Nonlinear Processes in Geophysics www.nonlin-processes-geophys.net 1023-5809 1607-7946 16 6 2009 10.5194/npg-16-641-2009 http://www.nonlin-processes-geophys.net/16/641/2009/ http://www.nonlin-processes-geophys.net/16/641/2009/npg-16-641-2009.html http://www.nonlin-processes-geophys.net/16/641/2009/npg-16-641-2009.pdf 641 653 2009-11-10 Improved moment scaling estimation for multifractal signals D. Veneziano P. Furcolo p.furcolo@unisa.it Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Dipartimento di Ingegneria Civile, Università degli Studi di Salerno, Fisciano (SA) 84084, Italy A fundamental problem in the analysis of multifractal processes is to estimate the scaling exponent K(q) of moments of different order q from data. Conventional estimators use the empirical moments <html><body> μ^</body></html>rq=⟨ | εr(τ)|q⟩ of wavelet coefficients εr(τ), where τ is location and r is resolution. For stationary measures one usually considers "wavelets of order 0" (averages), whereas for functions with multifractal increments one must use wavelets of order at least 1. One obtains <html><body> K^</body></html>(q) as the slope of log(<html><body> μ^</body></html>rq) against log(r) over a range of r. Negative moments are sensitive to measurement noise and quantization. For them, one typically uses only the local maxima of | εr(τ)| (modulus maxima methods). For the positive moments, we modify the standard estimator <html><body> K^</body></html>(q) to significantly reduce its variance at the expense of a modest increase in the bias. This is done by separately estimating K(q) from sub-records and averaging the results. For the negative moments, we show that the standard modulus maxima estimator is biased and, in the case of additive noise or quantization, is not applicable with wavelets of order 1 or higher. For these cases we propose alternative estimators. 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