Reconstruction of Holocene Northern Hemisphere precipitation fields using paleoclimate data assimilation

Abstract

Clarifying the spatiotemporal patterns of Holocene (~11.7 - 0 ka BP) precipitation variability over the Northern Hemisphere (NH) is essential for contextualizing modern hydroclimate variability. However, the scarcity of spatiotemporally-completed, temporally well-resolved and highly-reliable reconstructions has limited a full-field understanding of Holocene hydroclimate. Here, we present a reconstruction of NH annual precipitation spanning 12-0 ka BP at 3.75° spatial and 100-year temporal resolution. This dataset was generated using paleoclimate data assimilation (PDA) approach, integrating 2,421 Holocene precipitation records with two transient simulations via an Ensemble Optimal Interpolation algorithm. Validation suggested the PDA-based reconstructions have significantly improved reliability compared to model-only simulations, particularly at mid-high latitudes. As a core product of the Holocene Climate Reanalysis supported by Chinese Academy of Sciences (CAS-HCR), this dataset represents a substantial progress in applying the PDA approach to reconstruct Holocene precipitation fields at hemispherical scale, and provides a key resource for analyzing the spatiotemporal dynamics of Holocene hydroclimate, contextualizing modern precipitation changes and evaluating the performance of climate models in simulating long-term precipitation variability.

Background & Summary

The Holocene (~11.7 ka BP to the present; BP: before 1950 AD) is the current interglacial period, providing an ideal time window for investigating past climate change on orbital scale¹, assessing the current climate position since the last glacial maximum^2,3, informing future climate change⁴ and evaluating the skill of climate models for long-term simulations^5,6. Temperature and precipitation, the two most crucial variables in the energy and water cycles of the Earth system, have significant impacts on regional and global climates, making them the most critical variables in Holocene climate reconstruction. To date, Holocene temperature reconstructions have been produced on large scales - such as for the Northern Hemisphere and globally - using either regionally-composited approaches (e.g., refs. ^7–11) or spatially-resolved methods (refs. ^12,13). However, large-scale reconstructions of Holocene precipitation have received extremely less attention compared to large-scale temperature reconstructions. Most existing precipitation reconstructions are limited to time series at specific sites or in particular regions and continents (e.g., refs. ^14–21). The NH compilation of quantitative precipitation reconstructions during the Holocene was published as recently as 2023 by the authors of ref. ¹⁰. Additionally, precipitation generally exhibits high and heterogeneous variation across a wide range of spatial and temporal scales^11,22–25. This nature associated with precipitation presents a grand challenge for spatially-resolved reconstructions of Holocene precipitation at a hemispheric scale, consequently, spatiotemporally-completed, temporally well-resolved, highly-reliable, hemispheric-scale precipitation reconstruction data spanning the entire Holocene remain extremely scarce to date, which fundamentally limits full-field and accurate understanding on the spatiotemporal patterns of NH hydroclimate variability during this period.

The emerging paleoclimate data assimilation (PDA)^26,27 is a best-of-both-worlds method for large-scale climate field reconstruction²⁸ in paleoclimatology. The PDA fuses climate model simulations with paleoclimate records based on Bayesian principles, taking into account their respective uncertainties to produce a theoretically optimal and full-field estimate of past climate states⁴. To date, the PDA has been successfully applied to the Last Millennium Reanalysis (LMR) Project^27,29, reconstructions of global hydroclimate and dynamical variables over the Common Era³⁰, globally-resolved surface temperatures since the last glacial maximum^12,13,31 and Greenland temperature and precipitation over the last 20,000 years¹⁸. Currently, compared to the relatively numerous applications of the PDA for large-scale temperature reconstructions over long timescales, its use for reconstructing precipitation fields at similar spatial and temporal scales remains exceptionally rare. The LMR v2²⁹ employed PDA to reconstruct global precipitation anomaly fields during the Common Era, which is a landmark achievement in this domain. However, the Common Era represents only a narrow segment of the Holocene, making it unsuitable for studying global or hemispheric precipitation patterns during the Early, Middle, or early Late Holocene. More recently, the authors of ref. ³² made a significant advance by reconstructing global lake status and precipitation at 500-year resolution since 21 ka BP using the PDA approach. Nevertheless, this reconstruction relied on a limited global proxy network (only 216 lake-level records) and exhibited a relatively coarse temporal resolution of 500 years. Such low resolution risks smoothing out critical climatic processes, namely it may fail to capture centennial-scale events, obscure multidecadal-to-centennial precipitation variability and lose important signatures of internal climate variability. In this study, recognizing the importance of generating spatiotemporally-completed, temporally well-resolved and highly-reliable precipitation reconstructions during the Holocene across the NH to deepen our understanding of the spatiotemporal patterns of Holocene precipitation, and taking into account the substantial increase in both the quantity and temporal resolution of publicly available paleo-precipitation records across the NH in recent years^10,16,19,33, as well as the significant advantages of PDA in reconstructing large-scale climate fields, we reconstructed spatiotemporally-completed Holocene annual precipitation fields over the NH at spatial resolution of 3.75° and temporal resolution of 100 years using the PDA approach. This reconstruction was achieved based on 2,421 Holocene NH annual precipitation records from LegacyClimate v1.0 dataset¹⁰ and two transient climate simulations spanning the entire Holocene^34,35, using a time-averaged Ensemble Optimal Interpolation (EnOI) data assimilation algorithm³⁶. Cross-validation results indicated that the PDA-based Holocene precipitation reconstructions in this study were well aligned with site-specific precipitation reconstructions from the LegacyClimate v1.0 dataset¹⁰. Additionally, independent verification against Holocene annual precipitation records further demonstrated good consistency between our reconstructions and independent precipitation proxy records. This consistency was particularly evident in high and middle latitudes, where the PDA-based precipitation reconstruction exhibited better performance than model-only simulations. Moreover, spatial validation also suggested substantial skill improvement in our reconstructions relative to prior estimates across most NH regions, with the percentage of grids showing positive skill scores ranging from 74.00% to 88.54%.

Overall, this study represents a significant advancement in applying the PDA approach to reconstruct spatiotemporally-completed Holocene precipitation fields across the NH. Unlike most previous Holocene precipitation reconstructions, which were often limited to site-specific or regionally-composited series, the PDA-based precipitation reconstruction of this study is capable of providing a full-field perspective on the evolution of Holocene precipitation across the NH. Consequently, this dataset, as a core product of the Holocene Climate Reanalysis project (HCR) supported by the Chinese Academy of Sciences, can be employed to elucidate the spatiotemporal dynamics of Holocene precipitation in the NH, to situate current climatic conditions within the broader Holocene context and to evaluate the performance of climate models in simulating long-term precipitation patterns. Moreover, this study also suggests the robustness of the PDA approach for hemispheric-scale Holocene precipitation reconstructions and its potential for global application.

Methods

Paleoclimate data assimilation

As an emerging approach in paleoclimatology, the PDA aims to fuse paleoclimate information rooted in climatic proxy records (e.g., tree ring, ice core, speleothem) with model simulations to optimally estimate past climate states, following Bayes’ rule and incorporating error estimates from both the paleoclimate records and the prior states²⁶. The PDA is considered a best-of-both-worlds approach for estimating past climate^4,28. Theoretically, there are many data assimilation algorithms that can achieve the fusion of model simulations and paleoclimate records. However, in term of practical applications, Ensemble Square Root Filter (EnSRF, which is a variant of Ensemble Kalman filter)^27,30 and Particle Filtering (PF)^2,37 are the two most widely-used algorithms in PDA. The fundamental idea behind these two algorithms is the Bayesian principle. The differences between these two PDA algorithms lie in how they utilize the prior estimates of climate states provided by pre-existing long-term model simulations and the role of paleoclimate records in assimilation. For instance, the EnSRF constructs a static prior ensemble of climate states by randomly selecting N simulations from pre-existing long-term model simulations. This static prior ensemble is used to calculate the background error covariance matrix at all assimilation time intervals and represents the prior estimate of states at all assimilation time intervals. Consequently, all trends and temporal structures in the final EnSRF-based reconstructions are derived exclusively from the paleoclimate records, while the temporal trajectory from the model simulations is entirely excluded (see details in ref. ²⁷). For the PF, at each assimilation time interval, N simulations closest to the available paleoclimate records are selected from pre-existing long-term model simulations. These selected N simulations are treated as the state “particles”. The reconstruction at a given assimilation time interval is obtained by calculating the weighted average of these state “particles” based on their likelihoods to paleoclimate records. In other words, paleoclimate records serve merely as a reference for selecting N state “particles” from the long-term simulations. Although the temporal trajectory of climate states rooted in the paleoclimate records can be partly incorporated into the PF-based reconstruction by influencing the selection of state “particles” at each assimilation time interval, the spatial structure or patterns reflected in the PF-based reconstruction, essentially, still represent the spatial structure or pattern reflected by the model simulations. Because, fundamentally, the PF-based reconstruction at each assimilation time interval is merely the weighted average of the N selected simulations that most closely resemble the paleoclimate records at that assimilation time interval (see details in ref. ³⁷). In this study, we reviewed the existing data assimilation algorithms and then selected the time-averaged EnOI³⁶ data assimilation algorithm, which also is a variant of the Ensemble Kalman Filter. This choice was made to avoid specific limitations associated with the EnSRF and PF discussed above as much as possible. To clarify the workflow of the EnOI assimilation algorithm for readers, here, we use the EnSRF - a widely adopted method in PDA - as a comparative framework to elucidate EnOI’s operation process (see Fig. 1).

