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Spatial gradients from irregular, multiple-point spacecraft

configurations

C. Shen,

1

Z. J. Rong,

2,3

M. W. Dunlop,

4,5

Y. H. Ma,

1,6,7

X. Li,

8

G. Zeng,

1,7

G. Q. Yan,

1

W. X. Wan,

2,3

Z. X. Liu,

1

C. M. Carr,

5

and H. Rème

9,10

Received 29 June 2012; revised 28 September 2012; accepted 29 September 2012; published 14 November 2012.

[1]We present a generalized multipoint analysis of physical quantities, such as magnetic

field and plasma flow, based on spatial gradient properties, where the multipoint data

may be taken by irregular (distorted) configurations of any number of spacecraft.

The methodology is modified from a previous, fully 3-D gradient analysis technique,

designed to apply strictly to 4-point measurements and to be stable for regular spacecraft

configurations. Here, we adapt the method to be tolerant against distorted configurations and

to return a partial result when fewer spacecraft measurements are available. We apply the

method to a variety of important physical quantities, such as the electric current density and

the vorticity of plasma flows based on Cluster and THEMIS multiple-point measurements.

The method may also have valuable applications on the coming Swarm mission.

Citation: Shen, C., et al. (2012), Spatial gradients from irregular, multiple-point spacecraft configurations, J. Geophys. Res.,117,

A11207, doi:10.1029/2012JA018075.

1. Introduction

[2] The exploitation of the multipoint measurements from

the cluster spacecraft (S/C) have increasingly become the

manner in which the current exploration of space physics is

carried out, due to their ability to access spatial structure and

often separate the temporal and spatial variations of physical

quantities. The successful operation of multiple spacecraft

missions, as the earlier dual spacecraft ISEE1 and ISEE2

[Ogilvie et al., 1977; Russell and Elphic, 1978], AMPTE

missions [Bryant et al., 1985], Cluster (comprising four

identical S/C) [Escoubet et al., 1997, 2001], currently with

more than 10 year operation, and recently the THEMIS

mission (comprising five identical S/C) [Angelopoulos,

2009], has obtained many significant and original scientific

achievements. In the future, the Swarm mission (comprising

three S/C) [Friis-Christensen et al., 2006], is also due for

imminent launch. Nevertheless, it was the maintenance of the

close formation of the 4 S/C of Cluster which initially

allowed full 3-D spatial structure to be obtained regularly,

and the Magnetospheric Multiscale (MMS) mission [Curtis,

1999], due for launch in 2014, follows this close formation

flying, but at smaller spatial scales. Analysis tools which use

such close spacecraft configurations, however, need to be

selective in order to minimize errors as a consequence of

changing scales, evolution of the constellation shape and

complex or time dependent physical structure. Analysis of

the spacecraft data therefore benefits from robust methodol-

ogy which is tolerant to these factors as much as possible,

such as extremely distorted shapes of the S/C configuration,

and which is able to return particular results when fewer than

four point measurements are available.

[3] Previously developed analysis methods have demon-

strated furthermore that the radical separation of temporal-

spatial variation requires a cluster of at least four spacecraft.

As a paragon, the Cluster mission, which formed a small-

scale tetrahedron (hundreds km to thousands km) in the early

mission stages (2001–2004) can reveal well the local spatial

gradient of measured quantities. Taking the magnetic field B

as an example, the four-point measurements of Cluster tetra-

hedron, when accessed by a range of similarly developed

analysis methods [e.g., Harvey, 1998; Chanteur, 1998;

Chanteur and Harvey,1998;Shen et al., 2003, 2007; Shen and

Dunlop, 2008], allow the spatial gradients of B,including

magnetic field gradient rB, current density via m1

0rB,

and curvature of the magnetic field lines (MFLs) (B/Br)

(B/B), to be successfully derived in principle, although dif-

ferent methodology [Dunlop et al., 1988; Dunlop and

1

State Key Laboratory of Space Weather, National Space Science

Center and Center for Space Science and Applied Research, Chinese

Academy of Sciences, Beijing, China.

2

CAS Key Laboratory of Ionospheric Environment, Institute of

Geology and Geophysics, Chinese Academy of Sciences, Beijing, China.

3

Beijing National Observatory of Space Environment, Institute of

Geology and Geophysics, Chinese Academy of Sciences, Beijing, China.

4

Rutherford Appleton Laboratory, Didcot, UK.

5

Imperial College of Science, Technology and Medicine, London, UK.

6

Space Science Institute, Macau University of Science and Technology,

Macao, China.

7

College of Earth Science, University of Chinese Academy of Sciences,

Beijing, China.

8

Laboratory for Atmosphere and Space Physics, University of Colorado

Boulder, Boulder, Colorado, USA.

9

IRAP, UPS-OMP, University of Toulouse, Toulouse, France.

10

IRAP, CNRS, Toulouse, France.

Corresponding author: C. Shen, State Key Laboratory of Space

Weather, National Space Science Center and Center for Space Science

and Applied Research, Chinese Academy of Sciences, No.1 Nanertiao,

Zhongguancun, Haidian District, Beijing 100190, China. (sc@nssc.ac.cn)

©2012. American Geophysical Union. All Rights Reserved.

0148-0227/12/2012JA018075

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, A11207, doi:10.1029/2012JA018075, 2012

A11207 1of19

Woodward, 1998; Robert et al., 1998a] and complementary

techniques [Dunlop et al., 2002; Haaland et al., 2004a,

2004b; Shi et al., 2005, 2006] have treated the control of

errors in different ways or provided a variety of contextual

information for the interpretation of gradients.

[4] All these methods have advanced enormously our

knowledge of the dynamic magnetosphere. In the determina-

tion of spatial gradients, the inverse of the volumetric tensor

Rjl ¼1

NP

N

a¼1rajral,i.e.,R

jl

1

,(wherer

a

is the position vector of

the spacecraft arelative to the mesocenter of cluster, and j,

lare Cartesian components) plays an essential role in these

analysis methods [Harvey, 1998; Shen et al., 2003, 2007;

Shen and Dunlop, 2008, and references therein]. For the

regular Cluster tetrahedron, the direct operation of R

jl

1

works

well, whereas when the tetrahedron becomes much distorted

or irregular, especially when all the S/C are almost in the

same plane, R

jl

becomes an ill-matrix and the direct calcula-

tion of R

jl

1

then yields significant error. Thus, no useful

information about the gradients can be directly obtained for

that case.

[5] Nevertheless, in addition to the much distorted Cluster

tetrahedron, the plane- or line-like configuration of the

spacecraft cluster is common; such as the three closed-S/C

configuration within the first pass of the THEMIS mission

[Angelopoulos, 2009], and the planned operations of the

forthcoming Swarm mission [Friis-Christensen et al., 2006].

Thus, faced with the various cluster of spacecraft missions

having arbitrary configuration, a serious question is pre-

sented. That is, whether we can develop a new universal

method which cannot only avoid calculating the R

jl

1

directly,

but also may derive the correct information about the gra-

dients. For the three S/C array situation, some authors have

used spatial interpolation or reciprocal vectors method to

deduce the gradients of the physical quantities within the S/C

plane [Li and Chen, 2008; Vogt et al., 2009]. Vogt et al.

[2009], in particular, have made a systematic treatment of

the methodological framework of the planar reciprocal vec-

tors. Nevertheless, in this study, we aim to develop a general

method for determining the gradients of physical quantities

based on data from multiple-S/C with arbitrary shapes and

any number (1, 2, 3, 4, or more); suitable for linear or planer

S/C constellation. The three S/C situation is only one appli-

cation of this new approach.

[6] In the following, the approach for the new method is

presented in Section 2. In Section 3, we apply the new

method to the cases of a much distorted Cluster tetrahedron,

and to the cases of a three-point S/C array among the THE-

MIS and Cluster spacecraft. A discussion and summary are

given in Section 4.

2. Approach

2.1. General Theory

[7] We may deduce the gradient of a certain physical quan-

tity F(e.g., the density of plasmas, the components of magnetic

field or bulk plasma velocity) from its measurements by N

identical spacecrafts. At a specific time, the number aspace-

craft at the position r

a

(a=1,2,…,N) yields one measurement

of the physical quantity F,i.e.,F

a

(a=1,2,…,N).

