Introduction
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.
Methodology
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.
Traditional expressions
The scalar potential V of Earth's magnetic field in a source-free
region can be expanded in the truncated series of spherical harmonics at the
point P(r, θ, φ) with the geocentric distance r, co-latitude
θ and longitude φ (e.g., Backus et al., 1996):
V(r,θ,ϕ)=a∑l=1L∑m=0l(ar)l+1glmcosmϕ+hlmsinmϕP̃lmcosθ,
where a= 6371.2 km is the radius of Earth's magnetic reference sphere;
P̃lmcosθ (or P̃lm for simplification) is the SSALF of degree l and order m; L is the maximum
spherical harmonic degree; and glm and hlm are the geomagnetic
harmonic coefficients describing Earth's internal sources.
If considered in the LNORF {x,y,z} (e.g., Olsen et al., 2010), where the z axis points
downward in the geocentric radial direction, the x axis points to the north,
and the y axis towards the east (that is, a right-handed system). At the
poles, we define that the x axis points to the meridian of
180∘ E (or 180∘ W) at the North Pole and of 0∘ at
the South Pole, which will be discussed in Sect. 3. Therefore, the three
components of the MV can be expressed as
Bx(r,θ,ϕ)=-1r∂∂-θV(r,θ,ϕ)=∑l=1L∑m=0l(ar)l+2glmcosmϕ+hlmsinmϕddθP̃lmcosθ,By(r,θ,ϕ)=-1rsinθ∂∂ϕV(r,θ,ϕ)=∑l=1L∑m=0l(ar)l+2mglmsinmϕ-hlmcosmϕ1sinθP̃lmcosθ,Bz(r,θ,ϕ)=-∂∂-rV(r,θ,ϕ)=-∑l=1L∑m=0ll+1(ar)l+2glmcosmϕ+hlmsinmϕP̃lmcosθ.
The MGT can be written as (e.g., Kotsiaros and Olsen,
2012)
∇B=BxxBxyBxzByxByyByzBzxBzyBzz=
∂Bx/∂x∂Bx/∂y∂Bx/∂z∂By/∂x∂By/∂y∂By/∂z∂Bz/∂x∂Bz/∂y∂Bz/∂z,
where nine elements are expressed respectively as
Bxx=1a∑l=1L∑m=0l(ar)l+3glmcosmϕ+hlmsinmϕ×-d2dθ2P̃lmcosθ+l+1P̃lmcosθ,Bxy=Byx=1a∑l=1L∑m=0l(ar)l+3mglmsinmϕ-hlmcosmϕ×-1sinθddθP̃lmcosθ+cosθsin2θP̃lmcosθ,Bxz=Bzx=1a∑l=1L∑m=0l(ar)l+3l+2glmcosmϕ+hlmsinmϕddθP̃lmcosθ,Byy=1a∑l=1L∑m=0l(ar)l+3glmcosmϕ+hlmsinmϕ×l+1P̃lmcosθ+m2sin2θP̃lmcosθ-cosθsinθddθP̃lmcosθ,Byz=Bzy=1a∑l=1L∑m=0l(ar)l+3l+2mglmsinmϕ-hlmcosmϕ1sinθP̃lmcosθ,Bzz=-1a∑l=1L∑m=0l(ar)l+3l+1l+2glmcosmϕ+hlmsinmϕP̃lmcosθ.
The expressions for V, Bz and Bzz can be calculated stably even for
very high spherical harmonic degrees and orders by using the Holmes and
Featherstone (2002a) scheme. However, there exist the singular terms of
1/sinθ and 1/sin2θ in Eqs. (2b), (4b), (4d) and
(4e) when the computing point approaches to the poles. Moreover, some
expressions contain the terms of first- and second-order derivatives of
SSALFs, such as Eqs. (2a) and (4a)–(4d). Nevertheless,
the up to second-order derivatives for very high degrees and orders of SSALFs
can be recursively calculated by the Horner algorithm (Holmes and
Featherstone, 2002b). These algorithms are relatively complicated and thus
we want to use alternative expressions to avoid the singular terms and also
the partial derivatives of SSALFs. It should be stated that our work differs
from those presented by Petrovskaya and Vershkov (2006) and Eshagh (2009) in
the LNORF and also the associated Legendre functions (ALFs). Nonetheless,
the following mathematical derivations are carried out based on their
studies on gravity fields.
