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
—3—
Multi-Spacecraft Filtering:
General Framework
J EAN -L OUIS P INÇON
LPCE/CNRS
Orléans, France
U WE M OTSCHMANN
Institute for Geophysics and Meteorology
Braunschweig, Germany
3.1 Introduction
Measurements of the electric and magnetic fields in space plasmas commonly show
fluctuations in time and space on all observed scales. Single satellite measurements generally do not allow disentangling spatial and temporal variations. With two satellites the
ambiguity is removed only for simple motions of essentially one-dimensional structure.
The determination of the shape and dynamics of three-dimensional structures requires a
minimum of four spacecraft arranged in a tetrahedral configuration and equipped with instruments measuring fields and flows in three dimensions. Multi-satellite measurements
open fundamental new possibilities to analyse spatial plasma structures in the magnetosphere and in the solar wind.
Although the analysis of multipoint data has been an increasingly important activity,
there has been little effort to systematically analyse in situ data from multiple spacecraft
except in the context of event studies. In this chapter we describe a multi-spacecraft data
analysis technique allowing the identification of three-dimensional electromagnetic structures in the wave field. The organisation of the chapter is as follows: in Section 3.2 the
technique is presented; in Sections 3.3 and 3.4 the limitations related respectively to the
field and the experimental constraints are examined. Three applications using synthetic
data are presented in Section 3.5.
3.2 Multi-Spacecraft Filtering Technique
3.2.1
Spectral Representation of the Wave Field
Hereafter we consider a multi-spacecraft experiment composed of N satellites. We
denote as A(t, r α ) the column vector consisting of the N field vectors measured at the
satellite positions r α , α ∈ {1, . . . , N }. A(t, r) is assumed to be composed of L components. If we consider the case of electric and magnetic field measurements, then L = 6
65
66
3. F ILTERING : G ENERAL F RAMEWORK
and we have




A(t, r) = 



Ex (t, r)
Ey (t, r)
Ez (t, r)
Bx (t, r)
By (t, r)
Bz (t, r)








The spectral representation of A(t, r) in the (ω, k) domain is given by
Z Z
A(t, r) =
A(ω, k) ei(ωt−k·r ) dk dω
ω
(3.1)
(3.2)
k
The high time resolution of the experiments generally allows the determination of the spectral amplitude with respect to the angular frequency A(ω, r) by a Fourier transformation.
Equation 3.2 becomes
Z
A(ω, r) = A(ω, k)e−ik·r dk
(3.3)
k
The correlation matrix for two measurements is of particular importance for the characterisation of the field. It is constructed as the dyadic product of two measurement vectors
A(t, r α ) and A(t, r β ), where r α and r β refer to spacecraft positions. It is written
h
i
MA (ω, r α , r β ) = E A(ω, r α ) A† (ω, r β )
(3.4)
where E[. . .] stands for the mathematical expectation and indicates an ensemble average
over a large number of distinct realisations of the data set. The superscript † stands for
hermitian adjoint (transpose and complex conjugation). Assuming the field is homogeneous in space which means the statistics of the fluctuating field are translation invariant
in space, we have
Z
MA (ω, r αβ ) = SA (ω, k)e−ik·r αβ dk
r αβ = r α − r β
(3.5)
k
with
h
i
SA (ω, k) = E A(ω, k) A† (ω, k)
(3.6)
and the trace of SA (ω, k) is the spectral energy density P (ω, k) we are looking for.
3.2.2
Notations and Data Preparation
In order to cope with multi-spacecraft measurements, and for the sake of simplicity,
several definitions and notations are required. We define A(ω) the (NL by 1) single column vector containing the A(ω, r α ) measured by the N spacecraft, namely


A(ω, r 1 )
 A(ω, r 2 ) 


A(ω) = 
(3.7)

..


