Technical Note: Improved partial wavelet coherency for understanding scale-specific and localized bivariate relationships in geosciences

Abstract. Bivariate wavelet coherency is a measure of correlation between two variables in the location–scale (spatial data) or time–frequency (time series) domain. It is particularly suited to geoscience, where relationships between multiple variables differ with locations (times) and/or scales (frequencies) because of the various processes involved. However, it is well-known that bivariate relationships can be misleading when both variables are dependent on other variables. Partial wavelet coherency (PWC) has been proposed to detect scale-specific and localized bivariate relationships by excluding the effects of other variables but is limited to one excluding variable and provides no phase information. We aim to develop a new PWC method that can deal with multiple excluding variables and provide phase information. Both stationary and non-stationary artificial datasets with the response variable being the sum of five cosine waves at 256 locations are used to test the method. The new method was also applied to a free water evaporation dataset. Our results verified the advantages of the new method in capturing phase information and dealing with multiple excluding variables. Where there is one excluding variable, the new PWC implementation produces higher and more accurate PWC values than the previously published PWC implementation that mistakenly considered bivariate real coherence rather than bivariate complex coherence. We suggest the PWC method is used to untangle scale-specific and localized bivariate relationships after removing the effects of other variables in geosciences. The PWC implementations were coded with Matlab and are freely accessible (https://figshare.com/s/bc97956f43fe5734c784, last access: 14 January 2021).