The geomagnetic field gradient tensor

Transcription

1 DOI 0.007/s ORIGINAL PAPER The geoagnetic field gradient tensor Properties and paraetrization in ters of spherical haronics Stavros Kotsiaros Nils Olsen Received: 3 April 202 / Accepted: 25 June 202 Springer-Verlag 202 Abstract We develop the general atheatical basis for space agnetic gradioetry in spherical coordinates. The agnetic gradient tensor is a second rank tensor consisting of 3 3 = 9 spatial derivatives. Since the geoagnetic field vector B is always solenoidal B = 0 there are only eight independent tensor eleents. Furtherore, in current free regions the agnetic gradient tensor becoes syetric, further reducing the nuber of independent eleents to five. In that case B is a Laplacian potential field and the gradient tensor can be expressed in series of spherical haronics. We present properties of the agnetic gradient tensor and provide explicit expressions of its eleents in ters of spherical haronics. Finally we discuss the benefit of using gradient easureents for exploring the Earth s agnetic field fro space, in particular the advantage of the various tensor eleents for a better deterination of the sall-scale structure of the Earth s lithospheric field. Keywords Geoagnetic field Gradient tensor Space agnetic gradioetry Matheatics Subject Classification Priary 86A25; Secondary 33C55 Introduction Although an initial global apping of the Earth s agnetic field intensity has already been perfored in the 960s by the POGO satellite series Cain 2007, the first global vector apping of the Earth s agnetic field fro space was achieved by the Magsat satellite that flew for 6 onths in 979 and 980. More recently, the satellites S. Kotsiaros B N. Olsen DTU Space, Technical University of Denark, Kgs. Lyngby, Denark e-ail: skotsiaros@space.dtu.dk N. Olsen e-ail: nio@space.dtu.dk

2 Ørsted launched in 999, CHAMP and SAC-C provide a continuous apping and onitoring of the geoagnetic field. In the near future, the Swar satellite ission that is presently under construction by the European Space Agency ESA for a scheduled launch in 202, will provide easureents not only of the agnetic field vector B but also an estiate of its East West gradient. This additional paraeter contains inforation especially about North South oriented features of crustal agnetization, see e.g. Olsen et al. 2004, Fig It is therefore expected that Swar will provide ajor advances in exploring the geoagnetic field fro space. The derivatives of the agnetic field vector B in each of the three directions of three-diensional space define the gradient tensor, consisting of 3 3 = 9 spatial derivatives and foring a second rank tensor. Each eleent of the gradient tensor represents a directional filter and thereby ephasizes certain structures of the agnetic field Schidt and Clark The agnetic gradient tensor is therefore a powerful tool for detecting hidden geoagnetic structures by enhancing certain features of the field and suppressing specific undesirable contributions. Due to the siilarity of the agnetic and gravity field under certain conditions the gravity gradient tensor has siilar properties. Satellite gradioetry and especially its application to gravity field deterination has been investigated since the early 970s. Reed 973 studied the use of satelliteborne gravity gradient devices for deterining the Earth s gravity field. Ruel and Colobo 985 discuss ways of investigating the gravity field with an orbiting gradioeter easuring the gravity gradient tensor. In 988 ESA started a series of studies with the goal of preparing the geodetic user counity for a dedicated gravity field satellite ission. In the eantie, Ruel et al. 993 established a coprehensive atheatical basis for satellite gravity gradioetry. In 999 ESA decided to build and launch the GOCE satellite ission which easures the full gravity gradient tensor. GOCE was successfully launched in March 2009 and has contributed the ost detailed odels of Earth s gravity field to date. In the context of inverse probles of atheatical geodesy, Freeden and Nutz 20 studied the calculation of the gravitational potential at the Earth s surface fro the gravity gradient tensor at the height of a low Earth orbiting satellite. Magnetic gradioetry did not follow the sae trend as gravity gradioetry, partly due to the ore coplicated nature of the geoagnetic field B which is always solenoidal B = 0, whereas the gravity field g is always solenoidal and additionally irrotational g = 0, see Olsen and Kotsiaros 20. However, agnetic gradioetry has been used for regional studies based on near surface data for decades, see e.g. Pedersen and Rasussen 990; Christensen and Rajagopalan 2000; Schidt and Clark Soe advantages of easuring agnetic field gradients fro satellites have been discussed by Harrison and Southa 99. Purucker et al studied the lithospheric field using gradient inforation constructed fro agnetic field easureents collected by the ST-5 ission that consists of three icro satellites flying in a string of pearls constellation. A ajor step toward agnetic space gradioetry will be undertaken by the Swar satellite ission which provides estiates of the East West gradient of the vector agnetic field. However, a satellite ission providing global easureents of the full geoagnetic gradient tensor

