Analysis of magnetic satellite data to infer the mantle electrical conductivity of telluric planets in the solar system

Planetary and Space Science 84 (2013) 102–111

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Analysis of magnetic satellite data to infer the mantle electrical conductivity of telluric planets in the solar system F. Civet n, P. Tarits University of Brest, European Institute for Marine Studies, CNRS, Place Nicolas Copernic, 29280 Plouzané, France

art ic l e i nf o

a b s t r a c t

Article history: Received 6 June 2012 Received in revised form 12 April 2013 Accepted 7 May 2013 Available online 23 May 2013

Space missions launched to study the solar system planets generally involve a magnetometer in the scientific payload. The magnetic data may be used to infer the electrical conductivity of a planet's mantle using electromagnetic induction theory. The application of induction analysis on terrestrial bodies of the solar system other than the earth is challenging because of little information available about the external inducing sources. Here, we present a method to analyze magnetic data from these space missions that determines the geometry of the dominant external inducing magnetic field and deals with the inherent gaps in the satellite magnetic time series. We tested the approach on Earth synthetic satellite data generated to prepare the ESA magnetic mission Swarm and demonstrated the feasibility for recovering the 1-D conductivity part of the model used to generate these data. The analysis of real data from the Danish Ørsted magnetic mission provided satisfactory conductivity profiles of the Earth's mantle. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Electromagnetic induction Satellite data Magnetic field Transients fields Planetary internal structure Electrical conductivity

1. Introduction Electromagnetic induction analysis of magnetic data recorded by satellites over a planet may be used to characterize the electrical conductivity of the planet's mantle (see Olsen, 1999a; Saur et al., 2009; Kuvshinov, 2012, and references therein). The electrical conductivity is closely related to physical parameters, such as mineralogical composition, pressure, temperature or the water content (e.g. Tyburczy and Schubert, 2007), and can help us to understand the planet's thermodynamic processes. For the Earth, magnetic satellite data can be combined with the magnetotelluric and geomagnetic data collected on the ground, to provide images of the electrical conductivity distribution of the geological structures down to mantle depth. For other solar system bodies, data are scarce and, apart from the Moon, only obtainable from spacecraft (Sonett et al., 1972; Dyal et al., 1976; Wiskerchen and Sonett, 1977; Hood et al., 1982; Hood, 1986). In electromagnetic (EM) induction studies, data consist of continuous time series of the three components of the timevarying magnetic field. The time-varying field is the superimposition of an external inducing field and an internal field arising from the induction process in the electrically conductive planet. The induced field is a function of the planet conductivity down to a depth, controlled by the skin effect. The penetration depth (δ) is a

n

Corresponding author. Tel.: +33 674570278. E-mail address: [email protected] (F. Civet).

0032-0633/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.pss.2013.05.004

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi function of the frequency f considered, δ ¼ 1=πf μ0 s where μ0 is the magnetic permeability of the vacuum and s the conductivity of the medium. Data at low frequencies (less than a cycle per day) are necessary to infer the deep conductivity distribution in a planet that is deeper than several 100 km, hence the need for long time series. The length of the time series needed is difficult to determine precisely. A simple estimate assuming a 0.2 S m−1 mantle conductivity (e.g., Kelbert et al., 2009) leads to time series of a hundred days or more. Satellites measuring the Earth's magnetic field provide an excellent spatial coverage. Nevertheless, the induction study of magnetic satellite data is complicated because the satellites sample the time-varying field both in space and time. Furthermore, the series have gaps because of various technical issues during orbiting adjustments, periods of incompatible orbital configurations, software updates or blindness of the stellar camera. These gaps limit the ability of the data analysis to cover the full spectrum of the time-varying field necessary for EM induction study (e.g., Carbonell et al., 1992). On Earth, the source field is dominated by magnetospheric activities which are fairly well known. The magnetospheric time-varying magnetic field is reasonably well approximated by a uniform external source field for periods of more than a day (e.g., Olsen, 1999a). Authors take advantage of this rather simple geometry of the main part of the external magnetic field to infer the onedimensional (1-D) conductivity distribution globally from MAGSAT, SAC-C, Ørsted or CHAMP magnetic satellites (e.g., Olsen, 1999a; Constable and Constable, 2004; Kuvshinov and Olsen,

