Uncertainty Loops in Travel-Time Tomography from Nonlinear Wave Physics

Estimating image uncertainty is fundamental to guiding the interpretation of geoscientific tomographic maps. We reveal novel uncertainty topologies (loops) which indicate that while the speeds of both low- and high-velocity anomalies may be well constrained, their locations tend to remain uncertain. The effect is widespread: loops dominate around a third of United Kingdom Love wave tomographic uncertainties, changing the nature of interpretation of the observed anomalies. Loops exist due to 2nd and higher order aspects of wave physics; hence, although such structures must exist in many tomographic studies in the physical sciences and medicine, they are unobservable using standard linearized methods. Higher order methods might fruitfully be adopted.

Figure 1

Ambient-noise Love-wave tomography of the British Isles at 10 s period. (a) Average velocity model; (b) standard deviation map, obtained from an ensemble of 160 000 models, with two details highlighted in magnified plots; (c) standard deviation map for the same data inverted using linearized tomography.

Figure 2

The effect of a low velocity anomaly (dark to light red) on synthetic rays and travel times. (a) The velocity of the circular anomaly increases from left to right. (b) Both size and velocity of the anomaly decrease from left to right. (c) Three anomalies with similar velocity and size but different shape. The green dashed box encloses velocity models that are all compatible with a given travel time $t_1$; models on the right produce different travel times $t_2$ and $t_3$. The standard deviation across all models within the box is shown in the fourth plot of row (c). (d) Synthetic tomography of the low-velocity anomaly located at the center of an array of receivers: true velocity field and ray paths are shown in the first two plots; ensemble average and uncertainty map from fully nonlinear rjmcmc tomography are in the right two plots.

Figure 3

Posterior probability densities for velocities at different geographical points in the U.K. Histograms are calculated at positions marked by the gray (a) diamond ($-4.875°\mathrm{E}$, $54°\mathrm{N}$), (b) square ($-3.5°\mathrm{E}$, $54.375°\mathrm{N}$), (c) circle ($-4.25°\mathrm{E}$, $54.0625°\mathrm{N}$) in magnified plots in Fig. 1. Points (a) and (b) may be either inside or outside of the anomaly; hence, each has a probability maxima at two velocities and a large range. Point (c) is inside the anomaly and bounded above.