# A comprehensive model of Earth’s magnetic field determined from 4 years of Swarm satellite observations

- 674 Downloads

**Part of the following topical collections:**

## Abstract

*Swarm*, launched in November 2013, has provided unprecedented monitoring of Earth’s magnetic field via a unique set of gradiometric and multi-satellite measurements from low Earth orbit. In order to exploit these measurements, an advanced “comprehensive inversion” (CI) algorithm has been developed to optimally separate the various major magnetic field sources in the near-Earth regime. The CI algorithm is used to determine

*Swarm*Level-2 (L2) magnetic field data products that include the core, lithospheric, ionospheric, magnetospheric, and associated induced sources. In addition, it has become apparent that the CI is capable of extracting the magnetic signal associated with the oceanic principal lunar semidiurnal tidal constituent \(M_2\) to such an extent that it has been added to the L2 data product line. This paper presents the parent model of the

*Swarm*L2 CI products derived with measurements from the first 4 years of the

*Swarm*mission and from ground observatories, denoted as “CIY4,” including the new product describing the magnetic signal of the \(M_2\) oceanic tide.

## Keywords

Geomagnetism Field modeling*Swarm*satellites Tides

## Abbreviations

- CI
comprehensive inversion

- CM
comprehensive model

- CMB
core–mantle boundary

- CRF
common reference frame

- DI
dedicated inversion

- DIFI
Dedicated Ionospheric Field Inversion

- DISC
Data Innovation and Science Center

- ESA
European Space Agency

- EW
east–west

- ECEF
Earth-Centered Earth Fixed

- GN
Gauss–Newton

- LS
least squares

- LSLE-GN
least squares linear equality Gauss–Newton

- L1b
level-1b

- L2
level-2

- LT
local time

- MMA
magnetic-magnetospheric

- NS
north–south

- NEC
north, east, center

- OHM
observatory hourly means

- QD
quasi-dipole

- SCARF
Satellite Constellation Application and Research Facility

- SIVW
Selective Infinite Variance Weighting

- SA
secular acceleration

- SH
spherical harmonic

- SV
secular variation

- Sq
solar-quiet

- UT
universal time

- VMF
vector magnetometer frame

- 1D
one-dimensional

- 3D
three-dimensional

## Introduction

It has been over 4 years since the launch of the European Space Agency (ESA) *Swarm* mission on November, 22, 2013 whose objective is to provide the best-ever survey of Earth’s magnetic field. The constellation of the polar-orbiting trio of satellites was designed to provide north–south gradient information from each spacecraft and unique east–west gradient information from its low-altitude pair of fliers. The orbital planes of the high-altitude flier, known as “*Swarm Bravo*,” and the low pair, known as “*Swarm Alpha*” and “*Swarm Charlie*,” simultaneously sweep out different local times for improved determination of time-varying external fields. In order to best extract the signals from the various magnetic field sources, a modeling approach called “Comprehensive Inversion” (CI) (see Sabaka et al. 2013) has been developed over the years which basically parameterizes all of the major sources and subsequently co-estimates them in order to obtain a proper separation while taking into account systematic errors or biases, which are often more detrimental than random errors. This approach has led to the well-known series of “Comprehensive Models” (CMs) (e.g., Sabaka et al. 2002, 2004, 2015) and has been selected for deriving a consistent set of *Swarm* Level-2 (L2) magnetic data products. The latest CI model, denoted as “CIY4,” is derived from 4 years of *Swarm* magnetic measurements as well as ground-based observations and serves as the source of the fourth version of the L2 data products.

The *Swarm* “Satellite Constellation Application and Research Facility” (SCARF) has been established with the goal of deriving L2 products by combination of data from the three satellites and of the various instruments (Olsen 2013). SCARF uses Level-1b (L1b) data products (which are calibrated time series of magnetic field observations) and auxiliary data in order to determine specific L2 data products. The magnetic data products include models of the core, lithospheric, nonpolar ionospheric and large-scale magnetospheric fields derived using two independent chain branches: several Dedicated Inversion (DI) chains (e.g., Rother et al. 2013; Thébault et al. 2016; Chulliat et al. 2016) in which the various sources are determined in a sequential approach after removing models describing the other sources, and the CI chain where the various data products are co-estimated.

Tyler et al. (2003) were the first to determine the magnetic signal of the oceanic principal lunar semidiurnal constituent \(M_2\) from CHAMP satellite measurements, after filtering the data on an orbit-by-orbit basis (which unfortunately removes a significant part of the signal). However, such data pre-processing is not necessary in the CI approach, which successfully extracted \(M_2\) from CHAMP data in the CM5 model (Sabaka et al. 2015). Subsequently, CI was used to extract \(M_2\) again from the first 20.5 months of *Swarm* data within the context of a model denoted as “CI1” (Sabaka et al. 2016). Encouraged by these results, the SCARF CI software was updated to include \(M_2\) extraction that was consequently used to produce the second, third, and fourth year CI *Swarm* L2 data product versions. The original list of L2 products does not include the oceanic \(M_2\) field; however, the *Swarm* “Data, Innovation and Science Cluster” (DISC), an international consortium of expert partners with the goal of enhancing the scientific return of the *Swarm* satellite mission by identifying and deriving new, innovative data products, considered the \(M_2\) field determined by CI mature enough to be distributed to the broader scientific community. Thus, the CI \(M_2\) product is now part of the L2 portfolio and is also described in this paper. It should be noted that unlike the other CI products that have DI redundancy, the \(M_2\) product is only produced under the CI chain.

This paper reports on the CIY4 model and the associated L2 magnetic field products, including the new \(M_2\) field. Although there have been reports on the DI products and the \(M_2\) tidal portion of the CI1 model in the literature (see references above), this is the first complete description of a CI parent model derived from *Swarm* satellite constellation data. This paper first presents a description of the data selection procedure in section “Data selection” followed by a brief overview of the CI algorithm in section “Methodology,” including model parameterization and the estimation procedure, and ends with a discussion of the results in section “Results and discussion,” focusing in particular on the new \(M_2\) magnetic field product.

## Data selection

The *Swarm* data used in the CIY4 model is from the *Swarm* Mag-L L1b data product, version 0503, and its selection follows that of previous modeling efforts (e.g., Olsen et al. 2014; Olsen 2015; Finlay et al. 2016). Regarding magnetic activity level, data were chosen only when \(Kp \le 3^0\) and \(\left| {dDst/dt}\right| \le 3\,\hbox {nT/h}\). Gross outliers were controlled by selecting only those scalar and vector measurements for which the scalar \(\Delta F\) and vector \(\Delta {\mathbf {B}}\) residuals with respect to the CHAOS-6-x4 model (Finlay et al. 2016) satisfy \(\left| {\Delta F}\right| \le 100\,\hbox {nT}\) and \(\left| {\Delta \mathbf {B}}\right| \le 500\,\hbox {nT}\). The vector field measurements were further restricted to regions where the sun was more than \(10^{\circ }\) below the horizon and whose quasi-dipole (QD) latitude was equatorward of \(55^{\circ }\). Interestingly, the vector field measurements have been limited to the quieter conditions of \(Kp \le 2^+\) and \(\left| {dDst/dt}\right| \le 2\,\hbox {nT/h}\) in other studies (e.g., Sabaka et al. 2016), but the potential negative impact in CIY4 of the additional data from the relaxed selection criteria has been found to be negligible and in fact they may be beneficial since data coverage is improved. Temporal selection of *Swarm* data was between December 1, 2013, to December 31, 2017, at a \(15\,\hbox {s}\) sampling rate with north–south (NS) sums and differences being taken between every other pair. The east–west (EW) sums and differences are produced between *Alpha* and *Charlie* from April 17, 2014, to December 31, 2017, when the satellite pair were in a proper configuration. The EW measurements are constructed when *Alpha* and *Charlie* are at equal geographic latitude at slightly different times, usually within \(10\,\hbox {s}\).

