*Swarm* SCARF Dedicated Ionospheric Field Inversion chain

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## Abstract

The geomagnetic daily variation at mid-to-low latitudes, referred to as the geomagnetic Sq field, is generated by electrical currents within the conducting layers of the ionosphere on the dayside of the Earth. It is enhanced in a narrow equatorial band, due to the equatorial electrojet. The upcoming ESA Swarm satellite mission, to be launched end of 2013, will consist of three satellites in low-Earth orbit, providing a dense spatial and temporal coverage of the ionospheric Sq field. A Satellite Constellation Application and Research Facility (SCARF) has been set up by a consortium of research institutions, aiming at producing various level-2 data products during the *Swarm* mission. The Dedicated Ionospheric Field Inversion (DIFI) chain is a SCARF algorithm calculating global, spherical harmonic models of the Sq field at quiet times. It describes seasonal and solar cycle variations, separates primary and induced magnetic fields based upon advanced 3D-models of the mantle electrical conductivity, and relies on core, lithospheric and magnetospheric field models derived from other SCARF algorithms for removing non-ionospheric fields from the data. The DIFI chain was thoroughly tested on synthetic data during the SCARF preparation phase; it is now ready to be used for deriving models from real *Swarm* data.

### Key words

*Swarm*ionosphere inversion geomagnetism space magnetometry

## 1. Introduction

A small fraction of the Earth’s magnetic field is generated by electrical currents within the conductive layers of the ionosphere, near 110 km altitude. On geomagnetically quiet days, at mid-to-low latitudes, the ionospheric magnetic field has an amplitude of 10 to 50 nT on the ground and is referred to as the “Sq” magnetic field (see, e.g., Richmond and Thayer, 2000). It undergoes a characteristic daily variation, visible in geomagnetic observatory recordings, where the Z component increases (in absolute value) from sunrise to noon and decreases from noon to sunset. In a ±5? latitudinal band centered on the geomagnetic dip-equator, the amplitude of this daily variation is even larger and can reach up to 100 nT on the ground around noon. The ionospheric field in this band is caused by the equatorial electrojet, a thin current flowing eastward along the geomagnetic dip-equator.

Global spherical harmonic models of the Sq field have classically been determined from observatory data, relying on the global observatory network. Such models (e.g., Schmucker, 1999a, b; Takeda, 2002) provide a description of the large-scale current system generating the Sq field, as well as its seasonal and solar cycle variability. Use of magnetic measurements from low-Earth orbiting satellites such as ϕrsted and CHAMP makes it possible to model the Sq field up to higher spherical harmonic degrees, and to also model the equatorial electrojet field (Sabaka et al., 2002, 2004). However, satellite data are collected above the ionosphere. They cannot separate the primary ionospheric field from the secondary, induced field generated by electrical currents within Earth’s mantle. Such a separation is possible, however, if a pre-determined mantle conductivity model is available.

The upcoming ESA Swarm satellite mission, to be launched in 2013, will provide measurements of the Earth’s magnetic field with unprecedented precision from three identical satellites orbiting at different altitudes and local times (Friis-Christensen et al., 2006). Several research institutions have teamed up with ESA to form the Swarm Satellite Constellation Application and Research Facility (SCARF), a distributed processing facility that will produce geomagnetic field models during the Swarm mission (Olsen et al., 2013). Two of these models will be spherical harmonic descriptions of the ionospheric magnetic field at mid-to-low latitudes, i.e., below 55? dipole latitude. One of them will be calculated as part of a comprehensive inversion, where all major sources of the geomagnetic field will be calculated simultaneously (Sabaka et al., 2013); the other will be calculated through a dedicated inversion, after removing the magnetic fields generated by other sources from the data.

In the present paper, we introduce the Swarm Dedicated Ionospheric Field Inversion (DIFI) algorithm (Section 2), which will be used for calculating the dedicated ionospheric field from Swarm data. The DIFI chain takes into account the seasonal and solar cycle variability of the field. It was thoroughly tested during its development phase using synthetic data. Results of these tests are presented in Section 3. The DIFI chain is one of the three chains developed by IPGP within SCARF; other chains are the Dedicated Lithospheric Field Inversion (DLFI) chain (Thébault et al., 2013) and the Equatorial Electric Field (EEF) inversion chain (with NOAA, Alken et al., 2013).

