Poloidal and Toroidal Field Modeling in Terms of Locally ... - CiteSeerX

Poloidal and Toroidal Field Modeling in Terms of Locally Supported Vector Wavelets W. Freeden, C. Gerhards Abstract This paper deals with multiscale modeling of poloidal and toroidal fields such as geomagnetic field and currents. The wavelets are developed from scale-dependent regularizations of the Green function with respect to the Beltrami operator. They are constructed as to be locally compact, thus, allowing a locally reflected (zooming-in) reconstruction of the geomagnetic quantities. Finally, a reconstruction algorithm is indicated in form of a tree algorithm. Key Words Magnetic field, Mie representation, Green’s function, regularization, locally supported wavelets

1

Introduction

Geomagnetic satellites collect their data within the ionosphere and, therefore, within a source region of the geomagnetic field. In consequence, satellite data do not meet the prerequisites for the application of the standard Gauss representation. Usually they need to be carefully preselected prior to the modeling procedure (Langel and Hinze (1998) and Hulot et al. (2007) give a good overview of the various contributions of the magnetic field, their modeling and data selection). An alternative approach to resolve this problem is provided by the so-called Mie representation for solenoidal fields, i.e., by splitting the geomagnetic field into poloidal and toroidal parts (cf. Gerlich (1972), Backus (1986) and Backus et al. (1996)). In fact, for the magnetic field, the Mie representation can be regarded as a canonical generalization of the Gauss representation that is also valid within magnetic source regions, i.e., in regions where the electric current densities are no longer negligible. Since typical time scales and length scales in satellite magnetometry are such that the the system velocity is much smaller than the speed of light, it is justified to regard the quasi-static description of electrodynamics in form of the pre-Maxwell equations. Thus, the electric current densities allow a Mie representation, too, that depends on the Mie representation of the magnetic field. In other words, the ’direct source problem’ of determining the magnetic effects of a given current distribution as well as the ’inverse source problem’ of calculating current systems corresponding to a given magnetic field can both be handled by use of the Mie representation. This is of great significance, since modeling the ionospheric and magnetospheric current distributions and the resulting magnetic effects are more and more subject of recent research. Nevertheless, there remains the problem of how to numerically obtain – by means of suitable base functions – the Mie representation for a given set of vectorial data. The Mie approach has been applied intensively in terms of orthogonal (Fourier) expansions using spherical harmonics to the computation of magnetic field as well as to the inverse problem of determining the currents (e.g. Backus (1986), Engels and Olsen (1998) and Olsen (1997)). Those polynomial structures such as spherical harmonics, however, refer to a certain frequency of the geomagnetic quantity under consideration. But it should be pointed out that the spectrum of the geomagnetic field itself evolves over space in a significant way. Therefore, the frequencies themselves are spatially changing. This space-evolution of the frequencies is not (directly) reflected in the orthogonal (Fourier) coefficients in terms of the non-space localizing spherical harmonics, which was the reason why in Bayer et al. (2001), Maier (2005) and Mayer (2003), as a remedy,

University of Kaiserslautern, Geomathematics Group, 67663 Kaiserslautern, PO Box 3049, Germany e-mail: [email protected]

1

frequency packages were proposed to gain a certain amount of space localization (for an overview on spherical approximation and wavelet methods the reader is referred to Freeden et al. (1998) and the references therein; further recent wavelet methods with application to geomagnetics can, e.g., be found in Holschneider et al. (2003) and Chambodut et al. (2005)). The superposition of spherical harmonics to so-called scaling and wavelet (kernel) functions is, however, resulting in a reduction of frequency (more accurately, momentum) localization. More explicitly, the uncertainty principle (cf. Freeden (1998)) enables us to establish a quantitative classification in a canonically defined hierarchy of space/frequency localization of kernel functions, be they of scalar, vectorial, or tensorial nature. It tells us that space localization is in any case at the price of frequency localization. Seen from the numerical point of view, space-limited kernel functions (i.e., locally supported kernels) are more suited than band-limited kernels (i.e., finite frequency packages) because of their efficiency and economy, at least when local ’zooming-in’ approximation is reflected. Until now, however, locally supported wavelets that are both (i) physically consistent with the poloidal as well as toroidal character of the differential equations involving the operators ∇∗ , L∗ , ∇∗ ·, L∗ ·, and Δ∗ and (ii) simultaneously provide locally oriented ’zooming-in’ approximations are not available. To our knowledge, this is an essential gap in geomagnetic modeling. It is the objective of this paper to fill this gap. In more detail, locally supported wavelets are based on scale-dependent regularizations of a zonal kernel function suited for solving the differential equations involving the surface curl and the surface curl gradient on the sphere, viz. the Green function with respect to the Beltrami operator (cf. Freeden (1980)). As a matter of fact, locally supported poloidal-toroidal wavelet kernels are constructed in such a way that ’zooming-in’ modeling becomes applicable to both magnetic field and currents. Roughly speaking, if the regularized Green function gets close to a small scale parameter ρ > 0, the wavelet kernel becomes locally highly concentrated, while the space localization decreases for a growing parameter. In consequence, the filtering of a signal (magnetic field or current) by convolutions using wavelet kernels displays information in the signal at various levels of spatial resolution, namely information about coarse features at large scaling parameter, information about local features at small parameter. The layout of this paper is the following: Section 2 presents the constituting properties of the Green function with respect to the Beltrami operator. The differential equations involving the surface curl and the surface curl gradient are solved by use of the Green function. Section 3 deals with the scale-dependent regularization of the Green function. Central for the considerations is the comparison of potentials by means of the Green and the regularized Green function. Section 4 introduces locally supported wavelets for poloidal and toroidal currents, while Section 5 gives the wavelet concept for the poloidal and toroidal magnetic field. Finally, Section 6 explains the reconstruction scheme for multiscale modeling and discusses the advantages of locally supported kernels.

