On the 1/f noise in the UV solar spectral irradiance

Abstract

A major shortcoming of many climate studies on long-range dependence is the inference of long-term memory from climate data without considering that the power-law scaling needs to be established and a simple exponential decay of the autocorrelation function has to be rejected (Efstathiou and Varotsos 2010; Maraun et al. 2004; Varotsos 2005a; Varotsos and Kirk-Davidoff 2006; Varotsos et al. 2003a, b, 2005, 2006, 2007, 2008, 2009, 2013). Very recently, Varotsos et al. (2013), studying the highresolution observations of the spectral solar incident flux (SIF) reaching the ground and the top of the atmosphere, concluded that SIF versus ultraviolet wavelengths (WL) exhibit 1/f-type power-law correlations. This result was based on the slope (i.e., 1.02±0.02) of the log–log plot of the root mean square fluctuation function Fd(τ) of SIF versus the WL segment size τ, after the application of the detrended fluctuation analysis (DFA). However, the reader of the paper by Varotsos et al. (2013) may argue that the afore-mentioned power-law scaling should be established by employing the two criteria suggested by Maraun et al. (2004), notably the rejection of the exponential decay of the autocorrelation function and the constancy of “local slopes” in a certain range towards the low frequencies. In this respect, we herewith plot the profile of the power spectral density (i.e., the distribution of the variance over frequency) versus logτ for the detrended SIF-WL data set (depicted in Fig. 1a). This plot shows that in a WL scale range from logτ≈0.3 (given that the measurement step is 0.05, the value of τ≈10×0.05≈0.1 nm) to maximally logτ≈1.46 (τ≈10×0.05≈1.4 nm), the power spectral density is better fitted exponentially, while for larger scales, it turns into an algebraically (power-law) fit (i.e., satisfying thus the first criterion of Maraun et al. (2004) for the rejection of the exponential fit). On the other hand, we apply the method of local slopes, suggested by Maraun et al. (2004), in order to examine the establishment of the 1/f noise in the UV solar spectral irradiance. Since the single straight line of the DFA plot for the detrended SIF-WL data set that established in the whole range of scales may be biased, we evaluate the local slopes of logFd(τ) versus logτ detecting for constancy in a sufficient range. Along these lines, we first fit a straight line to logFd(τ) versus logτ within a specific window of WL scale range. This window is then shifted successively over all calculated scales τ. At this point, it should be noted that Maraun et al. (2004) state verbatim: “Choosing the optimal window size, one has to trade bias for variance: For small windows, the bias is small, but the variability renders the interpretation difficult, whereas for large windows, the variance is reduced at the cost of a biased estimate of a.” Figure 1b illustrates the local slope a versus logτ for two different window sizes. In addition, we compute the standard deviations, sa, of each estimated local slope over all the scales, along with the error bounds of each local slope a, defined as a±1.96⋅sa. Inspection of Fig. 1b shows that for small scales, the variance of the local slopes is low, while for large scales, the variance is increased with a values reaching to a constant threshold (a≈0.92) at a range that coincides with that in Fig. 1a, where the power-law behavior prevails (establishing thus the long-range dependence). Similar results were obtained by considering other windows. C. A. Varotsos (*) :M. N. Efstathiou :C. Tzanis Climate Research Group, Division of Environmental Physics and Meteorology, Faculty of Physics, University of Athens, University Campus Bldg. Phys. V, Athens 15784, Greece e-mail: covar@phys.uoa.gr