.
A(ω, r N )
67
3.2. Multi-Spacecraft Filtering Technique
then the relation between A(ω) and the spectral amplitude A(ω, k) is given by,
Z
A(ω) = H(k) A(ω, k) dk
(3.8)
k



H(k) = 


with
Ie−ik·r 1
Ie−ik·r 2
..
.
−i
k·r
N
Ie






(3.9)
I is an (L by L) unit matrix and thus H(k) is an (LN by L) matrix. Hereafter we use the
symbol I for all unit matrices whatever their rank. Now we introduce the (LN by LN )
covariance matrix MA (ω). It contains all the correlation matrices that can be estimated
from the N spacecraft experiment and is defined by
h
i
MA (ω) = E A(ω) A† (ω)
(3.10)
This matrix contains all the measured data, and thus is a known quantity in our context.
Theoretically, the determination of MA requires the computation of the mathematical expectation, that is, the averaging of different realisations of the measured field vector A(ω).
Assuming the process is ergodic, this ensemble average may be replaced by averaging in
time, and one obtains
Q
1 X
Aq (ω) Aq† (ω)
(3.11)
MA (ω) =
Q q=1
To get Q convenient measurements Aq (ω), the time series of measurements A(t) is divided in Q subintervals. Fourier transformation in each subinterval q yields Aq (ω). According to the Fourier transformation theory, the length of a subinterval determines the
frequency resolution.
From equation 3.5, the two matrices MA and SA are related by
Z
MA (ω) = H(k) SA (ω, k) H† (k) dk
(3.12)
k
The next section describes the multi-spacecraft filtering technique that allows us to obtain
an optimal estimation of the field spectral energy density P (ω, k) from the matrix MA (ω).
3.2.3
Filter Bank Approach and P (ω, k) Estimation
We shall adopt a filter-bank approach to find the appropriate combination of multispacecraft measurements for an optimum description of the field in the angular frequency
and wave vector domains. Thus, the main objective is the determination of filters, each
one being related to a different (ω, k) pair and characterised by a matrix F(ω, k). For each
(ω, k) the purpose is to design the corresponding filter in such a way that having multispacecraft measurements A(ω) as input, it provides an optimum A(ω, k) estimation given
by
A(ω, k) = F† (ω, k) A(ω)
(3.13)
68
3. F ILTERING : G ENERAL F RAMEWORK
Consequently F† is an (L by LN ) rectangular matrix. Taking the dyadic product of equation 3.13 with its hermitian adjoint, and then taking the expectation value, we connect the
three matrices SA , MA , and F.
SA (ω, k) = F† (ω, k) MA (ω) F(ω, k)
(3.14)
If no additional a priori information is available, a unique determination of the rectangular
matrix F(ω, k) is achieved by requiring that the filter shall absorb all energy contained in
MA except that related with the angular frequency ω and the wave vector k. This requirement is satisfied by minimising the trace of the matrix SA (ω, k) with the constraint that
any plane wave whose angular frequency is ω and wave vector is k is passed undisturbed
through the filter. Then the filter determination can be formulated as
P (ω, k) = Tr F† (ω, k) MA (ω) F(ω, k) = minimum
(3.15)
with
F† (ω, k) H(k) A(ω, k) = A(ω, k)
This problem can be solved using the Lagrange multiplier technique. The detailed derivation of the spectral energy density P (ω, k) is presented in the Appendix. We obtain the
following expression
h
i−1 P (ω, k) = Tr H† (k) M−1
(ω)
H(k)
(3.16)
A
If no a priori information is available, this expression is an optimum estimator of the field
spectral energy density. It is optimum in the sense that it provides a maximum likelihood
estimate of the frequency wave-vector power spectrum, provided the noise associated with
the field measurements is a multi-dimensional Gaussian process. If any given specific
knowledge about the signal is available, one can take it into account during the determination of the filter matrix F. This leads to a modification of the filter constraint and therefore
a new expression of the optimum estimator. The derivation of P (ω, k) estimators, including such additional a priori information, is presented in Section 3.5 for three different
applications.
Estimators obtained using the filter bank approach generally do not require much computation time. For instance, for the estimator corresponding to equation 3.16, once the
MA (ω) matrix has been estimated from the measured waveforms, the computations consist of multiplications and inversions of complex matrices. The largest matrix to invert is
MA (ω) whose rank is L × N. For a multi-spacecraft mission consisting of N = 4 satellites and with L = 6 (3 electric plus 3 magnetic wave-field components), the rank of MA
is 24. Thus we are not in the case of a large matrix for which specific algorithms must be
developed.
3.3 Limitations Related to the Field