The barycenter coordinates [Harvey, 1998; Chanteur, 1998]

are used here, i.e., the mesocenter, c, is taken as the origin

point, thus the position vector of the mesocenter rc¼

1

NP

N

a¼1ra¼0.

[8] Generally, the physical quantity Fis spatially varying

in space around the Nspacecraft. So that we may expand F

around the mesocenter cas

Fa¼Fcþr

nFðÞ

cranþ1

2rnrkFðÞ

cranrakþ…ð1Þ

where the subscripts n,kdenote Cartesian components. F

c

and rnFðÞ

care the value and the gradient of Fat the

mesocenter, and F

a

is the measured value at the position of

the spacecraft a.

[9] Averaging over the value of Fmeasured by the N

spacecraft yields

1

NX

a

Fa¼Fcþr

nFðÞ

c

1

NX

a

ranþ1

2rnrkFðÞ

c

1

NX

a

ranrakþ…

Then, F

c

can be obtained as

Fc¼1

NX

a

Fa1

2rnrkFðÞ

c

1

NX

a

ranrakþ…¼1

NX

a

Fa

L=DðÞ

2Oþ…ð2Þ

where, Dis the characteristic size of the physical structure,

and Lis typical scale of the spacecraft cluster. Thus, from

equation (2), F

c

is the average of all the measured quantities

Fby the Nspacecraft with the error of order of (L/D)

2

.

[10] In order to get the gradient rnFðÞ

c, we multiply both

sides of equation (1) by the lcomponent of r

a

, i.e., r

al

, and

further make average over Nspacecraft to yield

1

NX

a

Faral¼Fc

1

NX

a

ralþr

nFðÞ

c

1

NX

a

ranralþ1

2rnrkFðÞ

c

1

NX

a

ranrakralþ…;

or (since P

N

¼1

ra¼0)

1

NX

a

Faral¼r

nFðÞ

cRnl þ1

2NrnrkFðÞ

cX

a

ranrakralþ…;ð3Þ

where Rnl ¼1

NX

a

ranralis the volume tensor [Harvey,

1998].

[11] Generally, the gradient of Fcan be obtained by mul-

tiplying both sides of equation (3) by R

ln

1

as

rnFðÞ

c¼1

NX

N

a¼1

FaralR1

ln þL=DðÞO:ð4Þ

[12] The procedure to estimate equation (4) is the old

method that was used in the previous studies [Harvey, 1998;

Shen et al., 2003, 2007]. If the polyhedron of the spacecraft

cluster is distorted considerably, the corresponding volume

tensor R

nl

would become abnormal, however, and the above

formula fails to yield the gradient of Faccurately. In order to

overcome this difficulty, we may process the equation in the

space of the eigenvectors of R

nl

.

[13] The volume tensor R

nl

is symmetric, and has three

nonnegative eigenvalues w

1

,w

2

and w

3

(w

1

≥w

2

≥w

3

≥0)

SHEN ET AL.: TECHNIQUE A11207A11207

2of19

with three corresponding eigenvectors ^

k1ðÞ

,^

k2ðÞ and ^

k3ðÞ

(where ^

klðÞ^

kmðÞ

¼dlm)[Harvey, 1998]. ^

k1ðÞ

,^

k2ðÞand ^

k3ðÞ

are the three characteristic directions of the spacecraft clus-

ter, which can constitute an orthogonal coordinates with

^

k1ðÞ¼^

k2ðÞ^

k3ðÞ and form the eigenvector space. ﬃﬃﬃﬃﬃﬃ

w1

p,

ﬃﬃﬃﬃﬃﬃ

w2

pand ﬃﬃﬃﬃﬃﬃ

w3

pare the characteristic half widths of the

spacecraft cluster in these three characteristic directions,

respectively [Harvey, 1998]. It is noted here that the char-

acteristic size of the spacecraft cluster is defined as L¼

2ﬃﬃﬃﬃﬃﬃ

w1

p, the elongation as E¼1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

w2=w1

p, and the planarity

as P¼1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

w3=w2

pin Robert et al. [1998b]. The volume

tensor R

ij

can be rewritten as

Rij ¼wlklðÞ

iklðÞ

j:ð5Þ

[14]IfFis scalar, e.g., the density, temperature, pressure

etc., then in the eigenvector space of R

ij

, equation (3) can be

expressed as

1

NX

a

Fa~

ral¼~

rjF

cwldjl þ1

2N

~

rj~

rkF

cX

a

~

raj~

rak~

ralþ…;

ð6Þ

where, the quantity with wavy superscript represents the

projection component in the eigenvector space. For example,

~

rlis the lth component of the vector rin the eigenvector

space.

[15] From equation (6), ~

rlF

ccan be expressed easily as

~

rlF

c¼1

Nwl1X

a

Fa~

ral1

2Nwl1~

rj~

rkF

cX

a

~

raj~

rak~

ralþ…

¼~

GF

lþL=DðÞO;ð7Þ

where,

~

GF

l¼1

Nw

lX

a

Fa~

ral:ð8Þ

[16] Equations (7)–(8) are the formulae for calculating the

gradient of a scalar physical quantity (at the mesocenter of the

cluster polyhedron) which demonstrates the calculation has a

truncation error of order L/D. The formula (7)–(8) have a clear

meaning that, the gradient of a scalar physical quantity along

one eigenvector direction can be estimated by the weighted

average of the difference between the measurements at these

spacecraft over the corresponding characteristic length of

spacecraft cluster.

[17] For a vector physical quantity, e.g., the magnetic field

B, its gradient in the eigenvector space is

~

rl~

Bi

c¼1

Nwl1X

a

~

Bai~

ral1

2Nwl1~

rj~

rk~

Bi

cX

a

~

raj~

rak~

ralþ…

¼~

GB

il þL=DðÞO;ð9Þ

Where

~

GB

il ¼1

Nw

lX

a

~

Bai~

ral:ð10Þ

[18] The formulae (9)–(10) may also be used to calculate

the gradient of the bulk velocity of plasmas, for example. It is

implied distinctly by formulae (9)–(10) that, the vector field

gradient of one component along one eigenvector direction

can be estimated by the weighted average of the difference

between the measurements at these spacecraft divided by the

corresponding characteristic length of spacecraft cluster.

[19] Since the formulae (7)–(8) and (9)–(10) calculate the

gradients of scalar or vector physical quantities in the eigen-

space of the volumetric matrix, they avoid the difficulty of an

abnormal volume matrix when the polyhedron of the space-

craft cluster is severely distorted. If one of the eigenvalues

equals zero, then the gradient of Falong the corresponding

eigenvector cannot be determined. For example, if the four

S/C of Cluster are in a plane, then we can only determine the

components of gradient in that plane. Nevertheless, even if the

spacecraftcluster is aligned or is in a plane, the above formulas

are still valid in some directions, and moreover the eigenvec-

tors, being the principle directions for the configuration, nat-

urally order the calculation.

[20] In Appendix A, the errors of this method have been

evaluated and discussed. The advantage of this method is

that, it is very natural and general, and represents the essence

of multipoint data analysis. This approach is valid for situa-

tions with any number S/C and arbitrary kind of shape of the

S/C array.

[21] The transformation of the full gradients in the eigen-

vector space to the Cartesian coordinates can be expressed as

GB

ij ¼klðÞ

i

T~

GB

lmkmðÞ

j:ð11Þ

2.2. Linear Configuration Case

[22] When the spacecraft array is aligned, we may obtain

the gradient along the S/C line, as indicated by the formulas

(9)–(10). We may check the simplest case when there is only

two S/C. Assuming the distance between them is L, the

position vectors of the two S/C in the mesocenter coordinates

are r1¼1

2L^

k1ðÞ

,r2¼1

2L^

k1ðÞ

, respectively. Then the vol-

ume tensor of the two S/C array is

R¼1

2X

2

a¼1

rara¼L

2

2

^

k1ðÞ

^

k1ðÞ

:ð12Þ

[23] The first eigen vector ^

k1ðÞis along the S/C line. The 3

eigenvalues of the volume tensor are w1¼L

2

2,w

2

=w

3

=0,

respectively.

