Reprinted from Analysis Methods for Multi-Spacecraft Data
Götz Paschmann and Patrick W. Daly (Eds.),
ISSI Scientific Report SR-001 (Electronic edition 1.1)
c 1998, 2000 ISSI/ESA
— 15 —
Spatial Interpolation for Four Spacecraft:
Application to Magnetic Gradients
G ÉRARD C HANTEUR
C HRISTOPHER C. H ARVEY
CETP-CNRS
Vélizy, France
Observatoire de Paris-Meudon
Meudon, France
15.1 Introduction
The primary purpose of a multi-spacecraft mission such as Cluster is to distinguish
between temporal and spatial variations by means of four or more spacecraft. An obvious question is how, precisely, the gradients of the spatial variations should be determined
from such multipoint measurements. Some of the early studies related to the Cluster mission aimed to evaluate the components of the electric current density (see Chapter 16), i.e.
of peculiar combinations of various components of the gradient of the magnetic field. But
the first attempt to devise a method of determining the gradient of any field was due to
Chanteur and Mottez [1993], who introduced barycentric coordinates for the linear interpolation of scalar and vector field quantities. This work has been considerably extended
and a detailed and self-contained presentation has been made in Chapter 14; linear and
quadratic estimators of the gradient of a field have been defined. Chapter 12 introduced
two other linear estimators derived by least squares minimisation, constrained or not by
the solenoidal condition.
The present chapter aims to evaluate the accuracy of these estimators by applying them
to simulated data; it requires an understanding of Chapters 12 and 14. In Section 15.2
we prove the mathematical equivalence of the unconstrained least squares linear estimator with the linear barycentric estimator. In Section 15.3 a dipole field model is used to
illustrate analytically the nature and magnitude of the errors caused by using linear interpolation to approximate an non-uniform field. In Section 15.4 we compare the linear
barycentric estimator and the least squares estimator constrained by the solenoidality condition; the latter will be briefly referred to as the solenoidal estimator. The estimates of the
gradient of the magnetic field are computed by both methods for simulated observations
obtained by using a model of the Cluster orbit together with the Tsyganenko [1987] model
of the geomagnetic field. The exact gradient of the simulated observations is computed
directly from the model when preparing the simulated data sets. This comparison gives
further insight into the difficulties related to the determination of gradients, and their likely
variation as the tetrahedron geometry evolves along a typical spacecraft orbit. In the last
section we discuss the smoothing of the data which is required before using it to determine the gradient. We also recall the result demonstrated in Chapter 14 (equation 14.31 on
page 358) that the standard error in the determination of the gradient is intimately related
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15. S PATIAL I NTERPOLATION : A PPLICATION
not only to the properties of the reciprocal tensor (or of its inverse, the volumetric tensor),
but also to the properties of the covariance matrix of the error in the determination of the
spacecraft position.
15.1.1 Statement of the Problem
Cluster can determine gradients in two different ways, depending upon the scale size
of the spatial variation of the observed parameter.
1. Small-scale structures are spatial variations with scale size small compared with
the inter-spacecraft separation distances. They are identifiable in the sense that the
differences in their time of observation at the four spacecraft may be determined, for
example, by cross-correlation. Such observations may be interpreted in terms of a
one-dimensional spatial structure moving along the direction of its normal, and both
the direction of the normal and the speed of the motion may be determined from the
observations. This model is suitable for the study of shocks, discontinuities, and the
like.
2. Large-scale structures have a spatial scale large compared with the spacecraft separation distances. Cross-correlation does not yield any meaningful result. But if
there is a difference in the values of a parameter measured on the four spacecraft,
it is possible to determine the corresponding large-scale gradient. In particular, an
important objective of the Cluster project is to infer the mean current density from
magnetic measurements made at different points.
These two cases are relatively clear. The situation is more complicated when the spatial
scale is comparable with the inter-spacecraft separation, or when the tetrahedron is very
anisotropic, and further study is required.
