Robust, proxy-based reconstructions of relative sea-level (RSL) change are critical to distinguishing the processes that drive spatial and temporal sea-level variability. The relationships between individual proxies and RSL can be complex and are often poorly represented by traditional methods that assume Gaussian likelihood distributions. We develop a new statistical framework to estimate past RSL change based on nonparametric, empirical modern distributions of proxies in relation to RSL, applying the framework to corals and mangroves as an illustrative example. We validate our model by comparing its skill in reconstructing RSL and rates of change to two previous RSL models using synthetic time-series datasets based on Holocene sea-level data from South Florida. The new framework results in lower bias, better model fit, and greater accuracy and precision than the two previous RSL models. We also perform sensitivity tests using sea-level scenarios based on two periods of interest – meltwater pulses (MWPs) and the Holocene – to analyze the sensitivity of the statistical reconstructions to the quantity and precision of proxy data; we define high-precision indicators, such as mangroves and the reef-crest coral

Realistically projecting future rates of change in relative sea level (RSL) requires a better understanding of past rates of RSL change during warmer periods and times of abrupt climate change. Recent statistical models of past RSL change

Several recent studies have attempted to account for the empirical distributions of RSL proxies.

Here, we expand on previous work to incorporate modern elevation distributions of individual proxies from field observations in a new Bayesian statistical model used to estimate RSL. The model can use proxy data from any elevational distribution and can also explicitly incorporate data that provide only an upper or lower bound. We demonstrate the approach using coral and sedimentary proxies and build upon the analyses of

We run two distinct types of tests on the model: (1) validation tests to compare performance to previous RSL models and (2) sensitivity tests to evaluate the performance of the new framework for different types and combinations of data. The validation tests simulate Holocene sea level with synthetic data that mimic actual proxy data from South Florida. We evaluate the bias, precision, and accuracy of the new framework compared to two previous models

Although we use the example of mangroves and corals to implement the modeling framework, our method could be applied to any RSL proxies that have nonparametric relationships to RSL or other climate variables for which modern distributions have been measured. We do not attempt to make conclusions about RSL for any particular location or time period with real proxy data in this paper; rather, we aim to demonstrate the validity of this nonparametric modeling framework and how it could be used to answer open questions about variability in the past.

Geological RSL reconstructions are derived from sea-level proxies, such as sediments, fossils, and geomorphological and archeological features (e.g., mangrove peat, corals, and salt-marsh sediment cores), the formation of which were controlled by the past position of RSL

Whereas proxies with bounded indicative meanings produce sea-level index points, other proxies like freshwater peats and marine invertebrates have less precise relationships with RSL and only provide an upper (or lower) bound on past RSL; these proxies produce terrestrial (or marine) limiting data (Fig.

Schematic representation of the indicative meaning and a theoretical example of its application to reconstruct relative sea level (RSL) from radiocarbon-dated mangrove sediment and reef-crest cores:

Here, using the proxy estimates of RSL and age, we employ a hierarchical framework to characterize the variability in sea level through temporal correlations and predict the evolution of RSL through time with uncertainties. This framework partitions uncertainties among model levels. Although we develop the model with mangrove, coral, and marine- and terrestrial-limiting samples in mind, it is flexible and can accommodate other data types. In the following sections, we describe the statistical model structure and how we incorporate uncertainties into the hierarchical framework, which accommodates measurement and inferential data uncertainties in its different levels (Sect.

We use Bayesian analysis

Directed acyclic graph of the Bayesian hierarchical framework of our model. Square nodes represent observed quantities (data), and circular nodes represent unknown quantities (latent variables and model parameters), where the gray circle represents the ultimate variable of interest to be modeled. Arrows indicate conditionality. Each node in the graph is conditionally independent of all others, given the parameters in the previous level. Variables and conditional distributions are specified in the main text. The hyperparameters are represented in the lowest level, where

The goal of our analysis is to determine the probability distribution of RSL through time

Definitions of relevant notation in the model.

We construct a Bayesian hierarchical model to estimate the posterior distribution of RSL and parameters given observed data.

