Corals provide the most prominent Late Pleistocene sea level proxy. They can be radiometrically dated and provide especially accurate sea level estimates between 0 and 21 ka because of nearly continuous pristine coral specimens from several locations (Fairbanks, 1989; Bard et al., 1990, 1996; Edwards et al., 1993). Dated coral sea level estimates extend as far back as ∼ 600 ka (Stein et al., 1993; Stirling et al., 1995; Medina-Elizalde, 2013; Muhs et al., 2014; Andersen et al., 2008). However, coral data are increasingly discontinuous and inaccurate prior to 21 ka due to difficulty finding pristine and in situ older corals (particularly during sea level lowstands) and due to U–Th age uncertainties in older corals caused by isotope free exchange with the surrounding environment (e.g., Thompson and Goldstein, 2005; Blanchon et al., 2009; Medina-Elizalde, 2013). Interpretation of sea level from corals often requires a correction for rates of continental uplift, which may not be known precisely (Creveling et al., 2015). Glacial isostatic adjustment (GIA) and species habitat depth (up to 6 m below sea level) may also affect sea level estimates (Raymo and Mitrovica, 2012; Medina-Elizalde, 2013). Wave destruction and climate variations also alter coral growth patterns and may affect the height of colonies relative to sea level (Blanchon et al., 2009; Medina-Elizalde, 2013).

Organic proxies such as peat bogs and shell beds can also be used as sea level proxies and can be radiometrically dated (e.g., Horton, 2006). Geological formations indicating sea level such as abandoned beaches and sea cliffs can also be used as sea level proxies (Hanebuth et al., 2000; Boak and Turner, 2005; Bowen, 2010).

Corals and other coastal proxies are indicators of relative (local) sea level and, thus, are affected by in situ glacio-isostatic effects, ocean siphoning processes, and other local effects of sea level rise and fall. However, their wide spatial distribution, particularly corals in tropical regions, allows for modeling of glacio-isostatic adjustments (GIA) to create a global estimate of mean sea level change (e.g., Kopp et al., 2009; Lambeck et al., 2014; Dutton and Lambeck, 2012; Hay et al., 2014). GIA models constrained by these coastal indicators provide robust sea level change estimates of -130 to -134 m for the LGM (Clark et al., 2009; Lambeck et al., 2014). A compilation of dozens of corals and other sea level indicators also provide relatively well-constrained estimate of 8.7±0.7 m for peak global mean sea level at the last interglacial (Kopp et al., 2009). Estimates from multiple studies using different data are all in relatively good agreement yielding a consensus estimate of 6–9 m above modern (Dutton et al., 2015). Additionally, sea level during last interglacial likely experienced several meters of millennial-scale variability (Kopp et al., 2013; Govin et al., 2012). Uncertainties increase for older interglacials. GIA-corrected coastal sea level proxies for Marine Isotope Stage (MIS) 11 at ∼ 400 ka suggest a global mean sea level of 6–13 m above modern (Raymo and Mitrovica, 2012).

Global ice volume is a main control on the global mean of δ18O in seawater (δ18Osw), with global mean δ18Osw estimated to decrease by 0.008–0.01 ‰ m-1 of sea level rise (Adkins et al., 2002; Elderfield et al., 2012; Shakun et al., 2015). However, δ18Osw also varies spatially based on patterns of evaporation and precipitation and deep water formation processes. The δ18O of calcite (δ18Oc) is affected both by the δ18Osw and temperature. In the absence of any post-depositional alteration, subtracting the temperature-dependent fractionation effect from δ18Oc (Shackleton, 1974) should yield a good estimate of the δ18Osw in which the calcite formed. Pioneering studies for estimating time series of δ18Osw using independent measures of temperature include Dwyer et al. (1995), Martin et al. (2002), and Lea et al. (2002). Dwyer et al. (1995) used ostracod Mg/Ca ratios to determine temperature whereas Martin et al. (2002) and Lea et al. (2002) used benthic and planktonic foraminifera, respectively. The δ18Oc of benthic foraminifera reflects the temperature and δ18Osw of deep water, while the δ18Oc of planktonic foraminifera is affected by sea surface temperature (SST) and the δ18Osw of near-surface water.

