Validation of the k-filtering technique for a signal composed of random phase plane waves and non-random coherent structures

Recent observations of astrophysical magnetic ﬁelds have shown the presence of ﬂuc-tuations being wave-like (propagating in the plasma frame) and those described as being structure-like (advected by the plasma bulk velocity). Typically with single space-craft missions it is impossible to di ﬀ erentiate between these two ﬂuctuations, due to the 5 inherent spatio-temporal ambiguity associated with a single point measurement. However missions such as Cluster which contain multiple spacecraft have allowed temporal and spatial changes to be resolved, with techniques such as the k -ﬁltering technique. While this technique does not assume Taylor’s hypothesis as is necessary with single spacecraft missions, it does require weak stationarity of the time series, and that the 10 ﬂuctuations can be described by a superposition of plane waves with random phase. In this paper we test whether the method can cope with a synthetic signal which is composed of a combination of non-random phase coherent structures with a mean radius d and a mean separation λ , as well as plane waves with random phase.


Introduction
Understanding plasma physics processes in three dimensions is one of the major goals of the Cluster mission (Escoubet et al., 2001).The tetrahedral arrangement of the four identical spacecraft allows the disentanglement of temporal and spatial changes unambiguously in three dimensions.This property is essential to be able to distinguish between the two competing ideas for plasma heating in astrophysical plasmas, that of wave damping, and that of dissipation in coherent structures.In this regard single spacecraft measurements can be quite limiting, and Taylor's hypothesis needs to be invoked (Taylor, 1938).
The k-filtering technique (Pinçon and Lefeuvre, 1991) is a multi-spacecraft analysis technique, with the advantage that Taylor's hypothesis is not required.Here we use the term "k-filtering" to refer to all techniques which are based on the same mathematics Figures  (Pinçon and Motschmann, 1998) e.g. the wave-telescope technique (Neubauer and Glassmeier, 1990) or the multi-point signal resonator technique (Narita et al., 2011).The method takes the magnetic field (or magnetic and electric fields) measurements from the four spacecraft and is able to determine the full three dimensional power spectral density P (ω, k).A filter bank approach is used and the measured signals are filtered such that the only powers that pass through the filter correspond to a plane wave at the chosen frequency ω, and wavevector k, with all other powers in the signal being eliminated.The filtering can be performed for a number of points in wave space k = (k x , k y , k z ) and a fully three dimensional estimation of the power spectral density P (ω, k) in wavenumber space can be constructed.
The method itself has been tested extensively for a superposition of plane waves with random phases (e.g.Pinçon andLefeuvre, 1991, 1992;Motschmann, 1996;Pinçon and Motschmann, 1998;Sahraoui et al., 2010), and the effects of noise, time synchronization inaccuracies, spacecraft position inaccuracies and poor tetrahedral configurations are well understood.However, the random phase approximation is an assumption that has not been discussed thoroughly.The key assumption for this method is that the measured fluctuations can be described as a superposition of plane waves with random phases (Capon, 1969;Pinçon andLefeuvre, 1991, 1992;Motschmann et al., 1998;Tjulin et al., 2005).Here the terms "random phase plane waves" refer to waves having randomised (non-fixed) phase difference at each consecutive waveperiod.Practically for the application of the k-filtering method, this requires that at each frequency, waves with different wavevectors are incoherent.
The k-filtering method has been used in several near-Earth plasma environments with input coming exclusively from magnetic field measurements (e.g. the magnetosheath, the foreshock and the solar wind; Glassmeier et al., 2001;Eastwood et al., 2003;Sahraoui et al., 2003Sahraoui et al., , 2004;;Roberts et al., 2013), and has also been applied using a combination of both magnetic and electric field measurements (Tjulin et al., 2005(Tjulin et al., , 2008)).Introduction

