Data form Swarm Alpha and Charlie have been selected during 23 months, between April 2014 and March 2016, with a sampling rate of 15 seconds. To derive a lithospheric field model, predictions from CHAOS6 (Finlay et al. 2016) for both the core field (up to spherical harmonic degree \(N = 15\)) and the largescale magnetospheric field are subtracted from the vector data. Known disturbed days, for example associated with satellite maneuvers, are excluded. In addition, outliers for which the vector components exceed 300 nT the CHAOS6 model predictions are removed. Due to the failure of both absolute scalar magnetometers on Swarm Charlie after November 2014, its vector magnetometer is calibrated using scalar field values mapped over from Swarm Alpha. Moreover, vector field data are selected from dark regions (sun at least \(10^\circ\) below the horizon) and during relatively quiet geomagnetic conditions such as the change in the RCindex ≤3 nT/h and \(K_{p} \le 3^\circ\). For quasidipole (QD), Richmond (1995), latitudes polewards of \(\pm 55^\circ\) the horizontal vector components are excluded. Based on these data, the individual gradient elements \([\varvec{\nabla } \mathbf{B}]_{jk}\), with \(j,k= r, \theta , \varphi\), which make up the observables \(\varvec{\Gamma }^{(1)}=\{[\varvec{\nabla } {\mathbf{B}}]]_{r\theta },[\varvec{\nabla } {\mathbf{B}}]_{r\varphi }\}\) resp. \(\varvec{\Gamma }^{(2)}=\{[\varvec{\nabla } {\mathbf{B}}]_{\theta \theta }[\varvec{\nabla } {\mathbf{B}}]_{\varphi \varphi }, 2[\varvec{\nabla } {\mathbf{B}}]_{\theta \varphi }\}\) are estimated. Their expression as elements of the simplified gradient tensor is used, see equation 3.9 of Kotsiaros and Olsen (2012), which holds for smallscale field features, i.e., the lithospheric field. Their calculation is done by means of firstorder Taylor expansion. Specifically,
$$\begin{aligned} {[}\varvec{\nabla } {\mathbf{B}}]_{r\theta }&\approx \frac{{B_r}(r_1,\theta _1,\varphi _1)  {B_r}(r_2,\theta _2,\varphi _2)}{S_{\mathrm {NS}}},\quad {\hbox { with }}\;\theta _2\theta _1 \gg \varphi _2  \varphi _1 , \end{aligned}$$
(33)
$$\begin{aligned} {[}\varvec{\nabla } {\mathbf{B}}]_{r\varphi }&\approx \frac{{B_r}(r_1,\theta _1,\varphi _1)  {B_r}(r_2,\theta _2,\varphi _2)}{S_{\mathrm {EW}}}, \quad{\mathrm { with }}\;\varphi _2\varphi _1 \gg \theta _2  \theta _1 , \end{aligned}$$
(34)
$$\begin{aligned} {[}\varvec{\nabla } {\mathbf{B}}]_{\theta \theta }&\approx \frac{{B_\theta }(r_1,\theta _1,\varphi _1)  {B_\theta }(r_2,\theta _2,\varphi _2)}{S_{\mathrm {NS}}}, \quad {\hbox { with }}\;\theta _2\theta _1 \gg \varphi _2  \varphi _1 , \end{aligned}$$
(35)
$$\begin{aligned} {[}\varvec{\nabla } {\mathbf{B}}]_{\varphi \varphi }&\approx \frac{{B_\varphi }(r_1,\theta _1,\varphi _1)  {B_\varphi }(r_2,\theta _2,\varphi _2)}{S_{\mathrm {EW}}},\quad {\hbox { with}}\;\varphi _2\varphi _1 \gg \theta _2  \theta _1 , \end{aligned}$$
(36)
$$\begin{aligned} {[}\varvec{\nabla } {\mathbf{B}}]_{\theta \varphi }&\approx \frac{{B_\theta }(r_1,\theta _1,\varphi _1)  {B_\theta }(r_2,\theta _2,\varphi _2)}{S_{\mathrm {EW}}}, \quad {\hbox { with }}\;\varphi _2\varphi _1 \gg \theta _2  \theta _1. \end{aligned}$$
(37)
\(S_{\mathrm {NS}}\) is the North–South spherical distance, i.e., the distance between consecutive positions (alongtrack) of Swarm Alpha, whereas \(S_{\mathrm {EW}}\) is the East–West spherical distance, i.e., the distance between adjacent positions (crosstrack) of Swarm Alpha and Charlie. Figure 3 shows \(S_{\mathrm {NS}}\) (left panel) and \(S_{\mathrm {EW}}\) (right panel) calculated from the positions of Alpha and Charlie between April 2014 and March 2016.
