Ionospheric Multi-Spacecraft Analysis Tools pp 255-284 | Cite as

# Models of the Main Geomagnetic Field Based on Multi-satellite Magnetic Data and Gradients—Techniques and Latest Results from the Swarm Mission

## Abstract

Magnetic field observations from low-Earth-orbiting satellites provide a unique means of studying ionospheric current systems on a global scale. Such studies require that estimates of other sources of the Earth’s magnetic field, in particular, the dominant main field generated primarily in Earth’s core but also due to the magnetized lithosphere and large-scale magnetospheric currents, are first removed. Since 1999 multiple low-Earth-orbit satellites including Ørsted, CHAMP, SAC-C, and most recently the *Swarm* trio have surveyed the near-Earth magnetic field in increasing detail. This chapter reviews how models of the main magnetic field are today constructed from multiple satellites, in particular discussing how to take advantage of estimated field gradients, both along-track and across-track. A summary of recent results from the *Swarm* mission regarding the core and lithospheric field components is given, with the aim of informing users interested in ionospheric applications of the options available for high accuracy data reduction. Limitations of the present generation of main field models are also discussed, and it is pointed out that further progress requires improved treatment of ionospheric sources, in particular at polar latitudes.

## 12.1 Introduction

Ionospheric current systems produce magnetic fields that are measured by magnetometers on low-Earth-orbit satellites, together with the magnetic fields produced by a wide range of other natural sources. The largest of these sources is the so-called ‘main’ magnetic field, generated in Earth’s liquid metal outer core through motional induction in a process known as the geodynamo (e.g. Roberts and King 2013). For those interested in precise studies of ionospheric currents it is important to remove a high-resolution estimate of the internally-generated field, capturing as far as possible its small-scale structure and secular time dependence (see e.g. Stolle et al. 2016). In the context of this book, models of the main magnetic field can, therefore, be considered as important tools needed for studying ionospheric physics. Moreover, some of the data processing and modelling techniques used in main field studies are themselves of interest to ionospheric physicists, since they can easily be adapted to the study of ionospheric processes.

In this chapter, I begin by reviewing how models of the main geomagnetic field are constructed, focusing on recent developments that take advantage of magnetic field data collected by the low-Earth-orbit *Swarm* satellite constellation. The aim is to provide an easily accessible account of the construction of advanced main field models, so that users can make a well-informed decision about which models may be most suitable for their specific data processing and reduction tasks. Following this, a survey is given of the latest results regarding the structure and time-dependence of the internal geomagnetic field, as derived from data collected by the *Swarm* mission. The CHAOS series of field models (Olsen et al. 2006, 2009, 2010, 2014; Finlay et al. 2015, 2016) is a regularly updated, high resolution, main field model that covers the past one and half solar cycles. It will serve here as an illustrative example of an advanced field model that may be of interest for ionospheric studies.

The development of high-resolution geomagnetic field models is a community effort, in particular, facilitated by comparisons carried out within the framework of the International Geomagnetic Reference Field (IGRF) (Thébault et al. 2015a). Aside from the CHAOS model, that is the focus of this chapter, high-resolution field models are also available from a number of other groups, for example, the GRIMM series of models (e.g. Lesur et al. 2008, 2010, 2015b), the POMME series of models (e.g. Maus et al. 2005, 2006b) and the Comprehensive Model/Inversion series of models (e.g. Sabaka et al. 2004, 2015, 2018). Interested readers should consult these references for further details on these models. Limitations of all existing main field models, and opportunities to improve them using our expanding knowledge of ionospheric processes are discussed at the end of this chapter.

## 12.2 Fundamentals of Main Field Modelling

### 12.2.1 Calibration of Vector Magnetic Field Measurements

Modern main field geomagnetic reference models are derived primarily from magnetic field observations collected by low-Earth-orbit satellites. In particular, data from the *Swarm* satellite constellation, supplemented by measurements made on ground at geomagnetic observatories, are now crucial. For studies of the main field it is essential that the measurements, from both ground and satellite, have absolute accuracy—this is in contrast to the study of ionospheric processes, where often only rapid field variations are of interest. For satellite measurements this involves careful magnetometer design, strict magnetic cleanliness procedures when constructing the spacecrafts, pre-flight characterization of stray fields (Jørgensen et al. 2008) and in-flight calibration based on comparisons between fluxgate vector magnetometers and absolute scalar magnetometers that independently measure the field intensity (e.g. Olsen et al. 2003; Yin and Lühr 2011). Accurately orientated vector field measurements are also essential, since using scalar field intensity data alone there is a fundamental ambiguity arising from lack of knowledge perpendicular to the field (Backus 1970; Lowes 1975), particularly at the magnetic equator. In magnetic mapping missions, attitude information is today provided by high precision, non-magnetic, star trackers (Jørgensen et al. 2003). For *Swarm*, after application of models describing thermal fluctuations, attitude information is available at the arc-second level (Herceg et al. 2017).