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Full A recent study which applied the k-filtering technique to the solar wind was performed by Roberts et al. (2013) and concluded that the solar wind may be populated by linear Kinetic Alfvén waves (KAWs), as well as coherent structures which are advected solar wind bulk velocity (e.g.monopolar Alfénic vortices, Alexandrova et al., 2006;Alexandrova, 2008, pressure balanced structures, Burlaga andOgilvie, 1970) or those propagate with a small velocity v V sw in the plasma rest frame (e.g.dipolar Alfvén vortices Petviashvili andPokhotelov, 1985, 1992).This study concluded that techniques such as k-filtering may need to be further validated for analysis of a signal which is composed of a superposition of coherent structures and random phase plane waves.In relation to the signal we use the term "coherent structure" to denote an intermittent magnetic field signature which is characterised by a radius and a mean spacing between them.It is important to note that within the radius of a structure, the variation of the signal (magnetic field) is known and the phases are not random.This makes it similar to a coherent wave packet.However with respect to other structures they are likely to be incoherent, and a key goal of this work is to determine the effect of coherency on the determination of the power of waves or structures when the k-filtering technique is adopted.
Here we quote the comments of Pinçon and Lefeuvre (1991) when the k-filtering was first designed: "Measurements of the electric and/or magnetic field in space plasmas commonly show fluctuations in time and/or space on all observed scales; such fluctuations are mostly random in appearance and therefore are considered as turbulent phenomena."While we would expect waves in a turbulent plasma to have random phases, it is unclear whether this is satisfied for a population of coherent structures being advected past the spacecraft or for a signal with a coexistence of random phase plane waves and coherent structures (including short lived coherent wave packets).
The paper is also motivated by studies that discrete waves have been indeed observed in the solar wind (Jian et al., 2009(Jian et al., , 2010) ) and discrete waves can survive in a turbulent environment for a limited time at least theoretically (Ghosh et al., 2009) The goal of this work is to provide such a validation for the technique using synthetic magnetic field data.The k-filtering method will be presented in Sect. 2. Then we will present the various signals which we will input to the method, and how they were defined in Sect.3, followed by the results and conclusions.

The k-filtering method
Here we will provide a brief introduction to k-filtering, following the notation given in Pinçon and Motschmann (1998) and Tjulin et al. (2005).If we consider an array of detectors, in our case we have the Cluster spacecraft and the magnetometer measurements which yield 12 time series' (in our case four satellites yield three measurements each).We can represent our data as the following vector: These time series are then transformed into frequency space by using a windowed Fourier transform (Eq.2).The size of the window determines the frequency resolution.
When we have a window of a single data point (window size is zero) no frequency information can be obtained.In the other extreme case when we only have one window for an interval we are unable to resolve any temporal variations (Motschmann, 1996).Typically windows of 1024 points (which corresponds to 45 s for nominal mode Cluster FGM data sampled at 22 Hz, Balogh et al., 2001) are used (Sahraoui et al., 2003(Sahraoui et al., , 2006)).This corresponds to a frequency resolution of ∼ 0.024 Hz (Sahraoui et al., 2003), suitable for studying inertial and dissipation scales in the solar wind.It also should be noted that a key limitation due to the use of a windowed Fourier transform, is the requirement that the time series are at least weakly stationary so the goal of Introduction

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Full achieving ensemble average, a key requirement of the k-filtering method, can be realised by using time averages.This is an important restriction when selecting intervals for analysis.Intervals are chosen such that the magnetic field time series are devout of shocks, discontinuities or any trends so that the quality of the results are not impaired.
This is a 12 × 1 column vector containing three components of the Fourier transformed magnetic field for 4 satellites.From this the correlation matrix can be found, which is ensemble average of the matrix product of the Fourier transformed time series and its Hermitian adjoint (transposed and complex conjugated denoted by the superscript plus) can be written as: If a time series is weakly stationary and spatially homogeneous, then we can replace the ensemble average in Eq. ( 3), by a time average.This is an important limitation of the method, as the time series is not stationary, for example if there are trends in the data then the ensemble average and the time average will not be the same, and we cannot use a time average.We now need to constrain the data such that no unphysical results are obtained.This is done by ensuring that the distances between the spacecraft (and the associated phase difference), are accounted for in the estimation from k-filtering.To account for the relative spacecraft positions and the differences in phase in relation to this we define a matrix H where I is the identity matrix: Introduction

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Full When working solely with magnetic field data a second condition is also used, where we enforce the solution to be divergence free.When using both electric and magnetic fields Faraday's law can be used instead to constrain our solutions.We can ensure divergence free condition is satisfied by the estimated solutions by making k • δB = 0, and this is implemented by using the constraining matrix C where the matrix is given by (Pinçon and Motschmann, 1998): It follows that the full k-filtering equation is given by:

Simulated magnetic field data
To simulate magnetic field data we begin with a superposition of eight plane waves with equal power as described by: where k i = k i x e x +k i y e y is the wave vector of the i th wave, ω is the angular frequency, is the wavenumber defined, e x and e y are two unit vectors along the xand y-axis in a Cartesian coordinate system.A uniform background magnetic field can be in any direction.The N i is the noise term which is set at 1 % level and is required to ensure that there is a solution to the k-filtering method (Pinçon and Lefeuvre, 1991).
Here all waves are given a frequency of 0.2 Hz.The signal is divergence free.Spacecraft are at distances r in a perfect tetrahedral configuration such that the planarity and elongation (see Robert et al., 1998 for details of the Cluster tetrahedral configuration) are P ∼ E ∼ 0. The phase term φ is randomised between ±2π in each wave period in the case of a random phase plane wave.The randomisation process is a "jump" in the phase and not a continuous variation as a function of time φ(t).A continuous variation in the phase is not expected over short timescales.For a series of unrelated wave trains made up of wave packets (or a number of coherent structures) we would expect the phase to jump with each new wave packet in the train.We do not investigate changes in phase in the spatial domain, since these are assumed to be small on the scales of spacecraft separation.
This gives perhaps the simplest case we can investigate of a superposition of random phase plane waves.
We use a superposition of eight plane waves at a single frequency, since we are limited theoretically to recover NL−1 waves where N is the number of spacecraft, and L is the number of components we can resolve (Tjulin et al., 2005).Therefore theoretically, we are limited to eleven waves, however the resolution is too low to uniquely be able to identify more than eight waves.
To test the method we will define three different signals, and investigate the properties of the estimated power spectral density for each of the input signal.The first case we will test consists of a pure superposition of random phase plane waves.This case has been extensively tested previously, with both electric and magnetic field measurements (Pinçon and Motschmann, 1998), and only magnetic field measurements (Motschmann, 1996).

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Full The next case we will test is that of a signal composed of both random phase plane waves and non-random phase plane waves.This is implemented by some of the waves in Eq. ( 7) having a constant phase, so they are not randomised in each wave period.
The final cases to be tested will consist of various combinations of structure like signals, and random phase plane waves.
To model the advection of coherent structures past the spacecraft we define them to have a characteristic mean radius d , and a mean spacing between them λ as shown in Fig. 1.Both of these parameters vary with Poisson statistics about a mean for each structure which is advected over the spacecraft.It is important to note that for Poisson statistics, the mean and the varience are equal.The fluctuations due to the structure can be thought of as a "wave packet" which have a mean distance and separation, d and λ.The fluctuations themselves are defined as plane waves (Eq.7), each wave packet always begins at the same phase, and only the specific d and λ are random.One typical time series for a series of structures being convected across the spacecraft is shown in Fig. 2a.We can also investigate the case for when the fluctuations are modulated by a Gaussian envelope, which would make the fluctuations stronger closer to the centre which is the case with coherent structures such as the MHD Alfvén vortex (Petviashvili and Pokhotelov, 1992;Alexandrova et al., 2006;Alexandrova, 2008).This scenario is shown in Fig. 2b.
The total signal is given by the summation of the i individual components, where the summation can contain components that are random (or non random) phase plane waves or coherent structures.
The requirement of the k-filtering technique that waves have random phases may be illustrated here.For instance, two waves similar to those in Eq. ( 7) are given by sin(k 1 • r − ω 1 t − φ 1 ) and sin(k 2 • r − ω 1 t − φ 2 ), the combined wave field will be If φ 1 − φ 2 is fixed (the two waves are coherent), then the above equation basically produce a new wave field with a wavevector at (k 1 + k 2 )/2 at the same frequency ω 1 , 443 Introduction

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Full very different from the original two waves.However, if φ 1 − φ 2 is randomised (at each wave period), the ensemble average of the k-filtering technique will recover the two original random phased waves.

