### SOLA method for local estimation of CMB radial field SV

We now move on to investigate the behavior of the core field not at satellite altitude but down at the core–mantle boundary (CMB), on the edge of the region where it originates. To do this we use the Green’s functions of the Neumann boundary value problem that links the magnetic field at satellite altitude to the radial field at the CMB. Following Hammer and Finlay (2019), a localized estimate, \(\widehat{B}_r\), of the radial magnetic field at a target location and time, \((\mathbf {r}_0,t_0)\), at the CMB can be computed as a localized spatial average around the target location time-averaged over a specified interval. Because the CMB radial magnetic field is linearly related to the spherical polar components of the vector field at satellite altitude, we can write \(\widehat{B}_r\) as a weighted linear combination of the satellite magnetic measurements with weights \(q_n\) (Backus and Gilbert 1968, 1970; Hammer and Finlay 2019)

$$\begin{aligned} \widehat{B}_r(\mathbf {r}_0,t_0) = \sum _n^N q_n(\mathbf {r}_0,t_0)\, d_n(\mathbf {r}_n,t_n), \end{aligned}$$

(5)

where \(d_n\) are satellite magnetic measurements \((n=1,...,N)\) within a specified time window and \(q_n\) are weighting coefficients to be determined. Here the data \(d_{n}\), at positions \(\mathbf {r}_n\) and times \(t_n\), are taken from dataset #2 and for simplicity we consider using only observations of the radial component of the field. Corrections for the lithospheric field for SH degrees \(n \in [14,185]\) as given by the LCS-1 model (Olsen et al. 2017), for the magnetospheric and associated induced fields as given by the CHAOS-7.2 model (Finlay et al. 2020), and for the ionospheric and associated induced fields as given by the CIY4 model, (Sabaka et al. 2018) are removed from the observations in a pre-processing step. The radial magnetic field measurements, \(d_{n}(\mathbf {r}_n,t_n)\), are then related to the radial magnetic field \(B_r(\mathbf {r}',t_n)\), integrated over the CMB, by (Gubbins and Roberts 1983):

$$\begin{aligned} d_{n}(\mathbf {r}_n,t_n) = \oint _{S'} G_{r}(\mathbf {r}_n, \mathbf {r}')B_r(\mathbf {r}',t_n) dS', \end{aligned}$$

(6)

where the surface element is \(dS'=\mathrm{sin}\theta 'd\theta 'd\phi '\). The data kernel \(G_{r}(\mathbf {r}_n, \mathbf {r}')\) is the radial derivative with respect to \(\mathbf {r}\), of the Green’s functions for the exterior Neumann boundary value problem (e.g., Gubbins and Roberts 1983; Barton 1989):

$$\begin{aligned} G_{r} = \frac{1}{4\pi }\frac{h_n^2(1-h_n^2)}{f_n^3}, \end{aligned}$$

(7)

where \(h_n=r'/r_n\) and \(r'\) is the CMB radius, \(f_n=R_n/r_n\) where \(R_n=\sqrt{r_n^2+r'^2-2r_nr'\zeta _n}\) and \(\zeta _n = \mathrm{cos}\, \gamma _n=\mathrm{cos}\,\theta _n \, \mathrm{cos}\,\theta '+\mathrm{sin}\,\theta _n \mathrm{sin}\,\theta '\, \mathrm{cos(\phi _n-\phi ')}\), where \(\gamma _n\) is the angular distance between a measurement at position \((\theta _n,\phi _n)\) and a position on the CMB \((\theta ',\phi ')\). The data kernel describes how a particular measurement samples the CMB radial field; radial magnetic measurements sample the CMB radial field most strongly directly below the measurement position. Regarding the time-dependence, we use a first order Taylor expansion around a reference time \(t_0\), such that

$$\begin{aligned} d_{n}(\mathbf {r}_n,t_n) \approx \oint _{S'} G_{r}(\mathbf {r}_n,\mathbf {r}') \left[ B_r(\mathbf {r}',t_0)+\dot{B_r}(\mathbf {r}',t_0)\Delta t_n\right] dS'. \end{aligned}$$

(8)

The time difference, \(\Delta t_n = t_n-t_0\), is computed with respect to the target time, \(t_0\). Inserting Eq.(8) into Eq.(5), we obtain

$$\begin{aligned} \widehat{B}_r(\mathbf {r}_0,t_0) &= \oint _{S'} {\mathcal {K}}(\mathbf {r}_0, t_0, \mathbf {r}') \, B_r(\mathbf {r}',t_0)dS' \nonumber \\ & + \oint _{S'} \dot{{\mathcal {K}}} (\mathbf {r}_0, t_0, \mathbf {r}') \, \dot{B_r}(\mathbf {r}',t_0) dS', \end{aligned}$$