15.2 Relationship Between Homogeneous Least
Squares and Barycentric Methods
We now show that the unconstrained homogeneous least squares method and the linear
barycentric method described in Chapter 14 are equivalent. We do this by first showing
that the reciprocal tensor defined by equation 14.29 (page 356)
K=
4
X
k α k Tα
α=1
is equal to
15.2.1
1
4
of the inverse of the volumetric tensor.
Relationship Between the Reciprocal and Volumetric Tensors
The volumetric tensor is defined by equation 12.23 (page 315), where r b is the position
vector of the barycentre of the tetrahedron. The reciprocal and volumetric tensors are
15.3. Truncation Errors of LG for a Magnetic Dipole
373
symmetric. A short lemma is needed concerning the scalar product
1X
kα · r β − r b =
kα · r β − r γ
4 γ
with the help of equation 14.6 (page 350), we obtain
1
k α · r β − r b = r β − r b · k α = δαβ −
4
(15.1)
The dyadic notation allows us to compute the product of the two tensors, thus:
4KR =
XX
α
k α k Tα r β − r b
rβ − rb
T
β
The inner dyad k Tα r β − r b is the scalar product of equation 15.1, so that
4KR =
X
k α (r α − r b )T
α
Then equations 14.11 and 14.10 (page 352) , and the symmetry of both tensors, allow to
conclude that:
1
I = RK
(15.2)
KR =
4
15.2.2
Identity of the HLS and Linear Barycentric Estimators of the
Gradient
When the homogeneous least squares method is applied without imposing the condition of solenoidality (of Section 12.3.2), the resulting estimator HLSG[v] of the vector
field v minimises the scalar quantity (see equation 14.2 on page 350 for the definition of
GT [v])
XX
S=
w T w , where w = GT [v] r α − r β − v α − v β
α
β
It is obvious from definition 14.13 (page 352) that the linear barycentric estimator LGT [v]
satisfies:
LGT [v] r α − r β = v α − v β
hence LGT [v] exactly cancels w and S and
HLSG[v] = LG[v]
15.3 Truncation Errors of LG for a Magnetic Dipole
In this section we illustrate the inherent limitations of any linear (homogeneous least
squares, barycentric, or other) estimation of the gradient. We do this by evaluating analytically the linear estimator obtained from a simple dipole magnetic field sampled at four
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15. S PATIAL I NTERPOLATION : A PPLICATION
locations. Linear interpolation between these four field samples generally leads to an estimation of the gradient tensor which is asymmetric, with a trace different from zero; i.e.,
to the appearance of non-existent current and local violation of the conservation of the
magnetic flux. Detailed expressions will be given for a regular tetrahedron.
These results explain, for example, why forcing the divergence of the estimated magnetic field to be zero (the method of Section 12.3.2) does not necessarily yield the best
result, as will be discovered in Section 15.4.
15.3.1
The Dipole Field and its Gradient
The magnetic field created at location R = R ê by the magnetic dipole M = Mû
writes:
B dipole (R) = −∇ R −3 M · R = MR −3 3ê ûT ê − û
(15.3)
where ê and û are unit vectors. This is obtained from definition 14.1 (page 350) of the gradient of a scalar field and with the help of the identity R dR = R·dR. From definition 14.2
(page 350) of the gradient of a vector field we obtain:
R4 T
Gdipole = (ê · û) I − 5e eT + ê ûT + û êT
3M
(15.4)
where I is the unit tensor. The gradient of the dipole field is obviously symmetric and its
trace is equal to zero as it should.