Outliers for which the North–South and East–West spherical distances are out of range, namely \(S_{\mathrm {NS}} < 110\) km or \(S_{\mathrm {NS}}> 120\) km and \(S_{\mathrm {EW}} > 200\) km, are removed. In the particular case where Alpha and Charlie are crossing one another and change from ascending to descending orbit tracks, i.e., when \(\theta <  87.2^\circ\) or \(\theta > 87.2^\circ\), only data points with 4 km \(\le S_{\mathrm {EW}} \le 12\) km and are kept.
The model parameters \(\mathbf {m}\) are related to the observables \(\varvec{\Gamma }^{(1)}\) resp. \(\varvec{\Gamma }^{(2)}\) as
$$\begin{aligned} \varvec{\Gamma }^{(1)}&= {\mathbf {G}}^{(1)} {\mathbf {m}}, \end{aligned}$$
(38)
$$\begin{aligned} \varvec{\Gamma }^{(2)}&= {\mathbf {G}}^{(2)} {\mathbf {m}}, \end{aligned}$$
(39)
with the kernel matrices \({\mathbf {G}}^{(1)}=\{[\varvec{\nabla G}]_{r\theta },[\varvec{\nabla G}]_{r\varphi }\}\) and \({\mathbf {G}}^{(2)}=\{[\varvec{\nabla G}]_{\theta \theta }[\varvec{\nabla G}]_{\varphi \varphi }, 2[\varvec{\nabla G}]_{\theta \varphi }\}\). \([\varvec{\nabla G}]_{jk}\) (where \(j,k= r, \theta ,\varphi\)) are the individual gradient kernel matrices which can be approximated in a similar fashion to the individual gradient elements \([\varvec{\nabla } {\mathbf{B}}]_{jk}\) as
$$[\nabla {\varvec{G}}]_{r\theta } \approx \frac{{\mathbf {G}_r}(r_1,\theta _1,\varphi _1)  {\mathbf {G}_r}(r_2,\theta _2,\varphi _2)}{S_{\mathrm {NS}}}, \quad {\hbox { with }}\;\theta _2\theta _1 \gg \varphi _2  \varphi _1 ,$$
(40)
$$[\nabla {\varvec{G}}]_{r\varphi } \approx \frac{{\mathbf {G}_r}(r_1,\theta _1,\varphi _1)  {\mathbf {G}_r}(r_2,\theta _2,\varphi _2)}{S_{\mathrm {EW}}},\quad {\hbox { with }}\;\varphi _2\varphi _1 \gg \theta _2  \theta _1 ,$$
(41)
$$[\nabla {\varvec{G}}]_{\theta \theta }\approx \frac{{\mathbf {G}_\theta }(r_1,\theta _1,\varphi _1)  {\mathbf {G}_\theta }(r_2,\theta _2,\varphi _2)}{S_{\mathrm {NS}}}, \quad {\hbox { with }}\;\theta _2\theta _1 \gg \varphi _2  \varphi _1 ,$$
(42)
$$[\nabla {\varvec{G}}]_{\varphi \varphi }\approx \frac{{\mathbf {G}_\varphi }(r_1,\theta _1,\varphi _1)  {\mathbf {G}_\varphi }(r_2,\theta _2,\varphi _2)}{S_{\mathrm {EW}}},\quad {\hbox { with }}\;\varphi _2\varphi _1 \gg \theta _2  \theta _1 ,$$