(9)

where \(\mathcal {K}(\mathbf {r}_0, {t_0}, \mathbf {r}')\) and \(\dot{{\mathcal {K}}}(\mathbf {r}_0, t_0, \mathbf {r}')\) are spatial averaging kernels for the CMB field and secular variation, respectively, constructed from the weighting coefficients and the data kernels

$$\begin{aligned} {\mathcal {K}}(\mathbf {r}_0, t_0, \mathbf {r}')&= \sum _n^N q_n(\mathbf {r}_0,t_0) \, G_r(\mathbf {r}_n,\mathbf {r}') \\
\dot{{\mathcal {K}}}(\mathbf {r}_0, t_0, \mathbf {r}') &= \sum _n^N q_n(\mathbf {r}_0,t_0) \, G_r(\mathbf {r}_n,\mathbf {r}')\Delta t_n. \end{aligned}$$

(10)

By varying the weight coefficients, \(q_n\), the shape of the averaging kernels change. Notice that time differences \(\Delta t_n\), between the measurement times and the target time, are effectively additional temporal weights applied to the kernel \({\mathcal {K}}\) in order to obtain \(\dot{{\mathcal {K}}}\).

In order to obtain estimates of the secular variation of the radial field on the CMB, at the target location and time \(\widehat{\dot{B}}_r(\mathbf {r}_0,t_0)\), we minimize the following objective function:

$$\begin{aligned} \Theta &= \oint _{S'} [\dot{{\mathcal {K}}}(\mathbf {r}_0, t_0, \mathbf {r}') -\dot{{\mathcal {T}}}(\mathbf {r}_0, \mathbf {r}')]^2dS' \nonumber \\ & \oint _{S'} [{\mathcal {K}}(\mathbf {r}_0, t_0, \mathbf {r}')]^2dS' + \lambda ^2 \mathbf {q}^T \underline{\underline{\mathbf {E}}} \mathbf {q}, \end{aligned}$$

(11)

where \(\lambda\) is a trade-off parameter (units of \([\mathrm{nT}^{-1}]\)), \(\mathbf {q}\) is vector of the weighting coefficients, \(\underline{\underline{\mathbf {E}}}\) is the data error covariance matrix which we define below and \(\dot{{\mathcal {T}}}\) is an SV target kernel that we choose to be a Fisher distribution on the sphere (Fisher 1953):

$$\begin{aligned} \dot{{\mathcal {T}}}(\mathbf {r}_0, \mathbf {r}') =\frac{\kappa }{4 \pi \mathrm{sinh}\kappa } e^{\kappa \cos \, \gamma _0}, \end{aligned}$$

(12)

where \(\gamma _0\) is the angular distance on the CMB between the target position \((\theta _0,\phi _0)\) and another position \((\theta ',\phi ')\). On the basis of tests carried out by Hammer and Finlay (2019), we initially set \(\kappa =600\) corresponding to a target kernel width of \(15^{\circ }\); this is narrower than can be achieved for \(\dot{{\mathcal {K}}}\) with the available data, but it avoids excessive ringing associated with taking a Dirac delta function as the target kernel. When computing SOLA estimates for a given time window, we select a subset (\(n=1,...,N\)) of the measurements. Using this data subset the data error covariance matrix \(\mathbf {E}\) is defined as follows. Using all available measurements (\(m=1,...,M\)), for each satellite mission within 2 degree bins of quasi-dipole (QD) latitude (Richmond 1995), we first derived robust data error variances as a function of QD latitude:

$$\begin{aligned} \sigma ^2(\theta _{QD}) = { \sum \limits _{m=1}^M w_m\left( \epsilon _m-\mu \right) ^2 \over \sum \limits _{m=1}^M w_m }, \end{aligned}$$

(13)

where \(\epsilon _m\) are residuals with respect to predictions of the CHAOS-7.2 internal field model for SH degrees \(n \in [1,13]\), \(\mu\) are robust mean residuals within the considered bin and \(w_m\) are Huber weights (e.g., Constable 1988) for the data within each bin. Here we use QD coordinates, as this is appropriate for characterizing processes related to unmodeled ionospheric currents which we consider to be a likely source of contamination, especially at high latitudes. Figure 6 presents the resulting QD-latitude-dependent error estimates \(\sigma (\theta _{QD})\) for the radial field component used in this study, comparing the values for the Ørsted, CHAMP, CryoSat-2 and *Swarm* datasets. When computing SOLA estimates for a specified time window of, e.g., 2 years, we select a data subset of dataset #2. Using this data subset of N measurements, a specific data error covariance matrix E is computed. Diagonal elements of this data error covariance matrix \(\mathbf {E}\) are finally defined as:

$$\begin{aligned} E_{n,n}=\sigma ^2(\theta _{QD}) / w_n, \end{aligned}$$

(14)

where \(w_n\) are additional robust (Huber) weights determined a priori for each datum (\(n=1,...,N\)), based on their residual to CHAOS-7.2, in order to account for the expected long-tailed error distribution. Off-diagonal elements of \(\mathbf {E}\) are set to zero.