*Swarm*satellites, is carried out by minimizing the difference between the scalar magnetic field \(F_{ASM}\) measured by an absolute scalar magnetometer and the magnitude of the vector magnetic field \(|\mathbf {B}|\) measured by the vector fluxgate magnetometer (VFM) frame. Free parameters that can be adjusted during inflight calibration arise when relating \(\mathbf {B}\) to the vector field \(\mathbf {B}_\mathrm {pre-flight}\) determined using pre-flight determined fluxgate magnetometer calibration parameters, and after correction for the pre-flight determined stray magnetic fields (e.g. Olsen and Kotsiaros 2011), via the relation

*Swarm*satellites, a small solar-driven magnetic disturbance, thought to be due to currents flowing in the satellite body as a result of thermo-electric effects, was detected post-launch (Tøffner-Clausen et al. 2016). This has been successfully described using an empirical model that depends on the sun position relative to the satellite and it is applied as an additional offset factor during the Level 1b magnetic data calibration procedure (for more details see Tøffner-Clausen et al. 2016). Physics-based models of this disturbance have also been developed, and improvements in Swarm’s on-board calibration are ongoing.

### 12.2.2 Selection of Magnetic Field Data for Main Field Modelling

When constructing models of the main field, typically only data from geomagnetically quiet times are used, in an effort to reduce as far as possible the contaminating signatures arising from magnetospheric and ionospheric current systems. Of course this is the opposite mode of operation to that of the space physicist, who is often more interested in data collected during strongly disturbed conditions. Typical quiet-time data selection criteria are that the \(K_p\) index is less than 2*o*, that the rate of change of \(D_{st}\) or similar ring current indices is less than 2 nT/h, and that for data in the polar region that the merging electric field, as determined from solar wind and IMF conditions measured at the L1 point, is less than 0.8 mV/m [Olsen et al. (2006), Olsen et al. (2014)]; for a more detailed discussion of data selection in internal field modelling interested readers should consult the recent reviews by Finlay et al. (2017) and Kauristie et al. (2017).

### 12.2.3 Potential Field Modelling

*V*

*V*must be a solution of Laplace’s equation

*r*is the distance from the centre of the Earth, \(\vartheta \) is the co-latitude measured from the north pole, and \(\lambda \) is the longitude, all defined in an Earth-centred, Earth-fixed, geographic reference frame.

### 12.2.4 Representation of the Field Due to Internal Sources

*n*and order

*m*, and \(\left\{ g_{n}^{m},h_{n}^{m}\right\} \) are the Gauss coefficients describing the amplitude of the internal sources. In principle, the maximum degree \(N_\mathrm {int}\) would be infinity if one wished to represent all possible details of the field structure, but for practical reasons the expansion is truncated at some finite maximum degree, beyond which the smallest wavelengths cannot be reliably retrieved. Illustrative examples of spherical harmonic functions \(Y_{n}^{m}(\vartheta ,\lambda )\), i.e. \(\cos m\lambda \, P_{n}^{m}\left( \cos \vartheta \right) \) or \( \sin m\lambda \, P_{n}^{m}\left( \cos \vartheta \right) \) are presented in Fig. 12.1. These are fundamental building blocks that may be combined to represent global functions on a spherical surface.

In addition to spatial dependence, accurate models of the main field must take into account the slow temporal or secular variation of the internal field. In standard models such as the International Geomagnetic Reference Field (IGRF), this is accounted for by linear interpolation between Gauss coefficients \(g_n^m\) defined at reference epochs.

*K*(de Boor 1978; Bloxham and Jackson 1992) such that

*p*that span the time interval of interest. The B-spline basis functions \(\mathscr {B}_{K,p}\) are piecewise polynomials of order

*K*. Examples of B-spline basis functions of order \(K=4\) (i.e. cubic B-splines), that when combined with appropriate weights can reproduce the time-dependent signal of interest, are shown in Fig. 12.2. In advanced main models including the latest members of the CHAOS model series (Olsen et al. 2014; Finlay et al. 2016), order 6 B-splines are used, so that the resulting models can easily be differentiated twice in time to study the field acceleration.

