Linearize with the Jacobian
Around a current guess m_k, a first-order Taylor expansion replaces
G by its Jacobian
J_{ij} = \partial G_i/\partial m_j — the matrix of sensitivities of each
datum to each parameter:
G(m_k + \delta m) \approx G(m_k) + J\,\delta m.
Fitting the leftover misfit \Delta d = d - G(m_k) is now a
linear least-squares problem for the update \delta m, solved
with the very tools from earlier — normal equations plus regularization. Update, recompute the
Jacobian, repeat:
m_{k+1} = m_k + (J^{\mathsf T}J + \alpha^2 I)^{-1} J^{\mathsf T}\,\Delta d.
This is the regularized Gauss–Newton iteration (the Levenberg–Marquardt method
adapts \alpha as it goes). It is also exactly
gradient-based optimization
of the misfit — nonlinear inversion is nonlinear least squares.
- Nonlinear forward model d = G(m): no closed-form inverse; superposition fails.
- Linearize around the current model with the Jacobian J = \partial G/\partial m.
- Solve the regularized linear update (Gauss–Newton / Levenberg–Marquardt) and repeat to convergence.
- Nonlinearity brings the danger of local minima — a good starting model matters.