Fig. 1.

Schematic diagram comparing the operation processes of the EnSRF and EnOI assimilation algorithms. Arrows in the diagram indicate computational dependencies, pointing from source variables to derived quantities.

Before assimilation, the EnOI randomly draws N simulations from the pre-existing long-term model simulations to construct a static prior ensemble of climate states, which is then used to derive the background error covariance matrix at all assimilation time intervals, similar to the approach of the EnSRF. However, the key difference between the EnOI and the EnSRF is that the EnSRF uses this pre-selected, static prior ensemble to represent the prior estimate of states at all assimilation time intervals, whereas the EnOI dynamically utilizes the model simulation as the prior estimate of states at the corresponding assimilation time intervals, thus, the prior estimate of states across the assimilation time intervals are time-varying. As a result, the temporal trajectory of the prior estimate of states of the EnOI is the temporal trajectory of the model simulations within the reconstruction period. During the assimilation process, this temporal trajectory of the prior estimate of states is sequentially updated by the paleoclimate records, so that the final EnOI-based reconstruction reflects a temporal trajectory that leverages both the temporal trajectory reflected by the model simulations and the temporal trajectory rooted in the paleoclimate records. Meanwhile, the covariance relationships between climate variables at different locations represented by the background error covariance matrix also lead to the updating of the spatial structure or patterns reflected in the prior estimate of states within the assimilation process. Consequently, in principle, the EnOI algorithm can, to some extent, avoid the shortcomings of the PF and EnSRF algorithms discussed above. The key equation of the EnOI algorithm is typically expressed as follows:

$$x_a=x_b+\mathbf{K}[y_o-\mathbf{H}\left(x_b\right)]$$ 1

Where: $x_b$ and $x_a$ are prior and posterior estimate of climate states, respectively, which are $m\times 1$ vectors where m is the states size. If only precipitation fields are reconstructed then $m$ will be the number of grid points in the spatial precipitation field. The observation vector $y_o$ (i.e., paleoclimate records) is of dimension $p\times 1$, where p is the number of paleoclimate record values available in a given time interval. H(•) denotes observation operator used to map climate state variables from the prior state space to the observation space so that the prior state(s) can be quantitatively compared with the paleoclimate records in the same units. The difference $y_o-\mathbf{H}\left(x_b\right)$ is called the innovation representing the new information in the paleoclimate records not known already from the prior estimate of states. The $\mathbf{K}$ is called Kalman gain matrix playing two key roles in data assimilation, e.g., on the one hand, spatially spreading the climatic information of paleoclimate records to all model grids over the whole study area; on the other hand, weighing the extent to which the prior estimate of climate states can be updated by the paleoclimate records. The Kalman gain matrices $\mathbf{K}$ is calculated as follows:

$$\mathbf{K}={\mathbf{W}}_{\mathit{loc}}◦\mathbf{B}{\mathbf{H}}^{\mathrm{T}}{[{\mathbf{Y}}_{\mathit{loc}}◦\mathbf{HB}{\mathbf{H}}^{\mathrm{T}}+\mathbf{R}]}^{-1}$$ 2

$${\mathbf{A}}_b=\stackrel{\circledR }{{\mathbf{A}}_b}+{\mathbf{A}}_b^{\prime }$$ 3

$$\mathbf{B}{\mathbf{H}}^{\mathrm{T}}=\frac{1}{n-1}{\mathbf{A}}_b^{\prime }{\left[\mathbf{H}\right({\mathbf{A}}_b^{\prime }\left)\right]}^{\mathrm{T}}$$ 4

$$\mathbf{HB}{\mathbf{H}}^{\mathrm{T}}=\frac{1}{n-1}\mathbf{H}\left({\mathbf{A}}_b^{\prime }\right){\left[\mathbf{H}\right({\mathbf{A}}_b^{\prime }\left)\right]}^{\mathrm{T}}$$ 5

Where: T superscripts indicate a matrix transpose, −1 superscripts indicate a matrix inverse. $\mathbf{B}$ is a static background error covariance matrix. Note that, following recommendation in ref. ³⁶, when solving Eq. 2, there is no need to explicitly calculate $\mathbf{B}$; instead, the calculations directly focus on $\mathbf{B}{\mathbf{H}}^{\mathrm{T}}$ and $\mathbf{HB}{\mathbf{H}}^{\mathrm{T}}$. Before assimilation, a static prior ensemble ${\mathbf{A}}_b$ with dimension of $m\times N$ is constructed by randomly selecting N simulations from pre-existing long-term transient climate simulations. The ${\mathbf{A}}_b$ is then decomposed into an ensemble mean (i.e., $\stackrel{\circledR }{{\mathbf{A}}_b}$) and deviations (i.e., ${\mathbf{A}}_b^{\prime }$) from this mean (Eq. 3). The deviation ${\mathbf{A}}_b^{\prime }$ with dimension of $m\times N$ is subsequently used to calculate $\mathbf{B}{\mathbf{H}}^{\mathrm{T}}$ and $\mathbf{HB}{\mathbf{H}}^{\mathrm{T}}$ (Eqs. 4 and 5). $\mathbf{R}$ represents the $p\times p$ error covariance matrix for the paleoclimate records. It is typically assumed that these records are independent of one another, resulting in $\mathbf{R}$ being a diagonal matrix^27,29,38. Each diagonal element of $\mathbf{R}$ corresponds to the error associated with each paleoclimate record, estimated either through a proxy system model^27,29,39,40 or provided by the data contributors^8,10. Furthermore, Eq. 2 suggests that a smaller $\mathbf{R}$ indicates lower error in the paleoclimate records, leading to a larger Kalman gain $\mathbf{K}$. In this case, the prior estimate of states is significantly updated by the paleoclimate records. Conversely, a larger $\mathbf{R}$ corresponds to a smaller $\mathbf{K}$, resulting in minimal correction of the prior by the paleoclimate records. The operator $\mathrm{◦}$ denotes the Schur product. ${\mathbf{W}}_{\mathit{loc}}$ and ${\mathbf{Y}}_{\mathit{loc}}$ are weight matrices for covariance localization performing on the covariance matrix $\mathbf{B}$, acting as a distance-weighted filter to limit the extent of information spread from the paleoclimate records. As proposed by ref. ⁴¹, a fifth-order function with a specified cutoff radius (see detail in ref. ⁴²) is usually adopted to derive the weight matrices. This fifth-order function is similar to a Gaussian function in shape but compactly supported, that is, the weight decreases to zero when the Euclidean distance between a grid point and the observation location exceeds $2\sqrt{10/3}$ times the cutoff radius. The cutoff radius is a user-defined parameter, and the method for determining it will be explained in the section of experimental design.

We employed an off-line (or no-cycling) PDA strategy to reconstruct Holocene NH precipitation fields because, in the context of PDA, the off-line data assimilation strategy offers several advantages over the on-line data assimilation strategy, including time-savings, greater flexibility and ease of implementation^27,29,30. Moreover, the off-line strategy achieves reconstruction skill comparable to that of on-line data assimilation⁴³. Due to the above advantages, the off-line data assimilation strategy has been adopted in the LMR project^27,29 and many other PDA-based reconstructions^{12,13,30–32,38,44}.

