Toward more complete magnetic gradiometry with the Swarm mission
 Stavros Kotsiaros^{1}Email authorView ORCID ID profile
Received: 31 December 2015
Accepted: 27 June 2016
Published: 22 July 2016
Abstract
An analytical and numerical analysis of the spectral properties of the gradient tensor, initially performed by Rummel and van Gelderen (Geophys J Int 111(1):159–169, 1992) for the gravity potential, shows that when the tensor elements are grouped into sets of semitangential and puretangential parts, they produce almost identical signal content as the normal element. Moreover, simple eigenvalue relations can be derived between these sets and the spherical harmonic expansion of the potential. This theoretical development generally applies to any potential field. First, the analysis of Rummel and van Gelderen (1992) is adapted to the magnetic field case and then the elements of the magnetic gradient tensor are estimated by 2 years of Swarm data and grouped into \(\varvec{\Gamma }^{(1)} = \{[\varvec{\nabla } {\mathbf{B}}]_{r\theta },[\varvec{\nabla } {\mathbf{B}}]_{r\varphi }\}\) resp. \(\varvec{\Gamma }^{(2)} = \{[\varvec{\nabla } {\mathbf{B}}]_{\theta \theta }[\varvec{\nabla } {\mathbf{B}}]_{\varphi \varphi }, 2[\varvec{\nabla } {\mathbf{B}}]_{\theta \varphi }\}\). It is shown that the estimated combinations \(\varvec{\Gamma }^{(1)}\) and \(\varvec{\Gamma }^{(2)}\) produce similar signal content as the theoretical radial gradient \(\varvec{\Gamma }^{(0)} = \{[\varvec{\nabla } {\mathbf{B}}]_{rr}\}\). These results demonstrate the ability of multisatellite missions such as Swarm, which cannot directly measure the radial gradient, to retrieve similar signal content by means of the horizontal gradients. Finally, lithospheric field models are derived using the gradient combinations \(\varvec{\Gamma }^{(1)}\) and \(\varvec{\Gamma }^{(2)}\) and compared with models derived from traditional vector and gradient data. The model resulting from \(\varvec{\Gamma }^{(1)}\) leads to a very similar, and in particular cases improved, model compared to models retrieved by using approximately three times more data, i.e., a full set of vector, North–South and East–West gradients. This demonstrates the high information content of \(\varvec{\Gamma }^{(1)}\).
Keywords
Crustal field Swarm gradients Field modelingIntroduction
The Earth possesses an intrinsic magnetic field, the major part of which is produced by a selfsustaining dynamo operating in the outer core. However, what is measured at or near the Earth’s surface is a superposition of the core field, the lithospheric field due to magnetized rocks in the Earth’s lithosphere, external fields caused by electric currents in the ionosphere and the magnetosphere, and fields due to currents induced in the Earth by the timevarying external fields. More than 14 years of satellite measurements from Ørsted (Neubert et al. 2001) and CHAMP (Reigber et al. 2005) led to detailed and precise models of Earth’s magnetic field. The Swarm satellite mission (FriisChristensen et al. 2006) was launched by the European Space Agency (ESA) on November 22, 2013, and is the first multisatellite mission dedicated to the geomagnetic field exploration from space. Specifically, it consists of three identical spacecrafts two of which are flying sidebyside at lower altitudes (roughly at 450 km initial altitude) separated in longitude by \(1.4^\circ\), which allows for an instantaneous estimation of the East–West gradient of the magnetic field. The third flies at higher altitude (530 km) and at different local time compared to the lower pair. Kotsiaros et al. (2015) have recently showed that North–South gradients can be approximated by first differences along the orbit track and a model of the Earth’s magnetic field based on East–West and North–South gradients estimated from Swarm data differences has already been presented by Olsen et al. (2015).
On the other hand, Swarm with its current configuration is unable to directly estimate the radial gradient since this would require two satellites separated along the radial direction. However, the radial gradient seems to provide the highest information content (Kotsiaros and Olsen 2012) and could therefore be important for improving existing magnetic field models. In order to overcome this limitation of Swarm, we translate a tensorial analysis originally developed for the gravity field by Rummel and van Gelderen (1992) to the magnetic field case which shows that specific combinations of East–West and North–South gradients provide a similar signal content to the radial gradient. Thus, radial gradient information can indirectly be inferred using the Swarm constellation and therefore additional information on primarily the lithospheric field could possibly be extracted.