### 12.2.5 Representation of the Field Due to External Sources

Maus and Lühr (2005) and Olsen et al. (2005) developed more useful parameterizations of the near-Earth external field including (i) a component expressed in the *Solar Magnetic (SM)* coordinate system (with its *z* axis parallel to the Earth’s magnetic dipole axis and its *y* axis perpendicular to the plane containing the dipole axis and the Earth–Sun line) that represents well the geometry of the magnetospheric ring current, and (ii) a part in the *Geocentric Solar Magnetospheric (GSM)* coordinate system (with its *x* axis towards the Sun and its *z* axis being the projection of the Earth’s magnetic dipole axis (positive North) on to the plane perpendicular to the *x* axis) that is more suitable for studying magnetospheric phenomena strongly influenced by the interplanetary magnetic field direction including magnetotail and magnetopause currents. Further details of these and other magnetic coordinate systems, including how to convert between them, are described by Laundal and Richmond (2017).

*(SM)*coordinate system. The upper line is the SM dependent part; it is truncated at degree \(N_{SM}=2\), and includes a special treatment of the \(n=1\) terms (see below). The part in GSM coordinates on the lower line is truncated at degree \(N_{GSM}=2\), but is restricted to order \(m=0\)). Here the functions \(R^0_n(r, \vartheta , \lambda )\) are modifications of the Legendre functions that explicitly account for field contributions induced in the electrically conducting mantle due to the wobble of the GSM z-axis with respect to the Earth’s rotation axis. For a non-conducting Earth, these functions would be \(R^0_n= (r/a)^n P^0_n (\cos \vartheta _{GSM})\) where \(\vartheta _{GSM}\) is co-latitude in the GSM coordinate system; considering a plausible 1D model of mantle conductivity leads to a representation of \(R^0_n\) similar to the expansion described by Maus and Lühr (2005).

*SM*coordinates in an Earth-fixed, Earth-centred frame (which also results in an induced counterpart), the CHAOS model allows for additional time-dependence of the degree one field in SM coordinates, of the form

*RC*were a perfect description of the magnetospheric field at satellite altitude then the values of the regression coefficients would be \(\hat{q}_1^0 = -1\), \(\hat{q}_1^1 = \hat{s}_1^1 = 0\) and the ‘RC baseline corrections’ \(\varDelta q_1^0, \varDelta q_1^1\) and \(\varDelta s_1^0\) would vanish. The most recent version of the CHAOS model (Finlay et al. 2016) estimated such baseline corrections in bins of 5 days (for \(\varDelta q_1^0\)) and 30 days (for \(\varDelta q_1^1, \varDelta s_1^1\)).

### 12.2.6 Using Data in the Magnetometer Frame: Co-estimation of Magnetometer Attitude

*ICRF*) and is derived from the satellite position and time (Seeber 2003), \(\underline{\underline{\mathbf{R}}}_2\) rotates the magnetic field from the (

*ICRF*) system to the Common Reference Frame (

*CRF*) of the satellite’s star tracker and is constructed from the attitude data collected by the star tracker, and finally \({ \underline{\underline{\mathbf{R}}}}_1\) rotates from the star tracker

*CRF*to the orthogonal magnetometer (VFM) frame. \({\underline{\underline{\mathbf{R}}}}_3\) and \({ \underline{\underline{\mathbf{R}}}}_2\) are typically well known and in many cases it is also assumed that \({\underline{\underline{\mathbf{R}}}}_1\) is known exactly and in advance of collecting the data. However, in the most advanced main field models including the CHAOS series, rather than being assumed in advance, the Euler angles defining \({\underline{\underline{\mathbf{R}}}}_1\) are instead co-estimated as part of the modelling procedure (Olsen et al. 2006). In this case, the relation between the measurements in the VFM frame and the model parameters (coefficients of the internal potential from (12.1) and (12.9), of the external potential from (12.11) and (12.12), and the Euler angles defining \(\underline{\underline{\mathbf{R}}}_3\) from (12.13), is nonlinear, and model estimation becomes a nonlinear inverse problem.

### 12.2.7 Model Estimation: Solution of the Inverse Problem

Determination of the model parameters from the magnetic observations \(\mathbf {d}_{\mathrm {obs}}\) is a non-linear inverse problem. Furthermore, since it involves downward continuation of observations from satellite altitude, it is also an ill-conditioned problem. Moreover, since there are field sources that vary rapidly in space and time that cannot be captured by the model (i.e. the model is incomplete, for example, failing to account for auroral and polar-cap currents) the residuals between model predictions and the data are often long-tailed and not simply Gaussian distributed.