The existence of the spectral energy density estimator is based on the hypotheses of
time stationarity and space homogeneity of the measured field. Obviously, none of those
hypotheses is completely fulfilled in a real experiment. The conditions under which they
may be considered to be satisfied adequately are discussed below.
69
3.3. Limitations Related to the Field
The validity of this estimator is also limited by an additional hypothesis requiring that
the measured field be free of characteristic wavelengths smaller than the minimum interspacecraft distance. Otherwise an aliasing effect develops, which is similar to the one
observed in the case of under-sampling of time series.
3.3.1
Time Stationarity and Space Homogeneity
Actually, an approximate temporal stationarity of the field is sufficient to provide reasonable data statistics. For all practical purposes only time stationarity during time intervals much longer than the maximum period studied in the field is necessary. A well-known
statistical test [Bendat and Piersol, 1971] may be used to check the limited temporal stationarity of the data.
Strict conditions for homogeneity cannot be met in space, particularly in the vicinity
of geophysical boundaries such as the bow shock or the magnetopause. In our case a
statement of “limited homogeneity” is sufficient. It is fulfilled when the field is translation
invariant over distances much larger than the maximum wavelength studied in the field.
In the frame of a multi-spacecraft experiment, the translation invariance of the field can
only be tested along the direction of the spacecraft trajectory. Data sets measured at close
positions along the orbit may be used to check the invariance of the P (ω, k) solutions
obtained.
3.3.2
Spatial Aliasing
This problem is also discussed in Section 14.5.1 of Chapter 14. The origin of the
spatial aliasing comes from the fact that the spacecraft configuration does not distinguish
two plane waves differing only by their wave vectors in such a way that
1k · r α = 2π nα + φ
(3.17)
∀α ∈ {1, . . . , N}
where nα are signed integers. For N = 4 it can be shown [Neubauer and Glassmeier,
1990] that 1k has the solutions
1k =
l=3
X
(3.18)
nl 1k l
l=1
with the 1k l given by
1k 1 =
2π
r 31 × r 21 ;
V
1k 2 =
2π
r 41 × r 21 ;
V
1k 3 =
V = r 41 · (r 31 × r 21 )
2π
r 41 × r 31
V
(3.19)
The impossibility of distinguishing these plane waves is referred to as “spatial aliasing”. This spatial aliasing would be absent, and consequently the P (ω, k) estimation not
distorted, if the characteristic lengths of the field correspond to wave vectors included
inside the subvolume described by
k=
l=3
X
l=1
el 1k l
with
− 0.5 < el ≤ 0.5
(3.20)
70
3. F ILTERING : G ENERAL F RAMEWORK
This restriction corresponds roughly to the hypothesis of a field free of wavelengths smaller
than the minimum inter-spacecraft distance. In the absence of any a priori information related to the minimum characteristic length of the field, there is no satisfactory way to
identify spatial aliasing from multi-spacecraft data. Indeed, to insure the validity of the
P (ω, k) estimator, the solution would be to remove from the data the part of the field
related to the small wavelengths before performing the analysis. Unfortunately, such a
filtering process would require a large number of measuring points not compatible with
a multi-spacecraft mission. Actually, the motion of the satellites in space can be used to
identify the presence of short wavelength disturbances parallel to the main velocity (for instance one can rely on the consistency of successive analyses or on comparisons between
successive measurements at short distances).
3.4 Limitations Related to the Experimental Constraints
As real data were not available when we wrote this chapter, we used synthetic data
generated by numerical simulation to study the effects of multi-spacecraft experimental
constraints on the spectral energy density estimator. We applied the P (ω, k) estimator to
data sampled synchronously at different satellite position r α , α ∈ {1, . . . , N}. The satellite positions and the wave vectors were tuned to avoid spatial aliasing. By varying the
parameters of the simulations we were able to examine the effects related to the configuration geometry and to the accuracy in the measurements (distance between spacecraft, time
synchronisation).
3.4.1
The Spacecraft Configuration
An unambiguous determination of three-dimensional structures requires, as a minimum, four spacecraft in a three-dimensional configuration. Using simulated data, several
4-spacecraft configurations were studied. The best solutions are obtained with a tetrahedral
geometry [Pinçon and Lefeuvre, 1991, 1992]. The simulations demonstrate the necessity