3 i.e. a agnetic ission that is equivalent to what GOCE is for the gravity field has not yet been attepted. Magnetic space gradioetry is a scientifically coplicated and technically challenging task, for instance due to the existence of electric currents at satellite altitude. In this paper we develop a coprehensive atheatical basis for agnetic spacebased gradioetry. The properties of the gradient tensor are investigated, and various possible assuptions that siplify its deterination are presented. More specifically, we start fro the ost general case of a solenoid vector field B = 0. This condition reduces the nuber of independent eleents of the gradient tensor fro 9 to 8. According to the Mie representation of vector fields e.g. Backus 986, every solenoid field for instance the agnetic field B can be decoposed into poloidal and toroidal parts. Likewise, the gradient tensor can be constructed separately for the toroidal and poloidal parts of the agnetic field. Assuing slow tie variations of periods uch longer than s allows one to neglect displaceent currents in Maxwell s equations, leading to B = μ 0 J where J is electrical current density and μ 0 = 4π0 7 Vs/A is the agnetic pereability of free space. In current-free regions 0 = μ 0 J = B the toroidal part of the agnetic field vanishes and the reaining poloidal part ay be decoposed into internal and external field parts, an approach tered Gauss representation. In that case, the agnetic field vector B = V can be derived fro a scalar potential V that solves Laplace equation 2 V = 0. Due to B = 0 the agnetic gradient tensor is syetric which reduces the nuber of independent eleents to 5. Finally, the agnetic gradient tensor can be expanded in ters of Spherical Haronics SH using the sae set of Gauss coefficients as for expanding the agnetic scalar potential V. We investigate the inforation contained in each tensor eleent and their benefits and liitations for the deterination of the Gauss coefficients describing the core and lithospheric parts of the Earth s agnetic field. The outline of this paper is as follows: In Sect. 2, we present the atheatical basis for space-based agnetic gradioetry and the general expression of the agnetic gradient tensor in Euclidean Space is derived. In Sect. 3, the properties of the gradient tensor are investigated, whereas different assuptions that can siplify the deterination of the gradient tensor in space are shown. The gradient tensor is constructed separately for the toroidal and poloidal part of the agnetic field B in Sect. 4. In Sect. 5 we calculate the agnetic gradient tensor when the agnetic field B is a Laplacian potential field and in Sect. 6 the agnetic field gradient tensor is paraetrized in ters of Spherical Haronics. The inforation contained in each tensor eleent, its benefits and liitations in the deterination of the Gauss coefficients are investigated in Sect The atheatical basis To derive the agnetic gradient tensor in Euclidean space we define the basis vectors e p as e p =, 2. u p

4 with p =, 2, 3. r is the position vector in one set of coordinates, e.g. in geocentric Cartesian coordinates r = x, y, z, whereas u = u, u 2, u 3 is a set of coordinates, e.g. spherical coordinates u = r,θ,ϕ. In general the basis vectors e p do not have unit agnitude but they ay be noralized in order to obtain the unit vectors where the scale factors h p are given by ê p =, 2.2 h p u p h p = u. 2.3 p Following Talpaert 2002 the gradient of a vector like the agnetic field B at each point is the tensor field B which relates the differential of B to the position vector eleents dr such that where the position vector eleent dr is given by db = Bdr, 2.4 h du dr = h du ê h 2 du 2 ê 2 h 3 du 3 ê 3 = h 2 du 2, 2.5 h 3 du 3 see also Arfken and Weber Note that the second rank tensor B where is not an operator and denotes a tensor should be distinguished fro B, the divergence of B where is the gradient operator. Expanding the agnetic field vector B using the unit vectors defined in Eq. 2.2 yields the colun vector B = B, B 2, B 3 T where the superscript T stands for the transpose. Therefore, the differential db is db = B u du ê B ê u du B 2 u du ê 2 B 2 ê 2 u du B 3 u du ê 3 B 3 ê 3 u du B u 2 du 2 ê B ê u 2 du 2 B 2 u 2 du 2 ê 2 B 2 ê 2 u 2 du 2 B 3 u 2 du 2 ê 3 B 3 ê 3 u 2 du 2 B u 3 du 3 ê B ê u 3 du 3 B 2 u 3 du 3 ê 2 B 2 ê 2 u 3 du 3 B 3 u 3 du 3 ê 3 B 3 ê 3 u 3 du Using the position vector eleent, Eq. 2.5, then db = Bdr, Eq. 2.4, can be expressed with the help of Eq. 2.6 and the agnetic field gradient tensor B can be therefore expressed as