It should be mentioned that to the authors’ knowledge, only the CI algorithm incorporates data measurement sums (the complement of the differences) (see Sabaka et al. 2013, 2015, 2016) as opposed to field measurements alone (see Olsen et al. 2014), difference measurements alone (see Olsen et al. 2017), and field and difference measurements (see Olsen 2015; Finlay et al. 2016). The sums balance the influence of the differences in determining fields from sources such as the ionosphere.

To complement the *Swarm* measurements, and to provide surface data control, vector hourly mean measurements from permanent magnetic observatories have been included in CIY4. These “observatory hourly means” (OHMs) were selected under the activity conditions of \(Kp \le 2^+\) and \(\left| {dDst/dt}\right| \le 2\,\hbox {nT/h}\) at all geomagnetic latitudes from December 1, 2013, to October 16, 2017. Note that these criteria are currently more restrictive than those used for the satellites. However, the more relaxed criteria will be investigated for all data in future CI models. Further details on this OHM data set can be found in Macmillan and Olsen (2013).

*Swarm*and OHM data distributions over time used in CIY4 is shown in Fig. 1. Specifically, the plot shows the

*Swarm*NS \((\delta F_{NS})\) and EW \((\delta F_{EW})\) scalar difference/sum pairs, the NS \((\delta \mathbf {B}_{NS})\) and EW \((\delta \mathbf {B}_{EW})\) vector difference/sum pairs, the single scalar and vector

*Swarm*measurements, and the vector OHM measurements. The side-by-side constellation of the lower pair,

*Alpha*and

*Charlie*, has been maintained since April 17, 2014, and hence EW differences/sums are used only from this date onwards. Otherwise the data amounts and ratios are fairly consistent with natural variations due to the selection criteria, i.e., due to variations in the

*Kp*and

*Dst*indices as well as the drifts of the satellite orbital planes through local time;

*Alpha*and

*Charlie*cover all local times in about 19 weeks, whereas

*Bravo*covers all local times in 20 weeks. The OHMs are also absent during the last two months of the data envelope.

## Methodology

### Model parameterization

The CI algorithm considers several major field sources including the core, lithosphere, oceanic \(M_2\) tidal, ionospheric and magnetospheric and their associated induced fields, and observatory biases, which account for local baseline field levels, particularly in the local lithosphere. The parameterizations of the various sources are summarized in Table 1.

#### Core and lithospheric fields

*t*and position \(\mathbf {r}\), corresponding to Earth-Centered Earth-Fixed (ECEF) spherical coordinates of radius, colatitude, and longitude \((r,\theta ,\phi )\), is given by

*n*and order

*m*given by

*a*is the Earth mean-radius (\(6371.2\,\hbox {km}\)), \(P_n^m\) and \(\gamma _n^m\) are the Schmidt semi-normalized associated Legendre function and static complex Gauss coefficient of degree

*n*and order

*m*, respectively. The time-variable core field is a linear combination of basis functions \(Y_{nq}^m(t,\theta ,\phi )\) with associated multipliers \(\gamma _{nq}^m\) such that

*q*th cubic B-spline of the expansion and the epoch of the expansion is 2015.0. For \(n=1{-}16\) this is equivalent to the usual solid harmonic functions with time-dependent Gauss coefficient multipliers of the form

#### Oceanic \(M_2\) field

#### Ionospheric field

The CIY4 ionospheric and induced parameterization uses quasi-dipole (QD) symmetric basis function (Emmert et al. 2010; Richmond 1995) as in Sabaka et al. (2004, 2015) in order to conform to the conductivity structures found in the *E*-region ionosphere. As in Sabaka et al. (2015) the induced field now reflects a 3-dimensional (3D) conductivity model where a surface layer containing continents and oceans is underline by a 1-dimensional (1D) mantle known as “1D + oceans” (Kuvshinov 2008). The conductance of sea water has been taken from Manoj et al. (2006) and accounts for ocean bathymetry, ocean salinity, temperature and pressure. Conductance of the sediments is based on the global sediment thicknesses given by the map of Laske and Masters (1997) and calculated by a heuristic procedure similar to that described in Everett et al. (2003). The 1D mantle conductivity has been updated with satellite data by Kuvshinov and Olsen (2006).

As in Sabaka et al. (2015), the conductivity structure induces a secondary field in the spectral domain through transfer functions \(\mathbf {Q}(\omega )\) at frequency \(\omega\). If \(\varvec{\epsilon }(\omega )\) and \(\varvec{\iota }(\omega )\) are the vectors of complex SH coefficients for the inducing and induced fields, respectively, at frequency \(\omega\), then \(\varvec{\iota }(\omega )=\mathbf {Q}(\omega ) \varvec{\epsilon }(\omega )\). These complex matrices are dense owing to the fact that they reflect 3D conductivity, which means that a relatively smooth inducing field can create complicated induced field structure. Contrast this with a 1D conductivity where \(\varvec{\epsilon }_n^m\) can only induce \(\varvec{\iota }_n^m\), thus leading to a diagonal complex \(\mathbf {Q}(\omega )\) whose elements are functions of SH degree *n* only. The frequencies chosen correspond to the daily and sub-daily periods of 24, 12, 8, and 6 hours. In addition, these periods are modulated further by an annual and semiannual periodicity and by scaling from the 3-month running average of the \(F_{10.7}\) solar radiation index such that these \(\mathbf {Q}\) also reflect an infinite conductor at depth to approximate long-period variations.

#### Magnetospheric field

The CIY4 parameterization of the magnetosphere and associated induced fields also follows Sabaka et al. (2015) by discretizing time into bins within which the fields are treated as static external and internal SH expansions in dipole coordinates, respectively. These SH expansions are to degree \(N_{\max }=1\) and order \(M_{\max }=1\) for internal and external fields in 1 h bins for the selected quiet periods. This results in 27, 542 hourly bins covering 77% of the hours of the model time span from December 1, 2013, to December 31, 2017.

#### Alignment parameters

*x*-axis of the CRF followed about the new

*y*-axis and then the new

*z*-axis. The angles are treated as static in 10 day intervals.

CIY4 parameterization

Field source/effect | # Parms | Description | ||
---|---|---|---|---|

Core/lithosphere | 13,368 |
| ||

| ||||

\(M_2\) tidal | 2736 |
| ||

| ||||

Ionosphere/induced | 5520 |
| ||

| ||||

Magnetosphere/induced | 165,252 | Magnetosphere | ||

Induced | ||||

OHM biases | 465 | One vector bias for each station in local spherical system | ||

VFM-CRF alignment | 1350 | Three | ||

Total | 188,691 | − |

### Estimation procedure

*k*th step of the algorithm is given by

*j*th

*a priori*covariance matrix \({\mathbf {P}}_j^{-1}={\mathbf {F}}_j {\mathbf {F}}_j^{\mathrm{T}}\) that, along with the Lagrange multiplier \(\lambda _j\), specifies the deviation of the solution from the preferred

*a priori*model vector \(\mathbf {x}_j^{\prime }\). The matrix \(\mathbf {L}_k^{+}\) is the pseudo-inverse of \(\mathbf {L}_k\) which accounts for infinite variances in \(\mathbf {C}_k^{+}=\mathbf {L}_k^{+ \mathrm T} \mathbf {L}_k^{+}\). As will be seen, the system is subject to the linear equality constraints

*k*th step of LSLE-GN, denoted \(\varvec{\widetilde{\Delta x}}_k\), is given by

*Swarm*data. Table 2 shows the \(\ell _2\) norm of the adjustment vector \(\varvec{\Delta x}_k\) in Eq. 8 for each iteration

*k*computed with and without the magnetospheric/induced parameters. The size of the adjustments is three orders of magnitude smaller for \(k=3\) compared to \(k=0\) when the magnetospheric/induced parameters are excluded and one order of magnitude smaller when all parameters are considered. This, along with inspection of the fields at each iteration, leads to the conclusion that the CIY4 estimate has reasonably converged.

