SURFACE AND INTERFACE ANALYSIS Surf. Interface Anal. 2004; 36: 71–80 Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/sia.1650 Noise filtering and deconvolution of XPS data by wavelets and Fourier transform Catherine Charles, 1 Gervais Leclerc, 2 Pierre Louette, 2 Jean-Paul Rasson 1 and Jean-Jacques Pireaux 2* 1 Department of Mathematics, Facult ´ es Universitaires Notre-Dame de la Paix, B-5000 Namur, Belgium 2 Department of Physics, LISE Laboratory, Facult ´ es Universitaires Notre-Dame de la Paix, B-5000 Namur, Belgium Received 10 September 2002; Revised 15 October 2003; Accepted 15 October 2003 In experimental sciences, the recorded data are often modelled as the noisy convolution product of an instrumental response with the ‘true’ signal to find. Different models have been used for interpreting x-ray photoelectron spectroscopy (XPS) spectra. This article suggests a method of estimate the ‘true’ XPS signal that relies upon the use of wavelets, which, because they exhibit simultaneous time and frequency localization, are well suited to signal analysis. First, a wavelet shrinkage algorithm is used to filter the noise. This is achieved by decomposing the noisy signal into an appropriate wavelet basis and then thresholding the wavelet coefficients that contain noise. This algorithm has a particular threshold related to frequency and time. Secondly, the broadening due to the instrumental response is eliminated through a deconvolution process similar to that developed in the previous paper in this series for the analysis of HREELS data. This step mainly rests on least-squares and on the existing relation between the Fourier transform, the wavelet transform and the convolution product. Copyright  2004 John Wiley & Sons, Ltd. KEYWORDS: XPS; wavelets; deconvolution; Poisson noise INTRODUCTION With reference to the two other papers in this series (this issue), wavelet analysis of XPS data must concentrate on different areas than applications to High-resolution elec- tron energy-loss spectroscopy (HREELS) data, for differ- ent reasons. First, a simple consideration of the fundamental differ- ences in the excitation processes reveals that in HREELS the elemental vibrational excitations have in general an intrinsic linewidth close to zero, at least at the scale of probe elec- trons that have an energy (1–10 eV for example) defined to š1 meV ⊲š8 cm 1 ⊳: Fourier transform infrared spectra showing vibrational bands with a resolution of 0.1 cm 1 are not uncommon, justifying the fact that a deconvolution pro- cedure of HREELS data should consider the ‘true signal’ as a Dirac ‘delta’ function. In XPS, in contrast, intrinsic core-level linewidths are in the range of 0.1 (metals) to 0.4 eV (carbon), while stand-alone spectrometers have intrinsic resolution (monochromatized x-ray source and analyser contributions) in the same energy range: peaks are ‘naturally’ broad and should be approximated by finite Gaussian or Lorentzian shape (or a combination thereof). L Correspondence to: Jean-Jacques Pireaux, Department of Physics, LISE Laboratory, Facult´ es Universitaires Notre-Dame de la Paix, B-5000 Namur, Belgium. E-mail: Jean-Jacques.Pireaux@fundp.ac.be Contract/grant sponsor: Belgian FNRS. Contract/grant sponsor: Belgian Prime Minister’s Office; Contract/grant number: PAI/UIAP(4/10). Secondly, the literature is rich in contributions on spectra acquisition and data handling in XPS. All the basic principles are illustrated in Sherwood’s reference review, 1 while some state-of-the-art concerns have been listed and commented more recently; as a result of an international workshop entitled ‘X-Ray photoelectron Spec- troscopy: from Physics to Data’, a compilation of recent developments in available handbooks and databases, in data processing software and standard test data is pre- sented and well documented. 2 But, when processing XPS core-level spectra one should stay aware of the genuine algorithms hypotheses, and include easy-to-interpret sta- tistical diagnostics to judge objectively the quality of the regression. 3 The purpose of this contribution is therefore to explore what specific contribution(s) wavelet analysis could bring to XPS data: noise filtering and deconvolution (recovery of the ‘true’ signal) will be tested on synthetic (theoret- ical) spectra and then applied to real spectra. As in the previous paper in this series (this issue) for the work on HREELS data processing, care will be taken to try to recover real peak intensities, a prerequisite to keep quantification through data analysis. Note that this con- tribution will not discuss issues related to the choice and use of routines (linear, polynomial, Shirley, Tougaard, etc.) to estimate background, or the algorithms to allow com- positional depth profile data to be deduced from angle- resolved XPS measurements, which are beyond the scope of this work. Copyright  2004 John Wiley & Sons, Ltd.