(43)
$$[\nabla {\varvec{G}}]_{\theta \varphi }\approx \frac{{\mathbf {G}_\theta }(r_1,\theta _1,\varphi _1)  {\mathbf {G}_\theta }(r_2,\theta _2,\varphi _2)}{S_{\mathrm {EW}}},\quad {\hbox { with }}\;\varphi _2\varphi _1 \gg \theta _2  \theta _1.$$
(44)
\({\mathbf {G}_r}\), \({\mathbf {G}_\theta }\) and \({\mathbf {G}_\varphi }\) are the kernel matrices relating the model parameters \(\mathbf {m}\) to the field vector components \(\mathbf {B_r}\), \({\mathbf {B_\theta}}\) and \(\mathbf {B_\varphi }\) as
$$\begin{aligned} \mathbf {B_r}&= \mathbf {G}_r \mathbf {m}, \end{aligned}$$
(45)
$$\begin{aligned} \mathbf {B_\theta }&= \mathbf {G}_\theta \mathbf {m}, \end{aligned}$$
(46)
$$\begin{aligned} \mathbf {B_\varphi }&= \mathbf {G}_\varphi \mathbf {m} . \end{aligned}$$
(47)
The model parameters \(\mathbf {m}\) can be obtained by an iteratively reweighted leastsquares approach (Constable 1988; Huber 1964), e.g., in the ith iteration, the model parameters are determined as
$$\begin{aligned} \mathbf {m}_{i}&= \left[ {\mathbf {G}^{(\beta )}}^T\mathbf {W}_i\mathbf {G}^{(\beta )}\right] ^{1}\left[ {\mathbf {G}^{(\beta )}}^T\mathbf {W}_i\right] \varvec{\Gamma }^{(\beta )}, \end{aligned}$$
(48)
where the data weight matrix \(\mathbf {W}_i\) is updated by the residuals \(\mathbf {e} = \varvec{\Gamma }^{(\beta )}  \mathbf {G}^{(\beta )}\mathbf {m}_{i1}\) and \(\beta =1,2\). The variance \(\tilde{\sigma }_{(1)}^2\) resp. \(\tilde{\sigma }_{(2)}^2\) of the estimated model parameters for the observable \(\varvec{\Gamma }^{(1)}\) resp. \(\varvec{\Gamma }^{(2)}\) is
$$\begin{aligned} \tilde{\sigma }_{(1)}^2&= {\mathrm {diag}}\left[ {\mathbf {G}^{(1)}}^T\mathbf {W}\mathbf {G}^{(1)}\right] ^{1} \nonumber \\ {\mathrm {resp.}} \end{aligned}$$
(49)
$$\begin{aligned} \tilde{\sigma }_{(2)}^2&= {\mathrm {diag}}\left[ {\mathbf {G}^{(2)}}^T\mathbf {W}\mathbf {G}^{(2)}\right] ^{1} \end{aligned}$$
(50)
The selected Swarm data (Alpha and Charlie positions) have been used to approximate the kernel matrices \({\mathbf {G}^{(1)}}\) and \({\mathbf {G}^{(2)}}\) and compute the variances \(\tilde{\sigma }_{(1)}^2\) and \(\tilde{\sigma }_{(2)}^2\) which are shown in Fig. 4a, b in dependence of degree n and order m.