LSLE-GN convergence for CIY4, where \(\left| {\varvec{\Delta x}_k}\right|\) is the \(\ell _2\) norm of the adjustment vector \(\varvec{\Delta x}_k\) in Eq. 8 and “M/I” denotes magnetospheric/induced parameters

Iteration | \(\left| {\varvec{\Delta x}_k}\right| _2\) excluding M/I | \(\left| {\varvec{\Delta x}_k}\right| _2\) |
---|---|---|

0 | 4348.027 | 8351.771 |

1 | 135.642 | 3230.750 |

2 | 6.669 | 1556.125 |

3 | 1.997 | 642.987 |

#### Error-covariance

The data noise error-covariance matrix \(\mathbf {C}_k\) is designed to account for random, zero-mean error in the measurements and theory, but is also augmented, as will be discussed, to allow for bias mitigation which results from systematic error in the theory. In CIY4, the OHM vector components are in the (*North*, *East*, *Center*) or \(\textit{NEC}\) local spherical coordinate system and are given isotropic, i.e., the same for each component, uncertainties, \(\sigma\), of 7, 4, and \(15\,\hbox {nT}\) for observatories with QD latitudes equatorward of \(\pm \, 10^{\circ }\), poleward of \(\pm \, 10^{\circ }\) and equatorward of \(\pm\, 55^{\circ }\), and poleward of \(\pm \, 55^{\circ }\), respectively. Single satellite vector measurements are used in the \(\textit{BP3}\) orthogonal coordinate system where “*B*” is along the predicted magnetic field direction, “*P*” is in the \(\hat{\mathbf{n}}\times {\mathbf {B}}\) direction where \(\hat{\mathbf{n}}\) is the unit vector along the CRF *z*-axis, and “3” completes the system. The uncertainties are assumed isotropic at \(2.2\,\hbox {nT}\) and attitude error is assumed negligible. The satellite scalar measurements, *F*, are given the same uncertainty. As for the satellite vector sums and differences, they are computed in the \(\textit{NEC}\) system and are assigned isotropic uncertainties of \(2.2\,\hbox {nT}\) and \(0.3\,\hbox {nT}\), respectively. Therefore, the random error contribution to \(\mathbf {C}_k\) is diagonal.

*i*th scalar measurement is assigned a Huber weight at the

*k*th GN iteration according to

*i*th measurement, \(e_{i,k}\) is the current residual, and \(c=1.5\). Thus, the Huber distribution has a Gaussian core and Laplacian tails. These weights contribute to the diagonal elements of \(\mathbf {C}_k^{+}\).

CIY4 SIVW application, where the “x” indicate the QD latitude and sun position of the data type and which parameters it directly influences

Type | QD latitude | Sun position | Nominal | Nuisance | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Low | Mid | High | Light | Dark | Core | Lithosphere | Tide | Core | Lithosphere | Tide | |

OHM- | x | x | x | x | x | x | x | ||||

OHM- | x | x | x | x | x | x | x | ||||

Single- | x | x | x | x | x | x | x | ||||

Single- | x | x | x | x | x | x | |||||

Diffs- | x | x | x | x | x | x | x | ||||

Diffs- | x | x | x | x | x | x | |||||

Diffs- | x | x | x | x | x | ||||||

Diffs- | x | x | x | x | x | ||||||

Diffs- | x | x | x | x | x | x | |||||

Diffs- | x | x | x | x | x | ||||||

Sums- | x | x | x | x | x | x | x | ||||

Sums- | x | x | x | x | x | x | |||||

Sums- | x | x | x | x | x | x | x | ||||

Sums- | x | x | x | x | x | x |

#### Constraints

For the CIY4 model, the number of explicit quadratic constraints minimized is \(N_q=8\) in Eq. 8, although the linear equality constraints can also be expressed this way. They are distributed as five distinct smoothing constraints, i.e., \(\mathbf {x}_j^{\prime }=\mathbf {0}\), on the core and lithospheric fields, which includes the mean squared second and third time derivatives of the radial component of the magnetic field, \(B_r\), at the core–mantle Boundary (CMB) at \(3480\,\hbox {km}\) radius over the entire time domain of the model, denoted as “\(\mathcal{P}\langle |{\ddot{B}_r}|^2\rangle\)” and “\(\mathcal{P}\langle |{\dddot{B}_r}|^2\rangle\),” respectively, and additional customized smoothing of \(\dddot{B}_r\) applied to the \((n=1,m=0)\) and \((n=2,m=0)\) harmonics, denoted as “\(\mathcal{P}\langle {\dddot{B}_{r,n=1,m=0}^2}\rangle\)” and “\(\mathcal{P}\langle {\dddot{B}_{r,n=2,m=0}^2}\rangle\),” respectively. The inclusion of smoothing the third time derivative of the core field, with special treatment of the zonal harmonics, follows an approach previously applied with success in the CHAOS model series in order to study field accelerations (see Olsen et al. 2014; Finlay et al. 2016). The smoothing applied to the second time derivative is rather weak, and note the difference compared to the CHAOS model series, where the constraint on the second time derivative is applied only at the model end points whereas it is applied across the entire time domain here. The high-degree lithosphere (\(n\ge 85\)) is smoothed by minimizing the mean square \(B_r\) component over Earth’s mean surface at \(6371.2\,\hbox {km}\) and is denoted as “\(\mathcal{P}\langle \left| {\mathbf {B}_{n\ge 85}}\right| ^2\rangle\).”

Following Sabaka et al. (2004, 2015), the ionospheric field is smoothed using two constraints, where the first minimizes nightside *E*-region currents, denoted as “\(\mathcal{P}\langle \left| {\mathbf {J}_{\mathrm{eq},\mathrm{MLT}:21{-}05}}\right| _2^2\rangle\),” which measures the mean square magnitude of the *E*-region equivalent currents \({\mathbf {J}}_{\mathrm{eq}}\) flowing at \(110\,\hbox {km}\) altitude over the nighttime sector, defined as magnetic local time (MLT) \(21{:}00{-}05{:}00\) hours, through the year. The second, denoted as “\(\mathcal{P}\langle \left| {\nabla _s^2\mathbf {J}_{\mathrm{eq},p>0,\mathrm{mid{-}lat}}}\right| _2^2\rangle\),” measures the mean square magnitude of the surface Laplacian of the diurnally varying portion of \({\mathbf {J}}_{\mathrm{eq}}\) at mid-latitudes at all local times.

CIY4 damping parameter values

Norm | Damping parameter (\(\lambda\)) | |
---|---|---|

| ||

\(\mathcal{P}\langle {\dddot{B}_r^2}\rangle\) | \(1.0\times 10^1\,({\mathrm{nT}}\cdot {\mathrm{year}}^{-3})^{-2}\) | |

\(\mathcal{P}\langle {\dddot{B}_{r,n=1,m=0}^2}\rangle\) | \(3.0\times 10^2\,(\mathrm{nT}\cdot \mathrm{year}^{-3})^{-2}\) | |

\(\mathcal{P}\langle {\dddot{B}_{r,n=2,m=0}^2}\rangle\) | \(1.0\times 10^1\,(\mathrm{nT}\cdot \mathrm{year}^{-3})^{-2}\) | |

\(\mathcal{P}\langle {\ddot{B}_r^2}\rangle\) | \(4.0\times 10^{-2}\,(\mathrm{nT}\cdot \mathrm{year}^{-2})^{-2}\) | |

| ||

\(\mathcal{P}\langle \left| {\mathbf {B}_{n\ge 85}}\right| ^2\rangle\) | \(2.0\times 10^1\,(\mathrm{nT})^{-2}\) | |

| ||

\(\mathcal{P}\langle \left| {\mathbf {J}_{\mathrm{eq},\mathrm{MLT}:21{-}05}}\right| _2^2\rangle\) | \(4.0\times 10^7\,(\mathrm{A}\cdot \mathrm{km}^{-1})^{-2}\) | |

