The research, published in the Journal of Geophysical Research: Machine Learning and Computation, introduces a technique called Sparse Discrete Empirical Interpolation Method, or S-DEIM. It was developed by Mohammad Farazmand, associate professor of mathematics at NC State, together with graduate student Louisa Ebby and a team of undergraduate researchers from institutions across the United States. According to the authors, the method reconstructs high-resolution global sea surface temperature fields from as few as 100 in situ observations, a sampling density that amounts to just 0.2 percent of the full spatial grid. Even with so little direct information, more than 90 percent of the S-DEIM estimates landed within one degree Celsius of the true values in the team’s benchmark tests.

The problem the researchers set out to solve is one that oceanographers and climate scientists have wrestled with for decades. Federal agencies such as the National Oceanic and Atmospheric Administration have long relied on combinations of complicated differential equations to estimate temperatures across the vast unmonitored stretches of ocean. These model-based approaches are rigorous, but they are computationally demanding and depend on physical assumptions that may not hold perfectly across every ocean basin and season. Meanwhile, the explosive growth of machine learning has produced an alternative family of tools that can learn patterns directly from data, but at a steep price: deep neural networks often require hours of training on powerful hardware and enormous quantities of data before they can make a single useful prediction.

Somewhere in between sits the Discrete Empirical Interpolation Method, an established technique that the new work builds upon. Rather than modeling the ocean purely from physical first principles, DEIM specifies a basis, essentially a compact library of spatial patterns that jointly encode the structure of the temperature field being estimated. Given a handful of actual measurements, the method selects which of these patterns to activate and with what weights, producing a full-field estimate from sparse data. The approach is elegant and efficient, but it has a well-known weakness. When the available observations are truly sparse, as they typically are in the open ocean, the estimates it produces degrade considerably, because the method struggles to determine which patterns best explain a scattering of disconnected data points.

The NC State team’s insight was to bring historical information to bear on precisely this weakness. S-DEIM augments the classical framework with a so-called kernel vector, a quantity for which no closed-form mathematical formula exists, estimated instead from the long historical record of observations. In practice, the reconstruction produced by S-DEIM consists of two complementary terms. The first is computed from instantaneous in situ measurements using empirical interpolation, anchoring the estimate to what sensors are actually reporting right now. The second is learned from the historical time series using recurrent neural networks, which are particularly well suited to capturing how patterns in the data evolve over time. The marriage of the two allows the method to lean on decades of accumulated knowledge about ocean behavior while still respecting the fresh, if sparse, observations streaming in.

To train and test the method, the researchers used NOAA’s weekly high-resolution sea surface temperature dataset spanning 1989 through 2021, a record covering more than three decades of global ocean variability. The final year of the record, from January 2022 through January 2023, was withheld from the models entirely and reserved as a blind test. The team then asked S-DEIM, the classical DEIM method and a high-performing convolutional neural network to predict the sea surface temperatures for that unseen year, and compared their outputs against the actual historical data. This head-to-head design provided a rigorous measure of how each technique would perform under realistic conditions, where the future is genuinely unknown and the data available is sparse.

The results were striking. S-DEIM proved roughly 40 percent more accurate than DEIM, a substantial leap over the method it directly extends. More surprisingly, it also edged out the convolutional neural network, delivering estimates about 2 percent more accurate than the best AI model in the comparison. The efficiency gap was even more dramatic. Training the recurrent neural network at the heart of S-DEIM took approximately one minute, a one-time offline step, whereas the convolutional neural network required an hour and a half to train. Once trained, S-DEIM generates its full reconstructions in less than a second, making the approach practical for operational settings where forecasts must be produced continuously and quickly.

The method also displayed a robustness that matters greatly for real-world deployment. Sensor networks in the ocean are rarely arranged optimally; instruments drift, fail and are deployed wherever ships happen to travel. When the researchers distributed the sensors randomly rather than in favorable positions, the reconstruction error deteriorated by only 1 to 2 percent, suggesting that S-DEIM does not depend on carefully engineered measurement placements to deliver its accuracy. That resilience, combined with its computational thrift, makes the method attractive for agencies monitoring the ocean with limited and unevenly distributed instrumentation, and it opens the door to assimilating streaming observations in near real time.

