aa r X i v : . [ a s t r o - ph ] A p r Astrophysic and Space Science manuscript No. (will be inserted by the editor)
ALMA : Fourier phase analysis made possible
Franc¸ois Levrier · Edith Falgarone · Franc¸ois Viallefond
Received: date / Accepted: date
Abstract
Fourier phases contain a vast amount of infor-mation about structure in direct space, that most statisti-cal tools never tap into. We address ALMA’s ability to de-tect and recover this information, using the probability dis-tribution function (PDF) of phase increments, and the re-lated concepts of phase entropy and phase structure quan-tity. We show that ALMA, with its high dynamical range,is definitely needed to achieve significant detection of phasestructure, and that it will do so even in the presence of afair amount of atmospheric phase noise. We also show thatALMA should be able to recover the actual “amount” ofphase structure in the noise-free case, if multiple configura-tions are used.
Keywords
Instrumentation: interferometers · Methods:statistical · Methods: numerical · ISM: structure
Observations of the interstellar medium (ISM) reveal highlycomplex, fractal-like structures [8,7]. The self-similar hier-archy of these structures, over four decades, is thought tospring from the interplay of turbulent motions [12] and self-gravitation [2]. To understand this interplay, one thereforeneeds a quantitative description of the observed structures.Most of the statistical tools used to this end are more or lessderived from the power spectrum [6], which is given by the
F. LevrierLRA, ENS, 24 rue Lhomond 75231 Paris Cedex 05E-mail: [email protected]. FalgaroneLRA, ENS, 24 rue Lhomond 75231 Paris Cedex 05E-mail: [email protected]. ViallefondObservatoire de Paris, 61 Avenue de l’Observatoire, 75014 ParisE-mail: [email protected] squared amplitudes of Fourier components. Yet, a simplenumerical experiment performed by [5] shows that essentialstructural information lies in the Fourier-spatial distributionof the phases.In the following, we present some of the notions usedto exploit this information (section 2), and their practicalimplementation (section 3). We then consider the ability ofALMA and other arrays to detect and measure phase struc-ture information in real time (section 4). We conclude bygiving some future perspectives (section 5).
The importance of Fourier phases in terms of structure hasbeen recognized by various studies [15,13,5]. Since the in-formation sought lies in the Fourier spatial distribution ofphases, Scherrer et al. [15] suggested considering the statis-tics of phase increments ∆ δ φ ( k ) = φ ( k + δ ) − φ ( k ) betweenpoints separated by a given lag vector δ in Fourier space.In a field for which Fourier phases are uncorrelated, suchas fractional Brownian motions (fBm) [16], phase incre-ments are uniformly distributed over [ − π, π ], for any lag vec-tor δ . At the other end of the spectrum is the case of a singlepoint source, for which the PDF of phase increments is adelta function. In between those extremes, the PDF of phaseincrements presents a single wavelike oscillation (See Fig. 1for an example), which may be seen as a signature of phasestructure.A quantitative measure of the distribution’s departurefrom uniformity is phase entropy [13], S ( δ ) = − Z π − π ρ ( ∆ δ φ ) ln (cid:2) ρ ( ∆ δ φ ) (cid:3) d ∆ δ φ, These are random fields characterized by a power-law power spec-trum and random phases. which reaches its maximum value S = ln (2 π ) for fBms.It is therefore convenient to consider the positive quantity Q ( δ ) = S − S ( δ ), which we dub phase structure quantity ,and which may be directly computed on the histograms ofphase increments. Fig. 1
Top : Column density of a 512 weakly compressible hydrody-namical turbulence simulation obtained by Porter et al. [14], used hereas a model brightness distribution for phase structure analysis. Bottom : Histogram of phase increments for this field, with δ = e x (unit vec-tor along the k x axis in Fourier space) and n =
