  To analyze non-stationary harmonic signals typically contained in geophysical
observables is a quest that has seen continual advances in numerical techniques
over the decades. In this paper, based on transient z-pole estimation (in
Hankel matrices), a novel state-space analysis referred to as Hankel Spectral
Analysis (HSA), was developed. Depended on the Hankel total least square
(HTLS), the HSA incorporates truncated singular value decomposition (TSVD) and
its shift-invariant property in robustly decomposing the closely-spaced
sinusoids. Resorted to a sliding window processing, HSA can be used to analyze
non-stationary sequential structures, in the support of consecutive quaternary
parameters {Ai, {\alpha}i, fi, {\theta}i}. Based on a series of experiments
with special features commonly in real measurements, the availabilities of HSA
in complex harmonic constituents (e.g., the time-variant amplitude/frequency,
mutation, the episodic recording signals) with low Signal-to-Noise Ratio are
confirmed. In real applications, we use HSA to analyze both global geophysical
observables, including polar motion (PM) and earth's dynamic oblateness
({\Delta}J2), and some new findings are obtained. In the PM series since the
1900s, a total of triple jumps from Chandler wobble (CW) are firstly confirmed;
and all of them are synchronized by the sharp decrease of Chandler intensity
and period. In the {\Delta}J2 series, two decadal signals (18.6 yr, 10.5 yr)
are identified to be associated with the tide effect, and solar activity; and
its interannual-to-decadal oscillations contribute to multiple global gravity
anomalies. These findings implied the great potential of the HSA in searching
hitherto signals of geophysical observations.