The implications extend in two directions at once. In the short term, accurate and rapidly computed sea surface temperature fields feed directly into weather forecasting, where ocean conditions influence storm tracks, intensity and precipitation patterns on timescales of days to weeks. In the longer term, the same fields underpin climate models that track how the ocean absorbs and redistributes heat over decades. A tool that can deliver high-resolution temperature reconstructions from a sliver of the usual data, at a fraction of the computational cost, could meaningfully lower the barrier to both endeavors. The work also grew out of a National Science Foundation supported Research Experience for Undergraduates, with co-authors Cassidy All of the University of Colorado Boulder, Kevin Ho of Mississippi State University, Maya Magnuski of Bard College and Christopher Nicolaides of Indiana University contributing to the study alongside the NC State team.

Farazmand and his colleagues emphasize that this is not the end of the road. The team hopes to continue improving the accuracy of S-DEIM, and the framework’s flexibility suggests room for refinement, from richer historical models to better handling of measurement noise. For now, the study makes a compelling case that when data is scarce, a thoughtfully designed hybrid of classical interpolation and lightweight learning can beat brute-force deep learning on its own terms. In a field where every degree matters and every observation counts, S-DEIM offers a reminder that sometimes the smartest algorithm is not the biggest one, but the one that knows how to make the most of very little.

Subject of Research: A sparse-data interpolation method for rapidly reconstructing global sea surface temperatures from limited in situ observations.

Article Title: New method estimates sea surface temps quickly and accurately

Article References: New method estimates sea surface temps quickly and accurately. (n.d.). Original publication

Image Credits: AI Generated

DOI: Not provided

Keywords: sea surface temperature, S-DEIM, data assimilation, machine learning, recurrent neural networks, NOAA, climate modeling, oceanography, sparse data, empirical interpolation, weather forecasting, North Carolina State University

Phytoplankton are microscopic organisms that form the foundation of marine food webs and contribute more than half of the oxygen produced by Earth’s biosphere. Their distribution, however, is far from smooth. Ocean currents, eddies, internal waves, sunlight, nutrients, temperature and grazing by zooplankton can gather them into transient patches that shift through space and time. These structures may extend horizontally across kilometers but vary sharply with depth, sometimes forming narrow layers below the surface. Chlorophyll A is useful because its concentration generally tracks the amount of phytoplankton present, although the relationship depends on species and environmental conditions. Satellite ocean-color measurements can reveal broad surface patterns, but clouds, suspended particles and dissolved organic matter can obscure the signal. More importantly, satellites cannot reliably see blooms that begin deep underwater. An AUV carrying a fluorometer can instead measure chlorophyll directly while moving through the water column.

The new system treats the changing chlorophyll field as a four-dimensional problem: north-south position, east-west position, depth and time. Its statistical engine is a Gaussian random field, a mathematical model that represents how measurements at nearby locations are related. The researchers modeled the logarithm of chlorophyll A rather than the raw concentration, a transformation that helps accommodate strongly skewed biological data and allows the modeled quantity to vary across the full real-number line. Before the mission begins, the model is given a depth-dependent mean and correlation scales describing how quickly chlorophyll patterns change laterally, vertically and over time. The correlations in the horizontal plane and depth follow Matérn functions, which can represent moderately smooth environmental variation, while the time correlation follows an exponential form suited to less predictable fluctuations. Every new fluorometer reading updates the model, changing both the predicted chlorophyll level and the uncertainty at nearby unvisited locations.

The vehicle then evaluates possible future trajectories using a decision rule called expected improvement. At each candidate point, the algorithm estimates the probability that the vehicle will find a chlorophyll value higher than the best one observed so far, as well as the size of the potential gain. Mathematically, if the predicted log-chlorophyll value has mean (m), uncertainty (v), and the current maximum measurement is (x_{text{max}}), expected improvement combines the term ((m-x_{text{max}})Phi((m-x_{text{max}})/v)) with an uncertainty term involving the normal probability density. The result rewards both exploitation—returning to areas likely to contain intense chlorophyll—and exploration, where uncertainty is large enough that a previously unknown hotspot might be discovered. This balance is crucial. A strategy based only on predicted intensity can become trapped around a local maximum, while a strategy based only on variance may spend too little time sampling the biologically important regions.