[24] Only the gradient in the direction of the S/C line or

^

k1ðÞcan be obtained, according to the formula (9) and (10),

that is

^

k1ðÞr

B¼1

2w1X

2

a¼1

Ba~

ral¼B2B1

L;ð13Þ

where B

1

and B

2

are magnetic field measured by the two S/

C, respectively; obviously a reasonable result.

2.3. Planar Configuration Case

[25] If the spacecraft lie in a plane, the least eigenvalue of

R

jl

,w

3

equals zero, and the third eigenvector ^

k3ðÞ

is aligned to

SHEN ET AL.: TECHNIQUE A11207A11207

3of19

the normal ^

nof the spacecraft plane. Although it is impos-

sible to get the full gradient of physical quantities, we can still

obtain the gradient of physical quantities at the mesocenter in

the directions of the first eigenvector ^

k1ðÞand second eigen-

vector ^

k2ðÞfrom the equations (7)–(8) and (9)–(10), i.e.,

~

rlF≈1

Nwl1X

a

Fa~

ral;l¼1;2;ð14Þ

~

rl~

Bi≈1

Nwl1X

a

~

Bai~

ral;l¼1;2;ð15Þ

where the subscript cis omitted for clarity.

[26] When the spacecraft are in a plane, we are not able to

deduce the complete curl of magnetic field, but we can still

obtain the component of the curl of the magnetic field or the

current density along the normal of the spacecraft plane or

the third eigenvector ^

k3ðÞ

.

[27] The current density is the curl of the magnetic field, i.e.,

rBðÞ¼m0J:ð16Þ

The 3rd component of the current density is

m0~

J3¼^

k3ðÞm0J¼^

k1ðÞ^

k2ðÞ

m0J¼^

k2ðÞm0J

^

k1ðÞ

¼^

k2ðÞrB

^

k1ðÞ

;¼rBðÞ

^

k2ðÞ^

k2ðÞr

B

hi

^

k1ðÞ¼^

k1ðÞrBðÞ

^

k2ðÞ^

k2ðÞrBðÞ

^

k1ðÞ

:ð17Þ

The above equation is the formula for the normal component

of the current density, which may be further expressed as

m0~

J3¼~

r1~

B2~

r2~

B1¼1

Nw

1X

a

~

Ba2~

ra11

Nw

2X

a

~

Ba1~

ra2:

ð18Þ

Here, ~

ri~

Bj¼^

kiðÞrBðÞ

^

kjðÞ:The above formulas can

obviously be applied to deduce the normal component of the

current density from present THEMIS 5-point magnetic mea-

surements, where, in their string of pearls configuration during

the first operational pass, and later in the mission, often only 3

spacecraft formed a close array.

[28] For the case of the present Cluster and THEMIS

3-point plasma measurements, similar analysis may be done

with the gradient of the plasma velocity as is already carried

out for magnetic field. In the directions of the first eigen-

vector ^

k1ðÞand second eigenvector ^

k2ðÞ

, the gradients of the

plasma velocity are

~

rl~

Vi≈1

3wlX

a

~

Vai~

ral;l¼1;2:ð19Þ

Accordingly, the vorticity of the velocity along the normal ^

n

of the spacecraft plane or the third eigenvector ^

k3ðÞcan be

obtained as

~

W3¼~

r1~

V2~

r2~

V1¼1

Nw

1X

a

~

Va2~

ra11

Nw

2X

a

~

Va1~

ra2:

ð20Þ

The new approach developed here can yield as much useful

information on the gradient of the physical quantities for the

multiple point measurements as possible when the spacecraft

polyhedron becomes abnormal and the old method fails.

[29] In Appendix B, in order to check the correctness of the

new method, a test calculation has been made for obtaining

the normal current density from three S/C measurements in

the dipolar geomagnetic field, which has confirmed that the

new method is correct.

[30]Vogt et al. [2009] argued that, under certain con-

strains, for example, as the gradient becomes parallel or

perpendicular to a given vector, or when the stationarity

assumption is valid, the normal component of the gradient

can also be derived via the three-point analysis. Here, we

stress one common and important situation when the current

is field aligned or the force free condition is satisfied, which

has not been discussed by Vogt et al. [2009]. Under the force-

free condition, along with the solenoidal condition of the

magnetic field, our approach can also yield the normal

component of the magnetic gradient so that the full gradient

of the magnetic field is obtained. Assuming that the electric

current in the explored regions is field aligned, or the mag-

netic field is force-free; that is, the current density is J¼^

bJk,

we have

~

J3¼^

k3ðÞJ¼^

k3ðÞ^

b

Jk;Jk¼~

J3=^

k3ðÞ^

b

;ð21Þ

Then the current density is

J¼^

bJk¼^

b~

J3=^

k3ðÞ^

b

:ð22Þ

With equation (18), we get

J¼^

b1

m0^

k3ðÞ^

b

~

r1~

B2~

r2~

B1

:ð23Þ

So that the current density is completely determined when the

current is field-aligned.

[31] For 3-point S/C measurements, the gradient of the

magnetic field can be deduced in the S/C plane, as illustrated

in equation (15). Therefore, the following 6 components of

the gradient rBare already known:

~

r1~

B1;~

r1~

B2;~

r1~

B3

;and ~

r2~

B1;~

r2~

B2;~

r2~

B3

:

[32] Using the Ampere law and solenoidal condition of the

magnetic field (rB= 0), we may further determine the

other 3 component of the gradient rBalong the eigenvector

^

k3ðÞ

. That is

m0~

J1¼~

r2~

B3~

r3~

B2;ð24Þ

m0~

J2¼~

r3~

B1~

r1~

B3;ð25Þ

~

r1~

B1þ~

r2~

B2þ~

r3~

B3¼0:ð26Þ

SHEN ET AL.: TECHNIQUE A11207A11207

4of19

Then we further get

~

r3~

B1¼~

r1~

B3þm0~

J2¼~

r1~

B3þ^

k2ðÞ^

b

^

k3ðÞ^

b

~

r1~

B2~

r2~

B1

;ð27Þ

~

r3~

B2¼~

r2~

B3m0~

J1¼~

r2~

B3^

k1ðÞ^

b

^

k3ðÞ^

b

~

r1~

B2~

r2~

B1

;ð28Þ

~

r3~

B3¼~

r1~

B1~

r2~

B2:ð29Þ

where, in equation (27) and (28), the first and second com-

ponents of Jare already known, i.e., ~

J1¼^

k1ðÞJand ~

J2¼

^

k2ðÞJ, and equation (23) has been used. Thus, the gradient

of the magnetic field along the eigenvector ^

k3ðÞor the nor-

mal of the S/C plane is readily derived.

[33] When the magnetic field is non-curl or the electric

current is ignorable in the measured region, equations (27)

and (28) reduce to

~

r3~

B1¼~

r1~

B3;ð30Þ

~

r3~

B2¼~

r2~

B3;ð31Þ

So far all the components of the gradient of magnetic field

(rB) are determined, as shown in equations (15), (27), (28),

and (29), under the assumption that the electric current is

field-aligned. For the special situations when there is no or

only very weak electric current (magnetic field is non-curl),

the above equations (15), (29), (30), and (31) can readily

yield the complete gradient of the magnetic field.

[34] In Appendix C, a test calculation of a three-point S/C

crossing force-free flux rope is made. The test results have

demonstrated that our method can recover well the full

gradient of the magnetic field in the model.

[35] It is worth noting that Vogt et al. [2009], have yielded

a different formula for determining the gradient from three

S/C measurements, using reciprocal vectors. They have put

forward new kinds of reciprocals, qa¼srbg

s

jj

2a¼1;2;3ðÞ,

where sis the normal vector defined as s=r

12

r

13

, and the

gradient in the S/C plane is G¼P

3

a¼1

qaFa. We note that their

reciprocals q

a

are different from those for deducing the full

gradient from the 4-point measurements [Chanteur, 1998].

In Appendix D, we have discussed the relationship of these

two approaches. It is shown that the new method can yield

general formulae of the reciprocals for various spacecraft

arrays with any number and any shape. The uniqueness of

the reciprocal vectors has been verified. Therefore, for the

four spacecraft array forming one tetrahedron, the 4 recip-

rocal vectors as got in the new method are equivalent to

those of Chanteur [1998]. For the three spacecraft array, the

two methods are equivalent on obtaining the reciprocals as

well as the gradient.

[36] In the following Section, we will apply the new

method to several cases and explore its usefulness.

