When Climate History Rhymes: Finding Recurring Patterns Across 800,000 Years of Ice

The first time you look at an ice core pulled from Antarctica, you see a column of ancient ice. The second time, you notice the layers—dark bands of volcanic ash, lighter bands of snow, each one a year, a decade, a millennium compressed into a cylinder two inches wide. But Béatrice Désy and her colleagues saw something else: a message hidden in the shape of those layers, repeating itself across time. Not identical repetitions—that would be too simple for a system as chaotic as Earth's climate. But similar shapes, similar rhythms, recurring again and again over 800,000 years of ice ages and warm periods.

Their new method, published as "Wasserstein recurrence networks for multiscale time series pattern analysis," finds those repetitions by measuring similarity in a way that works whether you're comparing patterns lasting 30 years or 3,000. That's the key innovation: a single approach that spans two orders of magnitude without breaking. They call it a recurrence network, but it's really a map of when history rhymes with itself.

The Science

Climate data is messy. This isn't a minor inconvenience—it's the central challenge that makes paleoclimate research so difficult. When you drill an ice core in Antarctica, you're measuring deuterium ratios (a proxy for temperature), but those measurements come at irregular intervals, with gaps, with noise, with abrupt changes that might be real climate events or might be measurement artifacts. When marine sediment cores record ice-rafted debris—little bits of rock scraped from continents and dropped into the ocean by melting icebergs—the data spikes suddenly, stays flat for centuries, then spikes again. The temporal resolution isn't uniform. The signal isn't smooth. And you're trying to find patterns that repeat across scales from decades to millennia.

Existing methods struggle with this mess. Fourier transforms and wavelet analyses require preprocessing, detrending, assumptions about what counts as noise versus signal. Time-delay embeddings—which convert a time series into a higher-dimensional space to look for recurrences—work well for smooth, uniformly sampled data but degrade when the sampling is irregular or when the signal contains abrupt jumps. "Abrupt changes capture real climate processes," the researchers note, but many methods assume smoothness and erase those changes along with the noise.

Désy and colleagues wanted a method that works directly on raw, unevenly sampled data, that doesn't require interpolation or detrending, and that can detect patterns of vastly different durations using the same framework. Their solution builds on a branch of mathematics called optimal transport—the study of how to move "mass" from one distribution to another as efficiently as possible.

The key quantity is the 1-Wasserstein distance. Imagine you have two piles of sand on a line. The Wasserstein distance asks: how much work would it take to reshape one pile into the other, where work equals the amount of sand moved times the distance it travels? It's sometimes called the Earth mover's distance because that's exactly what it measures: the minimum effort required to transform one distribution into another.

For time series, this works beautifully. Take a subsequence of your data—a chunk of the ice core spanning 1,000 years, say. Transform it into a probability distribution: normalize the values so they sum to one, rescale the time axis to [0,1]. Now you have a histogram. Do the same for another subsequence. The Wasserstein distance between these two histograms tells you how similar they are in shape, regardless of when exactly the data points fall or how many there are.

The method has several properties that make it ideal for paleoclimate data. It doesn't require uniform sampling—you can compare a pattern with 30 data points to one with 3,000. It captures shape similarity rather than exact value matching, so a gradual warming looks like another gradual warming even if the absolute temperatures differ. And critically, it works across scales: the same distance metric applies whether you're looking at patterns lasting 30 years or 30,000.

To use this for recurrence detection, you pick a starting point in your time series and compute the Wasserstein distance from that pattern to all future patterns. Where the distance reaches a local minimum—a point where you can't find a closer match nearby—that's a candidate recurrence. But not every local minimum is meaningful. Random noise will produce local minima too. To separate signal from noise, you need a threshold.

That's where Brownian motion comes in.

Brownian motion is the simplest possible model for a random time series: a random walk where each step is independent of the last and normally distributed. It's not meant to represent actual climate data; it's a null model, a baseline. If even random noise can produce similar-looking patterns by chance, then detecting "recurrences" in real data means nothing. The question is: what Wasserstein distance corresponds to "similar by chance"?

The researchers generated millions of random patterns from Brownian motion trajectories, computed their Wasserstein distances, and analyzed the distributions. They looked at patterns of different durations—from 30 data points (ℓ=1) to 3,000 data points (ℓ=100), spanning two orders of magnitude. And they found something striking: the distributions were essentially identical.

This scale invariance is the heart of the method. For most distance functions, comparing a short pattern to a long one gives very different statistics—you'd need different thresholds for different scales. But the Wasserstein distance, applied this way, produces the same distribution of distances regardless of how long the patterns are. A Wasserstein distance of 0.019 has a p-value of 0.01 whether you're looking at 30-year patterns or 30,000-year patterns. The threshold is universal.

From there, building the full recurrence network is straightforward. For each point in the time series, identify patterns of various durations. Find local minima in the distance function that fall below the significance threshold. Remove duplicates where multiple time scales identify the same event. What remains is a set of statistically significant recurrences spanning multiple time scales—a network where each node is a moment in time, and edges connect moments when similar patterns occurred.

What They Found

The scale invariance is real. Across pattern durations ℓ from 1 to 500 (a 500-fold range in the number of data points), the Wasserstein distance distributions from Brownian motion collapse onto a single curve. At p=0.01, the threshold is $W=0.019\pm 0.001$. At p=0.05, it's $W=0.032\pm 0.002$. These aren't approximations or rules of thumb—they're precise statistical cutoffs derived from millions of random comparisons.

