Three‐dimensional seismic inversion of velocity‐ and density‐dependent reflectivity

Summary Assuming a model of the subsurface characterized by weak fluctuations in both velocity and density, an integral equation is derived for the case of zero‐offset reflection data which relates the surface measurements to a scalar reflectivity function given by f ( r ) = 1 c 2 / c ( r ) 2 + 2 In [ρ( r )/ρ 0 ]. In this relation, c ( r ) and ρ( r ) are, respectively, the variable velocity and density of the 3‐D medium, and c 0 and ρ 0 are corresponding reference values. Under the assumptions of the Born approximation, a non‐dissipative and non‐dispersive medium, and full bandwidth data, the derived integral equation can be solved in closed form for the reflectivity f ( r ). Using a WKBJ approximation for the Green's function in the depth direction only, a closed‐form solution is also derived for f ( r ) assuming a (known) variable depth‐dependent reference velocity. Finally, for the case of a constant background velocity, a band‐limited version of f ( r ) is shown to be recoverable from reflection data limited to a finite (temporal) frequency cutoff. Our ability to reconstruct a reflectivity function containing a variable density term results from the use of zero‐offset data, since, under these conditions, the directional dependence of the dipole‐like density term in the wave equation is removed, and the density variation manifests itself as a scalar contribution to the reflectivity.