*j*th data at the previous (i.e. ith) iteration (e.g. Constable 1988; Olsen 2002) and \(\sigma _j\) is an a priori estimate of data uncertainty for the

*j*th datum. \(H_{jj}\) are known as Huber weights, and they permit robust solutions to be obtained even in the presence of long-tailed error distributions.

## 12.3 Use of Field Gradients and Multi-satellite Data in Main Field Modelling

Above we have described the standard approach to geomagnetic field modelling, based on the observed vector or scalar field components. The launch of the *Swarm* multi-satellite constellation (Friis-Christensen et al. 2006; Olsen et al. 2016b), has opened exciting new possibilities for using approximate field gradients, estimated via along-track and across-track differences, in main field modelling. In this section, we outline these new techniques, focusing on how to construct suitable estimates of the field gradients and on how to deal with data from multiple satellites within a single inversion.

### 12.3.1 Estimates of Field Gradients: Approximation by Along-Track and Across Track Differences

The constellation of the *Swarm* trio of satellites, with two satellites (Alpha and Charlie) flying close together and a third (Bravo) flying at higher altitude and drifting in local time with respect to the lower pair, enables estimates both along-track and cross-track gradients of the geomagnetic field to be made from space. In particular, considering differences in the field recorded by the lower pair permits the cross-track field gradient to be estimated for the first time. This is extremely valuable for constraining small-scale east–west structures in the lithospheric field (Maus et al. 2006a; Thébault et al. 2016), although it provides no constraint on zonal (\(m=0\)) and little constraint on near zonal (small *m*) spherical harmonic components of the field.

Although the *Swarm* constellation does not have an along-track satellite pair (which would have an approximately north–south orientation at mid and low latitudes), one can instead use along-track differences from a single satellite to estimate the gradient in this direction, with the assumption the field does not change appreciably over the time taken to move between the locations differenced. A typical time between the locations differenced is 15 s (Olsen et al. 2015), much shorter than the time scale of large-scale magnetospheric field changes. This corresponds to an along-track spatial separation of about 115 km. (Kotsiaros et al. 2014) have explored the impact on lithospheric field models of using different time separation when computing along-track gradient estimates. There is clearly a trade-off between the signal amplitude (smaller for shorter time differences) and noise (larger for larger time-differences due to the breakdown of the stationarity assumption). The optimal time separation will also depend on the target wavelengths, on the geomagnetic latitude and on geomagnetic conditions (quiet or storm-time, dark or sunlit). There is certainly room to better optimize the calculation of field gradient estimates. Nonetheless, use of field gradients has already proven to be of great value in deriving high-resolution models of the core field (Olsen et al. 2015; Finlay et al. 2016) and lithospheric field (Sabaka et al. 2015; Kotsiaros 2016; Olsen et al. 2017).

How can one use field gradient estimates in the construction of main field models? The approach is a straightforward extension of standard procedure of performing least-square fits. One simply minimizes the square of the residuals between the observed and modelled field differences, which may be either differences of vector components or differences of scalar intensity values.

In the field models CHAOS-6 (Finlay et al. 2016), SIFM (Olsen et al. 2015) and SIFM+ (Olsen et al. 2016a), along-track (or approximately north–south at the equator so denoted north–south below) gradients were approximated by the differences \(\varDelta B_\mathrm {NS} = \pm [B_k(t_k, r_k, \theta _k, \phi _k) - B_k(t_k+15\,\mathrm {s}, r_k+\delta r, \theta _k + \delta \theta , \phi _k + \delta \phi )]\) of subsequent data, measured by the same satellite *k*, 15 s later, corresponding to an along-track distance of \({\approx }115\) km (\({\approx }1^\circ \) in latitude near the equator) for the *Swarm* satellites. Here *B* could be the scalar intensity *F* or any of the geocentric field components \((B_r, B_\theta , B_\phi )\). The sign of the difference was chosen positive if \(\delta \theta >0\), otherwise negative. The choice of 15 s was found to give a reasonable amplitude of internal field signal while being sufficiently short that much of the large-scale external field is unchanged, so therefore removed on taking the difference. 15 s differences have the advantage of involving differences over lengths similar to the East–West spacing between *Swarm*’s lower satellite pair.