F. Civet, P. Tarits / Planetary and Space Science 84 (2013) 102–111

2006; Velimsky, 2010). One approach consists of analyzing the magnetic field of every orbit to obtain a continuous time series (e. g., Olsen, 1999a; Constable and Constable, 2004). New sophisticated techniques are also developed to generalize the simple uniform source model using a complete three-dimensional parametrization of the source and conductivity distribution (Kuvshinov et al., 2009, 2010). Velimsky (2010) analyzed Earth magnetic satellite data in the time domain and used a geomagnetic index to fill the gaps in the time series. The methodology developed for the Earth does not apply easily to other terrestrial bodies because the external field is not sufficiently known and the data is scarce. However, the ionized environment of planets such as Mars, Venus or Mercury (e.g., Schubert and Soderlund, 2011, and references therein) are not known well enough to qualify a simple uniform source model, and this must be taken into account in the data analysis. A few studies have been carried out on telluric bodies other than Earth. Lunar missions provided magnetic data analyzed to study the electrical conductivity of the Moon's mantle (e.g., Schubert and Schwartz, 1971; Schubert et al., 1974; Dyal et al., 1974; Vanian and Egorov, 1975; Sonett, 1982), using the magnetometers from the Apollo (on the ground) and Explorer (in orbit) missions. Time series were compared for common locations and for appropriate orbital configurations to study the existence of conductive layers in which an induced electromagnetic field is generated in response to the inducing interplanetary magnetic field (IMF). On Jupiter's Gallilean satellites (Io, Europa, Callisto and Ganymede) induction studies were carried out (Khurana et al., 1998) using the Galileo spacecraft magnetometer to look for the presence of a global oceanic layer on Europa and Callisto, and of a global magma ocean on Io (Khurana et al., 2011). For these studies, magnetic time series are combined with magneto-hydro-dynamics simulations to infer the perturbation in the Jovian external field and identify the presence of conductive layers in the deep interior of the satellites. Finally, the most extensive magnetic dataset outside the Earth was obtained during the Mars Global Surveyor mission (Acuna et al., 2001). The data are comparable to Earth satellite data in length (several years) and quality, but with a complex time-varying external field (Dubinin et al., 2008; Kallio et al., 2008; Akalin et al., 2010). In this paper we studied a solution of the EM induction data analysis assuming a dominant source field without specifying its geometry. We introduced a general formulation for a proxy of the time variation of the magnetic field recorded by a satellite orbiting a planet. The analysis was carried out in both time and space spectral domains and the proxy provided the correct time spectra of the source field. The approach was tested and validated on synthetic satellite magnetic data generated by Kuvshinov et al. (2006) and Olsen et al. (2006), and on real Earth data recorded by the Danish satellite Ørsted (Neubert et al., 2001).

2. Method In the quasi-static approximation (e.g., Coulomb and Jobert, 1990), and assuming an insulating medium at the satellite altitude, the magnetic field may be written as the gradient of a scalar potential. Because the field is divergence-less, the potential verifies the Laplace equation. The solution contains both the external field potential and the internal field potential. Expanding the potentials in spherical harmonics (SH) and taking the gradient, we obtain the magnetic field BN ðt; r; θ; ϕÞ expressed in the time domain, as the sum of external and internal coefficients

103

in SH: 2

3

6 7 6  r l−1 alþ2 7 ∞ l 6 7 Nm N N m N m 7Y ðθ; ϕÞ B ðt; r; θ; ϕÞ ¼ ∑ ∑ 6 þ ιl βl ðtÞ 6εl αl ðtÞ a 7 l r 7 l ¼ 1 m ¼ −l6 4|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} 5 External

Internal

ð1Þ m In Eq. (1), the values αm l ðtÞ and βl ðtÞ are the SH coefficients of the magnetic potentials of the external and internal fields, respectively. The internal potential βm l is convolved to the external source term and is a function of the electrical conductivity in the planet. The parameter a is the radius of the planet and r is the orbital radius of the satellite. The angles θ and ϕ are the colatitude and longitude, respectively. The functions Y Nm l ðθ; ϕÞ are the generalized spherical harmonics with the generalized associated Legendre polynomials: P Nm l ð cos ðθÞÞ, and N (+1, 0, −1 defines the 3 vectorial magnetic components, e.g., Phinney and Burridge, 1973).

0 ð−Bθ þiBϕ Þ 1 pffiffi 2 C B C B B C BN ¼ @ B 0 A ¼ B r @ A ðB þiB Þ θ ϕ − ffiffi p B 0

Bþ

1

ð2Þ

2

The values εN and ιN correspond to the vector spherical l l harmonic components and are a function of degree l only 0

εþ ¼− l

qffiffiffiffiffiffiffiffiffiffi 10

lðlþ1Þ þ 2 CB ι l

¼−

qffiffiffiffiffiffiffiffiffiffi 1 lðlþ1Þ

2 C B B 0 CB 0 C B εl ¼ −l CB ι l ¼ l þ 1 C B B C C ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi q q @ A@ A lðlþ1Þ lðlþ1Þ − − εl ¼ − ιl ¼ − 2 2