Mathematical derivations
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 dP̃lm/dθ
Based on Eq. (Z.1.44) in Ilk (1983),dPlm/dθ=0.5l+ml-m+1Plm-1-Plm+1,and the relation between the ALFs and the SSALFs isP̃lm=Cml-m!/l+m!Plm;thus, the first-order derivative of the SSALFs can be deduced asdP̃lm/dθ=al,mP̃lm-1+bl,mP̃lm+1,al,m=0.5l+ml-m+1Cm/Cm-1,bl,m=-0.5l+m+1l-mCm/Cm+1,whereCm=2-δm,0=1,m=02,m≠0,and δ is the Kronecker delta.
Derivation of d2P̃lm/dθ2
According to Eq. (23) in Eshagh (2008),d2Plm/dθ2=0.25l+ml-m+1l+m-1l-m+2Plm-2-0.25l+ml-m+1+l-ml+m+1Plm+0.25Plm+2.The second-order derivative of the SSALFs can be written asd2P̃lm/dθ2=cl,mP̃lm-2+dl,mP̃lm+el,mP̃lm+2,cl,m=0.25l+ml+m-1l-m+2l-m+1Cm/Cm-2,dl,m=-0.25l+ml-m+1+l-ml+m+1,el,m=0.25l+m+2l+m+1l-ml-m-1Cm/Cm+2.
Derivation of P̃lm/sinθ
Using Eq. (Z.1.42) in Ilk (1983),Plm/sinθ=0.5l+ml+m-1Pl-1m-1+Pl-1m+1/m,m≥1,and using Eq. (6) we can obtain thatP̃lm/sinθ=fl,mP̃l-1m-1+gl,mP̃l-1m+1,m≥1,fl,m=0.5l+ml+m-1Cm/Cm-1/m,m≥1,gl,m=0.5l-ml-m-1Cm/Cm+1/m,m≥1.
Derivation of P̃lm/sin2θ
Employing Eq. (31) in Eshagh (2008),Plm/sin2θ=l+ml+m-1l-m+1l-m+2/m-1Plm-2+l+ml+m-1/m-1+l-ml-m-1/m+1Plm+1/m+1Plm+2/4m,,m≥2,and using Eq. (6) we haveP̃lm/sin2θ=hl,mP̃lm-2+kl,mP̃lm+nl,mP̃lm+2,m≥2,hl,m=0.25l+ml+m-1l-m+1l-m+2Cm/Cm-2/mm-1,m≥2,kl,m=0.25l+ml+m-1/m-1+l-ml-m-1/m+1/m,m≥2,nl,m=0.25l-ml-m-1l+m+2l+m+1Cm/Cm+2/mm+1,m≥1.
Derivation of dP̃lm/sinθdθ
Using Eq. (36) in Eshagh (2008),dPlm/sinθdθ=0.25l+ml+m-1l+m-2l-m+1/m-1Pl-1m-2+l+ml-m+1/m-1-l+m+1l+m/m+1Pl-1m-1/m+1Pl-1m+2,m≥2,and using Eq. (6) we can derivedP̃lm/sinθdθ=ol,mP̃l-1m-2+ql,mP̃l-1m+xl,mP̃l-1m+2,m≥2,ol,m=0.25l+ml+m-1l+m-2l-m+1Cm/Cm-2/m-1,m≥2,ql,m=0.25l-ml+ml-m+1/m-1-l+m+1/m+1,m≥2,xl,m=-0.25l+m+1l-ml-m-1l-m-2Cm/Cm+2/m+1.