15.3.2
Linear Estimation of the Gradient of the Dipole Field
The linear estimation of the gradient of the dipole field involves the magnetic field
vectors measured by the four spacecraft. We assume that the characteristic size a of the
tetrahedron is much less than R, the distance from the barycentre to the dipole, and we
develop measured (dipole) vector magnetic field in the vicinity of the barycentre up to
terms of second order in (a/R)2 . The vector position of spacecraft α is written r α =
R(ê + ε α ), which implies that the magnitude of ε α is of order a/R and 6ε α = 0. The
linear estimator of the magnetic field gradient writes up to second order in a/R:
LGT [B]
=
GTdipole +
a 3M n
(1 − 5ê · û)T
R R4
+ (û − 5ê ûT ê) C T0 − 5(û − 7ê ûT ê) C T1 − 5ê C T2
o
(15.5)
with the following definitions:
T
=
4
R2 X
(ê · ε α )ε α k Tα
a α=1
(15.6)
C0
=
4
R2 X
(ε α )2 k α
2a α=1
(15.7)
C1
=
4
R2 X
(ê · ε α )2 k α
2a α=1
(15.8)
15.4. Comparison of the Methods Using Simulated Data
C2
=
4
R2 X
(ê · ε α )(û · ε α )k α
a α=1
375
(15.9)
As expected the first order contribution to LGT [B] is equal to the exact gradient of the
dipole field at the barycentre, but the general second order contribution is not symmetric
and its trace is not equal to zero.
15.3.3 Truncation Errors for a Regular Tetrahedron
Consider the regular tetrahedron defined in Section 14.3.3 and let (ux , uy , uz ) and
(ex , ey , ez ) be respectively the cartesian components of û and ê in the reference frame of
the tetrahedron. With these notations, the auxiliary vectors defined by equations 15.7 to
15.9 write:
C0
C1
C2
0
−{ey ez x̂ + ez ex ŷ + ex ey ẑ}
= −{ ey uz + uy ez x̂ + (ez ux + uz ex ) ŷ + ex uy + ux ey ẑ}
=
=
(15.10)
(15.11)
(15.12)
where x̂, ŷ, ẑ are the unit vectors of the reference frame. The tensor T defined by equation 15.6 is equal to:


0 ez ey
T = −16  ez 0 ex 
(15.13)
ey ex 0
From all the second order contributions to the errors of truncation, this term is the only
one which does not contribute to false currents and to the non-conservation of the magnetic flux. Even in the simplest case of a regular cluster the truncation errors given by
equation 15.5 to 15.9 are too much intricate to be really useful for analytical considerations. Nevertheless it is possible to obtain a few simple results, for example when the
cluster lies in the equatorial plane of the dipole (in that case ê · û = 0), the linear estimator
of ∇ · B is, accordingly to equation 14.16 (page 353):
R
45a
LD[B] =
(ux ey ez + uy ez ex + uz ex ey )
B
R
(15.14)
Thus the linear estimation of the divergence of B can be equal to zero, for example for
uz = ex = 1, but that is usually not the case.
15.4 Comparison of the Methods Using Simulated Data
In this section we test the barycentric and solenoidal least-squares estimators of the
gradient of the magnetic field on simulated Cluster data obtained along predicted Cluster
orbits embedded in the Tsyganenko-87 magnetic field model [Tsyganenko, 1987]. The
orbits have been provided by the European Space Operations Centre (ESOC, Darmstadt,
Germany) in the geocentric equatorial inertial frame of reference. Taking into account
the Earth’s rotation and orbital motion [Coeur-Joly et al., 1995], the magnetic data were
computed in geocentric solar ecliptic (GSE) coordinates. An exhaustive investigation is
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15. S PATIAL I NTERPOLATION : A PPLICATION
beyond the scope of this article, so we have selected a single Cluster orbit, with apogee
in the geotail and hence the whole orbit lying inside the magnetosphere for the moderate
index Kp = 2 which we have chosen. For such quiet conditions the gradient of the magnetic field in the outer magnetosphere has a characteristic scale size which is larger than
the tetrahedron itself, and we may expect our determination to be a good approximation to
the true gradient; the situation is much less favourable near perigee, where steeper gradients are seen with a tetrahedron considerably elongated in the direction of the spacecraft
motion due to large orbital velocity. The precision of the gradient determination depends
also upon the quality of intercalibration of the relevant experiments, and for some Cluster
experiments, for example the magnetic field and the bulk flow velocity of the low energy
plasma, this is critical. This point has been investigated by Khurana et al. [1996] for the
dc magnetic field and will not be addressed here.
