Significance testing for cross correlation: A critical examination of correlations between ENSO and GRACE-derived terrestrial water storage variabilities

The cross correlation has a wide range of applications in geophysical fields for measuring linear connections or relationships among physical quantities. Nonetheless, there remains a dearth of comprehensive discourse regarding its statistical significance testing, which is crucial for differentiating meaningful outcomes from those merely stemming from fortuity or pure randomness. Conventional theoretical methods for significance testing, commonly used in statistical analysis tools such as SPSS and MATLAB, are only applicable when dealing with idealized circumstances such as white noise. In discretionary application of these methods to analyze geophysical signals with, say, red noise may result in potentially unjustified conclusions. This study aims to develop a comprehensive approach based on the $t$-distribution within a rigorous statistical context, aiming to facilitate significance tests of cross correlation for general signals with specified time shifts or ranges. Extensive Monte Carlo experiments substantiate its robustness, thereby paving the way for accurate and expeditious identification of statistically (hence potentially physically) meaningful correlations in general. As an example, we examine critically the previously purported significant correlations between ENSO (El Niño Southern Oscillation) and global terrestrial water storage variations derived from the GRACE (Gravity Recovery and Climate Experiment) satellite mission, demonstrating that they are subject to questioning in the absence of complete significance testing.