We present a Bayesian hierarchical model for reconstructing the continuous
and dynamic evolution of relative sea-level (RSL) change with quantified
uncertainty. The reconstruction is produced from biological (foraminifera)
and geochemical (

Paleoenvironmental reconstructions describe Earth's response to past climate changes and offer a context for
current trends and analogs for anticipated future changes (e.g.,

Relative sea-level (RSL) reconstructions can constrain the relationship
between temperature and sea level and reveal the long-term, equilibrium
response of ice sheets to climate forcing (e.g.,

We develop a Bayesian transfer function (B-TF) to reconstruct RSL using
foraminifera (expressed as raw counts) and measurements of bulk sediment

Transfer functions are empirically derived equations for reconstructing past
environmental conditions from the abundance of multiple species. The term
refers not to a single numerical method, but to a range of regression-based
techniques that are classified into two categories depending on whether the
underlying model maps environmental variables to species abundances
(classical calibration) or vice versa (inverse calibration). Classical
approaches are underpinned by the ecologically intuitive assumption that the
distribution of species is driven by environmental variables (e.g.,

Bayesian calibration methods are inherently classical and have recently been
given growing attention to produce paleoenvironmental reconstructions using
biological proxies (e.g.,

We now describe our statistical model, which produces
estimates of RSL and associated rates from raw inputs including foraminifera
counts and radiocarbon dates (or dates produced from recognition of pollution
and vegetation clearance markers of known age) from a sediment core. There
are two advances from existing approaches:

A B-TF model using a penalized spline (P-spline) as a non-parametric model of the multinomial response of foraminifera counts to tidal elevation. The multinomial distribution can take account of the fact that large counts give reduced uncertainty and the P-spline model allows for multi-modal and non-Gaussian species response to environmental variables;

A full hierarchical model which incorporates the B-TF, a chronology model accounting for time uncertainty, and a rich stochastic process for quantifying sea-level rate changes.

We start by outlining our notation:

Using the notation above we create a Bayesian hierarchical model to produce a posterior distribution of our parameters given data:

Following these assumptions, we obtain the three modules:

In this module we aim to estimate the parameters

The probability vectors

An error term,

The B-TF produces posterior estimates for the multinomial probability vector

We evaluate the performance of the B-TF via 10-fold cross-validation on the modern data, where the data are divided up into 10 randomly drawn equal size sections (known as folds) which are removed in turn. We create predictions for the removed sections repeatedly until every observation has an out-of-sample prediction value. To allow direct and meaningful comparison between models, we also cross-validated the WA-TF using the same approach on the same randomly drawn folds.

In cases where the secondary

The chronology module is concerned with estimating the ages

Once Bchron has been run, we obtain a joint posterior distribution of ages
for every layer in the sediment core, which we denote as

Our final step is to transform PME,

We use an integrated Gaussian process approach

On the Atlantic coast of southern New Jersey (Fig.

The transfer function approach is under-pinned by surveys that quantify the
modern, observable relationship between foraminifera and tidal elevation.
Modern (surface) samples are collected from depositional environments
analogous to those represented in the fossil material under investigation

The 12 sites were selected to span a wide range of physiographic settings
including brackish marshes located up to 25 km from the coast with a strong
fluvial influence. The sites share a common climate and oceanographic regime
and therefore constitute a regional-scale training set. The spatial scope of
modern training sets has frequently been discussed and it is widely accepted
that regional-scale training sets capture natural variability in the
distribution of foraminifera and provide a large suite of samples from which
to draw modern analogs (e.g., Horton and Edwards, 2005). The principal
advantage of local-scale training sets is that they produce more precise
reconstructions, but at the expense of offering only a narrow range of modern
analogs (e.g.,

Location of study sites in southern New Jersey, USA. The
distribution of modern foraminifera was described at 12 different salt
marshes including five in Great Egg Harbor (not located with symbols in the
figure). Bulk surface sediment