In addition to minimizing Eq.(11), we simultaneously impose the following constraint:

$$\begin{aligned} \oint _{S'}{\mathcal {K}}(\mathbf {r}_0, t_0, \mathbf {r}') dS' + \oint _{S'} \dot{{\mathcal {K}}}(\mathbf {r}_0, t_0, \mathbf {r}')dS' = 1, \end{aligned}$$

(15)

where the first term is in practice very small when estimating the SV, since it is minimized in Eq. (11). This constraint ensures that a valid averaging kernel is obtained.

Discretization of integrals over the CMB was carried out using Lebedev quadrature (Lebedev and Laikov 1999) and the system of equations was solved for the coefficients, \(q_n\), using a Lagrange multiplier method, see Hammer and Finlay (2019) for further details.

Once SOLA estimates of the CMB radial field SV at the chosen target location and epoch are obtained, by minimizing Eq. (11) subject Eq. (15), we are able to easily appraise them based on (i) their averaging kernel width, which we define as the angular distance between the points at which the averaging kernel first reaches zero amplitude moving away from its maximum value, and (ii) the variance of the SOLA estimate which we computed as:

$$\begin{aligned} \hat{\sigma }^2(\mathbf {r}_0,t_0) = \mathbf {q}^T \underline{\underline{\mathbf {E}}}\mathbf {q}. \end{aligned}$$

(16)

By changing the parameter \(\lambda\) of Eq.(11), a range of solutions can be computed which describes a trade-off between having an averaging kernel width as small as possible and the variance of the estimate being as small as possible (Parker 1977). Below we discuss the effect of changing \(\lambda\) on our results.

### Results: SOLA estimates of CMB SV and SA from Ørsted, CHAMP, CryoSat-2 and *Swarm* data

We begin by first comparing SOLA estimates for the CMB radial field SV obtained using separate data subsets from the *Swarm* and CryoSat-2 missions, respectively. First, *Swarm* and CryoSat-2 data subsets are extracted from the main dataset #2 described in Sect. 2, so that each cover the same 2-year time window from 2015.0 to 2017.0 Next, in order to obtain data subsets with suitable spatial and temporal coverage, we considered bins surrounding each point in an approximately equal-distance grid at satellite altitude of \(\approx 2.5^{\circ }\) spacing, based on the partitioning algorithm of Leopardi (2006), and randomly sampled one datapoint from each bin, resetting the bins every 2 months. Data subsets spanning the full 2-year window from 2015.0 to 2017.0 were produced by accumulating these 2-monthly globally-distributed subsets. The resulting *Swarm* and CryoSat-2 data subsets spanning 2015.0 to 2017.0 consisted of 62469 and 54685 radial field observations, respectively.

In Fig. 7, we compare maps collecting SOLA CMB radial field SV estimates centered on epoch 2016.0, derived using the CryoSat-2 and *Swarm* data subsets spanning 2015.0 to 2017.0. To ensure that we obtained SOLA estimates of comparable resolution, we first computed SOLA SV estimates using the *Swarm* data subset and taking \(\lambda =3 \times 10^{-3}\ \mathrm{nT}^{-1}\). This resulted in well-behaved averaging kernels with widths \(\approx 38^{\circ }\). Next, we used these averaging kernels as the target kernels in order to derive similar estimates using the *Swarm* and CryoSat-2 data subsets, thus effectively seeking *Swarm * and CryoSat-2 SV estimates with the same spatial resolution. The maps in the top row of Fig. 7 show the resulting global collections of SOLA SV estimates, obtained on a \(1^{\circ }\) grid of target locations at the CMB, based on the CryoSat-2 (left plot) and *S*warm (right plot) data subsets, respectively.