5 B = h h h B u 3 p= B 2 u 3 p= B 3 u h 3 h 3 h 3 3 p= B u 3 3 p= B 2 u 3 3 p= B 3 u 3 B p u ê B p u ê 2 B p u ê 3 3 p= A ore copact for for its eleents is h 2 h 2 h 2 B p u 3 ê B p u 3 ê 2 B p u 3 ê 3 [ B] ij = B i h j u j B u 2 3 p= B 2 u 2 3 p= B 3 u 2 3 p= B p u 2 ê B p u 2 ê 2 B p u 2 ê B p ê i, 2.8 u j p= with i, j =, 2, 3, whereas fro Eq. 2.2 we have = h p u j h 2 p u j u p h p 2 r u j u p. 2.9 In the case of spherical coordinates u = r,θ,ϕthe scale factors h p are h =, h 2 = r, h 3 = r sin θ, 2.0a 2.0b 2.0c and the position vector r and the position vector eleent dr are r sin θ cos ϕ r = r sin θ sin ϕ 2. r cos θ dr dr = rdθ r sin θdϕ 2.2 Arfken and Weber Therefore, aking use of Eqs. 2.9 and 2.8, the agnetic gradient tensor B in spherical coordinates is given by

6 B = B r B θ B ϕ B r r θ r B θ B θ r θ r B r B ϕ r θ B r r sin θ ϕ r B ϕ B θ r sin θ ϕ cot θ B ϕ r B ϕ r sin θ ϕ r B r cot θ B θ r Int J Geoath Properties of agnetic field gradient tensor Full characterization of the agnetic field gradient tensor, Eq. 2.3, requires the specification of nine independent eleents. However, aking use of Gauss law of agnetis, B = 0, 3. reduces the nuber of independent eleents fro 9 to 8. Physically, Eq. 3. eans that agnetic onopoles in analogy to electric charges do not exist. In spherical coordinates Eq. 3. reads 0 = B = r 2 r 2 B r θ sin θ B θ r sin θ r sin θ B θ θ cot θ B θ r r sin θ B ϕ ϕ 3.2 = B r 2 r B r B ϕ r ϕ. 3.3 Coparison with Eq. 2.3 shows that the divergence of the vector corresponds to the trace of the gradient tensor, which is always zero because B = 0: B = tr B = A powerful aspect of the gradient tensor can be recognized by starting fro Apere s law in the quasi-static approxiation B = μ 0 J 3.5 which, as entioned before, holds for processes with tie-scales longer than s, a condition that is very well satisfied in geoagnetis. In spherical coordinates the curl of B is B ϕ μ 0 J = B = r θ cot θ B ϕ r r sin θ B r r sin θ ϕ B ϕ r B ϕ ê θ Bθ r B θ B r r θ B θ ê r ϕ ê ϕ. 3.6

7 The difference of the gradient tensor with its transpose yields 0 B B T B θ = r B ϕ B r r sin θ ϕ r B B ϕ ϕ r θ r sin θ B r r sin θ ϕ r B ϕ B ϕ B θ r sin θ ϕ cos θ r sin θ B ϕ B ϕ r θ 0 B r r θ r B θ B θ B r θ r B θ 0 B θ ϕ cos θ r sin θ B ϕ 3.7 and with J = J r ê r J θ ê θ J ϕ ê ϕ Eqs. 3.5 and 3.7 can be cobined to 0 J ϕ J θ B B T = μ 0 J ϕ 0 J r. 3.8 J θ J r 0 The gradient tensor therefore provides inforation on the in-situ electrical current density J = J r, J θ, J ϕ T. An iportant aspect of Eq. 3.8 is that B = B T in current free regions J = 0, which results in a syetric gradient tensor. This reduces the nuber of independent eleents to 5 which significantly siplifies the deterination of the gradient tensor. The general expression of the gradient tensor in spherical coordinates, Eq. 2.3, indicates that all tensor eleents except those in the first colun contain, in addition to the field derivatives along the direction θ or ϕ, contributions fro the field coponents B r, B θ, B ϕ. Thus, they represent a ixture of contributions fro the field and its spatial derivatives. However, when studying sall-scale variations, e.g. the lithospheric field, it is possible to siplify the gradient tensor. Since the geoagnetic field is doinated by the large-scale ain field i.e. contributions described by spherical haronic degrees up to, say, n = 3 the ters of the gradient tensor that include the field rather than its spatial derivative can be neglected, as shown e.g. by Olsen and Kotsiaros 20. In that case Eq. 2.3 reduces to B B r B θ B ϕ B r r θ B θ r θ B ϕ r θ B r r sin θ ϕ B θ r sin θ ϕ B ϕ r sin θ ϕ 3.9 and the tensor eleents contain only contributions fro the field derivatives along each direction but not fro the vector field B itself.