\(\mathcal{P}\langle \left| {\nabla _s^2\mathbf {J}_{\mathrm{eq},p>0,\mathrm{mid{-}lat}}}\right| _2^2\rangle\) | \(1.0\times 10^0\,(\mathrm{A}\cdot \mathrm{km}^{-3})^{-2}\) | |

| ||

\({\mathcal{P}}\langle \left| {\mathbf{p}}_{\mathrm{mag/ind}} \right|_{2}^{2}\rangle\) | Variable \((\mathrm{nT})^{-2}\) | |

\(\mathcal{P}\langle \left| {\mathbf {p}_{\mathrm{ind}\perp \mathrm{core}}}\right| _2^2\rangle\) | \(\infty \,(\mathrm{nT})^{-2}\) |

## Results and discussion

### Residual statistics

*Alpha*and

*Bravo*satellites in Table 5 and continued in Table 6 for the

*Charlie*satellite, the EW sums and differences between

*Alpha*and

*Charlie*, and the field of the OHMs. For the satellite data, the categories reflect the SIVW application scheme. Weighted statistics are shown because these are more representative of how the estimator treats the data types in the IRLS framework. The weighted means and root-mean-squares, \(\mu _w\) and \(r_w\), respectively, are related to the Huber weights in Eq. 12 as

*K*is the number of measurements and \(e_i\) and \(w_i\) are the

*i*th residual and Huber weight for a particular component, respectively, at the final iterate.

*Alpha*and

*Charlie*show very similar residual statistics as expected since they constitute the low satellite pair, while

*Bravo*shows slightly higher residuals. The expected properties of larger residuals at higher QD latitudes and on the light versus nightside appear to hold. The differences tend to exhibit the best fits while the sums are the worst of all the measurement types, particularly the light-side sums in the

*E*component at low QD latitudes, probably due to radial currents (toroidal magnetic field) connected to the equatorial electrojet, and

*N*and

*E*components at mid QD latitudes, which is likely due to field-aligned currents. The EW residual differences also appear to be somewhat larger than in the NS direction, but this will at least be partly due to the differencing of two separate instruments that have slightly different biases. Although the

*B*and

*F*field components are in slightly different directions, i.e., in the computed and observed field directions, respectively, their residual statistics for a given satellite are very similar, as one would expect. The OHMs also exhibit the same property of larger residuals at high QD latitudes and during sunlit conditions.

CIY4 weighted residual statistics

Origin | Type | Sun | Comp | QD latitude | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Low | Mid | High | ||||||||||

| \(\mu _w\) | \(r_w\) | | \(\mu _w\) | \(r_w\) | | \(\mu _w\) | \(r_w\) | ||||

Alpha | Field | Dark | | 262,005 | − 0.664 | 1.834 | 1,176,307 | 0.034 | 1.506 | |||

| 262,005 | 0.193 | 1.914 | 1,176,307 | − 0.025 | 2.511 | ||||||

3 | 262,005 | 0.038 | 1.889 | 1,176,307 | 0.094 | 2.445 | ||||||

| 262,005 | − 0.645 | 1.825 | 1,176,307 | 0.045 | 1.497 | 710,017 | − 0.075 | 5.553 | |||

NS differences | Dark | | 130,908 | 0.002 | 0.201 | 587,671 | 0.003 | 0.313 | ||||

| 130,908 | 0.001 | 0.334 | 587,671 | 0.000 | 0.360 | ||||||

| 130,908 | − 0.011 | 0.335 | 587,671 | 0.001 | 0.257 | ||||||

| 130,865 | 0.007 | 0.163 | 587,645 | − 0.010 | 0.179 | 354,475 | − 0.023 | 0.937 | |||

Light | | 165,351 | − 0.004 | 0.760 | 807,365 | − 0.006 | 0.555 | |||||

| 165,351 | 0.002 | 0.807 | 807,365 | − 0.001 | 0.839 | ||||||

| 165,351 | − 0.007 | 0.917 | 807,365 | 0.004 | 0.520 | ||||||

| 165,349 | − 0.002 | 0.651 | 807,240 | 0.010 | 0.318 | 683,385 | − 0.037 | 1.095 | |||

NS sums | Dark | | 130,908 | − 1.175 | 3.165 | 587,671 | − 0.714 | 3.770 | ||||

| 130,908 | 0.270 | 3.194 | 587,671 | 0.018 | 4.069 | ||||||

| 130,908 | 0.016 | 3.110 | 587,671 | 0.105 | 2.936 | ||||||

| 130,865 | − 1.016 | 3.035 | 587,645 | − 0.010 | 2.520 | 354,475 | − 0.101 | 8.919 | |||

Light | | 165,351 | 2.154 | 7.117 | 807,365 | − 0.664 | 6.885 | |||||

| 165,351 | − 0.232 | 12.185 | 807,365 | 0.162 | 9.156 | ||||||

| 165,351 | 0.330 | 7.337 | 807,365 | − 0.005 | 5.360 | ||||||

| 165,349 | 2.113 | 6.798 | 807,240 | − 0.647 | 5.039 | 683,385 | − 3.831 | 12.903 | |||

Bravo | Field | Dark | | 259,916 | − 0.792 | 3.013 | 1,169,539 | 0.071 | 2.227 | |||

| 259,916 | 0.262 | 2.425 | 1,169,539 | 0.086 | 2.893 | ||||||

3 | 259,916 | − 0.131 | 1.999 | 1,169,539 | 0.067 | 3.231 | ||||||

| 259,916 | − 0.819 | 3.024 | 1,169,539 | 0.081 | 2.224 | 715,893 | 0.086 | 5.437 | |||

NS differences | Dark | | 129,850 | 0.005 | 0.195 | 584,275 | 0.006 | 0.316 | ||||

| 129,850 | − 0.002 | 0.325 | 584,275 | − 0.002 | 0.364 | ||||||

| 129,850 | − 0.002 | 0.330 | 584,275 | 0.001 | 0.256 | ||||||

| 129,820 | 0.011 | 0.164 | 584,270 | − 0.009 | 0.196 | 357,613 | − 0.023 | 0.850 | |||

Light | | 163,615 | − 0.002 | 0.671 | 795,240 | − 0.000 | 0.540 | |||||

| 163,615 | 0.004 | 0.741 | 795,240 | − 0.002 | 0.822 | ||||||

| 163,615 | 0.005 | 0.838 | 795,240 | 0.001 | 0.502 | ||||||

| 163,592 | − 0.001 | 0.565 | 795,286 | 0.009 | 0.303 | 673,198 | − 0.035 | 0.992 | |||

NS sums | Dark | | 129,850 | − 1.211 | 5.068 | 584,275 | − 0.792 | 5.476 | ||||

| 129,850 | 0.366 | 3.972 | 584,275 | 0.232 | 4.702 | ||||||

| 129,850 | − 0.220 | 3.104 | 584,275 | 0.016 | 3.406 | ||||||

| 129,820 | − 1.173 | 4.931 | 584,270 | 0.162 | 3.650 | 357,613 | 0.122 | 8.862 | |||

Light | | 163,615 | 2.341 | 8.171 | 795,,240 | − 0.241 | 7.909 | |||||

| 163,615 | − 0.506 | 12.955 | 795,,240 | − 0.253 | 9.850 | ||||||

| 163,615 | 0.093 | 7.026 | 795,,240 | − 0.121 | 5.684 | ||||||

| 163,592 | 2.199 | 7.662 | 795,286 | − 0.654 | 5.681 | 673,198 | − 3.838 | 12.618 |