Derivation of dP̃lm/sinθdθ-P̃lmcosθ/sin2θ
According to Petrovskaya and Vershkov (2006) and Eshagh (2009) we can writedPlm/sinθdθ-Plmcosθ/sin2θ=0.5m-1l+ml-m+1Plm-1/sinθ-m+1Plm+1/sinθ/m,m≥1,and using Eq. (36) in Eshagh (2008) we can obtainPlm-1/sinθ=0.5l-m+2l-m+3Pl+1m-2+Pl+1m/m-1,m≥2,Plm+1/sinθ=0.5l-ml-m+1Pl+1m+Pl+1m+2/m+1.
Substituting Eq. (17) into the right-hand side of Eq. (16), and after
simplification, we can derivedPlm/sinθdθ-Plmcosθ/sin2θ=0.25l+ml-m+1l-m+2l-m+3Pl+1m-2+2ml-m+1Pl+1m-Pl+1m+2/m,m≥1.
And combining Eq. (6) we obtain thatdP̃lm/sinθdθ-P̃lmcosθ/sin2θ=0.25l+ml-m+1l-m+2l-m+3Cm/Cm-2P̃l+1m-2+2ml-m+1l+m+1P̃l+1m-l+m+1l+m+2l+m+3l-mCm/Cm+2P̃l+1m+2/m,m≥1.
Derivation of l+1sin2θP̃lm+m2P̃lm- sinθcosθdP̃lm/dθ/sin2θ
Based on lemma 3 in Eshagh (2009),sinθcosθdPlm/dθ=mPlm+l+1sin2θPlm-sinθPl+1m+1and we can derivel+1sin2θPlm+m2Plm-sinθcosθdPlm/dθ/sin2θ=mm-1Plm/sin2θ+Pl+1m+1/sinθ.
According to Eq. (10) we can writePl+1m+1/sinθ=0.5l+m+2l+m+1Plm+Plm+2/m+1.
Inserting Eqs. (12) and (22) into Eq. (21), and after some
simplifications, we obtain thatl+1sin2θPlm+m2Plm-sinθcosθdPlm/dθ/sin2θ=0.25l+ml+m-1l-m+1l-m+2Plm-2+0.25l+ml+m-1+l-ml-m-1m-1/m+1+2l+m+2l+m+1/m+1Plm+0.25Plm+2.
And combining with Eq. (6) we can derivel+1sin2θP̃lm+m2P̃lm-sinθcosθdP̃lm/dθ/sin2θ=0.25l+ml+m-1l-m+1l-m+2Cm/Cm-2P̃lm-2+0.25l+ml+m-1+l-ml-m-1m-1/m+1+2l+m+2l+m+1/m+1P̃lm+0.25l+m+1l+m+2l-ml-m-1Cm/Cm+2P̃lm+2.
