The combination of GRACE and hydrological models is widely used for quantification and time-varying analysis of groundwater storage, and several signal-processing tools have been adopted in recent years. However, the popular empirical models constrained by a priori functions, such as least squares fitting, cannot comprehensively reveal the transient variation of nonlinear or nonstationary signal sequences. An emerging self-adaptive signal-processing tool named extreme-point symmetric mode decomposition (ESMD), used with independent component analysis (ICA), has been applied to investigate spatiotemporal characteristics of GRACE-derived groundwater storage (GWS) change in the Murray-Darling Basin, Australia. Although ESMD is firstly applied to GRACE signal analysis, the result is effective and credible. ESMD can explore finer periodic components than the least-squares fitting, and the adaptive ESMD method can more sensitively estimate transient trend change and anomalies in nonlinear or nonstationary signals compared with a priori models. These findings coincide well with hydrometeorological conditions, such as “the Millennium Drought” in Australia’s mainland and the 2010–2012 La Niña event. ICA can also separate the relative independent components of groundwater storage change and qualitatively investigate the spatial weights with corresponding time coefficients. The results suggest that rainfall may be the main input source or influencing factor of groundwater circulation. Contrasting long-term trends between the northern and southern parts of the basin are attributed to the diverse physical mechanism of discharge and recharge related to spatial distribution of surface-water bodies. Although with distinct working principles, the cooperative application of ESMD and ICA can provide cross-supported and complementary conclusions from different perspectives.