Holocene NH precipitation records

In this study, we utilized site-specific annual precipitation (Pann) reconstructions spanning the period of 12-0 ka BP from the LegacyClimate 1.0 dataset¹⁰ to serve as observational data (i.e., observation vector $y_o$ in Eq. 1) for updating the prior estimate of precipitation. The LegacyClimate 1.0 dataset is a pollen-based climate reconstructions from 2,594 NH sites covering the last 30 kyr and is publicly available⁴⁵. The fossil pollen records used to reconstruct the climate variables in the LegacyClimate v1.0 dataset had undergone rigorous screening by the original data creators. For example, as noted by the original data creators¹⁰: all reconstructed climate variables included in the LegacyClimate v1.0 dataset were effectively shown to explain most variance in the modern pollen data (July temperature; annual precipitation). Therefore, the LegacyClimate v1.0 dataset provides NH reconstructions of July temperature, mean annual temperature and annual precipitation (Pann). Of the 2,594 records, 2,421 are located within the 12-0 ka BP interval. The spatiotemporal distributions and temporal resolutions of the 2,421 records are illustrated in Fig. 2. The LegacyClimate v1.0 dataset employed three reconstruction methods: the modern analog technique (MAT), weighted averaging partial least squares regression (WA-PLS), and WA-PLS_tailored. We specifically selected the WA-PLS_tailored-based Pann reconstructions from the LegacyClimate v1.0 dataset as observations to be assimilated. This selection was made based on the recommendations of the original data creators, as this method minimizes the influence of annual and July temperatures on Pann reconstructions, helping to mitigate the issue of covariation observed in modern temperature and precipitation^10,11. In our study, all 2,421 records covering the period of 12-0 ka BP were included without filtering, as the LegacyClimate v1.0 dataset itself represents a publicly-published, peer-reviewed resource that had undergone rigorous screening, quality control, age-uncertainty estimation and reconstruction error assessment. This consideration both acknowledged the stringent quality assurance implemented by the original data creators and aligned with established practices in previous PDA studies^12,13,31. Each Pann record in the LegacyClimate 1.0 dataset comprises a time series along with pertinent metadata. All Pann records had been calibrated to mm/year, and reconstruction uncertainties for each record were provided as root mean square errors (RMSEs). This pre-calibration enabled us to utilize these pre-calibrated records and greatly facilitates assimilation, as done in previous PDA studies^13,37, rather than adopting a proxy system modeling approach^27,29,38. The reconstruction uncertainties, expressed as RMSEs, were converted to mean square errors (MSEs) to serve as the diagonal elements of the covariance matrix $\mathbf{R}$¹³. However, since the calibrated models were trained on a modern surface pollen dataset distributed across the NH¹⁰, these RMSEs might overestimate the error for individual proxy locations^12,31. Therefore, these RMSEs need to be systematically reduced by a specific factor, following the approach of two previous PDA studies^12,31. Please refer to experimental design on how to determine an optimal reduction factor in this study. Furthermore, the Pann records in the LegacyClimate 1.0 dataset are temporally discontinuous, meaning the time intervals between adjacent data points are not constant. In paleoclimate research, the binning approach is preferentially employed to handle temporally discontinuous paleoclimate records. This methodological choice is grounded in the need to: (1) preserve the inherent uncertainty of paleoclimate record chronologies, (2) avoid introducing artificial variability by assuming linearity between data points, and (3) maintain physical consistency with the time-averaged nature of most paleoclimate archives. This preference had been well-documented in previous reconstruction studies^{7,8,12,18,31,46,47}. Therefore, to facilitate the PDA-based reconstruction process, we grouped the original Pann records into uniform time intervals, known as bins, and replaced each bin with the mean value of the data points it contains. The median temporal resolution of the selected 2,421 Pann records is approximately 110 years, with 45.94% of records exhibiting this resolution or finer. This allowed us to set a bin size finer than those used in previous PDA studies (e.g., 200-year bin size in ref. ¹²). Thus, we chose a bin size of 100 years, resulting in 121 bins for the entire reconstruction period (i.e., middle ages of these bins are 0, 0.1, 0.2, …, 11.9 and 12 ka BP). All Pann records were binned accordingly before assimilation (i.e., for each bin of a single record, $y_o$ is the average of all data points contained in that bin), establishing a natural temporal resolution of 100 years for the final PDA-based precipitation reconstruction.

Fig. 2.

Spatiotemporal distributions and temporal resolution of the mean annual precipitation records utilized in the PDA experiments of this study, comprising a total of 2,421 records over the period of 12-0 ka BP. (a) Spatial locations of the 2,421 Pann records, different colors represent the data length associated with the records. (b) Temporal resolutions of all data points within the 2,421 records. (c) Temporal distribution of data points from the 2,421 records, binned into 100-year averages.

Prior ensemble

In the context of off-line PDA, the static prior ensemble of climate states generally is randomly drawn from pre-existing long-term transient climate simulations^{13,18,27,29,30,38,44} or from time-slice climate simulations of key epochs^12,31. Additionally, according to previous PDA research cases, most studies constructed the static prior ensemble of climate states from a single long-term transient simulation^{18,27,29,30,38,44}. However, PDA-based reconstruction of instrumental era implemented by ref. ⁴⁸ had demonstrated that using a static prior ensemble derived from multi-model simulations considerably improves reconstruction skill in PDA, as these ensembles better capture the covariance patterns between different locations and climate variables compared to ensembles drawn from single-model simulations. In this study, we established the following criteria for screening pre-existing transient climate simulation to construct the static prior ensemble of PDA: (1) it must be freely and publicly available to us; (2) it must fully cover the period of 12-0 ka BP; (3) it must be implemented based on fully-coupled climate model rather than intermediate-complexity earth system models with coarse spatial resolution and simplified physical processes; and (4) it must be non-accelerated simulation because the accelerated simulation lead to a partial delay in the feedback of climate processes⁴⁹. Only transient climate simulation meeting all four criteria were considered in this study. Finally, two pre-existing transient climate simulations could fully meet the criteria outlined above. The first simulation was the TraCE 21 ka simulation (accessed through the TraCE 21ka website: https://www.cesm.ucar.edu/working-groups/paleo/simulations/trace-21ka), which provided a transient climate evolution for the past 21 ka BP at horizontal resolution of about 3.75° × 3.75° for atmosphere³⁴. This simulation was conducted using the Community Climate System Model version 3 (CCSM3) with full forcing. The second simulation, the PMIP4 HadCM3 transient climate simulation⁵⁰ covered the past 23 ka BP and was conducted using the UK Met Office’s HadCM3 climate model at horizontal resolution of 2.5° × 3.75° for atmosphere³⁵. For clarity, we referred to these two transient simulations as TraCE and HadCM. Detailed descriptions of each transient climate simulation can be found in ref. ³⁴ and ref. ³⁵, respectively. Before implementing the PDA experiment, we averaged the monthly precipitation of the two simulations to a 100-years resolution (i.e., 100-years-averaged annual precipitation, in mm/year), ensuring that the time-scale represented by the prior ensemble aligns with that of the binned Pann records. Here, we emphasize that in PDA, exact time-scale matching between observations and prior estimates represents an perfect case. In previous PDA-based spatially-resolved global temperature reconstructions since the last glacial maximum^12,31 constrained by the limited number of time-slice model simulations - which generally could not provide a sufficient number of simulations - researchers typically averaged time-slice model simulations over shorter time intervals (e.g., 50 years) compared to the time-scale of paleoclimate records (e.g., 200 years) to generate a greater number of time-averaged simulations for constructing static prior ensemble ${\mathbf{A}}_b$ through random sampling. In addition, we employed a simple time-invariant bias correction approach⁵¹, following the methodology adopted by ref. ³⁰ for PDA-based hydroclimate reconstructions of the Common Era, to partly mitigate systematic biases inherent in the TraCE and HadCM precipitation simulations. We first quantified the model biases by calculating the differences between the 20^th-century-mean annual accumulated precipitations of the TraCE and HadCM (i.e., the bin of 0 ka BP) and that of the 20^th Century Reanalysis⁵² (accessed through the 20^th Century Reanalysis project website: https://psl.noaa.gov/data/20thC_Rean/). The 20^th Century Reanalysis was the sole reanalysis product that offers both complete spatial coverage of the NH and temporal coverage spanning the entire 20^th century, enabling a spatiotemporally-complete comparison with model simulations at the centennial scale. These stationary biases were subsequently removed from the model simulations to produce corrected precipitation fields. Moreover, in the PDA experiments conducted in this study, we constructed the static prior ensemble for PDA from both single-model simulations (e.g., either TraCE or HadCM) and multi-model simulations (such as a combination of both TraCE and HadCM). This approach allowed us to examine whether the conclusion drawn by ref. ⁴⁸ hold true for long-term paleoclimate scenarios like the Holocene. To enable the two simulations to be used together in a multi-model prior ensemble, the HadCM precipitation simulation was regridded to 3.75° × 3.75° using the bilinear interpolation method included in the Climate Data Operator (https://code.mpimet.mpg.de/projects/cdo), as done by ref. ¹³. It is crucial for a well-constructed model prior ensemble to be relevant to the period of interest, accurately capturing realistic relationships among different climate variables, and to be long enough to quantify these relationships on paleoclimate-relevant timescales^12,13. Thus, the static prior ensemble was drawn solely from the Holocene portion (12-0 ka BP) of the two transient simulations, yielding a total of 242 simulations of 100-year-averaged annual precipitation fields for its construction. These simulations were derived equally from TraCE and HadCM, with 121 from each model. Each simulation corresponded to a specific 100-year time bin, centered at 0, 0.1, 0.2, …, 11.9, and 12 ka BP.