Our analysis includes two benthic δ18Osw records from the North Atlantic and South Pacific, which use the Mg/Ca ratio of benthic foraminifera as a temperature proxy. The South Pacific benthic δ18Osw record (Elderfield et al., 2012) from Ocean Drilling Program (ODP) site 1123 (171∘ W, 41∘ S; 3290 m) reflects the properties of Lower Circumpolar Deep Water, which is a mix of Antarctic Bottom Water (AABW) and North Atlantic Deep Water (NADW). Mg/Ca ratios and δ18Oc were determined from separate samples of the same species of Uvigerina, which is considered fairly insensitive to the deep water carbonate saturation state (Elderfield et al., 2012). Elderfield et al. (2012) interpolate their data to 1 ka spacing, perform a 5 ka Gaussian smoothing, and convert from δ18Osw to sea level using a factor of 0.01 ‰ m-1. Elderfield et al. (2012) report measurement uncertainties for temperature and δ18Oc generate a δ18Osw uncertainty of ±0.2 ‰, corresponding to bottom water temperature range of ±1 ∘C or about 22 m of sea level.

The North Atlantic δ18Osw reconstruction is from Deep Sea Drilling Program (DSDP) site 607 (32∘ W, 41∘ N; 3427 m) and nearby piston core Chain 82-24-23PC (Sosdian and Rosenthal, 2009). These sites are bathed by NADW today but were likely influenced by AABW during glacial maxima (Raymo et al., 1990). Mg/Ca was measured using two benthic foraminiferal species, Cibicidoides wuellerstorfi and Oridorsalis umbonatus, which may be affected by changes in carbonate ion saturation state, particularly when deep water temperature drops below 3 ∘C (Sosdian and Rosenthal, 2009). The δ18Oc data come from a combination of Cibicidoides and Uvigerina species. Sea level was estimated from benthic δ18Osw using a conversion of 0.01 ‰ m-1 and then taking a three-point running mean. Combining the uncertainties for temperature (±1.1 ∘C) and δ18Oc (±0.2 ‰) reported by Sosdian and Rosenthal (2009) yields a sea level uncertainty of approximately ±20 m (1 standard error) for the three-point running mean.

A 49-core global stack uses the δ18Oc from planktonic foraminifera paired with SST proxies from the same core. The planktonic species in this reconstruction were G. ruber, G. bulloides, G. inflata, G. sacculifer, N. dutretriei, and N. pachyderma. Forty-four records span the most recent glacial cycle, and seven records extend back to 798 ka. Thirty-four records use Mg/Ca temperature estimates, and 15 use the alkenone U37k′ temperature proxy. Because U37k′ measurements derive from coccolithophore rather than foraminifera, there is some chance the temperature measured may differ slightly from that affecting δ18Oc (Schiebel et al., 2004). However, Shakun et al. (2015) observed no significant differences in δ18Osw estimated from the two SST proxies. An additional concern is that the surface ocean is affected by greater hydrologic variability and characterizes a smaller ocean volume than the deep ocean. Thus, planktonic δ18Osw may differ more from ice volume changes than benthic data. However, these potential disadvantages of using planktonic records may be largely compensated by the use of a global planktonic stack.

The first principal component (stack) of the planktonic records spanning the last glacial cycle represents 71 % of the variance in the records (n=44), suggesting a strong common signal in planktonic δ18Osw. However, the 800 ka planktonic δ18Osw stack appears to contain linear trends that differ from other sea level estimates. Therefore, Shakun et al. (2015) corrected their sea level estimate by detrending planktonic δ18Osw based on differences between planktonic and benthic δ18Oc. Standard errors reported by Shakun et al. (2015) for the δ18Osw stack increase from 0.05 ‰ for the last glacial cycle to 0.12 ‰ at 800 ka due to the reduction in the number of records. The equivalent sea level uncertainties are ±6 and ±18 m (1σ), respectively. All data were interpolated to even 3 ka time intervals.