Tensor harmonics
Spectral analysis
In the next section, I try to identify whether this conclusion could have a practical use in the Swarm case, e.g., can we make use of the North–South and East–West gradient combinations to get similar information as the radial gradient which is not measured by the Swarm constellation? To facilitate this, the observables \(\varvec{\Gamma }^{(1)}\) and \(\varvec{\Gamma }^{(2)}\) are estimated from Swarm data and their information content (regarding the recovery of the model parameters) is tested against the content of the theoretical radial gradient \(\varvec{\Gamma }^{(0)}\). Subsequently, lithospheric field models are derived using the observables \(\varvec{\Gamma }^{(1)}\) and \(\varvec{\Gamma }^{(2)}\), which are estimated from Swarm data.
Lithospheric field modeling with Swarm pseudoradial gradients

\(\it{{\mathrm {SM}}_{\mathrm {v}}}\) model derived by vector data from Swarm Alpha.

\({\mathrm {SM}}_{\mathrm {vNS}}\) model derived by Swarm Alpha vector data and North–South gradients estimated from Swarm Alpha vector data, see Eqs. (33) and (35).

\({\mathrm {SM}}_{\mathrm {vEW}}\) model derived by Swarm Alpha vector data and East–West gradients estimated from Swarm Alpha and Charlie vector data, see Eqs. (34), (36) and (37).

\({\mathrm {SM}}_{\mathrm {vNSEW}}\) model derived by Swarm Alpha vector, North–South and East–West gradients estimated from Swarm Alpha and Charlie vector data. For this model, North–South and East–West gradients (as well as their associated kernel matrices) are approximated by simple first differences (no division with the spherical distance), in a similar fashion to how the gradients are commonly treated for modeling, see, for example, Finlay et al. (2016), Olsen et al. (2015), Olsen et al. (2016), Sabaka et al. (2015) .

\({\mathrm {SM}}_{{\Gamma ^{(1)}}\delta }\) this model is equivalent to \({\mathrm {SM}}_{{\Gamma ^{(1)}}}\) except that the gradients involved in \(\varvec{\Gamma }^{(1)}\) (and the associated gradient kernel matrices) are approximated by simple data differences (no division with the spherical distance).
Conclusions
It has been shown that the gradient combinations \(\varvec{\Gamma }^{(1)}=\{[\varvec{\nabla } \mathbf{B}]_{r\theta },[\varvec{\nabla } \mathbf{B}]_{r\varphi }\}\) and \(\varvec{\Gamma }^{(2)}=\{[\varvec{\nabla } \mathbf{B}]_{\theta \theta }[\varvec{\nabla } \mathbf{B}]_{\varphi \varphi }, 2[\varvec{\nabla } \mathbf{B}]_{\theta \varphi }\}\) produce similar signal content as the radial gradient \(\varvec{\Gamma }^{(0)}=\{[\varvec{\nabla } \mathbf{B}]_{rr}\}\). That is a general theoretical conclusion from special tensorial analysis of potential fields and has been already shown for the gravity potential by Rummel and van Gelderen (1992). For the geomagnetic case, this has important implications because the gradients of the field are not measured instantaneously by a single satellite instrument. The radial gradient is currently not possible to be measured, as, for example, in the gravity case and the GOCE mission where the complete gradient tensor can be determined instantaneously by a gravity gradiometer (Rummel et al. 2011). Nevertheless, contrary to the radial gradient \(\{[\varvec{\nabla } \mathbf{B}]_{rr}\}\), which is not possible to be measured with the Swarm configuration, the gradient combinations \(\{[\varvec{\nabla } \mathbf{B}]_{r\theta },[\varvec{\nabla } \mathbf{B}]_{r\varphi }\}\) and \(\{[\varvec{\nabla } \mathbf{B}]_{\theta \theta }[\varvec{\nabla } \mathbf{B}]_{\varphi \varphi }, 2[\varvec{\nabla } \mathbf{B}]_{\theta \varphi }\}\) can be determined by the lowest satellite pair and contain similar information regarding the estimation of the model parameters as the radial gradient. Furthermore, the gradient combinations can be related to the magnetic field potential and the traditional spherical harmonic expansion coefficients with the help of simple eigenvalue relations, and therefore, they can be used for magnetic field modeling. For