Experimental design

We designed two types of PDA experiments. The first was a sensitivity experiment aiming at determining an optimal cutoff radius for covariance localization and an optimal scaling factor for the RMSEs associated with Pann records in the LegacyClimate 1.0 dataset. The second was a formal reconstruction experiment designed to produce Holocene annual precipitation fields over the NH. In these PDA experiments, the static prior ensemble size was set to 100, consistent with the static prior ensemble size of the LRM project^27,29. This size exceeded that used by ref. ³¹ for reconstructing global temperature fields of the Last Glacial Maximum. Moreover, previous PDA sensitive experiment had shown that PDA results are insensitive to this choice as long as the static prior ensemble includes at least 50 members²⁷.

The workflow of the PDA sensitivity experiment conducted in this study is illustrated in Fig. 3. In this experiment, we performed a series of PDA reconstruction (each referred to as a realization) with varying cutoff radius values ranging from 100 km to 20,000 km (e. g., 100 km, 250 km, 500 km, 1,000 km, 2,000 km, 4,000 km, …, 18,000 km, 20,000 km) and scaling factor values between 0.01 and 1.2 (e.g., 0.01, 0.05, 0.1, 0.2, …, 1.0, 1.1, 1.2). The cutoff radius was limited to 20,000 km because precipitation is highly heterogeneous across various spatial and temporal scales²⁴; therefore, a narrower cutoff radius for PDA-based precipitation reconstruction is theoretically more realistic than the broader cutoff radius typically used for temperature reconstructions (e.g., 12,000 km in ref. ³¹ and 25000 km in ref. ²⁹). The two transient simulations - TraCE and HadCM - were used to create three types of data sources: TraCE-only, HadCM-only and a mixed TraCE-HadCM combination. Accordingly, the PDA sensitivity experiment were categorized into three groups based on these data sources, designated as PDA (TraCE), PDA (HadCM) and PDA (Mixed), respectively. Each group comprised 210 distinct realizations (i.e., 15 cutoff radii and 14 scaling factors, resulting in 210 combinations of cutoff radius and scaling factor, each realization corresponds to a distinct combination of cutoff radius and scaling factor). Thus, each group produced 210 distinct reconstructions. Following the offline-PDA implementation steps, we constructed the static prior ensemble (e.g., ${\mathbf{A}}_b$ in Eq. 3) through randomly selecting 100 members from the 100-years-averaged annual precipitation simulations. Moreover, we also randomly selected 75% of the 2,421 Pann records (i.e., 1,816 records) for assimilation, while withholding the remaining 25% of the Pann records (i.e., 605 records) as independent data to validate the reconstructions from the PDA sensitivity experiment. This approach followed a standard cross-validation methodology to assess reconstruction skill. For each group of PDA sensitivity experiments, the same 100-member static prior ensemble and the same set of assimilated Pann records (75% of the total) were used in every realization. This consistency in the prior ensemble and assimilated data across all realizations within each group eliminated the potential influence of variations in these inputs on the final reconstructions. As a result, we were able to specifically quantify the sensitivity of the reconstructions to changes in the cutoff radius and scaling factor. Ultimately, the PDA sensitivity experiment results (see Fig. 4) revealed that a cutoff radius of 4,000 km and a scaling factor of 0.1 yielded the most optimal assimilation result for PDA (TraCE). For PDA (HadCM), a cutoff radius of 500 km and a scaling factor of 0.2 were found to be optimal. Lastly, for PDA (Mixed), a cutoff radius of 500 km and a scaling factor of 0.3 provided the best assimilation results. These relatively short cutoff radii revealed here were primarily attributed to the high spatiotemporal variability of precipitation. They also aligned with the established principle that using fewer proxy records in PDA generally corresponds to a broader cutoff radius, and vice versa (see details in ref. ¹²). Consequently, we applied these optimal cutoff radii and scaling factors to the PDA formal experiment.

Fig. 3.

The workflow of the PDA sensitivity experiment of this study.

Fig. 4.

The influence of varying cutoff radii and scaling factors on PDA performance derived from the PDA sensitivity experiment. For each group of experiment, we quantified PDA performance by computing the Pearson correlation coefficient between the PDA-based reconstructions (generated under different cutoff radius and scaling factor combinations) and the withheld 25% of Pann records, which served as independent validation data. Specifically, the correlation coefficients were obtained by comparing each PDA-based reconstruction pointwise with the withheld Pann records at their respective grid locations. The mean correlation coefficient for each parameter combination is shown in this figure and served as a quantitative metric for evaluating PDA performance.

Next, we outline the process of the formal reconstruction experiment (see Fig. 5). Based on the sources of the prior ensembles, the formal reconstruction also comprised three groups of PDA experiments: PDA (TraCE), PDA (HadCM) and PDA (Mixed). Here, we emphasize once more: for the EnOI-based PDA (TraCE) experiment in this study, the static prior ensemble ${\mathbf{A}}_b$ was constructed by randomly selecting N simulations from the 121 TraCE simulations. The prior estimate of states $x_b$ at each assimilation time interval directly utilized the TraCE simulation at the corresponding time interval. For the EnOI-based PDA (HadCM) experiment, the static prior ensemble ${\mathbf{A}}_b$ was constructed by randomly selecting N simulations from the 121 HadCM simulations. The prior estimate of states $x_b$ at each assimilation time interval directly adopted the HadCM simulations at the corresponding time interval. For the EnOI-based PDA (Mixed) experiment, the static prior ensemble ${\mathbf{A}}_b$ was constructed by randomly selecting N simulations from the 242 simulations of both TraCE and HadCM. The prior estimate of states $x_b$ at each assimilation time interval was obtained by averaging the TraCE and HadCM simulations corresponding to that time interval.

Fig. 5.

The workflow of the PDA formal experiment conducted in this study. The PDA formal experiment comprised three distinct configurations based on different simulation sources: (a) the first group used TraCE as the prior source; (b) the second group used HadCM as the prior source; (c) the third group incorporated both TraCE and HadCM as prior sources.

For each group of PDA experiment, we employed the optimal cutoff radius and scaling factor combination that had been identified for that specific group. We then performed the PDA reconstruction 200 times using a Monte Carlo approach (each iteration referred to as a realization), following the methodology of previous PDA studies^27,29,38,39. In each realization, we randomly selected a 100-member static prior ensemble ${\mathbf{A}}_b$ from the 100-years-averaged simulations and assimilated 75% of the 2,421 Pann records. Finally, each group of the PDA experiments thus produced 200 reconstructions of Holocene NH precipitation fields, covering the period from 12 to 0 ka BP at a spatial resolution of 3.75° and a temporal resolution of 100 years. These 200 reconstructions differed in the static prior ensemble ${\mathbf{A}}_b$ used (randomly drawn from the 100-years-averaged simulations) and the subset of Pann records used (75% of the 2,421 Pann records). The Monte Carlo approach ensured that the static prior ensemble ${\mathbf{A}}_b$ for each realization within a given group exhibited distinct variances and spatial covariance structures, while also capturing the diversity of the assimilated Pann records. This method further incorporated uncertainties associated with both the Pann records and model priors into the final assimilated results^27,29,38,39. Additionally, we performed 200 Monte Carlo realizations of reconstruction in each group of PDA experiments, each realization assimilating a randomly sampled 75% of the Pann records while using the remaining 25% for independent validation. This design ensured statistical independence between the assimilation and validation datasets in every realization.

Data Records

The Holocene NH annual precipitation reconstructions generated by the PDA (TraCE), PDA (HadCM) and PDA (Mixed) experiments are publicly available in the Zenodo data repository⁵³ as NetCDF4 files. The dataset includes: (1) 200 precipitation reconstructions derived from the Monte Carlo realizations for each experimental group, each realization represents an independent reconstruction using a randomly drawn 100 member prior ensemble and a random 75% proxy subset (with 25% withheld for validation); (2) the corresponding mean and ±1 standard deviation calculated from each set of 200 reconstructions. The NetCDF4 data file also includes all associated variable metadata. Figures 6 and 7 display the millennium-scale mean precipitation fields and precipitation anomaly fields, respectively, derived from the PDA (Mixed) reconstruction saved in the Zenodo data repository⁵³.