ð3Þ

In order to solve Eq. (1) and determine the αm and βm l l coefficients, a series of steps were necessary. First, we binned the data over a regular spatial grid to obtain time series distributed regularly over the sphere. The time series of the magnetic field were considered mesh centred. Then, we needed to calculate the spectral coefficients of the field components BN in each cell of mesh. However, once binned, satellite time series are not regular. To account for the gaps in the series, we introduced a function characterizing the time variability of the external source field. Finally, a SH analysis of the geographic distribution of BN was carried out to obtain the potential coeffim cients αm l , βl . The field BN splits into N BN ðr; tÞ ¼ BN ε ðr; tÞ þ Bι ðr; tÞ

ð4Þ

N where BN ε and Bι correspond to the external and internal fields, respectively, and r is the position vector (r, θ, ϕ). Let S(t) be a function that describes the dominant source of temporal variation in the external field. We assume further that this function describes the time fluctuation of a current sheet of constant geometry generating the source field. Be CN ε the spatial structure of the external field. We write N BN ε ðr; tÞ ¼ Cε ðrÞ  SðtÞ

ð5Þ

This approximation hold for a source of exosphere fields with a constant geometry over time such as the Earth ring current. The internal induced field is written as N BN ι ðr; tÞ ¼ Cι ðr; tÞnSðtÞ

ð6Þ

104

F. Civet, P. Tarits / Planetary and Space Science 84 (2013) 102–111

The term CN ι ðr; tÞ is the induced response. Hence, the total field becomes N

N

B ðr; tÞ ¼ C ðr; tÞnSðtÞ

ð7Þ

N

where C ðr; tÞ includes both the inducing field and the induced response of the Earth. For a regularly sampled time series of n samples measured during T days, the magnetic field may be defined by its discrete inverse Fourier transform with the Fourier N coefficients b ðωÞ, where ω is the pulsation (ω ¼ 2πf , f is the frequency) from −Ny to Ny (Ny is the Nyquist frequency): BN ðr; tÞ ¼

Ny

∑

ω ¼ −Ny

N

b ðr; ωÞ  eiωt

ð8Þ

The Fourier transform of Eq. (7) is N

b ðr; ωÞ ¼ cN ðr; ωÞ  SðωÞ

ð9Þ

and Eq. (8) becomes BN ðr; tÞ ¼

Ny

∑

ω ¼ −Ny

cN ðr; ωÞ  SðωÞ  eiωt

Ny

∑

ð10Þ

ω ¼ −Ny

N ^ c^ j ðω; r j ; θj ; ϕj Þ  SðωÞ  eiωt

ð11Þ

N c^ j

is the new spectral coefficient of the magnetic field. The where following relationship is thus confirmed: N ^ SðωÞ  c^ j ðω; r j ; θj ; ϕj Þ ¼ SðωÞ  cN j ðω; r j ; θ j ; ϕj Þ

ð12Þ

We need to solve Eq. (11) for all frequencies. In order to reduce the number of unknowns, we applied a band averaging method (e.g., Smirnov and Varentsov, 2004). The response of conductive layers to an induction source is a smooth function of frequency and does not vary much over small intervals of frequencies. We defined k bands of frequencies from 0 to Ny with an increasing number of frequencies as a function of the band number for nk bands. The value Ωk is the average frequency in the band k. The Eq. (10) becomes nk

N N ^  eiωt B~ j ðr; tÞ ¼ ∑ c^ j ðΩk ; r j ; θj ; ϕj Þ ∑ SðωÞ k¼1

ω∈k

∞

BN ðt; rðtÞ; θðtÞ; ϕðtÞÞ ¼ ∑

l ¼ 1 m ¼ −l

  r l−1 alþ2 m N ^m ^ εN  α ðΩ Þ  þ ι  β ðΩ Þ  Y Nm k k l l l ðθ j ; ϕj Þ l l a r

ð15Þ

^m The new potentials α^ m l ðΩk Þ and β l ðΩk Þ are proportional to the band averaged estimates of αðωÞ and βðωÞ. Eq. (15) was solved by LS with application of the L-curve method for the regularization to obtain the SH potential coefficients. The final step in the analysis ^m was to invert the α^ m l ðΩk Þ and β l ðΩk Þ values at the nk frequencies to obtain the conductivity distribution in the planet. Here, we considered only 1-D solutions for a spherically symmetric distribution of the conductivity. The inversion was carried out as follows. We minimized the misfit function M m 2 2 M ¼ ∑ W Ωk  F 2 ½β^ l ðΩk Þ; Q l ðΩk ; sÞ  α^ m l ðΩk Þ=V þ L