3. Applications of the New Method

[37] In order to exhibit best the value of the new method,

in this section we concentrate on cases of much distorted

tetrahedron, or planar configurations. The geocentric solar

magnetospheric (GSM) base coordinates are used throughout

this study and the spherical coordinates for the vector direc-

tion (q,8) in the frame of GSM have also been used. The

polar angle q(0≤q≤180) is the angle between the positive

Z-axis and the vector direction; the azimuth angle 8(0≤8≤

360) is the angle between the positive X-axis and the line

from the vector direction projected onto the XY-plane. For

example, the dawn direction or Y-direction is (90, 270),

while the dusk direction or +Y-direction is (90,90

).

3.1. Deducing the Current Density From

Cluster 4-Point Measurements

[38] In this section, with the multipoint magnetic field

measurements of Cluster [Balogh et al.,2001],wewillcal-

culate the current density in the tail lobe region by using the

new method developed above and also compare with that

given by the old method. As known from the last Section, if

the Cluster tetrahedron is planar, the old method will fail to

yield a reasonable current density; however, the new method

can still yield the gradient of the magnetic field in the space-

craft plane and thus obtain the component of the current den-

sity along the normal of the spacecraft plane. We compare

results from both methods below for the same observations.

[39] We consider the case during the period 11:00–15:00 UT

on 05 September 2005, when the average location of Cluster

tetrahedron was at X = 17.5 R

E

,Y=2.1 R

E

,Z=6.8 R

E

.

As shown in Figure 1, the Cluster array is stably located in the

southern lobe region as B

x

25 nT (Figure 1a). The mag-

netic field strengths measured by the four S/C have little dif-

ferences between them (Figure 1b). In the interval investigated,

the configuration of Cluster tetrahedron is highly distorted

because the minimum eigenvalue (w

3

) of the volumetric tensor

is much small (Figure 1c). In particular, at 13:08, the highly

distorted tetrahedron becomes planar as indicated by w

3

being

nearly zero and the dramatic change appears in the estimates of

current density.

[40] As shown in Figure 1d, if the old method is applied

(using equation (4)), the derived current density (m1

0rB)is

considerably enhanced when the distorted tetrahedron

becomes planar around 13:08 UT, that is obviously nonphys-

ical and unreal. Therefore, the old method fails to calculate the

magnetic gradients during planar S/C configuration period and

no correct information about the current density can be

obtained around this time. The reason for this is that, if the

separation of the S/C in one characteristic direction is so small

that the difference of the real magnetic field at along this

dimension is less than the absolute error of magnetic detector,

the gradient of the magnetic field along this direction cannot be

calculated correctly. Therefore, the validity of old method

requires that each of the three eigenvalues of volume tensor is

sufficiently large, i.e.,

ﬃﬃﬃﬃﬃ

wi

p≫dB

rB

jj

;i¼1;2and3 ð32Þ

SHEN ET AL.: TECHNIQUE A11207A11207

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where, |rB| is the measure of the gradient of the magnetic

field, and dBis the absolute error of the magnetic field

which is less than 0.1 nT [Dunlop et al., 1990]. It is noted

that, equation (32) is equivalent to (A11) in Appendix A.

However, if the critical condition (32) is violated, i.e.,

ﬃﬃﬃﬃﬃ

wi

p≤dB

rB

jj

,i= 1, 2 or 3, the old method will totally fail, and

the new method should be applied to draw useful information.

[41] Furthermore, we may re-examine this case with the

new method, which expresses the magnetic gradients in the

eigenspace of volume tensor as in equation (15), so that the

current density can be obtained in that eigenspace with

Figure 1. From top to bottom: (a) the magnetic field at the mesocenter of Cluster tetrahedron; (b) the

strength of magnetic field for Cluster four-S/C respectively; (c) the square roots of the three eigenvalues

of the volumetric tensor; (d) the current density calculated by the old method; (e) the component of current

density along ^

k1ðÞand the directional angles (q

1

,j

1

) of the first eigenvector ^

k1ðÞ

; (f) the component of cur-

rent density along ^

k2ðÞ

and the directional angles (q

2

,j

2

) of the second eigenvector ^

k2ðÞ

; (g) the component

of current density along ^

k3ðÞand the directional angle (q

3

,j

3

) of the third eigenvector ^

k3ðÞ

; (h) the com-

parison of the total current density between the old method and new method.

SHEN ET AL.: TECHNIQUE A11207A11207

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equation (18) (Figures 1e–1g). It is noted that the directions

of eigenvectors ^

k1ðÞ

,^

k2ðÞ

, and ^

k3ðÞare approximately along

the directions of Y, Z and +X axis, respectively, for the

whole interval. Around 13:08, although the derived J

1

(along ^

k1ðÞ

) and J

2

(along ^

k2ðÞ

) components are considerably

enhanced and obviously unreal, we still can reasonably

derive the J

3

(along ^

k3ðÞ

) component. It can be seen from

Figure 1g that the earthward component of current density

(J

3

) is very small and nearly zero, which is in consistence

with the traditional understanding on the current distribution

in the tail lobe region. On contrast, we cannot obtain any

component of the current density in this interval using the

old method (Figure 1d).

[42] However, when the above critical condition (32) is

satisfied, the old method and new method will yield the same

results, as shown in Figure 1h. Actually, throughout the whole

interval, the strength of total current density derived from the

old method and new method is the same (Figure 1h). Around

13:08, both methods yield unreal values of the strength of the

current density although the new method yields correct com-

ponent J

3

.

3.2. Deducing the Current Density From THEMIS

Three-Point Measurements

[43] THEMIS mission [Angelopoulos, 2009; Auster et al.,

2009], which are composed of five identical spacecraft and

have been launched into space on 17 Feb 2007, aims to

explore the global large scale evolution processes of the

magnetospheric substorms. During the early phase (in 2007)

of the mission, the spacecrafts were placed into a close ‘string

of pearls’configuration. From Sep 2007 through Sep 2009,

the outmost two spacecraft are far distant from the other three

space-craft. In addition, since 29 Dec 2009, the outside two

spacecraft (P1 and P2) have been placed near the lunar orbit,

and the remaining three spacecraft (P3, P4 and P5) have

separations of about 5003000 km. Therefore, we may apply

the new method to three point (P3, P4 and P5) measurements

and deduce the component of the current density along the

normal of the S/C plane from the 3-point magnetic field

measurements of THEMIS from equation (18).

[44] For illustration, we may make analysis on one case on

29 May 2010 when the THEMIS array is inbound from the

near-Earth tail (radial distance 11 R

E

) to the inner mag-

netosphere and at the same time a moderate geomagnetic

storm (the minimum Dst index is 85 nT) occurs. The

orbits of the three THEMIS spacecraft (P3, P4 and P5)

during the period 00:00–14:00 UT on 29 May 2010 are

demonstrated in Figure 2 where the separations of the three

S/C are amplified by a factor of five for the resolution.

[45] The measured magnetic field vector at the mesocenter

of the three S/C array, i.e., Bc¼1

3P

3

a¼1Ba, and the magnetic

strength measured by each S/C are shown in Figures 3a and

3b, respectively. For the interval 00:00–12:30, as indicated

by the B

z

component (Figure 3a, in blue) being rather small,

Figure 2. THEMIS’orbit (blue line for P5) during 03:00–14:00 UT on 29 May 2010, (a) projected in the

X-Y plane, and (b) projected in the Y-Z plane in GSM coordinates. In both panels, the separated size

between P3 (red dots), P4 (green dots) and P5 (blue dots) are amplified by a fact of 5. The nominal mag-

netopause with standoff distance r

0

= 10 RE and tail flaring level a= 0.5 provided by Shue’s model [Shue

et al., 1997] is shown as dashed lines.

SHEN ET AL.: TECHNIQUE A11207A11207

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Figure 3. From top to bottom: (a) the magnetic field at the mesocenter of the spacecrafts plane; (b) the

strengths of magnetic field at tha(P5), thd(P3) and the(P4), respectively; (c) the position of the mesocenter

in GSM spherical coordinates; (d) the square roots of the three eigenvalues of the volumetric tensor of P5,

P3 and P4; (e) the directional angle of the normal to the spacecraft (P5, P3 and P4) plane, i.e., the direction

of ^

k3ðÞ

; (f) the component of the current density along the normal of S/C plane; (g) the index of AE and

SYM_H.

