In the analysis of empirical signals, detecting correlations that capture genuine interactions between the elements of a complex system is a challenging task with applications across disciplines. Here we analyze a global data set of surface air temperature (SAT) with daily resolution. Hilbert analysis is used to obtain phase, instantaneous frequency and amplitude information of SAT seasonal cycles in different geographical zones. The analysis of the phase dynamics reveals large regions with coherent seasonality. The analysis of the instantaneous frequencies uncovers clean wave patterns formed by alternating regions of negative and positive correlations. In contrast, the analysis of the amplitude dynamics uncovers wave patterns with additional large-scale structures. These structures are interpreted as due to the fact that the amplitude dynamics is affected by processes that act in long and short time scales, while the dynamics of the instantaneous frequency is mainly governed by fast processes. Therefore, Hilbert analysis allows to disentangle climatic processes and to track planetary atmospheric waves. Our results are relevant for the analysis of complex oscillatory signals because they offer a general strategy for uncovering interactions that act at different time scales.
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