The sea level reconstruction of Waelbroeck et al. (2002) was developed by fitting polynomial regressions between benthic δ18Oc from North Atlantic cores NA 87-22/25 (55∘ N, 15∘ W; 2161 and 2320 m) and equatorial Pacific core V19-30 (3∘ S, 83∘ W; 3091 m) to sea level estimates for the last glacial cycle, primarily from corals. Quadratic polynomials were fit during times of ice sheet growth and during the glacial termination in the North Atlantic whereas a linear regression was fit to the Pacific glacial termination. A composite sea level curve was created from the most reliable sections of several cores, primarily from the Pacific. Waelbroeck et al. (2002) interpolated the composite time series to an even 1.5 ka time window and estimated the uncertainty associated with this technique to be ±13 m of sea level. Transfer functions between benthic δ18Oc and coral sea level estimates have also been estimated at lower resolution and applied to 10 different benthic δ18O records spanning 0–5 Ma (Siddall et al., 2010; Bates et al., 2014).

The inverse model of Bintanja et al. (2005) is based on the concept that Northern Hemisphere (NH) subpolar surface air temperature plays a key role in determining both ice sheet size and deepwater temperature, which are the two dominant factors affecting benthic δ18Oc. A three-dimensional thermomechanical ice sheet model simulates ice sheet δ18O content, height, and volume for NH ice sheets (excluding Greenland) as forced by subpolar air temperature, orbital insolation, and the modern spatial distributions of temperature and precipitation. Antarctic and Greenland ice sheets are assumed to account for 5 % of ocean isotopic change and 15 % of sea level change. Deep water temperature is assumed to scale linearly with the 3 ka mean air temperature. At each time step air temperature is adjusted to maximize agreement between predicted δ18Oc and the observed value 0.1 ka later in a benthic δ18Oc stack (Lisiecki and Raymo, 2005). The model solves for ice volume, temperature, and sea level changes since 1070 ka in 0.1 ka time steps; however, the δ18Oc stack used to constrain the model has a resolution of 1–1.5 ka. Bintanja et al. (2005) report the uncertainty of their sea level model to be approximately ±12 m (1σ).

Two sea level reconstructions use hydraulic control models to relate planktonic δ18Oc from the Red Sea and Mediterranean Sea to relative sea level. In these semi-isolated basins, δ18Osw is strongly affected by evaporation and exchange with the open ocean as affected by relative sea level at the basin's sill.

Red Sea RSL (Rohling et al., 2009) from 0 to 520 ka is estimated using the δ18Oc of planktonic foraminifera from the central Red Sea (GeoTü-KL09). Because extremely saline conditions killed foraminifera during MIS 2 and MIS 12, δ18Oc data for these time intervals were estimated by transforming bulk sediment values. Sea level is estimated using a physical circulation model for the Red Sea combined with an oxygen isotope model (Siddall et al., 2004). The physical circulation model simulates exchange flow through the Straits of Bab el Mandeb, which depends strongly on sea level. The current sill depth is 137 m, and its estimated uplift rate is 0.2 mka-1. The isotope model assumes steady state with exchange through the sill and evaporation/precipitation. Assumptions of the isotope model include (1) modern evaporation rates and humidity, (2) open ocean δ18Osw scales as 0.01 ‰ m-1, and (3) SST scales linearly with sea level. A 5 ∘C change in SST between Holocene and LGM is used to optimize the model's LGM sea level estimate. Steady-state model solutions for different sea level estimates are used to develop a conversion between δ18Oc and sea level, which is approximated as a fifth-order polynomial. Rohling et al. (2009) performed sensitivity tests using plausible ranges of climatic values to produce a 2-σ uncertainty estimate of ±12 m.

A Mediterranean RSL record (Rohling et al., 2014) is derived from a hydraulic model of flow through the Strait of Gibraltar (Bryden and Kinder, 1991) combined with evaporation and oxygen isotope fractionation equations for the Mediterranean (Siddall et al., 2004). Runoff and precipitation are parameterized based on present-day observations, humidity is assumed constant, and temperature is assumed to covary with sea level. The δ18Osw of Atlantic inflow is scaled using 0.009 ‰ m-1, and net heat flow through the sill is assumed to be zero. The combined models yield a converter between δ18Oc and sea level, which is approximated as a polynomial. This polynomial conversion is applied to an eastern Mediterranean planktonic δ18Oc stack (Wang et al., 2010) after identification and removal of sapropel layers. Model uncertainty is evaluated using random parameter variations, which yield 95 % confidence intervals of ±20 m for individual δ18Oc values. By performing a probabilistic assessment of the final sea level reconstruction with 1 ka time steps, Rohling et al. (2014) estimate that these uncertainties are reduced to ±6.3 m. Additionally, the authors propose that RSL at this location is linearly proportional to eustatic sea level.