To approximate the East–West gradients the above studies used the difference \(\delta B_\mathrm {EW} = \pm [B_A(t_1, r_1, \theta _1, \phi _1) - B_C(t_2, r_2, \theta _2, \phi _2)]\), between field components, or the scalar field, measured by the two satellites *Swarm* Alpha and *Swarm* Charlie. Here \(t_i, r_i, \theta _i, \phi _i, i=1-2\) are time, radius, geographic co-latitude and longitude of the two observations. The sign of the difference was chosen such that \(\delta \phi = \phi _1 - \phi _2 > 0\). For each observation \(B_A\) (from *Swarm* Alpha) the corresponding value \(B_C\) (from *Swarm* Charlie) was chosen to be that closest in co-latitude \(\theta \), with the requirement that \(|\delta t| = |t_1 - t_2|<\) 50 s. Note there is a time delay between the ground-tracks of *Swarm* Alpha and *Swarm* Charlie to avoid collisions at the pole, so simultaneous values from each satellite do not provide an estimate of the east–west gradient. Taking *Swarm* Charlie values slightly delayed, typically by around 10 s, allows differences to be taken that are very close to the desired East–West configuration. Note that this again requires that large-scale field be stationary over approximately 10 s if they are to cancel on taking the difference.

### 12.3.2 Information Content of Field Gradient Estimates

*Swarm*data in the context of studying the lithospheric field. Figure 12.3, reproduced from their study, shows theoretical model variances of spherical harmonic coefficients as a function of degree (y-axis) and order (x-axis), based on the positions of CHAMP and

*Swarm*data, and considering the impact of both vector data and vector gradients, i.e. north–south and east–west gradients. Blue colours show well determined coefficients, yellow colours indicate poorly determined model coefficients.

The results in Fig. 12.3 are based on a simplified linearized version of the inverse problem, where the Euler angles are assumed known and only the internal potential is considered. The plots show diagonal entries of the formal model covariance matrix \((\mathbf {G}^T \mathbf {W} \mathbf {G})^{-1}\), where \(\mathbf {G}\) here is simply the matrix for the linear forward problem connecting the satellite data (either vector field or vector field differences from either CHAMP and *Swarm*) to the spherical harmonic model coefficients, and where \(\mathbf {W}\) is a matrix of data weights.

North–south or along-track gradients of CHAMP data clearly provide a much improved resolution (i.e. lower model variances) for the high degree spherical harmonic coefficients, especially at degrees 120–140 which are poorly constrained by field data alone. North–south differences do not, however, constrain well the sectorial and near sectorial spherical harmonics. East–West differences from *Swarm* provide very valuable new constraints on these coefficients. The information provided by *Swarm* on the high degree field will, of course, be enhanced as the satellites descend to lower altitudes (the greater information provided by CHAMP north–south differences compared to *Swarm* north–south differences is purely due to the present higher altitude of the *Swarm* satellites). The poorer determination of high degree near-zonal coefficients is a consequence of the polar gap in the data distribution. Field differences are seen to provide useful information, not just on high degree coefficients associated with the lithospheric field signal, but also on low degree coefficients describing the core field; moreover, unmodelled large-scale magnetospheric fields are effectively suppressed when considering field gradient estimates, which increases the signal to noise ratio.

### 12.3.3 Examples of Field Gradient Data and Their Interpretation

*Swarm*satellites (north–south difference, labelled NS and east–west differences, labelled EW, divided by the separation distance in kilometers between the measurements) on example day-side (Fig. 12.4) and night-side (Fig. 12.5) half orbits. We present the signal remaining after the progressive removal of the estimates of the core field (top), core and lithospheric fields (middle), and core, lithospheric and magnetospheric fields (bottom) as a function of quasi-dipole (QD) latitude. These examples are taken from Olsen and Stolle (2017).

The selected day-side orbit shown in Fig. 12.4 has an equatorial Local Time crossing at 12:12 LT, and is from 2 May 2014, which was a geomagnetically quiet day (Kp < 1+ and Dst >-13 nT). The left column presents observations from *Swarm* Alpha; the middle column presents estimates of the East–West field difference measured between *Swarm* Alpha and *Swarm* Charlie, divided by the distance between the two spacecraft; the right column shows an estimate of the North–South gradient, obtained from 15-second north–south differences of *Swarm* Alpha divided by d \(=\) 141 km (which is the distance of two satellite measurements taken 15 s apart). For each of the three columns, the blue curves show the difference \(F = F_{obs} - F_{mod}\) between the observed magnetic intensity \(F_{obs}\) and various model values \(F_{mod}\) predicted by CHAOS-6 model, whereas the red curves show predictions of some additional parts of the model. The yellow curve in the bottom left panel shows \(F_{obs}\) from *Swarm* Charlie. See figure caption for more details. For this day-side half orbit, the signature of the equatorial electrojet is clearly seen at the magnetic equator, especially in the north–south gradient (right column), resulting from the depression it causes in the field intensity (see left column). The signature of the Sq ionospheric current system is also seen as minima in field intensity close to 30\(^\circ \) QD latitude. Large signals remain in the polar regions even after removal of estimates of the core, lithospheric, magnetospheric and Sq fields, especially in sunlit northern hemisphere, where large gradients are seen in both the east–west and north–south differences.