1.1

Spherical Parametrization of Magnetic Fields and Currents

In what follows the spherically oriented parametrization is recapitulated, i.e., the Mie scalars of the geomagnetic field and of the corresponding currents (cf. Backus (1986), Backus et al. (1996)) are formulated in spherical framework such that we can fall back to the theory of Green’s function with respect to the Beltrami operator on the (unit) sphere. Subsequently, regularizations of the Green function serve as the essential tools for introducing locally supported vector wavelets for multiscale approximation of poloidal and toroidal fields. To be more concrete, in order to treat electromagnetic fields in spherical shells, surface differential operators such as the surface gradient, ∇∗ , the surface curl gradient, L∗ , the surface divergence, ∇∗ ·, the surface curl, L∗ ·, and the surface Laplacian, Δ∗ , play an essential role. For example, the surface gradient and the surface curl gradient together with the radial (unit) vector behave much like three mutually orthogonal basis vectors such that any vector field f on a sphere Ωr with radius r > 0 around the origin can be represented in the form of the Helmholtz decomposition f (rξ) = ξF1 (rξ) + ∇∗ξ F2 (rξ) + L∗ξ F3 (rξ), 2

ξ∈Ω

(Ω = Ω1 is the unit sphere). In other words, any (sufficiently smooth) vector field f : Ωr → R3 admits an expression as linear combination of the system ξ, ∇∗ξ , L∗ξ with scalar components F1 , F2 , F3 : Ωr → R. The vector fields ξ → ξF1 (rξ), ξ → ∇∗ξ F2 (rξ), and ξ → L∗ξ F3 (rξ) are called radial, consoidal, and toroidal parts of f , respectively. F1 is uniquely determined by f , and so are F2 and F3 if they are subjected to the additional condition that their integral averages on Ωr vanish, i.e.,   1 1 F2 (rξ)dω(ξ) = F3 (rξ)dω(ξ) = 0 4π Ω 4π Ω (dω is the surface element). A vector field b (such as the geomagnetic field) is solenoidal in the spherical shell Ω(α,β) with inner radius α and outer radius α < β, if its divergence vanishes on Ω(α,β) and  1 b(rξ)dω(ξ) = 0 4π Ω for all r ∈ (α, β). From the Helmholtz representation on each Ωr , r ∈ (α, β), it can be easily deduced that there are scalar fields Pb and Qb such that the Mie representation pb qb

= (∇ ∧ L)Pb , = LQb

and b = pb + qb holds true in Ω(α,β) . The vector fields pb and qb are uniquely determined on Ω(α,β) by the (magnetic) field b. They are called the poloidal and toroidal parts of b. If subjected to the additional condition   1 1 Pb (rξ)dω(ξ) = Qb (rξ)dω(ξ) = 0, 4π Ω 4π Ω Pb and Qb are uniquely determined on Ω(α,β) by b. The fields Pb and Qb are called the principal poloidal and toroidal scalars for b (or briefly, Mie scalars of b). The Helmholtz representation of the Mie decomposition is given by b(rξ)

=

ξ

Δ∗ξ Pb (rξ) ∂r rPb (rξ) − ∇∗ξ + L∗ξ Qb (rξ) r r

(1.1)

for ξ ∈ Ω and r ∈ (α, β). The vectorial current density j (multiplied by the permeabilty μ0 of the vacuum) is the source field for the magnetic field b. The field j is solenoidal in Ω(α,β) . Consequently it also allows a Mie representation pj qj

= (∇ ∧ L)Pj , = LQj

with j = pj + qj and 1 4π



1 Pj (rξ)dω(ξ) = 4π Ω

 Ω

Qj (rξ)dω(ξ) = 0.

In connection with the Helmholtz decomposition into tangential and radial components we find j(rξ)

= ξ

Δ∗ξ Pj (rξ) ∂r rPj (rξ) − ∇∗ξ + L∗ξ Qj (rξ), r r 3

ξ ∈ Ω, r ∈ (α, β).

(1.2)

Due to the pre-Maxwell equations, the principal poloidal and toroidal scalars of b and j are coupled by μ0 Pj

=

Qb ,

(1.3)

−μ0 Qj

=

ΔPb .

(1.4)

The identities (1.3) and (1.4) are the usual point of departure in the geomagnetic literature to model ionospheric currents and their resulting magnetic fields, and in connection with 1.1, 1.1 deliver as well the crucial connection between the magnetic field and the corresponding currents in our considerations.

2

Green Function with Respect to the Beltrami Operator

Of basic importance for our considerations is the Green function with respect to the Beltrami operator on the unit sphere. Our point of departure is the list of its constituting properties. G(Δ∗ ; ·, ·) : (ξ, η) → G(Δ∗ ; ξ, η), −1 ≤ ξ · η < 1 (ξ · η denoting the scalar product of ξ and η), is called Green’s function on Ω with respect to the operator Δ∗ , if it satisfies the following properties: (i) (differential equation) for every point ξ ∈ Ω, η → G(Δ∗ ; ξ, η) is twice continuously differentiable on the set {η ∈ Ω| − 1 ≤ ξ · η < 1}, such that Δ∗η G(Δ∗ ; ξ, η) = −

1 , 4π

−1 ≤ ξ · η < 1,

(ii) (characteristic singularity) for every ξ ∈ Ω, the function η → G(Δ∗ ; ξ, η) −

1 ln(1 − ξ · η) 4π

is continuously differentiable on Ω, (iii) (rotational symmetry) for all orthogonal transformations t G(Δ∗ ; tξ, tη) = G(Δ∗ ; ξ, η), (iv) (normalisation) for every ξ ∈ Ω, 1 4π

 Ω

G(Δ∗ ; ξ, η) dω(η) = 0.

G(Δ∗ ; ·, ·) is uniquely determined by these properties. An easy calculation shows that, for ξ, η ∈ Ω with −1 ≤ ξ · η < 1, we have G(Δ∗ ; ξ, η) =

1 1 ln(1 − ξ · η) + (1 − ln(2)). 4π 4π

Throughout this paper we usually write G(Δ∗ ; ξ · η) instead of G(Δ∗ ; ξ, η) to point out the zonal nature (rotational symmetry) of the Green function. This indicates that G(Δ∗ ; ξ · η) only depends on the scalar product of ξ and η, i.e., G(Δ∗ , ·) may be understood as a function on the interval [−1, 1). More details on Green’s function can be found in Freeden and Schreiner (2009). After the definition of Green’s function we next come to integral formulas on the unit sphere under explicit formulation of the remainder term in integral form between functional value and integral.

4

Theorem 2.1. (Green Surface Theorem for ∇∗ ) Let ξ be a fixed point of the unit sphere Ω. Suppose that F is a continuously differentiable function on Ω. Then    ∗    1 ∇η G(Δ∗ ; ξ · η) · ∇∗η F (η) dω(η). F (ξ) = F (η) dω(η) − 4π Ω Ω Proof. Let ξ be a fixed point of the unit sphere Ω. Then, for each sufficiently small ρ > 0, Green’s first surface identity gives      F (η)Δ∗η G(Δ∗ ; ξ · η) + ∇∗η F (η) · ∇∗η G(Δ∗ ; ξ · η) dω(η) (2.1) 1−ξ·η≥ρ,|η|=1

 F (η)

=

1−ξ·η=ρ,|η|=1

∂ G(Δ∗ ; ξ · η) dσ(η), ∂ν(η)

