Outcrops and cores are primary sources of information about the Earth's past. Quantitative analyses rely on geochronologies that take into account highly variable sedimentation and erosion rates as well as gaps from missing strata. Using 23 geochronologies from the Holocene, Quaternary, Phanerozoic and Precambrian, we apply Haar fluctuation analysis to statistically characterize the number of measurements per unit time - the measurement densities. The analysis determines the densities' (multifractal) scaling regimes and exponents; collectively, the analyses span over nine orders of magnitude in time scale. The measurement density is a new paleoindicator that we show is typically correlated with the primary paleoindicator, biasing and complicating its statistical interpretation. We also analyze the distribution of gaps linking the latter's (probability) scaling with series incompleteness and the length Sadler effect. The density characteristics are needed to unbias spectra and other statistical characterizations.

Geological time is punctuated by events that define biostrata and the Geological Time Scale’s (GTS) hierarchy of eons, eras, periods, epochs, ages. Paleotemperatures and macroevolution rates, have already indicated that the range ≈ 1 Myr to (at least) several hundred Myrs) is a scaling (hence hierarchical) “megaclimate” regime. We apply analysis techniques including Haar fluctuations, structure functions trace moment and extended self-similarity to the temporal density of the boundary events (r(t)) of two global and four zonal series. We show that r(t) itself is a new paleoindicator and we determine the fundamental multifractal exponents characterizing the mean fluctuations, the intermittency and the degree of multifractality. The strong intermittency allows us to show that the (largest) megaclimate scale is at least ≈ 0.5 Gyr. We also analyze a Precambrian series going back 3.4Gyrs directly confirming this limit and allowing us to quantatively compare the Phanerozoic with the Proterozoic eons.We find that the probability distribution of the intervals (“gaps”) between boundaries and find that its tail is also scaling with an exponent qD≈ 3.3 indicating huge variability with occasional very large gaps such that it’s third order statistical moment barely converges. The scaling in time implies that record incompleteness increases with its resolution (the “Resolution Sadler effect”), while scaling in probability space implies that incompleteness increases with sample length (the “Length Sadler effect”). The density description of event boundaries is only a useful characterization over time intervals long enough for there to be typically one or more events. In order to model the full range of scales (and low to high r(t)), we introduce a compound Poisson-multifractal model in which the multifractal process determines the probability of a Poisson event. The model well reproduces all the observed statistics.Scaling changes our understanding of life and the planet and it is needed for unbiasing many statistical paleobiological and geological analyses, including unbiasing spectral analysis of the bulk of geodata that are derived from cores.

Climate models and paleoclimate proxies have temperature variability that diverge from each other locally and at long timescales. It is unknown to what extent these divergences also apply to hydroclimate and whether long-term hydroclimate variability is fundamentally different than temperature variability. Here we evaluate the long-term variability of near surface air temperature (tas) and hydroclimate (Palmer Drought Severity Index [PDSI]) using a climate model (the Community Earth System Model-Last Millennium Ensemble [CESM-LME]) and a paleoclimate reconstruction based on this model (the Paleo Hydrodynamics Data Assimilation product [PHYDA]); this framework allows us to see how a model's long-term climate variability is affected by informing it with proxy data. Using power-scaling exponents, we find universally higher scaling values in PHYDA (except for global mean tas) compared to the CESM-LME model. Thus, PHYDA's global PDSI, local PDSI, and local tas are more dominated by low-frequency variability than CESM-LME's. Additionally, PDSI is spectrally flatter than tas in CESM-LME, whereas scaling values of tas and PDSI are comparable in PHYDA. These results indicate that the paleoclimate reconstruction process adds low-frequency variability that CESM-LME otherwise would not have. Based on a range of null reconstruction experiments, we attribute the origin of low-frequency variability in PHYDA to proxy information and not the mathematical properties of the data assimilation methodology. This implies that long-term variability in PHYDA is dependent on the selection of assimilated proxy data.

