  The gravitational radiation from the ringdown of a binary black hole merger
is described by the solution of the Teukolsky equation, which predicts both the
temporal dependence and the angular distribution of the emission. Many studies
have explored the temporal feature of the ringdown wave through black hole
spectroscopy. In this work, we further study the spatial distribution, by
introducing a global fitting procedure over both temporal and spatial
dependences, to propose a more complete test of General Relativity. We show
that spin-weighted spheroidal harmonics are the better representation of the
ringdown angular emission patterns compared to spin-weighted spherical
harmonics. The differences are distinguishable in numerical relativity
waveforms. We also study the correlation between progenitor binary properties
and the excitation of quasinormal modes, including higher-order angular modes,
overtones, prograde and retrograde modes. Specifically, we show that the
excitation of retrograde modes is dominant when the remnant spin is
anti-aligned with the binary orbital angular momentum. This study seeks to
provide an analytical strategy and inspire the future development of ringdown
test using real gravitational wave events.