of checking for the shape of the spacecraft configuration before interpreting the obtained
solutions. This can be easily done using parameters describing the tetrahedron geometry
(we refer the reader to Chapter 13).
3.4.2
Inaccuracy in the Time Synchronisation
The accuracy in the time synchronisation between the measurements performed on the
different spacecraft is of prime importance. For a given frequency ω, a time inaccuracy
δt introduces a phase shift φ = ω δt in the estimation of the power spectra. Phase shifts
greater than a few degrees can theoretically distort a P (ω, k) estimation. Actually, an
exact threshold in the required time accuracy is not easy to define. It depends on the other
errors in the data, and on the satellite configuration. A series of simulations, not shown
here, has been performed by Pinçon and Lefeuvre [1992] to provide guidelines. From
them, it seems reasonable to fix an empirical threshold at 30◦ . At and above this value,
the fit with the model becomes very poor. The practical consequence of this is to limit
the validity domain of the P (ω, k) estimator to the low frequency range. For instance,
on board Cluster the electric and magnetic field waveform data will be available with an
3.5. Examples
71
accuracy of about 50 µs. For such a value, if we fix the maximum phase shift at 5◦ degree,
the corresponding maximum frequency is equal to 280 Hz.
3.4.3
Inaccuracy in the Inter-Spacecraft Distances
The effect of errors related to the inaccuracy in the inter-spacecraft distances is similar
to the one produced by the time synchronisation inaccuracy. For a given wave vector k
in the measured wave field, a distance inaccuracy δr αβ introduces a phase shift k · δr αβ
in the estimation of the power spectra. As previously, this effect can only be empirically
evaluated from simulations. It has been shown by Pinçon and Lefeuvre [1992] that the
upper limit of the relative error in the distance has to be fixed between 10% and 20%.
Cluster would easily meet this requirement since the relative distance error is expected to
be less than or equal to 10%.
3.5 Examples
The data analysis technique described in this chapter can be used to identify a large
class of three-dimensional structures. To illustrate this point, three different applications
are presented in this section. In the first example we applied the P (ω, k) estimator to
synthetic multi-spacecraft measurements composed of the three magnetic plus the three
electric components. In the second example a similar analysis is performed but now the
data are the three magnetic components. The last example demonstrates that relatively
slight modifications of the P (ω, k) estimator allow us to deal with the detection of surface
waves. For all examples the specific nature of the signal analysed is taken into account.
We used synthetic wave fields consisting of superpositions of plane waves and incoherent noise. All plane waves propagate with different wave vectors at one frequency ω.
The satellite positions and the wave vectors are tuned to avoid spatial aliasing. For reasons of presentation, the wave vectors are chosen in the plane kz = 0. To avoid a loss
of generality, a wave-field simulation including wave vectors distributed over the whole
three dimensional domain would have been preferable. But presentation of the corresponding P (ω, k) would have required several plots. Moreover, limited trials showed that
the kx and ky resolutions of the solutions obtained in the kz = 0 case are similar to the kx
and ky resolutions obtained in the three dimensional case by integrating P (ω, k) over all
kz . Hereafter the solutions are represented in the plane kz = 0 where the field energy is
concentrated. The crosses indicate the location of the k vectors related to the modes introduced in the simulation. The P (ω, k) solutions are expected to present significant peaks
at these locations. The actual solutions, obtained from the synthetic multi-spacecraft data,
are represented by contour lines linearly scaled between the maximum and the minimum
value.
3.5.1 P (ω, k) Estimator Related to the Electromagnetic Wave Field
First we determined the optimum filter related to the identification of electromagnetic
plane waves [Pinçon and Lefeuvre, 1991]. We assumed that the signals A(ω, r α ) measured
72
3. F ILTERING : G ENERAL F RAMEWORK
by the spacecraft are the 6 electromagnetic components of the wave field.
A(ω, r α ) =
E(ω, r α )
B(ω, r α )
∀α ∈ {1, . . . , N }
(3.21)
From the Maxwell-Faraday relation, the electric and magnetic components in the (ω, k)
domain are related through k and ω by k × E(ω, k) = ωB(ω, k). As a consequence we
have A(ω, k) = C1(ω, k) E(ω, k) with