15.4.1 Evolution of the Tetrahedron along the Orbit
The geometry of the cluster of spacecraft at a given time can be characterised by various geometrical factors which have been introduced and discussed in Chapter 13. For this
study we have selected six of these geometrical factors: the elongation E and planarity
P defined in Section 13.3.3, and the four parameters QRR , QSR , QR8 , QGM which are
defined in Section 13.3.1.
Figure 15.1 shows the variations of these geometrical factors of the tetrahedron along
one orbit between two successive apogees. It is worth noticing that the data are not periodic
owing to the drift of the orbits in the GSE frame of reference. Time is normalised to the
orbital period, and perigee has been chosen to occur at t = 0.5 so that apogees correspond
to t = 0 and 1. This particular satellite configuration is isotropic at t = 0.975, shortly
before apogee; at that time E and P are equal to zero and the four Q factors are equal
to one. The top panel shows the elongation E (dotted curve) and the planarity P (solid
curve). The centre panel shows the variations of QSR (dotted curve) and QRR (solid
curve), and the bottom panel displays QGM (dotted curve) and QR8 (solid curve). The
factors QRR and QR8 have values close to zero, meanwhile the planarity P is close to one,
at times t = 0.14, 0.42, 0.53 and 0.67; this is indicative of a degenerate configuration
of the spacecraft which are coplanar. The variations of QSR and QGM are smoother.
The low values of QRR and QR8 , as well as the high value of the elongation in the time
interval extending from 0.5 to .7 indicate an unfavourable configuration of the cluster
(either flattened when P is close to one, or elongated when P significantly differs from
one). The shaded bars emphasise the time intervals during which the numerical accuracy
is degraded due to the flattening of the cluster; this point will be made more precise at the
end of the next section, especially by equation 15.16 (page 379) and the following lines.
15.4.2 Estimations of G[B]
In this section we compare the different estimations of the gradient G[B] along the
selected orbit with the “true” gradient of the Tsyganenko magnetic field model. This
latter gradient was calculated at the same time as the field itself. We compared two finite
difference methods, of respectively second and fourth order with respect to the spatial
increment, which was 200 km. The two methods gave results which are indistinguishable
15.4. Comparison of the Methods Using Simulated Data
377
Elongation and Planarity of the cluster
Geometric Factors QSR and QRR of the cluster
Geometric Factors QGM and QR8 of the cluster
Figure 15.1: Characterisation of the tetrahedron along the orbit. The upper panel shows
the variations with normalised time of the elongation E (dotted curve) and planarity P
(solid curve) defined in Section 13.3.3. The centre panel shows the variations of QSR
(dotted curve) and QRR (solid curve). The lower panel shows the variations of QGM
(dotted curve) and QR8 (solid curve). See Section 13.3.1 for the definitions of these geometrical factors. The shaded bars delimit intervals of degraded numerical accuracy (see
Section 15.4.3).
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15. S PATIAL I NTERPOLATION : A PPLICATION
for the present purposes. The fourth order method was used to compute what we will call
the “true” gradient.
