  In this paper, we consider the approximate acoustic cloaking in inhomogeneous
isotropic background space. By employing transformation media, together with
the use of a sound-soft layer lining right outside the cloaked region, we show
that one can achieve the near-invisibility by the `blow-up-a-small-region'
construction. This is based on novel scattering estimates corresponding to
small sound-soft obstacles located in isotropic space. One of the major
novelties of our scattering estimates is that one cannot make use of the
scaling argument in the setting of current study due to the simultaneous
presence of asymptotically small obstacle components and regularly sized
obstacle components, and one has to decouple the nonlinear scattering
interaction between the small obstacle components and, the regular obstacle
components together with the background medium.
