Lab · Exploration 5

Finding Structure in the Pacific

EOFs, ENSO, and What a Reduced Representation Keeps

The Pacific Ocean is not one number. But one number can still be useful. This Studio explores how climate scientists compress huge temperature maps into a few patterns, what is gained by doing so, and what may be left out.

First, what is El Niño?

El Niño is the warm phase of the El Niño–Southern Oscillation (ENSO), a recurring coupled change in tropical Pacific ocean temperatures, winds, and circulation. During El Niño, unusually warm surface water extends across the central and eastern equatorial Pacific; during La Niña, the pattern is broadly reversed. These Pacific-wide changes can influence weather far beyond the tropics.

Here the goal is not to reduce ENSO to a single number, but to see how indices and EOFs reveal recurring structure in full temperature-anomaly maps—and what that compression can miss.

Start exploring ↓The monthly 1950–2025 fields are a reproducible, historical-event-calibrated teaching dataset, not an observational archive.

Watch first · short video

How We Find Patterns in Noisy Climate Data

Meet the idea of extracting a small number of meaningful patterns from a large, noisy climate dataset.

Watch video ↗Applied Mathematics in Geosciences · Episode 8

Then explore it yourself ↓

Start with one number

From a map to an index

An index compresses a complicated spatial pattern into one number. That makes climate variability easier to track, but different maps can still share a similar index.

Month 1997-12Notable event

The outlined Niño-3.4 region is only one part of a much larger, changing Pacific pattern.

Niño-3.4 teaching index+1.75 °C1997-12

What you are seeing

Niño-3.4 is the average SST anomaly in a fixed equatorial region. It turns one map into one trackable number.

Why it matters

Tracking becomes easier, but the shape, location, and extent of the anomaly outside the box are no longer preserved.

Try this

Compare 1997–98 with 2015–16. Similar peak indices do not require identical maps.
Find two months with similar index values and inspect what differs outside the box.

Dimensionality reduction

Finding dominant patterns with EOFs

Imagine every monthly anomaly map as one point in a high-dimensional space. EOF analysis finds a few spatial patterns that capture as much variance as possible.

1 · Form anomaliesRemove the 1991-2020 monthly climatology

January is compared with typical Januaries, February with typical Februaries, and so on.

2 · Separate changeLinear trend retained

Detrending asks EOFs to describe variability after a fitted gradual change is taken out.

3 · Respect grid area√cos(latitude) weighting applied

Area weighting prevents denser latitude rows from receiving disproportionate influence.

Analysis periodAnalysis domainRemove linear trendSeparate gradual change from variabilityApply area weightingUse √cos(latitude) before EOF analysisSelected pattern · EOF 1

EOF means empirical orthogonal function: a spatial pattern learned from the changing maps.

PC means principal component: the coefficient telling us how strongly this pattern appears each month.

EOF 1 explains 63.2% of weighted variance in this particular setup.

SST anomaly map≈mean state+PC₁ × EOF₁+PC₂ × EOF₂+ ···

The sign is arbitrary. Multiplying an EOF and its PC by −1 changes their displayed colors and orientation, but it leaves every reconstructed map unchanged.

Math behind this

Arrange the anomaly maps as columns of a matrix X. A singular value decomposition writes X = UΣV^T. The columns of U are EOF patterns; ΣV^T gives their time coefficients.

The first pattern maximizes captured variance, the second captures the largest remaining variance while staying orthogonal to the first, and so on. Orthogonal means the weighted spatial dot product between different patterns is zero.

What you are seeing

Every monthly map is represented by a small set of spatial patterns and changing coefficients. The bars show how much variance each pattern captures.

Why it matters

EOFs are a systematic compression, but the answer depends on the region, period, anomaly definition, trend treatment, and area weighting.

Try this

Does EOF1 look ENSO-related? Describe the resemblance without declaring the two identical.
Keep EOF2 and EOF3. How much additional variance do they capture?
Shorten the analysis period and watch the leading patterns adjust.

Low-rank reconstruction

How much structure survives compression?

Keeping only a few modes compresses the data. Broad basin-scale structure often survives; finer details are the first to disappear.

Retained modes r = 3Month 1997-12Notable event

Show metricsRetained variance and RMSE

All resolved spatial detail in the selected processed map.

The broad structure retained by a rank-3 representation.

Spatial information not represented by the retained modes.

Variance retained85.9%

This month’s spatial RMSE0.15 °C

Average RMSE in this period0.18 °C

What you are seeing

The same processed map appears as the original, a reconstruction using only the first r modes, and the information left behind.

Why it matters

A few modes often preserve basin-scale structure while filtering smaller details. Whether those details matter depends on the task.

Try this

Set r = 1 during a strong El Niño. Name what survives and what disappears.
Increase to r = 3 or r = 5. Which spatial details return?
Try a quieter month. Is rank 1 equally successful?

