Hundreds of Ocean Colors, One Simple Equation: How AI Is Rewriting the Rules of Ocean Observation

The quiet revolution in ocean science right now is happening at wavelengths we can't see. NASA's PACE satellite, launched in early 2024, photographs the ocean in hundreds of shades of color instead of the usual handful. And a new study asks a deceptively simple question: does all that extra color actually buy us better estimates of how much plant life is drifting in the world's seas?

The answer, from a team led by Abed Hammoud at Princeton, is a yes with real nuance attached. Using an AI technique called symbolic regression, they sifted through millions of possible equations to find compact formulas that turn satellite color into chlorophyll-a — the green pigment that marks phytoplankton, the microscopic plants that produce roughly half the oxygen on Earth. Their best formula nailed chlorophyll estimates with a root mean square error of 0.253 in log space, against 0.307 for the best conventional algorithm running on the same data (Hammoud et al., 2026). That's an almost 12% improvement in skill. And the equation that achieved it uses only a handful of wavelengths plucked from a possible 119.

The striking thing is that the machine's best work looks almost nothing like the algorithms oceanographers have used for three decades.

The Science

Chlorophyll-a concentration is the workhorse metric of ocean biology. Phytoplankton absorb blue light and reflect green, which means the ratio of blue to green light bouncing off the ocean tells you roughly how much chlorophyll is in the water. The standard "OCx" family of algorithms — the formulas that have underpinned global ocean color products since the 1990s — exploit exactly this. Pick the brightest blue band, divide by the green band, feed the resulting ratio into a polynomial, and out pops a chlorophyll estimate (Hammoud et al., 2026).

It's elegant, simple, and inspectable. But it has known weaknesses: a fixed band-ratio shape may not capture everything in a rich spectrum, and the polynomial coefficients often misbehave near the edges of the data they were trained on. Turbid coastal waters, where sediment and dissolved organic matter muddy the signal, have always been a trouble spot.

Hammoud and colleagues took a different route. They fed what are called remote sensing reflectances — the light leaving the ocean's surface — into a machine learning tool called PySR, which performs symbolic regression. Rather than prescribing a functional form in advance, symbolic regression evolves candidate equations the way natural selection evolves organisms: through mutation, recombination, and culling, balancing prediction accuracy against expression complexity. Each candidate is an equation tree, with reflectance values at specific wavelengths as leaves and mathematical operators like addition, division, and logarithms as branches. Out of millions of expressions, the algorithm surfaces a Pareto front — a family of equations trading off simplicity against skill.

The team ran the approach on two datasets. The first was the massive global bio-optical compilation of Valente et al. (2019), which stitches together 27 independently assembled datasets spanning SeaWiFS, MODIS, MERIS, VIIRS, and OLCI satellite matchups. The second was a global hyperspectral dataset — expanded from work by Sun et al. (2023) — of in situ underwater spectra with reflectance measured at 1-nanometer intervals, representing chlorophyll concentrations from 0.012 to 130.165 mg·m⁻³, spanning everything from the crystal-clear open ocean to turbid estuaries (Hammoud et al., 2026). This is the dataset that let them simulate PACE-like observations and test how much spectral resolution actually matters.

The first experiment was a sanity check: give symbolic regression the same multispectral bands that OC3, OC4, OC5, and OC6 use, and ask whether it rediscovers the band-ratio logic.

The second was the real question: hand the algorithm 119 PACE-like bands spanning 402.5 to 697.5 nm, and see which wavelengths it actually needs.

What They Found

Start with the sanity check, because its results are quietly profound. Given only the multispectral inputs, symbolic regression did converge on a blue-green contrast structure — a physics-consistent answer that mirrors the logic human oceanographers had derived decades earlier. Figure 1 traces the emergence of this structure: a complexity-1 equation depending only on green reflectance performs poorly, with correlation of just 0.435 and an RMSD of 0.745. But introducing a normalized contrast among the 443, 490, and 560 nm bands at complexity 5 jumps the correlation to 0.899 and halves the RMSD to 0.367 (Hammoud et al., 2026).

The machine independently walked to the same place several generations of ocean color scientists had arrived at by hand. That's a strong validation that symbolic regression is learning real optical physics, not just memorizing noise.

Then comes the part that changes the game. When given the full hyperspectral PACE-like spectrum, the best discovered expression achieved a held-out RMSD of 0.253 in log₁₀ chlorophyll, compared to 0.307 for a fitted OC6 polynomial on the same data split (Hammoud et al., 2026). The chart below shows the stability of that advantage across ten independent data splits — the symbolic regression models consistently land below the OC6 baseline.

