On the Statistical Decorrelation of the Wavelet Packet Coefficients of a Band-Limited Wide-Sense Stationary Random Process

International audienceThis paper is a contribution to the analysis of the statistical correlation of the wavelet packet coefficients resulting from the decomposition of a random process, stationary in the wide-sense, whose power spectral density is bounded with support in [-\pi,\pi]. Consider two quadrature mirror filters (QMF) that depend on a parameter r, such that these filters tend almost everywhere to the Shannon QMF when r increases. The parameter r is called the order of the QMF under consideration. The order of the Daubechies filters (resp. the Battle-Lemarié filters) is the number of vanishing moments of the wavelet function (resp. the spline order of the scaling function). Given any decomposition path in the wavelet packet tree, the wavelet packet coefficients are proved to decorrelate for every packet associated with a large enough resolution level, provided that the QMF order is large enough and above a value that depends on this wavelet packet. Another consequence of our derivation is that, when the coefficients associated with a given wavelet packet are approximately decorrelated, the value of the autocorrelation function of these coefficients at lag $0$ is close to the value taken by the power spectral density of the decomposed process at a specific point. This specific point depends on the path followed in the wavelet packet tree to attain the wavelet packet under consideration. Some simulations highlight the good quality of the ''whitening'' effect that can be obtained in practical cases

Analyse asymptotique de la fonction d'autocorrélation des coefficients des paquets d'ondelettes associés à un processus aléatoire stationnaire au sens large et à bande limitée

Dans ce papier, on analyse la corrélation statistique des coefficients des paquets d'ondelettes associés à un processus aléatoire, stationnaire au sens large et dont la densité spectrale est à support dans [-π;π]. Soient deux filtres miroirs en quadrature (FMQ) dépendant d'un paramètre r tel que ces filtres tendent presque partout vers les FMQ idéaux de Shannon lorsque r croît. Le paramètre r est appelé ordre des filtres FMQ considérés.L'ordre des filtres de Daubechies est le nombre de moments nuls de la fonction d'ondelette. L'ordre des filtres de Battle-Lemarié est l'ordre de la spline associée à la fonction d'échelle. Étant donné un chemin de l'arbre de décomposition en paquets d'ondelettes, nous montrons que les coefficients des paquets d'ondelettes tendent à se décorréler pour chaque paquet associé à un niveau de résolution suffisamment grand, à condition que l'ordre des FMQ soit lui-aussi suffisament grand et supérieur à une valeur qui dépend du paquet d'ondelettes considéré. Une autre conséquence de ce résultat est la suivante : lorsque les coefficients associés à un paquet d'ondelettes sont approximativement décorrélés, la valeur de la fonction d'auto-corrélation en 0 est proche de la valeur de la densité spectrale du processus en un point que l'on sait déterminer. Ce point dépend du chemin suivi dans l'arbre de décomposition pour atteindre le paquet d'ondelettes considéré. Quelques simulation mettent en évidence le bonne qualité de l'effet de blanchiment que l'on obtient en pratique. (english version on the document

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Measurement of the cosmic ray spectrum above 4×10184{\times}10^{18} eV using inclined events detected with the Pierre Auger Observatory

A measurement of the cosmic-ray spectrum for energies exceeding $4{\times}10^{18}$ eV is presented, which is based on the analysis of showers with zenith angles greater than $60^{\circ}$ detected with the Pierre Auger Observatory between 1 January 2004 and 31 December 2013. The measured spectrum confirms a flux suppression at the highest energies. Above $5.3{\times}10^{18}$ eV, the "ankle", the flux can be described by a power law $E^{-\gamma}$ with index $\gamma=2.70 \pm 0.02 \,\text{(stat)} \pm 0.1\,\text{(sys)}$ followed by a smooth suppression region. For the energy ($E_\text{s}$) at which the spectral flux has fallen to one-half of its extrapolated value in the absence of suppression, we find $E_\text{s}=(5.12\pm0.25\,\text{(stat)}^{+1.0}_{-1.2}\,\text{(sys)}){\times}10^{19}$ eV.Comment: Replaced with published version. Added journal reference and DO

Energy Estimation of Cosmic Rays with the Engineering Radio Array of the Pierre Auger Observatory

