  We study drop impacts on immiscible viscous liquid pool and investigate the
formation of droplet surface craters using experimental and theoretical
analysis. We attribute the formation of air craters to the rapid deceleration
of the droplet due to viscous drag force. The droplet response to the external
impulsive decelerating force induces oscillatory modes on the surface exposed
to the air forming capillary waves that superimpose to form air craters of
various shapes and sizes. We introduce a non-dimensional parameter
(${\Gamma}$), that is, the ratio of drag force to the capillary force acting on
the droplet. We show that ${\Gamma}$ is directly proportional to the capillary
number. We show that droplets forming air craters of significant depths have
${\Gamma}>1$. Further, we demonstrate that Legendre polynomials can locally
approximate the central air crater jet profile. We also decipher that the air
crater response time scale ($T$) varies as the square root of impact Weber
number ($T{\sim}We^{1/2}$). Further, we generalize the local droplet response
with a global response model for low-impact energies based on an eigenvalue
problem. We represent the penetrating drop as a constrained Rayleigh drop
problem with a dynamic contact line. The air-water interface dynamics is
analyzed using an inviscid droplet deformation model for small deformation
amplitudes. The local and global droplet response theory conforms with each
other and depicts that the deformation profiles could be represented as a
linear superposition of eigenmodes in Legendre polynomial basis. We unearth
that the droplet response in an immiscible impact system differs from the
miscible impact systems due to the presence of such a dynamic contact line.