50. The dotted linerepresents the uniform distribution.
For a finite-sized image, histograms of phase increments donot perfectly sample the underlying PDFs. Phase structurequantities Q associated with these distributions should bedistinguished from those ˜ Q found by numerical integrationof the histograms , which depend on the number p of avail- To give an idea, for the histogram shown on Fig. 1, we have˜ Q ( e x ) = . × − . able increments and the number n of bins. The di ff erencebetween the two can become significant for Q ≪ Q , Q , Q above which there is a given probability (say 0.99) that animage deviates significantly from a “structureless” field.The procedure is described in [11] and is largely basedon results from [4]. In short, the end result is that the thresh-old of ˜ Q depends on n and p , and may be found using well-known χ statistics.The influence of n and p on the reliability of ˜ Q may alsobe studied numerically, using fractional Brownian motions.Unsurprisingly, while Q = Q increases as thesize of the image decreases, and as the number of bins in-creases. In the ideal case, interferometers sample the Fourier trans-form of observed brightness distributions, and allow directmeasurement of phase increments. Since this can be doneas the Earth rotates, we may look for the minimum observ-ing time required to detect a significant phase structure inthe data. To focus on the problem of statistical estimationdescribed in the previous section, we shall not consider pri-mary beam attenuation nor regridding issues. These simpli-fications are discussed in more detail in [11].To estimate the ability of ALMA to detect and measurephase structure, we proceed as follows: A model brightnessdistribution is taken as input to a simple interferometer sim-ulator, which is based on the characteristics of ALMA anduses the array configurations optimized by Boone [1]. Theinstrument tracks the source as long as it remains above aminimum elevation of 10 ◦ . The output maps, for which nodeconvolution is performed, yield values of ˜ Q as a functionof integration time, with δ and n fixed.The model brightness distributions used are the one ofFig. 1, and a field with the same power spectrum, but withrandom phases. For comparison, we have also consideredconfigurations taken from current arrays, such as the Plateaude Bure (PdB) and the VLA, fictitiously located at the samegeographical coordinates as ALMA, and observing the samesource.As the observation is carried out, more and more Fourierphases are measured and p increases. The question is whetherthis allows to bring down the upper limit discussed in sec-tion 3, below the measured phase structure quantities, to en-sure positive detection. The results are summarized on Fig-ures 2 to 4, which show the evolution of ˜ Q ( e x ) as a func-tion of integration time. Fig. 2 shows that the number ofphase increments measured by the Plateau de Bure in its Fig. 2
Evolution of measured ˜ Q ( e x ) with integration time, for the Bconfiguration of the Plateau de Bure. The black solid line correspondsto the turbulent brightness distribution, and the grey solid line to therandom-phase brightness distribution. The dotted line represents ˜ Q ( e x )for the complete turbulent brightness distribution, and the dashed linerepresents the evolution of the theoretical upper limit (lying above theplotted range here). B configuration is insu ffi cient to detect phase structure, asthe curves for turbulent and random-phase brightness dis-tributions are indistinguishable from one another. The sameconclusion prevails for other configurations of this instru-ment and other lag vectors. On the contrary, Fig. 3 shows Fig. 3
Same as Fig. 2, but for the D configuration of the VLA. that the VLA allows such a detection, since the measured ˜ Q becomes larger than the theoretical upper limit, after about6 hours of integration. Long before that, however, we geta hint that phase structure is present in the field, since thecurves for both model brightness distributions go apart afterless than twenty minutes. This diagnosis can be performedin real time by drawing random phases for the visibilities asthey are measured. ALMA gives even better results (Fig. 4).In its E configuration and in our case, a short integration Fig. 4
Same as Fig. 2, but for the E configuration of ALMA. time of about twenty minutes is enough to conclude on thepresence of phase structure. However, the final value of ˜ Q obtained is not equal to the phase structure quantity mea-sured on the model brightness distribution. This is due tothe fact that only 24% of the 512 ×