ð16Þ

l

∑

Ny

∑

l ¼ 1 m ¼ −l ω ¼ −Ny

iωt Y Nm l ðθðtÞ; ϕðtÞÞe

βm l ðωÞ

where Q l ðΩk Þ is the model induced response for a unitary source potential. The sum goes over all degrees, orders and frequencies available (l; m; Ωk ). The theoretical response Ql is obtained for a given conductivity model s and is multiplied by the observed external potential α^ m l ðΩk Þ. The result is the calculated internal m m potential β~ l that will be compared to the observed β^ l . The 2 function F is defined as the square modulus of the difference between the complex logarithm of the β values in order to balance the moduli and phases: m m F 2 ¼ jclogðβ^ l ðωÞÞ−clogðβ~ l ðωÞÞj2

ð17Þ

with clogðaÞ ¼ logðjajÞ þ iArgðaÞ where a is a complex number. The coefficient W Ωk is a weight function taken here equal to the square of the frequency Ωk to homogenize the amplitudes of the time spectra. The uncertainties are difficult to estimate and are considered similar for all coefficients. Hence, we assigned the same error floor V ¼1 to all values. The value L2 is a regularization term defined to minimize the electrical conductivity variations with regard to depth L2 ¼ λ ∑ jlogðsi =sj Þj2

ð18Þ

i;ji≠j

where s is the conductivity of layer i or j, and λ a damping coefficient set, so that the regularization term is close to the misfit value. We used the 1-D spherical minimization algorithm by Tarits and Mandea (2010) to obtain an optimal 1-D conductivity model.

3. The Swarm synthetic dataset

    # rðtÞ l−1 a lþ2 N m N m εl  αl ðωÞ  þ ιl  βl ðωÞ : a rðtÞ

αm l ðωÞ

l

∑

ð13Þ

The least square (LS) solution of Eq. (13) provides the N c^ j ðΩk ; r j ; θj ; ϕj Þ. The L-curve method (e.g., Hansen, 1992; Velimsky, 2010) is used for the LS regularization. m Expanding the magnetic potentials αm l ðtÞ and βl ðtÞ (Eq. (1)) according to the definition of Eq. (8), we obtain

"

∞

N c^ j ðΩk ; r j ; θj ; ϕj Þ ¼ ∑

l;m;Ωk

In a mesh j, relation (10) is verified for each element of the N binned time series of the magnetic field B~ j ðt; r j ; θj ; ϕj Þ, where θj and ϕj are the coordinates of the mesh centre jth. This time series is not continuous, both because of possible gaps in the satellite data and because of the binning process. In order to be able to calculate the Fourier spectra of the field cN , we need to provide a continuous ^ proxy SðtÞ proportional to the function S(t) and its Fourier trans^ form SðωÞ. Using Eq. (10), the field in the mesh j is N B~ j ðt; r j ; θj ; ϕj Þ ¼

follows:

ð14Þ

where and are now the finite spectra coefficients of αðtÞ and βðtÞ at frequency ω. N Using Eqs. (10), (11) and (13), the c^ j ðΩk ; θj ; ϕj Þ values are related ^m to external and internal magnetic potentials α^ m l ðΩk Þ and β l ðΩk Þ as

Magnetic datasets available for planets other than Earth have been obtained by single spacecraft, generally with a variable sampling rate since the magnetometer experiments are not designed to perform EM induction studies. In order to test the approach described in the previous section, we found it useful to use synthetic data generated for the Earth. These data were produced by Kuvshinov et al. (2006) for a simulation of the Swarm constellation magnetic mission (Olsen et al., 2006). The authors calculated synthetic time series of hourly spherical harmonic external and internal coefficients of the Earth's magnetospheric field from 1998 to 2002, interpolated every 15 s along the orbits of three satellites. For the purpose of this study, we used data from only one satellite. The synthetic external SH coefficients were determined using the comprehensive model from Sabaka et al. (2002). The internal coefficients were computed from a 3-D conductivity model of the Earth. The 3-D conductivity model considered consists of an inhomogeneous conducting surface shell, three local conductors of

F. Civet, P. Tarits / Planetary and Space Science 84 (2013) 102–111

105

Fig. 1. Comparison of a binned 60-day synthetic satellite time series of the three magnetic components Br, Bθ and Bϕ (red) with the true synthetic times series (black). The data presented are located between 301–451 in colatitude and 01–151 in longitude. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

100 80

True time serie Satellite time series Calculated time series

Bθ

60 40 20 0 −20 5

10

15

20

25 30 35 Time (day)

40

45

50

55

Fig. 2. Comparison between true synthetic (blue) and recalculated (red) synthetic Swarm time series on a mid-latitude mesh j (latitude 45–601 and longitude 0–151 mesh) of the Bθ component. Black dots represent gapped binned satellite data of one of the Swarm simulated time series. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