*Swarm*satellites, not shown). Ionisation anomalies after sunset are frequently affected by plasma density irregularities (sometimes called ‘bubbles’) close to \({\pm }10^\circ \) QD latitude that produce magnetic signatures of a few nanotesla; these anomalies have small length scale, so can be different between

*Swarm*Alpha and Charlie, that results in clear signatures in the EW differences. Once again the largest unmodelled signals after the removal of the core, lithospheric, magnetospheric and Sq-induced parts, both for field data and field gradient estimates, are found in the polar regions, particularly in the summer (northern) hemisphere in the example presented here. These unmodelled polar signals are presently the major challenge facing main field modellers (e.g. Finlay et al. 2017).

### 12.3.4 Simultaneous Inversion of Data from Multiple Satellites

Separate uncertainty estimates for CHAMP north–south gradients, *Swarm* north–south gradients and *Swarm* east–west gradients are specified. Uncertainties for the scalar field are shown separately for sunlit and dark regions. These uncertainties were derived by binning the data residuals relative to the CHAOS-6 field model, and estimating the standard deviation \(\sigma \) using a robust (Huber weighting) approach within bins of 5\(^\circ \) QD latitude. The largest uncertainties occur in the polar regions, in particular for the sunlit parts, due to the enhanced ionospheric conductivity in these regions, and related magnetosphere-ionosphere coupling. The estimated uncertainties in *Swarm* inter-satellite East–West differences are generally larger than the north–south single satellite differences, and there is also interesting evidence for distinct unmodelled signals in the East–West differences at low quasi-dipole latitudes.

Using these uncertainties within the main or lithospheric field inverse problem essentially downweights data from the polar regions that is more likely to be contaminated by the signature of currents not included in the main or lithospheric field model. Data uncertainties allocated from CHAMP and *Swarm* turn out to be rather similar, showing similar trends as a functions of quasi-dipole latitude. It is also possible to define data and uncertainties from other satellite missions (Ørsted, SAC-C, DMSP etc.), where the uncertainties can be much larger, particularly with regard to how attitude errors influence our determination of the vector field (Olsen et al. 2006).

The error budgets presented here are still rather crude. In reality data errors with respect to internal field modelling are correlated in both space and time due to the structured nature of the unmodelled magnetospheric and ionospheric currents. In particular, the data are correlated along track (Lowes and Olsen 2004) which is why considering along-track or north–south field differences (which decorrelates this error source) is such an advantage. In addition, measurements from similar quasi-dipole latitudes and similar local times are likely to have correlated errors that are not presently taken into account.

## 12.4 The Internal Field as Seen by the Swarm Multi-satellite Mission

Having now set out the techniques used to construct advanced models of the internal field from multi-satellite data and gradient estimates, we now move on to give a brief summary the latest knowledge from such models.

For many interested in the Earth’s magnetic field, either at present its present state or its changes over the past century, the IGRF is a well known and reliable source of information. It is an IAGA/IUGG endorsed model, produced by a international group of scientists every five years from candidate models. It describes the main field up to spherical harmonic degree \(n=13\) and the linear rate of change of the field for the upcoming five years up to degree \(n=8\). The most recent 12th-generation update of IGRF (Thébault et al. 2015b) used data from CHAMP, *Swarm* and ground observatories to provide estimates of the field in 2010, 2015 and a predicted field change for 2015–2020. The advantage of IGRF is that it is an internationally agreed reference. However, it fails to describe the small scale lithospheric field, and it does not catch nonlinear secular variation, including geomagnetic jerk events.

