Least-squares inversion

Although I have already written a Fortran77 program to solve a system of equations similar to (3) by the conjugate gradient method, Figure 2 shows that this program needs improvement in order to perform the inversion automatically without running into problems of edge effects.

The general procedure for estimating the velocity model will be to iterate until an acceptable velocity model is calculated, as follows:

• The mapping of the seismic velocity data onto the model space gives an initial guess of the model.
• The standard damped least-squares inversion is applied. At iteration k, I
  • first invert for the model perturbation

    $$\delta m_k \; = \; \left[\epsilon^2 A^T A\right]^{-1} L^T C_w^{-1} (d_w \, - \, L m_k)$$ (6)

  • then update the velocity model

    $$m_{k+1} \; = \; m_k \; + \; \delta m_k$$ (7)

The convergence is achieved when the mistie between the well velocity and the calculated velocities has been eliminated.