomagnetism, vol. 1, pp. 249–512. Academic Press, London (1987)
Lowes F.J.: Mean-square values on sphere of spherical harmonic vector fields. J. Geophys. Res. 71, 2179 (1966)
Mie G.: Considerations on the optic of turbid media, especially colloidal metal sols. Ann. Phys. 25, 377–442 (1908)
Olsen N.: Ionospheric F region currents at middle and low latitudes estimated from Magsat data. J. Geophys. Res. 102(A3), 4563–4576 (1997)
Olsen, N., Kotsiaros, S.: Magnetic satellite missions and data. In: Mandea, M., Korte, M. (eds.) Geomagnetic Observations and Models, IAGA Special Sopron Book Series, Chap. 2, vol. 5, pp. 27–44. Springer, Heidelberg. doi:10.1007/978-90-481-9858-0_2 (2011)
Olsen, N., Friis-Christensen, E., Hulot, G., Korte, M., Kuvshinov, A.V., Lesur, V., Lühr, H., Macmillan, S., Mandea, M., Maus, S., Purucker, M., Reigber, C., Ritter, P., Rother, M., Sabaka, T., Tarits, P., Thomson, A.: Swarm: End-to-End Mission Performance Simulator Study, ESA Contract No 17263/03/NL/CB. DSRI Report 1/2004, Danish Space Research Institute, Copenhagen (2004)
Olsen N., Hulot G., Sabaka T.J.: Measuring the Earth’s magnetic field from space: concepts of past, present and future missions. Space Sci. Rev. 155, 65–93 (2010). doi:10.1007/s11214-010-9676-5
Pedersen L.B., Rasmussen T.M.: The gradient tensor of potential field anomalies: Some implications on data collection and data processing of maps. Geophysics 55(12), 1558–1566 (1990)
Purucker M., Sabaka T., Le G., Slavin J.A., Strangeway R.J., Busby C.: Magnetic field gradients from the st-5 constellation: improving magnetic and thermal models of the lithosphere. Geophys. Res. Lett. 34(24), L24,306 (2007)
Reed, G.B.: Application of kinematical geodesy for determining the short wave length components of the gravity field by satellite gradiometry. Reports of the Department of Geodetic Science (Report No. 201) (1973)
Rummel R., Colombo O.: Gravity field determination from satellite gradiometry. J. Geodesy 59, 233–246 (1985)
Rummel, R., van Gelderen, M., Koop, R., Schrama, E., Sanso, F., Brovelli, M., Miggliaccio, F., Sacerdote, F.: Spherical harmonic analysis of satellite gradiometry. Technical Report (1993)
Sabaka, T.J., Hulot, G., Olsen, N.: Mathematical properties relevant to geomagnetic field modelling. In: Freeden, W., Nashed, Z., Sonar, T. (eds.) Handbook of Geomathematics. Springer, Heidelberg (2010)
Schmidt P., Clark D.: Advantages of measuring the magnetic gradient tensor. Preview 85, 26–30 (2000)
Schmidt P., Clark D.: The magnetic gradient tensor: its properties and uses in source characterization. Lead. Edge 25(1), 75–78 (2006)
Talpaert Y.: Tensor analysis and continuum mechanics. Kluwer Academic Publishers, Dordrecht (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kotsiaros, S., Olsen, N. The geomagnetic field gradient tensor. Int J Geomath 3, 297–314 (2012). https://doi.org/10.1007/s13137-012-0041-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13137-012-0041-6