(2.2)

where ν is the unit normal to the circle consisting of all points η ∈ Ω with ξ · η = 1 − ρ, tangential to Ω, and directed exterior to the set of all points η ∈ Ω with ξ · η ≤ 1 − ρ. Explicitly, written out 1 we have νη = −(1 − (ξ · η)2 )− 2 η ∧ (η ∧ ξ). In the identity (2.2) we first observe the differential equation of Green’s function   1 F (η)Δ∗η G(Δ∗ ; ξ · η) dω(η) = − F (η) dω(η). (2.3) 4π 1−ξ·η≥ρ,|η|=1 1−ξ·η≥ρ,|η|=1 By virtue of the logarithmic singularity of the Green function G(Δ∗ ; ·), we get in analogy to well known results of potential theory  F (η)∂νη G(Δ∗ ; ξ · η)dσ(η) 1−ξ·η=ρ,|η|=1

  ξ − (1 − ρ)η 1  (ξ − (1 − ρ)η) dσ(η) F (η) · − 4πρ 1 − (1 − ρ)2 1−ξ·η=ρ,|η|=1   1 − (1 − ρ)2 1 − dσ(η). F (η) 4π 1−ξ·η=ρ,|η|=1 ρ 

= =

From the Mean Value Theorem we are able to deduce that   1 − (1 − ρ)2 1 F (η) dσ(η) − 4π ρ 1−ξ·η=ρ,|η|=1  1 − (1 − ρ)2  1 2π 1 − (1 − ρ)2 F (ηρ ) = − 4π ρ 1 = − (2 − ρ)F (ηρ ) 2 for some ηρ lying on the circle {η ∈ Ω|1 − ξ · η = ρ}. The continuity of F yields F (ηρ ) → F (ξ) as ηρ → ξ for ρ → 0 such that  F (η)∂νη G(Δ∗ ; ξ · η)dσ(η) = −F (ξ). (2.4) lim ρ→0

1−ξ·η=ρ,|η|=1

The statement of the Theorem then follows from (2.2), (2.3), (2.4) and taking the limit ρ → 0. 2 Corollary 2.2. (Green Surface Theorem for L∗ ) Under the assumptions of Theorem 2.1   1 F (ξ) = F (η) dω(η) − (L∗η G(Δ∗ ; ξ · η)) · (L∗η F (η)) dω(η). 4π Ω Ω 5

The above corollary is a direct consequence of Theorem 2.1 for the surface gradient. To achieve a similar integral formula for the Beltrami operator, we start from Green’s second identity and get for sufficiently small ρ > 0,  G(Δ∗ ; ξ · η)Δ∗η F (η) − F (η)Δ∗η G(Δ∗ ; ξ · η) dω(η) ξ·η≤1−ρ,|η|=1  = G(Δ∗ ; ξ · η)∂νη F (η) − F (η)∂νη G(Δ∗ ; ξ · η)dσ(η), ξ·η=1−ρ,|η|=1

provided that F is twice continuously differentiable on Ω. Observing the defining properties of the Green function with respect to Δ∗ we can use the same arguments as known from potential theory. In fact, the continuous differentiability of F on Ω leads us to  lim G(Δ∗ ; ξ · η)∂νη F (η)dσ(η) = 0. ρ→0

ξ·η=1−ρ,|η|=1

Together with (2.4) this shows us the following result. Theorem 2.3. (Green Surface Theorem for Δ∗ ) Let ξ be a fixed point of the unit sphere Ω. Suppose that F is a twice continuously differentiable function on Ω. Then   1 F (ξ) = F (η) dω(η) + G(Δ∗ ; ξ · η)Δ∗η F (η) dω(η). 4π Ω Ω In other words, the Green theorems as stated above compare the value of a function at a point ξ ∈ Ω with the integral mean of F relative to the unit sphere Ω under explicit representation of the error term in integral form. Essential tool is the Green function with respect to the Beltrami operator Δ∗ . Furthermore, the stated integral formulas allow explicit representations of the solutions to differential equations involving the operators ∇∗ , L∗ and Δ∗ . This is summarized in the next theorems. Theorem 2.4. (Differential Equation for ∇∗ ) Let f : Ω → R3 be a continuously differentiable vector field on Ω with ξ · f (ξ) = 0, L∗ξ · f (ξ) = 0, ξ ∈ Ω. Then  F (ξ) = − (∇∗η G(Δ∗ ; ξ · η)) · f (η) dω(η) Ω  1 1 (ξ − (ξ · η)η) · f (η) dω(η) = 4π Ω 1 − ξ · η is the uniquely determined solution of the differential equation ∇∗ξ F (ξ) = f (ξ), satisfying 1 4π

ξ ∈ Ω,

 Ω

F (η) dω(η) = 0.

Theorem 2.5. (Differential Equation for L∗ ) Let f : Ω → R3 be a continuously differentiable vector field on Ω with ξ · f (ξ) = 0, ∇∗ξ · f (ξ) = 0, ξ ∈ Ω. Then  F (ξ) = − (L∗η G(Δ∗ ; ξ · η)) · f (η) dω(η) Ω  1 1 (η ∧ ξ) · f (η) dω(η) = 4π Ω 1 − ξ · η is the uniquely determined solution of the differential equation L∗ξ F (ξ) = f (ξ), satisfying 1 4π

ξ ∈ Ω,

 Ω

F (ξ)dω(η) = 0.

6

Theorem 2.6. (Differential Equations for Δ∗ ) Let H be a continuous function on Ω with van 1 ishing integral mean value, i.e., 4π Ω H(ξ) dω(ξ) = 0. Then  F (ξ) = G(Δ∗ ; ξ · η)H(η) dω(η), ξ ∈ Ω. Ω

is the uniquely determined solution of equation Δ∗ξ F (ξ) = H(ξ), satisfying 1 4π

3

ξ ∈ Ω,

 Ω

F (ξ) dω(ξ) = 0.

Regularized Green Function with Respect to the Beltrami Operator

Regularizing the Green function G(Δ∗ ; ·), that shows its characteristic (logarithmic) singularity for ξ · η = 1, enables us later on to introduce wavelet kernels with locally compact supports. To this end, we define what we mean by a regularized Green function: For ρ > 0 and all ξ, η ∈ Ω we let Gρ (Δ∗ ; ξ · η) =

1 4π ln(1 ρ

− ξ · η) +

1 4π (1

− ln(2)), 1 − ξ · η > ρ

R (ξ · η) ,

1 − ξ · η ≤ ρ,

where Rρ : t → Rρ (t), t ∈ [−1, 1], is a scalar function such that Gρ (Δ∗ , ·) : t → Gρ (Δ∗ , t), t ∈ [−1, 1], is sufficiently smooth (at least continuously differentiable on the interval [−1, 1]), and additionally satisfying

 1

 k

k d



ρ R (t) dt = 0, k = 0, 1. lim ρ 2 (3.1)