## 2. DIFI Algorithm

### 2.1 Data pre-processing

During the exploitation phase, the DIFI algorithm will be used to calculate two types of models: ionospheric field models calculated over at least one year of data, which will include a description of the seasonal variation of model coefficients, and ionospheric field models calculated over less than one year of data (typically three to six months), with coefficients assumed constant with respect to season. The second type of model will be calculated only during the first year of the mission, to provide a first model within a few months after the commissioning phase. In what follows, we describe the algorithm to be used for the calculation of a full model, including seasonal variation, assuming one full year of level 1b magnetic data is available as input.

By default, the DIFI algorithm reads all level 1b vector magnetic data from Swarm A, one of the two satellites orbiting side-by-side at about 460 km altitude, and from Swarm C, the satellite orbiting at a higher altitude of about 530 km, in a different local time sector. Several failure cases have been investigated as part of the preparation phase within SCARF, for example the complete failure of one satellite or the lack of vector data from one of the satellites. If such an unhappy event were to occur, the DIFI algorithm would still be able to ingest reduced level 1b dataset. In what follows, we describe the nominal scenario where level 1b vector magnetic data are available from both Swarm A and C satellites.

The next step of the pre-processing block is data correction. It aims at removing non-ionospheric contributions from the total vector field recorded by each satellite. Three main sources are considered: the core, the lithosphere and the electrical currents in the magnetosphere. By default, SCARF dedicated (Hamilton, 2013; Rother et al., 2013; Thébault et al., 2013) and/or comprehensive (Sabaka et al., 2013) models are used to remove contributions from these sources. If needed, non-SCARF, auxiliary models will be considered during the exploitation phase to improve the quality of the final model.

Observatory data (hourly mean values, see Macmillan and Olsen, 2013) can also be used by the DIFI algorithm, although not in the default mode. These data are pre-processed in a similar way: first, they are selected according to geomagnetic Kp and Dst indices, as well as IMF By and Bz components; second, they are corrected for the core, lithospheric and magnetospherie fields using the same dedicated or comprehensive SCARF models. Observatory biases, i.e., the small-scale lithospheric field not described by these models, are left as variables to be determined by the inversion.

### 2.2 Model parameterization

The DIFI algorithm relies on the same ionospheric field model parameterization as the comprehensive models (see, e.g., Sabaka et al., 2000, 2002, 2013). In what follows, we briefly summarize the main features of this parameterization, using similar matrix notations as Sabaka et al. (2000).

Primary sources of the ionospheric magnetic field at mid-to-low latitudes are electrical currents flowing in the E-region of the ionosphere, at about h = 110 km altitude. The time-varying magnetic field generated by these currents induces secondary currents in the upper layers of the Earth’s electrically conducting mantle, which in turn contribute to the total ionospheric magnetic field. The spherical harmonic modeling of the ionospheric magnetic field relies on the assumption that *Swarm* satellites fly above these ionospheric sources. As a consequence, the ionospheric magnetic field * B* (i.e., observations minus contributions from the core, lithosphere and magnetosphere) may be expressed as

*= −*

**B****Δ**

*V*, where

*V*is a magnetic potential.

*r*is the radius, θ

_{ d }the dipole colatitude, ϸ

_{ d }the dipole longitude,

*t*(expressed in yrs) the season counted from January 1st, at 00:00 universal time,

*t*

_{ m }(expressed in hrs) the magnetic universal time, a the mean Earth radius (

*a*= 6371.2 km), Open image in new window the Schmidt normalized associated Legendre function of degree

*n*and order

*m*,

*ω*

_{ s }= 2π rad/yr the fundamental angular frequency for seasonal variation,

*ω*

_{ p }= 2À/24 rad/hr the fundamental angular frequency for diurnal variation,

*s*and

*p*the associated wavenumbers. The magnetic universal time is defined aswhere ϸ

_{ d,s }is the dipole longitude of the sub-solar point (defined as the point on the Earth’s surface closest to the sun), expressed in degrees. Then the potential

*V*may be uniquely expressed in the dipole reference frame as:where ϵ, ι and ϵ′ are vectors of complex coefficients Open image in new window and (

*∈*

_{ nsp }

^{′m})*, respectively, and

*S*

_{ e }and

*S*

_{ i }are the vectors of the Open image in new window and Open image in new window respectively. Here

*Re*{

*z*} denotes the real part of complex number

*z*,

*z** denotes its conjugate and

*A*

^{ H }denotes the conjugate transpose of vector or matrix

*A*. At ground altitude, the Open image in new window basis functions and e coefficients describe the primary ionospheric field, while the Open image in new window basis functions and ι coefficients describe the secondary (i.e., induced) ionospheric field. At satellite altitude, all sources are internal and therefore the Open image in new window basis functions describe the total field. The ι coefficients are the same at ground and at satellite altitude.