The criteria for record inclusion in our stack were availability, a temporal resolution of at least 5 ka, and a length of at least 430 ka. The five records which extended to 798 ka were also included in a longer stack. Some available records were too short for inclusion (e.g., Dwyer et al., 1995; Martin et al., 2002; Lea et al., 2002). The record of Siddall et al. (2010) was not included because it was based on the same technique as Waelbroeck et al. (2002) but with lower resolution. Bates et al. (2014) extended this technique to many benthic δ18O records but advocated against placing them all on a common age model; therefore, we include a summary of that study's lowstand and highstand estimates in Table 2 rather than aligning them for inclusion in the stack.

To create an average (or stack) of sea level records, all of the time series must be placed on a common age model (Fig. 1). Here we use the age model of the orbitally tuned “LR04” benthic δ18Oc stack (Lisiecki and Raymo, 2005), which has an uncertainty of 4 ka in the Late Pleistocene. An age model for the Red Sea reconstruction based on correlation to speleothems is generally similar to LR04 with smaller age uncertainty but only extends to 500 ka (Grant et al., 2014) and, thus, does not provide an age framework for the entire 798 ka stack. Due to age model uncertainty, our interpretation focuses on the amplitude of sea level variability rather than its precise timing.

We do not assume that sea level varies synchronously with benthic δ18Oc. Age models for three of the reconstructions are based on aligning individual δ18Oc records to the LR04 δ18Oc stack, and one reconstruction (Bintanja et al., 2005) was derived directly from the LR04 stack. The other three sea level reconstructions were dated by aligning their sea level estimates to a preliminary stack of the four sea level records that were dated using δ18Oc alignments. Alignments were performed using the Match graphic correlation software package (Lisiecki and Lisiecki, 2002).

The three records which use δ18Oc alignments to the LR04 stack are site 607, site 1123, and the planktonic δ18Osw stack. For site 607 we perform our own alignment of benthic δ18Oc to the LR04 stack, whereas for the other two we use the same age models published by Elderfield et al. (2012) and Shakun et al. (2015). One potential concern about aligning benthic δ18Oc records is that the timing of benthic δ18Oc change at different sites may differ by as much as 4 kyr during glacial terminations (Skinner and Shackleton, 2005; Lisiecki and Raymo, 2009; Stern and Lisiecki, 2014). The potential effects of lags in benthic δ18Oc are evaluated using bootstrap uncertainty analysis (Sect. 4.2).

For three reconstructions (Waelbroeck et al., 2002; Rohling et al., 2009, 2014) we aligned the individual sea level records with a preliminary sea level stack based on the other four sea level records on the LR04 age model. This was necessary because the local δ18Oc signals in semi-isolated basins (Rohling et al., 2009, 2014) differ substantially from global mean benthic δ18Oc. In the coral-regression reconstruction, Waelbroeck et al. (2002) pasted together portions of individual cores to form a preferred global composite. Although each core has benthic δ18Oc data, generating new age estimates for these cores could alter their δ18Oc regression functions or create gaps or inconsistencies in the composite. The procedure of aligning these three sea level records (Waelbroeck et al., 2002; Rohling et al., 2009, 2014) to a preliminary sea level stack should be approximately as accurate as the δ18Oc alignments. However, the direct sea level alignments do have a slightly greater potential to align noise or local sea level variability.

After age models were adjusted, five of the records ended within the Holocene. Therefore, we appended a value of 0 m (i.e., present-day sea level) at 0 ka. In the two records which did end at 0 ka, modern sea level estimates were slightly below zero: -1.5 m (Bintanja et al., 2005) and -1.3 m (Rohling et al., 2014).