ρ→0+



dt 1−ρ This last condition is satisfied for any polynomial regularizing function Rρ . Next we turn to limit relations between the Green function with respect to Δ∗ and its corresponding regularization, which are crucial for the definition of locally supported wavelets. Lemma 3.1. Suppose that Gρ (Δ∗ , ·) is continuously differentiable. Then, for every F ∈ C (0) (Ω), we have







(3.2) lim sup

Gρ (Δ∗ ; ξ · η)F (η)dω(η) − G(Δ∗ ; ξ · η)F (η)dω(η)

= 0, ρ↓0 ξ∈Ω Ω Ω







(3.3) lim sup

∇∗ξ Gρ (Δ∗ ; ξ · η)F (η)dω(η) − ∇∗ξ G(Δ∗ ; ξ · η)F (η)dω(η)

= 0, ρ↓0 ξ∈Ω Ω Ω







∗ ρ ∗ ∗ ∗

lim sup Lξ G (Δ ; ξ · η)F (η)dω(η) − Lξ G(Δ ; ξ · η)F (η)dω(η)

= 0. (3.4) ρ↓0 ξ∈Ω

Ω

Ω

Proof. We verify the limit relation only for the surface gradient. The relation for the surface curl gradient is then a direct consequence and the case (3.2) can be shown in a completely analogous way.

7

For ξ ∈ Ω we have

 



∗ ρ ∗ ∗ ∗

∇ξ G (Δ ; ξ · η)F (η)dω(η) − ∇ξ G(Δ ; ξ · η)F (η)dω(η)



Ω Ω





 ∗ ρ ∗  ∗ ∗

=

∇ξ G (Δ ; ξ · η) − ∇ξ G(Δ ; ξ · η) F (η)dω(η)

1−ξ·η≤ρ,|η|=1 

∗ ρ ∗

∇ξ G (Δ ; ξ · η) dω(η) ≤ sup |F (η)| η∈Ω 1−ξ·η≤ρ,|η|=1







1

+ sup |F (η)| (η − (ξ · η)ξ)

dω(η)

η∈Ω 1−ξ·η≤ρ,|η|=1 4π(1 − ξ · η)





d

 1



ρ R (t) = sup |F (η)|

1 − (ξ · η)2 2 dω(η) η∈Ω 1−ξ·η≤ρ,|η|=1 dt t=ξ·η





1

1



2 + sup |F (η)| 1 (1 + ξ · η) dω(η).

η∈Ω 1−ξ·η≤ρ,|η|=1 4π(1 − ξ · η) 2 Both integrals exist and vanish as ρ tends to zero, observing the limit property (3.1) of Rρ . Due to the zonality of the integrands this convergence is uniform as to ξ ∈ Ω. Furthermore, the above estimates yield the identity   ∗ ∗ ∗ ∇ξ G(Δ ; ξ · η)F (η)dω(η) = ∇ξ G(Δ∗ ; ξ · η)F (η)dω(η), Ω

Ω

which proves (3.3). Imposing further (smoothness) assumptions on the regularized Green function an analogous limit relation also holds true for the Beltrami operator. Lemma 3.2. Let Gρ (Δ∗ , ·) be twice continuously differentiable. Then for F ∈ C (1) (Ω),







G(Δ∗ ; ξ · η)F (η)dω(η)

= 0. lim sup

Δ∗ξ Gρ (Δ∗ ; ξ · η)F (η)dω(η) − Δ∗ξ ρ↓0 ξ∈Ω Ω

(3.5)

Ω

Proof. Since F is continuously differentiable, we get from Gauss’ theorem on the sphere,   Δ∗ξ G(Δ∗ ; ξ · η)F (η) dω(η) = L∗ξ · L∗ξ G(Δ∗ ; ξ · η)F (η) dω(η) Ω

Ω

=

3

L∗ξ

·ε

 i

i=1

=

3

L∗ξ · εi

i=1

 =

Ω

Ω

 Ω

L∗ξ G(Δ∗ ; ξ · η) · (εi F (η)) dω(η) G(Δ∗ ; ξ · η)L∗η · (εi F (η)) dω(η)

L∗ξ G(Δ∗ ; ξ · η) ·

3

εi L∗η · (εi F (η)) dω(η).

i=1

With this reformulation we get the desired statement of (3.5) in a very analogous way as in the previous Lemma 2

4

Reconstructing Currents by Regularization

In this section we assume a magnetic field b = pb +qb to be given on the shell Ω(α,β) with a poloidal part pb and a toroidal part qb . Then it is possible to approximate the correspondig currents j through an expansion by means of locally compact wavelets. Throughout this section we assume Gρ (Δ∗ , ·) to be three times continuously differentiable. 8

4.1

Regularized Poloidal Currents

From qb (rξ) = L∗ξ Qb (rξ) we get with Theorem 2.5  Qb (rξ) = − (L∗η G(Δ∗ ; ξ · η)) · b(rη)dω(η), Ω

ξ ∈ Ω, r ∈ (α, β).

Observing the Equations (1.3) and (1.2), the poloidal current reads as follows pj (rξ)

= ξ

Δ∗ξ Qb (rξ) ∂r rQb (rξ) − ∇∗ξ , μ0 r μ0 r

ξ ∈ Ω, r ∈ (α, β),

and substituting the Green’s function with respect to the Beltrami operator by its regularized version, this leads us for ρ > 0 and ξ ∈ Ω, r ∈ (α, β), to pρj (rξ)

=

=

Z Z ´ ` 1 1 ξ (Δ∗ξ L∗η Gρ (Δ∗ ; ξ · η)) · b(rη)dω(η) + ∇∗ (L∗η Gρ (Δ∗ ; ξ · η)) · b(rη) dω(η) μ0 r Ω μ0 r Ω ξ Z ´ ` 1 + ∇∗ξ (L∗η Gρ (Δ∗ ; ξ · η)) · ∂r b(rη) dω(η) (4.1) μ0 Ω Z Z “ ” ` ∗ ∗ ´T ρ ∗ 1 1 ξ (Δ∗ L∗ Gρ (Δ∗ ; ξ · η)) · b(rη)dω(η) + ∇ξ Lη G (Δ ; ξ · η) b(rη)dω(η) − μ0 r Ω ξ η μ0 r Ω Z “ ” ` ∗ ∗ ´T ρ ∗ 1 ∇ξ Lη G (Δ ; ξ · η) ∂r b(rη)dω(η). + μ0 Ω −

By use of Gauss’ theorem and the limit relations from section 3, the following lemma holds true. Lemma 4.1. For a magnetic field b ∈ c(2) (Ω(α,β) ) and a fixed r ∈ (α, β) we have



lim sup pρj (rξ) − pj (rξ) = 0. ρ↓0 ξ∈Ω

4.2

Regularized Toroidal Currents Δ∗ Pb (rξ)

Having a poloidal magnetic field pb (rξ) = ξ ξ r − ∇∗ξ ∂r rPrb (rξ) , we get Δ∗ξ Pb (rξ) = rξ · pb (rξ) = rξ · b(rξ), and thus, with Theorem 2.6,  Pb (rξ) = r G(Δ∗ ; ξ · η)η · b(rη)dω(η), ξ ∈ Ω, r ∈ (α, β). Ω

Applying (1.4) and (1.2), the toroidal current reads as qj (rξ)

=

−

1 ∗ L Δrξ Pb (rξ), μ0 ξ

ξ ∈ Ω, r ∈ (α, β).