We follow

The data level includes the noisy, observed elevation

The model can accommodate both parametric and nonparametric fitted distributions as likelihoods (

Mangrove peats are assumed to form between MTL (mean tide level) and HAT (highest astronomical tide) in a normal distribution with mean

Marine-limiting and terrestrial-limiting data define the lower and upper limits of sea level, respectively. We assume that the RSL likelihood decreases with the distance from the measured elevation of the limiting data. In order for the distribution to be normalized (continuous probability distribution function integrating to 1), we assign these a triangle distribution, which we conservatively bound with a lower (upper) limit of

We maintain the simplifying assumption of conditional independence among observed elevations (

In the implementation of the process-level model, we treat RSL

We choose to employ Gaussian process (GP) priors, which define the relationship among any arbitrary set of points in time (can be extended, without loss of generality, to a higher dimension such as space) as a multivariate normal distribution defined by a mean vector and a covariance matrix. In a GP model, the sea-level function,

At the parameter level,

For each parameter, we employ uniform prior distributions with bounds set to separate the terms in the covariance function according to patterns in RSL and to exclude unrealistic interpretations of the data, based on prior knowledge (see Fig.

We define three modules – a distribution-fitting module, a sampling module, and a sample-wise prediction module – which are implemented in succession. The distribution-fitting module estimates data parameters

Empirical elevation distributions of the nine coral taxa used in the statistical model:

The distribution-fitting module (Fig.

We fit nonparametric, empirical probability distributions to the modern coral depth data, using the “fitdist” function in MATLAB (Statistics and Machine Learning Toolbox), to assign the likelihood distribution of each coral taxon for use in the statistical model (Fig.

In some cases the nonparametric distributions are bimodal because the depth distributions of corals are controlled both by light, which decreases exponentially with depth, and wave energy, which decreases with depth but is also relatively low in shallow “back reef” lagoon environments behind the wave-breaking reef crest

In the sampling module (Fig.

In the sample-wise prediction module (Fig.

Each sample

More precise data have more weight in the model. For example, the less precise

We performed a model validation of the nonparametric model by comparing its predictions to synthetic “truths” and examined potential improvements over a previous Gaussian model

We created five random iterations of a synthetic Holocene RSL dataset to compare model performance to two previously published models. The elevation and temporal uncertainties in the dataset are designed to mimic characteristics of an extensive coral-based Holocene RSL archive from South Florida

The synthetic “true” RSL curve was created from the ICE-6G_C VM5a GIA model pairing of

We perturbed the true RSL heights with measurement error and to reflect the distribution (indicative meaning) of the indicator elevation relative to RSL:

We ran the new nonparametric model, a Gaussian model, and an “uncorrelated model” using the synthetic data described above. The Gaussian model samples hyperparameters instead of using single maximum-likelihood point estimates and can only handle data with normal distributions (i.e., the only coral taxa included is

We ran additional validation tests comparing the performance of the nonparametric and Gaussian models to determine which portions of potential improvements were due to the more realistic interpretation of the data using kernel densities and which were due to including more data in the model. In these tests, we approximate both

The sensitivity analysis tested the performance of the new model framework using different combinations of synthetic time-series data of various precision and quantity (Table

To test the sensitivity of the model to different types of data, we applied the model to a total of 48 synthetic datasets for each of the two RSL time series. These 48 datasets represent all possible combinations of three factors: (1) the number of data points per 1 kyr period (1, 5, or 10), (2) the precision of the data (combinations of normally distributed sedimentary data, limiting data, and fitted kernel distributions for the two most common coral taxa:

We applied the nonparametric model to the synthetic data, taking 20 000 MCMC samples for each of the 48 tests for each sea-level scenario (SL1 and SL2). We ran each set of tests with five starting seeds (random numbers for replication) in order to maximize randomness and increase sample size for accurate statistical evaluation of the models. Each model run produced a posterior predictive distribution of RSL and its rates of change at 100-year intervals in addition to summary statistics (e.g., 67 % and 95 % credible intervals and coverage, cross-model mean RSL, and root-mean-square errors; Sect.

