The uncertainty of model parameters obtained by full-waveform inversion can be determined from the Hessian of the least-squares error functional. A description of uncertainty characterisation is presented that takes the null space of the Hessian into account and does not rely on the Bayesian formulation. Because the Hessian is generally too costly to compute and too large to be stored, a segmented representation of perturbations of the reconstructed subsurface model in the form of geological units is proposed. This enables the computation of the Hessian and the related covariance matrix on a larger length scale. Synthetic two-dimensional isotropic elastic examples illustrate how conditional and marginal uncertainties can be estimated for the properties per geological unit by themselves and in relation to other units.

The accuracy of a model obtained by multi-parameter full-waveform inversion can be estimated by analysing the sensitivity of the data to perturbations of the model parameters in selected subsurface points. Each perturbation requires the computation of the seismic response in the form of Born scattering data for a typically very large number of shots, making the method time consuming. The computational cost can be significantly reduced by placing sources of different types at the Born scatterer, the point where the subsurface parameters are perturbed. Instead of modelling each shot separately, reciprocity relations provide the wavefields from the shot positions to the scatter point in terms of wavefields from the scatterer to the shot positions. In this way, the Born scattering data from a single point in the isotropic elastic case for a marine acquisition with pressure sources and receivers can be expressed in terms of the wavefields for force and moment tensor sources located at the scatterer and only a small number of forward runs are required. A two-dimensional example illustrates how the result can be used to determine the Hessian and local relative covariance matrix for the model parameters at the scatterer at the cost of five forward simulations. In three dimensions, that would be nine.