1‐D, 2‐D, and 3‐D Monte Carlo Ambient Noise Tomography Using a Dense Passive Seismic Array Installed on the North Sea Seabed

Abstract

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.

Probabilistic neural network-based 2D travel-time tomography

“…In a tomographic setting, this could be useful for monitoring purposes, where data collected periodically from the same set of sources and receivers can be inverted with the same network(s) each time new data arrive. However it should be noted that despite the longer computational time, Monte Carlo methods can be used to produce higher resolution 2D or 3D models [10,44]. Mixture density networks have also been shown to give conservative uncertainty estimates compared to Monte Carlo methods [12].…”

“…To find solutions with minimal computation, the physics relating local wave speed to measured travel times is usually simplified by linearisation [4], but this creates large differences between linearised and true probabilistic solutions [5,9]. Increases in compute power now allow fully nonlinear Monte Carlo sampling solutions to be found without linearisation, to solve problems in 2D [5,6] and 3D [7][8][9][10]. Using Bayesian methods, such solutions provide samples (example tomographic models) that fit the data to within their measurement uncertainties, are consistent with available prior information and are distributed according to the posterior probability density function (pdf) across the parameter space; this pdf constitutes the full solution of tomographic problems.…”

Bayesian Geophysical Inversion Using Invertible Neural Networks

“…We added Gaussian noise with a standard deviation of 5 m/s to those calculated dispersion curves, which is a typical noise level in near surface ambient noise studies (Zhang, Hansteen, et al, 2020). Note that to ensure the computed dispersion curves are fundamental mode Rayleigh waves, within the prior pdf we ensured that the top layer has smallest shear velocity-otherwise the wave recorded on the Earth's surface would be a higher mode Rayleigh wave (Zhang, Hansteen, et al, 2020). We use 90 percent of those model and disper-ZHANG AND CURTIS 10.1029/2021JB022320 8 of 27 sion curve pairs as training data, and the remaining 10 percent as test data used for independent evaluation of network performance.…”

“…We generate 100,000 models from the prior pdf and calculate Rayleigh wave dispersion curves corresponding to each model using a modal approximation method (Herrmann, 2013) over the period range 0.7–2.0 s with 0.1 s spacing (Figure 4b). We added Gaussian noise with a standard deviation of 5 m/s to those calculated dispersion curves, which is a typical noise level in near surface ambient noise studies (Zhang, Hansteen, et al., 2020). Note that to ensure the computed dispersion curves are fundamental mode Rayleigh waves, within the prior pdf we ensured that the top layer has smallest shear velocity—otherwise the wave recorded on the Earth's surface would be a higher mode Rayleigh wave (Zhang, Hansteen, et al., 2020).…”

Imaging of a magma system beneath the Merapi Volcano complex, Indonesia, using ambient seismic noise tomography

“…However, the approach of directly inverting dispersion data for 3-D shear wave velocity structure in a single step is gradually becoming more popular (e.g. Zhang et al 2019). In the following study, we apply the more traditional two-step method outlined above to image the mid-upper crustal structure beneath the Merapi-Merbabu volcanic complex, by exploiting ambient noise data from a temporary seismic deployment that spanned the two volcanoes.…”