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Instantaneous Local Wave Vector Estimation from Multi-Spacecraft Measurements Using Few Spatial Points T Instantaneous local wave vector estimation from multi-spacecraft measurements using few spatial points T. D. Carozzi, A. M. Buckley, M. P. Gough To cite this version: T. D. Carozzi, A. M. Buckley, M. P. Gough. Instantaneous local wave vector estimation from multi- spacecraft measurements using few spatial points. Annales Geophysicae, European Geosciences Union, 2004, 22 (7), pp.2633-2641. hal-00317540 HAL Id: hal-00317540 https://hal.archives-ouvertes.fr/hal-00317540 Submitted on 14 Jul 2004 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Annales Geophysicae (2004) 22: 2633–2641 SRef-ID: 1432-0576/ag/2004-22-2633 Annales © European Geosciences Union 2004 Geophysicae Instantaneous local wave vector estimation from multi-spacecraft measurements using few spatial points T. D. Carozzi, A. M. Buckley, and M. P. Gough Space Science Centre, University of Sussex, Brighton, England Received: 30 September 2003 – Revised: 14 March 2004 – Accepted: 14 April 2004 – Published: 14 July 2004 Part of Special Issue “Spatio-temporal analysis and multipoint measurements in space” Abstract. We introduce a technique to determine instan- damental assumptions, namely stationarity and homogene- taneous local properties of waves based on discrete-time ity. Unfortunately, in space physics most phenomenon are sampled, real-valued measurements from 4 or more spatial both nonstationary and inhomogeneous. Despite some re- points. The technique is a generalisation to the spatial do- cent techniques based on wavelets (Balikhin et al., 2003) few main of the notion of instantaneous frequency used in sig- multipoint techniques are available in these cases. In what nal processing. The quantities derived by our technique are follows we will present a novel analysis method which is closely related to those used in geometrical optics, namely applicable to non-stationary and inhomogeneous wave phe- the local wave vector and instantaneous phase velocity. Thus, nomena. This is accomplished by basing the analysis on the this experimental technique complements ray-tracing. We instantaneous and local properties of the observed field. The provide example applications of the technique to electric method is based on the concepts of instantaneous frequency field and potential data from the EFW instrument on Cluster. (Cohen, 1995) and local wave number (Beall et al., 1982) Cluster is the first space mission for which direct determina- and is implemented via the Hilbert transform. We present tion of the full 3-dimensional local wave vector is possible, the method from the perspective of multipoint measurements as described here. in space physics but the treatment can easily be generalised to other subjects areas. Key words. Electromagnetics (wave propagation) – Space plasma physics (experimental and mathematical techniques) – General or miscellaneous (techniques applicable in three or 2 Continuum theory of the instantaneous frequency more fields) and local wave vector The method presented in this paper is a multivariate, multi- 1 Introduction dimensional generalisation of the instantaneous frequency (IF), (see Cohen, 1995; Boashash, 1992), extended to include There are many general multipoint spatio-temporal algo- the three-dimensional spatial domain. In the construction rithms in use in experimental space physics for the study of of a spatial domain analogue of the IF, we rely heavily on waves and structures. A survey of some of these techniques the concept of the local wave vector from geometrical optics is given in the analysis methods book (Paschmann and Daly, (Whitham, 1974). We first present the necessary background 1998). There are time domain-based methods, like the dis- and concepts in terms of a continuum field before dealing in continuity analyser which measures structures using timing the next section with the discrete nature of actual multipoint differences between measurement points. Then there are fre- measurements. quency domain-based methods such as two point interferom- Consider a physical quantity defined on a time-space con- etry proposed in Kintner et al. (1984). There are techniques tinuum, (t, x), which exhibits wave-like phenomena, fluctu- which take advantage of the vectorial or polarimetric aspects ations, or structures. Although physically direct measure- of waves (Storey and Lefeuvre, 1974), such as the k-filtering ments of the quantity are normally real valued, it is fruitful (Pinc¸on and Motschmann, 1998) or wave telescope tech- to consider the measurement of the physical quantity sr as nique (Glassmeier et al., 2001). Basically