Fig. 6.

The spatial maps of mean annual precipitation fields at millennium intervals, derived from the PDA(Mixed) reconstruction.

Fig. 7.

The spatial maps of precipitation anomaly fields at millennium intervals with reference to the 12-0 ka BP mean, derived from the PDA(Mixed) reconstruction.

Technical Validation

For verification, the ideal approach would be to compare the PDA-based reconstruction with instrumental observation to assess their spatiotemporal consistency^27,29,30,38. However, this was not feasible for PDA-based Holocene climate reconstruction due to the limitations posed by the temporal resolution of PDA-based Holocene climate reconstruction (i.e., 100-year averages) and the extreme scarcity of observation from the early instrumental era. Consequently, in this study, we employed three alternative approaches to validate the PDA-based Holocene precipitation reconstructions. First, in each of the 200 Monte Carlo realizations in the formal reconstruction experiments, we randomly withheld 25% of the Pann records (i.e., 605 records) from assimilation. These withheld records served as independent validation data within each realization. After completing each reconstruction experiment (i.e., one realization), we extracted precipitation series from the PDA-based reconstruction at the grids corresponding to the locations of the withheld Pann records. These extracted precipitation series were then compared pointwise with the corresponding withheld Pann records at the same locations.

Additionally, we validated the PDA-based precipitation reconstructions against 70 independent annual precipitation proxy records, including δ¹⁸O from speleothems, ice accumulation from glacier ice and other proxy types sourced from various archives (see Fig. 8). These 70 precipitation proxy records were selected from a Holocene precipitation proxy compilation⁵⁰ published by ref. ¹⁹ (hereinafter referred to as Hancock2023). This compilation consists of 813 proxy records, with 508 indicating changing precipitation and 305 reflecting effective moisture (precipitation minus evaporation) across the globe. These records represent precipitation or effective moisture during different seasons (e.g., annual, summer, growing season and winter). Notably, 38% of Hancock2023 records are pollen-based precipitation reconstructions from the LegacyClimate 1.0 dataset¹⁰. Some records in Hancock2023 had been calibrated to precipitation (either absolute values or anomalies), while uncalibrated records are reported in their native proxy variables (e.g., ice accumulation in ice glacier, δ¹⁸O in speleothem). The selection of records from Hancock2023 for validating our PDA-based Holocene precipitation reconstructions followed these criteria: (1) records must be explicitly identified as annual precipitation by the original author(s), ensuring consistency with the temporal representativeness of our PDA-based reconstruction; (2) records must be geographically located within the NH, aligning with the spatial coverage of our PDA-based reconstruction; (3) records after temporal binning must contain at least 10 data points spanning the 12-0 ka BP period to meet the basic requirement for robust correlation analysis; and (4) records must not be sourced from the LegacyClimate 1.0 dataset, as these have already been assimilated in our PDA framework, thereby ensuring independent validation. Ultimately, 70 precipitation records (see Fig. 8) met these criteria and were used as an independent validation dataset. Moreover, some readers may question why the Hancock2023 dataset was not assimilated in this study. Our decision was based on the following considerations: (1) the spatial coverage of Hancock2023 significantly overlaps with that of the LegacyClimate 1.0 across most NH regions, and the latter contains a substantially larger number of records; (2) we required a completely independent validation dataset to objectively evaluate the spatial performance of the PDA-based reconstruction. Hancock2023 includes several records in North Africa - a region not covered by the LegacyClimate 1.0 - which enables validation in data-sparse areas distant from assimilated records.

Fig. 8.

The spatial distribution and types of the 70 annual precipitation records used to verify the PDA-based reconstruction. The selected records include various types such as δ¹⁸O from speleothems, ice accumulation from glacier ice and other proxy data sourced from different archives.

Moreover, we compared annual precipitation derived from the TraCE and HadCM simulations against the same validation dataset. This comparison aimed to evaluate whether assimilating Holocene Pann records improved precipitation estimates over those provided by the TraCE and HadCM simulations alone. In this study, we adopted the Pearson correlation coefficient (r) to measure the strength and direction of a linear relationship between the reconstructed (or simulated) series and the observational series. The coefficient of efficiency (CE) was used to evaluate the agreement between the reconstructed/simulated series and the reference data. Assuming that two time series X and Y of length N represent the reconstruction (or simulation) series and the reference series at the same grid, respectively, then the r and CE values were calculated as follows:

$$r=\frac{\sum _{i=1}^N(X_i-\stackrel{\circledR }{X})(Y_i-\stackrel{\circledR }{Y})}{\sqrt{\sum _{i=1}^N{(X_i-\stackrel{\circledR }{X})}^2}\sqrt{\sum _{i=1}^N{(Y_i-\stackrel{\circledR }{Y})}^2}}$$ 6

$$CE=1-\frac{\sum _{i=1}^N{(Y_i-X_i)}^2}{\sum _{i=1}^N{(Y_i-\stackrel{\circledR }{Y})}^2}$$ 7

Where: the overbar denotes the mean of one series.

To further assess the spatial skill of the PDA-based precipitation reconstruction, we calculated the continuous ranked probability skill score (CRPSS)⁵⁴, which is generally a more stringent skill metric than correlation³⁰. This metric is derived from the continuous ranked probability score (CRPS), a strictly proper scoring rule that evaluates the skill of posterior reconstruction distributions by penalizing inaccuracies in bias, variance, phasing and inappropriate ensemble spread (either too wide or overconfident). For CRPSS calculation, we calculated the 20^th-century-averaged annual accumulated precipitation (corresponding to the reconstruction bin of 0 ka BP or 1900-1999 AD) from the 20^th Century Reanalysis dataset⁵², which was served as the observational data. Given the approximately normal distribution of the 200 PDA-based precipitation reconstructions yielded from the 200 realizations within each group at a given reconstruction time interval, the CRPSS values for all three reconstruction groups were derived at each grid using Eqs. (8, 9)⁵⁴.

$$\mathit{CRPSS}=1-\frac{{\mathit{CRPS}}_{\mathit{rec}}}{{\mathit{CRPS}}_{\mathit{ref}}}$$ 8

$$\mathit{CRPS}=\sigma \left\{y_n\left[2f\left(y_n\right)-1\right]+2g\left(y_n\right)-\frac{1}{\sqrt{\pi }}\right\}$$ 9

Where: $y_n=(y-\mu )/\sigma$, with $y$ being the observational value (specifically, the 20^th-century-averaged annual accumulated precipitation from the 20^th Century Reanalysis in this study), and $\mu$ and $\sigma$ representing the mean and standard deviation, respectively, of the 200 PDA-based precipitation reconstructions yielded from the 200 realizations in the 0 ka BP reconstruction bin. Here, $f\left(y_n\right)$ and $g\left(y_n\right)$ denote the normal cumulative distribution function and the normal probability density function of $y_n$, respectively. This formulation assumes observation error is negligible. The reference distribution ${\mathit{CRPS}}_{\mathit{ref}}$ corresponds to the initial uninformed prior. We prefered CRPSS over CRPS because its bounded range [−∞, 1] explicitly quantifies skill improvement (positive values indicate reconstructed distribution is more skillful for this metric than the uninformed prior), whereas CRPS has an unbounded range [0, ∞).

Cross-validation across all realizations

In each of the 200 Monte Carlo realizations of the formal reconstruction experiment, we randomly withheld 25% of the Pann records from assimilation. After competing each realization, we extracted precipitation series from the three groups of PDA-based reconstructions and the two transient simulations for the grids corresponding to the locations of the withheld Pann records. The r values and CE values between these extracted precipitation series and the corresponding withheld Pann records at the same grids were then calculated using Eqs. (6, 7) for each realization. Grids with fewer than 10 overlapping data points between the extracted series and withheld records were excluded from statistical analysis due to insufficient sample size. Figure 9 illustrates the variation in r and CE values. To facilitate analysis, we organized the r and CE values from all 200 realizations within each group into ascending row vectors, defining the first and third quantiles as Q1 and Q3. The Q1, mean, median and Q3 values of the r and CE values associated with the five validated data are summarized in Table 1. We also performed pairwise t-tests (a statistical method used to determine whether there is a significant difference between the means of two samples) on the r and CE values samples. The pairwise t-tests revealed the following: the r value samples corresponding to the five validated datasets differed from each other with statistical significance (p < 0.05). For the CE values, statistically significant differences were observed between all pairs of CE samples, except between those of the PDA (HadCM) and PDA (Mixed).

Fig. 9.

Variation in r and CE values, as well as both their means with ±1 standard deviation (±1σ) ranges across all realizations, along with corresponding boxplots depicting the statistical distributions of r and CE values. The r and CE values were obtained from the cross-validation conducted across all realizations.