In addition, \(\tilde{\sigma }_{(0)}^2={\mathrm {diag}}\left[ {\mathbf {G}^{(0)}}^T\mathbf {W}\mathbf {G}^{(0)}\right] ^{1}\), which is the model variance for the observable \(\varvec{\Gamma }^{(0)}=\left\{ [\varvec{\nabla } \mathbf{B}]_{rr}\right\}\), is computed by evaluating the theoretical kernel matrix \(\mathbf {G}^{(0)}\) at Alpha positions. \(\mathbf {G}^{(0)}\) relates the model parameters \(\mathbf {m}\) to the radial gradient \([\varvec{\nabla } \mathbf{B}]_{rr}\) as
$$\begin{aligned}{}[\varvec{\nabla } \mathbf{B}]_{rr}&= \mathbf {G}^{(0)} \mathbf {m}, \end{aligned}$$
(51)
and can easily be constructed by looking at Eqs. (2) and (3). \(\tilde{\sigma }_{(0)}^2\) is shown in Fig. 4c. The approximated model variances \(\tilde{\sigma }_{(1)}^2\) and \(\tilde{\sigma }_{(2)}^2\) for the observables \(\varvec{\Gamma }^{(1)}\) and \(\varvec{\Gamma }^{(2)}\) bear a good resemblance and are very similar to the calculated theoretical variance \(\tilde{\sigma }_{(0)}^2\) which corresponds to the observable \(\varvec{\Gamma }^{(0)}\). Therefore, \(\varvec{\Gamma }^{(1)}\) and \(\varvec{\Gamma }^{(2)}\), which are approximated based on Swarm Alpha and Charlie data, carry similar information content (regarding the estimation of the model parameters) to the (ideal) radial gradient (\(\varvec{\Gamma }^{(0)}\)), which is not possible to be measured or estimated directly from the Swarm configuration. \({\tilde{\sigma }}_{(2)}^2\) shows some instabilities in the zonal terms \((m=0)\), which is most probably due to the Swarm orbit and in particular the polar gap in combination with the sensitivity of \(\varvec{\Gamma }^{(2)}\) to the East–West gradients \([\varvec{\nabla } \mathbf{B}]_{\theta \varphi }\), \([\varvec{\nabla } \mathbf{B}]_{\varphi \varphi }\), which do not constrain the zonal terms sufficiently (Kotsiaros and Olsen 2012). A quantitative comparison between \(\tilde{\sigma }_{(0)}^2\), \(\tilde{\sigma }_{(1)}^2\) and \(\tilde{\sigma }_{(2)}^2\) with the analytically calculated theoretical variances \(\sigma _{(0)}^2\), \(\sigma _{(1)}^2\) and \(\sigma _{(2)}^2\) shown in Fig. 2 cannot be made since the scales are so different. However, one recognizes similar features appearing in the same degree ranges for \(\tilde{\sigma }_{(0)}^2\), \(\tilde{\sigma }_{(1)}^2\), \(\tilde{\sigma }_{(2)}^2\) and \(\sigma _{(0)}^2\), resp. \(\sigma _{(1)}^2\), resp. \(\sigma _{(2)}^2\). Notice, for example, the relatively lower variances at degrees \(n \approx 1040\) in both cases. This band of lower variances (higher information) is relative to the satellite altitude. In particular, lower altitude expands this band to higher degrees. In case of \(\tilde{\sigma }_{(0)}^2\), \(\tilde{\sigma }_{(1)}^2\) and \(\tilde{\sigma }_{(2)}^2\), there seems to be, in addition to degree n, a dependence on the order m, which is more prominent for \(\tilde{\sigma }_{(2)}^2\). This is attributed to the nonperfect distribution of Swarm Alpha and Charlie data on the contrary to the theoretical case where perfect data distribution is assumed.
Lithospheric field models are derived from the pseudoradial gradients \(\varvec{\Gamma }^{(1)}\) and \(\varvec{\Gamma }^{(2)}\), which are estimated from the selected Swarm Alpha and Charlie data. The radial component of the lithospheric field resulting from the model obtained by \(\varvec{\Gamma }^{(1)}\) resp. \(\varvec{\Gamma }^{(2)}\) is shown in Fig. 5a resp. Fig. 5b.
