Rytov image perturbation

An alternative to the wavefield perturbation definition in equation (13) is given by the Rytov approximation. If we can estimate the accumulated phase differences between the two wavefields at every depth level

$$\Delta \Phi_r= \Phi- \Phi_o\;,$$ (15)

we can compute another wavefield perturbation using the relation:  

$$\Delta \mathcal U_r= \mathcal U_o\;i \Delta \Phi_r$$ (16)

which is directly derived from equation (9). With this definition of the wavefield perturbation, we can compute another slowness perturbation which corresponds to the Rytov approximation  

$$\Delta s_r= {\bf B}^* \left(\mathcal U_o\right)\left[\Delta \mathcal U_r\right]\;$$ (17)

using the same backprojection operator.