VALIDITY CONDITION RESULTS

In this section I will state the validity condition results of the mathematical analysis that follows in the main body of the paper. A valid acoustic wave equation governing fluid particle motion for a pressure wavefield $P({\bf \underline{x}},t)$ is

 

$$\{ \partial_{tt}- v^2({\bf \underline{x}})\nabla^2 \} P({\bf \underline{x}},t) = 0\,.$$ (1)

1. It is perfectly rigorous to use (1) in a linear isotropic non-viscous fluid with a constant mass density $\rho_o$ and a spatially variable bulk modulus $\lambda({\bf \underline{x}})$ such that

 

$$v_1^2({\bf \underline{x}}) = \lambda({\bf \underline{x}}) / \rho_o \,.$$ (2)

2. In ``reasonable'' background models where $\rho({\bf \underline{x}})$ varies smoothly with respect to pressure wavelength, 1 can be extended to $\rho_o = \rho({\bf \underline{x}})$ such that

 

$$v_2^2({\bf \underline{x}}) = \lambda({\bf \underline{x}}) / \rho({\bf \underline{x}}) \,,$$ (3)

when the following validity condition is satisfied:

 

$$\frac{ \vert \d_i \rho\vert }{ \vert \rho\vert } \ll \frac{ \vert \d_i \ddot{u}_i \vert}{ \vert \ddot{u}_i \vert} \,.$$ (4)

Condition (4) is invalid near regions of discontinuities in $\rho({\bf \underline{x}})$ (e.g., reflecting boundaries), but is valid in regions of smooth-varying spatial properties (e.g., a smoothed migration model).