512 Fourier phases aremeasured by this configuration.Using more extended configurations, one should be ableto measure the Fourier components lying outside the radiuscovered by the E configuration, and therefore hope to re-cover the correct value of the phase structure quantity bycombining visibilities from multiple configurations. Fig. 5shows the evolution of the measured ˜ Q ( e x ) with integrationtime, using this approach . It appears that the Fourier planecoverage achieved by ALMA will allow measurement of theactual value of the phase structure quantity for the observedfield, while the VLA fails.Finally, to assess whether atmospheric phase noise wouldprevent detection of phase structure, we introduced a maskgiving the refractivity field above the instrument. We as-sumed this mask to be a 200-m thick layer of frozen Kol-mogorov turbulence being transported along the east-westdirection at 2 m.s − , and normalized it so that the rms phasenoise σ for a pair of antennae observing the zenith and sep-arated by a baseline d =
100 m should be one of a fewspecific values, namely 15 ◦ , 45 ◦ and 90 ◦ . According to [3]and using the scaling relation given by [9], noise levels atChajnantor vary typically from σ ∼ ◦ to σ ∼ ◦ .Integration of the refractivity field along the di ff erentlines of sight for each antenna as the observation is per-formed yields phase delays, which are then correlated to ob-tain the atmospheric phase noise for each pair of antennae, atall times. Fig. 6 shows the evolution of the measured ˜ Q ( e x )for ALMA in its E configuration. The integration time τ is to be understood per configuration, andthe total time of integration is N configurations × τ . Fig. 5
Evolution of measured ˜ Q ( e x ) with integration time for an ob-servation using all configurations of the instrument in turn. The blacksolid line corresponds to the six configurations of ALMA, and the greysolid line to the four configurations of the VLA. The dotted line repre-sents the value of ˜ Q ( e x ) for the whole field. Fig. 6
Evolution of measured ˜ Q ( e x ) with integration time in the pres-ence of atmospheric phase noise (solid lines, with σ specified next toeach curve). The array used is the E configuration of ALMA. The dot-ted line represents ˜ Q ( e x ) for the whole field, and the dashed line showsthe theoretical upper limit. It appears that in this case, the presence of phase struc-ture can be easily detected in the presence of a fair amountof atmospheric phase noise. Indeed, even a rms phase fluctu-ation of σ = ◦ is insu ffi cient to bring the measured phasestructure quantity below the upper limit. Consequently, phasestructure will undoubtedly be detected by ALMA withoutany phase correction, although the use of dedicated watervapor radiometers, as is planned, should allow for an e ff ec-tive decrease of the atmospheric phase noise by a substantialfactor [10], making it possible to actually measure the phasestructure quantity for the observed field. In the context of interferometry, a more elaborate use ofphase information would be to keep track of the phase mea-sured by each baseline as a function of time, and to computephase increments along the baseline’s track. This should re-duce contamination by atmospheric phase noise, but wouldrequire a shift in the phase structure information formalism,since, in this approach, the lag vector δ is no longer a controlparameter, but a function of time and of the baseline.Another possible extension of this work is the inclu-sion of the kinematic dimension, which is accessible throughALMA’s high spectral resolution receivers. It may well bethat phase analysis applied to individual channel maps shouldprove a valuable tool for assessing the three-dimensionalstructure of velocity fields. References
1. Boone, F.: Interferometric array design: Optimizing the locationsof the antenna pads. A&A , 368 (2001)2. Burkert, A., Hartmann, L.: Collapse and Fragmentation in FiniteSheets. ApJ , 288–300 (2004). DOI 10.1086 / ffi ths, N.M., Stanimirovic, S.,Gaensler, B.M., Green, A.J.: Southern galactic plane surveymeasurements of the spatial power spectrum of interstellar H i inthe inner galaxy. ApJ , 264 (2001)7. Elmegreen, B.G., Falgarone, E.: A fractal origin for the mass spec-trum of interstellar clouds. ApJ , 816 (1996)8. Falgarone, E., Phillips, T.G., Walker, C.K.: The edges of molecu-lar clouds - fractal boundaries and density structure. ApJ , 186(1991)9. Lay, O.P.: Phase calibration and water vapor radiometry formillimeter-wave arrays. Astronomy & Astrophysics SupplementSeries , 547–557 (1997)10. Lay, O.P.: The temporal power spectrum of atmospheric fluctua-tions due to water vapor. Astronomy & Astrophysics SupplementSeries , 535–545 (1997)11. Levrier, F., Falgarone, E., Viallefond, F.: Fourier phase analysis inradio-interferometry. A&A , 205 (2006)12. Miesch, M.S., Bally, J.: Statistical analysis of turbulence in molec-ular clouds. ApJ , 645 (1994)13. Polygiannakis, J.M., Moussas, X.: Detection of nonlinear dynam-ics in solar wind and a comet using phase-correlation measures.Sol. Phys. , 159 (1995)14. Porter, D.H., Pouquet, A., Woodward, P.R.: Kolmogorov-likespectra in decaying three-dimensional supersonic flows. Ph. Fl. , 2133 (1994)15. Scherrer, R.J., Melott, A.L., Shandarin, S.F.: A quantitative mea-sure of phase correlations in density fields. ApJ , 29 (1991)16. Stutzki, J., Bensch, F., Heithausen, A., Ossenkopf, V., Zielinsky,M.: On the fractal structure of molecular clouds. A&A336