CHALLENGES TO THE SWARM MISSION: ON DIFFERENT INTERNAL SHA MAGNETIC FIELD MODELS OF THE EARTH IN DEPENDENCE ON SATELLITE ALTITUDES

Transcription

1 CHALLENGES TO THE SWARM MISSION: ON DIFFERENT INTERNAL SHA MAGNETIC FIELD MODELS OF THE EARTH IN DEPENDENCE ON SATELLITE ALTITUDES Wigor A. Webers Helmholtz- Zentrum Potsdam, Deutsches GeoForschungsZentrum, Germany, fax: /88-3, phone: /88-37 ABSTRACT The usual internal magnetic field model in the form of the spherical harmonic analysis (SHA) is the wellknown model for the internal magnetic field sources within the Earth body. Up to now this internal magnetic field model as spherical harmonic expansion (SHA) is calculated from multi-altitude magnetic observations (ground, balloon and satellite altitudes). According to the author s paper [] from the mathematics of modelling the internal magnetic field there results that the different physical properties of magnetic field data in dependence on the altitudes cause that only a mean internal magnetic field model is derived when field data from different altitudes are commonly used without any reasonable potential field continuation procedure. From the mathematical point of view the SHA internal field model is an infinite series expansion using a spherical coordinate system. The SHA expansion is convergent for all field points external to a sphere representing the Earth s surface where all the field sources are situated within this sphere. In dependence on the reference sphere there is a special convergence quality of the special SHA expansions. Consequently, a definite truncation index N used for different reference spheres results in different approximation qualities by finite partial sums. Hereby, there are new challenges to the forthcoming SWARM mission that enables to receive more physical information by comparing different internal SHA magnetic field models in dependence on the satellite altitudes.. INTRODUCTION The magnetic field recorded at irregularly distributed observatories and stations, contains internal and external field contributions. The internal includes components dominated by the Earth s main or core field - the global reference field - as well as relatively smaller contributions from the Earth s mantle and lithosphere / crust. All these magnetic field constituents are differently represented in observations taken at the Earth s surface and satellite altitudes due to the different measurement errors and physical and mathematical properties of the field model. The external magnetic field effects, for example, tend to contaminate the lithospheric constituents of the internal magnetic field much more severely at satellite altitudes than at the Earth s surface. Consequently, field models of the two data sets will reflect fundamentally different source effects.. ON THE MATHEMATICAL FIELD MODEL From potential field theory, the internal magnetic field of the Earth may be represented as the gradient of the potential V given by () B int = - V, where the spherical harmonic expansion (SHA) of the potential is. and (r, ϑ, λ) are the spherical polar coordinates a is the radius of the Earth s surface (nominally 6,37. km) m P n is a Schmidt quasi-normalized associated Legendre function of degree n and order m, and g n m ; h n m are the Gauss coefficients. In the space external to the source region (i.e. in free space), the potential V satisfies the Laplace equation, so that () (3) Proc. th ESA Symposium on European Rocket and Balloon Programmes and Related Research Hyère, France, 6 May (ESA SP-7, October )

2 V= for r a (4) n = f = a P f = a P In practice, the series expansion () is customarily referred to an Earth sphere of mean radius a = 6,37.3 km, or an ellipsoidal or other appropriate reference surface of the Earth. The originally infinite series expansion of Eq. () must be approximated by the partial sum with the truncation index N. Using the common index k instead of the indices n and m for the respective degree and order of the associated Legendre functions P m n (cos ) allows Eq. () to be expressed in terms of the orthogonal functional system {f k } and the coefficients {C k } = {g m n ; h m n }. f = a. P f = a. cos λ P f 3 = a. sin λ P f 4 = a. P f 5 = a. cos λ P f 6 = a. sin λ P f 7 = a. cos λ P f 8 = a. sin λ P.. for all k =,,..., N (N+). (5) f = a cosλp f 3 = a sinλp n = f 4 = a P f 5 = a cosλp f 6 = a sinλp f 7 = a cosλp f 8 = a sinλp n = 3 f 9 = a P 3 f = a cosλp 3 f = a cosλ P f 3 = a sinλp f 4 = a P f 5 = a cosλp f 6 = a sinλp f 7 = a cosλp f 8 = a sinλp f 9 = a P 3 f = a cosλp 3 The least squares method is applied to the derivatives of the potential V (Eq. ()) for numerically calculating the Gauss coefficients {g n m ; h n m }= {C k } of the SHA field model. For field components taken at the Earth s surface the least squares determination references the functional system and Gauss coefficients to the sphere of radius r=a. However, the determination for observations taken at satellite altitude h are referenced to r=a+h. Obviously, the two data models are based on different functional systems and Gauss coefficients as can be seen by considering the first few terms of the potential V (Eq. ()) shown below. for r = a for r = a + h (6) etc. with the set of Gauss coefficients C k and C k, respectively, being different due to the relevant functional systems in dependence from its reference sphere because any series expansion is determined for its convergence by the reference point and the reference sphere. To explain these differences there is to take into account that the SHA is a transcription of a 3- dimensional Taylor power series expansion using a three-dimensional Cartesian coordinate system with its origin in the centre of the sphere of the mean Earth radius. This transcription is performed by using the interrelation between the Cartesian and the spherical coordinate systems so that Eq. () has the threedimensional Cartesian form

