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 first block of the DIFI algorithm consists in reading all level 1b Swarm magnetic field data available during the considered time interval, selecting data at magnetically quiet times, and correcting these data for non-ionospheric fields. This block is refered to as “Data pre-processing” in Fig. 1. Level 1b magnetic data are selected using standard geomagnetic indices: Kp, Dst and the interplanetary magnetic field (IMF) By and Bz components. The minimum and maximum acceptable values for these parameters will be adjusted for each real dataset, in order to maximize spatial, local time and seasonal data coverage, while minimizing the overall magnitude of disturbance fields. Also, in order to minimize the size of the dataset to be inverted, level 1b data are decimated to one sample every 15 s prior to the inversion.
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 = − ΔV, where V is a magnetic potential.
Let us introduce the following set of basis functions:
where 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),
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 as
where 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
and (∈
′m
nsp
)*, respectively, and S
e
and S
i
are the vectors of the
and
respectively. Here Re{z} denotes the real part of complex number z, z* denotes its conjugate and AH denotes the conjugate transpose of vector or matrix A. At ground altitude, the
basis functions and e coefficients describe the primary ionospheric field, while the
basis functions and ι coefficients describe the secondary (i.e., induced) ionospheric field. At satellite altitude, all sources are internal and therefore the
basis functions describe the total field. The ι coefficients are the same at ground and at satellite altitude.
The induced field is related to the primary field through a transfer function, which may be expressed in matrix form as
For a 1D mantle conductivity, the 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
and
at r = a + h)
where C is a diagonal real matrix with the following elements
Therefore, (4) and (5) become
It is advantageous to model the ionospheric field in the quasi-dipole (QD) coordinate system (Richmond, 1995; Emmert et al, 2010), which follows the geometry of the Earth’s main magnetic field. This is mostly because the equatorial electrojet flows along the geomagnetic dip-equator, which is bended with respect to the dipole equator in the South American sector. Using QD coordinates helps minimizing the total number of parameters to be determined. The DIFI algorithm relies on the same QD basis function as Sabaka et al. (2000),
where (θ
q
, ϸ
q
) are the QD colatitude and longitude, h = 110 km is the altitude of the ionospheric electrical currents, and
are coefficients of the matrix relating spherical harmonics in dipole and quasi-dipole coordinates:
Here Nmax and Mmax 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
and
respectively, D is the matrix of
coefficients, 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
and
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 as
where
and
are vectors of complex coefficients
and ∊̃
′l
ksp
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 F10.7 (expressed in solar flux units, or SFU, where 1 SFU = 10−22 W m−2 Hz−1). Then
is replaced by
, 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
in (21)–(22) is truncated for 1 ≤ k ≤ Kmax,− min(k, Lmax ) ≤ l ≤ min(k, Lmax ), 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 ≤ Nmax, − min(n, Mmax ) ≤ m ≤ min(n, Mmax ), s
min
≤ s ≤ s
max
and p
min
≤ p ≤ p
max
. This leads to a total of N
q
= Lmax ( Lmax + 2) + (Kmax − Lmax )(2 Lmax + 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 Kmax = 45 and order Lmax = 5 (hence N
q
= 475 coefficients), with diurnal variations from p
min
= 0 to pmax = 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 Nmax = 60 and Mmax = 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
The “building and solving of normal equations” block of the DIFI algorithm (Fig. 1) uses a standard iterative least squares technique to find a model solution that minimizes the following objective function:
where is γ the data vector,
the data estimate vector calculated from the model, 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
In order to facilitate the distribution and use of the ionospheric field models produced by the 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 V1 of the primary ionospheric field is expressed as:
where
and
are real coefficients. Note that the
(resp. the
are obtained by taking the real part of the product ϵHS
e
in Eq. (4) (resp. the product ϵ′HS
i
in Eq. (5)).
The scalar potential V2 of the secondary (induced) ionospheric field is expressed as in Eq. (25), but with real coefficients
; these are obtained by taking the real part of the product ιHS
i
in Eqs. (4)–(5).
Equations (7)–(8) relate the coefficients below and above the ionosphere in Eqs. (24)–(25). Therefore, only coefficients
and
are provided in the MIO-SHA product files. The exact format of these files is given in Table 1 (and can also be found in the product specification document, Swarm Level 2 Processing System Consortium, 2013).