TFAW: Noise filtering Through the use of the Wavelet Transform in Astronomy Photometric Data

Abstract

The first confirmed detection of an exoplanet orbiting a main-sequence star was made in 1995, when a giant planet was found by radial velocity measurements in a four-day orbit around the nearby star 51 Pegasi by Mayor and Queloz in 1995. This finding encouraged the development of the method known as transit method that detects distant planets by measuring the small darkening of a star light curve as an orbiting planet passes between it and the Earth. The first detection of a transiting exoplanet, HD 209458 b, in 1999 by D. Charbonneau and collaborators and G. W. Henry and collaborators, and the discoveries obtained for this planet during follow- up observations (first planet with a detectable atmosphere containing oxygen and carbon, first detection of an evaporating hydrogen atmosphere and being one of the first two exoplanets to be directly observed spectroscopically) demonstrated the high scientific potential of planets discovered with this method.

Transit photometry is currently the most effective and sensitive method for detecting extrasolar planets. Several surveys have taken this approach, such as the ground-based MEarth, SuperWASP, KELT, HAT-South, TFRM-PSES, NGTS or the Evryscope, as well as the space-based CoRoT, Kepler, the recently commissioned, TESS and the future PLATO missions.

The photometric precision and accuracy achieved by an astronomical survey is a key factor in detecting a transiting signal or any other kind of variability. Many of the systematic variations in a given light curve are shared by light curves of other stars in the same data set. In order to remove those systematics, one can identify the objects in the field that suffer from the same kind of variations as the target (correlated noise) and then build and apply a filter based on the light curves of these comparison stars.

Wavelets have unique properties that make them an ideal tool for analyzing signals of non- stationary nature. In comparison to the sine wave used in the Fourier transform, which is smooth and of infinite length, the wavelet is irregular in shape and compactly supported. Their irregular shape allows to analyze signals with discontinuities, transients, singularities and sharp changes, while their compactly supported nature allows temporal localization of the signal's features.

Along this work we lay out the framework from which the main goal of this thesis, the Wavelet-based Trend Filtering Algorithm (TFAW) will be built from. TFAW is a wavelet-based modification of the Trend Filtering Algorithm developed by Kovács, Bakos and Noyes (2005). TFAW is a totally generic, Python-based, parallelized algorithm useful to improve the performance of signal detection, reconstruction and characterization, provided that a set of comparison light curves sharing the same systematics and trends as the target time series is available. differs from other wavelet-based noise-filtering algorithms in that it does not require any parametric model fitting or any extra computational method. TFAW estimates the noise contribution of the signal from its Stationary Wavelet Transform (SWT) at each iteration step and the de-noising is done through the subtraction of this contribution from the signal. TFAW de-noises the signal without modifying any of its intrinsic properties contrary to wavelet coefficient thresholding that can lead to distortions of the signal and introduce artificial oscillations or ripples around discontinuities. Tests conducted on simulated and real (coming from the TFRM-PSES, Evryscope, CoRoT and Kepler surveys) TFAW-filtered light curves show an improvement of 40% (although it can be higher) in their standard deviations with respect to

the ones detrended with TFA, leading to a better characterization of the signal, without modifying its features. It improves the transit detection rate a factor 2-5 for low SNR signals with respect TFA. We demonstrate that the TFAW-filtered light curve yields better MCMC posterior distributions, diminishes the bias in the fitted transit parameters and their uncertainties and narrows the credibility intervals up to a factor 10 for simulated transits. Finally, TFAW is able to isolate the different underlying signals within a light curve with multiple periodic signals, such as multi-transit signals, transients, modulations or other kinds of stellar variabilities.

El descubrimiento del exoplaneta gigante 51 Pegasi b (detectado por Mayor y Queloz en 1995), mediante el método de las velocidades radiales, promovió el desarrollo de una nueva técnica de detección. Esta técnica, conocida como el método del tránsito, detecta exoplanetas midiendo el pequeño oscurecimiento del flujo estelar cuando el planeta pasa entre la estrella y el observador. El método del tránsito es, actualmente, el modo más eficiente y sensible para detectar planetas extrasolares. Muchas misiones han seguido este modo de observación, aquellas basadas en tierra, como MEarth, SuperWASP, KELT, HAT-South, TFRM-PSES, NGTS o el Evryscope; así como aquellas misiones espaciales como COnvection ROtation and planetary Transits (CoRoT), Kepler, TESS y la futura misión PLATO.

La precisión fotométrica y la exactitud conseguida por una misión es un factor clave en la detección y caracterización de una señal correspondiente a un tránsito o a cualquier otro tipo de variabilidad.

Las wavelets tienen unas propiedades que las hacen ideales para analizar señales de naturaleza no estacionaria. La forma irregular de las wavelets permite analizar señales con discontinuidades, singularidades o cambios bruscos mientras que su naturaleza compacta permite la localización temporal de las características de la señal.

En este trabajo presentamos el Wavelet-based Trend Filtering Algorithm (TFAW). TFAW es un algoritmo totalmente genérico, desarrollado y paralelizado en Python basado en el Trend Filtering Algorithm (TFA) desarrollado por Kovács, Bakos y Noyes (2005). El algoritmo es útil para mejorar el rendimiento en la detección, reconstrucción y caracterización de señales astrofísicas. TFAW difiere de otros métodos de filtrado de ruido basados en wavelets en que no requiere ningún modelo de ajuste paramétrico o cualquier otro método computacional. TFAW estima la contribución de ruido de la señal a partir de su Stationary Wavelet Transform (SWT) y el filtrado se realiza eliminando esta contribución de la señal. Además, TFAW es capaz de hacer el filtrado de la señal sin modificar ninguna de sus características intrínsecas a diferencia de otros métodos como los umbrales calibrados de los coeficientes wavelet que pueden dar lugar a distorsiones de la señal o introducir oscilaciones o perturbaciones artificiales alrededor de discontinuidades. En curvas de luz simuladas, TFAW mejora la detección de tránsitos en un factor 2.5 para señales con bajo SNR. TFAW proporciona una mejor representación y caracterización de señales simuladas y reales (provenientes de las misiones TFRM-PSES, Evryscope, CoRoT y Kepler) afectadas por ruido. Así mismo, proporciona mejores probabilidades a posteriori y una disminución del bias y las incertidumbres de los parámetros ajustados mediante MCMC.