3 (7) being identical to for the Earth s surface. There is a special Theorem from the theory of infinite power series expansions that if there is a convergent power series expansion for a function then this expansion is unique. The proof that both forms of the potential V of Eq. (7) are identical is given by using the rearrangement Theorem for a convergent power series expansion (cp. []). Furthermore, any power series expansion is determined for its reference point and its reference surface, respectively. This reference essentially governs the convergence area and the convergence quality. The usual SHA field model is referred to the ground in form of the sphere of the mean Earth radius. This reference governs the convergence quality of this SHA series expansion as a relative slow convergent expansion. In practice, the Gauss coefficients of the SHA are calculated for a finite functional system referred to the Earth s surface. An internal SHA magnetic field model for a satellite altitude and derived from these satellite data has a different convergence quality in comparison to the ground because of the other reference sphere (cp. Eq. (6)). There is another special Theorem from the theory of infinite power series expansions that gives the analytical relations between power series expansions of different reference points / reference surfaces for the one-dimensional case []. The infinite sets of coefficients of both expansions are in relation to each-other by an infinite set of linear equations between them. As generalization there are more complicated generalized linear interrelations for the threedimensional case. According to the transcription to the spherical coordinate system there results an infinite set of linear equations between the separate relevant Gauss coefficients of the SHA model referred to the ground and the SHA model referred to a satellite altitude, respectively. And both these infinite sets of Gauss coefficients are essentially different because they are calculated for different functional systems (cp. Eq. (6)). In order to demonstrate for Eq. (8) the effect in the three-dimensional case it seems remarkable to mention a special example when a one-dimensional translation of the reference point in the vertical z-direction had been calculated [3]. For the relevant terms of Eq. () (for the potential V and its derivatives, respectively) this transformation of the spherical harmonic functions gives (8)

4 when Using the ground-based SHA model for a satellite altitude means to neglect the different convergence qualities and all its consequences. For this simplified procedure of using the ground-based Gauss coefficients as SHA model for the ground as well as for all satellite altitudes means that the linear relations between the Gauss coefficients are reduced to the lowest approximations. In Cartesian coordinates and for the one-dimensional case of Eq. (8) this simplification means that for all the coefficients a=b for all indices n and k being the worst approximations. 3. PRACTICAL CONSEQUENCES When in the forthcoming SWARM mission simultaneously in time magnetic field data are available for the ground and for different satellite altitudes there is the challenge - to prove the altitude dependence of the relevant separate SHA field models - to determine altitude dependent details of the magnetic field - to prove any approximations of field continuations, e.g. that method of the author []. Eq. (8) and its 3-dimensional generalization demonstrate that there are essential differences for the relevant Gauss coefficients in dependence from the satellite altitude h and significant differences in the related SHA field models as the consequence. The generalization of Eq. (8) is a large infinite set of linear equations that is difficult to treat with. Moreover, this infinite set is to be approximated by a reasonable finite approach by well-defined criteria. From the viewpoint of theoretical mathematics to relate SHA field models referred to reference spheres of different altitudes means to determine upward and downward potential field continuations being an ill-posed inverse problem that generally cannot be solved by a unique solution [4], [5], [6]. Moreover, there is a nonlinear dependence of the internal magnetic field sources on the distances from its sources, i.e. from the geometrical distances from the Earth s surface. Describing the nonlinear functional dependence in physical terms makes obvious that different wavelengths of the internal magnetic field reach different satellite altitudes. And there is to emphasize that the mentioned mathematical characteristics of the SHA field models correspond well with the physical model. According to these physical as well as mathematical properties the SHA field models can be improved significantly when separate SHA models are calculated separately for the different altitudes and exclusively only from field data of that altitude. To compare these different, i.e. altitude dependent SHA field models requires a reasonable field continuation upward or downward that can only be determined by a reasonable approximation. For this an approximation for the infinite threedimensional generalization of the interrelation between the systems of coefficients in Eq. (8) can be used but its consequences for the physical properties have to be evaluated for an upward or downward continuation, respectively. Another suitable procedure is to approximate the dependence of the mathematical property of the SHA series convergence on the satellite altitudes. In this way the author introduced a form of regularization to calculate a numerical downward continuation of the relevant Gauss coefficients of the SHA model to a lower satellite altitude []. C kreg (n, k, C k, r, r, γ, c () ) = (9)

