Choosing Appropriate Regularization Parameters by Splitting Data into Training and Validation Sets—Application in Global Surface-Wave Tomography

Abstract

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.