New expressions
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
Bx=∑l=1L∑m=0l(ar)l+2glmcosmϕ+hlmsinmϕal,mxP̃lm-1+bl,mxP̃lm+1,By=∑l=1L∑m=0l(ar)l+2glmsinmϕ-hlmcosmϕal,myP̃l-1m-1+bl,myP̃l-1m+1,Bz(r,ϕ,λ)=∑l=1L∑m=0l(ar)l+2glmcosmλ+hlmsinmλal.mzP̃lm,
Bxx=1a∑l=1L∑m=0l(ar)l+3glmcosmϕ+hlmsinmϕal,mxxP̃lm-2+bl,mxxP̃lm+cl,mxxP̃lm+2,Bxy=1a∑l=1L∑m=0l(ar)l+3glmsinmϕ-hlmcosmϕal,mxyP̃l+1m-2+bl,mxyP̃l+1m+cl,mxyP̃l+1m+2,Bxz=1a∑l=1L∑m=0l(ar)l+3glmcosmϕ+hlmsinmϕal,mxzP̃lm-1+bl,mxzP̃lm+1,Byy=1a∑l=1L∑m=0l(ar)l+3glmcosmϕ+hlmsinmϕal,myyP̃lm-2+bl,myyP̃lm+cl,myyP̃lm+2,Byz=1a∑l=1L∑m=0l(ar)l+3glmsinmλ-hlmcosmλal,myzP̃l-1m-1+bl,myzP̃l-1m+1,Bzz=1a∑l=1L∑m=0l(ar)l+3glmcosmλ+hlmsinmϕal,mzzP̃lm,
where the corresponding coefficients of the SSALFs are given as follows:
al,mx=0.5l+ml-m+1Cm/Cm-1bl,mx=-0.5l+m+1l-mCm/Cm+1,al,my=0.5l+ml+m-1Cm/Cm-1bl,my=0.5l-ml-m-1Cm/Cm+1,al,mz=-l+1,al,mxx=-0.25l+ml+m-1l-m+2l-m+1Cm/Cm-2bl,mxx=0.25l+ml-m+1+l-ml+m+1+l+1cl,mxx=-0.25l+m+2l+m+1l-ml-m-1Cm/Cm+2,al,mxy=-0.25l+ml-m+1l-m+2l-m+3Cm/Cm-2bl,mxy=-0.5ml-m+1l+m+1cl,mxy=0.25l+m+1l+m+2l+m+3l-mCm/Cm+2,al,mxz=0.5l+2l+ml-m+1Cm/Cm-1=l+2al,mxbl,mxz=-0.5l+2l+m+1l-mCm/Cm+1=l+2bl,mx,al,myy=0.25l+ml+m-1l-m+1l-m+2Cm/Cm-2bl,myy=0.25l+ml+m-1+l-ml-m-1m-1/m+1+2l+m+2l+m+1/m+1cl,myy=0.25l+m+1l+m+2]l-ml-m-1Cm/Cm+2,al,myz=0.5l+2l+ml+m-1Cm/Cm-1=l+2al,mybl,myz=0.5l+2l-ml-m-1Cm/Cm+1=l+2bl,my,al,mzz=-l+1l+2=l+2al,mz.
Lithospheric magnetic potential, magnetic vector and its gradient fields and
third-order partial derivatives of the magnetic potential field around the
North Pole (0∘ ≤θ≤ 30∘) at the altitude of
300 km as defined by the lithospheric magnetic field model GRIMM_L120
(version 0.0) (Lesur et al., 2013) for spherical harmonic degrees 16–90.
(a) is magnetic potential (V); (b), (c) and
(d) are three components (Bx, By and Bz) of the magnetic
vector; (e), (f), (g), (h), (i)
and (j) are six elements (Bxx, Bxy, Bxz, Byy,Byz and Bzz) of the magnetic gradient tensor; (k),
(l), (m), (n), (o) and (p) are
six elements (Bxxz, Bxyz, Bxzz, Byyz,Byzz and Bzzz) of third-order partial derivatives of the magnetic potential field,
respectively. The dark green lines are the plate boundaries by Bird (2003).
All maps are shown in polar stereographic projections.
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Statistics of the magnetic potential, MV, MGT and third-order
partial derivatives of the magnetic potential field around the North Pole
(0∘ ≤θ≤ 30∘) at the altitude of 300 km
using the lithospheric magnetic field model GRIMM_L120
(version 0.0) (Lesur et al., 2013) for spherical harmonic degrees
16–90.