In order to compare the different estimations of G[B] with the true gradient, a series
of plots (Figures 15.2–15.10), present the variations of the components of G[B], or of
various estimates of these components, versus the normalised time in the GSE coordinate
system. The various components of G[B] have large variations, therefore, instead of the
components ∂p Bq themselves (with p, q = x, y or z), we have plotted the quantities
Rmean
∂p Bq
Bmean
where Rmean and Bmean are the respective magnitudes of
R mean =
4
4
1X
1X
r α , and B mean =
Bα
4 α=1
4 α=1
Figures 15.2–15.4 compare the linear estimates, with and without the condition ∇·B =
0, with each other and with the “true” components of G[B]. The “true” gradient is plotted
as a solid grey curve, while the solenoidal least-squares and the barycentric linear estimates
are respectively represented by the dotted and solid black curves. The unconstrained leastsquares and barycentric linear estimations being identical, as shown in Section 15.2.2,
the solid black lines represent estimates computed from both methods. It can be seen
that large deviations of the estimated quantities occur during time intervals marked by
the shaded bars and that the enforcement of the solenoidality condition generally does
not improve the estimation of the gradient of the magnetic field. The deviations from the
exact components are due to the errors of truncation as demonstrated by computing the
quadratic barycentric estimator of the gradient. The exact quadratic estimator defined in
Section 14.2.2 will not be computable with a four spacecraft mission but the simulation
of the data allows one to compute the supplementary magnetic field vectors required to
build the approximate estimator discussed in the same section. Figures 15.5–15.7 display
the results provided by quadratic interpolation, which show a significant improvement of
the estimates even throughout most of the singular time intervals marked by shaded bars.
The light grey curves represent the exact components, and the black curves their quadratic
estimates. The predicted components of G[B] are quite accurate along the whole orbit
except near perigee for components involving partial derivatives with respect to coordinate
y (Figure 15.6).
The large deviations from the exact components around times of planarity of the cluster are due to the amplification of the truncation errors by singular configurations of the
spacecraft. When the tetrahedron flattens, the largest of the reciprocal vectors is normal to
the singular plane configuration and diverges as the inverse of the thickness of the tetrahedron. In order to demonstrate this amplification we subtract from the reciprocal vectors
their singular components (parallel to the largest reciprocal vector) in the vicinity of the
planar configurations according to the following formulas:
k α,corrected = k α − γ (k α · k̂ max )k̂ max
(15.15)
where k̂ max is the unit vector parallel to the largest reciprocal vector and γ is a numerical
factor which should be equal to zero almost everywhere except in the vicinity of the singular configurations where it should be equal to one. For any tetrahedron the sum of the
15.4. Comparison of the Methods Using Simulated Data
379
reciprocal vectors is equal to zero, but as will be shown below the numerical inaccuracy
increases in the vicinity of a planar cluster and the magnitude of this sum is peaked around
the singular times, although remaining small. Thus a practical choice for the function γ
is:
h
i2
γ = tanh (k 1 + k 2 + k 3 + k 4 )2 /S0
(15.16)
where S0 is a reference value chosen after inspection of the numerical values of (k 1 +
k 2 + k 3 + k 4 )2 ; in the present case S0 = 10−18 , when measuring the components of
the reciprocal vectors in km−1 . The time intervals during which γ ≥ 0.1 are marked by
vertical shaded bars in all figures of this section. This corrected reciprocal base has been
used to compute a corrected linear barycentric estimator of the gradient of the magnetic
field. Figures 15.8–15.10 are to be compared with Figure 15.2–15.4; the large errors
affecting the ∂x and ∂y components around t = 0.53 have been drastically reduced by this
correction, as well as errors affecting ∂x and ∂z components near t = 0.67. This correction
greatly reduces the amplification of the truncation errors by singular reciprocal bases, but
not enough to provide reliable estimates during the singular intervals marked by shaded
bars.
15.4.3 Quality of the Estimations
Numerical Precision
Although we do not know any a priori criterion of quality for the linear estimations,
it is possible to define useful indicators of accuracy for both methods. In the barycentric
method, the sum of the four reciprocal vectors of the tetrahedron formed by the spacecraft
is theoretically equal to zero, but as this property is not used by the computational algorithm, it can be used as a numerical check. Hence an indicator of numerical accuracy is
defined by:
Qbary = log10 (k 1 + k 2 + k 3 + k 4 )2
(15.17)
and is represented by the solid black curve in the centre and bottom panels of Figure 15.11
versus time. Measuring components of the reciprocal vectors in km−1 and using simple
precision IEEE arithmetic with words of 32 bits the background value of this indicator is
slightly above −20, but near the critical times the indicator increases and the peak values
are greater than −15, up to −9, at the critical times for which the spacecraft are coplanar.