Regarding Δrξ = ∂r2 + 2r ∂r + r12 Δ∗ξ , we define for ξ ∈ Ω, r ∈ (α, β) and ρ > 0 the regularization   1 2 (L∗ξ Δ∗ξ Gρ (Δ∗ ; ξ · η))η · b(rη)dω(η) − (L∗ Gρ (Δ∗ ; ξ · η))η · b(rη)dω(η) qjρ (rξ) = − μ0 r Ω μ0 r Ω ξ    1 (4.2) − (L∗ξ Gρ (Δ∗ ; ξ · η))η · r∂r2 + 4∂r b(rη)dω(η), μ0 Ω which, by section 3, converges again to the toroidal current, so that we end up with the following lemma. Lemma 4.2. For a magnetic field b ∈ c(2) (Ω(α,β) ) and a fixed r ∈ (α, β) we have



lim sup qjρ (rξ) − qj (rξ) = 0. ρ→0 ξ∈Ω

9

4.3

Locally Compact Wavelets for Currents

For ξ, η ∈ Ω and ρ > 0 we choose, for example, the auxiliary regularization function as follows   1 3 3 1 3 2 Rρ (ξ · η) = 12πρ ln(ρ) − 56 − ln(2) . 3 (1 − ξ · η) − 8πρ2 (1 − ξ · η) + 4πρ (1 − ξ · η) + 4π The associated regularized Green function then fulfills the smoothness demanded at the beginning of this section. For brevity, we set for ξ, η ∈ Ω with ξ = (ξ1 , ξ2 , ξ3 )T , η = (η1 , η2 , η3 )T `

M(ξ, η) = ∇∗ξ (ξ ∧ η)

0

−ξ1 ξ2 η3 + ξ3 ξ1 η2 = @ η3 − ξ22 η3 + ξ3 ξ2 η2 −ξ3 ξ2 η3 − ξ12 η2 − ξ22 η2

´T

1 η2 − ξ12 η2 + ξ2 ξ1 η1 −η1 − ξ1 ξ2 η2 + ξ22 η1 A . −ξ3 ξ1 η2 + ξ3 ξ2 η1

−η3 + ξ12 η3 − ξ3 ξ1 η1 ξ1 ξ2 η3 − ξ3 ξ2 η1 ξ22 η1 + ξ12 η1 + ξ3 ξ1 η3

Now, let 0 < ρ +1 < ρ , ∈ Z, be a sequence of scales with lim →∞ ρ = 0. A family of tensorial scaling kernels S and vectorial scaling kernels s ,1 , s ,2 for the current determination can then be introduced by S j (ξ, η)

 ∗ ∗ T ρ ∇ξ Lη G  (Δ∗ ; ξ · η) ⎧ 1 1 1 − ξ · η > ρ , ⎪ ⎪ 4π(1−ξ·η) M(ξ, η) + 4π(1−ξ·η)2 (η − (ξ · η)ξ) ⊗ (ξ ∧ η), ⎪ ⎨   1 3 3 2 M(ξ, η) = 3 (1 − ξ · η) − 4πρ2 (1 − ξ · η) + 4πρ 4πρ   ⎪    , 1 − ξ · η ≤ ρ , ⎪ ⎪ 1 3 ⎩ + − 3 (1 − ξ · η) + (η − (ξ · η)ξ) ⊗ (ξ ∧ η) 2πρ 4πρ2

=



s ,1 j (ξ, η)



= L∗η Δ∗ξ Gρ (Δ∗ ; ξ · η) ⎧ ⎨ 0,   = ⎩ − 3 3 (ξ · η)2 + 3 3 − πρ πρ 

s ,2 j (ξ, η)



= L∗ξ Gρ (Δ∗ ; ξ · η) ⎧ 1 (ξ ∧ η), ⎨ − 4π(1−ξ·η)  = ⎩ − 1 3 (1 − ξ · η)2 + 4πρ 

9 2πρ2

3 (1 4πρ2



(ξ · η) +



− ξ · η) −

3 2πρ2

3 4πρ



−

3 2πρ



1 − ξ · η > ρ , (ξ ∧ η),

1 − ξ · η ≤ ρ ,

1 − ξ · η > ρ , (ξ ∧ η),

1 − ξ · η ≤ ρ .

The corresponding (difference) wavelet kernels are given by Wj (ξ, η)

=

=

S+1 (ξ, η) − Sj (ξ, η) j 8 0, > > > > 1 − ξ · η > ρ > > > > > “ ” > > 1 1 > − ρ13 (1 − ξ · η)2 + ρ32 (1 − ξ · η) − ρ3 M(ξ, η) > 4π >  >   “1−ξ·η ” > > 1 1 1 3 > > < + π 4(1−ξ·η)2 + 2ρ3 (1 − ξ · η) − 4ρ2 (η − (ξ · η)ξ) ⊗ (ξ ∧ η), 

> > > > > > > > > > > > > > > > > > > :

1 4π

„“

− π1

1 ρ3

„“ +1



ρ+1 < 1 − ξ · η ≤ ρ , ” ” “ ”« “ 3 M(ξ, η) − ρ13 (1 − ξ · η)2 − ρ23 − ρ32 (1 − ξ · η) + ρ+1 − ρ3  +1  ” “ ”« − 2ρ13 (1 − ξ · η) − 4ρ23 − 4ρ32 (η − (ξ · η)ξ) ⊗ (ξ ∧ η),

1 2ρ3 +1



+1

10



1 − ξ · η ≤ ρ+1 ,

wj,1 (ξ, η)

=

=

wj,2 (ξ, η)

=

=

s+1,1 (ξ, η) − s,1 j j (ξ, η) 8 0, > > > > > > > “ “ > > 2 3 3 1 > > − 3 (ξ · η) − > π ρ ρ3 >   > < > > > > > > > > > > > > > > :

− π1 + π1

„“ “

3 2ρ2 +1

s+1,2 (ξ, η) j

8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > :

3 ρ3 +1

−

−

−

3 ρ3 

”

9 2ρ2 

” “ (ξ · η) − 2ρ32 −

(ξ · η)2 −

3 2ρ+1

−

3 2ρ2 

+



“

3 ρ3 +1

3 2ρ

”

