  We develop a linearized boundary control method for the inverse boundary
value problem of determining a density in the acoustic wave equation. The
objective is to reconstruct an unknown perturbation in a known background
density from the linearized Neumann-to-Dirichlet map. A key ingredient in the
derivation is a linearized Blagovescenskii's identity with a free parameter.
When the linearization is at a constant background density, we derive two
reconstructive algorithms with stability estimates based on the boundary
control method. When the linearization is at a non-constant background density,
we establish an increasing stability estimate for the recovery of the density
perturbation. The proposed reconstruction algorithms are implemented and
validated with several numerical experiments to demonstrate the feasibility.