8 4 Toroidal poloidal decoposition According to the Mie representation, a solenoid vector like the agnetic vector B can be written uniquely as the su of a toroidal and a poloidal part cf. Mie 908; Backus 986; Backus et al. 996; Sabaka et al Each part is derived via curl operations: B = B tor B pol 4. B = r r 4.2 where and are the toroidal and poloidal scalar potentials, respectively. In spherical coordinates this decoposition is given by 0 s r B = sin θ ϕ r θ r 4.3 θ r sin θ ϕ r }{{}}{{} toroidal part poloidal part with r = dr dr and s = r 2 sin θ θ sin θ θ 2 is the horizontal part r 2 sin 2 θ ϕ 2 of the Laplacian. An expansion of the scalar potentials and in a series of spherical haronics is given in Olsen 997. The gradient tensor of the toroidal, resp. poloidal, part follows as 0 r sin θ ϕ r θ B tor = 2 cos θ sin θ ϕ r θ sin θ ϕ r sin θ θ 2 r sin 2 θ ϕ θ r θ 2 r θ sin θ ϕ sr s r r r θ r B pol = r r θ r sr 2 r 2 θ 2 r r r sin θ r ϕ θ r 2 sin θ ϕ s r r r sin θ ϕ r r θ r 2 sin θ ϕ. 4.5 r 2 sr r 2 sin θ sin θ ϕ 2 cos θ r θ The general condition that the trace of the gradient tensor is zero, Eq. 3.4, holds also for the gradient tensor of the toroidal and poloidal parts separately.

9 5 Laplacian potential approxiation In case of vanishing currents J = 0 in a shell where the agnetic easureents are taken, i.e. a source-free region, the toroidal field B tor vanishes and the poloidal field B pol can be decoposed into a part B i caused by currents inside the shell and another part B e caused by currents outside the shell cf. Backus et al. 996, chap. 5: Since the agnetic vector B within the shell is curl-free, B = B i B e. 5. μ 0 J = B = 0, 5.2 it can be written as the negative gradient of a scalar potential V, B = V 5.3 and because of B = 0 the potential V has to obey Laplace s equation: 2 V = 0. Siilar to Eq. 5., the scalar potential V can be split into an internal part V i and an external part V e V = Vi Ve, 5.4 each of which ay be expanded in series of spherical haronics, as discussed in the next section. The agnetic field vector B follows as V i V e B = V i r θ V e r θ. 5.5 V i V e r sin θ ϕ r sin θ ϕ }{{}}{{} B i B e According to Eq. 2.3 the gradient tensor is then given by 2 V 2 V r θ B = V V r θ r 2 V r 2 θ 2 V V r sin θ ϕ θ r 2 sin θ ϕ V r sin θ ϕ V θ r 2, 5.6 sin θ ϕ V r cot θ V r 2 θ 2 V r 2 sin 2 θ ϕ 2

10 where V = V i V e. Equation 5.6 can be written independently for the internal part V i and the external part V e, respectively. The trace of this tensor is equal to 2 V which is zero, as expected. Additionally, the gradient tensor is syetric, which is consistent with what was stated in Sect. 3. Hence only five tensor eleents have to be specified in order to fully deterine the agnetic gradient tensor. 6 The spherical haronic representation The scalar potential V can be expanded in series of spherical haronics: { N n a n V = R a γn exp iϕ P r n cos θ n= =0 N e a n= =0 } n r n δn exp iϕ P a n cos θ, 6. where R{ }denotes the real part of the series and i is the iaginary nuber, a is a reference radius typically Earth s ean radius a = 6,37.2 k is chosen, Pn cos θ are the associated Schidt sei-noralized Legendre functions of degree n and order,γn = g n ih n and δ n = q n is n are the coplex Gauss coefficients describing sources internal and external to the Earth, respectively. Theoretically, the series extends up to infinite degree, but since in practice there is only a liited nuber of easureents at discrete points, the series are truncated at soe largest degree N and N e, respectively. The agnetic field vector follows as N n a n2 n γn exp iϕ P n= =0 r n cos θ B i = R N n a n2 dp γ n exp iϕ n cos θ, n= =0 r dθ N n a n2 n= =0 sin θ iγ n exp iϕ P r n cos θ 6.2a N e n r n nδn exp iϕ P n= =0 a n cos θ B e = R N e n r n dp δ n exp iϕ n cos θ. n= =0 a dθ N e n r n sin θ iδ n exp iϕ P a n cos θ 6.2b n= =0 Given Eq. 2.3, the gradient tensor eleents of the internal part [ B i ] jk and external part [ B e ] jk of the agnetic field can be expanded in series of spherical haronics. Here the first subscript, j, stands for the vector coponent for which the derivative is taken, while the second, k, indicates the direction of the spatial derivative,