CIY4 weighted residual statistics *continued*

Origin | Type | Sun | Comp | QD latitude | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Low | Mid | High | ||||||||||

| \(\mu _w\) | \(r_w\) | | \(\mu _w\) | \(r_w\) | | \(\mu _w\) | \(r_w\) | ||||

Charlie | Field | Dark | | 259,643 | − 0.586 | 1.846 | 1,167,024 | 0.100 | 1.604 | |||

| 259,643 | 0.110 | 1.912 | 1,167,024 | − 0.101 | 2.578 | ||||||

3 | 259,643 | − 0.159 | 2.241 | 1,167,024 | 0.076 | 2.662 | ||||||

| 259,643 | − 0.604 | 1.851 | 1,167,024 | 0.097 | 1.604 | 705,882 | 0.002 | 5.599 | |||

NS differences | Dark | | 129,773 | 0.003 | 0.213 | 582,975 | 0.002 | 0.332 | ||||

| 129,773 | − 0.001 | 0.342 | 582,975 | − 0.001 | 0.372 | ||||||

| 129,773 | − 0.006 | 0.355 | 582,975 | − 0.001 | 0.278 | ||||||

| 129,723 | 0.006 | 0.174 | 582,999 | − 0.012 | 0.186 | 352,487 | − 0.024 | 0.941 | |||

Light | | 164,450 | − 0.004 | 0.764 | 802,108 | − 0.006 | 0.571 | |||||

| 164,450 | − 0.006 | 0.811 | 802,108 | − 0.004 | 0.848 | ||||||

| 164,450 | − 0.001 | 0.926 | 802,108 | 0.001 | 0.533 | ||||||

| 164,346 | 0.000 | 0.652 | 802,180 | 0.009 | 0.320 | 678,530 | − 0.037 | 1.096 | |||

NS sums | Dark | | 129,773 | − 1.033 | 3.141 | 582,975 | − 0.577 | 3.965 | ||||

| 129,773 | 0.172 | 3.198 | 582,975 | − 0.107 | 4.147 | ||||||

| 129,773 | − 0.312 | 3.569 | 582,975 | 0.086 | 3.299 | ||||||

| 129,723 | − 0.940 | 3.049 | 582,999 | 0.204 | 2.662 | 352,487 | 0.028 | 8.983 | |||

Light | | 164,450 | 2.129 | 7.095 | 802,108 | − 0.682 | 7.054 | |||||

| 164,450 | − 0.502 | 11.935 | 802,108 | 0.045 | 9.276 | ||||||

| 164,450 | 0.104 | 7.662 | 802,108 | − 0.020 | 5.562 | ||||||

| 164,346 | 2.108 | 6.780 | 802,180 | − 0.659 | 5.051 | 678,530 | − 3.739 | 12.857 | |||

Alpha/Charlie | EW differences | Dark | | 234,862 | 0.108 | 0.407 | 1,057,587 | 0.081 | 0.495 | |||

| 234,862 | 0.015 | 0.905 | 1,057,587 | 0.009 | 0.910 | ||||||

| 234,862 | − 0.122 | 0.617 | 1,057,587 | 0.014 | 0.427 | ||||||

| 235,259 | − 0.095 | 0.373 | 1,060,137 | − 0.069 | 0.341 | 654,422 | − 0.065 | 0.555 | |||

Light | | 297,697 | − 0.033 | 0.648 | 1,451,527 | − 0.027 | 0.788 | |||||

| 297,697 | − 0.023 | 2.146 | 1,451,527 | − 0.009 | 1.862 | ||||||

| 297,697 | − 0.069 | 1.285 | 1,451,527 | 0.018 | 0.746 | ||||||

| 298,285 | 0.028 | 0.548 | 1,453,701 | 0.008 | 0.491 | 1,218,886 | − 0.072 | 0.616 | |||

EW sums | Dark | | 234,862 | − 1.175 | 3.086 | 1,057,587 | − 0.677 | 3.703 | ||||

| 234,862 | 0.280 | 2.984 | 1,057,587 | − 0.050 | 4.007 | ||||||

| 234,862 | − 0.262 | 2.822 | 1,057,587 | 0.096 | 2.797 | ||||||

| 235,259 | − 1.063 | 2.982 | 1,060,137 | 0.168 | 2.495 | 654,422 | − 0.005 | 9.166 | |||

Light | | 297,697 | 2.270 | 7.029 | 1,451,527 | − 0.630 | 6.772 | |||||

| 297,697 | − 0.367 | 11.680 | 1,451,527 | 0.057 | 8.920 | ||||||

| 297,697 | 0.099 | 7.038 | 1,451,527 | − 0.057 | 5.194 | ||||||

| 298,285 | 2.256 | 6.720 | 1,453,701 | − 0.566 | 4.945 | 1,218,886 | − 3.858 | 13.066 | |||

OHM | Field | Dark | | 30,040 | − 0.001 | 4.422 | 363,714 | − 0.004 | 4.133 | 103,441 | 0.009 | 14.686 |

| 30,040 | 0.000 | 5.246 | 363,714 | 0.002 | 4.847 | 103,441 | 0.000 | 11.599 | |||

| 30,040 | 0.001 | 4.499 | 363,714 | 0.003 | 3.675 | 103,441 | 0.005 | 14.850 | |||

Light | | 38,022 | 2.688 | 10.602 | 533,305 | 0.631 | 6.861 | 222,753 | 3.384 | 18.633 | ||

| 38,022 | − 1.961 | 9.344 | 533,305 | − 1.750 | 7.882 | 222,753 | − 3.384 | 15.666 | |||

| 38,022 | − 0.073 | 9.957 | 533,305 | − 0.180 | 5.452 | 222,753 | − 0.927 | 17.620 |

*Swarm*vector data are chosen during times when \(Kp \le 3^0\) and so it is interesting to see the effect of this activity level on how well the data are fit. To this end Fig. 2 shows the field and difference residuals for the scalar and vector components of

*Alpha*measurements as a function of

*Kp*activity level. The residuals of the

*Bravo*and

*Charlie*satellites show patterns similar to that of

*Alpha*and so are not included. One can see diminished ranges of scalar difference residuals compared to vector field residuals, as the former increase slightly and the latter more profoundly with

*Kp*level. The ranges of the vector differences in the \(\textit{NEC}\) frame are much smaller than the field in the \(\textit{BP3}\) frame with the former appearing to be almost invariant to

*Kp*activity level in contrast to the latter, which also increase with

*Kp*level. This is intriguing since it means that more liberal bounds may be placed on

*Kp*selection levels for differences (as previously reported by Olsen et al. 2016), allowing for better data coverage. Finally, the scalar residuals appear to increase asymmetrically (skewed toward more positive values) with increasing

*Kp*as opposed to the symmetric increase seen in vector residuals, which is due to the presence of high latitude scalar data.

### Parameter separability

The main advantage of the CI algorithm is that it co-estimates the parameters from all considered sources in order to obtain optimal separation. However, one must be aware of co-linearities between the parameters that can amplify noise in the measurements and contaminate the estimate. Though this may appear to be a weakness in the co-estimation compared to the sequential approach, these co-linearities may be present in both, but are only readily detectable in the former. The co-linearities may be measured by inspecting the classic correlation coefficient, \(\rho\), between parameter pairs.