For reference, the radial lithospheric field resulting from MF7 and CM5 (Sabaka et al. 2015) is also shown in Fig. 5c resp. Fig. 5d. The fields are calculated at the Earth’s surface from coefficients of degrees \(16\le n \le 90\). Green lines indicate the boundaries of the major tectonic plates, whereas red lines locate the dip equator (\(0^\circ\) QD latitude) and \(\pm 55^\circ\) isoQD latitudes. The model obtained by the pseudoradial gradient \(\varvec{\Gamma }^{(2)}\) seems to be dominated by a strong signal at QD latitudes polewards of \(\pm 55^\circ\) which also leaks to lower latitudes. Remember that \(\varvec{\Gamma }^{(2)}\) is built exclusively from gradients of the horizontal components, \([\varvec{\nabla } \mathbf{B}]_{\theta \theta }\), \([\varvec{\nabla } \mathbf{B}]_{\varphi \varphi }\) and \([\varvec{\nabla } \mathbf{B}]_{\theta \varphi }\) (and therefore horizontal components cannot be excluded from QD latitudes polewards of \(\pm 55^\circ\) otherwise we would end up with no data in those regions), which are sensitive to disturbing effects due to electrical currents of ionospheric/magnetospheric origin, such as field aligned currents. Therefore, if \(\varvec{\Gamma }^{(2)}\) is to be used for lithospheric field modeling, it is recommended that \(\varvec{\Gamma }^{(2)}\) is excluded in the regions polewards of \(\pm 55^\circ\) QD latitude and data of the radial vector and/or gradients of the radial vector and/or scalar data are used instead. In the following, we will concentrate on the model retrieved from \(\varvec{\Gamma }^{(1)}\) (Fig. 5a), which from now on will be called \({\mathrm {SM}}_{{\Gamma ^{(1)}}}\) (Swarm Model). This model exhibits a stronger signal over the oceanic regions and the regions associated with the magnetic lineations that arise from the seafloor spreading. Moreover, it exhibits sharper lithospheric field features, which is a characteristic property of the radial gradient, compared to MF7 and CM5. Note, for example, the crisp definition of the features in the polar regions, northwest Africa and the Bangui anomaly.
In a similar fashion to \({\mathrm {SM}}_{{\Gamma ^{(1)}}}\), five additional models are obtained for comparison by vector data as well as by North–South and East–West gradient estimates. Specifically, the following models are derived

\(\it{{\mathrm {SM}}_{\mathrm {v}}}\) 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).
No special data filtering nor regularization has been applied in \({\mathrm {SM}}_{{\Gamma ^{(1)}}}\) or in any of the additional models as opposed to CM5 where regularization is applied above degree \(n=60\) (Sabaka et al. 2015) and MF7 where data filtering and line leveling is applied to the data (Maus et al. 2008).
Figure 6a shows Mauersberger–Lowes spectra of the lithospheric field models \({\mathrm {SM}}_{\mathrm {v}}\), \({\mathrm {SM}}_{\mathrm {vNS}}\), \({\mathrm {SM}}_{\mathrm {vEW}}\), \({\mathrm {SM}}_{\mathrm {vNSEW}}\), \({\mathrm {SM}}_{{\Gamma ^{(1)}}}\) and \({\mathrm {SM}}_{{\Gamma ^{(1)}}\delta }\).
For reference, the power spectra of MF7 and CM5 are also presented. Up to degree \(n\approx 40\), all derived models agree relatively well with MF7 and CM5, whereas above that degree, \({\mathrm {SM}}_{\mathrm {v}}\), \({\mathrm {SM}}_{\mathrm {vNS}}\) and \({\mathrm {SM}}_{\mathrm {vEW}}\) have considerably more power. On the other hand, \({\mathrm {SM}}_{\mathrm {vNSEW}}\), \({\mathrm {SM}}_{{\Gamma ^{(1)}}}\) and \({\mathrm {SM}}_{{\Gamma ^{(1)}}\delta }\) follow the power of MF7 and CM5 up to degree \(n=83\) where they start to deviate. Looking at the spectra of model differences (dashed lines in Fig. 6a) and the degree correlation, \(\rho _n\), cf. Langel and Hinze (1998), Fig. 6b, \({\mathrm {SM}}_{{\Gamma ^{(1)}}\delta }\) and \({\mathrm {SM}}_{\mathrm {vNSEW}}\) are very similar and agree better with MF7 and CM5 than \({\mathrm {SM}}_{{\Gamma ^{(1)}}}\).
Figure 7a–f resp. 7g–l presents the relative difference (in \(\%\)) between each coefficient of the various derived models and MF7 resp.
