Seismic surface wave tomography is a tried and tested method to reveal the subsurface structure of the Earth. However, the conventional 2-step scheme of inverting first for 2-D maps of surface wave phase or group velocity and then inverting for the 3-D spatial velocity structure preserves little information about lateral spatial correlations, and introduces additional uncertainties and errors into the 3-D result. We introduce a 1-step 3-D non-linear surface wave tomography method that removes these effects by inverting for 3-D spatial structure directly from frequencydependent traveltime measurements. We achieve this using the reversible jump Markov chain Monte Carlo (McMC) algorithm with a fully 3-D model parametrization. Synthetic tests show that the method estimates the velocity model and associated uncertainties significantly better than the conventional 2-step McMC method, and that the computational cost seems to be comparable with 2-step McMC methods. The resulting uncertainties are more intuitively reasonable than those from the 2-step method, and provide directly interpretable uncertainty on volumetrics of structures of interest.

The permanent ocean-bottom array at the Valhall Field in Norway provides an excellent source of passive seismic data to test what might be accomplished with seismic interferometry. The array was installed in 2003 (Kommedal et al., 2004) and data can be recorded for long periods in all weather conditions. The subsurface structure is well known, both from numerous wells and from seismic imaging. During periods without active seismic acquisition at Valhall, there is abundant passive energy in the data over a wide range of fre-quencies.

In a variety of geoscientific applications we require 3‐D maps of properties of the Earth's interior and the corresponding map of uncertainties to assess their reliability. On the seabed it is common to use Scholte wave dispersion data to infer these maps using inversion‐based imaging theory. Previously we introduced a 3‐D fully nonlinear Monte Carlo tomography method that inverts for shear velocities directly from frequency‐dependent travel time measurements and which improves accuracy of the results and better estimates uncertainties. Here for the first time we apply that method to real data and compare it to two previous methods. We cross correlated 6.5 hr of ambient noise data recorded on a dense seismic array over Grane, North Sea, and observed two Scholte wave modes. For each mode, phase velocity maps are estimated using Eikonal tomography, which are in turn used to study the shear‐wave velocity structure of the subsurface. We applied three nonlinear inversion methods to the Grane data: standard 1‐D depth inversions, a 2‐D joint inversion along a vertical cross section, and a fully 3‐D inversion. We compare the shear‐velocity and uncertainty structures estimated along the same 2‐D cross section. Thus we show that the standard 1‐D inversion method creates significant errors in the results due to the independence of those 1‐D inversions, whereas the 2‐D and 3‐D inversions improve results by accounting for lateral spatial correlations. The 3‐D inversion bypasses the initial seabed Eikonal tomography step, thus avoiding the errors that the initial step introduces into subsequent 1‐D and 2‐D inversions.

Summary Marchenko methods are based on integral representations which express Green’s functions for virtual sources and/or receivers in the subsurface in terms of the reflection response at the surface. An underlying assumption is that inside the medium the wave field can be decomposed into downgoing and upgoing waves and that evanescent waves can be neglected. We present a new derivation of Green’s function representations which circumvents these assumptions, both for the acoustic and the elastodynamic situation. These representations form the basis for research into new Marchenko methods which have the potential to handle refracted and evanescent waves and to more accurately image steeply dipping reflectors.

Quality factor (Q) or equivalently attenuation α=1Q describes the amount of energy lost per cycle as a wave travels through a medium. This is important to correct seismic data amplitudes for near‐surface effects, to locate subsurface voids or porosity, to aid seismic interpretation, or for characterizing other rock and fluid properties. Seismic attenuation can be variable even when there are no discernible changes in seismic velocity or density (Yıldırım et al., 2017, https://doi.org/10.1016/j.jappgeo.2016.11.010) and so provides independent information about subsurface heterogeneity. This study uses ambient noise recordings made on the Ekofisk Life of Field Seismic array to estimate Q structure in the near surface. We employ the method of X. Liu et al. (2015, https://doi.org/10.1093/gji/ggv357), which uses linear triplets of receivers to estimate Q—ours is the first known application of the method to estimate the Q structure tomographically. Estimating Q requires an estimate of phase velocity which we obtain using the method of Bloch and Hales (1968, https://pubs.geoscienceworld.org/ssa/bssa/article-abstract/58/3/1021/116607/) followed by traveltime tomography. The Q structure at Ekofisk has features which can be related to local geology, showing that surface ambient noise recordings may provide a new and robust method to image Q. Our results suggest that there is a nonlinear relationship between Q and compression. They also may explain why it has been found that in the period range of 1 to 2 s considered here, ambient noise cross correlations along paths that span the North Sea Basin are unreliable: Such Q values would attenuate almost all ambient seismic energy during such a traverse.