the first time, magnetic field models are derived exclusively from estimated gradient observations as opposed to the standard technique of using a combination of vector and estimated gradients (Finlay et al. 2016; Olsen et al. 2015, 2016; Sabaka et al. 2015). Moreover, one of the main models (\({\mathrm {SM}}_{{\Gamma }^{(1)}}\)) presented here is built by gradient estimates using firstorder Taylor approximation (dividing by the spherical distance), on the contrary to the standard approach of estimating gradients simply with vector differences (Kotsiaros et al. 2015). It seems that when gradients are estimated by simple data differences, such as the model \({\mathrm {SM}}_{{\Gamma ^{(1)}}\delta }\) presented here, instead of dividing by the spherical distance, the derived models are slightly improved. The models are derived using 23 months of Swarm data with a sampling rate of 15 seconds. Tests have also been performed deriving equivalent models to the ones presented here using 5 s sampling rate, but no particular improvements were detected. The pseudoradial gradient \(\varvec{\Gamma }^{(1)}=\{[\varvec{\nabla } \mathbf{B}]_{r\theta },[\varvec{\nabla } \mathbf{B}]_{r\varphi }\}\), estimated with simple vector differences, leads to a very similar lithospheric model to the model retrieved by using approximately three times more data, i.e. a full set of vector, North–South and East–West gradients. This demonstrates the high information content of \(\varvec{\Gamma }^{(1)}\). Moreover, despite not accounting for the polar gaps, models resulting from \(\varvec{\Gamma }^{(1)}\) do not suffer from ringing and seem to agree better in the polar gap regions to the reference models MF7 and CM5 than the model built from the full set of vector, North–South and East–West gradients.
In this paper, the performance of lithospheric field models, such as the \({\mathrm {SM}}_{{\Gamma ^{(1)}}}\) determined exclusively by gradient combinations has been studied in detail. However, tests have also been performed extending the \({\mathrm {SM}}_{{\Gamma ^{(1)}}}\) to include the static main field (degrees n = 1–15). The extended \({\mathrm {SM}}_{{\Gamma ^{(1)}}}\) is derived in a similar fashion to \({\mathrm {SM}}_{{\Gamma ^{(1)}}}\), which is presented here, except that model predictions of the core field are not subtracted from the Swarm data. The initial tests show that, similarly to the lithospheric field case, \(\varvec{\Gamma }^{(1)}\) can lead to highquality models also for the static part of the core field. Further tests regarding the performance of \(\varvec{\Gamma }^{(1)}\) on the determination of the static core field as well as its timedependent part can be an extension to the current work and the topic of a separate paper.
Declarations
Acknowledgements
The author would like to thank ESA STSE Changing Earth Science Network for financing the SGC project, within the framework of which this work was completed. The author also wishes to express his gratitude to Rainer Rummel for his ideas on tensorial analysis and for providing influential motivation in the realization of this work. Chris Finlay and Nils Olsen are also thanked for their support and for the helpful discussions on the various aspects of geomagnetic field modeling. Terry Sabaka is thanked for providing CM5 model.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Backus G, Parker R, Constable C (1996) Foundations of geomagnetism. Cambridge University Press, CambridgeGoogle Scholar
 Constable CG (1988) Parameter estimation in nonGaussian noise. Geophys J 94:131–142View ArticleGoogle Scholar
 Finlay CC, Olsen N, Kotsiaros S, Gillet N, TøffnerClausen L (2016) Recent geomagnetic secular variation from Swarm and ground observatories in the CHAOS6 geomagnetic field model. Earth Planets Space. doi:https://doi.org/10.1186/s4062301604861 Google Scholar
 FriisChristensen E, Lühr H, Hulot G (2006) Swarm: a constellation to study the Earth’s magnetic field. Earth Planets Space 58(4):351–358View ArticleGoogle Scholar