Table 1.

Statistics of r and CE values obtained from the cross-validation.

Item	PDA (TraCE)	PDA (HadCM)	PDA (Mixed)	TraCE	HadCM
r mean	0.3217	0.3247	0.3305	0.2478	0.2341
r median	0.3126	0.3181	0.3223	0.2550	0.2288
r Q1	0.0512	0.0573	0.0580	−0.0244	−0.0402
r Q3	0.6182	0.6165	0.6333	0.5430	0.5149
CE mean	−0.4652	−0.4978	−0.4967	−0.5870	−0.6293
CE median	−0.4007	−0.4395	−0.4388	−0.4729	−0.5043
CE Q1	−0.8974	−0.9603	−0.9544	−0.9608	−1.0305
CE Q3	0.0793	0.0627	0.0866	−0.0370	−0.0879

Cross-validation suggested that the correlations between the three PDA-based reconstructions and the withheld Pann records were markedly greater than those for the two model simulations (i.e., TraCE and HadCM). Specifically, compared to TraCE simulations, the three PDA-based reconstructions showed mean correlation improvements of 29.84%, 31.02% and 33.39%, with median improvements of 22.58%, 24.75% and 26.39%. While compared to HadCM simulations, they exhibited mean correlation improvements of 37.42%, 38.66% and 41.17%, with median improvements of 36.59%, 39.00% and 41.17%. Furthermore, positive correlation rates with withheld records differed substantially among the datasets. The PDA (TraCE), PDA (HadCM) and PDA (Mixed) reconstructions showed positive correlations for 79.13%, 79.94% and 80.12% of the withheld Pann records, respectively, compared to 72.77% for TraCE and 71.98% for HadCM simulations. The above r statistics demonstrated that the PDA-based reconstructions exhibited statistically superior consistency with the withheld Pann records in terms of trends and directional variations, confirming enhanced capability to reproduce Holocene precipitation trends through data assimilation.

Additionally, unlike the r values, both the mean and median values of the CE were negative across all the three PDA-based reconstructions. However, we emphasize that this does not imply a lack of skill in data assimilation process. As noted in previous, the CE metric is sensitive to amplitude and bias, and negative CE values can occur even in skillful reconstructions if the means of the compared datasets differ^27,55. In the cross-validation, the CE values for all three PDA-based reconstructions were consistently higher than those of the TraCE and HadCM simulations. Specifically, relative to TraCE, the three PDA-based reconstructions showed mean CE improvements of 20.76%, 15.20% and 15.39%, with median improvements of 15.26%, 7.07% and 7.21%. Compared to HadCM, the mean CE improvements increased to 26.08%, 20.90% and 21.08%, while median improvements reaching 20.54%, 12.86% and 12.99%. The negative CE statistics observed here primarily aroused from amplitude and bias differences between the two datasets to be compared with each other. For example, the annual precipitation range in the two transient climate simulations was 748.82 ± 699.12 mm/year, while the range from the 2,421 pollen-based Pann records was 835.27 ± 365.42 mm/year. This highlighted substantial differences in both amplitude and bias of annual precipitation between the prior estimates and the assimilated Pann records. We used these transient climate simulations to construct the prior ensemble, although these prior estimates were updated by the Pann records during assimilation, the differences in amplitudes and biases between the PDA-based reconstructions and the Pann records were not fully eliminated, contributing to the poor CE performance. Although more sophisticated bias-correction methods than the one employed in this study might improve CE values, we refrained from adopting this step for two main reasons. First, our objective was to reconstruct absolute precipitation rather than precipitation anomalies or the fraction of precipitation relative to the reference period mean. Second, and more importantly, no universal bias-correction method currently exists that can adequately remove biases in climate simulations^56,57. It remains uncertain whether existing bias-correction methods developed for short-term historical simulations⁵⁸ or future projections⁵⁹ can be effectively applied to Holocene simulations⁶⁰, given the vastly different climate contexts. Overall, the cross-validation demonstrated substantial improvements in both r and CE across all three PDA-based reconstructions compared to model simulations alone, thereby confirming improved data quality achieved through data assimilation. Furthermore, the PDA (Mixed) reconstruction exhibited superior performance over both PDA (TraCE) and PDA (HadCM) reconstructions, as evidenced by greater improvements in both r and CE metrics. This finding corroborates the conclusion drawn by ref. ⁴⁸ that multi-model prior ensembles enhance PDA-based reconstruction skill, demonstrating its applicability to PDA-based reconstructions over longer timescales such as the Holocene.

Verification against Holocene precipitation records

We conducted an independent verification to assess the temporal consistency between the 70 independent annual precipitation records and the PDA-based precipitation reconstruction, as well as the TraCE and HadCM simulations. Since more than half of the 70 independent annual precipitation records are uncalibrated (expressed in their original units), this limitation precludes the calculation of the CE and CRPSS values. Therefore, we focused exclusively on the correlations between these 70 independent annual precipitation records and the corresponding precipitation series extracted from either the PDA-based reconstructions or the two model simulations at the grids of the 70 annual records situated. This approach allowed us to assess the capacity of PDA-based reconstructions to accurately reflect the variation trends in Holocene precipitation. All of the 70 independent annual precipitation records were binned using the same method applied to the 2,421 Pann records from the LegacyClimate 1.0 dataset. The spatial distributions of r values obtained from independent verification are shown in Fig. 10. Similarly, pairwise t-tests were conducted on the r value samples corresponding to the five validated datasets. There were no statistically significant differences among the r value samples corresponding to the three PDA-based reconstructions. However, statistically significant differences were detected between the r value samples of the PDA-based reconstructions and those of the model simulations, with the exception of the comparison between the PDA (HadCM) reconstruction and the TraCE simulation.

Fig. 10.

Independent verification results. (a) Spatial distribution of r values between the PDA (TraCE) precipitation reconstruction and the 70 annual precipitation records. (b–e) Same as (a) but for the PDA (TraCE) and PDA (Mixed) precipitation reconstructions, as well as the TraCE and HadCM simulations, respectively. The filled circles in (a–e) are colored according to their r values, with specific ranges detailed in the legend. (f) Boxplot of the r values associated with the five validated datasets.

The three PDA-based reconstructions showed positive correlations with 78.57%, 72.86% and 80.00% of the 70 precipitation records, respectively, whereas the TraCE and HadCM simulations showed positive correlations with 65.71% and 67.14%, respectively. Specifically, compared to the TraCE simulation, the three PDA-based reconstructions showed mean correlation improvements of 34.74%, 24.94% and 53.71%, with median improvements of 68.57%, 41.51% and 87.98%. While compared to the HadCM simulation, they exhibited mean correlation improvements of 127.26%, 110.73% and 159.26%, with median improvements of 180.72%, 135.65% and 213.05%. The details of the mean, median, Q1 and Q3 of r values are summarized in Table 2. Together, Fig. 10 and Table 2 indicate that the three PDA-based precipitation reconstructions exhibit better temporal consistency with the 70 precipitation records than the TraCE and HadCM simulations. Thus, if these 70 precipitation records are considered representative of the “true trajectory” of Holocene precipitation at their respective sites, the improved consistency of the three PDA-based reconstructions suggests that data assimilation improves the accuracy of precipitation estimates compared to model simulations alone, further confirming the effectiveness of the PDA approach.

Table 2.

Statistics of r values obtained from the independent verification.

Item	PDA (TraCE)	PDA (HadCM)	PDA (Mixed)	TraCE	HadCM
r mean	0.3048	0.2826	0.3477	0.2262	0.1341
r median	0.3214	0.2698	0.3584	0.1907	0.1145
r Q1	0.0448	−0.0203	0.0433	−0.0478	−0.1151
r Q3	0.5419	0.6389	0.6559	0.5735	0.3520

Since we obtained the r values at all of the sites of the 70 annual precipitation records from the independent verification, we organized the 70 r values into row vector in an ascending order, respectively, the Q1 and Q3 were the first and third quantiles in the row vector.