11 with j, k = r,θ,ϕ.since we are dealing with the potential case i.e. absence of electrical currents, the tensor is syetric and therefore only six out of nine eleents are presented. [ B i ] jk = R { a N n γn n= =0 exp iϕ a r } n3 P, i n, jk cos θ 6.3 with: Pn,, rr i cos θ = n n 2P n cos θ 6.4a and with P, i n,θθ cos θ = n P n cos θ d2 Pn cos θ dθ 2 [ Pn,ϕϕ, i 2 ] cos θ = sin 2 n Pn θ cos θ cot θ dp n cos θ dθ n, rθ cos θ = n 2dP n cos θ dθ P, i P, i n, rϕ n 2 cos θ = ipn cos θ sin θ 6.4e P, i cos θ n,θϕ cos θ = sin 2 θ ip n cos θ [ B e ] jk = R { a N e n= =0 sin θ i dp n cos θ dθ n r δn exp iϕ a n 2 P, e n, jk cos θ } 6.4b 6.4c 6.4d 6.4f 6.5 Pn,, rr e cos θ = nn P n cos θ 6.6a P, e n,θθ cos θ = np n cos θ d2 Pn cos θ dθ 2 6.6b [ Pn,ϕϕ, e 2 ] cos θ = sin 2 θ n Pn cos θ cot θ dp n cos θ 6.6c dθ P, e n, rθ cos θ = n dp n cos θ 6.6d P, e n, rϕ dθ cos θ = n ipn cos θ sin θ 6.6e P, e cos θ n,θϕ cos θ = sin 2 θ ip n cos θ sin θ i dp n cos θ. 6.6f dθ

12 As entioned earlier, the trace of the agnetic gradient tensor is always zero and therefore Pn,, rr i P, i n,θθ P, n,ϕϕ i = 0 and siilarly Pn,, rr e P, e n,θθ P, n,ϕϕ e = 0. 7 The gradient tensor as a tool for agnetic field deterination After having introduced in the previous sections a atheatical representation of the agnetic gradient tensor, we now discuss the proble of using agnetic gradient observations to explore the geoagnetic field. The goal of geoagnetic field odelling is to estiate, fro geoagnetic field observations, the spherical haronic expansion coefficients gn, h n and q n, s n of Eq. 6. describing sources of internal, resp. external, origin. These observations can be the coponents of the agnetic vector B, the agnetic field intensity B and/or eleents of the gradient tensor. In this section we investigate geoagnetic field odelling using the various eleents of the agnetic gradient tensor. For this task we will only consider a deterination of the Gauss coefficients gn, h n of internal origin. In addition, we concentrate on a deterination of the sall-scale lithospheric field. Deterination of Gauss coefficients fro observations of the agnetic field vector or the gradient tensor eleents is a linear proble, and therefore the odel vector containing the unknown Gauss coefficients the odel paraeters gn, h n is related to the data vector d containing observations of the vector coponent or gradient tensor eleents as d = G 7. where G is the data kernel atrix. An estiate ˆ of the odel vector in the weighted Least-Square sense results in ˆ = G T C d G G T C d d 7.2 = G T G G T d, 7.3 where C d is the data covariance atrix which is assued to be diagonal with eleents σ 2 d, corresponding to independent observations with coon variance σ 2 d. The diagonal eleents of the odel covariance atrix C = σ 2 d GT G 7.4 are the variances σg 2 of the estiated odel paraeters, i.e. of the Gauss coefficients gn, h n. C is independent of the data vector d, i.e. of the actual agnetic data, but depends on the satellite positions and on the type of observation, i.e. which vector coponents or gradient tensor eleents are considered in the inversion process. Therefore, given a satellite orbit configuration and a certain type of observations, aps of the variances σg 2 give inforation on the perforance of the agnetic field deterination for that given orbit and observation type. Approxiate analytic expressions for the covariance atrix C and thereby for the variances σg 2 of the estiated odel paraeters have been derived by Lowes 966