Scale Invariance of Wasserstein Distance Distributions

Label	Value
ℓ=1	0.32
ℓ=5	0.31
ℓ=10	0.3
ℓ=30	0.28
ℓ=50	0.27
ℓ=100	0.26
ℓ=500	0.25

Data: Désy et al., 2026. Distribution peak values shown; full distributions converge to similar probability density functions.

When applied to real paleoclimate data, the method finds recurrences that span two orders of magnitude in duration. In the EPICA Dome C ice core—a record of Antarctic climate spanning 800,000 years—recurrences range from events lasting a few thousand years to events lasting over 100,000 years. The ice age cycle itself is roughly 100,000 years, and the method detects it as a recurrence: the pattern of a glacial period beginning, deepening, and ending looks similar from one ice age to the next, even though the details differ.

The ice-rafted debris (IRD) records tell a complementary story. IRD spikes indicate periods when icebergs carried rock debris from continents into the ocean—when ice sheets were unstable, calving icebergs at their margins. The researchers compared IRD records from both hemispheres and found local minima in their Wasserstein distance that aligned in time across records. During the last five glacial terminations (the transitions from ice ages to warm periods), both hemispheres show similar patterns of ice sheet instability. But the alignment isn't perfect—during the inceptions that began glacial cycles (transitions from warm periods to ice ages), only some events produced synchronized behavior in both hemispheres.

Distribution of Recurrence Durations in Paleoclimate Record

Label	Value
1-5 kyr	35
5-20 kyr	28
20-50 kyr	18
50-100 kyr	12
>100 kyr	7

Data: Désy et al., 2026. Based on 800,000-year EPICA Dome C deuterium record.

The recurrence network for the Dome C record reveals structure in the timing of recurrences. Most recurrences connect moments that are close in time—natural enough, because similar patterns tend to cluster. But there are recurrences spanning tens of thousands of years: moments in the distant past that rhyme with moments in the equally distant future. The network isn't just capturing the obvious 100,000-year ice age cycle. It's finding shorter repetitions too—rhythms within rhythms.

Why This Changes Things

The method matters for three reasons.

First, it solves the multiscale problem. Most time series analysis tools work at one scale or require you to specify which scale you care about. Fourier transforms decompose a signal into frequencies, but you have to choose which frequencies to focus on. Wavelets zoom in and out, but each zoom level is a separate analysis. The Wasserstein approach lets you search for recurrences at every scale simultaneously, using the same statistical threshold throughout. For a system like climate that operates on seasonal, decadal, centennial, and millennial scales all at once, that's a significant advance.

Second, it respects the data. Many paleoclimate methods require interpolation—filling in the gaps between unevenly spaced measurements with assumed values. This makes an implicit choice about what counts as noise: short-term variability gets smoothed away, including potentially real abrupt changes. The new method never interpolates. It takes the data as it comes—sparse, noisy, irregular—and finds patterns directly. "We are not making ad hoc choices about what is noise and what is signal," the researchers note.

Third, it provides a principled threshold. Recurrence detection has always faced the threshold problem: how similar does a pattern need to be to count as a recurrence? Too strict, and you miss real repetitions. Too loose, and everything looks like a recurrence. Previous approaches often used fixed recurrence rates—the top 5% of local minima, regardless of what they represent. But a 5% recurrence rate means you're finding 5% of patterns recurring by chance alone. The new method ties the threshold directly to a statistical model. When you say a recurrence is significant at p=0.01, you mean exactly that: there's a 1% chance of finding a pattern this similar by chance alone.

For paleoclimatology, this opens new questions. The ice age cycle has been known for decades, but the method reveals structure within it—times when the climate rhymed with itself at shorter intervals, moments when patterns from different regions aligned. For climate science more broadly, it's a new tool for comparing records: finding similarities between datasets without forcing them into a common framework. For time series analysis in general, it's a way to handle the messiness that real data always contains.

What's Next

The method has limits. Brownian motion is a simple null model; real systems have structure that affects pattern distributions. For data like ice-rafted debris—sparse, spikey, with long zeros—a uniform random walk may not be the right baseline. The researchers acknowledge this and suggest alternative null models for specific applications. But they also show that even with imperfect null models, the method works: the IRD analysis finds meaningful recurrences, aligned across hemispheres, without requiring the null hypothesis to be exactly right.

The network analysis is descriptive rather than predictive. The method tells you when history rhymed with itself, but not what happened next. Causal relationships between recurrences—which events influenced which—are beyond the scope. That would require additional assumptions about the system dynamics.

Several open questions remain. Can the method be extended to spatial data—maps of climate rather than time series? Can it handle missing data more gracefully, or incorporate uncertainty in the age model (the mapping from ice depth to calendar year)? Can the scale invariance be proven analytically rather than just demonstrated numerically? These are technical challenges, not fundamental barriers.

What the method already provides is a new lens on old questions. Earth's climate has always contained repeated patterns—similar ice ages, similar warm periods, similar sequences of abrupt change and gradual stability. The Wasserstein approach finds those repetitions without forcing them into predefined categories, and it quantifies how significant each match is. History doesn't exactly repeat, the saying goes, but it rhymes. This method finds the rhymes.