Many linear(ized) geophysical inverse problems cannot be solved without regularization. Finding the regularization parameter that best balances the model complexity and data misfit is often a key step in the inversion problem. Traditionally, this is done by first plotting the measure of model complexity versus data misfit for different values of regularization parameter, which manifests as an L-shaped curve, and then choosing the regularization parameter corresponding to the corner point on the L-curve. For this approach, the difference in units between model complexity and data misfit must be considered, otherwise the result will be strongly affected by the scaling between these two quantities. Inspired by the machine learning literature, we here propose an extension to the traditional L-curve method. We first split the raw dataset into training and validation sets, obtain a solution by performing inversion on the training set only, and calculate data misfits on the validation set. We demonstrate the efficacy of this approach with a toy example and with two synthetic datasets. In realistic global surface-wave tomography studies where sampling of the Earth is nonuniform, we devise a procedure to generate a validation dataset with sampling as uniform as possible. We then show that the regularization parameter can be determined using this validation set, and this determination is apparently robust to the ratio of data split between training and validation sets. For both synthetic tests and realistic inversions, we find that our procedure can produce a minimal point that can be easily identified on the misfit curves calculated on the validation sets, and avoids the nuances encountered in the traditional L-curve analysis.

A controlled experiment was performed to investigate the influence of different assumptions made about the propagation of surface waves in surface wave tomography. Synthetic seismograms were generated using the spectral element method for 42 earthquakes and 190 stations and two different 3-D earth models. Fundamental-mode Rayleigh and Love wave traveltimes were measured using a cluster analysis technique for periods 35-250 s. We predicted Rayleigh and Love wave traveltimes for these paths using the great-circle ray approximation, exact ray theory and 2-D finite-frequency theory. Prediction approaches were evaluated with respect to the measurements by calculating misfit for all paths and with a 'winner' approach, which identifies the fraction of paths that are best fit by a prediction approach. Misfits indicated little difference between prediction approaches for Rayleigh waves at periods >50 s and that finite-frequency theory is superior for Love waves at periods >125 s. The 'winner' approach indicated that exact ray theory is superior at short periods, whereas finite-frequency theory is superior at long periods. However, when only the great-circle ray and finite-frequency approaches were considered, we found that each approach was superior as often as it was not. We solved for phase velocity maps with the measurements using two inversion approaches: the great-circle ray approximation and finite-frequency theory. While output maps generated using both approaches were generally similar, analysis of the variance reduction and the correlation with the input map showed that the great-circle ray theory inversion yielded phase velocity maps that better explained the measurements and matched the input maps. On the basis of the tests we have performed and the results we have obtained, we recommend using great-circle ray theory instead of 2-D finite-frequency theory for the analysis of surface wave phase delays, at least on the global scale.

(1) Northwestern University, Department of Earth and Planetary Sciences, United States (ecthompson@northwestern.edu), (2) University of Kiel, Institute of Geosciences, Germany, (3) Yale University, Department of Geology and Geophysics, United States, (4) Brown University, Department of Earth, Environmental and Planetary Sciences, United States, (5) Harvard University, Department of Earth and Planetary Sciences, United States, (6) Case Western Reserve University, Department of Earth, Environmental, and Planetary Sciences, United States, (7) Rice University, Department of Earth, Environmental, and Planetary Sciences, United States, (8) ETH Zurich, Institute of Geophysics, Switzerland, (9) University of Nevada Las Vegas, Department of Geoscience, United States, (10) University of Louisiana at Lafayette, Department of Physics, United States, (11) University of California Davis, Department of Earth and Planetary Sciences, United States

Variations in the phase velocity of fundamental-mode Rayleigh waves across the Indian Ocean are determined using two inversion approaches. First, variations in phase velocity as a function of seafloor age are estimated using a pure-path age-dependent inversion method. Second, a two-dimensional parameterization is used to solve for phase velocity within 1.25 degrees x 1.25 degrees grid cells. Rayleigh wave travel time delays have been measured between periods of 38 and 200 s. The number of measurements in the study area ranges between 4139 paths at a period of 200 s and 22,272 paths at a period of 40 s. At periods<100 s, the phase velocity variations are strongly controlled by seafloor age and shown to be consistent with temperature variations predicted by the half-space-cooling model for a mantle potential temperature of 1400 degrees C. The inferred thermal structure beneath the Indian Ocean is most similar to the structure of the Pacific upper mantle, where phase velocities can also be explained by a half-space-cooling model. The thermal structure is not consistent with that of the Atlantic upper mantle, which is best fit by a plate-cooling model and requires a thin plate. Removing age-dependent phase velocity from the 2-D maps of the Indian Ocean highlights anomalously high velocities at the Rodriguez Triple Junction and the Australian-Antarctic Discordance and anomalously low velocities immediately to the west of the Central Indian Ridge.