13 and Langel 987 for the case that the observations consist of N d easureents of all three coponents B r, B θ, B ϕ of the agnetic vector equally distributed by area on the sphere or weighted by sin θ if they are taken by a polar orbiting satellite and equally spaced in tie. In that case G T G, and hence also C, approach in the liiting case of an infinite nuber of observations diagonal atrices due to the orthogonality of spherical haronics, and the variances of the Gauss coefficients are given by σ 2 g σ 2 d r a 2n4 N d n, if d ={B r, B θ, B ϕ } 7.5 cf. Eq. 24 of Langel 987. Note that these variances do not depend on the order of the spherical haronics, i.e. they are equal for all Gauss coefficients of a given degree n. If only easureents of the radial vector coponent B r are used the corresponding variances of the Gauss coefficients are σ 2 g σ 2 d r a 2n4 2n N d n 2, if d ={B r } 7.6 which is larger than Eq. 7.5 by a factor between 3/2 for n = and 2 for n. Analytic expressions for σg 2 do not exist if the data consist of the horizontal field coponents B θ or B ϕ since in those cases G T G is not a diagonal atrix and it is not possible to analytically derive its inverse. In order to study these cases we therefore rely on nuerical siulations, siilar to the ones perfored in Olsen et al For those we used synthetic data fro a satellite in a polar circular orbit at 400 k altitude. They span a period of one onth with a sapling rate of 30 s, which gives us N d = 89,280 agnetic easureents in total. Since the data are equally spaced in tie we applied weights w = sin θ in order to siulate an equal area distribution. The diagonal eleents, σg 2, of the nuerically coputed odel covariance atrix C are shown in Fig. in dependence on degree n and order where 0 belongs to the coefficients gn while < 0 belongs to h n. The left part of the figure represents the variances for the case that observations of B r are used; this ap corresponds to the analytic expression for σg 2 given in Eq The other parts of the figure show the variances σg 2 for the cases that observations consist of the agnetic field coponents B θ iddle or B ϕ right. Note that σg 2 in these cases depend on both degree n and order. Overall, the aps show the inforation contained in each vector coponent. White represents low inforation content, whereas green represents high inforation content. If the data consist of the radial gradient of the radial vector coponent, [ B] rr, a siilar analytic approach as in the case of B r as data can be used to derive the variances There is a typo in the right-hand side of equation 22a and 22c of Langel 987 which correctly should read δ k, j [nn /2n /2] and δ k, j [n 2 /2n ], respectively.

14 Fig. Variance of the estiated internal Gauss coefficients fro 89,280 data points of each field coponent, a B r, b B θ and c B ϕ, obtained by a polar orbiting satellite at 400 k altitude. White represents low inforation content, whereas green represents high inforation content colour figure online of the Gauss coefficients. In that case G T G can be approxiated by G T G N d 4πa 2 2π π 0 0 { cosϕcos } ϕ sinϕsin [n n 2][n n 2] ϕ a n3 a n 3 P r r n cos θpn cos θsin θdθdϕ = a 2n6 Nd [n n 2] 2 a 2 δ, δ n,n 7.7 r 2n due to the orthogonality of the spherical haronics. The variances of the Gauss coefficients follow as σg 2 aσ d 2 r 2n6 2n a N d [n n 2] 2, if d ={[ B] rr}. 7.8 In this case the relative variance of the estiated Gauss coefficients, σ g, is again independent of the order. Note that the variance of the observations, σ d, has the units nt k, since it refers to gradient observations, in contrast to the variance σ d in Eqs. 7.5 and 7.6 where it has the units nt, since it refers to vector observations. For large n the variances of the Gauss coefficients obtained fro vector data, cf. Eq. 7.6, is proportional to /n while those obtained using gradient data, cf. Eq. 7.8 behaves like /n 3. This indicates the advantage of gradient inforation for deterination of the sall-scale i.e. high-degree structure of the lithospheric field. No analytic forula exist for the cases when the data consist of the other tensor eleents since they depend on a ore coplex cobination of Legendre functions, see Eqs. 6.4b, 6.4c and 6.4f, and orthogonality cannot be assued. We therefore have to rely on nuerical siulations siilar to the ones presented in Fig.. Maps of the obtained variances σd 2 are presented in Fig. 2 for the various type of agnetic gradient observations. Figure 2a corresponds to the eleent [ B] rr

15 Fig. 2 Variance of the estiated internal Gauss coefficients fro 89,280 data points of each tensor eleent, a [ B] rr, b [ B] rθ, c [ B] rϕ, d [ B] θθ, e [ B] θϕ, f [ B] ϕϕ, obtained by a polar orbiting satellite at 400 k altitude. White represents low inforation content, whereas green represents high inforation content colour figure online i.e. the Gauss coefficients are estiated by easureents of [ B] rr only and shows the advantage of knowing the radial gradient of the radial agnetic field coponent B r ; the analytic expression of these variances is given by Eq Figure 2b corresponds to the eleent [ B] rθ and shows the advantage of knowing the North South gradient of the radial agnetic field coponent B r. Figure 2c corresponds to the eleent [ B] rϕ and shows the benefit of having easureents of the East West gradient of the radial field coponent B r. Figure 2d, e illustrate the benefit of easuring the North South, resp. East West, gradient of the co-latitudinal field coponent B θ. At last, Fig. 2f shows the advantage of having easureents of the East West gradient of the longitudinal field coponent B ϕ. Note that due to the syetry of the agnetic gradient tensor in current-free regions the inforation contained in [ B] rϕ is identical to that contained in [ B] ϕr. Likewise, [ B] ϕθ and [ B] θϕ contain the sae inforation, and also [ B] θr and [ B] rθ. The lower left triangle part of Fig. 2 is therefore identical to its upper right part.