Related, not identical

Is the leading EOF the same as ENSO?

The leading tropical-Pacific SST pattern often looks ENSO-related. Compare the map, time series, and index before deciding what that resemblance means.

3-month smoothingReduce month-to-month wigglesStandardize seriesCompare shape rather than unitsLag · Niño-3.4 relative to PC1 0 monthsComposite threshold ±1.0 SD

Correlation at this lag: 0.97. High similarity is evidence of a close relationship, not identity.

Average processed map when PC1 is strongly positive.

Average processed map when PC1 is strongly negative.

What you are seeing

PC1 and Niño-3.4 often move together in this setup, and strong positive and negative PC1 months produce familiar basin-scale composites.

Why it matters

The resemblance is useful, but an SST variance pattern is not a complete causal description of ENSO.

Try this

Change the lag. Does simultaneous comparison remain strongest?
Change the analysis period or domain above and inspect the new relationship.
Flip the sign. Notice that the physical reconstruction has not changed.

Variance versus usefulness

A pattern can be large without being the most useful

Now change the question: instead of reconstructing today’s map, use retained PCs to predict a future index on data the model never saw while fitting.

Lead time 6 monthsRetained PCs 3Train / test split

3-PC linear model0.740Test RMSE

Persistence baseline0.834Test RMSE

Held-out comparisonmodel improves2003-05 to 2025-12

The model is fitted before 2003-04; every displayed score comes from later, held-out months.

This is a simple demonstration, not a state-of-the-art ENSO forecast model. It uses transparent linear regression so the representation, not model complexity,remains the focus.

What you are seeing

A linear model learns from an earlier part of the record and is evaluated only on later months. Persistence assumes the current index simply continues.

Why it matters

EOFs are optimal for variance-based reconstruction. That does not guarantee that the leading PCs are optimal features for every forecast target or lead time.

Try this

At one-month lead, see why persistence is difficult to beat.
At six months, add PCs one at a time. Does every added mode help?
Switch to the threshold task. A representation can perform differently for a different question.

Break universality

Change the question, change the representation

EOF results are answers to a specified data question, not fixed structures waiting unchanged inside every dataset.

Analysis regionTime periodMonthly climatologyDetrendRemove fitted linear changeArea weightingApply √cos(latitude)Modes retained 3Prediction target

Representation depends on the question.
Compression is always selective.
The best reconstruction is not automatically the best forecast representation.
Reduced coordinates are useful maps, not complete substitutes for the full system.

What you are seeing

The same teaching record is reanalyzed whenever you change the domain, period, climatology, trend treatment, or spatial weighting.

Why it matters

A reported EOF is inseparable from the choices that produced it.

Try this

Switch to the equatorial band and compare EOF1’s explained variance.
Turn detrending on. Which part of broad change moves out of the EOFs?
Change the climatology period and inspect how anomaly baselines shift.

Closing synthesis

What should you remember?

One number can be useful

A climate index turns a complex field into something easy to track and compare.

EOFs reveal dominant spatial patterns

They provide a systematic way to summarize large datasets.

Dominant variance is not the whole story

A pattern can explain a lot of variance without being the most useful feature for every task.

The right description depends on the question

There is no single best reduced representation for every purpose.

In climate science, as in many areas of applied mathematics, reduction is not about throwing information away carelessly. It is about choosing what to keep, knowing what is lost, and matching the representation to the scientific question.

Open the plain-language glossary

SST: Sea surface temperature: the temperature of the ocean’s uppermost layer.
Anomaly: A departure from the usual value for that location and calendar month.
Niño-3.4: Average SST anomaly in a fixed central-equatorial Pacific region.
EOF: A spatial pattern chosen to capture as much variance as possible.
Principal component: The time-varying coefficient attached to one EOF pattern.
Explained variance: The fraction of the dataset’s total variation represented by a mode.
Reconstruction: An approximate map rebuilt from a chosen set of EOFs and PCs.
Detrending: Removing a fitted gradual change before studying variability.
Area weighting: Adjusting grid values so unequal physical cell areas are represented fairly.

Sources, method, and data note

National Oceanic and Atmospheric Administration, definitions of Niño regions and sea-surface-temperature climate indices.
I. T. Jolliffe and J. Cadima, “Principal component analysis: a review and recent developments,” Philosophical Transactions of the Royal Society A, 374 (2016).
H. von Storch and F. W. Zwiers, Statistical Analysis in Climate Research, Cambridge University Press.

This page uses a deterministic, built-in monthly teaching dataset for 1950–2025. Its broad tropical-Pacific patterns and notable ENSO event timing are historically calibrated, but its grid values are synthetic and must not be used as an observational climate record. Monthly climatology removal, optional detrending, area weighting, EOF decomposition, reconstruction, and held-out regression are computed in the browser from that dataset.