Results of validation tests. Comparison of three models using a synthetic dataset based on the modern distributions of

We evaluated the ability of the nonparametric, Gaussian, and uncorrelated models to predict RSL at each point where we generated synthetic data. The log likelihood, credible intervals (CIs), average bias, absolute mean error, root-mean-square error (RMSE), and the coverage of each of the CIs for each model are provided in Table

Validation test results. These metrics are used to validate the new nonparametric model against the Gaussian and uncorrelated models when applied to equivalent synthetic datasets:

The nonparametric model also had less bias than the previously published models. The nonparametric model tended to slightly underpredict RSL (negative bias measured by the mean error; cross-model means of

Comparison of several metrics to validate the new nonparametric (NP) model against the Gaussian model with the same exact data using sea-level curve SL1. The Gaussian model includes the same quantity of data as the nonparametric model but approximated with normal distributions instead of with kernel densities for both

Comparison of several metrics to validate the new nonparametric (NP) model against the Gaussian model with the same exact data using sea-level curve SL2. The Gaussian model includes the same quantity of data as the nonparametric model but approximated with normal distributions instead of with kernel densities for both

To account for the possibility that the differences in the performance of the nonparametric and Gaussian models were due to the fact that the Gaussian model excludes some (nonparametric) data that the nonparametric model includes, we conducted an additional comparison of the model where the number of data points and uncertainties were equivalent. In 10 of the 12 data cases tested, the nonparametric model had a lower RMSE (Fig.

These results indicate that the nonparametric model performs slightly better than the Gaussian model when the same amount of data are used in each. Thus, the majority of the improvement stems from the new model's ability to realistically handle nonparametric proxies, allowing the use of more data.

SL1 sensitivity test results. The quantity of data is shown on the left, and the type of proxy is displayed on the bottom of each panel. The numbers in the boxes represent the cross-model averages of each metric, and the circles represent each of the individual tests. The specific values of these individual metrics are shown in Table

The sensitivity tests analyzed how the nonparametric model performs with various amounts and types of data (see Sect.

The precision of the data directly affected the precision of the RSL and rate predictions as well as the average error of the predictions for the first sea-level scenario, SL1 (Fig.

For tests with low-precision data, uncertainties increased to an average of

The largest factor in the accuracy, precision, and bias of all models was the number of data points (Fig.

For reconstructing RSL with rates observed during the early Holocene (

SL2 sensitivity test results. The quantity of data is shown on the left, and the type of proxy is displayed on the bottom of each panel. The numbers in the boxes represent the cross-model averages of each metric, and the circles represent each of the individual tests. The specific values of these individual metrics are shown in Table

In sensitivity analysis of the second sea-level scenario, SL2, which includes abrupt accelerations, results varied (Fig.

Twelve examples of sensitivity tests performed on “true” synthetic relative sea-level (RSL) curves: panels

The sensitivity tests show that high-precision proxies (sedimentary or

Eight examples of sensitivity tests showing various precision and error in rates for sea-level curve 2 (SL2).
The red line in each plot shows the true rate, and the black and gray plots show the median and the 67 % and 95 % credible intervals (CIs), respectively. The quantity, precision, and age error of each dataset is labeled above the corresponding plot. Note that panels

Tests with different amounts of data varied greatly in their ability to predict true RSL for SL2. For example, Fig.

The precision of the data also influences model results. For example, the models in Fig.

In general, all of the models result in a smoothing effect, but a large amount of the most precise data are required to detect abrupt rates of change like MWPs. We find that employing the model with 5 to 10 high-precision data points per kiloyear enables the constraint of rapid rates of RSL (like those represented in SL2) to within

We develop a new technique for integrating nonparametric likelihoods into a hierarchical statistical framework to allow for a more realistic treatment of proxy uncertainties in probabilistic models of past RSL change. It is more flexible than past methods, with the ability to employ parametric, nonparametric, and limiting distributions. The framework provides a robust method for incorporating fitted empirical elevation distributions of a variety of proxies into RSL models, which we have illustrated with coral taxa.

Validation tests of the new framework show that it performed better than previous (Gaussian and uncorrelated) models based on model fit, accuracy, precision, bias, and overall error, although some of the differences in the model results were small (Fig.

