||Astronomical forcing (obliquity and precession) has been thought to modulate Dansgaard-Oeschger (DO) events, yet the detailed quantification of such modulations has not been examined. In this study, we apply the novel Holo-Hilbert Spectral Analysis (HHSA) to five polar ice core records, quantifying astronomical forcing's time-varying amplitude modulation of DO events and identifying the preferred obliquity phases for large amplitude modulations. The unique advantages of HHSA over the widely used windowed Fourier spectral analysis for quantifying astronomical forcing's nonlinear modulations of DO events is first demonstrated with a synthetic data that closely resembles DO events recorded in Greenland ice cores (NGRIP, GRIP, and GISP2 cores on GICC05 modelext timescale). The analysis of paleoclimatic proxies show that statistically significantly more frequent DO events, with larger amplitude modulation in the Greenland region, tend to occur in the decreasing phase of obliquity, especially from its mean value to its minimum value. In the eastern Antarctic, although statistically significantly more DO events tend to occur in the decreasing obliquity phase in general, the preferred phase of obliquity for large amplitude modulation on DO events is a segment of the increasing phase near the maximum obliquity, implying that the physical mechanisms of DO events may be different for the two polar regions. Additionally, by using cross-spectrum and magnitude-squared analyses, Greenland DO mode at a timescale of about 1400 years leads the Antarctic DO mode at the same timescale by about 1000 years. (C) 2018 Elsevier Ltd. All rights reserved.