Towards a theoretical understanding of multiscalar drought indices based on the relationship between precipitation and standardized precipitation index

This study reports a theoretical understanding of multiscalar drought indices based on the relationship between precipitation and Standardized Precipitation Index (SPI). To unveil the multiscalar structure of precipitation, the advanced technique of wavelet decomposition is systematically applied to dissect the precipitation into a number of orthogonal components according to different time scales. A case study over Southwest China demonstrated a time lag or a synchronous correlation, depending on the time scale, between precipitation and the SPI, with precipitation always leading the SPI. The delayed response of the SPI to precipitation becomes more significant as the temporal scale increases, while the lead-lag effect vanishes at the shortest time scales. Most importantly, the SPI at a specific time responds primarily to the corresponding precipitation component, regardless of the contribution of its variance to the total variability. The conclusions obtained in the case study are further strengthened by global analysis. Moreover, the lag time between the SPI and precipitation at longer time scales has great geographic diversity worldwide, in contrast to shorter time scales, which have spatially uniform response times irrespective of site. In addition, we also clarify two core concepts that are easily confused, time scale and lag time. Finally, our study highlights the prominent utility of a multiscalar drought index to detect drought for a wide range of time scales compared to other metrics with rigid time scale, owing to the multistructural property of precipitation that results in multiscalar drought.