Optimization of Cost Functions in Absolute Plate Motion Modeling

This paper proposes an improved objective function for the optAPM code, featuring a simplified hotspot cost formulation and pre-interpolation of trail data, to reduce modeling errors and enhance the accuracy of absolute plate motion reconstructions over geological timescales.

The Gist

Imagine the Earth's surface isn't a solid, unchanging shell, but rather a giant, slow-moving jigsaw puzzle made of massive plates. These plates drift, collide, and slide past each other over millions of years. Scientists want to build a "time machine" to see exactly where these puzzle pieces were 80 million years ago. This is called Absolute Plate Motion (APM) modeling.

The paper you provided is about two researchers, James and Steve, who took a leading computer program designed to run this time machine (called optAPM) and gave it a major software update. They found that the original program was using a slightly broken ruler to measure the past, and they fixed it to make the history much more accurate.

Here is the breakdown of their work using simple analogies:

1. The Problem: The "Noisy" Time Machine

To figure out where plates were in the past, scientists use clues. One of the best clues is Hotspots.

• The Analogy: Imagine a giant, stationary blowtorch (the hotspot) poking up from deep inside the Earth. As a piece of paper (the tectonic plate) slides over it, the blowtorch burns a hole. As the paper keeps moving, you get a trail of burn marks.
• The Goal: If we know where the burn marks are today, we can work backward to figure out how the paper moved.
• The Issue: The original computer program (optAPM) was trying to match the plate's movement to these burn marks, but it was doing the math in a clunky way. It was like trying to trace a winding road by only looking at the road every 10 miles and guessing what happened in between. This led to "jittery" and inaccurate predictions.

2. The Three Rules of the Game

The computer program tries to satisfy three different "rules" to make sure its guess is realistic:

1. The Hotspot Rule: The plate must move in a way that matches the burn marks (hotspot trails).
2. The Trench Rule: The edges of the plates (where they dive under each other) shouldn't be moving too wildly.
3. The Spin Rule: The entire Earth shouldn't be spinning around its own axis in a weird way just because of the model.

The program tries to find a path that satisfies all three rules at once. It does this by minimizing a "Cost Function."

• The Analogy: Think of this as a video game score. Every time the model makes a mistake (e.g., the plate moves too fast, or the burn marks don't line up), it gets "points" (cost). The goal is to get the lowest possible score.

3. The Flaw: The Broken Ruler

The authors discovered that the original program's way of calculating the "Hotspot Score" was flawed.

• The Old Way: The program looked at the burn marks, then tried to guess the plate's position between the marks. It was like trying to draw a smooth curve by only connecting the dots with jagged, straight lines. This caused the model to "wobble" and accumulate errors over time.
• The New Way: James and Steve realized they should do the opposite. Instead of guessing the plate's path between the dots, they should smooth out the burn marks themselves first.
• The Fix: They pre-interpolated the data. Imagine taking those jagged burn marks and smoothing them into a perfect, continuous line before you start measuring the plate's movement. This removes the "noise" and lets the computer see the true path.

4. The Results: A Smoother Ride

When they applied this new, smoother method:

• Less Wobble: The predicted path of the African plate (the reference point they used) became much straighter and more logical.
• Slower Speed: The original model thought the plates were zooming around at an average of 22 cm per year (about 8 inches). The new model corrected this to 2.6 cm per year (about 1 inch). This is a huge difference! The old model was essentially hallucinating a much faster, more chaotic world.
• Better Alignment: The new model's predictions for how the trenches and the Earth's spin behave matched real-world geology much better.

5. The Big Picture: Why This Matters

The authors aren't saying the original program was useless; it was actually a very sophisticated piece of code. However, they showed that how you set up the math matters just as much as the code itself.

• The Metaphor: Imagine a GPS navigation app. The original app had great hardware but a slightly buggy map algorithm that told you to take a detour every 5 miles. The new paper didn't build a new car; they just fixed the map algorithm. Now, the GPS gives you a direct, smooth route instead of a zig-zagging nightmare.

In summary: This paper is about cleaning up the data and refining the math behind a computer model of Earth's history. By smoothing out the "burn marks" (hotspots) before measuring the movement, the researchers created a much more reliable time machine, proving that the Earth's plates moved more steadily and slowly than previously thought.

Technical Summary

Here is a detailed technical summary of the paper "Optimization of Cost Functions in Absolute Plate Motion Modeling" by James Unwin and Steve Zhang.

1. Problem Statement

The reconstruction of Earth's tectonic plate motions over geological timescales (O(100) million years) is a complex inverse problem. While Relative Plate Motion (RPM) models describe how plates move relative to one another, Absolute Plate Motion (APM) models are required to understand dynamic topography, subduction flux, and net lithosphere rotation by fixing plates to a reference frame (e.g., the mantle).

