Summary By specifying a discrete matrix formulation for the frequency–space modelling problem for linear partial differential 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 differential 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 misfit function using only three forward modelling steps (one to compute the residuals, one to backpropagate the residuals, and a final 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 efficiently 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 first‐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 effects. In a numerical test, the Gauss–Newton and full Newton methods prove effective in helping to solve the difficult non‐linear problem of extracting a smooth background velocity model from surface seismic‐reflection data.

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Wavefield-based tomographic methods are idoneous for recovering velocity models from seismic data. The use of wavefields rather than rays is more consistent with the bandlimited nature of seismic data. Image domain methods seek to improve the focusing in extended images, thus producing better seismic images. However, image domain methods produce low resolution models due to the fact their objective functions are smooth, particularly in the vicinity of the global minimum. In contrast, data-domain methods produce high resolution models but suffer from strong non-linearity causing cycle skipping if certain conditions are not met. By combining the characteristics of each method, we can obtain models that produce better images and contain high resolution features at the same time. We demonstrate the strength of the workflow that combines both methods with an application to a marine 2D dataset with variable streamer depth.

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