  We propose a new approach toward reconstructing the late-time near-horizon
geometry of merging binary black holes, and toward computing gravitational-wave
echoes from exotic compact objects. A binary black-hole merger spacetime can be
divided by a time-like hypersurface into a Black-Hole Perturbation (BHP)
region, in which the space-time geometry can be approximated by homogeneous
linear perturbations of the final Kerr black hole, and a nonlinear region. At
late times, the boundary between the two regions is an infalling shell. The BHP
region contains late-time gravitational-waves emitted toward the future
horizon, as well as those emitted toward future null infinity. In this region,
by imposing no-ingoing wave conditions at past null infinity, and matching
out-going waves at future null infinity with waveforms computed from numerical
relativity, we can obtain waves that travel toward the future horizon. In
particular, the Newman-Penrose $\psi_0$ associated with the in-going wave on
the horizon is related to tidal deformations measured by fiducial observers
floating above the horizon. We further determine the boundary of the BHP region
on the future horizon by imposing that $\psi_0$ inside the BHP region can be
faithfully represented by quasi-normal modes. Using a physically-motivated way
to impose boundary conditions near the horizon, and applying the so-called
Boltzmann reflectivity, we compute the quasi-normal modes of non-rotating ECOs,
as well as gravitational-wave echoes. We also investigate the detectability of
these echoes in current and future detectors, and prospects for parameter
estimation.