The method of least-squares requires the inversion of the volumetric tensor, it is thus
natural to check how close to the unit tensor is the product of the volumetric tensor by its
numerically computed inverse. Hence an empirical indicator of the numerical accuracy is
defined by:
X −1
QLS =
(15.18)
RR
i,j i6=j
the base 10 logarithm of the sum of the absolute values of the non-diagonal elements of this
product. The variations of this indicator along the orbit are represented by the solid black
curve on the upper panel of Figure 15.11; the minimal value is of the order of −16 and
peak values reach −11 exactly at the same critical times found by the former indicator.
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15. S PATIAL I NTERPOLATION : A PPLICATION
−1 ∂ B along
Figure 15.2: Variations of the linear estimations of the components Rmean Bmean
x
a “tail orbit” of Cluster versus time normalised to the orbital period. The least-squares
estimates, taking into account the solenoidal constraint ∇ · B = 0, are represented by the
dotted black curves, while the barycentric, identical to the least-squares estimate without
the solenoidal condition, are displayed by solid black curves. The solid grey curves are
the true components of G[B] for the Tsyganenko-87 model. The time resolution is 20
minutes.
15.4. Comparison of the Methods Using Simulated Data
−1 ∂ B.
Figure 15.3: As Figure 15.2 but for the components Rmean Bmean
y
381
382
15. S PATIAL I NTERPOLATION : A PPLICATION
−1 ∂ B.
Figure 15.4: As Figure 15.2 but for the components Rmean Bmean
z
15.4. Comparison of the Methods Using Simulated Data
383
−1 ∂ B estimated through a quadratic interpolation of
Figure 15.5: Components Rmean Bmean
x
B for a time resolution of 20 minutes. The quadratic barycentric estimations are displayed
as black curves, while the true components for the Tsyganenko-87 model, represented by
the grey curves, are most of the time hidden by the estimated ones.
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15. S PATIAL I NTERPOLATION : A PPLICATION
−1 ∂ B estimated by quadratic interpolation of B. OthFigure 15.6: Components Rmean Bmean
y
erwise similar to Figure 15.5.
15.4. Comparison of the Methods Using Simulated Data
385
−1 ∂ B estimated by quadratic interpolation of B. OthFigure 15.7: Components Rmean Bmean
z
erwise similar to Figure 15.5.
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15. S PATIAL I NTERPOLATION : A PPLICATION
−1 ∂ B estimated by the corrected linear estimator in
Figure 15.8: Components Rmean Bmean
x
order to reduce the amplification of the truncation errors by singular reciprocal bases. The
exact components are plotted as solid grey curves, and the estimated components as dotted
black curves for the solenoidal estimator and solid black curves for the corrected linear
barycentric estimator.
15.4. Comparison of the Methods Using Simulated Data
−1 ∂ B.
Figure 15.9: As Figure 15.8 but for the components Rmean Bmean
y
387
388
15. S PATIAL I NTERPOLATION : A PPLICATION
−1 ∂ B.
Figure 15.10: As Figure 15.8 but for the components Rmean Bmean
z
15.5. Future Developments
389
Neither indicator requires diagonalisation of the volumetric tensor, both predict equally
well the critical times, but neither describes what occurs to the tetrahedron.
Physical Precision
At the beginning of these investigations it was somewhat naturally thought that the
estimation of ∇ · B, the trace of the estimated G[B], should be indicative of the quality
of the linear interpolation for a solenoidal field, such as B. This quantity ignores the offdiagonal components, but nevertheless the plots of log10 (|∇ · B|) for the Tsyganenko field
model and the three estimates of the gradient deserve some comments.