Comparison of zonal-averaged precipitation series

To evaluate the consistency of Holocene precipitation variations across latitudinal zones, we calculated the zonal-averaged precipitation series from the 70 independent annual precipitation records, the three PDA-based reconstructions and the TraCE and HadCM simulations (see Figs. 11 and 12). Since all 70 independent precipitation records are located over land or coastal areas (see Fig. 8), the calculation of zonal-averaged precipitation in this study was strictly limited to land grid points in both the PDA-based reconstructions and model simulations. This consideration ensured consistent spatial representativeness between datasets in comparative analyses, as emphasized by ref. ⁶¹. In addition, since more than half of the 70 annual precipitation records from the Hancock2023 are uncalibrated (expressed in their original units), we converted each record to Z-scores before calculating zonal averages, following the method of ref. ⁸. This conversion facilitated the mathematical aggregation of diverse precipitation records with different proxy types and units into coherent zonal-averaged series. We first calculated the correlations between the NH-averaged precipitation series derived from the PDA-based reconstructions and model simulations and the NH-averaged Z-score series derived from the 70 Hancock2023 records. The r values between the three PDA-based reconstructions (i.e., PDA (TraCE), PDA (HadCM) and PDA (Mixed)) and the Hancock2023 series were 0.7312 (p < 0.01), 0.8432 (p < 0.01) and 0.8595 (p < 0.01), respectively, while those between the two model simulations (i.e., TraCE and HadCM) and the Hancock2023 were 0.8170 (p < 0.01) and 0.8423 (p < 0.01). In addition, all series (except PDA (TraCE)) in Fig. 11 commonly display a rapid increase from the Early Holocene to the Middle Holocene, peaking around 6 ka BP, followed by a gradual decline. This trend aligns with the findings of previous studies^11,19,21 regarding the trend in NH-averaged annual precipitation during the Holocene. Although the PDA (TraCE) reconstruction (see Fig. 11a) also shows a general increase and subsequent decrease, it peaks later, at approximately 4.5 ka BP, diverging from the other series. Overall, the series shown in Fig. 11 display positive and high correlations (e.g., ranging from 0.7312 to 0.8595, p < 0.01). These correlations indicate good agreement in both the direction and overall trend of NH-averaged Holocene annual precipitation among the PDA-based reconstructions, model simulations, and precipitation proxy records. This consistency enhances the high reliability of the three PDA-based Holocene NH reconstructions in capturing the NH-averaged precipitation trend during the Holocene. Furthermore, a comparison of Fig. 11a with Fig. 11b,c further reveals that the PDA (TraCE) reconstruction exhibits greater interannual variability of precipitation (measured as the standard deviation of the detrended series, following ref. ⁶²) and wider uncertainty ranges (represented by the 95% confidence interval, following ref. ⁶³), compared to the PDA (HadCM) and PDA (Mixed) reconstructions. Specifically, the PDA (TraCE) reconstruction has an interannual variability of 12.99 mm/year and a mean 95% confidence interval of ±17.13 mm/year, compared to 7.51 mm/year and ±4.76 mm/year for the PDA (HadCM), and 7.09 mm/year and ±3.88 mm/year for the PDA (Mixed), respectively. We propose that the observed discrepancies likely originate from the inherently higher interannual variability and broader ranges in the TraCE precipitation simulation, which has an interannual variability of 8.72 mm/year and variation range of 58.58 mm (from 411.62 mm/year to 470.20 mm/year). This is in comparison to the HadCM precipitation simulation, which has an interannual variability of 7.02 mm/year and variation range of 41.57 mm (from 485.50 mm/year to 527.07 mm/year). According to the principle of EnOI data assimilation³⁶, EnOI-based reconstructions that utilize prior estimate of states (i.e., $x_b$) derived from pre-existing model simulations with higher variability and employ static prior ensembles (i.e., ${\mathbf{A}}_b$) randomly drawn from a broader range of simulations, tend to yield reconstructions with greater variability and wider uncertainty estimates, and vice versa. Furthermore, the above analysis underscores that while PDA-based reconstructions using different model priors may agree on the overall direction of precipitation change, they can differ substantially in estimated variability and uncertainty. Therefore, when exploring the sensitivity of PDA-based reconstructions to different model simulations used for randomly sampling the static prior ensemble, it is essential to evaluate not only the consistency of overall trends but also the differences in variability and uncertainty ranges among the resulting reconstructions.

Fig. 11.

The NH-averaged precipitation series derived from PDA-based reconstructions and model simulations, along with the NH-averaged precipitation Z-scores derived from the 70 precipitation records after binning. The gray shading in (a–c) represents the 95% confidence interval derived from the PDA-based 200 reconstructions within each group of the PDA-based reconstruction experiments. Note: since all 70 precipitation records are located over land or coastal areas, the NH-averaged precipitation series presented specifically represents precipitation across the NH land (here and throughout).

Fig. 12.

Holocene zonal-averaged precipitation variability and inter-series correlations across different latitude bands. The three columns present the zonal-averaged annual precipitation series over high-latitudes (60–90°N), mid-latitudes (30–60°N) and low-latitudes (0–30°N), respectively.

We further compared the Holocene precipitation series across different NH latitude bands using the datasets described above (see Fig. 12). In high-latitudes, the PDA (Mixed) reconstruction demonstrated the highest correlation with the Hancock2023 (r = 0.7518, p < 0.01), followed by the PDA (HadCM) (r = 0.7348, p < 0.01) and the PDA (TraCE) (r = 0.7144, p < 0.01). The two model simulations exhibited lower correlations: TraCE (r = 0.7057, p < 0.01) and HadCM (r = 0.6564, p < 0.01). A similar performance hierarchy was observed in mid-latitudes, where the PDA (Mixed) again achieved the highest correlation (r = 0.7867, p < 0.01), outperforming both the PDA (HadCM) (r = 0.7677, p < 0.01) and the PDA (TraCE) (r = 0.7516, p < 0.01). The model simulations showed consistently lower correlations: TraCE (r = 0.7498, p < 0.01) and HadCM (r = 0.7175, p < 0.01). Low-latitudes results revealed a different pattern. The PDA (Mixed) maintained the strongest correlation (r = 0.7454, p < 0.01), both the PDA (HadCM) (r = 0.6437, p < 0.01) and especially the PDA (TraCE) (r = 0.2648, p < 0.01) underperformed relative to the PDA (Mixed). Moreover, their correlations fell below those of the two model simulations (TraCE: r = 0.7289, p < 0.01; HadCM: r = 0.7143, p < 0.01). Notably, the PDA (TraCE)’s correlation was substantially lower than all other datasets. Collectively, all three PDA-based reconstructions exhibited their highest correlations with the Hancock2023 in mid-latitudes, followed by high-latitudes, with relatively lower correlations in low-latitudes. However, we would like to clarify that in low-latitudes, only the PDA(TraCE) reconstruction demonstrated notably poor skill (e.g., r = 0.2648). In fact, both the PDA(Mixed) and PDA(HadCM) still achieved good reconstruction performance (with r values of 0.7454 and 0.6437, respectively), while these two r values were slightly lower than those obtained in mid- and high-latitudes, they nevertheless represented meaningful skill. Additionally, in low-latitudes, although the PDA-based reconstructions based on single model priors underperformed compared to model simulations alone, the performance of PDA-based reconstruction could still be improved by selecting priors using multi-model simulations. This finding further validates the conclusion of ref. ⁴⁸ regarding the benefit of multi-model ensembles in PDA, as the PDA (Mixed) consistently outperformed both single-model assimilations and raw model simulations across all latitude bands. Moreover, Fig. 12k–o reveal pronounced discrepancies in precipitation peak magnitudes among the different series in low latitudes, highlighting the complexity of accurately reconstructing and simulating Holocene precipitation in these regions and underscoring the need for future efforts to improve the accuracy and consistency of these datasets in low-latitudes. Another notable feature in Fig. 12 is the pronounced decrease in precipitation around 8.2 ka BP shown in the TraCE simulation. In contrast, this feature is absent from the three PDA-based reconstructions, the Hancock2023 series and the HadCM simulation. The absence in the PDA reconstructions can be attributed to the lack of a corresponding signal in the assimilated LegacyClimate 1.0 dataset¹¹. Similarly, the Hancock2023 compilation shows no clear evidence of such a sudden precipitation decrease around 8.2 ka BP¹⁹. The reasons for its absence in the HadCM simulation have not been documented in the literature.

