Summary We present a new technique in surface wave tomography that takes the finite frequency of surface waves into account using first‐order scattering theory in a SNREI Earth. Physically, propagating surface waves with a finite frequency are diffracted by heterogeneity distributed on a sphere and then interfere at the receiver position. Paradoxically, surface waves have the largest sensitivity to velocity anomalies off the path of the geometrical ray. The non‐ray geometrical effect is increasingly important for increasing period and distance. Therefore, it is expected that the violation of ray theory in surface wave tomography is most significant for the longest periods. We applied scattering theory to phaseshift measurements of Love waves between periods of 40 and 150 s to obtain global phase velocity maps expanded in spherical harmonics to angular degree and order 40. These models obtained with scattering theory were compared with those constructed with ray theory. We observed that ray theory and scattering theory predict the same structure in the phase velocity maps to degree and order 25–30 for Love waves at 40 s and to degree and order 12–15 for Love waves at 150 s. For reasons of spectral leakage, a smoothness condition was included in the phaseshift inversions to construct the phase velocity maps, so we could not access the small length‐scale structure in the obtained Earth models. We carried out a synthetic experiment for phase velocity measurements to investigate the limits of classical ray theory in surface wave tomography. In the synthetic experiment, we computed, using the source–receiver paths of our surface wave data set, the discrepancy between ray theoretical and scattering theoretical phase velocity measurements for an input model with slowness heterogeneity for increasing angular degree. We found that classical ray theory in global surface wave tomography is only applicable for structures with angular degrees smaller than 25 (equivalent to 1600 km) and 15 (equivalent to 2700 km) for Love waves at 40 and 150 s, respectively. The synthetic experiment suggests that the ray theoretical great circle approximation is appropriate to use in present‐day global surface wave tomography. On the other hand, in order to obtain reliable models with a higher resolution we must take the non‐ray geometrical effect of surface waves into account.

The increase in number and strength of shallow induced seismicity connected to the Groningen gas field since 2003 and the occurrence of a M L 3.6 event in 2012 started the development of a full probabilistic seismic hazard assessment (PSHA) for Groningen, required by the regulator. Densification of the monitoring network resulted in a decrease of the location threshold and magnitude of completeness down to ∼ M L = 0.5. Combined with a detailed local velocity model, epicentre accuracy could be reduced from 0.5-1 km to 0.1-0.3 km and a vertical resolution ∼0.3 km. Time-dependent seismic activity is observed and taken into account into PSHA calculations. Development of the Ground Motion Model for Groningen resulted in a significant reduction of the hazard. Comparison of different implementations of the PSHA, using different source models, based on either a compaction model and production scenarios or on extrapolation of past seismicity, and methods of calculation, shows similar results.

In seismic imaging experiments, it is common to use a geometric ray theory that is an asymptotic solution of the wave equation in the high‐frequency limit. Consequently, it is assumed that waves propagate along infinitely narrow lines through space, called rays, that join the source and receiver. In reality, recorded waves have a finite‐frequency content. The band limitation of waves implies that the propagation of waves is extended to a finite volume of space around the geometrical ray path. This volume is called the Fresnel volume. In this tutorial, we introduce the physics of the Fresnel volume and we present a solution of the wave equation that accounts for the band limitation of waves. The finite‐frequency wave theory specifies sensitivity kernels that linearly relate the traveltime and amplitude of band‐limited transmitted and reflected waves to slowness variations in the earth. The Fresnel zone and the finite‐frequency sensitivity kernels are closely connected through the concept of constructive interference of waves. The finite‐frequency wave theory leads to the counterintuitive result that a pointlike velocity perturbation placed on the geometric ray in three dimensions does not cause a perturbation of the phase of the wavefield. Also, it turns out that Fermat's theorem in the context of geometric ray theory is a special case of the finite‐frequency wave theory in the limit of infinite frequency. Last, we address the misconception that the width of the Fresnel volume limits the resolution in imaging experiments.

Summary Small‐scale heterogeneity alters the arrival times of waves in a way that cannot by explained by ray theory. This is because ray theory is a high‐frequency approximation that does not take the finite frequency of wavefields into account. We present a theory based on the first‐order Rytov approximation that predicts well the arrival times of waves propagating in media with small‐scale inhomogeneity with a length scale smaller than the width of Fresnel zones. In the regime for which scattering theory is relevant we find that caustics are easily generated in wavefields, but this does not influence the good prediction of finite frequency arrival times of waves by scattering theory. The regime of scattering theory is relevant when the characteristic length of heterogeneity is smaller than the width of Fresnel zones. The regime of triplications is independent of frequency but it is more significant the greater the magnitude of slowness fluctuations.

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S U M M A R YWe compared surface wave tomography models obtained using finite-frequency kernels and ray theory. We systematically changed regularization in both cases and plotted data misfit against the number of independent parameters in the solution. Our tests show that models from finite-frequency kernels and ray-theoretical kernels are statistically similar. This means that any model obtained using one forward theory can be obtained using the other one by appropriately changing the damping constant. It is clear that finite-frequency theory is a better forward theory to represent the wavefield, but the associated inverse problem is not less ill posed. Indeed, current data coverage is such that the solution is dominated by the chosen regularization. This prevents us from achieving a resolution of the order of the Fresnel zone, or beyond, and noticing the benefits of a better forward theory.

Abstract. The ray theoretical great circle approximation in global surface wave tomography is found to be limited to Earth models with a maximum degree 1 _< 30 for surface waves at 40 s and 1 < 20 for surface waves at 150 s.This result holds for both phase velocity and group velocity maps. The highest resolution in present-day global surface wave tomography is close to these limits of ray theory. In order to obtain higher degree resolution models of the Earth in future surface wave tomography, it is necessary to take the scattering of surface waves into account. Increasing the data coverage in seismological networks will not improve the details of tomographic images if ray theory is still applied. It is essential to include the finite-frequency effects as well.

Ultrasonic measurements of acoustic wavefields scattered by single spheres placed in a homogenous background medium ͑water͒ are presented. The dimensions of the spheres are comparable to the wavelength and the wavelength and represent both positive ͑rubber͒ and negative ͑teflon͒ velocity anomalies with respect to the background medium. The sensitivity of the recorded wavefield to scattering in terms of traveltime delay and amplitude variation is investigated. The results validate a linear ͑first-order͒ diffraction theory for wavefields propagating in heterogeneous media with anomaly contrasts on the order of ±15%. The diffraction theory is compared further with the exact results known from literature for scattering from an elastic sphere, formulated in terms of Legendre polynomials. To investigate the 2D case, the firstorder scattering theory is tested against 2D elastic finite-difference calculations. As the presented theory involves a volume integral, it is applicable to any geometric shape, and the scattering object does not need to be spherical or any other specific symmetrical shape. Furthermore, it can be implemented easily in seismic data inversion schemes, which is illustrated with examples from seismic crosswell tomography and a reflection experiment.