General expressions of magnetic vector (MV) and magnetic gradient tensor (MGT) in terms of the first- and second-order derivatives of spherical harmonics at different degrees/orders are relatively complicated and singular at the poles. In this paper, we derived alternative non-singular expressions for the MV, the MGT and also the third-order partial derivatives of the magnetic potential field in the local north-oriented reference frame. Using our newly derived formulae, the magnetic potential, vector and gradient tensor fields and also the third-order partial derivatives of the magnetic potential field at an altitude of 300 km are calculated based on a global lithospheric magnetic field model GRIMM_L120 (GFZ Reference Internal Magnetic Model, version 0.0) with spherical harmonic degrees 16–90. The corresponding results at the poles are discussed and the validity of the derived formulas is verified using the Laplace equation of the magnetic potential field.

Compared to the magnetic vector and scalar measurements, magnetic gradients lead to more robust models of the lithospheric magnetic field. The ongoing Swarm mission of the European Space Agency (ESA) provides measurements not only of the vector and scalar data but also an estimate of their east–west gradients (e.g., Olsen et al., 2004, 2015; Friis-Christensen et al., 2006). Kotsiaros and Olsen (2012, 2014) proposed to recover the lithospheric magnetic field through magnetic space gradiometry in the same way that has been done for modeling the gravitational potential field from the satellite gravity gradient tensor measurements by the Gravity field and steady-state Ocean Circulation Explorer (GOCE). Purucker (2005), Purucker et al. (2007), Sabaka et al. (2015) and Kotsiaros et al. (2015) also reported efforts to model the lithospheric magnetic field using magnetic gradient information from the satellite constellation. Their results showed that, by using gradient data, the modeled lithospheric magnetic anomaly field has enhanced shorter wavelength content and a much higher quality compared to models built from vector field data. This is because the gradient data can remove the highly time-dependent contributions of the magnetosphere and ionosphere that are correlated between two side-by-side satellites.

The second-order magnetic gradient tensor consists of spatial derivatives highlighting certain structures of the magnetic field (e.g., Schmidt and Clark, 2000, 2006). It can be used to detect the hidden and small-scale magnetized sources (e.g., Pedersen and Rasmussen, 1990; Harrison and Southam, 1991) and to investigate the orientation of the lineated magnetic anomalies (e.g., Blakely and Simpson, 1986). Quantitative magnetic interpretation methods such as the analytic signal, edge detection, spatial derivatives, Euler deconvolution, and transformations, all set in a Cartesian coordinate system (e.g., Blakely, 1995; Purucker and Whaler, 2007; Taylor et al., 2014) also require calculating the higher-order derivatives of the magnetic anomaly field and need to be extended to regional and global scales to handle the curvature of Earth and other planets. Ravat et al. (2002) and Ravat (2011) utilized the analytic signal method and the total gradient to interpret the satellite-altitude magnetic anomaly data. Therefore, both the magnetic field modeling and also the geological interpretations require the calculation for the partial derivatives of the magnetic field, possibly at the poles for specific systems of coordinates. Spherical harmonic analysis, established originally by Gauss (1839), is generally used to model the global magnetic internal fields of Earth and other terrestrial planets (e.g., Maus et al., 2008; Langlais et al., 2010; Thébault et al., 2010; Finlay et al., 2010; Lesur et al., 2013; Sabaka et al., 2013; Olsen et al., 2014). Series of spherical harmonic functions themselves made of Schmidt semi-normalized associated Legendre functions (SSALFs) (e.g., Blakely, 1995; Langel and Hinze, 1998) are fitted by least squares to magnetic measurements, giving the spherical harmonic coefficients (i.e., the Gaussian coefficients) defining the model. Kotsiaros and Olsen (2012, 2014) presented the MV (magnetic vector) and the MGT (magnetic gradient tensor) using a spherical harmonic representation and, of course, their expressions are singular as they approach the poles. Even if there are satellite data gaps around the poles, it is advisable to use non-singular spherical harmonic expressions for the MV and the MGT in case airborne or shipborne magnetic data are utilized (e.g., Golynsky et al., 2013; Maus, 2010). A rotation of the coordinate system is always possible to avoid the polar singularity, but this solution is very ineffective for large data sets.

In this paper, following Petrovskaya and Vershkov (2006) and Eshagh (2008, 2009) for the gravitational gradient tensor in the local north-oriented, orbital reference and geocentric spherical frames, the non-singular expressions in terms of spherical harmonics for the MV, the MGT and the third-order derivatives of the magnetic potential field in the specially defined local-north-oriented reference frame (LNORF) are presented. In the next section, the traditional expressions of the MV and the MGT are first stated, some necessary propositions are then proved and, lastly, new non-singular expressions are derived. In Sect. 3, the new formulae are tested using the global lithospheric magnetic field model GRIMM_L120 (GFZ Reference Internal Magnetic Model, version 0.0) (Lesur et al., 2013) and compared with the results by traditional formulae. Finally, some conclusions are drawn and further applications are also discussed.