5 where for r < r and γ = α - α, - the different physical properties of SHA field models can be determined approximately by the difference field that is calculated for the altitudes in comparison with the downwardly continued field models using Eq. (9). r surface r is the radius of the satellite altitude reference is the radius of the ground reference surface. The reg index in the downward continued Gauss coefficients C kreg [Eq. (9)] indicates the fact that the downward field continuation mathematically means an ill-posed inverse problem that is solved by a regularization process. Details of the procedure and its parameters are given in the mentioned paper []. Here, the regularization parameters γ, c (), c j () for j =,,, have been derived from the inherent mathematical characteristics, i.e. the different convergence property at the ground in comparison to the satellite altitude. In this respect no further explanations are necessary, what the consequences of these regularizing parameters would mean. In this way a downward continued SHA field model can be calculated and compared with that SHA model being determined from data of its reference surface. Such comparisons enable to evaluate the different physical field properties of the different altitudes. As examples Fig. and Fig. demonstrate the quality and the amounts of such non-linear differences of the SHA fields in dependence on the altitudes. According to the Eq. (9) the differences of the SHA fields depend on the parameters of the used regularization, i.e. the convergence properties and the finite approximations. Practically, these are the following parameters: - truncation index N of the SHA field model - satellite altitudes - approximation quality for evaluating the convergence property of the SHA - field data quality and data distribution for the reference spheres (i.e. ground and satellite altitudes) The figures and make obvious that the forthcoming SWARM mission will give excellent suppositions to study the non-linear altitude dependence of the internal magnetic field when simultaneously in time field data from different satellite altitudes will be available. Moreover, in this way more details of the field with respect to the field sources can be studied. 4. CONCLUSIONS The mathematical SHA field model proves that there are essential different mathematical and physical properties in dependence on the satellite altitude. Consequently, - the SHA field models are to be calculated separately for the different altitudes. - a common SHA field model based on the ground as reference surface and using only the geometrical coordinates of the ground points and of the satellite points for the ground as well as for the field data in Eq. () produces only a mean SHA field model because the different functional systems (cp. Eq. (6) are neglected. - separate SHA field models referred to the relevant satellite altitude as reference surface of the SHA series expansion enable to determine the different mathematical as well as the physical properties in dependence on the satellite altitudes. 5. REFERENCES. Webers, W. A., On different properties of internal magnetic field models at the Earth s surface and at satellite altitudes, J. Geodyn. 43, 39-47, 7.. Knopp, K., Theorie und Anwendung der unendlichen Reihen, Springer Verlag, Berlin, Kautzleben, H., Kugelfunktionen in Geomagnetismus und Aeronomie Bd. / Ergänzungsband, Teubner Verlag, Leipzig, Huestis, S.P., Parker, R.L. Upward and downward continuations as inverse problems, Geophys. J. R. Astr. Soc. 57, 7-88, Anger, G., Inverse problems in differential equations, Akademie/Plenum Press, Berlin/London, Webers, W. A., Downward field continuation in combining satellite and ground-based internal magnetic field data, J. Geodyn. 33, -6,.

6 Fig.. DGRF 99: difference chart DGRF 99 DGRFsd (4) 99, Z-component in nt: DGRF 99 mathematically upward continued to the satellite altitude of h = 4 km as DGRFs (4) 99, downward continued to the ground as DGRFsd (4) 99. Fig.. DGRF 99: difference chart DGRF 99 DGRFsd (75) 99, Z-component in nt: DGRF 99 mathematically upward continued to the satellite altitude of h = 75 km as DGRFs (75) 99, downward continued to the ground as DGRFsd (75) 99.