Magnetic effects
Minimum
Maximum
Mean
Standard deviation
V (mT × m)
-5.1554771
+4.7867519
+0.0828017
±1.7377648
Bx (nT)
-14.7389250
+17.6917740
-0.0890689
±4.9797007
By (nT)
-15.1297000
+13.6053000
+0.0010738
±4.8239313
Bz (nT)
-19.8715270
+25.3666030
-0.1988485
±6.7066701
Bxx (pT m-1)
-0.1054684
+0.0621351
+0.0001872
±0.0215871
Bxy (pT m-1)
-0.0410371
+0.0491030
+0.0000003
±0.0115018
Bxz (pT m-1)
-0.0929498
+0.1082861
+0.0006867
±0.0247522
Byy (pT m-1)
-0.0726248
+0.0505990
-0.0004789
±0.0186580
Byz (pT m-1)
-0.0868184
+0.0826627
+0.0000058
±0.0228174
Bzz (pT m-1)
-0.1015986
+0.1511038
+0.0002917
±0.0336965
Bxx+Byy+Bzz (pT m-1)
-2.012 × 10-15
+2.026×10-15
+8.085×10-19
±5.101 × 10-16
Bxxz (aT m-2)
-0.7589853
+0.4794999
+0.0002436
±0.1537058
Bxyz (aT m-2)
-0.2628265
+0.3734132
-0.0000004
±0.0734794
Bxzz (aT m-2)
-0.7067652
+0.8470055
+0.0140820
±0.1752880
Byyz (aT m-2)
-0.5259662
+0.4076568
-0.0134321
±0.1370902
Byzz (aT m-2)
-0.6058631
+0.6396412
+0.0000341
±0.1448002
Bzzz (aT m-2)
-0.7609268
+1.1697371
+0.0131885
±0.2421663
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
Bxxz=∂Bxx∂z=∂2Bx∂x∂z=∂2Bx∂z∂x=1a2∑l=1L∑m=0l(ar)l+4glmcosmϕ+hlmsinmϕal,mxxzP̃lm-2+bl,mxxzP̃lm+cl,mxxzP̃lm+2,Bxyz=∂Bxy∂z=∂Byx∂z=∂2Bx∂y∂z=∂2Bx∂z∂y=∂2By∂x∂z=∂2By∂z∂x=1a2∑l=1L∑m=0l(ar)l+4glmsinmϕ-hlmcosmϕal,mxyzP̃l+1m-2+bl,mxyzP̃l+1m+cl,mxyzP̃l+1m+2,Bxzz=∂Bxz∂z=∂Bzx∂z=∂2Bx∂z2=∂2Bz∂x∂z=∂2Bz∂z∂x=1a2∑l=1L∑m=0l(ar)l+4glmcosmϕ+hlmsinmϕal,mxzzP̃lm-1+bl,mxzzP̃lm+1,Byyz=∂Byy∂z=∂2By∂y∂z=∂2By∂z∂y=1a2∑l=1L∑m=0l(ar)l+4glmcosmϕ+hlmsinmϕal,myyzP̃lm-2+bl,myyzP̃lm+cl,myyzP̃lm+2,Byzz=∂Byz∂z=∂Bzy∂z=∂2By∂z2=∂2Bz∂y∂z=∂2Bz∂z∂y=1a2∑l=1L∑m=0l(ar)l+4glmsinmλ-hlmcosmλal,myzzP̃l-1m-1+bl,myzzP̃l-1m+1,Bzzz=∂2Bz∂z2=1a2∑l=1L∑m=0l(ar)l+4glmcosmϕ+hlmsinmϕal,mzzzP̃lm,
where the corresponding coefficients of the SSALFs are presented as
al,mxxz=l+3al,mxxbl,mxxz=l+3bl,mxxcl,mxxz=l+3cl,mxx,al,mxyz=l+3al,mxybl,mxyz=l+3bl,mxycl,mxyz=l+3cl,mxy,al,mxzz=0.5l+2l+3l+ml-m+1Cm/Cm-1=l+2l+3al,mx=l+3al,mxzbl,mxzz=-0.5l+2l+3l+m+1l-mCm/Cm+1=l+2l+3bl,mx=l+3bl,mxz,al,myyz=l+3al,myybl,myyz=l+3bl,myycl,myyz=l+3cl,myy,al,myzz=0.5l+2l+3l+ml+m-1Cm/Cm-1=l+2l+3al,my=l+3al,myzbl,myzz=0.5l+2l+3l-ml-m-1Cm/Cm+1=l+2l+3bl,my=l+3bl,myz,al,mzzz=-l+1l+2l+3=l+3al,mzz=l+2l+3al,mz.