Knowledge on natural climate variability is pivotal for making future climate projections. Previous studies demonstrated that centennial to millennial temperature variability is lacking in climate model simulations and that this bias is spatially heterogeneous. Various mechanisms have been proposed that might be important to modulate this low-frequency variability such as the ocean circulation, the meridional temperature gradient or external forcing and climate sensitivity to that forcing, but the evidence to identify the main driver(s) is still debated. Here, we provide preliminary insights on the respective importance of those mechanisms in driving long-term climate variability by investigating spatial patterns of low-frequency climate variability. Low-frequency variability beyond multi-decadal timescales cannot be studied using only instrumental data due to data limitations and the confounding impact of anthropogenic forcing. Consequently, noisy and biased palaeoclimate proxy observations have to be utilised in order to investigate spatio-temporal patterns of climate change. Using a multi-archive and -proxy approach, we characterise the first-order spatial pattern of low-frequency climate variability of interglacial periods. By combining information on the spatio-temporal fingerprint derived from various archives and proxies with different characteristics, we aim to identify the common climate variability signal and assess the ability of climate models to explain the proxy-based spatial pattern of low-frequency variability.

With increased pressure from anthropogenic climate change, boreal forests are suspected to be approaching tipping points which could cause large-scale changes in tree cover and affect global climatic feedback. However, evidence for this proposed tipping is sparse and relies heavily on observations on short time scales from remote sensing data and space-for-time substitutions. Here we make use of an extensive pollen data set including 239 records of large lakes to investigate the existence of alternative stable forest cover states in the boreal forest and its adjacent biomes during the last 8000 years. By using a multimodality measure on time series of reconstructed tree cover we find very little multimodality in pollen data. To test whether this lack of multimodality is caused by limitations in the paleo data set we perform surrogate experiments. Surrogate data with alternative stable states based on the paleo vegetation-climate relationship were generated and significant multimodality was found more often than for the pollen-based tree cover (24.7% and 5.3% respectively). The response of tree cover to climate may, therefore, be more gradual and not as abrupt as would be expected from remote sensing analyses on stability. The apparent alternative stability hypothesized in the analyses of climate-vegetation relationships could be due to the strong spatial heterogeneity of environmental factors and vegetation responses as an artifact of space-for-time substitutions. Even though current and upcoming shifts in the boreal forest are indisputable and a reason for strong concern, these changes could happen gradually without going through large-scale tipping between alternative stable states. To aid adaptation and conservation measures, more knowledge is needed about boreal forest drivers and their spatial heterogeneity.

Turbidity currents on continental margins are often attributed to cyclic climate variability and sea-level change, while the causes of deep ocean turbidites are as yet to be tested. The Atlantic Iberian margin provides a unique setting to contrast deep ocean and continental environments, including depression features that further protect from resuspension and erosion by along-slope bottom currents. We present records of low-frequency, non-periodic, climate-independent turbidites from three deep cores covering up to 426,000 years in the Tore seamounts area. By evaluating a range of physical oceanographic mechanisms, the breaking of internal waves and mesoscale Mediterranean-eddies against unstable slopes in the seamounts area arises as the most likely triggers that precondition the recurrence pattern of the observed deep ocean turbidites.

Regional climate change projections over the course of the 21st century require the accurate simulation of both anthropogenic and natural variability. The spatial patterns of natural variability are relatively well constrained on sub-decadal timescales based on instrumental data evidence, and climate models can simulate them. For longer (supra-decadal) timescales, however, the spatial patterns of natural (temperature) variability are poorly constrained because of the shortness of the instrumental record and the overlap with anthropogenic influences. Insights gained from paleoclimate data over land in mid-latitudes suggest that oceanic influence was the main driver of increased low-frequency natural variability, in contrast to its stabilizing role on sub-decadal timescales. Here, by studying the spatial imprint of multi-decadal climate variability, we show that the instrumental data is consistent with this hypothesis. While the pattern is also observed in climate models, it is much weaker and seems to rely solely on forced variability. Therefore, while climate models can simulate anthropogenic warming, our evidence indicates, particularly over the northern land mid-latitudes, that they are not simulating supra-decadal natural variability (forced and internal) consistent with instrumental observations in terms of the spatial pattern and its amplitude.