1
0
0
 0
1
0 


 0
0
1 


ky 
C1(ω, k) = 
(3.22)
−kz
 0
ω
ω 
 kz
−kx 
0
 ω
ω 
−ky
kx
0
ω
ω
Taking into account the a priori information coming from the electromagnetic nature of
the field analysed and the Maxwell-Faraday relation, the new filter constraint is
F† (ω, k) H(k) C1(ω, k) = C1(ω, k)
(3.23)
Solving the minimisation we find
P 1(ω, k) =
h
i−1
†
†
†
Tr C1(ω, k) C1 (ω, k) H (k) M−1
(ω)
H(k)
C1(ω,
k)
C1
(ω,
k)
A
(3.24)
Figure 3.1a shows the P 1(ω, k) solution obtained with the four satellites arranged in
a tetrahedral geometry. The wave vectors of the nine plane waves used for the simulation
have been chosen in a way to avoid spatial aliasing. These wave vectors are quite well
identified, the discrepancies between the peaks and the crosses are very small.
An illustration of the effects of multi-spacecraft experimental constraints on the
P 1(ω, k) solution is given by the Figures 3.1b, c, and d. Figure 3.1b shows the solution obtained with the four satellites arranged in a linear geometry parallel to the x axis. In
this case the resolution following the y axis is very low and the fit with the model becomes
very poor. Figure 3.1c shows the solution obtained with the four spacecraft arranged in a
tetrahedral configuration. To simulate inaccuracy in time synchronisation, random phase
shifts between −30◦ and +30◦ were introduced in the data. Figure 3.1d shows the solution
obtained with the four spacecraft arranged in a tetrahedral configuration. The inaccuracy
in the distance measurements is taken into account by imposing a relative error (|δr αβ | /
|r αβ |) equal to 20%.
Figure 3.2 demonstrates the effects of aliasing on the P 1(ω, k) solution obtained with
four satellites arranged in a tetrahedral geometry. The wave field analysed is the one used
to obtain the Figure 3.1 with one more electromagnetic plane wave. The corresponding
tenth k vector has been deliberately chosen to introduce aliasing (k †10 = [−10, −1, 0]).
As pointed out in Section 3.3.2, the P 1(ω, k) solution is not valid any more. This is clearly
illustrated by the presence of a spurious peak located at kx = −0.7 and ky = −1.2.
3.5. Examples
73
Figure 3.1: P 1(ω, k) solutions obtained from four spacecraft for a wave field consisting of
a superposition of nine electromagnetic plane waves. The crosses indicate the location of
the nine k vectors related to the plane waves. The satellite positions and the wave vectors
are tuned to avoid spatial aliasing. The solutions are represented by contour lines linearly
scaled between the maximum and the minimum. Solutions obtained using: a) a tetrahedral
geometry; b) a linear geometry (parallel to the x axis; c) a tetrahedral geometry with