SHEN ET AL.: TECHNIQUE A11207A11207

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the three S/C are in the stretched magnetotail. Particularly

for the short interval 10:5012:30, the total field reaches the

minimum and B

x

component reverses the sign, the three S/C

are crossing the inner plasma sheet. Then, for the interval

12:30–14:00, the three S/C transit from the tail region to the

dipolar field region as indicated by the gradually growing B

z

component and the decreasing radial distance (Figure 3d).

[46] As the three THEMIS S/C are coplanar, the volume

tensor is abnormal and one eigenvalue is zero, so that the old

method (equation (4)) is not fitful for analysis as indicated by

the violation of the criteria (32). Nevertheless, the new method

is still applicable for obtaining the component of the current

density along the normal of the S/C plane (equation (18)).

[47] During the whole interval of 00:00–14:00 UT, as

indicated by the eigenvalues of volume tensor in Figure 3d,

the separation of the three S/C is generally less than 3000 km.

By using the new method (equation (18)), the component of

current density along the normal of the three-S/C plane

(Figure 3e), i.e., J

3

, is calculated (Figure 3f). As it happens, in

this case, a moderate geomagnetic storm occurs as indicated

by the SYM_H index (Figure 3g) and the minimum SYM_H

index is about 75 nT. The investigated period spans the

whole main phase of the magnetic storm, and there is per-

sistent substorm activities with the maximum AE index being

about 1800 nT. In the initial stage during 00:00–06:00 UT of

the main phase, the three THEMIS S/C are operating around

their apogee in the near-Earth tail lobe region (r 11 R

E

,B

x

50 nT), and the eigenvector ^

k3ðÞ

, that is also the normal to

the S/C plane, is about along the direction of positive Yor

duskward (Figure 3e). The calculated J

3

, which is about the

duskward component of the current density, is constantly

around 3 nA/m

2

with small fluctuations (Figure 3f). In the

period of 08:00–14:00 UT, which is the most active stage of

the main phase of geomagnetic storm, the three S/C move

into the inner magnetosphere while their radial distance is

decreasing from 10 R

E

to 5R

E

(Figure 3c). During this

stage, the eigenvector ^

k3ðÞis roughly along the direction of

negative Yor dawnward (Figure 3e), and the corresponding

J

3

derived was duskward or westward with great enhance-

ment. It is noted that, the enhanced J

3

reaches the maximum

value 25 nA/m

2

, for the short interval of 10:5012:30

when THEMIS are in the center part of inner plasma sheet.

[48] One may argue that, the enhanced J

3

is possible induced

by the spatial effect, i.e., S/C moves from one region to the other

region, instead of the result induced by the magnetic storm. To

check this argument, the THEMIS orbits before or after the case

for several days are also surveyed, but similar trend of enhanced

J

3

is not observed for those orbits. Therefore, the enhancement

of the westward J

3

for this case is directly driven by the mag-

netic storm. So that it is yielded that, during the main phase of

this geomagnetic storm, the electric current density enhances

greatly in the inner plasma sheet. It is unclear about the prop-

erties of the inner plasma sheet with strong current density, and

the roles it would play in the evolution of magnetic storms.

Theseissuesdeservetobeexploredinthefuture.

[49] Anyway, the enhancement of the westward J

3

obtained from THEMIS 3-point measurements during geo-

magnetic storm is consistent with the traditional picture of

well-known partial ring current, and also consistent with the

statistical observation of ring current as obtained from

Cluster 4-point analysis [Zhang et al., 2011].

[50] In this section, it has been confirmed that the new

method developed here may be used to derive the current

distribution in the near earth tail region, as well as in the ring

current region, based on THEMIS 3-point magnetic field

measurements.

3.3. Deducing the Vorticity of Plasma Flows From

THEMIS Three-Point Measurements

[51] We may also apply this method to obtain the gradient

of the velocity of plasma flows. Particularly for the three-

point observations of plasma flow, with the formula (19) and

(20) the gradients of flow in the plane of S/C cluster can be

calculated, and further the component of flow vorticity along

the normal of S/C plane can be derived. This is beyond the

ability of the old methods.

[52] One case analysis on THEMIS observations will be

made to show the usefulness of the method developed above.

During the period 10:22–10:32 UT on 17 March 2008, the

five S/C of THEMIS are located in the near-Earth magneto-

tail with geocentric distances of about 8 R

E

to 14 R

E

. With

THEMIS simultaneous measurements of plasma flows for

this case, Panov et al. [2010] have claimed that the earthward

or tailward flow bursts can lead to the formation of flow

vortices. To check this argument of Panov et al. [2010] and

also to exhibit the ability of the new method developed

here, we may re-examine quantitatively this case with the

three-S/C measurements of THEMIS mission.

[53] During this interval, the locations of the three-S/C of

THEMIS in GSM are as follows: P1 at (12.7, 3.3, 0.2) R

E

,

P2 at (11.1, 2.7, 1.2) R

E

, and P4 at (10.2, 3.3, 1.6) R

E

.

The data from ESA (the reduced mode with 3-s resolution)

[McFadden et al., 2009] have been used to obtain the plasma

velocity in the frame of GSM coordinates. The plasma velocity

for P1, P2 and P4 are shown in Figures 4a–4c, respectively. In

these panels, the V

x

,V

y

components and the total speed are

presented by the blue line, green line and black line, respec-

tively. As indicated by the eigenvalues of the volume tensor,

the separation size of these S/C exceeds 1 R

E

(Figure 4d). The

normal direction of the S/C plane, i.e., ^

k3ðÞ

(150,174

), points

roughly toward the southward (Figure 4e). With equation (20),

the flow vorticity along the direction of ^

k3ðÞ

,i.e.,W

3

,isesti-

mated and shown in Figure 4f. It can be seen from Figure 4f

that, the derived W

3

is about always negative, implying the

vorticity ofthe flow velocity has a significant component at the

northern direction.

Figure 4. From top to bottom: (a) the velocity of plasma flow of P1; (b) the velocity of plasma flow of P2; (c) the velocity of

plasma flow of P4; (d) the square roots of the three eigenvalues of the volumetric tensor; (e) the directional angles of the nor-

mal to the spacecraft plane, i.e., the direction of third eigenvector ^

k3ðÞ

; (f ) the component of the flow vorticity along the normal

of S/C plane (^

k3ðÞ

). The four vertical black lines mark the times when the calculated vorticity W

3

reach the maximum values.

For the lower part, the orientations of plasma flows at the locations of the three S/C are showed in the XY plane corresponding

to the marked times, where, P1, P2 and P4 are labeled as red, green and blue dots, respectively.

SHEN ET AL.: TECHNIQUE A11207A11207

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Figure 4

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Figure 5. Evolution of the K-H waves observed by Cluster S/C (C1, C3 and C4) during the period

20:26–20:42 UT on 20 Nov. 2001. From top to bottom: (a) the magnetic field at the mesocenter of triangle

plane of C1, C3 and C4; (b) the B

z

component of magnetic field measured by the three S/C; (c) the V

x

com-

ponent of CODIF H

+

flows measured by the three S/C; (d) the V

y

component of CODIF H

+

flows mea-

sured by the three S/C; (e) the V

z

component of CODIF H

+

flows measured by the three S/C; (f ) the

square roots of the three eigenvalues of the volumetric tensor; (g) the directional angles of the normal

to the spacecraft plane, i.e., ^

k3ðÞ

; (h) the component of current density along ^

k3ðÞ

; (i) the component of flow

vorticity along ^

k3ðÞ

. Five vertical black lines mark the time when the calculated J

3

reaches the extremes.

SHEN ET AL.: TECHNIQUE A11207A11207

11 of 19

[54] The four vertical black lines in Figure 4 mark the times

when W

3

reaches its extreme. To check whether the derived

W

3

is reasonable, we have plotted the orientations of plasma

flows at the locations of the three S/C in XY plane at the four

marked times in the lower part of Figure 4. It can be clearly

seen from all the four plots that, the orientations of plasma

flows at the locations of the three S/C consistently satisfy the

expected swirl of vortex (denoted as the black arrowhead

circles) which has a northward flow vorticity. This means

that the derived W

3

by the new method is reasonable and can

be the estimation of the component of flow vorticity along

the normal of S/C plane.

