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Disentangling Forced and Internal Climate Variability in Single Realizations using Dynamic Mode Decomposition with Control

This paper introduces PullbackDMDc, a novel method grounded in non-autonomous dynamical systems theory that successfully decomposes a single climate realization into forced and internal variability components, thereby enabling skillful climate projection and model evaluation without requiring large ensemble datasets.

Original authors: Nathan Mankovich, Andrei Gavrilov, Gustau Camps-Valls

Published 2026-07-22

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CC BY 4.0

Original authors: Nathan Mankovich, Andrei Gavrilov, Gustau Camps-Valls

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to understand the weather by watching a single, unbroken movie of the sky. You see clouds swirling, storms brewing, and sunshine breaking through, but you can't pause the film to rewind and ask, "Was that storm caused by the sun heating the ocean, or just because the wind decided to change its mind?" This is the ultimate puzzle of climate science. The Earth's climate is a giant, chaotic machine driven by two main forces: the "forced" response, which is the planet's reaction to external pushes like greenhouse gases, volcanic eruptions, and the sun; and "internal variability," which is the climate's own chaotic jitter, like a dog shaking off water after a bath. Scientists need to separate these two to know if the planet is warming because of human activity or just because of natural wobbles. The problem is, we only have one movie of Earth's history, so we can't run the same scene twice to see what happens if we change the script.

To solve this, researchers have tried various tricks. Some use giant computer models that run the same movie hundreds of times with slightly different starting points to find the "average" story (the forced part) and treat the differences as the "noise" (internal variability). Others try to guess the forced part by looking at simple patterns, like how the global average temperature rises. But these methods often miss the mark: the computer models are expensive and sometimes wrong, while the simple guesses ignore the complex dance of the atmosphere. The big question remains: Can we take our single, messy movie of Earth's climate and mathematically split it into the "script" (what the external forces are doing) and the "improvisation" (what the climate is doing on its own) without needing a time machine or a thousand computer simulations?

This paper introduces a clever new tool called PullbackDMDc to answer that question. Think of the climate system as a car driving down a bumpy road. The "forced" part is the driver pressing the gas pedal (external forcing), while the "internal" part is the car bouncing over potholes (internal variability). PullbackDMDc is like a super-smart mechanic who looks at the car's single trip and, knowing exactly how the engine responds to the gas pedal, can mathematically separate the smooth acceleration from the bumpy shaking. The authors treat the climate as a linear system where the external forces act as a "control" term. By using a concept from mathematics called a "pullback attractor"—which is essentially a way of saying "if we rewind time far enough, the car's path is determined only by the road and the driver, not by where it started"—they can mathematically peel apart the two components from just one record of data.

The researchers tested this new tool on real-world data and four different giant climate models (ESMs) that have been run with dozens of different starting conditions to create a "ground truth" for comparison. They found that PullbackDMDc is surprisingly good at guessing the forced response, matching or even beating existing methods that rely on massive computer simulations. It successfully identified that the "script" of global warming is clearly visible in the data, especially when the tool is fed detailed information about different types of forcing, like volcanic ash and greenhouse gases.

However, the tool also revealed some interesting flaws in how our current climate models behave. When the researchers used PullbackDMDc to look at the "improvisation" (internal variability) in the models, they found a mismatch. In real-world observations, the climate has a slow, deep "heartbeat" that beats every 60 to 80 years (related to patterns like the Atlantic Multidecadal Oscillation), and this slow beat carries most of the warming signal. But in the computer models, this slow heartbeat is weak or missing. Instead, the models seem to rely too much on faster, shorter-term rhythms (like the 2-to-7-year El Niño cycle) to carry the warming signal. The paper suggests that while our models are good at capturing the fast, jittery parts of the climate, they are systematically underestimating the slow, deep, long-term swings that dominate the real world.

In short, PullbackDMDc offers a new, practical way to read a single climate record and separate the human-made signal from the natural noise. It confirms that we can do this without needing a thousand computer simulations, but it also warns us that our current models might be missing the slow, deep rhythms of the Earth, potentially making our future projections less accurate than we hope. The authors suggest that while their method is powerful, it relies on linear math, and the real climate might be more complex and non-linear, meaning future versions of the tool will need to get even smarter to capture the full story.

Technical Summary: Disentangling Forced and Internal Climate Variability in Single Realizations using Dynamic Mode Decomposition with Control

Problem Statement
Climate science faces a fundamental methodological challenge: the observed climate record is a single realization where the forced response (driven by external agents like greenhouse gases and aerosols) and internal variability (generated by the system's coupled dynamics) are superposed. Controlled experimentation on the observational record is impossible. While Earth System Models (ESMs) can generate large ensembles to define a forced response via the ensemble mean, observational analysis requires disentangling these components from a single time series. Existing statistical approaches fall into two categories with complementary limitations: dynamical system models (e.g., Linear Inverse Models, LIMs) often ignore external forcing predictors, while regression-based methods incorporate forcing but omit the system's intrinsic dynamics. Furthermore, many advanced methods rely on supervised learning using ESM ensembles as "ground truth," which limits their applicability when ESM fidelity is uncertain or computational resources are constrained.

Methodology: PullbackDMDc
The authors introduce PullbackDMDc, an unsupervised, linear method grounded in non-autonomous dynamical systems theory and Dynamic Mode Decomposition with Control (DMDc). The core theoretical framework treats the observed climate trajectory as lying on the pullback attractor of a linear stochastic system driven by external forcing.

