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    Illustration of the method used in Adkins et al. (2002) to reconstruct the LGM salinity and δ18O. Changes in bottom water salinity were assumed to scale to changes in the mean sea level. The scaling factor was adjusted until the pore fluid data produced by the hypothetical bottom water salinity time series closely matched the measured pore fluid data. Low sea level corresponds to high salinity and vice versa. (a) Bottom water salinity histories using three different scaling factors. (b) Forward model output in response to the bottom water histories, overlaid on measured [Cl] data {black circles, converted from [Cl] to S using the relation S = (19.354/35)[Cl]} in sediment pore fluids at ODP site 1093 (49°58.6′S, 5°51.9′E; Shona Ridge). Each color corresponds to a different LGM–modern scaling factor.

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    Reconstructions of past sea level relative to present (black circles) and the points we use for sea level in computing the prior mean salinity and δ18O (red triangles). See also Table 1.

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    Modern deep ocean measurements below 2000 m, excluding the Mediterranean Sea from the GISS Global Seawater Oxygen-18 gridded database (Schmidt et al. 1999; LeGrande and Schmidt 2006). Red curves are Gaussian distribution fits to data. (a) Modern S below 2000 m and (b) modern δ18O below 2000 m.

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    Prior distribution for is lognormal centered on 5.05 × 10−5 cm2 s−1, with standard deviation of the natural logarithm equal to 1.5. (a) Prior probability density plotted as a function of the natural logarithm of . (b) Prior probability density plotted as a function of .

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    Linear stretched sea level synthetic example with uninformative prior variance = 100 g2 kg−2. Red line with triangular markers is the known synthetic SBW(t). The black open circles are the mean posterior values of SBW(t) (i.e., the salinity values estimated by inversion). Gray shading denotes the standard deviation of the posterior distribution. Black dashed line with star markers is the prior mean. (a) Case where no temporal covariance in bottom water salinity is assumed (T = 0 years), ϵ = 0.66. (b) Case with a covariance time scale T = 6000 years, ϵ = 0.92.

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    Histograms of the prior (blue) and posterior (red) distributions of bottom water salinity. Histograms are directly proportional to the marginal distributions of each variable (i.e., the absolute probability of the variable not conditioned on any other variable). The prior distribution is characterized by a variance = 100 g2 kg−2. (a) Case where no temporal covariance in bottom water salinity is assumed (T = 0 years), ϵ = 0.66 (b) Case with a covariance time scale T = 6000 years, ϵ = 0.92.

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    Lomb–Scargle periodogram of the posterior mean SBW(t) for the stretched sea level example using a Gaussian prior with variance 100 g2 kg−2. Spectral power units are variance of the time series (g2 kg−2). The p value is the probability (at the 99% significance level) that the power detected at a given frequency is generated by N independent random Gaussian variables (noise), where N is the number of frequencies sampled in computing the periodogram. The black dashed line is the periodogram of the prior mean that is equivalent to the periodogram of the true synthetic SBW(t). Each color is the periodogram of the posterior mean with a different prior covariance time scale T, from 0 to 6000 years. The vertical lines overlain show the peak frequencies for the prior and those of the posterior for the example T = 6000 years.

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    Linear stretched sea level synthetic example with modern prior variance ( = 0.05 g2 kg−2). Red line with triangular markers is the known synthetic SBW(t). The black open circles are the mean posterior values of SBW(t) (i.e., the salinity values estimated by inversion). Gray shading denotes the standard deviation of the posterior distribution. Black dashed line with star markers is the prior mean. (a) Case where no temporal covariance in bottom water salinity is assumed (T = 0 years), ϵ = 0.28. (b) Case with a covariance time scale T = 4000 years, ϵ = 0.13.

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    As in Fig. 8, but with intermediate prior variance. (a) Case where no temporal covariance in bottom water salinity is assumed (T = 0 years), ϵ = 0.23. (b) Case with a covariance time scale T = 4000 years, ϵ = 0.16.