There is concern that climate change might lead to abrupt and irreversible changes in parts of the Earth system at so-called tipping points. Theoretical considerations suggest that statistical measures can be used to detect early warning signals (EWSs) for reduced resilience, which could be interpreted as an increased proximity to climate tipping points. Here we discuss limitations of commonly used EWSs and their detection and discuss how alternative explanations can lead to resilience loss in the absence of tipping points. We argue for better testing of the existence of tipping points, beyond the application of EWSs, and propose a method to better quantify the probability of approaching tipping points using EWSs.

With rapid anthropogenic climate change, future vegetation trajectories are uncertain. Climate–vegetation models can be useful for predictions, but they require extensive data on past vegetation for validation and improving systemic understanding. Even though pollen data provide a great source of this information, they are compositionally biased due to differences in taxon-specific relative pollen productivity (RPP) and dispersal. Here, we present a Northern Hemisphere reconstruction of quantitative regional vegetation cover from a sedimentary pollen dataset for the last 14 kyr using the REVEALS model to correct for taxon- and basin-specific biases. For the reconstruction, we expanded on a previously published synthesis of continental RPP values. The datasets include the taxonomic composition and reconstructed tree cover for each original pollen sample. Additional metadata include modeled ages, age model sources, basin locations, types, and sizes. The improvements in tree cover reconstructions with the REVEALS reconstruction using continental RPP values range from 22 % (Asia) to 67 % (Europe) relative to the mean absolute error (MAE) of the pollen-based tree cover. The dataset can be used as a grid with binned and aggregated samples (adjustable script provided on Zenodo at https://doi.org/10.5281/zenodo.13902976, Schild, 2024) or as individual time series if the record's basin size exceeds 50 ha. This alternative quantitative reconstruction of vegetation cover is beneficial for the investigation of past vegetation dynamics and modern model validation when varying spatial and temporal resolutions may be required. By collecting more RPP estimates, especially in North America, and adding more records to existing pollen data syntheses, reconstructions may be improved even further. The new REVEALS dataset is freely available on PANGAEA (see the Code and data availability section; https://doi.org/10.1594/PANGAEA.974798, Herzschuh et al., 2025).

Geological time is punctuated by events that define biostrata and the Geological Time Scale's (GTS) hierarchy of eons, eras, periods, epochs, ages. Paleotemperatures and macroevolution rates, have already indicated that the range ti 1 Myr to (at least) several hundred Myrs is a scaling (hence hierarchical) "megaclimate" regime. We apply analysis techniques including Haar fluctuations, structure functions, trace moment and extended self similarity to the temporal density of the boundary events (rho(t)) of two global and four zonal series. We show that rho(t) itself is a new paleoindicator and we determine the fundamental multifractal exponents characterizing the mean fluctuations, the intermittency and the degree of multifractality. The strong intermittency allows us to show that the (largest) megaclimate scale is at least ti 0.5 Gyr. We find that the tail of the probability distribution of the intervals ("gaps") between boundaries is also scaling with an exponent qD ti 3.3 indicating huge variability with occasional very large gaps such that it's third order statistical moment barely converges. The scaling in time implies that record incompleteness increases with its resolution (the "Resolution Sadler effect"), while scaling in probability space implies that incompleteness increases with sample length (the "Length Sadler effect"). The density description of event boundaries is only a useful characterization over time intervals long enough for there to be typically one or more events. In order to model the full range of scales and densities, we introduce a compound multifractal-Poisson process in which the subordinating multifractal process determines the probability of a Poisson event and that this new process is close to the observed statistics. Scaling changes our understanding of life and the planet and it is needed for unbiasing many statistical paleobiological and geological analyses, including unbiasing spectral analysis of the bulk of geodata that are derived from paleoclimatic and paleoenvironmental archives.