*Q*matrix is diagonal, while for a 3D conductivity it is dense for one single frequency, and block-diagonal for multiple frequencies. The DIFI algorithm can deal with both the 1D and 3D cases. Noting that the radial component of the ionospheric field is continuous through the current sheet at

*r*=

*a + h*, we necessarily have (by taking the radial derivative of Open image in new window and Open image in new window at

*r*=

*a*+

*h*)where

*C*is a diagonal real matrix with the following elementsTherefore, (4) and (5) become

_{ q }, ϸ

_{ q }) are the QD colatitude and longitude,

*h*= 110 km is the altitude of the ionospheric electrical currents, and Open image in new window are coefficients of the matrix relating spherical harmonics in dipole and quasi-dipole coordinates:Here

*N*

_{max}and

*M*

_{max}are chosen so that the convergence of the above summation is sufficient (see numerical values in the text below and Fig. 2). Equations (11) and (12) may be expressed in matrix form:where

**T**_{ e }and

**T**_{ i }are the vectors of the Open image in new window and Open image in new window respectively,

*is the matrix of Open image in new window coefficients,*

**D**

**U**_{ e }is the diagonal matrix of (

*a*/(

*a*+

*h*))

^{n−1}values, and

**U**_{ i }is the diagonal matrix of ((

*a*+

*h*)/

*a*)

^{n+1}values. It is worth noting that the spherical harmonics in QD coordinates are not orthogonal. By construction, the Open image in new window and Open image in new window functions have an exact QD geometry only on the sphere

*r*=

*a*+

*h*. By analogy with Eqs. (4)–(5), the DIFI algorithm expresses the ionospheric magnetic potential in quasi-dipole coordinates aswhere Open image in new window and Open image in new window are vectors of complex coefficients Open image in new window and

*∊̃*

_{ ksp }

^{′l}respectively. This leads to the following constraints on the ϵ, ι and ϵ′ vector of coefficients:Combining (18)–(20) with (9)–(10) then leads to

It is also assumed that the ionospheric magnetic field responds linearly to solar activity, parameterized by the solar radio flux index *F*_{10.7} (expressed in solar flux units, or SFU, where 1 SFU = 10^{−22} W m^{−2} Hz^{−1}). Then Open image in new window is replaced by Open image in new window , where the so-called Wolf ratio *N* = 14.85 × 10^{−3} SFU^{−1} was determined by Olsen et al. (1993). It is worth noting that the Wolf ratio actually varies with season (see, e.g., Penquerc’h and Chulliat, 2009). This effect will be investigated during the *Swarm* mission and could lead us to develop a more sophisticated parameterization of the response to solar activity in later DIFI models.