3.4. Deducing the Vorticity of Plasma Flows From

CLUSTER Three-Point Measurements

[55] In contrast to the four-point measurements of mag-

netic field, Cluster mission can only obtain 3-point observa-

tions of the plasma measurement due to the failure of the CIS

onboard C2 [Rème et al., 2001]. It is impossible, therefore,

to derive the plasma-related gradients by the previous methods.

With the method developed in this study, we can obtain the

plasma-related gradients in the plane of three-S/C cluster as

having been carried out in section 3.3. In this section, to

further show such unique ability of the new method, we apply

it to the analysis of Cluster plasma data.

[56] Due to the strong flow shear at the flank magneto-

pause, the Kelvin-Helmhotz (K-H) instability can be driven

by the down-streamed magnetosheath flow as manifested by

the rolled-up K-H waves. One well-known case of K-H

waves observed by Cluster is the event occurred during the

period of 20:26–20:42 UT on 20 Nov. 2001, which has been

previously investigated by Hasegawa et al. [2004]. Here, we

may apply the new method to re-examine this case, so that

some quantitative features of K-H waves can be revealed.

[57] In this case, Cluster is at the dusk flank of the mag-

netosphere, and the average location of Cluster tetrahedron is

at (3.6, 18.7, 2.7) R

E

in GSM. As shown in Figure 5, the

B

z

component of the magnetic field is enhanced periodically

while the magnetic strength has only small variations

(Figures 5a and 5b). The plasmas are flowing mainly anti-

sunward (Figures 5c–5e). It is well known that, within the

magnetopause, the plasma temperature is higher and the

density is smaller, while in the magnetosheath the opposite is

true with down-streaming sheath flows. So repetitive varia-

tions of magnetospheric-like plasmas to magnetosheath-like

plasmas [see Hasegawa et al., 2004, Figure 2], as well as the

anti-sunward plasma flow, demonstrates that the Cluster

spacecraft are located in the low latitude boundary layer

region and crossing the flank magnetopause repetitively. The

Figure 6. Variations of the plasma properties of the K-H waves observed by Cluster S/C (C1, C3 and

C4) during the period 20:26–20:42 UT on 20 Nov. 2001. From top to bottom, the B

z

component of mag-

netic field, the density of CODIF H

+

, and the plasma temperature, respectively. Five vertical black lines

mark the time when the calculated J

3

reaches the extremes.

SHEN ET AL.: TECHNIQUE A11207A11207

12 of 19

separation size between C1, C3, and C4 is about 1400 km

(Figure 5f), much less than the width of the K-H vortex and

the thickness of the boundary layer [Hasegawa et al. 2004].

Therefore, such three-point measurements can be well used

to determine the local gradients of the magnetic field and

plasma moments with the new method. The direction of ^

k3ðÞ

,

i.e., the normal of the three S/C plane, is (48.2, 6.0),

pointing about northward (Figure 5g). By using equation (18)

and equation (20), the component J

3

of the current density

and the component W

3

of the plasma flow (CODIF H

+

) vor-

ticity along ^

k3ðÞ are derived, which are illustrated in

Figures 5h and 5i, respectively.

[58] As shown from Figure 5 by the marked vertical black

line, both the derived J

3

and W

3

are periodically pulse-

enhanced simultaneously along with the jumps of B

z

. Based

on a detailed check of the plasma density and temperature as

shown in Figure 6, it is found that, the pulse-enhanced J

3

and W

3

occurred when S/C transits from magnetopause to

magnetosheath (see Table 1). Such periodical pulse-

enhanced flow vorticity can be regarded as a direct indictor

of the rolled-up K-H waves. In addition, the wave period

implied by Table 1 is about 34 min, and the detected

sheath flow is about 220 km/s, so the wavelength can

roughly be estimated as 68R

E

.

4. Summary and Discussions

[59] Multiple spacecraft measurements have increasingly

become the mainstream manner of space exploration. The 4

and 5 Cluster and THEMIS spacecraft have been successfully

launched into orbits, operating satisfactorily for more than a

decade in the case of Cluster. Fruitful results have been

achieved from the multipoint observation data, but the

methodology applied depends on the close arrays of space-

craft achieved. In the near future, Swarm (3 S/C) and MMS

(4 S/C) will also been launched into the magnetospheric

space. The satellite configurations of these missions are

diverse. The existent methods for multiple-S/C data analysis

are primarily defined for determining the full gradients from

the four-point measurements of the phased Cluster array in a

regular tetrahedron configuration.

[60] In this research, we have deduced the gradients in the

coordinates of the eigenvector space of the volume tensor of

satellite cluster. This new approach cannot only be applied to

the analysis of data from a S/C cluster with the regular tet-

rahedron configuration, as in the previous methods, but also

can be used successfully to draw the gradients from the data

observed by S/C cluster with the irregular (distorted) con-

figurations, e.g., in a planar or linear configuration. If the S/C

cluster is planar, or there is only 3 S/C, the gradients of the

physical quantities in the S/C plane can still be obtained,

although the gradient along the normal of the S/C plane is not

available; and furthermore, the component of the curl of

magnetic field along the normal of the spacecraft plane can

be deduced. If the S/C cluster is aligned, only the gradient

along the S/C line can be determined. It is also shown that,

under the force free assumption along with the divergence

free condition of the magnetic field, the full magnetic gradi-

ent can be obtained based on three-spacecraft magnetic

measurements. In Appendix C, a test calculation of three-

point S/C crossing force-free flux rope is made, which has

confirmed the new approach provides high accuracy.

[61] To demonstrate the abilities of the new approach

developed in this research, four case analyses have been

carried out. First we have studied the tail current density

calculation based on the 4 point magnetic field data of Cluster

with abnormal tetrahedron. Even if the Cluster tetrahedron

becomes planar, the component of the current density along

the normal of the S/C plane can be deduced and here its value

is almost zero in the tail lobe. In the second case, with the

3-point THEMIS data, we try to deduce the component of the

current density along the normal of the THEMIS 3-S/C

plane, and the enhanced near-earth duskward or westward

current density are readily yielded in the main phase of a

geomagnetic storm. In the last two cases, we have investi-

gated the flow vorticity determinations with 3-point obser-

vations. In the third case, with THEMIS 3-point plasma

measurements, the component of the flow vorticity along the

normal of the S/C plane has been estimated and the derived

results are reasonable. Last, in the fourth case, by using the

new method, we have quantitatively investigated the varie-

ties of the vortices created by the severe K-H instability at the

dusk flank of the magnetopause. Therefore, one component

of the current density has been obtained based on 3 S/C

magnetic field measurements for the first time; and also, one

component of the flow vorticity has been first obtained based

on 3-satellite plasma measurements.

[62] The new method can find more applications on data

analysis for Cluster, THEMIS and Swarm missions. For

Cluster, we may calculate the gradient of the magnetic field in

the inner magnetosphere as the spacecraft tetrahedron has

abnormal configurations. And furthermore, with the Cluster

C1, C2 and C4 CIS measurements, the gradient of the flow

velocity in the 3 S/C plane and the component of the vorticity

along the normal to the S/C plane can readily be deduced in

the regions with plasma flows. For THEMIS, the new method

may find similar usages to those for Cluster, e.g., it may be

applied to deduce the ring current and field aligned current

distributions in the inner magnetosphere. As for Swarm

mission (planned to be launched on July 2012), only if the

three S/C are sufficiently near to each other, the gradient of

the magnetic field in the 3 S/C plane can be determined, and

further one component of the current density can also be

obtained, as well as the field aligned current density.