−

9 2ρ2 +1

3 2ρ

−

3 ρ3 

””

+

1 − ξ · η > ρ (ξ ∧ η),

9 2ρ2 

”

ρ+1 < 1 − ξ · η ≤ ρ , « (ξ · η) (ξ ∧ η)

(ξ ∧ η), 1 − ξ · η ≤ ρ+1 ,

s,2 j (ξ, η)

0, “

1 − 4π

1 − 4π 1 − 4π

1 1−ξ·η

„“ “

1 ρ3 +1

3

ρ+1

−

−

− 3 ρ

1 (1 ρ3 

2

3 (1 ρ2 

− ξ · η) +

”

(1 − ξ · η)2 − ” (ξ ∧ η),

1 ρ3 

“

− ξ · η) −

3 ρ2 +1

3 ρ

”

1 − ξ · η > ρ (ξ ∧ η), ρ+1 < 1 − ξ · η ≤ ρ ,

« ” − ρ32 (1 − ξ · η) (ξ ∧ η) 

1 − ξ · η ≤ ρ+1 .

It is obvious that the wavelet kernels Wj (ξ, ·), wj ,1 (ξ, ·) and wj ,2 (ξ, ·) have locally compact support which, for growing , is localizing better and better around a fixed point ξ ∈ Ω (cf. Figure 1). Thus we get the announced ’zooming-in’ capability, offering the possibility of handling certain regions at a higher resolution. Let K denote one of the tensorial kernels from above and k one of the vectorial ones. Then we define for sufficiently smooth functions F : Ω → R, f : Ω → R3 the following convolutions,  (k ∗ f )(rξ) = k(ξ, η) · f (rη)dω(η), Ω (k  F )(rξ) = k(ξ, η)F (rη)dω(η), Ω (K  f )(rξ) = K(ξ, η)f (rη)dω(η), Ω

for ξ ∈ Ω and r ∈ (α, β). Keeping this setting in mind, we can rewrite (4.1) as pρj L (rξ)

 1  0,1  1  0 ξ sj ∗ b (rξ) + S  (b + r∂r b) (rξ) μ0 r μ0 r j L−1 L−1  1  ,1  1 

ξ − wj ∗ b (rξ) + Wj  (b + r∂r b) (rξ). μ0 r μ0 r

= −

=0

=0

Hence, with Brad (rη) = η · b(rη), equation (4.2) turns into qjρL (rξ)

=

  1  0,2  1  0,1 sj  Brad (rξ) − sj  2Brad + 4r∂r Brad + r2 ∂r2 Brad (rξ) μ0 r μ0 r L−1 L−1   1  ,2  1  ,1 wj  Brad (rξ) − wj  2Brad + 4r∂r Brad + r2 ∂r2 Brad (rξ). + μ0 r μ0 r

=0

=0

From Lemma 4.1 and 4.2 we then get the following theorem as immediate consequence. 11

|s0,2 j (ξ, ·)| with ρ0 = 1

|wj ,2 (ξ, ·)| at scales = 1 (left) and = 2 (right) with ρ = 2−

|wj ,2 (ξ, ·)| at scales = 4 (left) and = 6 (right) with ρ = 2−

,2 Figure 1: Absolute values of s0,2 j (ξ, ·) and wj (ξ, ·) at different scales

12

Theorem 4.3. Let b ∈ c(2) (Ω), then then the poloidal and toroidal currents read for ξ ∈ Ω and r ∈ (α, β) as,  1  0,1  1  0 pj (rξ) = − ξ sj ∗ b (rξ) + Sj  (b + r∂r b) (rξ) μ0 r μ0 r ∞  ∞ 

 1 

1 wj ,1 ∗ b (rξ) + ξ − Wj  (b + r∂r b) (rξ), μ0 r μ0 r

=0

=0

and qj (rξ)

=

  1  0,2  1  0,1 sj  Brad (rξ) − sj  2Brad + 4r∂r Brad + r2 ∂r2 Brad (rξ) μ0 r μ0 r ∞ ∞    1 ,1 1  ,2  + wj  Brad (rξ) − wj  2Brad + 4r∂r Brad + r2 ∂r2 Brad (rξ). μ0 r μ0 r

=0

5

=0

Reconstructing the Magnetic Field by Regularization

In this section we assume a current j = pj + qj to be given on the shell Ω(α,β) with a poloidal part pj and a toroidal part qj . Similar as done before for the currents, the corresponding magnetic field can be expanded in locally supported wavelets. Throughout this section we need Gρ (Δ∗ , ·) to be only continuously differentiable.

5.1

Regularized Toroidal Magnetic Fields

Given a poloidal current pj (rξ) = ξ ξ

Δ∗ ξ Qb (rξ) μ0 r

−

b (rξ) ∇∗ξ ∂r rQ μ0 r

Δ∗ ξ Pj (rξ) r

− ∇∗ξ

∂r rPj (rξ) , r

we have with (1.3) that pj (rξ) =

and, therefore, get Δ∗ξ Qb (rξ) = μ0 rξ · pj (rξ) = μ0 rξ · j(rξ).

Hence, we have with Theorem 2.6 that  Qb (rξ) = μ0 r G(Δ∗ ; ξ · η)η · j(rη)dω(η), Ω

ξ ∈ Ω, r ∈ (α, β).

The toroidal magnetic field then reads as qb (rξ) = L∗ξ Qb (rξ), and by means of regularization we set for ρ > 0 and ξ ∈ Ω, r ∈ (α, β)  ρ (5.1) qb (rξ) = μ0 r (L∗ξ Gρ (Δ∗ ; ξ · η))η · j(rη)dω(η). Ω

Limit relation (3.4) again yields that the regularized field converges towards the toroidal magnetic. Lemma 5.1. For a current j ∈ c(0) (Ω(α,β) ) and a fixed r ∈ (α, β) we have lim sup |qbρ (rξ) − qb (rξ)| = 0.

ρ→0 ξ∈Ω

5.2

Regularized Poloidal Magnetic Fields

Because qj (rξ) = L∗ξ Qj (rξ), we have with (1.4) that μ0 qj (rξ) = −L∗ξ Δrξ Pb (rξ) and get from Theorem 2.5  Δrξ Pb (rξ) = μ0 (L∗η G(Δ∗ ; ξ · η)) · j(rη)dω(η), ξ ∈ Ω, r ∈ (α, β). Ω

13

To be able to solve this equation via the developed Green function approach, we have to assume the radial dependence of the magnetic field to be negligible, i.e.,   2 1 1 Δrξ Pb (rξ) = ∂r2 + ∂r + 2 Δ∗ξ Pb (rξ) = 2 Δ∗ξ Pb (rξ) r r r for ξ ∈ Ω and r ∈ (α, β). Under this restriction it follows from the above and Theorem 2.6 that   Pb (rξ) = μ0 r2 G(Δ∗ ; ξ · η) (L∗ζ G(Δ∗ ; η · ζ)) · j(rζ)dω(ζ)dω(η), ξ ∈ Ω, r ∈ (α, β), Ω

Ω

and from (1.1) and the negligible radial dependence we get the poloidal magnetic field pb (rξ) = ξ

Δ∗ξ Pb (rξ) Pb (rξ) − ∇∗ξ , r r

ξ ∈ Ω, r ∈ (α, β).