Although the new model outperformed the previous models when applied to time-series data, its utility for large spatiotemporal datasets may be limited. The flexible, new framework can be adapted to spatial datasets and uses geographic correlations among data, but the computational expense of running fully Bayesian models may be time-prohibitive with large amounts of data (

The quantity, precision, and temporal resolution of the data used in the model were important factors in the accuracy and precision of RSL reconstructions in validation experiments and sensitivity tests. Although including some data (5 or 10 points per kiloyear – 60 or 120 points total) with high precision assisted in creating the most accurate models of RSL change, there was generally a trade-off between the quality (precision or uncertainty) and quantity of data. Using the most data possible, from a variety of RSL proxies with well-characterized likelihood distributions, provided the most accurate and precise estimates of past RSL variability as shown in sensitivity tests (Tables

Our results also have implications for how low-precision data should be interpreted and used in the new framework. We found that models that used precise, sedimentary data and the low-precision kernel distributions of

The new framework developed in this study has broad applicability to the analysis of past RSL. Many previous studies do not appropriately incorporate data uncertainties nor do they estimate rates in a statistically robust way. As discussed above, the flexibility of the new framework enables integration of proxies previously unused in Gaussian models (e.g., massive corals, limiting data). This provides an important improvement over previous RSL models, particularly in places or for periods of time where sedimentary indicators may not accumulate or preserve

Application of the new framework to sea-level datasets from a variety of locations and time periods has the potential to reveal more precise and accurate estimates of rates of RSL because the new framework has less bias, especially in rates. The nonparametric framework also allows the incorporation of limiting data, which could be valuable when there are long hiatuses of other data. In addition, the framework can statistically incorporate information that refines coral indicative meaning (e.g., by incorporating location-specific proxy depth distributions) that produce nonparametric likelihoods. At a minimum, this new framework could serve as a check on previous results when models similar to the Gaussian and uncorrelated models have been used. Although we have demonstrated the model’s potential using synthetic datasets, the future application of the new framework to real datasets could include previously unincorporated data to better estimate past RSL and rates of RSL change.

Age uncertainties are incorporated using the NIGP method of

The MCMC samples of

Proposed samples of

The whole model is summarized in three modules:

initializing

sequentially sampling new

every

if a ratio

if a ratio

The published uncorrelated method samples in both age and elevation measurement uncertainty, whereas the age is assumed to be constant in our approximation of that method. In this way, the log likelihood of the method achieves a higher value than it would if we assumed a bivariate distribution of uncertainties for each data point.

There is no correlation assumed between the data. Instead each nonparametric distribution is directly applied to each indicator, which individually determines the “posterior” probability of RSL at each point.

The Gaussian model is similar to the implementation in

We analyzed the results of various combinations of hyperparameters in order to restrict the bounds within the model framework. Figure

Examples of various hyperparameter combinations and the way they influence how the same data are interpreted and reflect the predicted relative sea level (RSL).

The average results of the analyses are shown in Tables

Monte Carlo Markov chains are strongly autocorrelated, which produces clumps of samples that may not be representative of the true underlying posterior distribution. Analyzing these autocorrelation plots leads us to accept every 20th MCMC sample (thinning by discarding the other 19 samples) so that samples are approximately independent of one another (Fig.

Temporal data density, age uncertainty, and proxy distributions used to create synthetic data to perform the sensitivity tests of the nonparametric model. Runs 1–12 are also applied in the validation tests comparing the nonparametric and Gaussian models (Sect.

Validation tests results: model comparison of nonparametric, Gaussian, and uncorrelated models.

Summary of validation test results for the Gaussian model, which are compared to the nonparametric model results (Table

Summary of validation test results for the Gaussian model, which are compared to the nonparametric model results (Table

Sensitivity test results for sea-level curve 1 (SL1).

Sensitivity test results for sea-level curve 2 (SL2).

Model diagnostics.

The MATLAB code and data required to run the models can be found on GitHub:

ELA developed the methodology and models (with help from REK) and designed, implemented, and tested the software and models. REK conceived the project (with help from AD), developed the original base code, and acquired funding. LTT and NSK curated, synthesized, and performed preliminary analysis on the data. All co-authors contributed to writing, reviewing, and editing the manuscript.

The contact author has declared that neither they nor their co-authors have any competing interests.

Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government. Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Robert E. Kopp and Erica L. Ashe (grant nos. OCE-1831450, OCE-1702587, and OCE-2002437) and Andrea Dutton (grant no. 2041325) were supported by the National Science Foundation. Lauren T. Toth was supported by the U.S. Geological Survey Coastal and Marine Hazards and Resources Program. This work is a contribution to IGCP Project 639 “Sea Level Change from Minutes to Millennia”, INQUA Project CMP1601P “HOLSEA”, and “PALSEA3”.

This research has been supported by the National Science Foundation (grant nos. OCE-1702587 and OCE-2002437).

This paper was edited by Reik Donner and reviewed by three anonymous referees.