Linear Stochastic Model with Forcing: The method models the climate state $x(t)$ using a discrete-time equation:
$x(t) = Ax(t-1) + By(t) + \xi(t)$
where $A$ represents the autonomous dynamical core, $B$ maps the external forcing $y(t)$ (e.g., radiative forcing) to the state, and $\xi(t)$ represents internal stochastic noise.
Pullback Attractor Estimation: By assuming the system is globally stable (eigenvalues of $A$ $A$ have magnitude $<1$ $< 1$ ), the authors define the pullback attractor as the limit of the system's evolution as the initial time $t_0 \to -\infty$ $t_{0} \to - \infty$ . In this linear limit, the attractor collapses to a single trajectory that uniquely separates into:
- Forced Response ( $x^{(f)}(t)$ ): The deterministic component driven by the history of external forcing $y(t)$ .
- Internal Variability ( $x^{(i)}(t)$ ): The stochastic component driven by the history of noise $\xi(t)$ .
Spatiotemporal Decomposition: The method projects the data onto the eigenmodes of the matrix $A$ . This yields a decomposition where the full spatiotemporal record is expressed as a sum of dynamic modes (spatial patterns), each accompanied by two distinct time series: a forced component and an internal component.
Implementation: The authors apply this to Principal Components (PCs) of near-surface air temperature (SAT), ocean SAT (OSAT), and sea-level pressure (SLP). They test various forcing inputs, including global mean temperature, effective radiative forcing, and specific components like volcanic and greenhouse gas forcing.

Key Contributions

Theoretical Unification: PullbackDMDc bridges the gap between dynamical system models and regression by embedding external forcing directly into the evolution operator of a linear stochastic system, allowing for the exact separation of forced and internal components in a single realization.
Interpretable Decomposition: Unlike standard EOF or DMD, which provide a single time series per mode, PullbackDMDc provides a physically distinct pair of time series (forced and internal) for every spatial mode, enabling direct interrogation of how specific patterns respond to forcing versus internal dynamics.
Unsupervised Framework: The method operates without requiring large ESM ensembles for training, relying instead on the mathematical properties of the pullback attractor and available forcing data.

Results
The method was evaluated against four ESM large ensembles (CanESM5, MIROC6, CESM2, MPI-ESM) and 20th Century Reanalysis (20CRv3), using the ensemble mean as the ground truth for forced response.

Forced Response Estimation: PullbackDMDc variants achieved skill matching or exceeding established baselines (including Linear Regression and LIM-opt). Specifically:
- Methods incorporating forcing predictors (PullbackDMDc and Linear Regression) significantly outperformed those that did not (LIM, RegGMST).
- Forcing Dimensionality: Performance was sensitive to the choice of forcing predictors. For temperature variables (OSAT/SAT), a 3D forcing representation (Greenhouse gases, aerosols, volcanics) yielded the best results. For Sea Level Pressure (SLP), a 2D representation (Volcanics + GHGs) performed best, suggesting that higher-dimensional inputs may lead to overfitting in low signal-to-noise regimes.
- PullbackDMDc variants showed lower uncertainty (variance across ensemble members) compared to baselines that ignored forcing, particularly for SLP.
ESM Evaluation via Decomposition: The decomposition revealed systematic structural differences between reanalysis and ESMs:
- Multidecadal Variability: In reanalysis, the leading modes captured slow, multidecadal variability (resembling the Atlantic Multidecadal Oscillation, AMO), and the forced response projected predominantly onto these slow modes. In contrast, ESMs systematically prioritized faster, ENSO-like dynamics in their leading modes and underrepresented the multidecadal structure.
- Forced Response Distribution: In reanalysis, the forced signal was concentrated in the slowest modes. In ESMs, the forced response was distributed across a broader set of modes, including faster ENSO- and PDO-like dynamics.
- Seasonal Cycle Non-Stationarity: The method detected non-stationary changes in the seasonal cycle (retained in anomalies) in reanalysis for SAT and SLP, a feature that ESMs largely failed to reproduce, appearing as weaker or absent annual periods in the internal modes of the models.

Significance and Claims
The paper claims that PullbackDMDc provides a practical, physically interpretable tool for single-realization climate analysis and ESM evaluation. Its primary significance lies in its ability to:

Disentangle Components: Successfully separate forced and internal variability in a single record without relying on large ensemble means for the decomposition itself.
Diagnose Model Deficiencies: Reveal that current ESMs may systematically misrepresent the hierarchy of climate variability timescales, specifically underestimating the persistence of multidecadal modes and the amplitude of the forced response's projection onto these slow structures.
Guide Forcing Selection: Demonstrate that the choice of forcing predictors is critical and variable-dependent, offering a framework for selecting optimal predictors based on model ground truth.

The authors maintain a modest stance, acknowledging that the method is currently linear and that the exact separation of forced and internal components is a property of the linear approximation. They note that the estimation of the dynamical operator $A$ from finite, noisy data remains a source of uncertainty, particularly for long-timescale structures, and that future work should explore nonlinear extensions and improved estimation strategies.

Technical Summary: Disentangling Forced and Internal Climate Variability in Single Realizations using Dynamic Mode Decomposition with Control

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