the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 11 Jun 2026
Extracting coherent spatio-temporal modes of simulated multi-centennial AMOC variability under constraints that reflect sparsity of proxy data
Abstract. A mechanistic understanding of internal variability of the climate is crucial as internal variability has a strong influence on regional to local climate projections throughout the 21st century. The Atlantic Meridional Overturning Circulation (AMOC) strongly impacts regional to local climate and geological evidence suggests (multi-)centennial AMOC variability (mCAV) is a feature of the Mid- to Late Holocene climate. However, our understanding of the spatio-temporal aspects and underlying mechanisms of mCAV is very limited. Understanding the mechanisms behind Holocene mCAV requires a methodology that isolates spatio-temporal patterns of variability and is applicable to both climate model output and the geological archive. Multi-channel singular spectrum analysis (MSSA) has been successfully applied to climate model output to identify and isolate basin-wide spatio-temporal modes of variability. However, it remains unclear if the correct modes can be identified, and if these modes retain their spatio-temporal coherence, when based on input data that is constrained by relatively sparse locations where proxy records are available. Here, we explore this issue in a transient Late Holocene simulation of an earth system model of intermediate complexity that is known to contain mCAV. Our results show that under constrained input data MSSA can be used to identify robust modes of simulated mCAV and that the modes retain their spatio-temporal coherence within at least the northern and eastern North Atlantic. These findings suggest MSSA can be a suitable tool to extract basin-wide modes of variability and associated spatio-temporal patterns from geological reconstructions. This motivates further work that incorporates uncertainties associated with geological reconstructions into the MSSA methodology. Furthermore, our findings motivate the identification and clustering of temperature based phase-relationships in different climate models that contain different mechanisms of mCAV.
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Status: open (until 02 Oct 2026)
- RC1: 'Comment on egusphere-2026-2794', Anonymous Referee #1, 16 Aug 2026 reply
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RC2: 'Comment on egusphere-2026-2794', Anonymous Referee #2, 30 Aug 2026
reply
The manuscript applies M-SSA to a transient iLOVECLIM simulation and asks whether modes of multi-centennial AMOC variability survive when the input is restricted to proxy locations. The question is relevant and the design is sound. One sentence on the origin of the method would help: Broomhead and King (1986a, b) introduced the delay-coordinate approach, including its multichannel form, within dynamical systems theory; Vautard and Ghil (1989) recognized its spectral-analysis aspect and applied it to climatic and paleoclimatic problems; Ghil et al. (2002) reviews the history and the methodology.
Please revise the following points:
The automated identification of oscillatory pairs, on which Sections 3.2 to 3.4 rest, uses an equal-power tolerance of 2.5 percent ("To meet the power criterium, both members of the T-EOF pair must fall within the interval") that is below the eigenvalue splitting a finite window imposes on a standing oscillation. Running the authors' own pipeline (GitHub repository, src) on synthetic data with standing 140- and 280-year oscillations in red noise, the 280-year pair meets frequency and quadrature in 10 of 10 runs and power in 0 of 10; the 140-year pair meets power in 5 of 5 runs with a 500-year window and 0 of 5 with a 400-year window. The authors note themselves that "the automated method to determine the power criterium (Sect. 2.2.3) may be too strict", and their sensitivity test confirms it: "The identification rate of the 140-yr mode is between ∼12-50%" with the criterion, against "higher identification rates (∼22-72%) (Fig. A10a)" without it. Can the authors make the tolerance depend on window and period, or drop the criterion and keep the significance test and quadrature?
The abstract states that "under constrained input data MSSA can be used to identify robust modes of simulated mCAV", and Section 4.5 that "it is possible to identify robust modes of simulated (multi-)centennial AMOC variability under strongly constrained input data". In Section 3.4, however, "There is no evidence of T-EOF pairs associated with the 200-yr mode" for the full proxy grid, and the subsampling recovers the most robust mode in 12 to 50 percent of the realizations. The text should say what a single real proxy network can deliver. The code standardises each grid cell and applies no area weighting on a regular three-degree grid, where cells north of 70°N are about 40 percent of the domain for about 15 percent of its area. This over-represents high-latitude variance (e.g., the Arctic signal). If the analysis is not recomputed, it is requested that this methodological limitation be explicitly acknowledged in the discussion.