1. The Tsyganenko-87 model itself is not completely divergence free, as indicated by
the light-grey curves in the three panels of Figure 15.11. Except when the sign
changes, the magnitude of ∇ · B is larger than 10−5 nT km−1 everywhere, and is
greater than 10−3 nT km−1 close to the Earth; in fact |∇ · B|/|B| is almost constant
along the orbit.
2. log10 (|∇ · B|) computed for the solenoidal least squares estimator is plotted as the
dotted curve of the upper panel of Figure 15.11; it varies between −20 and −16,
very small values indicating good respect of the imposed constraint. But, as already
mentioned (Section 15.3), this does not guarantee that the resulting linear estimate
is better than the one obtained without this constraint.
3. The linear barycentric estimation (dotted curve of the central panel) is greater than
the divergence of the field model (light-grey curve) by one to three orders of magnitude. The quadratic estimation represented by the dotted line in the bottom panel
follows more closely the divergence of the model, but nevertheless deviates from
the divergence of the model at the third critical time, very near the perigee.
These results emphasise that the crucial point is the applicability of linear interpolation
within the tetrahedron, but unfortunately there is presently no criterion to test this point.
A pseudo-quadratic estimator as been proposed in Chapter 14 in an attempt to reduce the
truncation errors by taking advantage of the orbital motion of the cluster; but, as shown
for a planar current sheet model, this estimator is linearly degenerate and is affected by
truncation errors which are of the same order as the truncation errors affecting the linear
estimator.
It is worth noticing once more that truncation errors affecting ∇ · B and ∇ × B are
independent: the simulations presented in this section illustrate this point especially during
the time interval between t = 0.85 and 1.0 for which, according to Figures 15.8–15.10, all
components of G[B] involved in ∇ × B are accurately estimated meanwhile the components involved in ∇ · B obviously deviate from the “true” components.
15.5 Future Developments
15.5.1
Filtering of the Data
Real data will require the use of low pass filtering. It is assumed, of course, that the
experimental data is already free from any effects of temporal aliasing, as explained in
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15. S PATIAL I NTERPOLATION : A PPLICATION
Figure 15.11: Estimated divergences and numerical accuracy indicators QLS , for the
solenoidal least-squares estimator (upper panel), and Qbary for the linear (central panel)
and quadratic (bottom panel) barycentric estimators. In all frames the black solid curves
represent these accuracy indicators defined by equations 15.18 (page 379) and 15.17
(page 379), respectively, and the light-grey curves represent the variation of log10 (|∇ · B|)
of the Tsyganenko-87 model along the orbit of the tetrahedron (where ∇ · B is measured
in nT km−1 ); near the perigee |∇ · B| is greater than 10−3 nT km−1 . The dotted curves
represent the variations of log10 (|∇ · B|) of the estimators. The time resolution is 20
minutes.
15.6. Conclusions
391
Chapter 2. The effects we are talking about now are spatial aliasing: it is necessary to
remove short wavelength (i.e., high k) fluctuations which would otherwise degrade the determination of the gradient on the scale of the size of the tetrahedron. The analysis of these
high spatial frequency fluctuations is the subject of Chapter 3. The condition of spatial
aliasing (Section 14.5.1) is easily derived with the barycentric formalism of Chapter 14.
Real data will require averaging over a characteristic time such that the displacement of
the tetrahedron during this time is comparable to, but less than, its size. Averaging over a
longer interval would degrade the resolution.
15.5.2 The Spacecraft Position
The discussion of Section 14.3 shows that the precision of the determination of the
satellite position is important. The probable error in the position of the spacecraft is generally not isotropic in space. This fact must be taken into account in the determination of
the probable errors affecting the experimentally determined gradients. Furthermore, it is
clear that for almost coplanar spacecraft the most important information is the precision
of the spacecraft position in the direction of planarity; for this component of the gradient estimate problems are most likely to be encountered. It is therefore essential that the
covariance matrix of the position error be computed along the orbit.