The relatively short observational record limits our ability to understand the long-term variability of key climate factors like temperature and hydroclimate. Climate models and paleoclimate proxies appear to have long-term temperature variabilities that diverge from each other at local and long time scales. But it is unclear whether these divergences also apply to hydroclimate and whether long-term hydroclimate variability is fundamentally different than temperature variability.Here we evaluate the long-term variability over the Common Era of temperature and hydroclimate using a climate model (the Community Earth System Model-Last Millennium Ensemble, CESM-LME) and a paleoclimate reconstruction based on this model (the Paleo Hydrodynamics Data Assimilation product, PHYDA); this framework allows us to see how a model’s long-term climate variability is affected by informing it with proxy data. We specifically focus our analyses on the continuum of temperature (tas) and the Palmer Drought Severity Index (PDSI) in four regions of low reconstruction uncertainty: the Western USA, the Eastern USA, Central Europe, and Scandinavia.Using the power-scaling exponents β from the relationship S(τ) ∼ τβ, where S denotes the power spectral density (PSD) and τ the period, we find universally higher values of β in PHYDA (except for tas globally); in these four regions PHYDA’s β values are 0.30 to 0.63 higher than CESM-LME. Thus, long range dependence behavior is more pronounced in PHYDA than in the CESM-LME model. We find that PHYDA has different spatial distributions of β than CESM-LME. We also find that hydroclimate is spectrally flatter than temperature in CESM-LME, whereas temperature and hydroclimate β-values are comparable in PHYDA. These results show that CESM-LME’s hydroclimate and temperature is less dominated by long timescales compared to PHYDA’s. The robustness of the low-frequency variability signal in PHYDA was verified by performing pseudoproxy experiments. Furthermore, preliminary results of other temperature DA reconstructions over the Holocene and since the Last Glacial Maximum also reveal spectral divergencies with model data. In particular, for the PSD of the global mean temperatures, higher beta values were obtained for the reconstructions compared to the model data, indicating a deficit in simulated low-frequency.

The complex biological and physical processes that preserve paleoclimate information over centuries or longer introduce variations in proxy time series that are unrelated to the true climate. These non-climatic variations act on different timescales and are often referred to as “noise” of a specific color, based on similarities between a time series' power spectrum and the electromagnetic spectrum of light. For example, “white noise” equally affects all timescales, where “red noise” dominates only on long timescales, similar to longwave red light. Noise spectra in proxy records have far-reaching implications in paleoclimate research, but noise characteristics are often assumed based on first principles rather than estimated directly, risking either inflating or underestimating error at particular frequencies. Here, we provide concrete definitions of the various types of timescale-dependent errors that are present in proxy data, and review the literature on methods for quantifying noise terms. We then synthesize the results of several published studies that use a common empirical approach for estimating the noise spectrum in ice-core, coral, and tree-ring data. We posit that the colors of proxy noise are archive-specific, with white noise dominating in depositional archives such as ice cores and marine sediment cores, while red noise is more common in biological archives such as tree rings and corals. Our synthesis supports assigning specific colored noise terms in proxy system models, data assimilations and other experiments.

Regional climate change in the $21^{st}$ century will result from the interplay between human-induced changes and internal climate variability. Competing effects from greenhouse gas warming and aerosol cooling have historically caused multidecadal forced climate variations overlapping with internal variability. Despite extensive historical observations, disentangling the contributions of internal and forced variability remains debated, largely due to the uncertain magnitude of anthropogenic aerosols. Here, we show that, after removing CO$_{2}$-congruent variability, multidecadal temperature variability in instrumental data is largely attributable to internal processes of oceanic origin. This follows from an emergent relationship, identified in historical climate model simulations, between the driver of variability in oceanic regions and the land-ocean variance ratio in the mid-latitudes. Thus, climate models with higher residual (non-CO$_{2}$) forced variability, largely linked to volcanic and anthropogenic aerosols, exhibit more spatially coherent and amplified temperature patterns over land compared to observations. In contrast, models with higher internal variability agree better with the instrumental data. Our results underscore that internal modes of ocean-driven variability may be too weak in many climate models, and that current projections may be underestimating the range of internal variability in regions with high oceanic influence.

