Newton's method and WEMVA

One can also consider the problem of estimating the slowness field from wavefields using WEMVA in the general non-linear inversion framework.

In particular, if $\mathcal W\left({z+\Delta z} \right)$ is the upgpoing wavefield at the bottom of a layer and $\mathcal W\left(z \right)$ is the upgoing wavefield at the top of the layer, the layer slowness s is constrained by the nonlinear equation  

$$F[s] = P[s]\,\mathcal W\left({z+\Delta z} \right)- \mathcal W\left({z } \right)= 0\;,$$ (18)

where $P[s] = e^{i\,\k_z\left[s \right]}$ is the wave propagation operator.

The Newton method applied to equation (18) amounts to inversion of the linear system  

$$F'[s_k]\,(s_{k+1} - s_k) = - F[s_k]\;,$$ (19)

where k is the nonlinear iteration counter (the iteration starts with some a priori slowness model s₀), and F'[s] is the Fréchet derivative of the wave propagation operator. Since F[s] is complex-valued, we can multiply both sides of system (19) by the adjoint (complex-conjugate) operator F'[s_k]^T to obtain the purely real system

$$\begin{eqnarray} F'[s_k]^T\,F'[s_k]\,(s_{k+1} - s_k) & = & - F'[s_k]^T\,F[s_k] ... ...s_k) & = & - \left(R'[s_k]^T\,R[s_k] + I'[s_k]^T\,I[s_k]\right)\;,\end{eqnarray}$$ (20)

where R[s] and I[s] are the real and imaginary parts of F[s]. Algorithm (21) is equivalent to the Gauss-Newton method applied to the least-squares solution of

$$\begin{eqnarray} R[s] & \approx & 0\;, \\ I[s] & \approx & 0\;.\end{eqnarray}$$ (21) (22)

It is well-known that the Newton and Newton-Gauss methods possess fast convergence provided that the original estimate s₀ is sufficiently close to the solution. They may diverge otherwise. To guarantee convergence, the norm (spectral radius) of the Fréchet derivative G'[s] for the operator  

$$G[s] = s - \frac { R'[s]^T\, R[s] + I'[s]^T\, I[s] } { R'[s]^T\,R'[s] + I'[s]^T\,I'[s] }\,$$ (23)

must be strictly smaller than one in the vicinity of the solution that contains the starting value s₀. Convergence follows then from the contraction mapping theorem. The speed of convergence is higher for smaller norms.

It is important to realize that modifying the original nonlinear Equation (18) may change the convergence behavior and lead to faster convergence and wider convergence area. A particularly meaningful way to modify Equation (18) is to multiply it by $P[s]^{-\xi}$, where $\xi$ is a scalar between and 1. The modified equation takes the form  

$$F_\xi[s] = P[s]^{-\xi}\,F[s] = P[s]^{1-\xi}\,\mathcal W\left... ...lta z} \right)- P[s]^{-\xi}\,\mathcal W\left({z } \right)= 0\;.$$ (24)

The case of $\xi=0$ corresponds to the original system. Its linearization with the Newton method leads to the Born approximation. Analogously, the case of $\xi=1$ corresponds to the implicit method: the two wavefields are compared at the bottom of the layer rather than at the top. The case of $\xi=1/2$ leads to the bilinear method: both wavefields are continued to the middle of the layer for comparison. Many other intermediate results are possible,