Continuous and Differentiable Neural Network Representations of the Surfzone Sea-Surface from Lidar Point Clouds

The ocean's surface is never still. It is a restless, shifting landscape of rolling swells, crashing breakers, and churning foam, constantly reshaping itself as energy moves from the deep sea toward the shore. For scientists who study these waves, understanding exactly how the water moves is essential. They need to know not just the height of a wave, but how fast it is traveling, how steep its face is, and how it changes shape as it breaks. Traditionally, researchers have used buoys that bob in one spot or cameras that take pictures from a distance. While useful, these tools often struggle to capture the full, three-dimensional complexity of a breaking wave, especially in the turbulent zone where the water crashes onto the beach. To get a clearer picture, scientists have turned to lasers mounted on drones, which can scan the water's surface and create a cloud of millions of tiny points, each marking a specific location in space and time. However, this raw data is messy and scattered; to make sense of it, researchers usually have to force the points into a rigid grid, like fitting irregular puzzle pieces into a square frame, which can smooth out important details or introduce errors.

A team of researchers at the Scripps Institution of Oceanography has found a different way to handle this data, one that avoids the grid entirely. Instead of forcing the laser points into a boxy structure, they used a type of artificial intelligence called a neural network to learn the shape of the ocean surface directly from the scattered points. Think of this network as a student who is shown a million scattered dots and asked to draw a smooth, continuous line that connects them all, not just at the dots, but everywhere in between. The researchers tested this method over a rocky coast in Monterey, California, where a drone hovered to scan massive waves that were four meters high. The result was a digital model of the sea that is smooth and continuous, allowing the scientists to calculate the speed and slope of the water at any single point, even in places where the drone did not collect a direct measurement.

The experiment took place near a rocky headland known as China Rock, where the ocean floor drops off steeply and waves break with significant force. A drone equipped with a specialized laser scanner hovered at an altitude of 33 meters, firing beams of light at the water surface ten times every second. These beams bounced off the water and returned to the drone, creating a massive collection of data points that mapped the height of the waves in three dimensions. The researchers focused on a 60-second window of this data, capturing the moment a large wave rolled in, steepened, and eventually crashed. In this short burst of time, the drone gathered roughly 4.5 million individual points, but these points were not spread out evenly; they were densest directly under the drone and became sparser toward the edges of the scan.

To turn this scattered cloud of points into a useful picture of the ocean, the team trained a neural network, a computer program designed to recognize patterns. They fed the network the coordinates of every single point and the height of the water at that spot. The network's job was to find a mathematical function that could predict the water's height at any location and time, not just where the laser had hit. Unlike traditional methods that would average the points into a grid, this network learned to create a smooth, flowing surface that passed through the data points without being forced into a rigid structure. The researchers found that this approach worked remarkably well. The digital model it produced matched the actual wave shapes seen in the laser data, capturing the rise of the wave crest and the steep drop of its face. It also successfully filtered out the chaotic noise of sea spray and splashing water, which often confuses other measurement tools, leaving behind a clear picture of the main wave structure.

One of the most significant findings was that this method remained accurate even in areas where the data was thin. Because the network learned the overall shape of the wave, it could estimate the water's height in gaps where the drone's laser had few or no points. When the researchers tested the model against data it had not seen during training, the errors were small, comparable to the natural limitations of the laser instrument itself. They also tested the reliability of the method by training two separate networks on the exact same data. The two models produced nearly identical results, differing by only a few centimeters in most places. However, in the very sparsest areas where the drone had collected almost no data, the two models began to drift apart slightly, suggesting that while the method is robust, it still relies on having enough information to learn the wave's shape.

The true power of this new approach lies in its ability to calculate how the wave is changing. Because the digital model is a smooth, continuous function, the researchers could easily calculate how fast the water was moving up and down, or how steep the wave face was at any specific moment. In the 60-second window they studied, they were able to track a wave crest moving toward the shore at a speed of about 8.75 meters per second, a value that matched theoretical predictions for waves of that size and depth. They could also see how the wave changed shape as it moved, noting how the crest grew taller and steeper before breaking. This level of detail is difficult to achieve with traditional grid-based methods, which often struggle to calculate these rates of change accurately, especially on the steep, rapidly changing faces of breaking waves.

The researchers acknowledge that this method is not without its challenges. Training the neural network took about 30 minutes on standard computer hardware for just 60 seconds of data, which is significantly longer than the time it takes to simply sort data into a grid. If scientists wanted to analyze an hour of wave data, they would need to train the network for many hours, or run multiple computers in parallel. Additionally, the method works best when there is a decent amount of data to learn from; in areas where the water is very clear and the laser returns very few points, the model's accuracy might decrease. Despite these limitations, the study demonstrates that using neural networks to represent the ocean surface offers a powerful new tool. It provides a continuous, smooth, and highly detailed view of the sea that can be analyzed at any point in time and space, opening the door to a deeper understanding of how waves behave as they crash onto the shore.