With few exceptions, paleodata are irregularly sampled; this poses numerous challenges for the statistical characterization of paleoindicators, this includes the indicators needed to understand the climate and macroevolution. The key variable is the measurement density - the number of measurements per unit time (r(t)). Our study used 27 paleoindicators collectively spanning time scales from years to hundreds of millions of years. Using Haar fluctuation analysis and for all the series, we show that r(t) has two scaling regimes. At high frequencies, there is a low intermittency (quasi-Gaussian) scaling regime (intermittency parameter C1 ≈ 0). Over this regime, the fluctuation exponent H is negative implying that the chronologies become more uniform at longer time scales, r(t) is commonly close to a Gaussian white noise (H = -1/2). In contrast, at low frequencies, r(t) is highly intermittent (large C1), but it also has positive H so that fluctuations tend to grow with scale but in a highly intermittent fashion. In this this regime, “gaps” at all scales are important. The two regimes have simple physical interpretations: the high frequency behaviour can be explained by fairly smooth (but scaling) sedimentation rates, whereas the low frequencies can be explained by scaling erosion processes that introduce gaps over a wide range of scales (in conformity with the Sadler effect). To confirm this interpretation, we introduce a simple multiplicative sedimentation - erosion model that is close to the data. Finally, we empirically show that the gaps typically have extreme power law probability tails so that the series are not only scaling in time, but also in probability space. A key issue for paleontologists is the effect of variable r(t) on the paleoindicator estimates themselves (e.g. on paleotemperatures T(t)). Using Haar fluctuations we determined the fluctuation - fluctuation correlation R(Δt) = < Δ r(Δt) ΔT(Δt) >. When R(Δt) is small, the measurements and indicators are statistically independent so that the biases due to r(t) variability on paleoindicator statistics are easy to correct. However, at large Δt, the correlations are frequently large, and this poses additional difficulties in data interpretation. Strong correlations were observed in the Quaternary, but not the Holocene or Phanerozoic. Our study spans more than 8 orders of magnitude in time scale and it shows that it is wrong to theorize paleoseries as being fundamentally regularly sampled but interspersed with occasional data “holes” that can be dealt with using conventional techniques such as interpolation. While Haar fluctuation analysis is insensitive to the chronology variability - and if needed can easily be statistically corrected for any biases that it introduces - this is not true of existing spectral estimators that are extremely sensitive to scaling data gaps.

Data assimilation techniques, such as the Kalman Filter, have enabled the development of complete climate field reconstructions over the last millennium, commonly referred to as paleoclimate reanalysis. These techniques effectively integrate paleoclimate data, facilitating the understanding and attribution of past climate events. The resulting spatio-temporal fields are invaluable for studying teleconnections and exploring dynamical links between variables and locations in the past. However, when the observation network is sparse, or proxies exhibit high levels of non-climatic noise, the Kalman Filter tends to revert to the prior. These limitations often result in paleoclimate data assimilation products underestimating variability in earlier periods and overestimating spatial coherence compared to modern observations, reducing their reliability. We thus investigate the timescale-dependent variance and the spatio-temporal covariance of different paleoclimate data assimilation products: ModE-RA, LMR, and PHYDA, and relate differences primarily to the methodology and prior assumptions. The results from the data assimilation products were further assessed against instrumental data and CMIP6 pre-industrial control and fully forced simulations.

Abstract. With rapid anthropogenic climate change future vegetation trajectories are uncertain. Climate-vegetation models can be useful for predictions but need extensive data on past vegetation for validation and improving systemic understanding. Even though pollen data provide a great source of this information, the data is compositionally biased due to differences in taxon-specific relative pollen productivity (RPP) and dispersal. Here we reconstructed quantitative regional vegetation cover from a global sedimentary pollen data set for the last 50 ka using the REVEALS model to correct for taxon- and basin-specific biases. In a first reconstruction, we used previously published, continental RPP values. For a second reconstruction, we statistically optimized RPP values for common taxa with the goal of improving the fit of reconstructed forest cover from modern pollen samples with remote sensing forest cover. The data sets include taxonomic compositions as well as reconstructed forest cover for each original pollen sample. Relative pollen sources areas were also calculated and are included in the data set of the original REVEALS run. Additional metadata includes modeled ages, age model sources, basin locations, types and sizes. The improvements in forest cover reconstructions with the REVEALS reconstruction using original/optimized parameters range from 1/0 % (Australia and Oceania/Australia and Oceania) to 58/65 % (Europe/North America) relative to the mean absolute error (MAE) in the pollen-based reconstruction. Optimizations were considerably more successful in reducing MAE when more records and RPP estimates were available. The optimizations were purely statistical and only partly ecologically informed and should, therefore, be used with caution depending on the study matter. This improved quantitative reconstruction of vegetation cover is invaluable for the investigation of past vegetation dynamics and modern model validation. By collecting more RPP estimates for taxa in the Southern Hemisphere and adding more records to existing pollen data syntheses, reconstructions may be improved even further. Both reconstructions are freely available on PANGAEA (see Data availability section).