The quasi-dipole vector of coefficients Open image in new window in (21)–(22) is truncated for 1 ≤ *k* ≤ *K*_{max,}− min(*k*, *L*_{max} ) ≤ *l* ≤ min(*k*, *L*_{max} ), *s*_{ min } ≤ s ≤ *s*_{ max } and *p*_{ min } ≤ *p* ≤ *p*_{ max }. The dipole vector of coefficients ϵ in (9)–(10) (as well as (18)) is truncated for 1 ≤ *n* ≤ *N*_{max}, − min(*n*, *M*_{max} ) ≤ *m* ≤ min(*n*, *M*_{max} ), *s*_{ min } ≤ *s* ≤ *s*_{ max } and *p*_{ min } ≤ p ≤ *p*_{ max }. This leads to a total of *N*_{ q } = *L*_{max} ( *L*_{max} + 2) + (*K*_{max} − *L*_{max} )(2 *L*_{max} + 1) quasi-dipole coefficients and dipole coefficients for each pair of wavenumbers (*p*, *s*). In the tests reported in Section 3, the ionospheric field was modelled in quasi-dipole coordinates up to degree *K*_{max} = 45 and order *L*_{max} = 5 (hence *N*_{ q } = 475 coefficients), with diurnal variations from *p*_{ min } = 0 to *p*_{max} = 4 (i.e., down to a period of 6 hours) and seasonal variations from *s*_{ min } = −2 to *s*_{ max } = 2 (i.e., constant, annual and semi-annual variation). These parameters are the ones set in the original SCARF specifications (Swarm Level 2 Processing System Consortium, 2013). Note that *p* is arbitrarily taken positive, so that modes propagate westward for *l* > 0 (or *m* > 0) and eastward for *l* ≤ 0 (or *m* ≤ 0). Unlike Sabaka et al. (2000, 2002), we do not select only modes closest to the local time modes *l* = *p*. A numerical investigation of Eq. (13) shows that *N*_{max} = 60 and *M*_{max} = 12 (hence *N*_{ d } = 1368 coefficients) are sufficient to achieve convergence of the * D* matrix. The modulus of the obtained matrix is shown in Fig. 2.

### 2.3 Inversion

*C*

_{ e }the data covariance matrix, λ a damping parameter and

*C*

_{ m }a damping matrix. For the purpose of the tests based upon synthetic data, we used an identity matrix for the data covariance matrix, and set the damping parameter to zero. These choices will be revised during the exploitation phase, when real data will be considered. We anticipate that the actual data covariance matrix will be diagonal, assuming that the data noise is caused by stationary and uncorrelated processes. For each data type (satellite and observatory) and each component, variances will be iteratively determined, starting from an existing Sq field model. Regarding the damping matrix, a simple, diagonal matrix will first be used, to minimize those coefficients that will be found less constrained by the data. If needed, more sophisticated damping strategies will be used, such as minimizing night-time ionospheric currents (Sabaka et al., 2002).

### 2.4 Output

*Swarm*SCARF, it was decided to distribute them as sets of real Gauss coefficients in dipole coordinates. The same format is to be used for both the comprehensive and dedicated chains. Specifically, the scalar potential

*V*

_{1}of the primary ionospheric field is expressed as:where Open image in new window and Open image in new window are real coefficients. Note that the Open image in new window (resp. the Open image in new window are obtained by taking the real part of the product ϵ

^{ H }

*S*

_{ e }in Eq. (4) (resp. the product ϵ′

^{ H }

*S*

_{ i }in Eq. (5)).

The scalar potential *V*_{2} of the secondary (induced) ionospheric field is expressed as in Eq. (25), but with real coefficients Open image in new window ; these are obtained by taking the real part of the product ι^{ H }*S*_{ i } in Eqs. (4)–(5).

*Swarm*Level 2 Processing System Consortium, 2013).

## 3. Test Results

During the preparation phase, the DIFI algorithm was tested using synthetic data. The way the synthetic orbits and magnetic data were generated is described in Olsen et al. (2013). We used only one year of synthetic data, from January 1, 2000 to December 31, 2000, and relied on real *K*_{p},*D*_{st}, IMF and *F*_{10.7} values for the selected period.

The synthetic data were selected using the following criteria: *K*_{ p } ≤ 2_{ o }, −20 nT ≤ *D*_{st} ≤ 20 nT, −8 nT ≤+ IMF *B*_{ y } ≤ 8 nT and −2 nT ≤ IMF *B*_{ z } ≤ 6 nT. Starting from synthetic Level 1b 1 Hz data, the selection process lead to 583705 data triples for each satellite A and C. The data were further decimated so that the time difference between two successive data never fell below 15 s, which corresponds to a minimum distance of about 105 km (since *Swarm* satellites orbit at about 7 km/s).

Removal of non-ionospheric contributions were implemented in different ways, depending on the tests carried out. During the so-called “AR1 test”, data were corrected using the same reference field models as the ones used for generating the synthetic data (see Olsen et al., 2013, for details about these models). As the random noise added to the synthetic data had a very small standard deviation (between 0.1 and 0.7 nT, depending on the component) compared to the typical amplitude of ionospheric fields, the data after correction were very close to the synthetic ionospheric field data themselves. Thus, the AR1 test essentially amounted to a closed-loop simulation of both the pre-processing and inversion blocks. During the second, so-called “AR2 test”, data were corrected using core, lithospheric and magneto-spheric field models derived as part of the test by SCARF partners, thus imperfectly reproducing the contribution of each source to the synthetic data, as would be the case with real data.

