Toward more complete magnetic gradiometry with the Swarm mission

The Earth possesses an intrinsic magnetic field, the major part of which is produced by a self-sustaining 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 time-varying 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 (Friis-Christensen et al. 2006) was launched by the European Space Agency (ESA) on November 22, 2013, and is the first multi-satellite mission dedicated to the geomagnetic field exploration from space. Specifically, it consists of three identical spacecrafts two of which are flying side-by-side 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.

A similar spectrum to the Mauersberger–Lowes spectrum (Backus et al. 1996; Lowes 1974) can be calculated for the three different observables \(\varvec{\Gamma }^{(0)}\), \(\varvec{\Gamma }^{(1)}\) and \(\varvec{\Gamma }^{(2)}\) as

$$\begin{aligned} R_n^{(\beta )}(r)=\frac{1}{4\pi r^2}\int _\Omega \sum _{j,k}\left[ \varvec{\Gamma }_n^{(\beta )}\right] _{jk}\left[ \varvec{\Gamma }_n^{(\beta )}\right] _{jk}d\Omega \end{aligned}$$

(22)

where \(\beta = \{0, 1, 2\}\), [ \(]_{jk}\) stands for the tensor components and \(d\Omega = r^2\sin \theta d\theta d\phi\). Based on Eqs. 16, 17 and 18, the eigenvalue relations 19, 20 and 21 as well as the orthogonality of the tensor spherical harmonics, Eq. 9, we get

$$\begin{aligned} {R}_n^{(0)}(r)&= \frac{1}{a^2}\left( \frac{a}{r}\right) ^{(2n+6)} \frac{[(n+1)(n+2)]^2}{2n+1} \sum _{m=1}^n||\gamma _n^m||^2, \end{aligned}$$

(23)

$$\begin{aligned} {R}_n^{(1)}(r)&= \frac{1}{a^2}\left( \frac{a}{r}\right) ^{(2n+6)} \frac{(n+2)^2n(n+1)}{2n+1} \sum _{m=1}^n||\gamma _n^m||^2, \end{aligned}$$

(24)

$$\begin{aligned} {R}_n^{(2)}(r)&= \frac{1}{a^2}\left( \frac{a}{r}\right) ^{(2n+6)} \frac{(n-1)n(n+1)(n+2)}{2n+1} \sum _{m=1}^n||\gamma _n^m||^2. \end{aligned}$$

(25)

Figure 1 shows the spectra \({R}_n^{(0)}(r_h)\), \({R}_n^{(1)}(r_h)\) and \({R}_n^{(2)}(r_h)\) of the observables \(\varvec{\Gamma }^{(0)}\), \(\varvec{\Gamma }^{(1)}\) and \(\varvec{\Gamma }^{(2)}\), respectively, at \(r=r_h=a+h\), where \(h=0\) km (dashed lines) and \(h=460\) km (solid lines). The coefficients \(\gamma _n^m\) are taken from the MF7¹ model by Maus et al. (2008). One observes the slightly higher power of \(\varvec{\Gamma }^{(0)}\) and the slightly weaker power of \(\varvec{\Gamma }^{(2)}\); however, it can be seen that \(\varvec{\Gamma }^{(0)}\), \(\varvec{\Gamma }^{(1)}\) and \(\varvec{\Gamma }^{(2)}\) share in general almost identical spectral levels.

Following Lowes (1966) or Langel (1987), for the case that the observations consist of \(N_d\) measurements of \(\varvec{\Gamma }^{(0)}\), i.e., \({\mathbf {d}}=\{[\varvec{\nabla } {\mathbf{B}}]_{rr}\}\), the \({\mathbf {G}}^T{\mathbf {G}}\) can be approximated by

$$\begin{aligned} {\mathbf {G}}^T {\mathbf {G}} \approx&\frac{N_d}{4\pi a^2}\int _0^{2\pi }\int _0^\pi (n+1)(n+2)(n^\prime +1)(n^\prime +2)\left(\frac{a}{r}\right) ^{(n+3)}\left(\frac{a}{r}\right) ^{(n^\prime +3)} \nonumber \\&\times {\mathbf {Z}}_{nm}^{(0)}{\mathbf {Z}}_{n^\prime m^\prime }^{(0)}\sin \theta d\theta d\varphi \nonumber \\ =&\frac{1}{a^2}\left(\frac{a}{r}\right) ^{(2n+6)}\frac{N_d[(n+1)(n+2)]^2}{2n+1}\delta _{m,m^\prime }\delta _{n,n^\prime }, \end{aligned}$$

(27)

due to orthogonality of the tensor spherical harmonics, cf. Eq. (9). Therefore, the variances of the estimated model parameters are

$$\begin{aligned} \sigma _{(0)}^2 \approx (a\sigma _d)^2 \left(\frac{r}{a}\right) ^{(2n+6)}\frac{2n+1}{N_d[(n+1)(n+2)]^2}, \quad {\mathrm {if}}\, {\mathbf {d}} = \{ [\varvec{\nabla {B}}]_{rr} \}. \end{aligned}$$

(28)