Hyperspectral symbolic regression beats standard ocean color baselines

Label	Value
Gaussian PACE-like PySR	0.247 log10 CHL-a RMSD
Gaussian OC6-LASSO	0.307 log10 CHL-a RMSD

Mean and standard deviation of test RMSD across ten 90/10 splits, from the paper's robustness experiment.

The gap is largest where it matters most: in high-chlorophyll, turbid waters. Figuratively, this is the ocean's murky back alleys — the coastal and optically complex environments where the classic algorithms have always struggled. When the researchers broke down performance by environmental regime, they found the hyperspectral advantage concentrated precisely in these hard cases. In clean open-ocean waters the two approaches are close; in turbid conditions the symbolic regression models pull decisively ahead.

The wavelength-selection results are equally telling.

Across 100 equations selected in a stability analysis, the algorithm kept returning to a small set of wavelengths — certain bands in the blue and green regions appeared again and again, while others were almost never touched. The technique effectively performs automated feature selection, pointing to which parts of the spectrum carry the most chlorophyll information. This isn't just useful for designing algorithms; it's a clue about the optics — which spectral regions genuinely encode the pigment signal versus which are redundant.

Adding sea surface temperature to the models provided surprisingly little benefit. The analysis shown in Figure 8 included an SST term that only appeared at complexity 33, and its improvement over a complexity-10 reflectance-only model was marginal (Hammoud et al., 2026). For most practical purposes, the spectrum alone carries the information that matters.

More spectral resolution, better chlorophyll estimates

Label	Value
OC3 Ridge	0.36
OC4 Ridge	0.34
OC5 Ridge	0.33
OC6 Ridge	0.32
5nm SR	0.3
2.5nm SR	0.29
PACE-like PySR	0.247

Representative RMSD values on the common 85/15 split reported in Figure 7 of the paper; OCx Ridge baselines shown as approximate validation values.

Why This Changes Things

The PACE mission doesn't give scientists a nicer version of what they already had; it gives them a different kind of instrument. Conventional ocean color sensors record roughly ten reflectance bands — snapshots of color at ten discrete wavelengths. PACE's Ocean Color Instrument records hundreds, an almost continuous rainbow from the violet through the red. That's a leap in information, but information is useless if you can't turn it into insight.

This study is a proof that you can. The key contribution is demonstrating that dense, 119-band hyperspectral data can be compressed down to a handful of sparsely selected wavelengths with a closed-form equation — something compact enough to read, trust, and implement across sensors, not a black box with a million parameters.

That last point matters enormously for the scientific culture of oceanography. Neural network chlorophyll retrievals exist and often perform well, but they resist inspection: you can't look at a trained network and say "this band matters, that one doesn't." Symbolic regression produces equations you can hold in your hand, with variables and coefficients you can examine — features the paper argues are essential when an algorithm is to be evaluated across sensors and applied globally (Hammoud et al., 2026).

There's also a deeper scientific yield: the wavelength-selection results effectively do automated physics discovery. By repeatedly returning to the same optical features, the algorithm highlights which spectral signatures genuinely encode chlorophyll against the noise of sediment, yellow substances, and atmospheric interference. That's a hypothesis generator for ocean optics — a way to ask why certain wavelengths carry the signal, and whether the current understanding of why blue-to-green ratios work is complete.

What's Next

The most immediate next step is turning these discovered equations into operational products. The paper walks through applying the models to PACE scenes, diagnosing reflectances that fall outside the training domain — a crucial safeguard, since an equation trained on one range of water types may silently fail on another. The team's careful validation design, including held-out test subsets evaluated only after model selection, models the kind of rigor operational products will need.

Several open questions remain. The symbolic regression search space excluded arbitrary exponentiation and square roots — including them might surface even better equations. The current best models still carry errors, especially in extreme turbidity, and a 0.253 RMSD still means a factor-of-1.8 uncertainty at the high end. And while the hyperspectral advantage is real, the study notes that most of the improvement is captured by models with complexity around 8–12, suggesting diminishing returns from further elaboration — a useful calibration for algorithm designers.

The broader promise is bigger than chlorophyll. The same symbolic regression machinery could be pointed at other ocean color products — diffuse attenuation, inherent optical properties, particle size distributions — to derive interpretable equations for each. If ocean color remote sensing is entering its hyperspectral era, symbolic regression offers a principled way to make sure we actually understand what we're seeing, rather than just predicting it.

For now, the takeaway is a striking recalibration: three decades of band-ratio algorithms were not wrong — they were just running on a fraction of the information available. The future, it turns out, is written in hundreds of shades of blue.