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    Synthetic, prior mean, and posterior mean bottom water salinity from representative linear synthetic random ensembles. (a) Variance 0.05 g2 kg−2 and covariance T = 4000 years, ϵ = 0.10. (b) Variance 0.5 g2 kg−2 and covariance T = 0 years, ϵ = 0.60. Red is the target or true model from which the data were generated. Black circles are the mean of the posterior distribution. Black stars and dashed lines show the mean of the prior distribution.

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    Difference between the ensemble of random synthetic examples and the prior (black) compared to the difference between the ensemble of random synthetic examples and posterior (blue) distribution means. Values closer to zero indicate better estimates. Shading and dotted lines represent one standard deviation of the ensemble difference: (a) = 0.05 g2 kg−2, T = 4000 years, = 0.13 and (b) = 0.5 g2 kg−2, T = 0 years, = 0.58.

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    Prior and posterior distributions of ln for the nonlinear stretched sea level inverse problem with uninformative prior variance ( = 100 g2 kg−2). Gaussian noise of 0.05% is added to the synthetic salinity data before inversion. The black dot in each panel shows the true (target) value of that is used to generate the data. (a) Case where no temporal covariance in bottom water salinity is assumed (T = 0 years), ϵ = 1.3. (b) Case with T = 6000 years, ϵ = 0.90.

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    Correlation coefficients of vs salinity time series for the nonlinear stretched sea level synthetic example with different assumptions in the prior distribution: (a) = 0.05 g2 kg−2 and T = 4000 years covariance time scale, ϵ = 0.23 and (b) = 0.5 g2 kg−2 and T = 0 years, ϵ = 0.28.

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    Lomb–Scargle periodograms for the nonlinear stretched sea level synthetic example of bottom water salinity estimated by inversion (nonlinear problem) for different assumptions about the prior distribution. (a) Salinity variance = 0.05 g2 kg−2 and covariance time scale T = 0 years, ϵ = 0.34; (b) = 0.5 g2 kg−2 and T = 0 years, ϵ = 0.28; (c) = 0.05 g2 kg−2 and T = 4000 years, ϵ = 0.23; and (d) = 0.5 g2 kg−2 and T = 4000 years, ϵ = 0.26.

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    Lomb–Scargle periodograms for representative examples from the nonlinear random ensemble synthetic examples of bottom water salinity estimated by inversion (nonlinear problem) for different assumptions about the prior distribution. (a) Salinity variance = 0.05 g2 kg−2 and covariance time scale T = 0 years, ϵ = 0.23; (b) = 0.5 g2 kg−2 and T = 0 years, ϵ = 0.68; (c) = 0.05 g2 kg−2 and T = 4000 years, ϵ = 0.13; and (d) = 0.5 g2 kg−2 and T = 4000 years, ϵ = 0.47.

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    Prior and posterior distributions of ln from representative examples from the nonlinear synthetic random ensembles. (a) Salinity variance = 0.05 g2 kg−2 and covariance time scale T = 4000 years, ϵ = 0.13; (b) = 0.5 g2 kg−2 and T = 0 years, ϵ = 0.68.

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The Information Content of Pore Fluid δ18O and [Cl]

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  • 1 California Institute of Technology, Pasadena, California
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Abstract

Paleoceanographic proxies indicate that the ocean state during the Last Glacial Maximum (LGM) differed from the modern ocean state. Depth profiles of ocean sediment pore fluid δ18O and [Cl] have been used to reconstruct the δ18O and salinity at the LGM. Here, it is investigated whether pore fluid profiles can constrain ocean δ18O and salinity at other times and, simultaneously, their ability to constrain the LGM δ18O and salinity. An inverse framework is developed that relies on Bayesian parameter estimation, thus allowing formal separation of prior assumptions from the information in observations. Synthetic problems are used to explore the information about past ocean tracers that can be recovered from pore fluid profiles. It is concluded that prior knowledge of deep ocean mixing time scales is essential to an accurate inverse estimate of LGM ocean salinity and δ18O from modern pore fluid profiles. The most recent 10 000 years of ocean salinity and δ18O and the error in their estimates are better constrained by the pore fluid profiles than are the LGM values. The inverse estimate of salinity and δ18O is strongly correlated with the estimate of diffusivity of oxygen isotopes and [Cl] in sediment pore fluids. Uncertainty on the diffusivity of oxygen isotopes and chloride in sediments is reduced through inversion of the pore fluid profiles, but simultaneous estimation of δ18O or salinity and diffusivity increases the total uncertainty. The error reported in previous work may underestimate the true uncertainty of LGM deep ocean salinity and δ18O.