Maps collecting the formal standard deviations for each SOLA estimate (derived using Eq.16) and their averaging kernel widths are presented in middle and bottom rows of Fig. 7. The standard deviations of the *Swarm*-based SOLA SV estimates are fairly homogeneous with values of \(0.3-0.4\mu \hbox {T}/\hbox {yr}\); those for CryoSat-2 SOLA are somewhat larger, being in the range \(\approx 0.6-1.8\mu \hbox {T}/\hbox {yr}\). We note that the CryoSat-2 error estimates are slightly lower at higher latitudes. The same is true for the *Swarm* map, but the variations in errors estimates are in that case much smaller. Kernel widths in both cases are also fairly homogeneous except at auroral latitudes where distinct behavior of the kernels are found related to the data error estimates having increased amplitude, as seen in Fig. 6.

As seen from the kernel widths in the bottom row of Fig. 7, very similar resolution has been obtained (by construction) in the *Swarm* and CryoSat-2 SV maps. The same field features are clearly identified in both maps. For instance, we notice high latitude SV patches in the northern hemisphere which have been associated with a high latitude jet of core flow (Livermore et al. 2017), and there is increased amplitude of SV over the hemisphere centered on the Atlantic in comparison with the Pacific hemisphere. This first test gives us confidence that the CryoSat-2 measurements can indeed be used to reliably map SV features at the CMB on a timescale of 2 years, and with a spatial resolution down to \(\approx 38^{\circ }\) degrees.

Next, we go further and investigate the second time derivative or secular acceleration (SA) of the radial field at the CMB, which is of great interest for investigating the dynamics of the core (e.g., Chulliat et al. 2010; Finlay et al. 2016; Chi-Durán et al. 2020). Again we first compare maps based on CryoSat-2 and *Swarm* data. To obtain SA estimates we initially use the accumulated change between SV estimates 2 years apart. In particular, in Fig. 8 we show the SA in 2017 based on the difference between SOLA SV estimates in 2016.0 and 2018.0. Note, that this is not an instantaneous secular acceleration but a centered difference in SV estimates 2 years apart, each based on 2 years of data. To study the SA, we computed SOLA SV estimates from *Swarm* data taking \(\lambda =1 \times 10^{-2}\ \mathrm{nT}^{-1}\), and used the resulting associated averaging kernels as the target kernels for the SOLA estimates from both *Swarm* and CryoSat-2 data.

Figure 8 presents global grids of SOLA CMB radial field SA estimates centered on 2017.0, again with a \(1^{\circ }\) spacing, derived from CryoSat-2 (left plot) and *Swarm* (right plot) data subsets. Here the map is centered on the Pacific region where there has been interesting SA activity during the past 6 years (Finlay et al. 2016, 2020). As for the SV maps, we find error estimates, computed assuming the contributing SV estimates have independent errors, to be fairly homogeneous, with values ranging between \(0.11-0.27\mu \mathrm{T}/\mathrm{yr}^2\) for the estimates derived using CryoSat-2 data and \(0.06-0.08\mu \mathrm{T}/\mathrm{yr}^2\) for the estimates derived using *Swarm* data. In both cases kernel widths are close to \(\approx 42^{\circ }\), except in the auroral region. Comparing the CryoSat-2 and *Swarm*-based SA maps, similar features can be observed. In particular, this is the case for the features seen under Asia and Indonesia. A distinctive feature reproduced in both maps is the sequence of intense patches of SA at low latitudes in a localized region below central America, having amplitudes of approximately \(1.9\pm 0.3\mu \mathrm{T}/\mathrm{yr}^2\) and \(1.7\pm 0.1\mu \mathrm{T}/\mathrm{yr}^2\) for the CryoSat-2 and *Swarm* maps, respectively. The location and amplitude of these features are similar in the two maps, confirming that CryoSat-2 data can be used to track such SA structures. Because the SOLA estimates are local averages, distant high latitude measurements, where ionospheric electrical currents may be prominent even during dark quiet times, will have little influence on such low latitude SV and SA estimates. Finally, we note a strong patch under the Bering Sea of amplitude approximately \(1.4\pm 0.3\mu \mathrm{T}/\mathrm{yr}^2\) and \(1.0\pm 0.1\mu \mathrm{T}/\mathrm{yr}^2\) for the CryoSat-2 and *Swarm* maps, respectively.

These initial investigations of the CMB radial field SV and SA using the SOLA technique indicate, as seen earlier in the GVO time series, that low latitude regions experience significant sub-decadal core field variations. We are therefore motivated to study the field time-dependence in the equatorial region in more detail. We do this in Fig. 9 by constructing time-longitude (TL) plots of SOLA CMB radial field SA estimates along the geographic equator from 2002 to 2019, centered on the Pacific. We again compute our SA estimates based on differences of SOLA SV estimates 2 years apart, each derived from 2-year data windows, and sliding the windows in 2-month steps. Here we use radial field data from the Ørsted, CHAMP, CryoSat-2 and *Swarm* satellites.