In this section, the traditional expressions of MV and MGT are presented and their numerical problems are stated. Then, based on some necessary mathematical derivations, new expressions are given.

The scalar potential

If considered in the LNORF

The expressions for

To deal with the singular terms and first- and second-order derivatives of
the SSALFs, some useful mathematical derivations are introduced and proved
in the following.

Derivation of

Based on Eq. (Z.1.44) in Ilk (1983),

Derivation of

According to Eq. (23) in Eshagh (2008),

Derivation of

Using Eq. (Z.1.42) in Ilk (1983),

Derivation of

Employing Eq. (31) in Eshagh (2008),

Derivation of

Using Eq. (36) in Eshagh (2008),

Derivation of

According to Petrovskaya and Vershkov (2006) and Eshagh (2009) we can write

Substituting Eq. (17) into the right-hand side of Eq. (16), and after
simplification, we can derive

And combining Eq. (6) we obtain that

Derivation of

Based on lemma 3 in Eshagh (2009),

According to Eq. (10) we can write

Inserting Eqs. (12) and (22) into Eq. (21), and after some
simplifications, we obtain that

And combining with Eq. (6) we can derive

Inserting the corresponding mathematical derivations in the last section into
Eqs. (2) and (4), and after some simplifications, the new expressions for MV
and MGT can be written as

Lithospheric magnetic potential, magnetic vector and its gradient fields and
third-order partial derivatives of the magnetic potential field around the
North Pole (0

Statistics of the magnetic potential, MV, MGT and third-order
partial derivatives of the magnetic potential field around the North Pole
(0

Furthermore, some other higher-order partial derivatives and their
transformations are usually used to image geologic boundaries in magnetic
prospecting, such as the higher-order enhanced analytic signal (e.g., Hsu et
al., 1996). Therefore, we also give the third-order partial derivatives of
the magnetic potential field as

Statistics of the magnetic potential, MV, MGT and third-order
partial derivatives of the magnetic potential field around the South Pole
(150

Lithospheric magnetic potential, magnetic vector and its gradient fields and
third-order partial derivatives of the magnetic potential field around the
South Pole (150

Limit values of the magnetic potential (

In this way, we avoid computing recursively the SSALFs with singular terms,
their first- and second-order derivatives as in the traditional formulae.
The cost is only to calculate two additional degrees and orders for the
SSALFs at most. It should be noted that, in this study, we use the
conventional form of SSALF that if

We test the derived expressions and the numerical implementation in
C/C

Furthermore, the computed magnetic fields are smooth near the poles and
do not have the singularities, but some components have the dependence on the
direction of the reference frame at the poles. As shown in Fig. 3, the magnetic
potential

Compared with the traditional formulae in Sect. 2.1, there are two
advantages of our derived formulae in Sect. 2.3. On the one hand, the
traditional up to second-order derivatives are removed in the new formulae;
therefore, the relatively complicated method by Horner's recursive
algorithm (Holmes and Featherstone, 2002b) can be avoided. On the other
hand, the singular terms of

We develop in this paper the new expressions for the MV, the MGT and the third-order partial derivatives of the magnetic potential field in terms of spherical harmonics. The traditional expressions have complicated forms involving first- and second-order derivatives of the SSALFs and are singular when approaching the poles. Our newly derived formulae do not contain the first- and second-order derivatives of the SSALFs and remove the singularities at the poles. However, our formulae are derived in the spherical LNORF with specific definition at the poles. For a future application to the magnetic data of a satellite gradiometry mission (e.g., Kotsiaros and Olsen, 2014), it is necessary to describe the MV and the MGT in the local orbital or other reference frame, where the new MV and MGT are the linear functions of the MV and the MGT in the LNORF with coefficients related to the satellite track azimuth (e.g., Petrovskaya and Vershkov, 2006) or other rotation angles. The other main purpose of this paper is, in the future, to contribute to the signal processing and the geophysical and geological interpretation of the global lithospheric magnetic field model, especially near polar areas.

Supplementary software implementation is performed with the programming language C/C

This study is supported by the International Cooperation Project in Science and Technology of China (no. 2010DFA24580), the Hubei Subsurface Multi-scale Imaging Key Laboratory (Institute of Geophysics & Geomatics, China University of Geosciences, Wuhan) (grant no. SMIL-2015-06) and State Key Laboratory of Geodesy and Earth's Dynamics (Institute of Geodesy and Geophysics, CAS) (grant no. SKLGED2015-5-5-EZ). Jinsong Du is sponsored by the China Scholarship Council (CSC). We would like to thank Mehdi Eshagh and another anonymous reviewer for their constructive comments. All projected figures are drawn using the Generic Mapping Tools (GMT) (Wessel and Smith, 1991). The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.Edited by: L. Gross