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.

To estimate the depth errors in a subsurface model obtained from the inversion of seismic data, the stationary-phase approximation in a two-dimensional constant-velocity model with a dipped reflector is applied to migration with a time-shift extension. This produces two asymptotic solutions: one is a straight line, and the other is a curve. If the velocity differs from the true one, a closed-form expression of the depth error follows from the depth and apparent dip of the reflector as well as the position of the amplitude peak at a non-zero time shift, where the two solutions meet and the extended migration image focuses. The results are compared to finite-frequency results from a finite-difference code. A two-dimensional synthetic example with a salt diapir illustrates how depth errors can be estimated in an inhomogeneous model after inverting the seismic data for the velocity model.

Intersecting faults are often ignored in the geomechanical simulation of induced seismicity. To investigate the effects of fault intersection and the resulting reservoir geometry on induced seismicity, caused, for instance, by gas extraction, we have developed 3D geomechanical models considering two intersecting normal faults and the surrounding horst structure. We simulate the stress field and the dynamic fault reactivation in a uniformly depleted reservoir. We observe that a smaller intersection angle increases the incremental Coulomb stress at the lower reservoir juxtaposition, thus changing the temporal rupture pattern of the seismic event. In our dynamic simulation, the rupture propagates from the main fault to the secondary fault. We conclude that the fault intersection has important effects on the induced seismicity and should be taken into account when evaluating the seismicity risk in a specific region.

Temporal dispersion correction of second-order finite-difference time stepping for numerical wave propagation modelling exploits the fact that the discrete operator is exact but for the wrong frequencies. Mapping recorded traces to the correct frequencies removes the numerical error. Most of the implementations employ forward and inverse Fourier transforms. Here, it is noted that these can be replaced by a series expansion involving higher time derivatives of the data. Its implementation by higher-order finite differencing can be sensitive to numerical noise, but this can be suppressed by enlarging the stencil. Tests with the finite-element method on a homogeneous acoustic problem with an exact solution show that the method can achieve the same accuracy as higher-order time stepping, similar to that obtained with Fourier transforms. The same holds for an inhomogeneous problem with topography where the solution on a very fine mesh is used as reference. The series approach costs less than dispersion correction with the Fourier method and can be used on the fly during the time stepping. It does, however, require a wavelet that is sufficiently many times differentiable in time.

Finite elements with mass lumping allow for explicit time stepping when modelling wave propagation and can be more efficient than finite differences in complex geological settings. In two dimensions on quadrilaterals, spectral elements are the obvious choice. Triangles offer more flexibility for meshing, but the construction of polynomial elements is less straightforward. The elements have to be augmented with higher-degree polynomials in the interior to preserve accuracy after lumping of the mass matrix. With the classic accuracy criterion, triangular elements suitable for mass lumping up to a polynomial degree 9 were found. With a newer, less restrictive criterion, new elements were constructed of degree 5–7. Some of these are more efficient than the older ones. To assess which of all these elements performs best, the acoustic wave equation is solved for a homogeneous model on a square and on a domain with corners, as well as on a heterogeneous example with topography. The accuracy and runtimes are measured using either higher-order time stepping or second-order time stepping with dispersion correction. For elements of polynomial degree 2 and higher, the latter is more efficient. Among the various finite elements, the degree-4 element appears to be a good choice.

Finite elements with polynomial basis functions on the simplex with a symmetric distribution of nodes should have a unique polynomial representation. Unisolvence not only requires that the number of nodes equals the number of independent polynomials spanning a polynomial space of a given degree, but also that the Vandermonde matrix controlling their mapping to the Lagrange interpolating polynomials can be inverted. Here, a necessary condition for unisolvence is presented for polynomial spaces that have non-decreasing degrees when going from the edges and the various faces to the interior of the simplex. It leads to a proof of a conjecture on a necessary condition for unisolvence, requiring the node pattern to be the same as that of the regular simplex.

When solving the wave equation with finite elements, mass lumping allows for explicit time stepping, avoiding the cost of a lower-upper decomposition of the large sparse mass matrix. Mass lumping on the reference element amounts to numerical quadrature. The weights should be positive for stable time stepping and preserve numerical accuracy. The standard triangular polynomial elements, except for the linear element, do not have these properties. Accuracy can be preserved by augmenting them with higher-degree polynomials in the interior. This leaves the search for elements with positive weights, which were found up to degree 9 by various authors. The classic accuracy condition, however, is too restrictive. A sharper, less restrictive condition recently led to new mass-lumped tetrahedral elements up to degree 4. Compared to the known ones up to degree 3, they have less nodes and are computationally more efficient. The same criterion is applied here to the construction of triangular elements. For degrees 2 to 4, these turn out to be identical to the known ones. For degree 5, the number of nodes is the same as for the known element, but now there are infinitely many solutions. Some of these have a considerably larger stability limit for time stepping. For degree 6, two elements are found with less nodes than the known ones. For degree 7, one element with less nodes was found but with a negative weight, making it useless for time stepping with the wave equation. If the number of nodes is the same as for the classic element, there are now infinitely many solutions. Numerical tests for a homogeneous wave-propagation problem with a point source confirm the expected accuracy of the new elements. Some of them require less compute time than those obtained with the more restrictive accuracy criterion.