Statistics of the magnetic potential, MV, MGT and third-order
partial derivatives of the magnetic potential field around the South Pole
(150∘ ≤θ≤ 180∘) at the altitude of 300 km using the lithospheric magnetic field model GRIMM_L120
(version 0.0) (Lesur et al., 2013) for spherical harmonic degrees
16–90.
Magnetic effects
Minimum
Maximum
Mean
Standard deviation
V (mT × m)
-3.3267455
+4.6543369
+0.0801853
±1.2427083
Bx (nT)
-11.440070
+15.9109730
+0.3451248
±3.5403285
By (nT)
-9.1169009
+15.0436160
-0.0001605
±3.1560093
Bz (nT)
-22.202857
+14.5020010
-0.3022955
±4.7971494
Bxx (pT m-1)
-0.0579914
+0.0704617
+0.0000845
±0.0166266
Bxy (pT m-1)
-0.0364002
+0.0308075
-0.0000006
±0.0074702
Bxz (pT m-1)
-0.0741850
+0.0831062
+0.0019925
±0.0187492
Byy (pT m-1)
-0.0569493
+0.0706456
+0.0019055
±0.0143289
Byz (pT m-1)
-0.0599346
+0.0897167
-0.0000012
±0.0154623
Bzz (pT m-1)
-0.1367168
+0.0735795
-0.0019900
±0.0258066
Bxx+Byy+Bzz (pT m-1)
-1.027 × 10-15
+2.012×10-15
+1.113×10-18
±5.059 × 10-16
Bxxz (aT m-2)
-0.4605216
+0.5307263
+0.0011232
±0.1328515
Bxyz (aT m-2)
-0.2840344
+0.2947601
-0.0000015
±0.0526629
Bxzz (aT m-2)
-0.5686811
+0.5634376
0.0181792
±0.1497829
Byyz (aT m-2)
-0.4262850
+0.5819095
+0.0186968
±0.1169641
Byzz (aT m-2)
-0.6194116
+0.6520948
-0.0000118
±0.1085051
Bzzz (aT m-2)
-1.0199774
+0.5863084
-0.0198200
±0.2084566
Lithospheric magnetic potential, magnetic vector and its gradient fields and
third-order partial derivatives of the magnetic potential field around the
South Pole (150∘ ≤θ≤ 180∘) at the altitude of
300 km as defined by the lithospheric magnetic field model GRIMM_L120
(version 0.0) (Lesur et al., 2013) for spherical harmonic degrees 16–90.
(a) is magnetic potential (V), (b), (c) and
(d) are three components (Bx, By and Bz) of the magnetic
vector; (e), (f), (g), (h), (i)
and (j) are six elements (Bxx, Bxy, Bxz, Byy,Byz andBzz) of the magnetic gradient tensor; (k),
(l), (m), (n), (o) and (p) are
six elements (Bxxz, Bxyz, Bxzz, Byyz,Byzz and Bzzz) of third-order partial derivatives of the magnetic potential field,
respectively. The dark green lines are the plate boundaries by Bird (2003).
All maps are shown in polar stereographic projections.