16 Variances obtained fro [ B] rr, the radial gradient of the radial field coponent, cf. Fig. 2a, are independent of the order and resolve best the higher degrees n. There is a focus on higher degrees which is iposed by the factor r 2n6 a and therefore is characteristic of the orbit height. On the other hand, the relative variance σg 2/σ d 2 of the Gauss coefficients estiated by the North South gradients of B r, cf. Fig. 2b, depends also on the order. In particular, there is a degradation toward high-order coefficients. In case of the East West gradient of B r, cf. Fig. 2c, there is an iproved estiation of the sectoral n = and near-sectoral n coefficients. This also shows that relying on the East West gradient of the radial field coponent alone does not allow the deterination of the zonal ters = 0. This issue can however be solved by including other types of data, such as vector coponents or other tensor eleents, in addition to the East West gradient. Knowledge of the North South gradient of the co-latitudinal field coponent B θ iproves the estiation of the low-order coefficients, see Fig. 2d. Inforation on the East West gradient of B θ, i.e. Fig. 2e, iproves the estiation of the tesseral coefficients. However, the sae issue as in Fig. 2c appears, which is that East West gradient data alone do not allow the zonal ters = 0 to be deterined. Finally, inforation on the East West gradient of the longitudinal field coponent B ϕ iproves the estiation of the high-order coefficients, see Fig. 2f. Unlike the other two cases, zonal ters can now also be deterined with this particular East West gradient tensor eleent although the quality is degraded especially for the high degree values. The reason is that, unlike the other two cases, the East West gradient of the longitudinal field coponent [ B] ϕϕ does not only depend on ters that are sensitive to the order such as B ϕ, or B θ ϕ, but also on the ter r B r, which is sensitive only to the degree n, see Eq Therefore, the deterination of zonal ters, which was not possible by the other East West gradients, can be achieved fro [ B] ϕϕ. In general, Fig. 2 shows that each tensor eleent iproves the deterination of coefficients of certain degrees n and orders. The inforation provided by each tensor eleent, and thus the particular benefits in the deterination of certain coefficients, can be cobined and thereby a ore precise deterination of the coplete Gauss coefficients set can be achieved. The optial cobination of tensor eleent observations depends, however, on several factors, such as orbit configuration, attitude and instruent errors along certain directions. At high latitudes the disturbance agnetic field i.e. due to agnetospheric and ionospheric currents is uch stronger and ore coplex copared to low and iddle latitudes, ainly because of the occurrence of field-aligned currents. Therefore, present estiates of the high-latitude anoaly fields e.g. lithospheric field are not fully reliable since they ight be containated by external current contributions. Measureents of the individual gradient tensor eleents yield inforation on the in situ electrical current density, as indicated by Eq Consequently, the ipact of the field-aligned currents on each tensor eleent can be deterined and hence, accordingly cobining or even down-weighting and excluding these tensor eleents in the deterination of the lithospheric field has the potential of obtaining ore reliable estiates of the lithospheric field at high latitudes.

17 8 Conclusions The Swar satellite ission will provide a new diension of geoagnetic exploration fro space by easuring, in addition to the agnetic field vector, also an estiate of the East West gradient of the geoagnetic field. This contains valuable inforation on North South oriented features of crustal agnetization. However, a siilar resolution of East West oriented structures is not possible with Swar since this requires easureent of the North South agnetic field gradient. Going beyond Swar s liitations, a possible future way for exploring the Earth s agnetic field fro space would be to ove towards a full gradioetry ission. In this paper we have developed a coprehensive atheatical basis for agnetic space gradioetry. The agnetic gradient tensor B is a second rank tensor consisting of 3 3 = 9 spatial derivatives. Making use of the general constraint that the agnetic field is always solenoidal B = 0 reduces the nuber of independent tensor eleents fro 9 to 8. Furtherore, in a current free region J = 0 the agnetic gradient tensor becoes syetric, further reducing the nuber of independent eleents to 5. In that case the agnetic field B is a Laplacian potential field and the gradient tensor can be used to estiate the spherical haronic expansion coefficients Gauss coefficients of the agnetic potential. The possible benefit of using gradient tensor easureents is exploited by looking at the variances of the estiated odel paraeters i.e. the Gauss coefficients. For ideally sapled data, the variances depend only on the satellite orbit and the type of field observations e.g. which tensor eleents are easured. Maps of the variances of the estiated Gauss coefficients show the perforance of the agnetic field deterination for a given orbit and observation type. Each tensor eleent provides particular inforation and iproves the estiation of Gauss coefficients of specific degree and order. Therefore, the cobination of individual tensor eleents for specific ranges of degree and order has the potential for iproving the overall deterination of the Gauss coefficients and hence for obtaining a ore accurate description of the geoagnetic field. Present deterinations of the high-latitude lithospheric fields are for exaple not fully reliable ainly due to the existence of field aligned currents. However, the agnetic gradient tensor provides inforation on the in situ electrical current density. Therefore, the ipact of field aligned currents on each tensor eleent can be deterined, and hence these eleents can be suitably cobined to obtain ore accurate estiates of the high-latitude lithospheric field. In order to investigate the inforation content i.e. the variances of the Gauss coefficients in each vector coponent as well as in each gradient tensor eleent, we perfor a siulation based on synthetic data generated on a polar circular orbit using a basic input odel. That input odel provides the static agnetic field contribution fro the core and lithosphere. A full end-to-end ission siulation based on a ore sophisticated orbit and on a ore realistic input odel, which also accounts for the highly tie dependent agnetic field contributions due to currents in the ionosphere and agnetosphere, is the topic of a forthcoing paper.