Code and data: I could not download the Zenodo archive due to the size (15.3 GB); I downloaded the GitHub repository instead. The core (M-SSA, varimax, Procrustes test, pair criteria) runs on synthetic data. environment.yml omits statsmodels and cmocean, which the code imports; the Fig. 2 notebook uses a local path and HadISST is not listed in the archive; the Fig. 2 run retains 95 percent of the variance in the pre-PCA, which the text does not state. In mssakit_TB.py the varimax step rotates the singular values and squares them, whereas the original MSSAkit rotates the eigenvalues (Groth and Ghil 2011); the difference is below 1 percent in my tests.
Minor corrections:
"as a first order criteria to select records" → criterion.
"periods of ∼46 and 26 months" and "the 46 month period" in the text; "the T-EOF pair at 48 months" in the Fig. 2 caption.
"PCA is applied to a set of geospatial time-series to identify the spatial patterns that explain most of the variance": PCA is not inherently spatial; in T-mode (time steps as variables, locations as samples) it yields temporal patterns. Please state that the PCA used here is S-mode PCA (Richman, 1986).
"when a T-EOF pair is though of as a sine-cosine pair" → thought of.
Fig. 2d: the legend hides part of the plot; please move it outside the data area.
"criterium" (Sect. 2.2.3 onward) → criterion.
"these periodicities aligns well" → align.
"shows the the MSSA spectrum" → shows the MSSA spectrum.
"show a the Beaufort sector" → show the Beaufort sector.
"[0.25 ± 0.025 x Tav]" → (0.25 ± 0.025) x Tav.
"the S-GIN and East North Atlantic remain among the modes with highest accuracy" → sectors.Table A1 caption: "(Table is work in progress)". Is "work in progress" correct or a remnant of a draft version?
Citation: https://doi.org/10.5194/egusphere-2026-2794-RC2 -
RC3: 'Comment on egusphere-2026-2794', Anonymous Referee #3, 14 Sep 2026
reply
Review of the manuscript “Extracting coherent spatio-temporal modes of simulated multi-centennial AMOC variability under constraints that reflect sparsity of proxy data”, by: Toon Bense, Henk A. Dijkstra, and Pepijn Bakker
Overall assessment
In the manuscript is investigated whether multichannel singular spectrum analysis (MSSA) can recover multi-centennial modes of simulated Atlantic Meridional Overturning Circulation variability when model output is sampled only at locations where suitable marine temperature proxy records exist. The authors first diagnose variability in a 3000-year iLOVECLIM simulation, then compare MSSA results obtained from complete spatial fields with results from a 35-cell “Proxy Grid” and from Monte Carlo subsets of that grid. Thus, it is tested how spatial sparsity affects MSSA.
The manuscript is well structured, the MSSA introduction is accessible, and the attempt to evaluate both mode detection and phase-pattern recovery is valuable. The perfect-model framework is appropriate as an initial methodological step and the conceptual explanation of SSA, PCA, and MSSA is useful for paleoclimate readers. However, as presented below, some aspects need further clarifications. Therefore, I recommend major revision.
Major comments
1. The concept of “mode of variability” is often used. A definition for it is useful. How is defined a mode of variability?
2. Para 35: Why different frequencies identified with SSA can be considered modes? If an analyzed signal has no harmonic shape, then a spectral analyzes could provide spurious peaks which do not reflect physical modes, but only a mismatch between the non-harmonic shape of the input signal and the shapes of the functions of the basis (harmonics).