Skill assessment relative to the 20^th century reanalysis

We evaluated the spatial performance of the three PDA-based precipitation reconstructions by analyzing the geographic distribution of CRPSS at each grid points, complemented by a statistical summary via boxplot visualization (see Fig. 13). Figure 13a shows the spatial distribution of the CRPSS values for the PDA (TraCE). Positive CRPSS values were observed at 74.00% of grid points, indicating improved reconstruction skill relative to the prior in most NH regions. However, 26.00% of grid points showed negative values, particularly across the Arabian Peninsula and surrounding regions, Northern East Asia and Southern North America, where the reconstruction skill of the PDA (TraCE) was comparatively poor. Figure 13b illustrates the spatial distribution of the CRPSS values for the PDA (HadCM), with 75.35% of the grid points exhibiting positive CRPSS values. The remaining 24.65% showed negative values, most notably in the North African Sahara region, where the reconstruction skill of PDA (HadCM) was weak. Figure 13c depicts the spatial distribution of the CRPSS values for the PDA (Mixed), which achieved positive CRPSS values at 88.54% of the grid points - the highest proportion among the three reconstructions. Negative CRPSS values were limited to 11.46% of the grid points, mainly concentrated in North Africa and the Arabian Peninsula, where the reconstruction skill of the PDA (Mixed) was less effective. Collectively, all three PDA-based reconstructions showed enhanced skill over the prior estimates across most regions of the NH, with the proportion of improved grids ranging from 74.00% to 88.54%. Additionally, an encouraging finding was that although all assimilated Pann records are from land or coastal areas (see Fig. 2), the assimilation process (especially in PDA (Mixed)) yielded significantly improved skill over most ocean regions. This can be attributed to an inherent advantage of data assimilation⁶⁴, which leverages the covariance relationships between climate variables in different regions to propagate observational information spatially to areas without observational data, thereby updating the prior estimates in those areas. However, performance disparities were most pronounced over North Africa and the Arabian Peninsula. These discrepancies are likely attributable to substantial differences in precipitation simulations between TraCE and HadCM in these regions (not shown here), which were naturally propagated into the PDA-based reconstructions. Additionally, the boxplot in Fig. 13d indicates that the PDA (TraCE) achieved a higher median CRPSS value than both the PDA (Mixed) and PDA (HadCM), and that the PDA (Mixed) also outperformed the PDA (HadCM) in terms of median CRPSS. A higher median CRPSS value implies that, on average, the PDA (TraCE) reconstruction agrees more closely with the observations than the PDA (HadCM) and PDA (Mixed) reconstructions in terms of median CRPSS value. However, when considering other skill metrics, the PDA (Mixed) demonstrated superior performance compared to the PDA (TraCE), as evidenced by a greater proportion of positive CRPSS values (e.g., 88.54% vs. 74.00%) and a narrower interquartile range (IQR) of CRPSS values (IQR: 0.0081–0.0150 for PDA (Mixed) vs. -0.0014–0.0347 for PDA (TraCE)). A narrower IQR associated with the PDA (Mixed) indicates a more stable and reliable spatial distribution of reconstruction skill⁶⁵, underscoring the overall robustness of the PDA-based reconstruction using multi-model priors.

Fig. 13.

Spatial patterns and statistical summary of CRPSS values for the three PDA-based reconstructions during the 20^th century. (a) PDA (TraCE); (b) PDA (HadCM); (c) PDA (Mixed); (d) Boxplot of CRPSS statistics across all grid points. Positive CRPSS values (red tones) indicate superior reconstruction skill relative to the uninformed prior distribution, while negative values (gray-black tones) denote reduced skill.

In summary, as demonstrated in previous paleoclimate reconstruction studies, neither the PDA approach^{12,13,18,27,29,30,38} nor traditional spatially-resolved reconstruction methods^66,67 can achieve perfect one-to-one consistency between reconstructions and observations at every grid. This fundamental limitation partly stems from their reliance on natural proxy records rather than high-quality instrumental measurements. These proxy records are inherently sparse, unevenly distributed in space and time, and noisy, with typical signal-to-noise ratios of 0.25–0.5 (e.g., ref. ⁶⁸). Generally, the highly complex spatiotemporal heterogeneity and locality of precipitation, along with the complicated driving mechanisms associated with it, result in precipitation estimates being significantly less accurate than temperature estimates across various datasets, including station-based interpolation climate datasets (e.g., CRU TS dataset, ref. ⁶⁹), climate model simulations (e.g., CMIP6 outputs, ref. ⁷⁰) and data assimilation-based reanalysis products (e.g., ERA-Interim, ref. ⁷¹). This inherent challenge further complicates the large-scale, spatially-resolved reconstruction of past precipitation based on paleoclimate proxy records, a point also emphasized in a recent 2ka reanalysis study⁷². Furthermore, it should be noted that existing PDA approaches typically use a limited-member, static prior ensemble, which is generally randomly selected from pre-existing model simulations, to estimate the covariance relationships among climate variables at different locations across all assimilation time intervals. The covariance relationships estimated through this approach can not reflect time-varying characteristics (i.e., are not flow-dependent)²⁷ and are also prone to the drawback of under-sampling^73,74. Collectively, these limitations discussed are not conducive to improving the skill of PDA-based, large-scale precipitation reconstruction.

Nonetheless, in this study, although neither the PDA-based reconstructions nor the model simulations achieved perfect one-to-one correspondence with the Hancock2023 proxy records at every grid (as shown in Fig. 10), the validation analysis and comparative assessments collectively demonstrated that the three PDA-based precipitation reconstructions exhibited significantly improved reliability compared to model simulations alone. This finding confirms the effectiveness of assimilating Holocene precipitation records to improve the quality of Holocene precipitation estimates. Moreover, the aforementioned datasets show good agreement in terms of positive and high correlations in zonal-averaged series over the NH, high and middle latitudes (e.g., Figs. 11 and 12). We note that while the PDA (TraCE) exhibited very weak correlation (e.g., r = 0.2648) with Hancock2023 in low-latitudes, other PDA-based reconstructions - the PDA (Mixed) and PDA (HadCM) - demonstrated strong positive correlations with Hancock2023 in these same latitudes, especially for the PDA (Mixed). The overall agreement among PDA-based reconstructions, model simulations and precipitation proxy records (at least from the zonal-averaged perspective) aligns with the conclusion drawn from a previous study¹⁹, which reported better proxy-model agreement for precipitation during the Holocene. This finding further supports the reliability of the PDA-based precipitation reconstructions in this study.

Finally, while the reliability of the three PDA-based precipitation reconstructions generated in this study has been examined through cross-validation, independent verification and their capacity to capture large-scale precipitation changes, we emphasize that further efforts are necessary to enhance the accuracy of Holocene precipitation reconstruction using the PDA approach. These efforts encompass three key developments: (1) improved representation of dynamic covarying relationships among climate variables in PDA algorithms⁷⁵, (2) expanded spatial coverage of precipitation records - particularly over data-sparse oceanic regions, and (3) more robust bias-correction methods for debiasing model simulations. Such advances are particularly critical for improving reconstruction reliability in challenging regions such as North Africa, the Arabian Peninsula and adjacent areas, where the three PDA-based precipitation reconstructions of this study exhibit significant spatial limitations.

Usage Notes

The three PDA-based Holocene NH annual precipitation fields reconstructions in this study are based on an uneven network of noisy paleo-precipitation record series. The reconstructions may have significant uncertainty depending on the PDA experimental configuration, the geographical location and the time period of interest. Prior to using these reconstructions for any analysis, it is imperative for users to review the relevant verification and comparison presented in Figs. 6–13 and Tables 1–2. This review will assist in determining whether the reconstructions can provide useful information and which group of reconstruction is more suitable for one’s particular research area. For general users, we recommend directly using the mean and ±1 standard deviation derived from the 200 reconstructions, particularly those from the PDA (Mixed) experiment. The mean and standard deviation associated with each experiment are also provided in the Zenodo data repository⁵³. For users interested in exploring more detailed information, we recommend working directly with the full ensemble of 200 reconstructions for each group. We also recommend that users read the methods section in detail to gain a complete understanding of the methods and procedures used to generate the data.

Author contributions

M.F. developed the code, designed the experiments, ran the experiments, analyzed the experimental results, and prepared the Data Descriptor. J.L.W. and H.C. reviewed the experimental results and helped prepare the Data Descriptor.

Data availability

The TraCE 21ka simulations is available at the TraCE 21ka website (https://www.cesm.ucar.edu/working-groups/paleo/simulations/trace-21ka). The 20^th Century Reanalysis is available at the 20^th Century Reanalysis project website (https://psl.noaa.gov/data/20thC_Rean/). The LegacyClimate 1.0 dataset is available at the PANGAEA repository (10.1594/PANGAEA.930512)⁴⁵. The Hancock2023 proxy records and HadCM simulations are available at the Zenodo data repository (10.5281/zenodo.7939488)⁵⁰. The Holocene NH annual precipitation reconstructions generated by the PDA (TraCE), PDA (HadCM) and PDA (Mixed) experiments of this study have been deposited to the Zenodo data repository (10.5281/zenodo.17355050)⁵³.

Code availability

The MATLAB code used to produce the PDA-based Holocene precipitation reconstructions presented in this data descriptor, along with all necessary input data required to execute it, has been made publicly available in the Zenodo repository⁷⁶. The code released was developed using MATLAB version 2016a and is compatible with later versions of MATLAB.

Competing interests

The authors declare no competing interests.