CM5 in a degree versus order matrix. The model \({\mathrm {SM}}_{\mathrm {v}}\), built only from vector data, shows significant differences with respect to MF7 and CM5 after degree \(n\approx 55\). Inclusion of North–South gradients in the \({\mathrm {SM}}_{\mathrm {vNS}}\) improves the agreement in the highdegree nearzonal coefficients \((m \approx 0 \ll n)\), whereas inclusion of East–West gradients in the \({\mathrm {SM}}_{\mathrm {vEW}}\) improves the agreement in the nearsectoral terms \((m \approx n \gg 0)\). The model \({\mathrm {SM}}_{\mathrm {vNSEW}}\), built including both North–South and East–West gradients, improves the determination of both the nearzonal and nearsectoral terms leading to an overall improved model. The model \({\mathrm {SM}}_{\mathrm {\Gamma^{(1)}}}\), which is built exclusively from pseudoradial gradient \(\varvec{\Gamma }^{(1)}\), results in better agreement with MF7 and CM5 than \({\mathrm {SM}}_{\mathrm {vNSEW}}\) in the highdegree zonal coefficients \((m=0)\). However, the highdegree nearsectoral \((m \approx n \gg 0)\) coefficients show a worse agreement. This is not the case if we look at \({\mathrm {SM}}_{{\Gamma ^{(1)}}\delta }\) which is also derived, similarly to \({\mathrm {SM}}_{{\Gamma ^{(1)}}}\), exclusively from \(\varvec{\Gamma }^{(1)}\) but gradients are estimated by simple data differences instead of dividing by the spherical distance. The agreement of \({\mathrm {SM}}_{{\Gamma ^{(1)}}\delta }\) with MF7 resp. CM5 is generally on the same level as \({\mathrm {SM}}_{\mathrm {vNSEW}}\) and improved in the zonal terms.
Maps of lithospheric field differences of \(B_r\) between MF7 and \({\mathrm {SM}}_{\mathrm {vNSEW}}\) (a), \({\mathrm {SM}}_{{\Gamma }^{(1)}}\) (b) and \({\mathrm {SM}}_{{\Gamma ^{(1)}}\delta }\) (c) as well as between CM5 and \({\mathrm {SM}}_{\mathrm {vNSEW}}\) (d), \({\mathrm {SM}}_{{\Gamma }^{(1)}}\) (e) and \({\mathrm {SM}}_{{\Gamma ^{(1)}}\delta }\) (f) are shown in Fig. 8.
The maps are produced at the Earth’s surface using degrees n = 16–70 of the respective models. Overall, the agreement of the derived models with MF7 is slightly better than with CM5. In particular, \({\mathrm {SM}}_{{\Gamma }^{(1)}}\) shows slightly higher differences with MF7 and CM5 than \({\mathrm {SM}}_{\mathrm {vNSEW}}\) or \({\mathrm {SM}}_{{\Gamma ^{(1)}}\delta }\), whereas the differences of \({\mathrm {SM}}_{\mathrm {vNSEW}}\) and \({\mathrm {SM}}_{{\Gamma ^{(1)}}\delta }\) with both MF7 and CM5 are very similar. On the other hand, compared to \({\mathrm {SM}}_{\mathrm {vNSEW}}\), the models \({\mathrm {SM}}_{{\Gamma }^{(1)}}\) and \({\mathrm {SM}}_{{\Gamma ^{(1)}}\delta }\) show a relatively good agreement with MF7 and CM5 at the poles despite the fact that the polar gap is not accounted for as opposed to MF7 and CM5. In MF7, the polar gap is filled with synthetic model values from MF5 (Maus et al. 2007) and in CM5 the lithospheric field is smoothened over the polar gap region above degree \(n=60\) (Sabaka et al. 2015). Despite leaving the polar gap untreated in \({\mathrm {SM}}_{{\Gamma }^{(1)}}\) and \({\mathrm {SM}}_{{\Gamma ^{(1)}}\delta }\), no particular ringing appears which is usually the case if the polar gaps are not accounted for (Sabaka et al. 2015; Maus et al. 2008).