Principal component analysis (PCA) is commonly used to create stacks of paleoclimate data (e.g., Huybers and Wunsch, 2004; Clark et al., 2012; Gibbons et al., 2014) and to quantify the common signal contained in core data. Synthesis is valuable because each record has its own assumptions and errors. If these records are all well-constrained measures of sea level, then PCA will reveal their respective levels of agreement or discrepancy. Additionally, PCA does not require the assumption that each sea level record represents an independent measure of common signal. In contrast, a sea level estimate based on the unweighted mean of records would imply that uncertainties are uncorrelated across individual reconstructions. While all records contain a strong ice volume signal, some of the non-ice volume signals are expected to correlate with one another. For example, as the δ18O of ice sheet changes as it melts or freezes, the conversion from the δ18Osw to ice volume will be systematically biased, whereas changes in the hydrological cycle may induce changes in the spatial variability of δ18Osw at different locations in the ocean.

We include both relative and eustatic sea level estimates in the analysis because PCA should identify the common variance that dominates both relative and eustatic sea level records. Three records are proxies for relative sea level at their respective locations: the strait of Gibraltar (Rohling et al., 2014), the Straits of Bab el Mandeb (Rohling et al., 2009), and tropical coral terraces (Waelbroeck et al., 2002). The inverse model generates eustatic sea level from a modeled ice volume estimate (Bintanja et al., 2005), and the three δ18Osw records (Elderfield et al., 2012; Sosdian and Rosenthal, 2009; Shakun et al., 2015) were scaled to eustatic sea level. However, for the planktonic stack we use the δ18Osw record rather than the eustatic sea level conversion because the sea level conversion involved detrending to make planktonic δ18Oc values agree with benthic δ18Oc. Because PCA is designed to identify the common variance between the sea level proxies, it is preferable to keep the planktonic and benthic δ18Osw records independent of one another.

In the Mediterranean RSL record we removed putative sapropel layers at 434–452, 543–558, and 630–663 ka as visually identified by Rohling et al. (2014). Because interpolating linearly across these gaps (Fig. 1) would bias sea level estimates towards higher lowstands for the glacial maxima occurring during these sapropel layers, we assumed that sea level remained constant at its pre-sapropel (glacial) level and then immediately jumped to the higher sea level values observed the ends of the sapropel layers (midway through the glacial terminations). Although this solution is not ideal, we must assume some sea level value at these times in order to include this record in the PCA.

Before PCA all seven records were interpolated to an even 1 ka time step. Then, to ensure equal weighting for each record in the PCA, each time series was normalized to a mean of zero and a standard deviation of one within each of the two time windows (0–430 and 0–798 ka). PCA was performed on seven records from 0 to 430 ka and five records from 0 to 798 ka (Fig. 2). Because PC1 produces similar loadings for each record (Table 1), the PC1 scores approximate the average of all records for each point in time, which we refer to as a sea level stack.

We scaled the short and long stacks to eustatic sea level using an LGM value of -130 m at 24 ka based on a GIA-corrected coral compilation (Clark et al., 2009) and a Holocene value of 0 m at 5 ka. We scale the Holocene at 5 ka because eustatic sea level has been essentially constant for the past 5 ka (Clark et al., 2009), whereas the sea level stacks display a trend throughout the Holocene perhaps due to bioturbation in the sediment cores. Scaling the sea level stack based on the mid-Holocene (rather than 0 ka) should more accurately correct for the effects of bioturbation on previous interglacials because those highstand values have been subjected to mixing from both above and below. Finally, a composite sea level stack was created by joining the 0–430 ka stack with the 431–798 ka portion of the long stack after each was scaled to sea level. Because the two scaled sea level stacks produce similar values for 0–430 ka (Fig. 2), no correction was needed to combine the records.

To test the effectiveness of using the scaled PC1 as a record of mean sea level, we compared our stack with highstand and lowstand values identified from individual records and with coral-based estimates where available (Tables 2, 3). We picked the relevant highstand or lowstand for each individual record by choosing the peak that lies within the age range of each Marine Isotope Stage (MIS) as identified in the sea level stack. Highstand or lowstand peaks which occurred outside of the age range of each particular glacial or interglacial stage were not used (e.g., extreme values at ∼ 250 ka from ODP sites 1123 and 607).

