Optimization of Cost Functions in Absolute Plate Motion Modeling

Abstract

We consider the implementation of optimization techniques within the study of tectonic plate motion. Specifically, we examine the optimization underlying optAPM, a leading code for modeling absolute plate motion. We highlight that modifications in the construction of the objective function, composed of individual cost functions, can improve modelling performance. In particular, we propose a simpler and more intuitive formulation of the hotspot cost function. A key part of the new hotspot analysis is the pre-interpolation of hotspot trail data, crucial geological markers for validating absolute plate motion over O(100) Myr timescales. By reducing the propagation of modeling errors, our refined model provides more precise reconstructions of historical plate movements. Our modified hotspot modelling improves the accuracy and reliability of the optAPM outputs.

Figures (8)

• Figure 1: A schematic visualization of a reconstruction tree. Green nodes and edges represent information present in RPM models. The pink node and its corresponding edge represents the additional reference frame provided by the APM model, linking the African plate to the mantle.
• Figure 2: Predicted motion paths when the point ($0^\circ$N, $27.5^\circ$E) in the African plate is reconstructed back to 80 Ma for 36 different models, each incorporating a unique subset of 7 hotspots from the 9 total hotspots utilized in optAPM. Color represents age in millions of years, and individual line segments between adjacent points represent absolute motion in a 5 Myr interval period. The aim of the plot is to highlight the compounding effect of small deviations.
• Figure 3: Reconstruction path of a reference point in Laurentia over the past 1000 Ma, derived from models with introduced variability. The intermediate rotation angle $\omega$ is adjusted at 5 Myr intervals using the function $\omega (1+0.1z)$, where $z$ is a random variable derived from a Gaussian distribution centered at $0$ with standard deviation $1$. The red lines depict the latitudinal and longitudinal paths from the unoptimized model, while the dark blue lines show the same for the optimized model. The shaded blue areas represent the variability among 1000 perturbed model.
• Figure 4: Workflow diagram illustrating the full optimization process (emulating the diagramatic style in Muller). The code optimizes absolute plate motion at 5-million-year (5Myr) intervals, dating back to 80 Ma. The African plate is used as the reference throughout the procedure, chosen for its central position and close relationship with other major plates. At each interval, the code propagates a collection of 105 seed Euler rotations, uniformly distributed within a 60$^\circ$ radius of the preceding optimal Euler pole. For each seed rotation, the algorithm creates a temporary rotation model. This model is then used to calculate the value of the objective function, guiding the algorithm through iterative adjustments toward a local minimum. The process then determines a global minimum from these local minima, incrementally refining the plate motion model.
• Figure 5: Predicted motion paths when the point at ($0^\circ$N, $27.5^\circ$E) in the African plate is reconstructed back to 80 Ma according to three different models, each isolating a different model constraint: trench migration (pink circles), net lithospheric rotation (green triangles), and hotspot trail misfit (orange squares). Additionally, the red plus signs indicate the integrated model that considers all three constraints simultaneously. The models operate on 5 Myr intervals, with varying colors denoting the age of the reconstructed position.