In terms of regularization we set for ξ ∈ Ω, r ∈ (α, β) and ρ > 0  ρ pb (rξ) = μ0 rξ (L∗η Gρ (Δ∗ ; ξ · η)) · j(rη)dω(η) Ω  −μ0 r ∇∗ξ Gρ (Δ∗ ; ξ · η) (L∗ζ Gρ (Δ∗ ; η · ζ)) · j(rζ)dω(ζ)dω(η). Ω

(5.2)

Ω

Lemma 5.2. For a current j ∈ c(0) (Ω) and a fixed r ∈ (α, β) we have lim sup |pρb (rξ) − pb (rξ)| = 0.

ρ→0 ξ∈Ω

Proof. The convergence of the first summand in (5.2) is clear from (3.4). For the second summand we have ˛ ˛Z Z ˛ ˛ ˛ (∇∗ξ Gρ (Δ∗ ; ξ · η)) (L∗ζ Gρ (Δ∗ ; η · ζ)) · j(rζ)dω(ζ)dω(η) − pb (rξ)˛ ˛ ˛ Ω Ω ˛Z „ ˛ «Z ˛ ˛ ≤ ˛˛ (L∗ζ G(Δ∗ ; η · ζ)) · j(rζ)dω(ζ)dω(η)˛˛ (∇∗ξ Gρ (Δ∗ ; ξ · η)) − (∇∗ξ G(Δ∗ ; ξ · η)) Ω ˛ΩZ ˛ Z ˛ ˛ ∗ ρ ∗ ∗ ρ ∗ ∗ ∗ ˛ +˛ (∇ξ G (Δ ; ξ · η)) (Lζ G (Δ ; η · ζ)) · j(rζ) − (Lζ G(Δ ; η · ζ)) · j(rζ)dω(ζ)dω(η)˛˛ Ω ˛Z Ω ˛ ˛ ˛Z ˛ ˛ ˛ ˛ ∗ ρ ∗ ≤ ˛˛ (∇ξ G (Δ ; ξ · η)) − (∇∗ξ G(Δ∗ ; ξ · η))dω(η)˛˛ sup ˛˛ (L∗ζ G(Δ∗ ; η · ζ)) · j(rζ)dω(ζ)˛˛ η∈Ω Ω Ω ˛Z ˛ ˛ ˛Z ˛ ˛ ˛ ˛ ∗ ρ ∗ ∗ ρ ∗ ∗ ∗ ˛ ˛ ˛ +˛ (∇ξ G (Δ ; ξ · η))dω(η)˛ sup ˛ (Lζ G (Δ ; η · ζ)) · j(rζ) − (Lζ G(Δ ; η · ζ)) · j(rζ)dω(ζ)˛˛ η∈Ω

Ω

Ω



The lemma then follows from the uniform boundedness of Ω (L∗ζ G(Δ∗ ; η · ζ)) · qj (rζ)dω(ζ) as to



η ∈ Ω, the uniform boundedness of Ω (∇∗ξ Gρ (Δ∗ ; ξ · η))dω(η) as to ρ > 0 and ξ ∈ Ω, and the properties (3.3) and (3.4).

5.3

Locally Compact Wavelets for the Magnetic Field

Since we only require G(Δ∗ ; ·) to be continuously differentiable, we can choose a linear regularization Rρ (ξ · η) =

1 1−ξ·η + (ln(ρ) − ln(2)), 4πρ 4π

ξ, η ∈ Ω, ρ > 0.

Unfortunately, it seems that the tangential part of the poloidal magnetic field cannot be reconstructed in a multiscale framework with locally compact wavelets that offers a ’zooming-in’ 14

capability. Trying to achieve a localization around a fixed ξ ∈ Ω, one would still need data on the whole sphere to calculate the inner integral in (5.2). Nonetheless the reconstructions of the toroidal field and the radial part brad of the (poloidal) magnetic field have the desired properties. For scales 0 < ρ < ρ +1 , ∈ Z, with lim →∞ ρ = 0 we define the vectorial scaling kernels s ,1 b (ξ, η)

= L∗ξ Gρ (Δ∗ ; ξ · η) 1 (ξ ∧ η), 1 − ξ · η > ρ , − 4π(1−ξ·η) = 1 − 4πρ (ξ ∧ η), 1 − ξ · η ≤ ρ ,

s ,2 b (ξ, η)

= ∇∗ξ Gρ (Δ∗ ; ξ · η) 1 − 4π(1−ξ·η) (η − (ξ · η)ξ), = 1 − 4πρ (η − (ξ · η)ξ)

1 − ξ · η > ρ , 1 − ξ · η ≤ ρ .

The corresponding locally supported wavelets are again defined as the differences wb ,1 (ξ, η)

=

=

wb ,2 (ξ, η)

=

=

s +1,1 (ξ, η) − s ,1 b (ξ, η) ⎧b 0, 1 − ξ · η > ρ , ⎪ ⎪ ⎪  ⎨  1 1 − 4π(1−ξ·η) (ξ ∧ η), ρ +1 < 1 − ξ · η ≤ ρ , − 4πρ  ⎪   ⎪ ⎪ 1 1 ⎩ − 1 − ξ · η ≤ ρ +1 4πρ+1 − 4πρ (ξ ∧ η), s +1,1 (ξ, η) − s ,1 b (ξ, η) ⎧b 0, ⎪ ⎪ ⎪  ⎨  1 1 − 4π(1−ξ·η) (η − (ξ · η)ξ), − 4πρ  ⎪   ⎪ ⎪ 1 1 ⎩ − 4πρ+1 − 4πρ (η − (ξ · η)ξ),

1 − ξ · η > ρ , ρ +1 < 1 − ξ · η ≤ ρ , 1 − ξ · η ≤ ρ +1 .

With Jrad (rη) = η · j(rη) we can rewrite (5.1) as qbρL (rξ) = μ0 r(s0,1 b  Jrad )(rξ) + μ0 r

L−1

(wb ,1  Jrad )(rξ),

=0

and the radial part of (5.2) turns into L bρrad (rξ) = −μ0 rξ(s0,1 b ∗ j)(rξ) − μ0 r

L−1

ξ(wb ,1 ∗ j)(rξ).