The paper critiques the leading APM optimization code, optAPM (Tetley et al., 2019), arguing that its implementation contains suboptimal aspects, particularly in the formulation of its objective function. The authors identify two primary issues:

1. Unjustified Weighting: The combination of different physical constraints (hotspot trails, trench migration, net lithospheric rotation) uses ad-hoc scaling factors ($\sigma$) without rigorous statistical justification.
2. Flawed Hotspot Cost Function: The original method for minimizing the misfit between predicted plate positions and observed hotspot trails introduces error propagation. It optimizes interpolation points between model outputs rather than the model outputs themselves, leading to unrealistic plate velocities and erratic trajectories when isolated.

2. Methodology

The authors adapted the publicly available optAPM code to implement a more rigorous optimization framework. Their methodology involved:

• Base Model: Utilizing a comprehensive RPM model (Merdith et al., 2017) dating back to 1000 Ma, with reconstructions limited to the last 80 Ma due to hotspot data reliability.
• Optimization Algorithm: Replaced the original search method with the COBYLA algorithm (Constrained Optimization BY Linear Approximations) from the NLopt library. This algorithm is gradient-free and effective for complex, constraint-rich environments.
• Seed Propagation: To avoid local minima, the code propagates $10^5$seed Euler rotations uniformly distributed within a$60^\circ$ radius of the preceding optimal pole at 5 Myr intervals.
• Constraint Isolation: The authors tested the three primary constraints individually by setting their weights to infinity (effectively disabling them) to isolate the behavior of each cost function:
  1. Hotspot Trail Misfit ($HSm$)
  2. Global Trench Migration ($TMk$)
  3. Net Lithospheric Rotation ($\omega_{net}$)
• Proposed Reformulation:
  • Hotspot Cost Function: Instead of interpolating model outputs to match raw data points (which causes oscillation), the authors proposed pre-interpolating the hotspot trail data to create a smooth function $\Phi(x,y,t)$. The optimization then minimizes the distance between the raw model output and this pre-interpolated data at the specific reconstruction times.
  • Objective Function: They moved away from the original ad-hoc weighted sum toward a formulation using averages for constraints, aiming for a more transparent and mathematically sound calibration.

3. Key Contributions

• Identification of Error Propagation: Demonstrated that the original hotspot cost function formulation causes the model to fluctuate significantly around data points, leading to an accumulation of errors over time.
• Novel Hotspot Formulation: Introduced a simpler, more intuitive cost function that pre-interpolates geological data. This shifts the optimization focus to the model outputs themselves, ensuring smoother trajectories that align better with the physical reality of hotspot trails.
• Quantification of Uncertainty: Provided a systematic analysis of how small deviations in rotation angles compound over time, showing that unoptimized models can result in position spreads of ~35° in latitude over 1000 Myr.
• Isolation of Constraints: Revealed that the original hotspot-only model predicted physically impossible plate velocities (22 cm/yr), whereas the trench migration and net lithospheric rotation constraints suggested much lower, realistic velocities (2 cm/yr).

4. Results

The modified model yielded significant improvements in physical plausibility and statistical consistency:

• Plate Velocity Reduction: The average absolute velocity of the African plate dropped from 22.1 cm/yr (original optAPM) to 2.6 cm/yr (modified model), bringing it in line with geodynamic expectations.
• Angular Variation: The angular variation over 5 Myr periods decreased from 57.5° to 17.0°, indicating a much smoother and more stable reconstruction.
• Trench Migration: The median trench migration velocity was reduced from 5.42 cm/yr to 0.87 cm/yr, with a standard deviation dropping from 8.23 cm/yr to 0.40 cm/yr. This aligns with geodynamic models suggesting low but positive migration rates.
• Net Lithospheric Rotation (NLR): The NLR rate decreased from 2.78°/Myr to 0.26°/Myr. While slightly higher than some geodynamic upper bounds, it falls within the range (0.10–0.50°/Myr) typical of historical hotspot-based models.
• Path Accuracy: In a full integrated model, the average deviation from the expected path improved from 379 km (original) to 315 km (modified).

5. Significance

This paper provides a critical refinement to the field of paleogeographic reconstruction. By correcting the mathematical formulation of the cost functions, the authors have:

1. Enhanced Reliability: The new model produces reconstructions that are statistically more robust and physically plausible, reducing the "noise" introduced by the optimization process itself.
2. Validated Constraints: It demonstrates that while hotspot data is crucial, it must be integrated with other constraints (trench migration, NLR) using a mathematically consistent framework to avoid bias.
3. Future Framework: The proposed pre-interpolation technique and the use of COBYLA offer a blueprint for future APM studies, ensuring that the "optimal" solution is not an artifact of the cost function's construction but a true reflection of geological history.

In conclusion, Unwin and Zhang argue that the "optimal" transport approach of optAPM is powerful but requires careful formulation of its objective function to achieve its full potential in modeling Earth's deep-time tectonic evolution.