For applications in detailed studies of ionospheric current systems, advantages have been documented in reducing data using more advanced field models that include the small-scale lithospheric field, estimates of the large magnetospheric field, and that follow fast changes in the core field (Stolle et al. 2016; Alken 2016). Such advanced field models include the POMME model developed by Maus and co-workers (Maus et al. 2005, 2006b, 2010), the GRIMM model produced by Lesur and co-workers (Lesur et al. 2008, 2010, 2015a) and the CHAOS model produced by Olsen and colleagues (Olsen et al. 2006, 2009, 2010, 2014; Finlay et al. 2015, 2016). The Comprehensive model series, developed by Sabaka and co-workers (Sabaka et al. 2002, 2004, 2013, 2015), takes an alternative approach and seeks to co-estimate not only the internal field but as far as possible all near-Earth field sources, including ionospheric and oceanic tidal sources. For further details on these models and their differences, the interested reader should consult the above references.

Here, we present the current state of knowledge of the core field, as determined from the latest *Swarm* data in the CHAOS-6 field model (Finlay et al. 2016), and a recent image of the global lithospheric field, from the LCS-1 field model (Olsen et al. 2017), based on the latest data from CHAMP and *Swarm*.

### 12.4.1 The Core Field

*Swarm*, as well as ground observatory vector field data (Finlay et al. 2016). Movies showing the time changes of such maps are available at www.spacecenter.dk/files/magnetic-models/CHAOS-6.

The radial field at the core-mantle boundary is characterized by high latitude flux concentrations over Canada and Siberia, and similarly in the Southern hemisphere under the edges of Antarctica towards South America and Australia. It is these features that give rise to the first-order dipolar structure of the geomagnetic field. Other striking features include a train of flux concentrations at low latitude under the Western hemisphere that have been observed to move westwards since the advent of continuous satellite observations in 1999, and the large concentration of reversed flux in the Southern hemisphere.

Turning to the time derivative of the field, known as the secular variation (SV), we find that regions of intense radial SV at the core surface occur close to edges of patches of strong radial field. Intense SV is observed in 2015 to lie in a broad band equatorward of \(30^\circ \) latitude between longitudes \(100^\circ \)E and \(90^\circ \)W and is particularly associated with the westward movement of the intense low latitude flux patches. There is also a well-localized negative–positive–negative series of three patches of radial SV visible under Alaska and Siberia; this appears to be a consequence of the rapid westward movement of intense high latitude radial field concentrations. The SV is also generally high in the Asian longitudinal sector \(60^\circ \)–\(120^\circ \)E.

The second time derivative of the radial field, or secular acceleration (SA) at the core-mantle boundary in 2015, displays a prominent positive–negative pair of foci under India–South East Asia, and a series of strong radial SA patches of alternating sign in the region under northern South America, as well as a positive–negative pair at high northern latitudes under Alaska–Siberia. In both the radial SV and SA, there is a striking absence of structure in the southern polar region (Holme et al. 2011; Olsen et al. 2014). Although the Pacific region shows weak radial SV (again see Holme et al. 2011; Olsen et al. 2014), in 2015, there was strong radial SA in the central Pacific, consistent with the aftermath of the jerk observed in 2014 at Hawaii. The SA also changes dramatically on sub-decadal time scales (Chulliat and Maus 2014; Chulliat et al. 2015; Finlay et al. 2015), exhibiting a series of pulses in amplitude. CHAOS-6 shows pulses of SA around 2006, 2009.5 and 2013. Maps and movies of the radial SA at the core surface also show recurring oscillations at particular locations, for example, under northern South America around \(40^\circ \)W close to the equator, and high amplitude SA is often found around longitude \(100^\circ \)E.

Figure 12.8 presents the radial field, its SV and SA, from CHAOS-6, at a typical low-Earth-orbit satellite altitude of 400 km. This is the internal field and its time changes that needs to be accurately accounted for when carrying out data reduction for ionospheric studies. The radial field at 400 km shows clear departures from a tilted dipole, with the high latitudes flux concentrations familiar from the core-mantle boundary again being evident. The radial field in the South Atlantic is distinctly weaker and there is a noticeable kink in the magnetic equator near South America. The radial field at satellite altitude is presently changing most rapidly at low latitudes in the American sector, while there are notable field accelerations taking place close to Indonesia and India–Pakistan, as well as in the mid-Pacific.