0.04 S m−1 at depths from the bottom of that shell down to 400 km, and a deep-seated regional conductor of 1 S m−1 located between 400 km and 700 km depth. These conductors are embedded in three radially symmetric shells: from 0 to 400 km, 0.004 S m−1, from 400 to 700 km, 0.04 S m−1 and an inner uniform layer of 2 S m−1. We binned the synthetic data obtained for one of the three satellites that is in a low altitude orbit onto a grid of 15  15 degrees. The choice of grid was guided by a compromise between having enough meshes on the sphere and enough data points in one mesh to build a relevant magnetic time series. An example of a binned time series is presented in Fig. 1 for 60 days of satellite data. For this dataset, we used the Fourier transform of the Disturbance ^ storm time index (Dst) series as a proxy SðωÞ in Eq. (13). The Dst index, first introduced by Chapman (1956), characterizes the interaction of the solar wind with the Earth's magnetosphere. The index is a good analogue of the magnetospheric field time variation (e.g., Langel and Estes, 1985; Constable and Constable, 2004; Olsen and

Kuvshinov, 2004) even though it does not reproduce all the variability of the external field. With this proxy, we solved Eq. N (15) to calculate the c^ j ðΩ; θj ; ϕj Þ coefficients in all meshes. Fig. 2 shows the true synthetic and the recalculated time series (here for the East component Bθ ) in a mid latitude box (latitude 45  601 and longitude 0  151) as well as the gapped data used in the calculation. The estimated time series was recalculated using N the set of c^ j ðΩ; θj ; ϕj Þ coefficients combined with the Dst time spectrum according to Eq. (13). We observe a good fit between the true synthetic and recalculated time series. Nevertheless, we could not recover the maximum amplitudes observed in the true synthetic data because the external synthetic coefficients calculated by Kuvshinov and Olsen (2006) were not derived from a simple Dst index (Olsen et al., 2006) and the effect is enhanced over short periods. We analyzed 730 days of data (2000–2002) and carried out a N SH analysis on the spatial distribution of the c^ ðΩ; θ; ϕÞ spectra coefficients at each frequency Ω to obtain the magnetic potential

F. Civet, P. Tarits / Planetary and Space Science 84 (2013) 102–111

Period (days)

α synthetic

2.0

100

100

1.8

10

10

1.6

1

1

1.4

0.1

Period (days)

α calculated

1

2

3

0.1

1.2 1

2

3

−1 0 1 −2 −1 0 1 2 −3 −2 −1 0 1 2 3

−1 0 1 −2 −1 0 1 2 −3 −2 −1 0 1 2 3

β synthetic

β calculated

100

100

10

10

1

1

0.1

0.1

1.0

0.8

Square root of energy

106

0.6

0.4

0.2 1

2

3

−1 0 1 −2 −1 0 1 2 −3 −2 −1 0 1 2 3 Spherical Harmonic (l,m)

1

2

3

−1 0 1 −2 −1 0 1 2 −3 −2 −1 0 1 2 3 Spherical Harmonic (l,m)

0.0

Fig. 3. Spectrogram of the external (upper part) and internal (lower part) distribution of energy in the synthetic spherical harmonic coefficient spectra, calculated for 730 days (2000–2002). The value presented for a given period and a SH degree l is the squared root of the amplitude of the Fourier coefficient multiplied by l+1 for the internal coefficient β and multiplied by l for the external coefficient α. The left-hand part corresponds to the true synthetic spherical harmonic coefficients and is compared to the calculated spherical harmonic coefficients from the gapped time series (right-hand part).

^m coefficients α^ m l ðΩÞ and β l ðΩÞ (Eq. (15)). We confined the SH expansion to degrees l ¼1,3 and orders m ¼ −l; l. Fig. 3 shows the spectrogram of the SH for each l; m at all frequencies. As expected, we observe that the energy in the SH coefficients is dominant for the first degree (l ¼ 1; m ¼ −l; l) and at periods greater than a day. The true synthetic and calculated energy distribution are similar. Fig. 4 presents the SH coefficients with respect to periods. We observe a very good agreement between the true synthetic and calculated coefficients, both for external and internal potentials at periods greater than 1 day when the source field is energetic at the SH degree considered (Fig. 3). In contrast, the similarity is less at degrees and orders for which the source amplitude is weak. This observation reveals that the method is able to retrieve SH coefficients for the most energetic SH ðl; mÞ of the source field. Consequently, we only inverted the SH coefficients for the first degree and orders (m ¼ −1; 0; 1) to obtain the electrical conductivity distribution. For the purpose of testing the effect of the length of the time series, we carried out several calculations with different series. Here we present two end members, a series of 60 days and one of ^m 730 days. The α^ m l ðΩÞ and β l ðΩÞ coefficients were inverted for periods greater than 1 day. Some of the recovered conductivity models are presented in Fig. 5. Here, the starting model has 10 layers of 200 km thick. The model the closest to the true synthetic one was obtained for the 730-day time series. The method tested on synthetic data for a known conductivity model was successful in recovering the SH spectral magnetic potentials of realistic satellite-like time series. We showed that the results are satisfactory for the SH degrees and orders for which the source field has energy. The true conductivity model was correctly determined with these SH spectral coefficients.