random phase shifts between −30◦ and +30◦ to simulate time synchronisation inaccuracy;
d) a tetrahedral geometry in the case of a 20% inter-spacecraft distance inaccuracy.
74
3. F ILTERING : G ENERAL F RAMEWORK
Figure 3.2: P1(ω, k) solutions obtained using a tetrahedral geometry for a wave field
consisting of a superposition of the wave field of Figure 3.1 and one more electromagnetic
plane wave. The additional k vector has been chosen to introduce aliasing. The solution
presents a spurious peak at kx = −0.7 and ky = −1.2.
3.5.2 P (ω, k) Estimator Related to the Magnetic Field
As a second illustration we determined the optimum filter related to the identification
of magnetic plane waves [Motschmann et al., 1996]. We assumed that the signals A(ω, r α )
measured by the spacecraft are the 3 magnetic components.
A(ω, r α ) = B(ω, r α )
∀α ∈ {1, . . . , N}
(3.25)
Since the magnetic field is divergence-free, the magnetic components in the (ω, k) domain are related to k by k · B(ω, k) = 0. As a consequence we have B(ω, k) =
C2(ω, k) B(ω, k), where
kk †
C2(ω, k) = I + 2
(3.26)
|k|
By taking into account the divergence-free nature of the measured wave field, the constraint can be rewritten as
F† (ω, k) H(k) C2(ω, k) = I
(3.27)
With this new constraint the minimisation described by equation 3.15 yields
h
i−1 †
−1
†
P 2(ω, k) = Tr C2 (ω, k) H (k) MA (ω) H(k) C2(ω, k)
(3.28)
The result of the application of P 2(ω, k) to synthetic data is shown in Figure 3.3. We
assumed a tetrahedron-like configuration, a perfect time synchronisation and no error in
75
3.5. Examples
Figure 3.3: P 2(ω, k) solution for the plane kz = 0 obtained using a tetrahedral geometry. The wave field consists of a superposition of seven plane waves. [Reproduced from
Motschmann et al., 1996.]
the inter-spacecraft distances. The P 2(ω, k) estimator has no problem finding the seven
magnetic plane waves used in the simulation.
3.5.3
Electromagnetic Surface Wave Detector (SWD)
The last example is somewhat different from the two previous. Instead of a plane
wave detector, we determined a filter allowing the identification of electromagnetic surface
waves. We assumed that the signals measured by the spacecraft are the six electromagnetic
components of the wave field.
A(ω, r α ) =
E(ω, r α )
B(ω, r α )
∀α ∈ {1, . . . , N }
(3.29)
A link with the previous P 1(ω, k) estimator can be found by noting that an electromagnetic surface wave can be described as an electromagnetic plane wave with complex wave
vector (k = k r + ik i ). The spatial inhomogeneity associated with the surface waves is
included in the filter by choosing one of the satellite position as a reference and rewriting
the H matrix as