Current affiliation: Department of Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts.

Corresponding author address: Madeline D. Miller, Department of Earth and Planetary Sciences, Harvard University, 20 Oxford St., Cambridge, MA 02138. E-mail: madelinemiller@fas.harvard.edu

Abstract

Paleoceanographic proxies indicate that the ocean state during the Last Glacial Maximum (LGM) differed from the modern ocean state. Depth profiles of ocean sediment pore fluid δ18O and [Cl] have been used to reconstruct the δ18O and salinity at the LGM. Here, it is investigated whether pore fluid profiles can constrain ocean δ18O and salinity at other times and, simultaneously, their ability to constrain the LGM δ18O and salinity. An inverse framework is developed that relies on Bayesian parameter estimation, thus allowing formal separation of prior assumptions from the information in observations. Synthetic problems are used to explore the information about past ocean tracers that can be recovered from pore fluid profiles. It is concluded that prior knowledge of deep ocean mixing time scales is essential to an accurate inverse estimate of LGM ocean salinity and δ18O from modern pore fluid profiles. The most recent 10 000 years of ocean salinity and δ18O and the error in their estimates are better constrained by the pore fluid profiles than are the LGM values. The inverse estimate of salinity and δ18O is strongly correlated with the estimate of diffusivity of oxygen isotopes and [Cl] in sediment pore fluids. Uncertainty on the diffusivity of oxygen isotopes and chloride in sediments is reduced through inversion of the pore fluid profiles, but simultaneous estimation of δ18O or salinity and diffusivity increases the total uncertainty. The error reported in previous work may underestimate the true uncertainty of LGM deep ocean salinity and δ18O.

Current affiliation: Department of Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts.

Corresponding author address: Madeline D. Miller, Department of Earth and Planetary Sciences, Harvard University, 20 Oxford St., Cambridge, MA 02138. E-mail: madelinemiller@fas.harvard.edu

1. Introduction

Combining constraints from sediment pore fluid profiles of δ18O and [Cl] with δ18O of carbonate of benthic foraminifera shells (δ18Oc), Adkins et al. (2002) inferred that there were larger-than-modern density differences between deep ocean water masses at the Last Glacial Maximum (LGM), a period lasting from 23 to 19 thousand years before present (ka BP; MARGO Project Members 2009), and that differences in density were due primarily to salinity. Of the ocean water masses considered, Adkins et al. (2002) concluded that Glacial Southern Source Bottom Water (GSSBW), deep water originating from the Southern Hemisphere, was the densest due to its high salinity. These results contrast strikingly with the distribution of today’s deep ocean water masses whose density differences are set primarily by temperature; modern southern source bottom water, Antarctic Bottom Water (AABW), is the densest deep ocean water mass because it is cold, while remaining less saline than overlying water masses [see discussion in Miller et al. (2012)].

Larger density differences between northern source deep water and southern source bottom water imply higher vertical stratification. A greater vertical stratification would help explain indications that there was a physically isolated reservoir of CO2 in the deep ocean at the LGM (Broecker and Barker 2007). While the LGM distribution of δ13C, Cd/Ca, and δ18O possibly indicates a different-than-modern ocean circulation (e.g., Marchal and Curry 2008; Lund et al. 2011), the LGM ocean distributions of these properties may also be consistent with a modern circulation and differences in surface forcing (e.g., Gebbie 2012; Gebbie and Huybers 2006; LeGrand and Wunsch 1995). Improved constraints on ocean temperature and salinity in the past would facilitate the disentanglement of physical changes from chemical and biological signals and better explain why tracer fields in the past ocean varied so strikingly from those of today’s ocean.

