Applying Machine Learning to Characterize and Transport the Relationship Between Seismic Structure and Surface Heat Flux

Geothermal heat flux beneath the Greenland and Antarctic ice sheets is an important boundary condition for ice sheet dynamics. Subglacial heat flux is rarely measured directly, so it has been inferred indirectly from proxies (e.g. seismic structure, magnetic Curie depth, surface topography). We seek to improve understanding of the relationship between heat flux and one such proxy---seismic structure---and determine how well heat flux data can be predicted from the structure (the \emph{characterization} problem). We also seek to quantify the extent to which this relationship can be transported from one continent to another (the \emph{transportability} problem). To address these problems, we use direct heat flux observations and new seismic structural information in the contiguous US and Europe, and construct three Machine Learning models of the relationship across a hierarchy of model complexity (Linear Regression, Decision Tree, Random Forest). The more complex models fit smaller scale variations in heat flux. We compare the models in terms of model interpretability, accuracy to predict heat flux, and transportability from one continent to another. To evaluate model accuracy, we divide data on the same continent into training and validation datasets, and then validate the model (trained from the training data) with validation data. We measure model transportability by cross-validating the US-trained models against European heat flux, and vice versa. We find that the Random Forest and Decision Tree models are the most accurate, while the Linear Regression and Decision Tree models are the most transportable. The Decision Tree model can uniquely illuminate the regional variations of the relationship between heat flux and seismic structure. From the Decision Tree model, uppermost mantle shear wavespeed, crustal shear wavespeed and Moho depth together explain about half of the observed heat flux variations in both the US ($r^2 \approx 0.6$ (coefficient of determination), $\mathrm{RMSE} \approx \hfu{8}$ (Root Mean Squared Error)) and Europe ($r^2 \approx 0.5, \mathrm{RMSE} \approx \hfu{13}$). Uppermost mantle wavespeed is a much stronger predictor than the other two variables combined. Transporting the US-trained models to Europe reveals that the geographical distribution of heat flux can be reasonably predicted ($\rho = 0.48$ (correlation coefficient)), but the absolute amplitude of the variations cannot ($r^2 = 0.17$), similarly from Europe to the US ($\rho = 0.66, r^2 = 0.24$). We attribute the transportability deterioration to differences between the continents in seismic structural imaging data and parameterization, and crustal radiogenic heat production. Despite these issues, our method has the potential to improve the reliability and resolution of heat flux inferences across Antarctica. Furthermore, our validation and cross-validation methods can be applied to heat flux proxies other than seismic structure, which may help resolve inconsistencies between existing subglacial heat flux inferences using different proxies.