=0

such that Lemma 5.1 and 5.2 directly provide the following wavelet expansion. Theorem 5.3. Let j ∈ c(0) (Ω), then we have for ξ, η ∈ Ω and r ∈ (α, β), qb (rξ) = μ0 r(s0,1 b  Jrad )(rξ) + μ0 r

∞

(wb ,1  Jrad )(rξ)

=0

and brad (rξ) =

−μ0 rξ(s0,1 b

∗ j)(rξ) − μ0 r

∞

ξ(wb ,1 ∗ j)(rξ).

=0

Just for the sake of completeness we indicate the reconstruction of the tangential part of the poloidal magnetic field in the above framework, reminding that it does not satisfy the desirable ’zooming-in’ capability. 15

Theorem 5.4. Let j ∈ c(0) (Ω), then we have for ξ ∈ Ω and r ∈ (α, β), pb,tan (r, ξ)

=

0,1 μ0 r(s0,2 b  sb ∗ j)(rξ) + μ0 r

+μ0 r

∞

,1 (s0,2 b  wb ∗ j)(rξ)

=0

(wb ,2  s0,1 b ∗ j)(rξ) + μ0 r

=0

6

∞

∞

(wbk,2  wb ,1 ∗ j)(rξ).

k, =0

Conclusion and Discussion

We have seen that the currents and parts of the magnetic field can be reconstructed by wavelet expansions with locally supported convolution kernels. As already mentioned, the local support is at the price of frequency localization, i.e., a physical interpretation in terms of multipole fields. On the other hand, the spatial localization offers the numerical advantages of being able to handle (i) unevenly distibuted data and (ii) large amounts of data. Using fixed bandlimited kernels always implies the necessity of to some extend equidistributed data. This is circumvented by the local support of the above constructed wavelet kernels which allow high resolution reconstructions in regions of higher data density without deterioration outside the modeling region. However, in case of modeling inospheric currents, where usually satellite measurements are used, one can assume sufficiently equidistributed data but is generally dealing with a large set of it. The lower scale approximations in the above developed multiscale framework require only sparse data for the reconstruction since they are dealing with coarse features of minor spatial variations, while the higher scale approximations can easily deal with larger amounts of information due to the evaluation of integrals over small regions of scale-dependent extend. This way we can achieve a reasonably fast reconstruction also for large amounts of data. In section 4 we see that in case of radial currents not only the wavelets but the scaling kernels have local support as well, such that (if one is not interested in the evolution over the different scales) one can directly approximate the modeled feature at a high scale by evaluating the occuring integrals only over very small regions, implying a very fast algorithm. Last, we want to indicate the general reconstruction scheme for such multiscale methods: Let S be one of the scaling kernels from section 4 or 5 and W the corresponding wavelet kernel. For a given scalar or vectorial quantity X (in our case, e.g., the toroidal or poloidal magnetic field with appropriate prefactors) we can apply the following ’zooming-in’ scheme for the determination of the desired quantity Y (in our case, e.g., the poloidal or toroidal part of the currents). Step 1: We start with a rough global aproximation (i.e., a large scaling parameter ρ0 ) of the coarse features of Y and get Y 0 = S 0 ∗ X (resp. Y 0 = S 0  X ). Step 2: A first improvement of this rough approximation is given by adding the detail information R0 = W 0 ∗ X (resp. R0 = W 0  X ). The new approximation then reads as Y 1 = Y 0 + R0 . Step 3: Iterating this procedure by adding detail information R = W ∗ X (resp. R = W  X ) at growing scales (i.e., a decreasing scaling parameter ρ ), we get the following multiscale approximation scheme. R

Y

@ @ @ R ⊕

R +1

@ @ @ R - Y +1 -⊕ 16

R +2

@ @ @ R - Y +2 -⊕

-

...

In consequence, an efficient and economical method is given to determine geomagnetic quantities in terms of a locally supported mutiscale analysis.

References Backus, G.E., 1986. Poloidal and Toroidal Fields in Geomagnetic Field Modeling. Rev. Geophys. 24, 75-109. Backus, G.E., Parker, R., Constable, C., 1996. Foundations of Geomagnetism. Cambridge University Press, Cambridge. Bayer, M., Freeden, W., Maier, T., 2001. A Vector Wavelet Approach to Iono- and Magnetospheric Geomagnetic Satellite Data. J. Atmos. Sol.-Terr. Phy. 65, 581-597. Chambodut, A., Panet, I., Mandea, M., Diament, M., Holschneider, M., 2005. Wavelet Frames an Alternative to Spherical Harmonic Representation of Potential Fields. J. Geoph. Int. 163, 875-899. Engels, U., Olsen, N., 1998. Computation of Magnetic Fields within Source Regions of Ionospheric and Magnetospheric Currents. J. Atmos. Sol.-Terr. Phy. 60, 1585-1592. Freeden, W., 1980. On Integral Formulas of the (Unit) Sphere and their Applications to Numerical Computation of Integrals. Computing 25, 131-146. Freeden, W., 1998. The Uncertainty Principle and its Role in Physical Geodesy. In: Progress in Geodetic Science, Shaker, Aachen. Freeden, W., Grevens, T., Schreiner, M., 1998. Constructive Approximation on the Sphere. Oxford Science Publications, Clarendon Press, Oxford. Freeden, W. Schreiner, M., 2009. Spherical Functions of Mathematical Geosciences. Springer, Heidelberg. Gerlich, G., 1972. Magnetfeldbeschreibung mit Verallgemeinerten Poloiloidalen und Toroidalen Skalaren. Z. Naturforsch. 8, 1167-1172. Holschneider, M., Chambodut, A., Mandea, M., 2003. From Global to Regional Anylsis of the Magnetic Field on the Sphere Using Wavelet Frames. Phys. Earth Planet. Inter. 135, 107-124. Hulot, G., Sabaka, T.J., Olsen, N., 2007. The Present Field. In: Treatise on Geophysics, Vol. 5 (Geomagnetism). Elsevier, Amsterdam. Langel, R.A., Hinze, W.J., 1998. The Magnetic Field of the Earth’s Lithosphere (The Satellite Perspective), Cambridge University Press, Cambridge. Maier, T., 2005. Wavelet-Mie-Representations for Solenoidal Vector Fields with Applications to Ionospheric Geomagnetic Data. SIAM Journal on Applied Mathematics 65, 1888-1912. Mayer, C., 2003. Wavelet Modeling of Ionospheric Currents and Induced Magnetic Fields from Satellite Data. PhD. Thesis, Geomathematics Group, University of Kaiserslautern. Olsen, N., 1997. Ionospheric F -Region Currents at Middle and Low Latitudes Estimated from Magsat Data. J. Geophys. Res. 102, 4563-4576.

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