4. Magnetic data from the Ørsted mission The application of the technique to real Earth satellite magnetic data was carried out on the Ørsted magnetic satellite dataset. This satellite was launched in early 1999 (Neubert et al., 2001). The satellite measured the Earth magnetic field at an orbit altitude from 539–766 km until 2006. We used 4 years of data measured by the on-board fluxgate magnetometer at a nominal sampling rate of 1.135 s, with a certain amount of missing data. Fig. 6 presents the time series of the daily percentage of available data. The geomagnetic field measured by the Ørsted satellite is the superimposition of fields of different origins: the main field from the core dynamics, the crustal field from rock magnetization, and the transient field from the ionized environment of the Earth. Induction in the Earth has its source in this transient field, whose origin is both magnetospheric and ionospheric. Before any processing, we removed the main field and the crustal field. We used the spherical harmonic coefficients of the 11th IGRF (International Geomagnetic Reference Field, Finlay et al., 2010). These coefficients take into account the secular variation. The crustal magnetic field was removed using the NGDC-720 spherical harmonic model (McLean, 2010). These models were both calculated at all positions of the satellite. However, for computational time reasons, the calculation of the crustal field was limited to SH degrees in the range 16–100. The resulting reduced field Bred corresponds to the sum of the magnetospheric and ionospheric magnetic field and their induced parts. The ionospheric magnetic field is coupled with the magnetospheric field by field-aligned currents (FAC) leading to the existence of the auroral electrojets (Iijima and Potemra, 1978; Erlandson et al., 1988). These currents generate

F. Civet, P. Tarits / Planetary and Space Science 84 (2013) 102–111

107

Fig. 4. Spherical harmonic potentials as function of the period (greater than 1 day) for a selection of degrees and orders for a 730-day window. The black line shows the true synthetic modulus and the phase coefficients (external α top and internal β bottom of each SH potential). The red line shows the coefficients calculated from the gapped time series. The blue line in the external α^ 01 coefficient corresponds to the Dst spectrum. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

an energetic high frequency and short wavelength magnetic signal in Bred . In many studies (e.g., Constable and Constable, 2004; Velimsky, 2010), authors isolated the magnetospheric

field from the FAC and the ionospheric effects. For this purpose, they used only night-side mid-latitude data (from magnetic latitude 151–501 for the two hemispheres) (e.g., Olsen, 1999a).

F. Civet, P. Tarits / Planetary and Space Science 84 (2013) 102–111

of 4000). The grid consisted of two 301 polar cap meshes, 12 highlatitude 15  601 meshes and 144 15  151 low-latitude meshes. Using the Dst index as proxy of the time-varying source field, as for the N synthetic Swarm data analysis, we calculated the c^ j ðΩk ; θj ; ϕj Þ coefficients from Eq. (15). The data available from the Ørsted mission consists of a non-continuous 1400-day time series. Finally, we m obtained the internal and external coefficients β^ l ðΩk Þ and α^ m l ðΩk Þ of the induced and inducing magnetic potential. Fig. 7 shows the spectrogram of the SH coefficient amplitude for each l; m at all frequencies. We observe on this figure that the major part of the energy is distributed on the first degree (l¼1) and α Oersted

1.0

100

10 0.8

1

0.6

0.1 1

2

−1 0

1 −2 −1 0

0

3 1

2 −3 −2 −1 0

1

2

3

β Oersted

−500 100

0.4

−1000 10 Period (days)

Depth (Km)