Appendix A: The Error of the Method

[63] Considering equations (7)–(8) for calculating the

gradient of a scalar physical quantity, the relative error of the

gradient of Fin direction of the l

th

eigenvector is

d~

@lF

~

@lF

≈dPaFa~

ral

PaFa~

ral

þdwl

wl

:ðA1Þ

Table 1. The Data of Pulse-Enhanced J

3

and W

3

Time J

3

(nAm

2

)W

3

(s

1

)

20:28:11 4.9 0.041

20:30:59 7.5 0.085

20:34:59 7.3 0.076

20:38:27 8.2 0.048

20:40:55 2.9 0.088

SHEN ET AL.: TECHNIQUE A11207A11207

13 of 19

The error of PaFa~

ralis

dX

a

Fa~

ral

jj

¼X

a

Fa

jj

d~

ral

jj

þ~

ral

jj

dFa

jj

ðÞ

¼X

a

Fa~

ral

jj

d~

ral

jj

~

ral

jj

þdFa

jj

Fa

jj

X

a

Fa~

ral

jj

dr

ﬃﬃﬃﬃﬃ

wl

pþdF

Fjj

dr

ﬃﬃﬃﬃﬃ

wl

pþdF

F

jj

NF ﬃﬃﬃﬃﬃ

wl

p;ðA2Þ

Where, Fis the typical value of the scalar physical quantity,

d

r

is the error of the position of the S/C, and Nis the number

of S/C. The relative error

dPaFa~

ral

PaFa~

ral

1

Nwl~

@lF

NF ﬃﬃﬃﬃﬃ

wl

pdr

ﬃﬃﬃﬃﬃ

wl

pþdF

F

≈D

ﬃﬃﬃﬃﬃ

wl

pdr

ﬃﬃﬃﬃﬃ

wl

pþdF

F

;ðA3Þ

Where, Dis the characteristic size of the structure, and dFis

the error of the measurement of the physical quantity.

Because the eigenvalues may be written as

wl¼1

NX

N

a

~

ral

ðÞ

2;ðA4Þ

So the errors of the eigenvalues are

dwl¼1

NX

N

a

2~

ral

jjd~

ral1

NX

N

a

2ﬃﬃﬃﬃﬃ

wl

pdr¼2ﬃﬃﬃﬃﬃ

wl

pdr:ðA5Þ

Then the relative errors the eigenvalues are

dwl

wl2dr

ﬃﬃﬃﬃﬃ

wl

p:ðA6Þ

Therefore, the total relative error is

d~

@lF

~

@lF

≈dPaFa~

ral

PaFa~

ral

þdwl

wlD

ﬃﬃﬃﬃﬃ

wl

pdr

ﬃﬃﬃﬃﬃ

wl

pþdF

F

þ2dr

ﬃﬃﬃﬃﬃ

wl

p

¼2þD

ﬃﬃﬃﬃﬃ

wl

p

dr

ﬃﬃﬃﬃﬃ

wl

pþD

ﬃﬃﬃﬃﬃ

wl

p

dF

F:ðA7Þ

The above formula will pose restraints on the measurement

errors of the S/C positions and the physical quantities. In

order to yield accurate results, i.e., d~

@lF

jj

~

@lF

jj

≪1, it requires that

2þD

ﬃﬃﬃﬃﬃ

wl

p

dr

ﬃﬃﬃﬃﬃ

wl

p≪1;ðA8Þ

and

D

ﬃﬃﬃﬃﬃ

wl

pdF

F≪1:ðA9Þ

Furthermore, the restraints on drand dFare

dr

ﬃﬃﬃﬃﬃ

wl

p≪ﬃﬃﬃﬃﬃ

wl

p

2ﬃﬃﬃﬃﬃ

wl

pþD≈ﬃﬃﬃﬃﬃ

wl

p

D;ðA10Þ

dF

F≪ﬃﬃﬃﬃﬃ

wl

p

D:ðA11Þ

Here, it has been assumed that ﬃﬃﬃﬃﬃ

wl

p≪D.

[64] On the other hand, if

dr

ﬃﬃﬃﬃﬃ

wl

p≥ﬃﬃﬃﬃﬃ

wl

p

D;ðA12Þ

or

dF

F≥ﬃﬃﬃﬃﬃ

wl

p

D;ðA13Þ

the errors are too large that we cannot effectively deduce the

gradient of the physical quantity F. E. g., when the space-

craft array is growing planar, the third eigenvalue w

3

is so

small that

dF

F≥ﬃﬃﬃﬃﬃﬃ

w3

p

D;ðA14Þ

then the gradient at the direction of the third eigenvector ^

k3ðÞ

cannot be obtained with reasonable accuracy. This can be

illustrated in section 3.1.

[65] Furthermore, if the spacecraft array is becoming lin-

ear, the second eigenvalue w

2

is so small that

dF

F≥ﬃﬃﬃﬃﬃﬃ

w2

p

D;ðA15Þ

the gradient along the directions ^

k2ðÞ and ^

k3ðÞ cannot be

deduced, and only the gradient along the first eigenvector

^

k1ðÞcan be obtained by the new method.

[66] Therefore, the equations (A14) and (A15) define

when the spacecraft array has an abnormal configuration, or

when the spacecraft array has to be regarded as planar and

linear, respectively.

Appendix B: Test on the New Method—Three S/C

Observations on the Dipolar Magnetic Field

[67] We assume the three S/C array are measuring the

geomagnetic field as illustrated in Figure B1. The geomag-

netic field is approximated as dipolar one. The orbit of the

three S/C are regular circles with the same geocentric dis-

tance R. The three S/C constitute one regular triangle.

[68] The volume tensor is

R¼1

3X

3

a¼1

rara

¼1

3"d2^

q^

qþﬃﬃﬃ

3

pd

2^

fþd

2^

q

ﬃﬃﬃ

3

pd

2^

fþd

2^

q

þﬃﬃﬃ

3

pd

2^

fþd

2^

q

ﬃﬃﬃ

3

pd

2^

fþd

2^

q

#

¼d2

2^

q^

qþ^

f^

f

;ðB1Þ

SHEN ET AL.: TECHNIQUE A11207A11207

14 of 19

So that, the eigenvectors: ^

k1ðÞ¼^

q,^

k2ðÞ¼^

f,^

k3ðÞ¼^

r, the

eigenvalues: w1¼w2¼d

2

2,w

3

= 0. With the formula (14),

the components of the magnetic field gradient

~

r1~

B2¼1

3w1X

3

a¼1

~

Ba2~

ra1¼1

3w1

~

Baf~

raqþ~

Bbf~

rbqþ~

Bcf~

rcq

;

ðB2Þ

~

r2~

B1¼1

3w2X

3

a¼1

~

Ba1~

ra2¼1

3w2

~

Baq~

rafþ~

Bbq~

rbfþ~

Bcq~

rcf

:

ðB3Þ

[69] It is noted that

~

raf¼ﬃﬃﬃ

3

p

2d;~

rbf¼ ﬃﬃﬃ

3

p

2d;~

rcf¼0;

~

Baf¼~

Bbf¼~

Bcf¼0;

~

Baq¼MR3sin qþd

2R

;~

Bbq¼MR3sin qþd

2R

;

then

~

r1~

B2¼~

r2~

B1¼0:ðB4Þ

Therefore, from equation (18), the normal component of the

current density is

m0~

J3≈0:ðB5Þ

So that the new method yields correct normal current

density.

Appendix C: Test on the New Method—Three S/C

Observations on the Force-Free Flux Rope

[70] As shown in Figure C1, we assume that a three

spacecraft array is crossing a stationary, force-free flux rope.

The spacecraft array is taken to be a regular triangle shape

with its side length being 0.1 R

E

(amplified by a factor of 5

in Figure C1). The flux rope can be described by the well-

known Lundquist-Lepping (L-L) model [Lundquist, 1950],

which is a particular solution of the force-free field condition

rB=aBwith the assumption of cylindrical symmetry.

The magnetic field in the force-free flux rope of the L-L

model can be expressed as

Bz¼B0J0arðÞ;Bj¼B0J1arðÞ:ðC1Þ

Figure B1. One ideal situation for three S/C measurements on the dipolar geomagnetic field. (left) The

orbits of the three S/C and (right) the configuration and orientation of the three spacecraft. The spherical

coordinates system is used. The center of the Earth is the origin, and ^

r,^

qand ^

fare the unit radial, polar

and azimuthal vectors, respectively.

SHEN ET AL.: TECHNIQUE A11207A11207

15 of 19

For the test, we arbitrarily adopt the parameters B

0

= 20 nT,

a¼1RE1. This can also be expressed in Cartesian coor-

dinates as

Bz¼B0J0arðÞ;Bx¼B0J1arðÞsinj;By¼B0J1arðÞcosjðC2Þ

Where, jis the azimuthal angle and ris the radial distance.