For both tests, we used the same *Q* matrix as the one used for generating the synthetic dataset (see Olsen et al., 2013, for details). This matrix was derived from a 3D electrical conductivity model of the mantle, including oceans, designed by Kuvshinov et al. (2006). We used the * D* matrix already mentioned in Section 2.2 and presented in Fig. 2.

### 3.1 AR1 test

Residual statistics (in nT) for AR1 and AR2 tests.

Component | AR1 test | AR2 test | |
---|---|---|---|

All data | |||

| 0.00 ± 0.15 | 0.01 ± 1.77 | |

| 0.00 ± 0.11 | 0.00 ± 2.88 | |

| −0.01 ± 0.11 | 0.00 ± 2.37 | |

ǀ90 − ¸ | |||

| 0.00 ± 0.15 | 0.01 ± 1.78 | |

| 0.00 ± 0.10 | −0.01 ± 2.75 | |

| −0.01 ± 0.10 | 0.00 ± 2.04 |

_{ d }À ≤ 55°). To produce this figure, differences were calculated on a regular grid, for 8 three-hour universal time bins and 40 seasonal bins. (Although one seasonal bin per day should be used to account for differences caused by day-to-day magneto-spheric field fluctuations, this number is not critical when using synthetic data and will be adjusted during the operational phase.) For each component, the standard deviation of these differences divided by the maximum field value (from all bins) is taken as a measure of the model performance. Numerical results are provided in Table 3. It can be checked that the performances are below 2%, which is well below the threshold requirement of 10% (see table 5 in Olsen et al., 2013). These numbers can be seen as the maximum performance achievable by the current version of the DIFI chain.

Model performances (as defined in the main text) for AR1 and AR2 tests.

Component | AR1 test | AR2 test | |
---|---|---|---|

| 0.75% | 8.93% | |

| 1.56% | 10.4% | |

| 1.90% | 14.6% | |

| 0.65% | 4.67% |

### 3.2 AR2 test

*B*

_{ r }distribution, around 4 nT, as well as some bias of a few nT in

*B*

_{¸}and the scalar field. The bumps are also present when plotting the output of the pre-processing block; i.e., they are not caused by the inversion. We found that they originate in the imperfect correction for the core field.

## 4. Concluding Remarks

In the present paper, we described the algorithm of the DIFI chain to be used during the upcoming *Swarm* mission to calculate global, spherical harmonic models of the quiet-time ionospheric field at mid-to-low latitudes. This algorithm relies on quasi-dipole coordinates to minimize the number of parameters, while describing the smallest spatial features of the average field such as the equatorial electro-jet. It can take advantage of a 3D conductivity model of the mantle and oceans when separating the primary and induced fields.

Closed-loop tests using synthetic data showed that the DIFI algorithm is able to reproduce the original primary and induced ionospheric fields to a very high accuracy, at ground and satellite altitudes. Tests relying on other SCARF models produced from the same dataset for data correction of the core, lithospheric and magnetospheric fields revealed that the pre-processing block of the algorithm is very sensitive to the quality of the correcting models. This problem is specific to the DIFI chain, as this chain is the last in the sequence of dedicated inversion chains to be processed. As a result, all errors made by previous chains in the sequence cascade down to the DIFI input. However, the inversion block proved remarkably stable with respect to these input errors; it is possible to calculate a meaningful ionospheric field, with performances within the threshold requirement, even when starting from imperfectly corrected data. During the exploitation phase, no reference field will be available to validate the DIFI chain output. The validation will be made by comparing the total field at ground with measurements from geomagnetic observatories.

## Notes

### Acknowledgments

The research reported here was financially supported by the Centre National d’Etudes Spatiales (CNES) through the “Travaux préparatoires et exploitation de la mission Swarm” project, and by the European Space Agency (ESA) through ESTEC contract 4000102140/10/NL/JA “Development of the *Swarm* Level 2 Algorithms and Associated Level 2 Processing Facility”. The authors thank the reviewers for their very helpful comments. This is IPGP contribution number 3423.

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