The modern data set comprised 172 paired observations of 18 foraminiferal
species (including many zeros) and tidal elevation. The foraminifera count
sizes ranged from 8 to 307 dead individuals. The highest occurrence of
foraminifera (HOF; Wright et al., 2011) in the modern data set is at 141.5 SWLI. Higher samples were
devoid of foraminifera and interpreted as being located above marine
influence. This modern training set demonstrates that foraminifera (like
plants) form distinct assemblages that correspond to elevation in the tidal
frame (e.g.,

Data set of modern foraminifera described from a total of 172 surface sediment samples from 12 different sites. The data are presented as relative abundances. Only the abundance of the eight most common species are shown. Modified from Kemp et al. (2013a). Red dashed lines and pink shading indicate the species optima and tolerances estimated by the weighted averaging transfer function. Tidal datums are indicated by blue dashed lines. HOF is highest occurrence of foraminifera, MHHW the mean higher high water, and MTL the mean tide level.

In the mid-Atlantic and northeastern USA, the low
salt-marsh and high salt-marsh zones are dominated by C

Samples with

Samples with

Samples with intermediate

Cores of salt-marsh sediment were recovered from two sites in
southern New Jersey (Cape May Courthouse and Leeds Point; Fig.

In the Cape May Courthouse core

The assemblages of foraminifera preserved in each core were compared to those
in the modern training set. If the dissimilarity between a core sample and
its closest modern analog (measured using the Bray–Curtis metric) exceeded
the 20th percentile of dissimilarity among all possible pairings of
modern samples then the core sample was deemed to lack a suitable modern
analog and was excluded from further analysis by

In the Cape May Courthouse core all samples were less depleted
than

Age–depth models for the Cape May Courthouse and Leeds Point cores were
previously developed by

The abundance of the three most common species (Tc

A tide gauge is an instrument that automatically measures the sea surface
height with reference to a control point on the land many times during a day.
These measurements are averaged to obtain annual values to minimize the
effects of weather and tidal variability. In New Jersey (Fig.

The Cape May Courthouse and Leeds Point RSL reconstructions were developed separately to one another and were then merged into a single, regional-scale data set for analysis using the EIV-IGP model to capture the continuous and dynamic evolution of RSL change while taking account of the quantified uncertainty in both sea-level and age reconstructions.

The B-TF estimated a response curve (mean with a 95 % credible interval) for
each species of foraminifera (expressed as raw counts) to tidal elevation
(expressed as SWLI) from the modern training set of 172 samples (Fig.

The response of foraminifera species to elevation estimated from the modern training set of raw counts using the Bayesian transfer function. The blue circles represent the probabilities of species occurrence as determined from the count data (empirical probabilities). The response probabilities of occurrence estimated by the Bayesian transfer function model are shown with a mean (heavy red line), a credible interval for the mean (light red line), and a prediction interval (blue line). The green vertical lines and shading represent the species optima and tolerances determined from the weighted average transfer function. Tidal datums are indicated by vertical dashed black lines. MTL is mean tide level and MHHW the mean higher high water. SWLI indicates the standardized water level index.

Broadly, we identify two forms of species-response curve in southern New
Jersey. First, a skewed, unimodal form describes the distribution of

Performance of the new B-TF and existing WA-TF was judged using
10-fold cross-validation (Fig.

Cross-validation of the modern training set for the weighted
averaging
transfer function (red) and the Bayesian transfer function (blue). Upper
panels are measured vs. predicted elevations in standardized water-level index (SWLI) units, with
lines representing

The pattern of residuals in the WA-TF displayed a structure in which the
elevation of low salt-marsh samples is overpredicted (negative residuals)
and the elevation of high salt-marsh samples is underpredicted (positive
residuals; Fig.

We reconstructed PME in the Cape May Courthouse and Leeds
Point cores using the WA-TF and B-TF models. At Cape May Courthouse, the B-TF
estimated an average PME close to MHHW (SWLI

On salt-marshes in the mid-Atlantic and northeast coast of the
USA measurements of

At Cape May Courthouse, using the downcore

We applied the EIV-IGP model to the RSL reconstructions
produced from the WA-TF, B-TF and multi-proxy B-TF to describe RSL trends
along the coast of southern New Jersey since

There are some differences among the three reconstructions. For example, the
B-TF shows the highest modern rate of rise at 4.1 mm yr