Results
The results for the case of random and non random plane waves are shown in Fig. 3, where the green crosses denote the random phase plane waves and the orange are the non-random components; the contours denote the estimation P (ω, k) from k-filtering.Note mathematically, the power of an individual wave is a singular point located at the crosses.As can be seen the method can easily recover eight random plane waves, and when we introduce one non-random wave the method can recover both random and non-random components.However when more than one component in the time series is non-random, k-filtering has difficulty resolving the non-random fluctuations.This is due to the requirement of random phases, since two or more waves (of the same frequency) with fixed phase differences will superpose which will result in a periodic resultant amplitude.In previous tests of the method the need for incoherent signals was recognized (Capon, 1969;Pinçon and Lefeuvre, 1991;Motschmann et al., 1998).
A key point is that recovery of random phase components is unaffected by the presence of non-random components in the time series.This result gives confidence that, provided that the phases of some waves is random, the method can recover the random component.
Following this test we now seek to understand the effects that intermittent coherent structures have on the estimation of P (ω, k).We introduce some structure elements into the signal as shown in Fig. 2. Figure 4a shows the cases where we have a combination of random phase plane waves (green) and coherent structures (orange).The method succeeds in identifying all components of the defined signal, alleviating the issues which were pointed out by Roberts et al. (2013).Figure 4b shows a case when the signal contains only coherent structures, while Fig. 4c  structures are modulated with a Gaussian.We note that while the technique can identify all of the structures present, the estimation P (ω, k) is broadened in wave space.This implies that the errors associated with a signal consisting mainly of coherent structures are larger than when the signal only consists of random phase plane waves.We also note that there is very little difference in the estimation of P (ω, k) when we modulate the structures using a Gaussian function, as was demonstrated in Fig. 2b.
To explore the issue of broadening associated with a signal composed of structures we investigate the choice of the structure parameters λ and d , and what effects these have on the estimation from k-filtering.If d λ a structure and a random phase plane wave will produce very similar fluctuations as seen by the satellite.Figure 5 shows the power spectral density P (ω, k) for various values of λ and d .A salient feature of these plots is that the estimation improves (contours are closer together) when d λ, and declines as λ d , but the peaks in all cases correspond well to where the structures are launched in k space.We suggest that a reasonable limit for k-filtering to be able to resolve these structures is that d /λ ≥ 0.1.When λ becomes very large there will be fewer structures present in a given interval.The necessary statistics required for the ensemble average of the k-filtering technique will not be met for such interval.On the other hand, as d increases the structures will be similar to non-random phase waves, and can not be identified as we have discussed at the end of Sect. 3.This criteria is often satisfied in space plasmas e.g. in the Earth's magnetosheath (Alexandrova et al., 2006;Alexandrova, 2008) where Alfvénic vortices were identified and the parameters d and λ were derived from their spectral shapes, and were shown to be approximately d = 600 km and λ = 1000 km respectively.

Conclusions
By investigating synthetic magnetic field data we have demonstrated that the k-filtering technique can be used to analyze a signal which consists of random phase plane waves and coherent structures.The presence of any non-random plane wave does not

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Full affect the ability to resolve random phase components, and can even resolve a random component when it is a minority.In the limit d λ there is very little difference between the wave and the structure paradigm and k-filtering has no difficulty in resolving structures in this limit.As λ d the estimation of P (ω, k) is broadened making errors in estimating k larger.We have also tested for a signal where the parameters are similar to those observed in the Earth's magnetosheath and the method resolves the structures with comparable accuracy to the random phase plane wave case.We have demonstrated that waves at any frequency are required to be incoherent, since the k-filtering technique is unable to resolve two waves with a fixed phase difference.A series of randomly located coherent structures arranged with characteristic mean radius and spacing can be assumed to be incoherent and can be resolved by the kfiltering technique.While they are "coherent" within their radius they are incoherent with respect to each other due to their random locations.For turbulent environments such as the solar wind, where waves and structures are both present the incoherency of the signal is likely to be satisfied, and the application of the technique justified.We conclude that there should be no issues in using this technique on a signal which contains both random phases and non-random phases.Introduction

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Full Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | H Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | shows the case when the Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Tjulin, A., Pinçon, J.-L., Sahraoui, F., Andre, M., and Cornilleau-Wehrlin, N.: The k-filtering technique applied to wave electric and magnetic field measurements from the Cluster satellites, J. Geophys.Res., 110, 1-17, 2005.437, 439, 442 Tjulin, A., Lucek, E. A., and Dandouras, I.: Wave activity inside hot flow anomaly cavities, J. Geophys.Res., 113, A08113, doi:10.1029/2005JA011125,2008. of structures with a mean separation λ and diameter d (or thickness) of the structures.These are being advected e solar wind bulk velocity

Figure 1 .
Figure 1.Schematic of a number of structures with a mean separation λ and diameter d (or thickness) of the structures.These are being advected over the spacecraft with the solar wind bulk velocity.

Figure 2 .Fig. 3 .
Figure 2. (a) Typical time series made up of structures with a mean spacing λ = 1000 time steps and d = 250 time steps.(b) Time series of structures which have been modulated by a Gaussian envelope.(c) time series with λ = 1000, d = 2000.The values of λ and d , vary with Poisson statistics where the mean value is equal to the variance.

Figure 3 .
Figure 3.Estimated P (ω, k) for (a) eight random phase plane waves, (b) seven random phase plane waves, one fixed phase plane wave (orange cross), (c) six random phase plane waves and two fixed phase plane waves.The contours denote 0.1, 0.5, 1, 10, 25, 30, 50, 80, 90 % of the maximum power, which cover the large range of powers estimated from the method well.