The search for reliable proxies of past deep ocean temperature and salinity has proved difficult. In theory, pore fluid reconstructions of past δ18O and [Cl] have a unique advantage over other paleoceanographic proxies of temperature and salinity in that they respond only to physical changes in the ocean. However, so far pore fluid profiles of δ18O and [Cl] have only been used to reconstruct the temperature and salinity at a single point in time: the LGM. To fully understand the changing ocean circulation over the most recent deglaciation, we must know the temporal evolution of temperature and salinity fields. Here we investigate whether pore fluid profiles can constrain the ocean δ18O and [Cl] at other points in time. Simultaneously, we investigate the extent to which they actually constrain the LGM δ18O and [Cl].

In their reconstructions of LGM δ18O and [Cl], Lado Insua et al. (2014), Malone et al. (2004), Adkins et al. (2002), Schrag et al. (2002), Adkins and Schrag (2001), Paul et al. (2001), Schrag et al. (1996), Schrag and DePaolo (1993), and McDuff (1985) relied on a number of restrictive assumptions that made it impossible to recover histories of δ18O and [Cl] at any time other than the LGM. They supposed that δ18O and [Cl] are both conservative tracers in ocean sediments subject to one-dimensional advection and diffusion.1 All of the parameters of the problem were assumed known except the bottom water histories of both δ18O and [Cl]. These histories determine the concentration boundary condition time series at the sediment–water interface. They further assumed that the basic shape of the bottom water histories was known up to a scaling constant; that is, the bottom water histories of δ18O and [Cl] primarily tracked changes in δ18O of benthic foraminiferal calcite, benthic δ18Oc, and sea level, but were able to scale relative to an LGM-to-modern difference in concentration. This scaling parameter was then varied in order to find a good fit between the modeled output and the measurements (Fig. 1). Three different salinity histories resulting from three choices for the LGM-to-modern scaling parameter are shown in Fig. 1a. The modern pore fluid profile data that would result from the application of each of these histories as the bottom water boundary condition are plotted on top of measurements [Ocean Drilling Program (ODP) site 1093] in Fig. 1b. The reported value of the LGM salinity was determined from the LGM-to-modern scaling parameter that yielded pore fluid profiles with the best fit to the measurements. Thus, the deglacial history at each site was fixed to a composite benthic δ18Oc record and sea level curve, scaled by a spatially variable (site specific) constant.

Fig. 1.
Fig. 1.

Illustration of the method used in Adkins et al. (2002) to reconstruct the LGM salinity and δ18O. Changes in bottom water salinity were assumed to scale to changes in the mean sea level. The scaling factor was adjusted until the pore fluid data produced by the hypothetical bottom water salinity time series closely matched the measured pore fluid data. Low sea level corresponds to high salinity and vice versa. (a) Bottom water salinity histories using three different scaling factors. (b) Forward model output in response to the bottom water histories, overlaid on measured [Cl] data {black circles, converted from [Cl] to S using the relation S = (19.354/35)[Cl]} in sediment pore fluids at ODP site 1093 (49°58.6′S, 5°51.9′E; Shona Ridge). Each color corresponds to a different LGM–modern scaling factor.

Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0203.1

Today the bulk of the ocean interior is weakly stratified, such that the deep ocean salinity and δ18O are everywhere very close to their mean ocean values. However, coupling the local time series to the global mean time series implicitly assumes that changes in total ocean water volume always produce the same local change in properties. Further, this approach requires that all sites coevolve in the same way at all times. Tying bottom water histories at all sites proportionally to the mean sea level curve prohibits them from expressing independent temporal evolution as they approach present day values. Finally, this assumption requires that there is a linear relationship between a site’s concentration and the global mean. Adkins et al. (2002) note that their best fit models generate systematic misfits between all of the data and model-generated profiles in the upper sediment column, implying that the pore fluid profiles contain information about the deglacial evolutions of temperature and salinity that could not be extracted with their methodology.