The EIV-IGP model results for reconstructions produced using the
Bayesian transfer function (B-TF), the weighted averaging transfer function (WA-TF) and the multi-proxy Bayesian transfer function. The upper
panel shows individual data points (represented by rectangular boxes that
illustrate the 95 % confidence region) and include age and relative sea-level
uncertainties. The middle panels show the posterior fit of the
Errors-In-Variables integrated Gaussian process model to the relative
sea-level reconstructions. Solid line represents the mean fit with the 68
and 95 % confidence intervals (C.I.) denoted by shading. The lower panels are
the rates of relative sea-level (RSL) change. Shading denotes 68 and 95 %
confidence intervals (C.I.) for the posterior mean of the rate process. The
average rate for each phase of the reconstruction is given (in mm yr

Comparison of the weighted average transfer function

The B-TF provides an alternative to the (non-Bayesian) regression-based
transfer functions commonly used for reconstructing RSL (e.g.,

The implication of the flexibility of the B-TF is illustrated in the cross
validation results. The WA-TF displayed edge effects (a tendency to bias PME
predictions towards the mean of the training data), which is a common
artifact of using weighted-average-based methods (e.g.,

Further motivation for the development of the B-TF lies in the quantification
of PME uncertainty. Non-Bayesian transfer function methods (e.g., the WA-TF
model) assume that model parameters are fixed and known. Therefore, they do
not incorporate uncertainty into the estimation of the PME reconstruction
itself, rather, the uncertainty is produced separately either before or after
PME was estimated. This uncertainty is the root mean square error from two
sources (S1 and S2;

Alternatively, Bayesian methods explicitly model the uncertainty associated
with individual reconstructions. Uncertainty for PME (and other unknown
parameters) is included in the probability model through prior distributions.
Assuming distributions for unknown parameters (in contrast to non-Bayesian
approaches that use point estimates) allows the parameter uncertainty from
the calibration step to be formally propagated into the reconstruction step.
Therefore, estimates of PME produced by the B-TF take fuller account of the
uncertainties related to the model and its parameters than non-Bayesian
approaches. Consequently, the uncertainties estimated by the B-TF (excluding
a secondary proxy) show more pronounced variability among core samples
(2

The variability of reconstructed PME from the B-TF may reflect a more
ecologically plausible reconstruction than the WA-TF model. For example, the
key, high salt-marsh species

The majority of quantitative RSL reconstructions employ a single proxy (e.g.,

To accurately reconstruct the continuous and dynamic evolution
of relative sea-level change, we developed a Bayesian hierarchical model
comprised of three formally interconnected modules. (1) A B-TF for the
calibration of foraminifera into tidal elevation, which is flexible enough to
formally accommodate additional proxies such as bulk sediment

Our new B-TF provides an alternative to existing transfer functions. The
relationship between species of salt-marsh foraminifera and tidal elevation
was described using a regional-scale modern training set (

We applied the transfer functions to cores of salt-marsh sediment that were recovered from two sites in southern New Jersey. The flexible approach utilized in the B-TF results in more variability in reconstructed PME and associated uncertainty among samples than the WA-TF model. This variability is consistent with observed changes in foraminiferal population in core samples and we propose that the B-TF produces a more complete evaluation of uncertainty than the WA-TF model.

The B-TF allows results from additional, independent sea-level proxies to be
formally incorporated alongside the primary biological proxy to produce a
multi-proxy reconstruction. In New Jersey, we used bulk sediment

We assessed the ability of the multi-proxy B-TF, B-TF and the WA-TF to
reconstruct RSL through comparison with observed tide-gauge data from New
Jersey. Results showed that the 2

We are grateful to the editor Eduardo Zorita, the anonymous reviewer and Robin Edwards (the second reviewer) for their comments that greatly improved the early version of the paper. This research was supported by the Structured PhD in Simulation Science which is funded by the Programme for Research in Third Level Institutions (PRTLI) Cycle 5 and co-funded by the European Regional Development Fund, and the Science Foundation Ireland Research Frontiers Programme (2007/RFP/MATF281) and also supported by the National Science Foundation awards EAR 1402017 and OCE 1458904. Edited by: E. Zorita