### 12.4.2 The Lithospheric Field

A recent map of the vertical field anomaly at the Earth’s surface, due to the magnetized lithosphere is shown in Fig. 12.9 (top). This is derived from the LCS-1 model (Olsen et al. 2017) determined from CHAMP and *Swarm* field differences data and synthesized for spherical harmonic degrees \(n=16-185\). The map shows the detailed structure of lithospheric field features throughout the world including the cratonic regions of the continents (especially Archean and Proterozoic domains) that show stronger anomalies, and the long wavelength features associated with and sub-parallel to the oceanic magnetic reversal stripes that are seen consistently on or near widely separated isochrones (green lines). In LCS-1, we see for the first time from the satellite data alone EW oceanic features associated with the reversal stripes formed during the last 50 Ma of separation history of Australia from Antarctica. A number of other features on the ocean crust are evident. For example, there are NS trending lows in the vertical component map associated with the NS trending \(85^\circ \)E ridge in the Bay of Bengal.

*Swarm*satellite-data-based lithospheric field model MF7 (Maus 2010), up to its truncation degree of 133. If one wishes to study smaller scales of the lithospheric field, satellite data must be combined with near surface Aeromagnetic or marine survey data, for example as collected in the World Digital Magnetic Anomaly Map (WDMAM, currently in its second addition Lesur et al. 2016). Note, however, that the amplitude of small scale lithospheric field signals at satellite altitude is tiny. The bottom part of Fig. 12.9 shows the vertical component of the lithospheric field at satellite altitude (400 km); the scale is then ten times smaller, and the lithospheric signal is on the order of a few to tens of nT, it is for this reason it must be accounted for when studying the magnetic signals due to ionospheric currents, especially when considering weaker current systems.

## 12.5 Limitations of Present Main Field Models

Although the present generation of main field models are rather impressive and extremely useful for studying ionospheric current systems, it is nonetheless important that users are aware of their limitations, and that space physicists realize that there is a clear opportunity for them to contribute in improving future main field models.

The major factor limiting the accuracy of the internal part of field models is the inability to correctly account for and remove all magnetospheric and ionospheric signals (Finlay et al. 2017). This includes the difficulty in modelling rapid changes in the magnetospheric field. Global coverage (requiring many days with the present satellite missions) are formally required to perform a separation into internal and external field components. Yet the magnetospheric field changes much faster on time scales of minutes to hours; present models try to account for this using activity indices based on ground-based observatory data but there are differences between ground and satellite data that remain poorly understood.

Another source of uncertainty is the internal signal due to currents induced in the electrically conducting mantle by the time-changing external field, which remains difficult to isolate. Present separations often rely on a priori models of the conductivity of the mantle and lithosphere, which although improving (e.g. Kuvshinov 2012) are subject to uncertainties.

## 12.6 Concluding Remarks

Modern models of the main geomagnetic field are derived from multi-satellite data, and increasingly make use of along-track and inter-satellite field differences (i.e. approximate gradients) in order to reduce the signatures of large-scale magnetospheric sources and to enhance the signal of small-scale internal fields. Examples from the construction of two such models, CHAOS-6 and LCS-1 have been presented here in detail. Such advanced main field models, including contributions from the small-scale lithospheric field and accounting for the near-Earth signature of magnetospheric sources, now enable more detailed study of relatively weak ionospheric current systems (Stolle et al. 2016).

Field gradient estimates can easily be incorporated within the conventional modelling framework as field differences, while also differencing the rows of the corresponding kernel matrices associated with each data point. It is very important to correctly specify the data uncertainty budget for vector, scalar and field gradient data, treating each satellite separately, in order for these to contribute appropriately during the field estimation procedure. Use of such procedures has enabled new details of the core field (in particular its time-dependent accelerations) and the lithospheric field (anomalies on scales down to 250 km) to be imaged using data from the *Swarm* and CHAMP missions.

Internal field models are nevertheless certainly imperfect, in particular in the polar region. New ideas on how best to parameterize quiet-time field variations in the polar region are much needed; this represents a clear opportunity for the space physics community to apply their expertise in another domain.

## Notes

### Acknowledgements

The author is grateful to Nils Olsen and Stavros Kotsiaros for collaboration on the work presented here, and for their help in preparing some of the material presented. Thanks also to Patrick Alken and Clemens Kloss for helpful comments on an early draft. The European Space Agency (ESA) is acknowledged for providing the Swarm data and for financially supporting the work on developing the Swarm Level 1 product ‘Magnet’. The CHAMP mission was sponsored by the Space Agency of the German Aerospace Center (DLR) through funds of the Federal Ministry of Economics and Technology. The author thanks the International Space Science Institute in Bern, Switzerland, for supporting the ISSI Working Group: ‘Multi-Satellite Analysis Tools, Ionosphere’, from which this chapter resulted. The Editors thank Vincent Lesur for his assistance in evaluating this chapter.

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