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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
SHEN ET AL.: TECHNIQUE A11207A11207
10 of 19
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
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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
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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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