The three components of the error in the spacecraft position will vary differently with
position along the orbit. To a first approximation, the component (of the error) parallel
to the spacecraft velocity will have a minimum at apogee (where the orbital velocity is
small) and maximum at perigee; this component depends mainly upon a timing offset. The
components which depend upon a difference of orbital eccentricity or orbital plane will
have the opposite behaviour, being larger near apogee than near perigee. These variations
are similar to the variations of the geometry itself of the Cluster tetrahedron. Therefore
the determination of gradients near apogee, when the tetrahedron is considerably flattened,
may be rather better than predicted by the simple rule that the position accuracy is within
a certain limit, e.g. ±5 km. Note that for the numerical demonstration of Section 15.4 the
orbit was chosen to be isotropic close to apogee, and consequently this flattening was not
obvious. But in general the scientific objectives will require isotropy to be targetted at
lower altitudes, leading to flattening of the tetrahedron near apogee.
It is essential to give more thought to this matter, because the Cluster experimenters
will probably often want to push to the limit their knowledge of the orbital position, especially when the spacecraft are far from that part of the orbit concerned by a primary
science objective.
15.6 Conclusions
The determination of gradients from measurements by four spacecraft is just the spatial
interpolation of data within the tetrahedron formed by the spacecraft. The homogeneous
least squares method presented in Section 12.2 complements the theoretical framework of
Chapter 14. The unconstrained least square method was proved in Section 15.2 to be mathematically identical to the linear barycentric method described in Section 14.2. In principle
the least squares approach allows account to be taken of the solenoidal constraint, but this
does not necessarily improve the estimation of the gradient because, as demonstrated in
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Section 15.3, the linearly interpolated gradient of a solenoidal field is generally not divergence free. The simulations presented in Section 15.4 show that in practice the two
methods may yield different results, especially in the vicinity of singular configurations of
the tetrahedron, with no indication as to which determination is better.
The errors of truncation may lead to completely erroneous estimates, especially for
nearly planar clusters. Presently, no criterion is available which could give a hint about the
validity of the linear interpolation of the field in between the spacecraft; this linear interpolation is the common basis of all methods designed to estimate the gradient of a given
field. Such a criterion is mandatory to give credit or not to these estimates; more investigations are required of this crucial point. Nevertheless, the accuracy indicators discussed
in section 15.4.3 are reliable warnings of inaccuracy in the vicinity of the planar clusters.
In some cases, a regularisation of the reciprocal base can be attempted as demonstrated
in the same section. The approximate quadratic estimator defined in Section 14.2 does
not improve the estimation of the gradient along realistic orbits in a Tsyganenko model of
the geomagnetic field. Note that the least squares method as described in Section 12.2 is
directly applicable to more than four spacecraft.
The Tsyganenko-87 model we have used does not include a magnetopause, hence the
simulations have been restricted to orbits completely within the magnetosphere, but from
the theoretical analysis of Section 14.4 we conclude that a correct determination of the
Chapman-Ferraro currents at the magnetopause will require a small size tetrahedron, with
an inter-spacecraft distance of the order of 100 km; otherwise, the truncation errors will
lead to meaningless estimations of the magnetic gradients and electric currents.
When the truncation errors are acceptably small, the accuracy of the estimated gradient
is determined by the geometrical errors related to the shape of the tetrahedron and to
the uncertainties on the spacecraft positions; to determine these errors it is mandatory
to know the reciprocal vectors and the covariance matrix of the errors of the spacecraft
positions. Lastly, it can be seen that the size, elongation and planarity of the tetrahedron,
together with the three Euler angles describing the orientation of the principle axes (i.e.,
the direction of the axes of elongation and of planarity), are essential parameters for all
multi-spacecraft science.
Acknowledgements
The authors are indebted to P. Robert for the files of simulated magnetic data in GSE coordinates
and for the computation of the geometrical factors. They are also indebted to J. Vogt for having
drawn their attention to the relationship 15.2 between the volumetric and reciprocal tensors.
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Coeur-Joly, O., Robert, P., Chanteur, G., and Roux, A., Simulated daily summaries of
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