Because the CI algorithm initially removes the magnetospheric and associated induced parameters through Gaussian elimination, only \(\rho\) between non-magnetospheric/induced parameters is directly available. Of these parameters, there are five groups with \(\left| {\rho }\right| \ge 0.7\) inter-correlations. The first group is between spline parameters of the core/SV basis functions that are mostly positive such that many are \(\rho \approx 1\). This was also detected in CM5 (Sabaka et al. 2015) and is due to the influence of the core/SV quadratic constraints that contain null spaces. The second group is between certain zonal pairs of coefficients in the nuisance crustal field of the form \(g_n^0\) and \(g_{n+2}^0\) that are positive and can reach 0.74. The third group is between ionospheric parameters which are mostly negative and can be as low as \(-\,0.96\). These were not detected in CM5, but are also due to the influence of the quadratic constraints applied to the ionosphere that have large ranges in their eigenvalues. This suggests that these constraints are relatively stronger in CIY4 compared to CM5. The fourth group is between OHM biases whose locations are in close proximity such that some \(\rho \approx 1\). This was also seen in CM5 and is due to the similarity of the crustal field at the two locations, which is discussed in more detail in Sabaka et al. (2015). The last group concerns the Euler alignment angles and are either negative correlations between the *x*- and *z*-axis rotations for the same satellite in a given bin that can reach \(-\,0.96\) due to the intermediate rotation of approximately \(76^{\circ }\) about the *y*-axis or positive correlations between similar rotation axes in adjacent bins between \(\textit{Alpha}\) and \(\textit{Charlie}\) that can reach 0.98. The first and third groups are the result of smoothing constraints and are shown in Appendix A of Sabaka et al. (2015) to not adversely affect the solution and despite some large positive and negative \(\rho\) values in last group, it appears, as in CM5, that there are no deleterious effects.

*i*th magnetospheric/induced parameter and another

*j*th parameter of interest such that \(\left| {\rho _{ij}}\right| \le \rho ^{\prime }_j\) for all

*i*. Figure 3 shows \(\rho ^{\prime }\) for all non-magnetospheric/induced parameters where the letters indicate the parameter regime. Most \(\rho ^{\prime }\) are well below the \(\rho =0.7\) threshold and all \(\rho ^{\prime }<0.94\). It peaks above 0.7 for nominal and nuisance core/SV and nuisance \(M_2\) tidal parameters and several Euler angles. The pattern is generally similar to that seen in CM5 and the correlations with the magnetospheric/induced parameters do not appear to be detrimental.

### Core field

*Swarm*data (L1b data product version 0503) up to the end of 2017 but were in addition constrained by annual differences of ground observatory monthly means. Despite the differences in their data selection and modeling techniques, the time dependence of the CIY4 and CHAOS-6-x5 SV coefficients between 2014 and 2018 are in good agreement. In particular, the almost linear slopes (corresponding to a constant secular acceleration (SA)) in the coefficients \(\dot{g^1_1}\), \(\dot{h^2_2}\) and \(\dot{g^2_3}\) match very well, while changes in the slope of the SV, corresponding to pulses in SA, are observed at similar times in both models in the time series of \(\dot{g^0_1}\), \(\dot{g^2_2}\), \(\dot{g^0_3}\), \(\dot{g^1_3}\). Overall, the time dependence of the SV in CIY4 is simpler than that in CHAOS-6-x5, and there are some differences in the starting levels since CHAOS-6-x5 contains other data sources at earlier times, but the major features are shared by the two models.

Considering the SA and its time changes in CIY4 provides a number of new insights concerning the latest changes in the core field. The third panel in Fig. 6 shows the estimated change in the radial SV (i.e., the accumulated radial SA) at the CMB over the first 4 years of the *Swarm* mission, between 2014 and 2018. With 4 years of data it is now possible to confirm that using *Swarm* data alone results in similar SA patterns to those inferred previously from CHAMP-only and mixed CHAMP-*Swarm* field models (Chulliat and Maus 2014; Finlay et al. 2015). In particular, between 2014 and 2018 significant field acceleration at the core surface has occurred (i) at low latitudes under northern South America and extending into the Eastern Pacific, (ii) under South-East Asia, and (iii) under Alaska and Siberia. The latter point confirms that CIY4 shows similar patterns of field accelerations at high northern latitudes to those highlighted by Livermore et al. (2017), indicating these features are not a consequence of the gap between CHAMP and *Swarm* or due to any differences in the observing capabilities of these missions. Localized changes in CMB field acceleration patterns have previously been linked to the occurrence of geomagnetic jerks (Olsen and Mandea 2007; Chulliat et al. 2010). The largest differences in the CIY4 radial field SA between 2014 and 2018 are found under northern South America and under the equatorial Pacific, consistent with a possible jerk-like features found toward the end of the most up-to-date ground observatory SV series from this region (see Fig. 5). The prospect of detailed magnetic field observations from the *Swarm* constellation during a jerk event is tantalizing, but a detailed assessment needs to await the accumulation of longer ground observatory series. It is in any case striking that large changes in field acceleration occur in the Pacific hemisphere, despite the lower amplitude of secular variation in this region.

### Lithospheric field

*a*and

*r*are the reference and evaluation radii, respectively, \(\gamma _n^m\) are the complex, and \(g_n^m\) and \(h_n^m\) are the real Gauss coefficients of the SH expansion. The second metric is the degree correlation between two models

*k*.” The last metric is the matrix of normalized coefficient differences (in %),

*S*(

*n*,

*m*), given by

*x*” is “

*e*” or “

*r*,” respectively.

*Swarm*data.

*Z*component are plotted and compared on Earth’s ellipsoidal surface (WGS84). The top panel shows the CIY4 lithospheric field for degrees \(n=16{-}100\) while the bottom shows the difference between the fields of CIY4 and LCS-1 for the same degree range. Red curves represent the QD latitudes of \(\pm\, 55^\circ\) and \(0^\circ\) and both maps use the same scale. The models appear to agree well overall with the largest discrepancies in the polar regions, as expected. There also appears to be a faint patchwork of differences in the proximity of low QD latitudes. This may be a result of including dayside differences in determining the nominal lithospheric part of the model. Overall, the quality of the CIY4 lithospheric model is quite encouraging, especially given the altitude of the

*Swarm*satellites and the level of magnetic activity compared to the LCS-1 and MF7 models, which include CHAMP data.

### Oceanic \(M_2\) tidal field

*Swarm*data in CI1 by Sabaka et al. (2016). Results were in all cases validated by comparison with forward models described by Tyler et al. (2003) and Kuvshinov (2008). Here the progression of models based on increasing amounts of

*Swarm*data are compared along with the CM5 results. It is useful to consider the power of the tidal magnetic field using a generalization of the classic \(R_n\) spectrum of Lowes (1966) introduced in Sabaka et al. (2015, 2016) and defined as the mean square magnitude of the \(M_2\) magnetic field at SH degree

*n*over a sphere of radius

*r*and over the \(M_2\) tidal period given by

*Swarm*data and from the entire CHAMP mission (CM5). All models show strong peak regions in the vicinity of degrees \(n=4{-}7\) and roughly similar patterns up to about \(n=20\). However, at higher degrees the

*Swarm*2nd year model and CM5 diverge with higher power, especially CM5, due no doubt to field contamination. The

*Swarm*3rd and 4th (CIY4) year models show much less power at higher degrees and the latter shows a prominent peak at degree \(n=5\). There is a clear reduction in power from the 2nd to 3rd years of

*Swarm*data, but not so much between the 3rd and 4th years, which is likely indicating some critical coverage threshold being achieved by the 3rd year or perhaps due to a decrease in solar activity.

*Swarm*data through the 2nd, 3rd, and 4th (CIY4) years of the mission and from the CM5 model at the bottom. The progression confirms what is seen in the power spectra in that small-scale spurious, often north–south trending, features are eliminated as more

*Swarm*data are available, culminating in the CIY4 model which is much less noise-prone then the CM5 model derived from CHAMP satellite data. It should be noted, however, that the mid-to-large (and several small) features in the

*Swarm*fields appear to be converging to those of CHAMP, thus validating the high-quality measurements of both missions.