all the techniques the real part of a complex field z. Explicitly, we model the to date for measuring wave properties are based on two fun- physical field as the sum of the measured real part sr and an imaginary part si, such that Correspondence to: T. D. Carozzi ([email protected]) z(t, x) = sr (t, x) + isi(t, x). (1) 2634 T. D. Carozzi et al.: Instantaneous local wave vector estimation x the geometrical optics approximation, one can show (Born phase field and Wolf, 1999) that it is given by κ (projected) inst. loc. 2 2 x wave 4−vector |n(t, x)| = |∇φ(t, x)| . (4) Ray path This equation is known as the eikonal equation and it im- isophase plies that the dynamics of wave propagation is determined contours = w through the dynamics of the phase. Surfaces with constant phase, φ(x) = const, are called wave fronts, whereas char- space wavefronts acteristics, or ray paths, are the curves which are everywhere 2 S/C worldlines orthogonal to the wave fronts. Thus, in geometrical optics, the rays follow the wave front normals ∇φ. In particular, the quantity time t κ(t, x) := −∇φ, (5) Fig. 1. Illustration of a 2-dimensional projection (time and one spa- is known as the local wave vector. These concepts are illus- tial component) of the instantaneous local wave 4-vector under the trated in Fig. 1. assumptions of geometrical optics and isotropy. From the phase The definition of the instantaneous local wave vector κ field of a physical quantity one can define wavefronts in this illustra- is the spatial compliment of the instantaneous local angular tion as isophase contour lines (solid curves) and a ray path (dashed frequency, w, defined as the temporal gradient of the wave curve) that are everywhere orthogonal to the wavefronts. The in- phase or stantaneous local wave vector κ is the spatial gradient of the phase field and, in the geometrical optics approximation, κ is tangent to ∂φ w(t, x) := . (6) the ray path. The problem addressed in this paper is how to estimate ∂t κ from real valued discrete measurements taken along world lines (dotted curves). The name for w is motivated from the fact that it is a gen- eralisation of frequency defined at instant t and local spatial point x. It can be shown that the temporal average of the in- Surprisingly, although we have not directly measured the stantaneous frequency is equal to the spectral average of the imaginary part, one can determine it from the real part as frequency spectrum, that is, under appropriate normalisation, long as the field is analytically well behaved. An example of Z Z this is the Kramers-Kronig relation, which in fact originally hwi = w(t)|z(t)|2 dt = ω|Z(ω)|2 dω = hωi, (7) related the imaginary part of the dispersion index to the real part by assuming causality. Mathematically, the imaginary where Z(ω) is the Fourier transform of z(t) (see Cohen, part is related to the real part through the Hilbert transform 1995). (HT), that is the HT of a real function is its imaginary part In the above we are assuming that there is only one wave. and the inverse HT of the imaginary part of a function gives By this we mean that the amplitude and phase of the signal its real part. Thus, by using the HT we can construct the in Eq. (1) corresponds to a unique wave as opposed to a su- analytic field, such as z, from a real function sr . perposition of independent waves such as Once we are in possession of an analytical field it is iφ1 iφ2 straightforward to define the amplitude A z = A1e + A2e + ... q The former case is known as a single component wave. In the A = |z| = s2 + s2 (2) r i current paper we will only consider single component waves, and the phase φ although we developed an extension of our analysis to treat multiple component waves. φ = arg z = arctan sr /si (3) To interpret w and κ, assume that the phase is slowly vary- ing around some point in space-time (t0, x0) and perform a of the field so that we can write the field in the polar form multivariate Taylor expansion with the result iφ(t,x) z(t, x) = A(t, x)e . ∂φ φ(t, x) = φ(t0, x0) + (t − t0) + (x − x0)∇φ + ... Note that both φ and A are real valued fields on the space- ∂t time continuum. so that close to any arbitrary point (t0, x0) the field of the The propagation of a wave mode m is ultimately deter- physical quantity can be approximated by mined by the local refractive index nm(x) which depends on ≈ iφ0 − · the local properties of the medium and characteristics of the z(t, x) Ae exp (i(wt κ x)) mode. Unfortunately, one does not always have direct mea- with surements of nm(x), but under the simplifying approximation that the medium varies slowly over a wavelength, known as φ0 := φ(t0, x0) − wt0 + κ · x0, T.
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