18 References Arfken, G.B., Weber, H.J.: Matheatical Methods for Physicists, 6th edn. Elsevier Acadeic Press, Asterda 2005 Backus, G.: Poloidal and toroidal fields in geoagnetic field odeling. Rev. Geophys. 24, Backus, G., Parker, R., Constable, C.: Foundations of Geoagnetis. Cabridge University Press, Cabridge 996 Cain, J.C.: POGO OGO-2, -4 and -6 spacecraft. In: Gubbins, D., Herrero-Bervera, E. eds. Encyclopedia of Geoagnetis and Paleoagnetis, pp , Springer, Heidelberg 2007 Christensen, A., Rajagopalan, S.: The agnetic vector and gradient tensor in ineral and oil exploration. Preview 84, Freeden, W., Nutz, H.: Satellite gravity gradioetry as tensorial inverse proble. GEM Int. J. Geoath. 2, Harrison, C., Southa, J.: Magnetic field gradients and their uses in the study of the Earth s agnetic field. J. Geoagn. Geoelectr. 43, Langel, R.A.: The ain field. In: Jacobs, J.A. ed Geoagnetis, vol., pp Acadeic Press, London 987 Lowes, F.J.: Mean-square values on sphere of spherical haronic vector fields. J. Geophys. Res. 7, Mie, G.: Considerations on the optic of turbid edia, especially colloidal etal sols. Ann. Phys. 25, Olsen, N.: Ionospheric F region currents at iddle and low latitudes estiated fro Magsat data. J. Geophys. Res. 02A3, Olsen, N., Kotsiaros, S.: Magnetic satellite issions and data. In: Mandea, M., Korte, M. eds. Geoagnetic Observations and Models, IAGA Special Sopron Book Series, Chap. 2, vol. 5, pp Springer, Heidelberg. doi:0.007/ _2 20 Olsen, N., Friis-Christensen, E., Hulot, G., Korte, M., Kuvshinov, A.V., Lesur, V., Lühr, H., Macillan, S., Mandea, M., Maus, S., Purucker, M., Reigber, C., Ritter, P., Rother, M., Sabaka, T., Tarits, P., Thoson, A.: Swar: End-to-End Mission Perforance Siulator Study, ESA Contract No 7263/03/NL/CB. DSRI Report /2004, Danish Space Research Institute, Copenhagen 2004 Olsen, N., Hulot, G., Sabaka, T.J.: Measuring the Earth s agnetic field fro space: concepts of past, present and future issions. Space Sci. Rev. 55, doi:0.007/s Pedersen, L.B., Rasussen, T.M.: The gradient tensor of potential field anoalies: Soe iplications on data collection and data processing of aps. Geophysics 552, Purucker, M., Sabaka, T., Le, G., Slavin, J.A., Strangeway, R.J., Busby, C.: Magnetic field gradients fro the st-5 constellation: iproving agnetic and theral odels of the lithosphere. Geophys. Res. Lett. 3424, L24, Reed, G.B.: Application of kineatical geodesy for deterining the short wave length coponents of the gravity field by satellite gradioetry. Reports of the Departent of Geodetic Science Report No Ruel, R., Colobo, O.: Gravity field deterination fro satellite gradioetry. J. Geodesy 59, Ruel, R., van Gelderen, M., Koop, R., Schraa, E., Sanso, F., Brovelli, M., Miggliaccio, F., Sacerdote, F.: Spherical haronic analysis of satellite gradioetry. Technical Report 993 Sabaka, T.J., Hulot, G., Olsen, N.: Matheatical properties relevant to geoagnetic field odelling. In: Freeden, W., Nashed, Z., Sonar, T. eds. Handbook of Geoatheatics. Springer, Heidelberg 200 Schidt, P., Clark, D.: Advantages of easuring the agnetic gradient tensor. Preview 85, Schidt, P., Clark, D.: The agnetic gradient tensor: its properties and uses in source characterization. Lead. Edge 25, Talpaert, Y.: Tensor analysis and continuu echanics. Kluwer Acadeic Publishers, Dordrecht 2002