*Swarm*gradiometric measurements Fig. 11 shows the altitudes of the CHAMP satellite over its mission, the

*Swarm*satellites through the end of the CIY4 data envelope, and the \(F_{10.7}\) solar radiation index. The CHAMP mission ran for over 10 years during which the final 4 occurred in a period of anomalously low solar activity, hence less magnetic disturbances, and allowed the satellite altitude to go below \(350\,\hbox {km}\). In contrast, the

*Swarm*mission began flying during a relative high in \(F_{10.7}\) for almost 2 years, which has now decreased through the fourth year of the mission. However, until now, the low-pair altitudes have not gone below \(450\,\hbox {km}\). Thus, in spite of higher altitudes during relatively longer disturbed times, the

*Swarm*constellation has extracted a high quality \(M_2\) tidal driven magnetic field.

*Swarm*L2 data product. For this it has been decided that \(M_2\) SH coefficients will be presented in real notation as opposed to the complex notation used above such that Eq. 6 may be rewritten as

### Ionospheric field

*E*-region current system is treated as a sheet current at an altitude of \(110\,\hbox {km}\) while the secondary system is induced by the primary system via the “1D+oceans” conductivity structure described in section “Ionospheric field.” Figure 12 shows the variability of the equivalent current, i.e., stream, function corresponding to the primary system in two aspects: variation with respect to local time during vernal equinox in the top four maps, and variation with respect to season in the bottom four maps. As the basis functions for the ionosphere in CIY4 have QD symmetry, the QD latitudes of \(\pm\, 55^{\circ }\) and \(0^\circ\) are shown in red and blue, respectively. As expected, the top four maps show opposing streamlines mostly following QD lines of latitude and the two major solar-quiet (Sq) foci remaining mostly aligned along the same meridian during vernal equinox.

The bottom four maps show Sq foci aligned along the same meridian during vernal (March) and autumnal (September) equinox, while the northern foci is stronger and lags the southern foci in local time during northern summer (June) and the opposite happening during northern winter (December). Hence, the maps are similar to those from Sabaka et al. (2002, 2015); Chulliat et al. (2016) and are realistic at low and mid-latitudes. At high latitudes, however, the field is probably over damped and does not show the fixed-local time cells related to the well-known current systems associated with plasma convection in the polar cap ionosphere, for example captured in the SIFMplus model of Olsen et al. (2016).

In order to further validate the CIY4 ionospheric field, a comparison of predictions of QD mid-latitude OHM values was performed between CIY4 and a hybrid model in which the CIY4 ionosphere was replaced by the *Swarm* L2 “Dedicated Ionospheric Field Inversion” (DIFI) product presented in Chulliat et al. (2016), but updated with data through 2017 (*Swarm* L2 product SW_OPER_MIO_SHA_2D_20131201T000000_20171231T235959_0402), and the magnetosphere was replaced by the *Swarm* L2 MMA product (SW_OPER_MMA _SHA_2C_20131201T000000_20180101T000000_0401, described in the next section). The weighted RMS fit, \(r_w\), in the *NEC* frame from the hybrid model is (5.903, 6.056, 5.024) nT for dayside data and (3.512, 3.809, 3.657) nT for nightside data. This can be compared to the values for CIY4 from Table 6 in which the dayside is (6.861, 7.882, 5.452) nT and nightside is (4.133, 4.847, 3.675) nT. The hybrid model is clearly out performing CIY4 for this data set on the basis of \(r_w\), which is due to the DIFI ionospheric field predicting these data more closely. However, it should be stressed that the goal of field modeling is not the fitting of data, but rather the extraction of the most plausible geophysical parameters.

### Magnetospheric and induced fields

The CIY4 model is based on magnetic field observations from geomagnetic quiet periods, and as described in section “Magnetospheric field,” degree-1 external (magnetospheric) and internal (induced) SH expansions are co-estimated in hourly bins for the selected quiet periods. However, in order to obtain a continuous time series of magnetospheric and induced expansion coefficients a subsequent non-comprehensive approach is used: first, remove the CIY4 models of core, lithosphere (including observatory biases when applicable) and ionosphere (and its secondary induced part) from magnetic observations taken by *Swarm* and ground observatories covering the whole period from December 2013 to December 2017, including the geomagnetic disturbed periods that were excluded from CIY4, and then perform a SH analysis of the residuals in bins of 1.5 and 6 h duration for degree-1 and higher degree coefficients, respectively. Details of this resulting *Swarm* MMA (“Magnetic-Magnetospheric”) L2 product will be described in a separate publication.

*RC*, an index of magnetospheric ring-current strength (Olsen et al. 2014) determined using 14 ground magnetic observatories (in the reference, 21 observatories were used to define

*RC*), and

*Est*, which is the external part of the

*Dst*index determined using data from four low-latitude magnetic observatories (Maus and Weidelt 2004). Agreement between CIY4 and \(-RC_e\) (the negative sign makes the value comparable with \(q_1^0\)) is also very good; their difference (dark red curve) is smaller than \(3\,\hbox {nT}\) after correction for an offset in \(-RC_e\) of \(12\,\hbox {nT}\). This offset accounts for the unknown absolute baseline level of ring-current indices such as

*RC*and

*Dst*, which are entirely determined from ground observatory data. There seems to be a small annual variation in the difference of about \(1\,\hbox {nT}\) amplitude, with minima in December and maxima in June, whose origin is unknown. The difference with \(-Est\) (light blue curve) reveals erratic variations of up to \(\pm \, 8\,\hbox {nT}\) and more, which reflects the well-known baseline-instabilities of the

*Dst*index (e.g., Olsen et al. 2014).

## Conclusions

The ESA *Swarm* L2 CI magnetic products have been extracted from the CIY4 parent model that was produced from 4 years of *Swarm* satellite and complementary observatory hourly means data. The core, lithospheric, ionospheric, and magnetospheric fields, as well as the new \(M_2\) tidal product, have been validated and are found to be of good quality. The core field is in good agreement with the CHAOS-6 model, and the modeled SV follows closely trends seen at ground observatories. The SV in CIY4 is stable at the CMB out to at least degree 13, with a region of rapid change in core field SA seen at low latitudes under the Eastern Pacific and South America between 2014 and 2018. The lithospheric field agrees quite well with the MF7 model and the new high-resolution LCS-1 model over the entire SH degree range \(n=15{-}100\). Maps of the radial field show good agreement, even at high latitudes. The power in the differences between CIY4 and these models is still well below the power of the actual lithospheric field over this same SH degree range. The ionospheric field at low-to-mid-latitudes is also plausible and exhibits the same large-scale structure as seen in previous CMs and the DI versions. The LT variability of its stream function as a function of UT and season is also what is expected. The estimated quiet-time magnetospheric field variation shows good agreement with independent estimates of magnetospheric ring-current activity like *RC* and *Dst*.

The new *Swarm* \(M_2\) magnetic field product has been introduced in this paper. Its field coefficients will be distributed in real rather than complex form, and thus, there will be 2 coefficients for \(m=0\) terms and 4 for \(m{>}0\) terms. The progression from CHAMP through 2, 3, and now 4 years of *Swarm* data, culminating in the CIY4 model, shows a clear evolution of improvement in resolving the oceanic \(M_2\) magnetic field signal. Given that the CHAMP and *Swarm* missions are independent and have flown at different times under different conditions, the agreement between their \(M_2\) fields in amplitude and phase is very impressive. The resolution achieved with *Swarm* also suggests that other major tidal constituents could be convincingly detected.

## Notes

### Authors’ contributions

TJS led the development of the CI algorithm, oversaw its running, and led the writing of this manuscript. LT-C produced the official 4th year version of the ESA *Swarm* L2 CI magnetic products. NO developed aspects of the CI algorithm and provided the scientific guidance for the external and associated induced magnetic fields. CCF designed the core field regularization and provided scientific guidance for the core field. All authors read and approved the final manuscript.

### Acknowlegements

We would like to thank two anonymous reviewers for their useful comments. We would also like to thank ESA for providing prompt access to the *Swarm* L1b data. The staff of the geomagnetic observatories and INTERMAGNET are thanked for supplying high-quality observatory data, and BGS is thanked for providing checked and corrected observatory hourly mean-values.