[71] The three-S/C array crosses the LL flux rope in a

straight line with a constant velocity. The mesocenter,

labeled by the asteroid marker, has a parallel motion from

the point (x = 0.55, y = 1.9711, z = 0) R

E

to the point (x =

0.55, y = 1.5289, z = 1) R

E

with a time interval 500 s. The

resolution of the magnetic field measurement is taken to be

1-s.

[72] The test results are shown in Figure C2, wherein the

thicker red lines and the thinner blues line in panels b to j are

the results based on equations (15), (27), (28), and (29) and

the exact analytic results at the mesocenter, respectively.

Obviously, the full derived magnetic field gradient compo-

nents are well consistent with the exact analytic results. As

shown in Figure C2, the absolute error of the components of

the calculated magnetic gradient from the exact ones is less

than about 0.2 nT/R

E

. In this case, the characteristic size of

the spacecraft array is L¼2ﬃﬃﬃﬃﬃﬃ

w1

p≈0:08RE, the characteris-

tic spatial scale of the flux rope is D≈1/a=1R

E

, the typical

magnetic gradient is about aB

0

= 20 nT/R

E

. Thus the trun-

cation error is at the order of (L/D)aB

0

≈1.6 nT/R

E

.

Therefore, the errors in the calculated components of the

magnetic gradient are well within the truncation errors. The

approach put forward in this paper therefore has the ability

to recover the full gradient components of magnetic field

with good accuracy based on three-point measurements and

if the current is field-aligned.

Appendix D: Discussion on the Relationship

Between the New Approach and the Reciprocal

Vector Method

[73] We consider the situation when only 3 satellites make

observations. Vogt et al. [2009] yielded the reciprocal vec-

tors as

qa¼srbg

sjj

2;ðD1Þ

where, r

bg

=r

g

r

b

,s=r

12

r

13

. Note that s¼s^

k3ðÞ

.

The value of sis twice the area of the triangle of the three

satellites. If the three spacecraft are aligned, sis zero, and

Vogt et al. [2009] formula (D1) fails.

[74] For the three satellite situation, we may consider only

the two dimensions in the spacecraft plane. The spatial

scales in the two independent directions ^

k1ðÞ and ^

k2ðÞ are

ﬃﬃﬃﬃﬃﬃ

w1

pand ﬃﬃﬃﬃﬃﬃ

w2

p, respectively. Thus, the area s/2 of the triangle

of the three satellites is proportional to ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

w1w2

p. Similarly to

the deduction in the paper of Harvey [1998], it is readily

obtained that

s¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

27w1w2

p:ðD2Þ

The coefficient in the above formula (D2) is easy verified as

shown in Appendix B in the situation of the regular triangle

spacecraft array.

[75] From the newly proposed approach in this paper, it is

straightforward to get the formulae of the reciprocal vectors

for various situations when the multiple spacecraft arrays

have shapes of either tetrahedrons, planes or lines.

[76] From section 2.1, we may get the gradient of the

quantity Ffrom the observations of N spacecraft as

rF¼~

rlF

^

klðÞ ≈1

NwlX

N

a¼1

Fa~

ral^

klðÞ

;ðD3Þ

Considering the gradient rF¼PN

a¼1Faqa, we may get the

formula of the N reciprocal vectors accordingly as

qa¼X

3

l¼1

1

Nwl

~

ral^

klðÞ

;a¼1;2;⋯;N:ðD4Þ

The reciprocal vectors q

a

(a=1,2,⋯,N) are determined by

the structure and orientation of the spacecraft array.

[77] There is one problem if there are other sets of recip-

rocal vectors for deducing the gradient rF. Here we may

verify that the uniqueness of the reciprocal vectors for a

spacecraft array. Assume that there are two different sets of

reciprocal vectors, q

a

(1)

(a=1,2,⋯,N) and q

a

(2)

(a=1,2,⋯,N),

for the same spacecraft array. Then

rF¼X

N

a¼1

Faq1ðÞ

a;

rF¼X

N

a¼1

Faq2ðÞ

a:

Figure C1. The three-S/C array is crossing the LL flux

rope in straight trajectory (thin black line). For this LL flux

rope, the characteristic magnetic field is B

0

= 2 nT and the

force free factor a¼1RE1. The three-S/C array, labeled

as the colored squares, constitutes one regular triangle with

side length 0. One R

E

(amplified by a factor of 5 in the plot),

The mesocenter is denoted by asteroids, which moves in a

straight line from (x = 0.55, y = 1.9711, z = 0) R

E

to

(x = 0.55, y = 1.5289, z = 1) R

E

with a time interval of

500 s.

SHEN ET AL.: TECHNIQUE A11207A11207

16 of 19

[78] The difference of the above two equations is

0¼X

N

a¼1

Faq1ðÞ

aq2ðÞ

a

:

Because F

a

(a=1,2,⋯,N) may be arbitrary, it is obvious

that q1ðÞ

aq2ðÞ

a¼0, or q1ðÞ

a¼q2ðÞ

a. Therefore, the reciprocal

vectors for a spacecraft array are unique. For the four

spacecraft array, the reciprocal vectors q

a

(a=1,2,⋯,4)as

shown in (D4) are equivalent to those in Chanteur [1998].

[79] If the spacecraft array becomes planar, w

3

equals

zero. Then the gradient in the spacecraft plane may be

expressed as

rFðÞ

P¼X

2

l¼1

~

rlF

^

klðÞ ≈1

Nw1X

N

a¼1

Fa~

ra1^

k1ðÞþ1

Nw2X

N

a¼1

Fa~

ra2^

k2ðÞ

;

ðD5Þ

and the corresponding N reciprocal vectors are

qa¼1

Nw1

~

ra1^

k1ðÞþ1

Nw2

~

ra2^

k2ðÞ

;a¼1;2;⋯;N:ðD6Þ

Figure C2. Comparison between the test results and the exact analytic results. (a) The time series of the

measured magnetic field strength and the Bx, By and Bz components along the trajectory of the mesocen-

ter. (b–j) The components of the magnetic gradient ∂

x

B

x

,∂

y

B

x

,∂

z

B

x

,∂

x

B

y

,∂

y

B

y

,∂

z

B

y

,∂

x

B

z

,∂

y

B

z

and

∂

z

B

z

, respectively, along the trajectory of the mesocenter, where, the thicker red lines illustrate the results

inferred from the new method and the thinner blue lines demonstrate the exact analytic results.

SHEN ET AL.: TECHNIQUE A11207A11207

17 of 19

[80] As demonstrated above, for the case of three space-

craft array, the reciprocal vectors q

a

(a= 1, 2, 3)from the

new approach as expressed by (D6) are equivalent to the

formula (D1) of Vogt et al. [2009]. Especially, for the situ-

ation when the three satellites form a regular triangle as

shown in Figure B1, applying the formula (D6) based on our

method or the formula (D1) of Vogt et al. [2009], we may

get the same reciprocal vectors as the follows.

qa¼2

3dra¼2

3d

1

2^

qþﬃﬃﬃ

3

p

2^

f

;ðD7Þ

qb¼2

3drb¼2

3d

1

2^

qﬃﬃﬃ

3

p

2^

f

;ðD8Þ

qc¼2

3drc¼2

3d

^

q:ðD9Þ

Obviously, in Appendix B, the two methods will yield the

same magnetic gradient with the assumed spacecraft array

configuration.

[81] Therefore, from the new method proposed in this

paper, we may obtain the general formulae of the reciprocal

vectors for various spacecraft arrays with any number and

any shape.

[82]Acknowledgments. This work was supported by Ministry of

Science and Technology of China Grant 2011CB811404, the National

Natural Science Foundation of China grants 40921063, 41231066, 40974101

and 41104114, China Postdoctoral Science Foundation Funded Project

(20100480446, 2012T50132), Chinese Academy of Sciences (CAS) visiting

Professorship for senior international scientists grant 2012T1G0018, and the

Specialized Research Fund for State Key Laboratories of the CAS. The authors

are thankful to Cluster II FGM team and ESA Cluster Active Archive for

providing Cluster data, appreciate the THEMIS team for providing the public

THEMIS scientific data, and also thank Q. H. Zhang for the valuable

suggestions.

[83]Masaki Fujimoto thanks the reviewers for their assistance in eval-

uating the paper.

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