Retrieving the Seismic Moment Release Rate of Small Seismic Events Using the Adjoint Method - Theory and Numerical Validation

When the Earth shakes, even a tiny tremor carries a hidden story about how the ground broke. Seismologists have long been able to pinpoint where an earthquake started and how much energy it released, but understanding exactly how that energy was released over time has remained a difficult puzzle. This story is told by something called the source time function, a record of the rupture's heartbeat from its first crack to its final stop. Knowing this rhythm is crucial because it reveals the speed of the rupture, the direction it traveled, and the stress changes that occurred deep underground. However, extracting this rhythm from the messy waves recorded at the surface is notoriously hard. Traditional methods either rely on finding a smaller, similar earthquake to use as a reference—a process that can be subjective and imprecise—or they attempt to run massive, computer-heavy simulations that are too slow for routine use.

In a new study, researchers Wojciech Dębski and Kamil Waśkiewicz from the Institute of Geophysics at the Polish Academy of Sciences have proposed a different way to listen to these hidden rhythms. They developed a technique based on the simple physical idea that sound and seismic waves can travel backward in time just as they travel forward. By taking the recordings from the surface, flipping them in time, and sending them back into the ground using a computer model, the waves naturally refocus on the spot where the earthquake began. As these reversed waves converge, they reconstruct the original pattern of energy release. The team tested this idea using computer simulations of small, human-caused earthquakes, such as those triggered by mining, using noise-free synthetic data. Their results show that this method can successfully recover the detailed timing of the rupture without needing complex mathematical inversions or finding a reference earthquake. It offers a faster, more robust way to understand the mechanics of small seismic events, turning a difficult mathematical problem into a straightforward physical process.

The core of this new approach relies on a fundamental property of wave physics: time reversibility. In a perfectly elastic medium, where energy is not lost to heat or friction, the equations that describe how waves move work the same way whether time is moving forward or backward. Imagine a video of a stone dropping into a pond; if you play the video in reverse, the ripples travel back inward and the stone leaps out of the water. While you cannot actually reverse time in the real world, the mathematics allows scientists to simulate this process. The researchers used this concept to create an algorithm they call TRSTF. The process begins by taking the seismic waves recorded at various stations on the surface. These waves are then reversed in time and fed back into a computer model of the Earth's crust. As the reversed waves travel through the model, they bounce off boundaries and interact with the complex geology, eventually converging back at the original source location.

When these reversed waves meet at the source, they interfere with each other. At the exact location of the earthquake, the waves line up perfectly, creating a strong, focused signal. At any other location, the waves arrive out of step and cancel each other out, leaving only a faint noise. This focusing effect is the key to the method. The signal that emerges at the focal point is a direct reconstruction of the source time function, the very rhythm of the earthquake's energy release. The researchers found that this technique works remarkably well in their noise-free synthetic experiments, even when the model of the Earth's interior is not perfectly known. In their computer experiments, they simulated three different scenarios to mimic real-world monitoring setups: one where sensors were placed near the level of the mining activity, one where sensors were on the surface above the event, and a third with a different surface arrangement. In every case, the method successfully recovered the shape of the original rupture pattern.

The study also explored how errors in the model of the Earth's speed of sound affect the results. In the real world, geologists do not know the exact speed of seismic waves at every point underground; they rely on estimates. The researchers tested their method by introducing random errors and block-like variations into their computer models, simulating a situation where the velocity model was only about 90 percent accurate. The results indicated that the use of inaccurate velocity models introduces systematic errors into the reconstructed source time function. The method proved more sensitive to organized, block-like errors than to random, scattered ones. However, despite these errors, the recovered signal remained recognizable and provided useful information, demonstrating that the method retains utility even with imperfect models, though the accuracy is not immune to model inaccuracies.

One of the most striking findings was how the arrangement of the sensors influenced the quality of the result. The researchers discovered that having sensors closer to the boundaries of the area being studied actually helped the reconstruction. This is because the reflections from these boundaries act like additional virtual sensors, providing more information to the system. In one of their test cases, where the sensors were placed near the top edge of the model, the reconstructed signal was sharper and more accurate than in the other setups, even though the raw data looked more distorted due to interference. This counterintuitive result suggests that the method benefits from the complexity of the wavefield, using reflections to its advantage rather than treating them as noise.

The researchers also measured specific characteristics of the reconstructed signals, such as how quickly the rupture started and how long it lasted. They found that while the reconstructed signals were slightly broader than the originals, likely due to the limited number of sensors used in the simulation, the overall shape and timing were preserved. The ability to estimate the duration of the rupture is vital because it helps scientists calculate the dynamic stress drop, a measure of how much stress was released during the event. This information is essential for understanding the mechanics of induced seismicity, such as earthquakes caused by mining or fluid injection. The study showed that the new method could estimate these parameters with an accuracy comparable to, or better than, existing techniques, but with far less computational effort.

Unlike full-waveform inversion, which requires running thousands of simulations and complex optimization loops to find the best fit, this new method requires only a single pass of the reversed waves through the model. This makes it computationally efficient, capable of running on standard desktop computers for small models or large supercomputers for regional scales. It also avoids the need to select a reference earthquake, a step that can introduce human bias and uncertainty in traditional methods. The researchers emphasize that their approach is particularly well-suited for small seismic events, where the source is small enough to be treated as a single point. They acknowledge that the method assumes the ground behaves like a perfect elastic material, meaning it does not absorb energy, and that the source is small compared to the wavelength of the waves. These are reasonable assumptions for many mining-related events, but the researchers note that further study is needed to understand how the method performs in more complex, real-world environments with significant energy loss.

The paper concludes by highlighting the potential of this technique to become a standard tool in seismology. By providing a fast, reliable, and physically intuitive way to retrieve the source time function, it opens the door to routine analysis of small seismic events that were previously too difficult to characterize in detail. The researchers suggest that this method could be integrated into existing monitoring systems to provide immediate insights into the nature of induced seismicity. While the current study was limited to computer simulations, the theoretical foundation is solid, and the results are promising. The next step, according to the authors, will be to test the method on real seismic data and to refine the understanding of its limitations and uncertainties. For now, the study stands as a proof of concept that time reversal can be a powerful lens for looking back into the Earth's past, revealing the hidden story of how the ground breaks.