I exp(−ik r · r 1 ) exp(k i · (r 1 − r 1 ))


..
H3(k) = 
(3.30)

.
I exp(−ik r · r N ) exp(k i · (r N − r 1 ))
76
3. F ILTERING : G ENERAL F RAMEWORK
Figure 3.4: SWD(ω, k) solution for the complex plane (kx , iky , kz = 0) obtained using a
triangular geometry; see text. [Reproduced from Pinçon, 1995.]
This way the H3 matrix contains not only the relative phase information at all spacecraft positions, but also a model of the inhomogeneity between spacecraft. The previous
hypothesis of spatial homogeneity is here replaced by the hypothesis of spatial inhomogeneity exclusively related to surface waves. Using the Maxwell-Faraday relation we still
can relate the electric and magnetic components in the (ω, k) domain through k and ω. We
have A(ω, k) = C3(ω, k) E(ω, k) with C3 similar to C1 but k is complex. The optimum
(ω, k)-filter is obtained by demanding that it absorbs all energy contained in MA (ω) except
that corresponding to a surface wave characterised by the frequency ω and the complex
wave vector k.
SW D(ω, k) = Tr F† (ω, k) MA (ω) F(ω, k) = minimum
(3.31)
with
F† (ω, k) H3(k) C3(ω, k) = C3(ω, k)
Solving the minimisation problem we find
SW D(ω, k) =
h
i−1
†
†
−1
†
Tr C3(ω, k) C3 (ω, k) H3 (k) MA (ω) H3(k) C3(ω, k)
C3 (ω, k)
(3.32)
Figure 3.4 shows the SWD solution obtained using data generated by a 2-D MHD
code. The simulated spacecraft are arranged in a triangular configuration, they are moving through the simulation domain with a constant velocity. We assumed a perfect time
synchronisation and no errors in the inter-spacecraft distances. The simulated wave field
is composed of two MHD electromagnetic surface waves with the same frequency and
different complex vectors. The solution is represented in the complex plane (kx real, ky
77
3.6. Summary
imaginary, kz = 0) where the energy of the simulated field is concentrated. The two
surface waves of the simulation are clearly identified by the surface wave detector.
3.6 Summary
The multi-spacecraft data filtering technique described in this chapter allows the identification of three-dimensional structures in the wave field. The two important cases of plane
waves- and surface waves-identification are examined. The characterisation is achieved
through an estimation of the wave field spectral energy density in the angular frequency
and wave vector domains. Using synthetic data we have validated the usefulness of this
approach.
The list of applications presented in this chapter is not exhaustive. For instance it is
also possible using this technique to develop a large scale current detector from magnetic
field measurement. Having in mind a precise objective, and relevant multi-spacecraft data
being available, the filter bank approach can be used to design filters specifically adapted
to the problem. The performance of the estimators derived using the multi-spacecraft
filtering technique is strongly related to the amount of a priori information included in the
filter design. Not much can be said about this point, each case being a particular case.
The main limitation is related to the characteristic field lengths which have to be
larger than the mean inter-spacecraft distance to avoid aliasing effects. To obtain a threedimensional characterisation, simultaneous measurements on a minimum of four points
in a tetrahedral configuration are required. Other limitations are related to the hypothesis
of space homogeneity and the accuracy with which the multi-spacecraft configuration is
defined in space and time.
The multi-spacecraft filtering technique does not require much computation time and
can be used to identify a large class of different structures. The applications studied have
shown that the resolution obtained in the k domain is sufficient to provide an accurate
characterisation of the wave field. This remains true even in the case of an experiment
consisting of only four measuring points in space.
Appendix: Derivation of P (ω, k)
The problem of the filter determination can be formulated as
P (ω, k) = Tr F† M F = minimum
with
F† H = I
(3.33)
Since F is a complex matrix each element has a real and an imaginary part, thus
Fij = Xij + iYij
(F † )j i = Xij − iYij
(3.34)
Xij and Yij are independent variables, consequently Fij and (F † )j i are independent variables too. In what follows we treat F and F† as independent matrices. Then, using the
Lagrange multiplier technique, to solve the filter determination problem formulated by
equation 3.33 is equivalent to minimise
n
o
P (ω, k) = Tr F† MA (ω) F + Λ I − H† F + I − F† H Γ
(3.35)
78
3. F ILTERING : G ENERAL F RAMEWORK
where Λ and Γ are the Lagrangian multiplier matrices related to constraints involving F
and F† , respectively. Equation 3.35 written in terms of components reads
P (ω, k) = (F † )ij Mj k Fki + 3ii − 3ij (H † )j k Fki + 0ii − (F † )ij Hj k 0ki
(3.36)
Equating to 0 the partial derivative of this expression with respect to Fki and (F † )ij , yields
the following results
∂P (ω, k)
= 0 −→ F† M = ΛH†
∂Fki
∂P (ω, k)
= 0 −→ MF = HΓ
∂(F † )ij
(3.37)
(3.38)
Then we multiply equation 3.37 by F and equation 3.38 by F† . From the definition of P
and the constraint of equation 3.33, we get
Λ=Γ
and
P (ω, k) = Tr {Γ}
(3.39)
Provided M is not singular, the expression of Γ is obtained by multiplying equation 3.38
by H† M−1 , thus
h
i−1
Γ = H† M−1 H
(3.40)
Finally, substituting this into equation 3.39, we obtain an expression for the spectral energy
density P
h
i−1 P (ω, k) = Tr H† M−1 H
(3.41)
Acknowledgements
We thank Steven J. Schwartz, Götz Paschmann, and the referee for helpful comments and valuable suggestions.
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