Respectively, when the observations consist of \(N_d\) measurements of \(\varvec{\Gamma }^{(1)}\), i.e., \({\mathbf {d}}=\{[\varvec{\nabla } {\mathbf{B}}]_{r\theta },[\varvec{\nabla } {\mathbf{B}}]_{r\varphi }\}\), the \({\mathbf {G}}^T{\mathbf {G}}\) reads

$$\begin{aligned} {\mathbf {G}}^T {\mathbf {G}} \approx&\frac{N_d}{4\pi a^2}\int _0^{2\pi }\int _0^\pi [(n+2)\sqrt{n(n+1)}][(n^\prime +2)\sqrt{n^\prime (n^\prime +1)}]\left(\frac{a}{r}\right) ^{(n+3)} \nonumber \\&\times \left( \frac{a}{r}\right) ^{(n^\prime +3)}{\mathbf {Z}}_{nm}^{(1)}{\mathbf {Z}}_{n^\prime m^\prime }^{(1)}\sin \theta d\theta d\varphi \nonumber \\ =&\frac{1}{a^2}\left( \frac{a}{r}\right) ^{(2n+6)}\frac{N_d[(n+2)^2n(n+1)]}{2n+1}\delta _{m,m^\prime }\delta _{n,n^\prime }. \end{aligned}$$

(29)

Therefore, the variances of the estimated model parameters are

$$\begin{aligned} \sigma _{(1)}^2 \approx (a\sigma _d)^2 \left( \frac{r}{a}\right) ^{(2n+6)}\frac{2n+1}{N_d[(n+2)^2n(n+1)]}, \quad {\mathrm { if }}\,{\mathbf {d}} = \{[\varvec{\nabla } {\mathbf{B}}]_{r\theta },[\varvec{\nabla } {\mathbf{B}}]_{r\varphi }\}. \end{aligned}$$

(30)

Finally, when the observations consist of \(N_d\) measurements of \(\varvec{\Gamma }^{(2)}\), i.e., \({\mathbf {d}}=\{[\varvec{\nabla } {\mathbf{B}}]_{\theta \theta }-[\varvec{\nabla } {\mathbf{B}}]_{\varphi \varphi }, 2[\varvec{\nabla } {\mathbf{B}}]_{\theta \varphi }\}\), the \({\mathbf {G}}^T{\mathbf {G}}\) is given by

$$\begin{aligned} {\mathbf {G}}^T {\mathbf {G}}\,\approx\,&\frac{N_d}{4\pi a^2}\int _0^{2\pi }\int _0^\pi \sqrt{(n-1)n(n+1)(n+2)}\sqrt{(n^\prime -1)n^\prime (n^\prime +1)(n^\prime +2)} \nonumber \\&\times \left( \frac{a}{r}\right) ^{(n+3)}\left( \frac{a}{r}\right) ^{(n^\prime +3)} {\mathbf {Z}}_{nm}^{(2)}{\mathbf {Z}}_{n^\prime m^\prime }^{(2)}\sin \theta d\theta d\varphi \nonumber \\ =\, &\frac{1}{a^2}\left( \frac{a}{r}\right) ^{(2n+6)}\frac{N_d[(n-1)n(n+1)(n+2)]}{2n+1}\delta _{m,m^\prime }\delta _{n,n^\prime }. \end{aligned}$$

(31)

Therefore, the variances of the estimated model parameters are

$$\begin{aligned} \sigma _{(2)}^2&\approx (a\sigma _d)^2 \left( \frac{r}{a}\right) ^{(2n+6)}\frac{2n+1}{N_d[(n-1)n(n+1)(n+2)]}, \nonumber \\&{\mathrm { if }}\;{\mathbf {d}} =\{[\varvec{\nabla } {\mathbf{B}}]_{\theta \theta }-[\varvec{\nabla } {\mathbf{B}}]_{\varphi \varphi }, 2[\varvec{\nabla } {\mathbf{B}}]_{\theta \varphi }\}. \end{aligned}$$

(32)

The variances \(\sigma _{(0)}^2\), \(\sigma _{(1)}^2\) and \(\sigma _{(2)}^2\) which are obtained when observations consist of \(\varvec{\Gamma }^{(0)}\), \(\varvec{\Gamma }^{(1)}\) and \(\varvec{\Gamma }^{(2)}\), respectively, are almost identical and are proportional to a factor of \(\frac{1}{n^3}\). If the model parameters are estimated by the observable \(\varvec{\Gamma }^{(2)}\), singularity emerges for degree \(n=1\), which was also pointed out by its eigenvalue Eq. (21). The analytically calculated model variances a) \(\sigma _{(0)}^2\), b) \(\sigma _{(1)}^2\) and c) \(\sigma _{(2)}^2\), cf. Eqs. (28), (30) and (32), are shown in Fig. 2 in dependence of degree n and order m. White represents high variance, i.e., low information content, whereas green represents low variance, i.e., high information content. One can see that the specific combinations \(\{[\varvec{\nabla } {\mathbf{B}}]_{r\theta },[\varvec{\nabla } {\mathbf{B]}}]_{r\varphi }\}\) and \(\{[\varvec{\nabla } {\mathbf{B}}]_{\theta \theta }-[\varvec{\nabla } {\mathbf{B}}]_{\varphi \varphi }, 2[\varvec{\nabla } {\mathbf{B}}]_{\theta \varphi }\}\), namely \(\varvec{\Gamma }^{(1)}\) resp. \(\varvec{\Gamma }^{(2)}\), which basically are combinations of North–South and East–West gradients, have in theory almost identical information content as the radial gradient \([\varvec{\nabla } {\mathbf{B}}]_{rr}\), i.e., the observable \(\varvec{\Gamma }^{(0)}\). For that reason and for simplicity, we refer to \(\varvec{\Gamma }^{(1)}\) and \(\varvec{\Gamma } ^{(2)}\) as pseudo-radial gradients.

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.

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 first-order 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 pseudo-radial 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 high-quality 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 time-dependent part can be an extension to the current work and the topic of a separate paper.