Previous pore fluid reconstructions of LGM δ18O and [Cl] have all assumed that the dominant diffusion coefficient at a site can be determined independently of the bottom water boundary condition. The coefficient was calculated as L2/T, where T = 20 000 years and L was the depth of the maximum value of δ18O and [Cl] in the sediments. For a maximum measured salinity at 40 m below sea floor (mbsf), this relation yields = 2.5 × 10−5 cm2 s−1. Recent studies have shown that the LGM occurred at different times for different mountain glaciers and ice sheets, reaching their maxima between 33 and 26.5 ka BP and remaining stable until 19–14 ka BP (Clark et al. 2009), which calls into question both the idea of a synchronous LGM in the ocean and the timing of the LGM at any site in the ocean, particularly given that the ocean equilibration time scales may be up to 10 000 years or even longer (Wunsch and Heimbach 2008).

We develop an inverse framework to reconstruct past ocean bottom temperature and salinity from ocean sediment core interstitial water profiles of δ18O and [Cl] (in combination with δ18O of carbonate of benthic foraminifera) that allows us to relax many of the assumptions listed above. In particular, we relax the assumption that bottom water histories scale to the sea level curve, allowing us to test the robustness of previous reconstructions and to examine the extent to which we can extract more information than the LGM salinity and δ18O from present-day interstitial water profiles. This new approach to the problem allows us to simultaneously reevaluate the information about the LGM yielded by the pore fluid profiles.

There are infinite bottom water histories that could have generated our observations; the inverse problem is nonunique. However, some of the mathematically acceptable histories may be physically implausible. For example, negative salinities in the inverse solution time series are mathematically possible but physically impossible. Therefore, we seek the full distribution of physically acceptable bottom water time series that could have generated our measurements. Our approach relies on a Bayesian Markov chain Monte Carlo (MCMC) method to reconstruct a distribution of bottom water histories that could generate modern pore fluid profiles of δ18O and [Cl]. A Bayesian approach allows us to easily restrict our solution space. The Bayesian approach also explicitly allows us to separate our prior assumptions and uncertainty from information contained in the measurements and to recover a distribution of inverse solutions with varying probabilities. MCMC techniques allow us to consider the fully non-Gaussian and nonlinear problem, in which we invert for physical parameters such as the diffusion coefficient with arbitrary prior distributions, in a manner robust to local minima and maxima of the inverse solution space. The particular algorithm we use is named the Cascading Adaptive Transitional Metropolis in Parallel (CATMIP; Minson et al. 2013), which is one example of the general class of Bayesian MCMC algorithms.

We first describe our methodological approach to the forward and inverse problems and explain how we encapsulate our knowledge about the deep ocean as probability distributions that can be supplied to a Bayesian MCMC sampler. With synthetic examples, we then evaluate what information about past bottom water histories can be recovered from modern pore fluid profiles of δ18O and [Cl]. In a separate manuscript, we use what we have learned from synthetic examples to invert real data and demonstrate the robust and nonrobust oceanographic constraints that can be recovered from existing pore fluid profiles.

For convenience in the text we often refer to [Cl] as salinity or S, using the conversion factor S = (19.354/35)[Cl]; however, all of the data measurements are derived from pore fluid [Cl]. In our synthetic examples, we consider the evolution of salinity, but our conclusions apply equally to reconstructions of both δ18O and S, as well as other conservative tracers.