Highstand sea level estimates vary widely between individual records with standard deviations of 11–26 m for each isotopic stage (Table 3). For example, individual estimates for MIS 11 at ∼ 400 ka vary between -5 and 57 m above modern, with a mean of 18 m and a standard deviation of 25 m. MIS 5e (119–126 ka) estimates range from -4 to 28 m above modern with a mean of 7 m and a standard deviation of 12 m. Generally, the highstand means have slightly greater amplitudes than our scaled stack; for example, the scaled stack estimates are 18 and 7 m for MIS 11 and MIS 5e, respectively. On the other hand, the mean of individual lowstands for the LGM (-123 m) underestimates eustatic sea level change, which is estimated to be -130 to -134 m (Clark et al., 2009; Lambeck et al., 2014).

The means of the individually picked highstands may be biased by the additive effects of noise. Conversely, the stack may underestimate sea level highstands if the individual age models are not properly aligned. The most definitive sea level estimates come from GIA-corrected coral compilations, which yield highstand estimates of 6–13 m above modern for MIS 11 (Raymo and Mitrovica, 2012) and 8–9.4 m for MIS 5e (Kopp et al., 2009). These values suggest that the stack may be more accurate for MIS 11 than MIS 5e, potentially because age model uncertainty would have less effect on the longer MIS 11 highstand. In contrast, MIS 5e may have consisted of two highstands each lasting only ∼ 2 ka separated by several thousand years with sea level at or below modern (Kopp et al., 2013). Thus, the stack's highstand estimates likely fail to capture short-term sea level fluctuations but rather reflect mean sea level during each interglacial.

To further test the sensitivity of our method, we compared the scaled PC1 with the unweighted mean of the seven interpolated sea level records (Fig. 2b). The unweighted-mean stack incorporates the same data as scaled PC1 except that it excludes Mediterranean estimates from sapropel intervals and uses the detrended sea level estimates from Shakun et al. (2015) instead of the raw δ18Osw data. The unweighted stack closely resembles PC1 because the loadings of PC1 are very similar for all seven records (Table 1). However, the unweighted stack underestimates LGM sea level, possibly because some records (e.g., Rohling et al., 2009) may contain brief gaps at the glacial maximum. Thus, we prefer to scale PC1 to agree with well-constrained LGM sea level estimates. The scaled PC1 is in better agreement with the glacial sea level estimates of the unweighted five-record stack from 430 to 798 ka.

We estimate uncertainty in the stack using a bootstrap technique instead of using the published uncertainty estimates for each sea level reconstruction, which are based on different assumptions and techniques and do not necessarily include all sources of uncertainty (e.g., uncertainty in benthic δ18Oc alignments). We ran 1000 bootstrap iterations while also performing random sampling to account for several of the uncertainties associated with our method. Before each iteration of the bootstrapped PCA, we simulate the effects of uncertainty associated with our age model alignments by applying an independent age shift of -2, -1, 0, +1, or +2 ka to each component record, with each potential value selected with equal probability. After performing each iteration of the PCA, we use random sampling to evaluate the effects of uncertainty associated with scaling PC1 to Holocene and LGM sea level. The particular Holocene point scaled to 0 m is randomly sampled from 0 to 6 ka with uniform distribution. The LGM age is identified as the minimum sea level estimate between 19 and 34 ka, and the sea level to which it is scaled is sampled with a normal distribution centered at 132 m with a standard deviation of 2 m. The bootstrap results for the scaled PC1 yield a mean standard deviation of 9.4 m with seven records (0–430 ka) and 12 m with five records (0–798 ka). Additionally, the inclusion of age uncertainty in the bootstrap analysis has the effect of systematically smoothing the record. Because many of the individual reconstructions are of low resolution relative to brief interglacial highstands such as MIS 5e and 7e, the bootstrapped median is biased towards underestimating these highstands (Fig. 2c). Therefore, in Table 3 we additionally describe the 95 % confidence interval for sea level maxima and minima in the bootstrapped samples.

The sea level stack is archived in the Supplement and at the World Data Center for Paleoclimatology operated by the National Climatic Data Center of the National Oceanographic and Atmospheric Association (https://www.ncdc.noaa.gov/paleo/study/19982).