Gauss–Newton and full Newton methods in frequency–space seismic waveform inversion

SUMMARY By specifying a discrete matrix formulation for the frequency^space modelling problem for linear partial diierential equations (‘FDM’ methods), it is possible to derive a matrix formalism for standard iterative non-linear inverse methods, such as the gradient (steepest descent) method, the Gauss^Newton method and the full Newton method. We obtain expressions for each of these methods directly from the discrete FDM method, and we refer to this approach as frequency-domain inversion (FDI).The FDI methods are based on simple notions of matrix algebra, but are nevertheless very general. The FDI methods only require that the original partial diierential equations can be expressed as a discrete boundary-value problem (that is as a matrix problem). Simple algebraic manipulation of the FDI expressions allows us to compute the gradient of the mis¢t function using only three forward modelling steps (one to compute the residuals, one to backpropagate the residuals, and a ¢nal computation to compute a step length). This result is exactly analogous to earlier backpropagation methods derived using methods of functional analysis for continuous problems. Following from the simplicity of this result, we give FDI expressions for the approximate Hessian matrix used in the Gauss^Newton method, and the full Hessian matrix used in the full Newton method. In a new development, we show that the additional term in the exact Hessian, ignored in the Gauss^Newton method, can be e⁄ciently computed using a backpropagation approach similar to that used to compute the gradient vector. The additional term in the Hessian predicts the degradation of linearized inversions due to the presence of ¢rst-order multiples (such as free-surface multiples in seismic data). Another interpretation is that this term predicts changes in the gradient vector due to second-order non-linear eiects. In a numerical test, the Gauss^Newton and full Newton methods prove eiective in helping to solve the di⁄cult non-linear problem of extracting a smooth background velocity model from surface seismic-re£ection data.