A method of spherical harmonic analysis in the geosciences via hierarchical Bayesian inference

Abstract

S U M M A R YThe problem of decomposing irregular data on the sphere into a set of spherical harmonics is common in many fields of geosciences where it is necessary to build a quantitative understanding of a globally varying field. For example, in global seismology, a compressional or shear wave speed that emerges from tomographic images is used to interpret current state and composition of the mantle, and in geomagnetism, secular variation of magnetic field intensity measured at the surface is studied to better understand the changes in the Earth's core. Optimization methods are widely used for spherical harmonic analysis of irregular data, but they typically do not treat the dependence of the uncertainty estimates on the imposed regularization. This can cause significant difficulties in interpretation, especially when the best-fit model requires more variables as a result of underestimating data noise. Here, with the above limitations in mind, the problem of spherical harmonic expansion of irregular data is treated within the hierarchical Bayesian framework. The hierarchical approach significantly simplifies the problem by removing the need for regularization terms and user-supplied noise estimates. The use of the corrected Akaike Information Criterion for picking the optimal maximum degree of spherical harmonic expansion and the resulting spherical harmonic analyses are first illustrated on a noisy synthetic data set. Subsequently, the method is applied to two global data sets sensitive to the Earth's inner core and lowermost mantle, consisting of PKPab-df and PcP-P differential traveltime residuals relative to a spherically symmetric Earth model. The posterior probability distributions for each spherical harmonic coefficient are calculated via Markov Chain Monte Carlo sampling; the uncertainty obtained for the coefficients thus reflects the noise present in the real data and the imperfections in the spherical harmonic expansion.

Strong, Multi-Scale Heterogeneity in Earth’s Lowermost Mantle

“…To investigate the effect of CMB topography on our results, we compare tomographic models derived from: 1) PKPab-PKIKP and 2) PcP-P datasets. The two independent datasets result in similar models of harmonic degrees 1 and 2 using the Bayesian hierarchical method of Muir and Tkalčić 19 , illustrating that large scale CMB topography does not have a dominant impact on travel times of these two datasets ( Figure S1 ). In the case that CMB topography had a strong degree 1 or degree 2 signal dominating the velocity heterogeneity, it would generate PKPab-PKIKP and PcP-P differential travel times of opposite sign, resulting in negatively correlated velocity anomalies between the two models, which we do not see.…”

Inversion of Structural Geology Data for Fold Geometry

“…This allows for the noise in the data to be properly accounted for without requiring subjective, and usually wrong, user input quantifying the uncertainty associated with the data. This technique has been applied extensively in the geophysical literature (e.g., Bodin, Sambridge, Rawlinson, & Arroucau, ; Bodin, Sambridge, Tkalcic, et al, ; Kolb & Lekić, ; Muir & Tkalčić, ). This is an improvement on Grose et al () where the data are smoothed using a radial basis interpolation scheme and no specific assumptions are made about the noise in the data.…”

“…Bayesian inference is a commonly used method for solving nonlinear problems with extensive application in geosciences particularly in solving inverse problems in geophysics (e.g. Kolb & Lekić, 2014;Mosegaard & Tarantola, 1995;Muir & Tkalčić, 2015). The main advantage of using Bayesian methods over standard statistical methods is in the ability to incorporate additional knowledge in the form of prior distributions.…”

“…When using structural observations to create a 3‐D model, or in this case characterize the fold geometry, we are actually interested in the combined misfit between the model and the structural observations, that is, how closely do we expect the model to fit the data. Rather than adding further subjectivity into the modeling process, uncertainty of structural observations can be incorporated using a hierarchical Bayesian approach where the data uncertainty is represented by an additional PDF, similar to Muir and Tkalčić (). The misfit ( σ ) between the observed fold rotation angle and the model (equation ) is represented using an uninformed hyperparameter, Jeffery's prior (Sivia, ).…”

Computer Model Calibration with Large Non-Stationary Spatial Outputs: Application to the Calibration of a Climate Model

“…In case the results have no statistically significant effects, then the number is sufficiently large. Muir and Tkalčić (2015) utilized the corrected Akaike information criterion AIC to choose an optimal maximum order of expansion for irregular data on the sphere in a hierarchical Bayesian setting. The results show that the third-fifth orders of expansion in SHs are generally a turning point from fast to slow reduction in AIC in terms of balancing explanatory power with simplicity (although not the smallest AIC).…”