 Groten E, Moritz H, P. Ohio State University. Institute of Geodesy, Cartography, and A. F. C. R. L. (U.S.) (1964) On the accuracy of Geoid heights and deflections of the vertical, report (Ohio State University. Institute of Geodesy, Photogrammetry and Cartography), Ohio State University Research FoundationGoogle Scholar
 Huber PJ (1964) Robust estimation of a location parameter. Ann Math Stat 35(1):73–101. doi:https://doi.org/10.1214/aoms/1177703732 View ArticleGoogle Scholar
 Kotsiaros S, Finlay C, Olsen N (2015) Use of alongtrack magnetic field differences in lithospheric field modelling. Geophys J Int 200(2):878–887View ArticleGoogle Scholar
 Kotsiaros S, Olsen N (2012) The geomagnetic field gradient tensor. GEM Int J Geomath 3:297–314. doi:https://doi.org/10.1007/s1313701200416 View ArticleGoogle Scholar
 Langel RA (1987) The main field. In: Jacobs JA (ed) Geomagnetism, vol 1. Academic Press, London, pp 249–512Google Scholar
 Langel RA, Hinze W (1998) he magnetic field of the Earth’s lithosphere. The satellite perspective. Cambridge University Press, Cambridge, pp 113–122View ArticleGoogle Scholar
 Lowes FJ (1966) Meansquare values on sphere of spherical harmonic vector fields. J Geophys Res 71:2179View ArticleGoogle Scholar
 Lowes FJ (1974) Spatial power spectrum of the main geomagnetic field, and extrapolation to the core. Geophys J R Astr Soc 36:717–730View ArticleGoogle Scholar
 Maus S, Lühr H, Rother M, Hemant K, Balasis G, Ritter P, Stolle C (2007) Fifthgeneration lithospheric magnetic field model from CHAMP satellite measurements. Geochem Geophys Geosyst 8(5):Q05,013View ArticleGoogle Scholar
 Maus S, Yin F, Lühr H, Manoj C, Rother M, Rauberg J, Michaelis I, Stolle C, Müller R (2008) Resolution of direction of oceanic magnetic lineations by the sixthgeneration lithospheric magnetic field model from CHAMP satellite magnetic measurements. Geochem Geophys Geosyst 9(7):Q07,021View ArticleGoogle Scholar
 Meissl P (1971) A study of covariance functions related to the Earth’s disturbing potential. Ohio State Univ Columbus Dept of Geodetic Science, report No. 151Google Scholar
 Neubert T et al (2001) Ørsted satellite captures highprecision geomagnetic field data. Eos Trans AGU 82(7):81–88View ArticleGoogle Scholar
 Olsen N et al (2015) The swarm initial field model for the 2014 geomagnetic field. Geophys Res Lett 42(4):2014GL062,659View ArticleGoogle Scholar
 Olsen N, Finlay CC, Kotsiaros S, TøffnerClausen L (2016) A model of earth’s magnetic field derived from two years of swarm satellite constellation data. Earth Planets Space. doi:https://doi.org/10.1186/s406230160488z Google Scholar
 Reigber C, Lühr H, Schwintzer P, Wickert J (2005) Earth observation with CHAMP: results from three years in orbit. Springer, BerlinView ArticleGoogle Scholar
 Richmond AD (1995) Ionospheric electrodynamics using magnetic apex coordinates. J Geomagn Geoelectr 47:191–212View ArticleGoogle Scholar
 Rummel R (1997) Spherical properties of the earth’s gravitational potential and its first and second derivatives. Lecture Notes in Earth Sciences. Springer, Berlin 65:359–404. doi:https://doi.org/10.1007/BFb0011699
 Rummel R, Yi W, Stummer C (2011) Goce gravitational gradiometry. J Geod 85(11):777–790. doi:https://doi.org/10.1007/s0019001105000 View ArticleGoogle Scholar
 Rummel R, van Gelderen M (1992) Spectral analysis of the full gravity tensor. Geophys J Int 111(1):159–169View ArticleGoogle Scholar
 Sabaka TJ, Olsen N, Tyler RH, Kuvshinov A (2015) CM5, a preSwarm comprehensive magnetic field model derived from over 12 years of CHAMP, Ørsted, SACC and observatory data. Geophys J Int 200:1596View ArticleGoogle Scholar
 van Gelderen M, Koop R (1997) The use of degree variances in satellite gradiometry. J Geod 71(6):337–343View ArticleGoogle Scholar
 Zerilli JF (1970) Tensor harmonics in canonical form for gravitational radiation and other applications. J Math Phys 11(7):2203–2208. doi:https://doi.org/10.1063/1.1665380 View ArticleGoogle Scholar