### Availability of data and materials

The *Swarm* L1B data is available from ESA. Observatory data is available from INTERMAGNET. The CIY4 versions of the various L2 magnetic field products are available from the ESA *Swarm* data product website at https://earth.esa.int/web/guest/swarm/data-access.

### Competing interests

The authors declare that they have no competing interests.

### Funding

TJS is supported through the NASA Earth Surface and Interior program. CCF acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 772561).

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## References

- Chulliat A, Maus S (2014) Geomagnetic secular acceleration, jerks, and a localized standing wave at the core surface from 2000 to 2010. J Geophys Res Solid Earth 119(3):1531–1543CrossRefGoogle Scholar
- Chulliat A, Thébault E, Hulot G (2010) Core field acceleration pulse as a common cause of the 2003 and 2007 geomagnetic jerks. Geophys Res Lett 37:L07301CrossRefGoogle Scholar
- Chulliat A, Vigneron P, Hulot G (2016) First results from the Swarm dedicated ionospheric field inversion chain. Earth Planets Space 68(1):112. https://doi.org/10.1186/s40623-016-0481-6 CrossRefGoogle Scholar
- Constable CG (1988) Parameter estimation in non-Gaussian noise. Geophys J 94:131–142CrossRefGoogle Scholar
- Emmert JT, Richmond AD, Drob DP (2010) A computationally compact representation of magnetic-apex and quasi-dipole coordinates with smooth base vectors. J Geophys Res 115(A8):a08322CrossRefGoogle Scholar
- Everett M, Constable S, Constable C (2003) Effects of near-surface conductance on global satellite induction responses. Geophys J Int 153:277–286CrossRefGoogle Scholar
- Finlay CC, Olsen N, Tøffner-Clausen L (2015) DTU candidate field models for IGRF-12 and the CHAOS-5 geomagnetic field model. Earth Planets Space 67:114. https://doi.org/10.1186/s40623-015-0274-3 CrossRefGoogle Scholar
- Finlay CC, Olsen N, Kotsiaros S, Gillet N, Tøffner-Clausen L (2016) Recent geomagnetic secular variation from Swarm and ground observatories as estimated in the CHAOS-6 geomagnetic field model. Earth Planets Space 68(1):112. https://doi.org/10.1186/s40623-016-0486-1 CrossRefGoogle Scholar
- Kuvshinov A (2008) 3-D global induction in the oceans and solid earth: recent progress in modeling magnetic and electric fields from sources of magnetospheric, ionospheric and oceanic origin. Surv Geophys 29(2):139–186CrossRefGoogle Scholar
- Kuvshinov AV, Olsen N (2006) A global model of mantle conductivity derived from 5 years of CHAMP, Ørsted, and SAC-C magnetic data. Geophys Res Lett 33:L18301Google Scholar
- Laske G, Masters G (1997) A global digital map of sediment thickness. EOS Trans AGU 78:F483Google Scholar
- Livermore PW, Hollerbach R, Finlay CC (2017) An accelerating high-latitude jet in Earth’s core. Nat Geosci 10:62–68CrossRefGoogle Scholar
- Lowes FJ (1966) Mean-square values on sphere of spherical harmonic vector fields. J Geophys Res 71:2179CrossRefGoogle Scholar
- Macmillan S, Olsen N (2013) Observatory data and the Swarm mission. Earth Planets Space 65:1355–1362. https://doi.org/10.1186/s40623-015-0228-9 CrossRefGoogle Scholar
- Manoj C, Kuvshinov AV, Maus S, Lühr H (2006) Ocean circulation generated magnetic signals. Earth Planets Space 58:429–437. https://doi.org/10.1186/BF03351939 CrossRefGoogle Scholar
- Maus S (2010) Magnetic field model MF7. www.geomag.us/models/MF7.html
- Maus S, Weidelt P (2004) Separating the magnetospheric disturbance magnetic field into external and transient internal contributions using a 1D conductivity model of the Earth. Geophys Res Lett 31:L12614Google Scholar
- Olsen N (2013) The Swarm Satellite Constellation Application and Research Facility (SCARF) and Swarm data products. Earth Planets Space 65:1189–1200. https://doi.org/10.1186/s40623-016-0481-6 CrossRefGoogle Scholar
- Olsen N (2015) The Swarm Initial Field Model for the 2014 geomagnetic field. Geophys Res Lett 42(4):1092–1098CrossRefGoogle Scholar
- Olsen N, Mandea M (2007) Investigation of a secular variation impulse using satellite data: the 2003 geomagnetic jerk. Earth Planet Sci Lett 255:94–105CrossRefGoogle Scholar
- Olsen N, Lühr H, Finlay CC, Sabaka TJ, Michaelis I, Rauberg J, Tøffner-Clausen L (2014) The CHAOS-4 geomagnetic field model. Geophys J Int 197:815–827CrossRefGoogle Scholar
- Olsen N, Finlay CC, Kotsiaros S, Tøffner-Clausen L (2016) A model of Earth’s magnetic field derived from two years of Swarm satellite constellation data. Earth Planets Space 68:124. https://doi.org/10.1186/s40623-016-0546-6 CrossRefGoogle Scholar
- Olsen N, Ravat D, Finlay CC, Kother LK (2017) LCS-1: a high-resolution global model of the lithospheric magnetic field derived from CHAMP and Swarm satellite observations. Geophys J Int 211(3):1461–1477CrossRefGoogle Scholar
- Richmond AD (1995) Ionospheric electrodynamics using magnetic Apex coordinates. J Geomagn Geoelectr 47:191–212CrossRefGoogle Scholar
- Rother M, Lesur V, Schachtschneider R (2013) An algorithm for deriving core magnetic field models from the Swarm data set. Earth Planets Space 65:1223–1231. https://doi.org/10.5047/eps.2013.07.005 CrossRefGoogle Scholar
- Sabaka TJ, Olsen N (2006) Enhancing comprehensive inversions using the Swarm constellation. Earth Planets Space 58:371–395. https://doi.org/10.1186/BF03351935 CrossRefGoogle Scholar
- Sabaka TJ, Olsen N, Langel RA (2002) A comprehensive model of the quiet-time near-Earth magnetic field: Phase 3. Geophys J Int 151:32–68CrossRefGoogle Scholar
- Sabaka TJ, Olsen N, Purucker ME (2004) Extending comprehensive models of the Earth’s magnetic field with Ørsted and CHAMP data. Geophys J Int 159:521–547CrossRefGoogle Scholar
- Sabaka TJ, Tøffner-Clausen L, Olsen N (2013) Use of the comprehensive inversion method for Swarm satellite data analysis. Earth Planets Space 65:1201–1222. https://doi.org/10.5047/eps.2013.09.007 CrossRefGoogle Scholar
- Sabaka TJ, Olsen N, Tyler RH, Kuvshinov A (2015) CM5, a pre-Swarm comprehensive magnetic field model derived from over 12 years of CHAMP, Ørsted, SAC-C and observatory data. Geophys J Int 200:1596–1626CrossRefGoogle Scholar
- Sabaka TJ, Tyler RH, Olsen N (2016) Extracting ocean-generated tidal magnetic signals from Swarm data through satellite gradiometry. Geophys Res Lett 43(7):3237–3245CrossRefGoogle Scholar
- Seber GAF, Wild CJ (2003) Nonlinear regression. Wiley-Interscience, New YorkGoogle Scholar
- Thébault E, Vigneron P, Langlais B, Hulot G (2016) A Swarm lithospheric magnetic field model to SH degree 80. Earth Planets Space 68(1):104. https://doi.org/10.1186/s40623-016-0481-6 CrossRefGoogle Scholar
- Tyler RH, Maus S, Lühr H (2003) Satellite observations of magnetic fields due to ocean tidal flow. Science 299:239–241CrossRefGoogle Scholar

## Copyright information

**Open Access**This 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.