Cross-Wavelet Bias Corrected by Normalizing Scales

The cross-wavelet transform (XWT) is a powerful tool for testing the proposed connections between two time series. Because of XWT's skeletal structure, which is based on the wavelet transform, it is suitable for the analysis of nonstationary periodic signals. Recent work has shown that the power spectrum based on the wavelet transform can produce a deviation, which can be corrected by choosing a proper rectification scale. In this study, it is shown that the standard application of the XWT can also lead to a biased result. A corrected version of the standard XWT was constructed using the scale of each series as normalizing factors. This correction was first tested with an artificial example involving two series built from combinations of two harmonic series with different amplitudes and frequencies. The standard XWT applied to this example produces a biased result, whereas the correct result is obtained with the use of the proposed normalization. This analysis was then applied to a real geophysical situation with important implications to climate modulation on the northwestern Brazilian coast. The linkage between the relative humidity and the shortwave radiation measurements, obtained from the 8 S, 30 W Autonomous Temperature Line Acquisition System (ATLAS) buoy of the Southwestern Extension of the Prediction and Research Moored Array in the Tropical Atlantic (PIRATA-SWE), was explored. The analysis revealed the importance of including the correction in order to not overlook any possible connections. The requirements of incorporating this correction in the XWT calculations are emphasized.