Formulation of the estimation problem as a conventional inverse problem

Equation 5 shows that using usual notations, the estimation problem can be formulated as a conventional inversion problem. For a given reflector at a given midpoint, we represent the RMO values for different aperture angles by the data vector $\bf d$. We represent the perturbation vector by the model vector ${\bf m}$and denote ${\bf m_0}$ the vector of correct migration velocity ($\rho=1$). The forward problem from equation 5 can thus be written

$$\mathbf{d} = \mathbf{F} \left( {\bf m} - {\bf m_0} \right) \... ...partial \rho_j} \right\vert _{\gamma=\gamma_i, \rho ={\bf 1}}$$ (6)

Under linearization around the correct migration velocity, the estimation of the anisotropic parameters is thus equivalent to a conventional inversion problem: it consists in inverting the forward modeling matrix ${\bf F}$ to estimate the anisotropic parameters ${\bf m}$.