Square root Energy

We carefully surveyed the Ørsted time series in order to select a maximum of data only weakly affected by either the FAC or the equatorial electrojet. This analysis was necessary because the data exclusion generated large gaps in the time series, which could have had an impact on our processing. For the equatorial region, we tested whether or not the data affected the results. We observed that the effect of the equatorial electrojet was most significant at periods of 24 and 12 h, in agreement with the results from Hutton (1972). Hence, we kept the equatorial band to maintain the dataset as complete as possible to properly recover the data spectrum at periods longer than a day. We also tested the day/night influence. The analysis showed that the major contributions were for periods shorter than a day, so we also kept day-side data in the subsequent calculations. At high latitudes, we sorted out the data with the help of the AL and Kp magnetic indices (Menvielle and Marchaudon, 2007) to remove data with large FAC signals. After a conversion from planetocentric to geomagnetic coordinates, for AL values lower than 100 nT, we kept data between −601 and 601 and between −551 and 551 for AL greater than or equal to 100 nT. We also used the Kp index to eliminate data during periods where Kp was greater than 4, which is when the Interplanetary Magnetic Field (IMF, originating from the Sun) was in a strong active period. After this preliminary processing, we binned the magnetic data on a spatial grid. We designed the grid to obtain a sufficiently large number of data in any mesh (of the order

Period (days)

108

−1500 Kuvshinov model 730 days − 3 layers 730 days − 10 layers 60 days − 3 layers 60 days − 6 layers

−2000

0.2

1

0.1

−3

−2

−1

0

1

2

1

2

−1 0

1 −2 −1 0

Log(σ)

3 1

2 −3 −2 −1 0

1

2

3

Fig. 5. Electrical conductivity model obtained from the inversion of the potential coefficients calculated for two time series windows: 60 days and 730 days. The true synthetic 1-D the model used to generate the data is given in black. For 730 days, we also used starting models of 3 and 10 layers.

Fig. 7. Spectrogram of the external (upper part) and internal (lower part) distribution of energy in the Ørsted spherical harmonic coefficient spectra, calculated for 500 days (since 01/01/2002), expressed as the square root of the energy.

100 90 available data (%)

80 70 60 50 40 30 20 10 0 Jan

Apr

Jul Oct 2002

0.0

Spherical Harmonic (l,m)

Jan

Apr

Jul 2003

Oct

Jan

Apr

Jul 2004

Oct

Jan

Apr

Jul 2005

Oct

Time Fig. 6. Time series of the daily percentage of available data during the Ørsted mission from 2002 to 2006, sampling rate 1.135 s.

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109

0 0.01

|Vi|

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Oersted 200 days multiT Oersted 500 days multiT

ϕ

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Velimsky, 2010 (space) Kuvshinov, 2006 (space) Semenov, 2006 (land) Olsen, 1999 (land)

−2.5 −2.0 −1.5 −1.0 −0.5 0.0 Log(σ)

0.5

1.0

1.5

100

160 120 80 40 0 −40 −80 −120 −160 10

2.0

100

Period (days)

Fig. 8. 1-D electrical conductivity profiles obtained using several lengths of temporal window (200 and 500 days), with or without window tapering. Models of previous land and space studies are represented from Velimsky (2010), Kuvshinov and Olsen (2006), Semenov and Jozwiak (1999) and Olsen (1999b).

|Vi|

0.1

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0.001 10

Degree 1 − Order 0

observed calculated

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10

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observed calculated

100

160 120 80 40

ϕ

particularly on the P 01 for periods greater than 1 day. We performed our calculations in geographic coordinates, hence some energy is also −1 present in P −1 1 and P 1 SH potentials. For degrees greater than 1 (l ¼ 2−3), the energy is much smaller and appear only for very long periods. We noticed during the test on the synthetic Swarm data that the best resolved potential coefficients were obtained for degrees and orders for which the external source had a significant amplitude. In the real data, while it is known that the degree 1 source field is dominant, other source terms may be significant (e.g., Sabaka et al., 2002). However, noise and gaps in the data may generate significant biased internal and external coefficients in the least square solution of systems such as Eq. (15), which could be taken erroneously for source field signals. This effect is observed in Fig. 7, in the internal potential at degrees and orders with little or no energy in the external coefficient. Only the data at degree l¼1 and m ¼ −l; l seem reliable, as seen in the synthetic Swarm data. We applied a 65% overlap on the full time series using a Blackman tapering window in order to maximize the number of degree of freedom (NDF) for each frequency (e.g., Smirnov and Varentsov, 2004). We split the time series into windows of 200 days and 500 days, which led to NDF¼ 38 and 14, respectively, in N the calculation of the c^ j ðΩk ; θj ; ϕj Þ values. ^m We carried out the inversion of α^ m l ðΩk Þ and β l ðΩk Þ to obtain a global 1-D conductivity model (see Section 3). We used the potentials α^ and β^ at l¼1 and m ¼ −l; l, at all available frequencies for either the 200-day or 500-day time windows. The Earth was parameterized with eight layers of increasing thickness down to 2800 km depth. The results of the inversion are presented in Fig. 8. For these models, the misfit function was decreased by 60–90%. An example of fit is presented in Fig. 9. Despite the difference in length of the time windows used to obtain these models (200 and 500 days), they are quite similar, except in the mid-mantle where the conductivity changes rapidly with depth. We ran a sensitivity analysis by varying the conductivity in each layer. We considered the conductivity value acceptable as long as the misfit function did not change more than 100%. We found that conductivities were constrained within 0.5 log level down to 1500 km, and to 1 log level at greater depths. The conductivity profiles obtained are consistent with existing profiles obtained from observatory and satellite data (e.g., Velimsky, 2010; Kuvshinov and Olsen, 2006; Semenov and Jozwiak, 1999; Olsen,

