Joint multifractal spectrum analysis for characterizing the nonlinear relationship among hydrological variables

Joint multifractal spectrum analysis (JMS) is an effective approach for investigation of relationship among hydrological variables and is regarded as one potential way to reveal nonlinear relationship in hydrology. In this study, the joint multifractal spectrum analysis was for the first time used to analyze soil moisture-soil temperature-precipitation (SM-ST-P) relationship. Pearson's linear correlation coefficient considering lag effects and scale effects was also applied for the aims of comparison. Soil data were collected at four sites in Tibet and three depths for each site (10 cm, 30 cm and 60 cm) at 2-hourly frequency from November 2016 to November 2017. Correlation analysis showed that the coefficients varied at every measurement point but failed to provide general and comprehensive analysis of SM-ST-P relationship. On the contrary, multifractal spectra of SM-ST-P confirmed that the relationship was strong for all measurement points. Furthermore, JMS analysis provided holistic results including verification of multifractal of SM-ST-P relationship, relationship evaluation for each part of variables separately (high value, middle value, low value), intuitive diagrammatic comparison of SM-ST and SM-P relationships, etc. Seasonal JMS analysis was adopted to investigate the freezing-thawing process in the study area as well. The multifractal spectra of frozen soil site and no frozen soil site are totally different for dry cold period but similar for rainy warm period. The difference in multifractal of SM-ST-P relationship with and without frozen soil presented the latent application of JMS to quantitatively detect the process of frozen soil. This study implied the capability of JMS analysis in hydrology, especially for multivariable relationship like SM-ST-P relationship.