Path selection is divided into two linked stages designed to match the limitations of an underwater robot. First, while near the surface, the AUV chooses among seven possible lateral directions arranged like the spokes of a spider web. It selects the direction whose prospective transect offers the greatest expected improvement. The second stage chooses depths along that route. The vehicle’s diving angle limits how rapidly it can move vertically; in the Norwegian trials, a 10-degree limit allowed roughly 17 meters of vertical movement for every 100 meters traveled laterally. Nine possible depth profiles were evaluated during each transect. The vehicle also returned to the surface after 800 meters or 15 minutes, whichever came first, so that it could obtain a GPS position and correct accumulated navigation error. This surface reset sacrifices some sampling time, but it prevents uncertainty in dead-reckoned position from growing too large, particularly in strong currents.

A major engineering challenge was making the calculations fast enough for a relatively small onboard computer. Conventional spatio-temporal models often rely on dense grids covering an entire survey area. Updating a Gaussian model on such a grid can require matrix operations whose computational cost rises approximately as the cube of the number of conditioning measurements. The researchers therefore used a grid-free design. The AUV retained observed locations and values rather than maintaining a permanent high-resolution map, and it generated only the small sets of points needed to compare candidate paths immediately ahead. The system also thinned the stored data when the mission became computationally demanding, removing redundant nearby observations and measurements far from the vehicle. Because spatially distant data have limited influence on local predictions—a property related to the screening effect in kriging—this reduction was designed to preserve useful accuracy while keeping response times manageable. The onboard platform was a Light Autonomous Underwater Vehicle equipped with an NVIDIA Jetson TX2 and integrated with robotic software used to exchange sensor and navigation data.

Before going to sea, the team tested the strategy in 100 simulated chlorophyll landscapes, each covering a 4-by-4-kilometer area and extending to 75 meters depth. The virtual vehicle had four hours to survey, traveled at 1 meter per second and periodically surfaced. Expected improvement was compared with maximum variance, maximum expected intensity, probability of improvement and a systematic lawnmower pattern. The principal test classified a hotspot as a location above the 90th percentile of chlorophyll values across the simulated field and mission. Expected improvement increased the fraction of time spent in these top-concentration areas more rapidly than the other adaptive methods and eventually stabilized at the highest level. It also explored hotspot clusters more effectively than the strategy based on maximum expected intensity, which sometimes remained focused on one region after finding a promising signal. Maximum variance visited slightly more clusters overall, but did not examine them as thoroughly. The results indicate that the best strategy depends on the goal: broad uncertainty reduction across an entire field favored systematic or variance-driven paths, whereas locating and characterizing intense patches favored expected improvement.

The field demonstration took place in the Frohavet region near Mausund, roughly 100 kilometers from Trondheim, during two missions on June 6 and 7, 2024. The vehicle used a RBR Tuner Cyclops7 fluorometer to guide its decisions and carried additional instruments for offline comparison, including a conductivity-temperature-depth sensor and a SilCam imaging system for zooplankton. After an initial dive to 70 meters, the adaptive controller directed the AUV mainly toward depths between about 10 and 30 meters, where chlorophyll readings were highest. On the first day, the strongest layer occupied approximately 0 to 25 meters; on the second, it was centered slightly deeper, around 10 to 30 meters, and appeared narrower. The observations also revealed a sharp transition in temperature and salinity near 40 meters, consistent with a seasonal thermocline separating warmer, fresher surface water from colder, saltier water below. Chlorophyll declined rapidly beneath the well-mixed upper layer, suggesting that the robot was tracking a biologically distinct near-surface structure rather than simply responding to a gradual vertical trend.

The researchers also found evidence that phytoplankton-rich water was associated with concentrations of the copepod Calanus finmarchicus, a common zooplankton grazer and an important food source for larger marine animals. The SilCam photographed a small illuminated volume of water at one frame per second, and images were later segmented and classified with a convolutional neural network. The clearest relationship appeared at depths of roughly 10 to 25 meters and at chlorophyll readings around 2 to 4 in the study’s measurement scale, where images contained more suspected Calanus individuals. The result is consistent with copepods gathering where phytoplankton is abundant, although it does not yet provide a calibrated estimate of population size or biomass. Motion blur caused by the AUV’s operating speed made species identification difficult, and copepods may have avoided the vehicle’s hydrodynamic disturbance. The authors therefore describe the relationship as suggestive rather than definitive. Future versions could assimilate chlorophyll, temperature, salinity and image-derived plankton data simultaneously, allowing robots to seek regions that satisfy several biological objectives at once. For now, the work shows how an underwater robot can turn sparse observations into an adaptive biological survey, seeking not merely to pass through the ocean but to follow its most important living signals.