3. What are the advantages of using SSA relative to other spectral methods? What are the advantages of using PCA?
4. Yes, PCA provides a pattern associated only with a phase of a mode, whereas MSSA provides a dynamical picture of the mode’s evolution. However, one could derive such a dynamical picture, for example. by constructing lagged regression maps based on the time component of the mode. This is much simpler than MSSA. What would be the advantage of using MSSA over PCA and lagged regressions?
5. Para 345: In SSA and MSSA is common than two consecutive eigenvalues are associated with an oscillatory signal. In the manuscript on oscillatory mode was associated with eigenvalues 3 and 5. What about eigenvalue 4?
6. The manuscript identifies approximately 140-, 200-, and 280-year modes in temperature and salinity fields and calls them modes of multi-centennial AMOC variability. This is suggestive but insufficient to establish that the MSSA modes are AMOC modes rather than ocean-temperature modes that share similar periods. The authors should present more evidence that every retained MSSA mode is essentially linked with AMOC variability.
Minor comments
1. Lines 27–30: The model–proxy variance discrepancy is more nuanced than an absence of internal modes. Proxy noise, temporal smoothing and spatial representativeness should also be acknowledged.
2. Lines 107–110: The omission of volcanic and solar forcing may be acceptable for a methodological test, but it makes the simulated spectral composition less representative of the real Holocene record.
3. Lines 117–121: Replace “criteria” with “criterion.” Please explain how duplicates were defined and how records were mapped onto coarse grid cells.
4. Lines 190–198: The statement that T-EOFs “capture nonlinear behaviour” could be misunderstood. MSSA is a linear decomposition that can represent non-sinusoidal temporal structures; it does not infer nonlinear dynamics.
5. Equations 2–3: Please mention it the series have already been centered and standardized.
6. Lines 326–329: Please explain the reason for defining AMOC as the maximum between 45–65°N rather than using a conventional subtropical index. This definition may emphasize deep-water-formation variability rather than basin-scale overturning.
7. Lines 493–503: Please define the sign convention for “lead” more explicitly. A lead between 0 and one period is equivalent to a negative lag over the complementary part of the cycle.
8. Lines 548–555: The failure to identify one particularly harmful grid cell does not demonstrate that the lower signal-to-noise ratio is the explanation. Interactions among locations and phase cancellation may matter.
9. The manuscript would benefit from a concise table listing each proposed mode, its period uncertainty, variance explained, detection rate, variables in which it occurs, robustness across windows, and relationship with AMOC.
10. A thorough language edit is advisable. Examples include “the more known” -> “better-known,” “has previously shown” -> “has previously been shown,” “with a measures” -> “with measures,” and several instances of singular/plural disagreement.
Data sets
Data for publication: Extracting coherent spatio-temporal modes of simulated multi-centennial AMOC variability under constraints that reflect sparsity of proxy data Toon Bense https://doi.org/10.5281/zenodo.20122668
Interactive computing environment
tbense/Bense_ea_2026_Extracting_coherent_spatio: Notebooks for submission Bense et al. (2026) Clim. of the Past Toon Bense https://doi.org/10.5281/zenodo.20201057
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This paper uses M-SSA to analyze the output of a paleoclimate model that exhibits multi-centennial variability. It aims to "introduce M-SSA to paleoclimate community" and addresses an important issue of a limited data coverage in relation to the method's ability to elucidate the true lead-lag relationships (that presumably underly the oscillation dynamics), in both constrained and unconstrained setups, in the perfect-model setting.
The authors do a decent job in summarizing the gist of the M-SSA methodology and demonstrate it to be a viable tool for the task at hand, and can provide useful information even in the data-scarce situations mimicking the actual proxy data. They also convey an important point of making sure that the oscillatory modes identified by M-SSA are robust with respect to the pre-processing choices (fields considered, embedding dimension etc.); these considerations are exceptionally well illustrated by the analyses described in the paper.
Overall, I think this paper presents a useful contribution to the field and recommend publication.