Fractional-Order Viscoelastic Modeling of Seismic P-, S-, and Rayleigh-Wave Attenuation: Marmara Seismotectonic Context, Fault Geometry, and a Reproducible Strong-Motion Framework

Deep beneath the surface of the Earth, the ground does not behave like a simple, solid block of rock. When an earthquake strikes, the energy travels outward as waves, but as these waves move through the crust, they lose energy and change shape. This process, known as attenuation, is the reason a shaking ground feels different at a distance from the source than it does right next to the rupture. For decades, scientists have tried to describe this energy loss using standard rules of physics, treating the Earth's crust as either a perfectly elastic spring or a simple, sticky fluid. However, the real world is often more complex than these two extremes allow. The ground is a messy mixture of rock, fluid, and cracks that remembers its past movements in a way that standard physics struggles to capture. This is where a branch of mathematics called fractional calculus becomes useful. Instead of using whole numbers to describe how materials react, fractional calculus allows for values in between, offering a way to model the "memory" of the rock with far fewer assumptions.

In a new study focused on the Sea of Marmara, a region of intense geological activity just south of Istanbul, researchers Taylan Demir and Atakan Koçyiğit have built a more precise mathematical framework to describe how seismic waves fade away. The Marmara Sea sits on top of the North Anatolian Fault, a massive crack in the Earth's crust where tectonic plates grind past one another. Because this fault runs directly beneath a major city, understanding exactly how earthquake waves travel and weaken in this specific area is critical for engineering safer buildings and predicting the strength of future shaking. The researchers set out to test whether a fractional model, which accounts for the complex, history-dependent behavior of the rock, could explain the fading of seismic waves better than traditional models. They did not attempt to predict when the next big earthquake would happen; instead, they focused on how to accurately reconstruct the shaking that occurs after an event is recorded.

The team began by mapping the geological stage. They examined the specific segments of the fault that run under the sea, including the Tekirdağ, Central Basin, and Kumburgaz branches, and looked at the history of earthquakes in the region, such as the major 1999 İzmit quake and the smaller 2019 Silivri event. This geological context was essential because the math they were developing needed to be grounded in the actual shape and structure of the fault zone. They then introduced a new way of describing the rock's behavior. Imagine the Earth's crust not as a simple spring that snaps back instantly, but as a material that slowly relaxes over time, holding onto the memory of how it was stretched or squeezed. The researchers used a mathematical tool called the Caputo derivative to describe this memory effect. This tool allowed them to create a model that connects the physical properties of the rock to the way waves lose energy as they travel.

Using this new model, the researchers derived how three different types of seismic waves—P waves, which are the fastest and arrive first; S waves, which shake the ground side-to-side; and Rayleigh waves, which roll along the surface—should behave. They proved mathematically that their model always results in a loss of energy, which is a necessary condition for any realistic description of an earthquake. If a model suggested that waves could gain energy as they traveled, it would be physically impossible. Their equations showed that the fractional approach produces a smooth, continuous curve of energy loss that matches the way real rocks behave across a wide range of frequencies. This is a significant improvement over older models, which often struggle to fit the data across different frequencies without becoming overly complicated.

To test if this theory actually works, the team ran a controlled experiment using computer-generated data. They created a synthetic earthquake scenario where they knew the exact rules of how the waves should fade. They then asked their fractional model to figure out those rules based only on the wave patterns. The results were striking. The fractional model successfully recovered the correct parameters with very high accuracy, reducing the error in its predictions by nearly 87 percent compared to a traditional model that used whole-number rules. In the traditional model, the researchers had to force the numbers to fit, resulting in a poor match to the actual wave behavior. The fractional model, by allowing for that "in-between" memory effect, captured the physics of the wave loss much more naturally.

The study also looked at how this framework could be applied to real-world data from the Marmara region. The researchers identified the 2019 Silivri earthquake as a perfect candidate for a first real-world test. They outlined a clear plan for how engineers and seismologists could download actual recordings from this event and use their new equations to reconstruct the shaking intensity. This would allow them to calculate precise measures of ground motion, such as the maximum acceleration or velocity, which are the numbers architects use to design earthquake-resistant buildings. The authors were careful to state that this tool is for analyzing what has already happened, not for predicting the future. It does not tell us when the next big rupture will occur, but it provides a much sharper lens for understanding the violence of the shaking once it happens.

By combining a rigorous mathematical proof with a clear geological map and a plan for real data analysis, this paper moves the conversation from abstract theory to practical application. It shows that the fractional approach is not just a mathematical curiosity but a viable, more accurate way to describe the complex reality of seismic waves in the Marmara Sea. The work bridges the gap between the messy, layered geology of the fault zone and the clean, predictable equations needed for engineering safety. While the full calibration against thousands of real earthquake records is a task for the future, this study has laid the necessary groundwork, proving that the method works in theory and in simulation, and is ready to be tested against the real ground beneath Istanbul.