Abstract. Uncertainty in paleoclimate time series is inherent to the complex biological and physical processes involved in forming and archiving them in the environment for centuries or longer. The timescale-dependency of this uncertainty is often referred to as "noise" of a particular color based on similarities between the power spectrum of a timeseries and the electromagnetic spectrum of light. For example, "white noise" equally affects all timescales, where "red noise" dominates only on long timescales, similar to longwave red light. In paleoclimate research, the frequency characteristics of proxy noise are often assumed based on first principles rather than estimated directly, which risks either inflating or underestimating error at particular frequencies. Here, we synthesize several studies that use a common method to estimate the spectrum of error in ice core, coral, and tree-ring data. We conceptualize how time-scale dependent noise in proxy time series is created through the archive formation and data processing. Our results suggest that the colors of proxy noise are archive- specific, with white noise dominating in depositional archives such as ice-cores and marine sediment cores, while red noise is likely more common in biological archives such as tree rings and corals. Our aim is to clarify these concepts and provide tools for assigning noise terms in proxy system models, data assimilations, and other experiments.

&lt;p&gt;Whether tree-rings faithfully archive the low-frequency variability (LFV) in climate remains debated. In theory, trees are fundamentally limited by being relatively short-lived and therefore unable to capture variations in the climate that are longer than their own lifespans. In addition, near universal practices of &amp;#8220;detrending&amp;#8221; tree-ring records to remove individualistic age-growth trends place further constraints on the amount of LFV that is maintained in final chronologies. Detrending methods designed to boost LFV may increase low-frequency signals, but how well those patterns reflect true variations in climate as opposed to long growth trends is still unclear. In this study, we first compared the spectral properties of the PAGES North America 2k dataset of temperature-sensitive tree-ring records against long temperature records to determine how much variability is retained in tree-rings after detrending, and how detrending method influences agreement in tree-ring power spectra across space. Then, we compare the spectral properties of tree-rings to the CMIP6 last millennium simulation to validate climate models against long proxy records. This research works to resolve discrepancies between temperature proxies and climate models on long timescales in order to improve our understanding of centennial-scale variability in the Earth&amp;#8217;s climate system.&lt;/p&gt;

&lt;p&gt;The amplitude and spatial distribution of low-frequency natural variability is determinant for regional climate projections, but it is still poorly understood.&amp;#160;&lt;/p&gt; &lt;p&gt;&amp;#160;&lt;/p&gt; &lt;p&gt;In a previous study, pollen-based temperature reconstructions were used to quantify spatial patterns of millennial temperature variability. This showed an inverse relationship across timescales with sub-decadal variability from instrumental data in extra-tropical regions over land (H&amp;#233;bert et al., 2022, under review). We concluded that due to varying marine influence, regions characterized by stable oceanic climate at sub-decadal timescales experience stronger long-term variability while continental regions with higher sub-decadal variability show weaker long-term variability. Indications of this relationship could also be inferred from instrumental data alone as regions of low sub-decadal variability were more likely to exhibit a steeper increase of variability over multi-decadal timescales and vice versa.&amp;#160;&lt;/p&gt; &lt;p&gt;&amp;#160;&lt;/p&gt; &lt;p&gt;In the current work, the relationship found in the instrumental data was further investigated using different instrumental products. In addition, a large multi-model ensemble of CMIP6 models, as well as single-model ensembles, were considered for analysis and it was found that they do not systematically reproduce the relationship found in the instrumental data. This indicates a fundamental deficiency in the model simulations with regard to the mechanism driving the emergence of low-frequency climate variability. This characteristic being related to multi-decadal variability thus has important significance for multi-decadal regional climate projections and might be used as an emergent constraint in model evaluation and inter-comparison.&lt;/p&gt;

Weather station records are too short and sparse to effectively detect the signature of climate change in Antarctica. Using the isotopic composition of ice cores as a temperature proxy suggests that Antarctica is warming faster than the global average temperature and expectations from climate models for the region.