The top left plot in Fig. 9 presents estimates based on averaging kernels obtained from *Swarm* data using \(\lambda =3 \times 10^{-2}\ \mathrm{nT}^{-1}\). These have associated error estimates between \(0.02-0.08\mu \mathrm{T}/\mathrm{yr}^2\) and kernel widths \(\approx 50^{\circ }\). Their resolution is lower than that shown in Fig. 8 and corresponds to approximately SH degree 8. For comparison the SA predicted by the CHAOS-7.2 model for SH degrees \(n \in [1,8]\) are shown on the top right plot. Note here that the CHAOS-7.2 model makes use of uncalibrated vector magnetic data from CryoSat-2 between 2010 and 2014, and co-estimates magnetometer calibration parameters (Finlay et al. 2020). The SOLA and CHAOS TL plots in the top row of Fig. 9 show largely the same SA features, illustrating the convergence of the two techniques at long wavelengths of the SA and when constructing SOLA SA estimates from SV differences between consecutive 2-year time windows. For instance, the evolution of the SA features observed in Fig. 8 under the central Americas, can be identified ranging from longitudes \(240^{\circ }\) to \(320^{\circ }\) centered on 2017. In addition, we find strong SA patches in the CryoSat-2 data around 2013 at longitudes \(70^{\circ }\) to \(160^{\circ }\) and \(280^{\circ }\) to \(320^{\circ }\). Notice, that there seems to be a sign changing sequence occurring at longitudes \(240^{\circ }\) to \(320^{\circ }\) going from 2005 to 2019.

Next, we increase the spatial resolution by instead deriving SV estimates using \(\lambda =1 \times 10^{-2}\ \mathrm{nT}^{-1}\), which leads to slightly larger error estimates in the range \(0.1-0.5\mu \mathrm{T}/\mathrm{yr}^2\) and kernel widths \(\approx 42^{\circ }\), i.e., similar to Fig. 8. This is shown in the bottom left plot while the CHAOS-7.2 model predictions for SH degrees \(n \in [1,10]\), matching approximately the kernel width, are shown on the bottom right plot. Although the SOLA TL-plot looks somewhat noisier than the CHAOS plot, similar coherent evolving structures having higher amplitudes can clearly be identified. The noisier appearance in the interval 2010-2014 likely indicates the limitations of the CryoSat-2 data, but they clearly provide useful information during this period.

With data from the *Swarm* mission, it is possible to go further and also increase the temporal resolution of the SA by taking 1-year differences of SOLA SV estimates derived from 1-year data windows and sliding in 1-month steps. The result of applying this procedure to obtain SA estimates on the geographic equator between 2015.0 and 2019.5 is shown in the left plot of Fig. 10. These SOLA estimates have associated errors of \(0.3-0.6\mu \mathrm{T}/\mathrm{yr}^2\) and kernel widths \(\approx 42^{\circ }\). The right plot shows similar CHAOS-7.2 model predictions for SH degrees \(n \in [1,10]\). Both TL-plots shows the similar large-scale features, for instance, the features under central America from longitudes \(240^{\circ }\) to \(320^{\circ }\), which are elongated compared with Fig. 9 due to the change in scale of the y-axis (time). However the SOLA results show significantly more time-dependence, revealing features that were smoothed out by the temporal regularization of CHAOS-7.2. Changes of sign in the SA within about 1 year can be observed.

Particularly interesting is the appearance in the Pacific region around 2017, at longitudes \(150^{\circ }\) to \(220^{\circ }\), of side-by-side positive and negative intense SA features, that have subsequently drifted westwards. This SA change coincides with the peak in the radial SV field observed in the Pacific region during *Swarm* time seen in the GVO map (Fig. 5). We note the presence of features in Fig. 10 that appear to drift rapidly both eastwards and westwards, for example from \(160^{\circ }\) East in 2015 to \(220^{\circ }\) East in 2017. Such rapidly drifting behavior of low latitude SA patches is difficult to explain in terms of simple core flow advection processes. They may instead be a signature of wave propagation close to the core surface. A range of possible candidates for fast waves in the core have recently been described, some requiring only a strong magnetic field and rotation (Aubert and Finlay 2019; Gerick et al. 2020) while others rely on the presence of a possible stratified layer at the top of the core (Buffett and Matsui 2019). Though tempting, it may be dangerous to interpret such features that are at the limit of the present spatial resolution and temporal resolution (Gillet 2019). It will be important to assess whether such features remain coherent in the future, as the resolution of the SA increases.