Summary
The classical Backus-Gilbert method seeks localized Earth-structure averages at the shortest length scales possible, given a dataset, data errors, and a threshold for acceptable model errors. The resolving length at a point is the width of the local averaging kernel, and the optimal averaging kernel is the narrowest one such that the model error is below a specified level. This approach is well suited for seismic tomography, which maps three-dimensional Earth structure using large sets of seismic measurements. The continual measurement-error decreases and data-redundancy increases have reduced the impact of random errors on tomographic models. Systematic errors, however, are resistant to data redundancy and their effect on the model is difficult to predict. Here, we develop a method for finding the optimal resolving length at every point, implementing it for surface-wave tomography. As in the Backus-Gilbert method, every solution at a point results from an entire-system inversion, and the model error is reduced by increasing the model-parameter averaging. The key advantage of our method stems from its direct, empirical evaluation of the posterior model error at a point. We first measure inter-station phase velocities at simultaneously recording station pairs and compute phase-velocity maps at densely, logarithmically spaced periods. Numerous versions of the maps with varying smoothness are then computed, ranging from very rough to very smooth. Phase-velocity curves extracted from the maps at every point can be inverted for shear-velocity (VS) profiles. As we show, errors in these phase-velocity curves increase nearly monotonically with the map roughness. We evaluate the error by isolating the roughness of the phase-velocity curve that cannot be explained by any Earth structure and determine the optimal resolving length at a point such that the error of the local phase-velocity curve is below a threshold. A 3D VS model is then computed by the inversion of the composite phase-velocity maps with an optimal resolution at every point. The estimated optimal resolution shows smooth lateral variations, confirming the robustness of the procedure. Importantly, the optimal resolving length does not scale with the density of the data coverage: some of the best-sampled locations display relatively low lateral resolution, probably due to systematic errors in the data. We apply the method to image the lithosphere and underlying mantle beneath Ireland and Britain. Our very large dataset was created using new data from Ireland Array, the Irish National Seismic Network, the UK Seismograph Network, and other deployments. A total of 11238 inter-station dispersion curves, spanning a very broad total period range (4–500 s), yield unprecedented data coverage of the area and provide fine regional resolution from the crust to the deep asthenosphere. The lateral resolution of the 3D model is computed explicitly and varies from 39 km in central Ireland to over 800 km at the edges of the area, where the data coverage declines. Our tomography reveals pronounced, previously unknown variations in the lithospheric thickness beneath Ireland and Britain, with implications for their Caledonian assembly and for the mechanisms of the British Tertiary Igneous Province magmatism.