The Auger Engineering Radio Array (AERA) is part of the Pierre Auger Observatory and is used to detect the radio emission of cosmic-ray air showers. These observations are compared to the data of the surface detector stations of the Observatory, which provide well-calibrated information on the cosmic-ray energies and arrival directions. The response of the radio stations in the 30 to 80 MHz regime has been thoroughly calibrated to enable the reconstruction of the incoming electric field. For the latter, the energy deposit per area is determined from the radio pulses at each observer position and is interpolated using a two-dimensional function that takes into account signal asymmetries due to interference between the geomagnetic and charge-excess emission components. The spatial integral over the signal distribution gives a direct measurement of the energy transferred from the primary cosmic ray into radio emission in the AERA frequency range. We measure 15.8 MeV of radiation energy for a 1 EeV air shower arriving perpendicularly to the geomagnetic field. This radiation energy -- corrected for geometrical effects -- is used as a cosmic-ray energy estimator. Performing an absolute energy calibration against the surface-detector information, we observe that this radio-energy estimator scales quadratically with the cosmic-ray energy as expected for coherent emission. We find an energy resolution of the radio reconstruction of 22% for the data set and 17% for a high-quality subset containing only events with at least five radio stations with signal.Comment: Replaced with published version. Added journal reference and DO

Measurement of the Radiation Energy in the Radio Signal of Extensive Air Showers as a Universal Estimator of Cosmic-Ray Energy

We measure the energy emitted by extensive air showers in the form of radio emission in the frequency range from 30 to 80 MHz. Exploiting the accurate energy scale of the Pierre Auger Observatory, we obtain a radiation energy of 15.8 \pm 0.7 (stat) \pm 6.7 (sys) MeV for cosmic rays with an energy of 1 EeV arriving perpendicularly to a geomagnetic field of 0.24 G, scaling quadratically with the cosmic-ray energy. A comparison with predictions from state-of-the-art first-principle calculations shows agreement with our measurement. The radiation energy provides direct access to the calorimetric energy in the electromagnetic cascade of extensive air showers. Comparison with our result thus allows the direct calibration of any cosmic-ray radio detector against the well-established energy scale of the Pierre Auger Observatory.Comment: Replaced with published version. Added journal reference and DOI. Supplemental material in the ancillary file

A search for ultra-high-energy photons at the Pierre Auger Observatory exploiting air-shower universality

The Pierre Auger Observatory is the most sensitive detector to primary photons with energies above ∼0.2 EeV. It measures extensive air showers using a hybrid technique that combines a fluorescence detector (FD) with a ground array of particle detectors (SD). The signatures of a photon-induced air shower are a larger atmospheric depth at the shower maximum (X$_{max}$) and a steeper lateral distribution function, along with a lower number of muons with respect to the bulk of hadron-induced background. Using observables measured by the FD and SD, three photon searches in different energy bands are performed. In particular, between threshold energies of 1-10 EeV, a new analysis technique has been developed by combining the FD-based measurement of X$_{max}$ with the SD signal through a parameter related to its muon content, derived from the universality of the air showers. This technique has led to a better photon/hadron separation and, consequently, to a higher search sensitivity, resulting in a tighter upper limit than before. The outcome of this new analysis is presented here, along with previous results in the energy ranges below 1 EeV and above 10 EeV. From the data collected by the Pierre Auger Observatory in about 15 years of operation, the most stringent constraints on the fraction of photons in the cosmic flux are set over almost three decades in energy

Study on multi-ELVES in the Pierre Auger Observatory

Since 2013, the four sites of the Fluorescence Detector (FD) of the Pierre Auger Observatory record ELVES with a dedicated trigger. These UV light emissions are correlated to distant lightning strikes. The length of recorded traces has been increased from 100 μs (2013), to 300 μs (2014-16), to 900 μs (2017-present), to progressively extend the observation of the light emission towards the vertical of the causative lightning and beyond. A large fraction of the observed events shows double ELVES within the time window, and, in some cases, even more complex structures are observed. The nature of the multi-ELVES is not completely understood but may be related to the different types of lightning in which they are originated. For example, it is known that Narrow Bipolar Events can produce double ELVES, and Energetic In-cloud Pulses, occurring between the main negative and upper positive charge layer of clouds, can induce double and even quadruple ELVES in the ionosphere. This report shows the seasonal and daily dependence of the time gap, amplitude ratio, and correlation between the pulse widths of the peaks in a sample of 1000+ multi-ELVES events recorded during the period 2014-20. The events have been compared with data from other satellite and ground-based sensing devices to study the correlation of their properties with lightning observables such as altitude and polarity