observed calculated

0 −40 −80 −120 10

100

Period (days) Fig. 9. Example of fit between the observed and recalculated internal SH potentials −1 0 1 for a 200-day time series. The internal potentials β^ 1 , β^ 1 and β^ 1 are presented as a function of period in days (modulus, jV i j) and phase, φ).

1999b). The transition zone between the 410 and 660 km seismic discontinuities (e.g., White, 2005) is not clearly identified in any models, while all agree that there is a rapid increase in conductivity in the 800–1000 km range. This increase is well marked in both our models and those of Velimsky (2010). This consistency suggests that

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our data have good resolution in this mantle depth range. The conductivity increase may be related to changes in the mineralogical composition in the uppermost part of the lower mantle (e.g., Xu et al., 1998).

5. Conclusion Magnetometers are almost always part of the scientific payload in space exploration of the solar system bodies. One of the objective with the satellite magnetic data is to obtain information about the electrical conductivity of a body's interior. This objective is often difficult to reach because the electromagnetic induction analysis of the magnetic data must take into account the inducing magnetic ionized environment and its interaction with the planetary body. On Earth, the analyzes are based on a fairly good knowledge of the magnetic source field. Here, we developed an approach to analyze satellite magnetic data with limited a priori knowledge on the inducing magnetic field. The method also takes into account the inherently gapped nature of the satellite time series. A proxy of the time-varying source field is explicitly introduced in the analysis in order to help recover the internal induced field. The latter carries the information about the planet conductivity distribution. We used both synthetic and real Earth satellite data to test the approach. The analyzes performed on these data led to 1-D mantle conductivity models in good agreement with the results from studies using Earth observatory and satellite data. The method will now be applied to study the internal conductivity of telluric bodies in the solar system with data from missions such as Mars Global Surveyor or MESSENGER for Mercury and future missions, such as Bepi-Colombo on Mercury and MAVEN on Mars. However, the method requires the definition of a proxy of the induction source field in the studied planet, which is beyond the scope of the present paper. Our results on Mars are presented in another article.

Acknowledgment The work was supported by grants from the French Ministry of Research and CNRS-INSU (program PNP). We thank Alexei Kuvshinov for providing the Swarm synthetic magnetic dataset. We also thank Jackub Velimsky and an anonymous referee for their useful remarks and advice on improving the manuscript. References Acuna, M., Connerney, J., Wasilewski, P., Lin, R., Mitchell, D., Anderson, K., Carlson, C., McFadden, J., Reme, H., Mazelle, C., Vignes, D., Bauer, S., Cloutier, P., Ness, N., 2001. Magnetic field of mars: summary of results from the aerobraking and mapping orbits. Journal of Geophysical Research 106 (E10), 23,403–23,417. Akalin, F., Morgan, D., Gurnett, D., Kirchner, D., Brain, D., Modolo, R., Acuna, M., Espley, J., 2010. Dayside induced magnetic field in the ionosphere of mars. Icarus 206 (March (1)), 104–111. Carbonell, M., Oliver, R., Ballester, J., 1992. Power spectra of gapped time series—a comparison of several methods. Astronomy and Astrophysics 264 (October), 350–360. Chapman, S., 1956. The morphology of geomagnetic storms and bays: a general review. Vistas in Astronomy 2, 912–928. Constable, S., Constable, C., 2004. Observing geomagnetic induction in magnetic satellite measurements and associated implications for mantle conductivity. Geochemistry Geophysics Geosystems 5. Coulomb, J., Jobert, G., 1990. Traite de geophysique interne. vol. 2. Dubinin, E., Fraenz, M., Woch, J., Winnigham, J., Frahm, R., Lundin, R., Barabash, S., 2008. Suprathermal electron fluxes on the nightside of mars: ASPERA-3 observations. Planetary and Space Science 56 (May (6)), 846–851. Dyal, P., Parkin, C., Daily, W., 1974. Magnetism and the interior of the moon. Reviews of Geophysics and Space Physics 12 (November), 568–591. Dyal, P., Parkin, C., Daily, W., April 1976. Structure of the Lunar Interior from Magnetic Field Measurements. vol. 7, pp. 3077–3095.

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