![]()
Limit values of the magnetic potential (V), magnetic vector (Bx, By
and Bz) and its gradients (Bxx, Bxy, Bxz, Byy,Byz andBzz) and third-order partial derivatives of the magnetic
potential field (Bxxz, Bxyz, Bxzz, Byyz,Byzz and Bzzz) at the poles when the local reference frames vary from different
meridians (the direction of xP axes changing from different meridians to
the poles). Red and blue lines indicate the magnetic effects at the North Pole
and at the South Pole, respectively. The reference frame is specially defined
that the xP axis points to the meridian of 180∘ E (or
180∘ W) at the North Pole and 0f 0∘ at the South Pole and the
yP axis points to the meridian of 90∘ E at both poles. The
values at both poles shown by black dashed arrows are used to plot the maps
in Figs. 1 and 2.
![]()
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 m<0, then P̃lm=-1mP̃lm
and if m>l, then P̃lm=0.
Numerical investigation and discussion
We test the derived expressions and the numerical implementation in
C/C++ by calculating the magnetic potential, magnetic vector and its gradients
and also the third-order partial derivatives of the magnetic potential field
on a grid with 0.125∘ × 0.125∘ cell size at the
altitude of 300 km relative to Earth's magnetic reference sphere
using the lithospheric magnetic field model GRIMM_L120
(version 0.0) defined by Lesur et al. (2013). The magnetic potential, MV,
MGT and the third-order partial derivatives of the magnetic potential field
in the two polar regions mapped by the lithospheric field model with
spherical harmonic degrees 16–90 are shown in Fig. 1
and Fig. 2, respectively. The corresponding statistics around the North Pole and
South Pole are, respectively, presented in Tables 1 and 2. A simple
test is that the MGT meets Laplace's equation of the potential field;
that is, the trace of the MGT should be equal to zero. Our numerical results
show that the amplitudes of Bxx+Byy+Bzz in the North Pole and
South Pole regions are in the range of [-2.012×10-15 pT m-1 to +2.026 × 10-15 pT m-1] (1 Tesla = 103 mT = 109
nT = 1012 pT = 1018 aT), respectively. The relative error is
almost equal to the machine's accuracy. Therefore, this feature proves the
validity of our derived formulae. In addition, as shown in Figs. 1 and
2, it is obvious that the MGT and also the third-order partial derivatives
of the magnetic potential field enhance the lineation and contacts at the
satellite altitude. It also reveals some small-scale anomalies, which is
very helpful for further geological interpretation. A core field model
with spherical harmonic degrees/orders 1–15 is also used for
testing and the results, not shown here, indicate the correctness of the formulae
in the full range of the spherical harmonic degrees/orders, where the
computational stability of the Legendre function with ultrahigh-order is not
considered.
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 V, the Bz, Bzz and Bzzz components at the poles are
independent of the direction of the xP and yP axes;
while changing with the direction of the xP and yP
axes at the poles, the Bx, By, Bxz, Byz, Bxzz and
Byzz components have a period of 360∘ and the Bxx,
Bxy, Byy, Bxxz, Bxyz and Byyz components have a period
of 180∘. These variations can be accurately described by a sine
or cosine function relating to the horizontal rotation of the reference
frame and the differences among these magnetic effects are magnitude, period
and initial phase. Therefore, the Bx, By, Bxz, Byz, Bxx,
Bxy, Byy, Bxzz, ByzzBxxz, Bxyz and Byyz
components are not smooth at/across the poles. Moreover, to determine the
single value at the poles (Figs. 1, 2) we specially define that the
x axis points to the meridian of 180∘ E (or 180∘ W) at the North Pole and of 0∘ at the South Pole, that is, the LNORF
moving from Greenwich meridian to the poles.
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 1/sinθ and 1/sin2θ are
removed in the new formulae; consequently, the scale factor of e.g.,
10-280 (Holmes and Featherstone, 2002a, b) is not required when the
computing point approaches the poles, and the magnetic fields at the poles
can also be calculated in the defined reference frame. In fact, there are
differences between the results by our expressions and those by Horner's
recursive algorithm; for instance, if using the same model and the
parameters as those in Figs. 1 and 2, the differences of the three
components Bx, By and Bz are at a level of [-3×10-11 nT to +3×10-11 nT].