2. Problem framework and inverse approach

a. Forward problem

In the following section, we outline the derivation of the advection–diffusion equation for the time evolution of pore fluid concentration. For more complete treatment of the assumptions and equations, we refer the reader to Berner (1980) and Boudreau (1997). We assume that [Cl] and δ18O are chemically inert and that there are no significant local horizontal gradients in the sediment properties. The vertical distribution of each individual tracer c then evolves in time according to a one-dimensional advection–diffusion equation:
e1
where ϕ is the sediment porosity (water volume/bulk volume), θ is the sediment tortuosity (traveled distance of diffused ion/direct distance between spaces), and D0 is the diffusivity in water of the chemical species of interest. The variable D0 is a function of temperature and ionic strength, which both can be functions of depth and time in ocean sediments. The variability of D0 with temperature is well-predicted by the Stokes–Einstein relation , where η is the temperature-dependent viscosity of the water and T is the absolute temperature. Thus, D0 = rD0,T=20°C = r, where r is a function of temperature. Here we take to be an unknown constant and use the temperature measurements logged downhole at the time of drilling to compute r, assuming that the vertical gradient of temperature is steady. The variable u is the vertical velocity of the pore water with respect to the sediment–water interface, which is predominantly due to sediment compaction, and z is the depth in the sediment, defined as increasing downward.
If we assume steady-state compaction and incompressible flow , Eq. (1) simplifies to
e2
where . A typical model for θ2 is θ2 = ϕf, where f is the nondimensional formation factor:
e3
Laboratory measurements suggest that a good approximation for f is ϕn, where an average value of n over various sediments is 1.8 (Berner 1980). Our final equation describing concentration evolution with time and space is
e4
where ϕ in our model is a smoothed version of the measured porosity in each core. Assuming that compaction ceases at some depth in the sediment and that compaction is at steady state , u can be derived from the constant sedimentation rate and porosity data. At the depth of no compaction H, the fluid velocity u and the sediment burial velocity w are the same, that is, uH = wH and . Then, by continuity, , or . The variable wH is the constant sedimentation rate, determined by dating the sediment core. The vertical concentration profile is modeled according to Eq. (4), with a concentration profile as the initial condition and with two boundary conditions. The top (ocean–sediment interface, at z = 0 m) boundary condition is the ocean bottom water history overlying the core site that we seek to constrain. At the lower boundary (z = 350 m), we assign a gradient boundary condition. The evolution of Eq. (4) is heavily dominated by diffusion, making it a stiff partial differential equation. We numerically solve the equation with a second-order implicit–explicit (IMEX) method, in which the advection terms are solved explicitly while the diffusion terms are solved implicitly (Ascher et al. 1995). The time step in the solver is 10 years and the spatial resolution is 50 cm. The total integration time is 125 000 years. Over the integration time of 125 000 years and 350 m of sediment, these choices lead to a two-norm combined round-off and truncation error in the solution of O(10−4) g kg−1. This error estimate is derived from integration of an analogous one-dimensional advection–diffusion equation with constant coefficients to which there is a known analytical solution.

b. Inverse problem

In practice, we measure the modern concentration depth profile c(z)|t=mod, and from this set of observations we infer the past time evolution of water properties at the ocean sediment–water interface, c0(t). We can write this problem in general as
e5
where m represents the unknown boundary conditions; dobs is the measured data [c(z)|t=mod], the pore fluid profile; and G is the forward model of advection and diffusion in the sediments. When we solve for boundary conditions only, Eq. (5) is a linear equation that can be written in terms of discrete Green’s functions as a matrix–vector multiplication:
e6
where the bold, Arial font indicates that is a matrix rather than a function. We cannot invert the matrix directly for m, as the solution for m in both the continuous and discretized framework is an ill-posed problem (Hansen 1998).

We find traditional regularized or damped inverse methods insufficient for solving our particular ill-posed inverse problem. It is difficult to bound the bias in the inverse solution because of the choice of regularization. Further, inversion for the diffusion coefficient requires a nonlinear technique. Finally, as we know a priori from the structure of the problem that there exist infinite possible solutions, we seek to recover a distribution of physically reasonable solutions that can serve as an error estimate (Miller 2014).

c. Bayes’ theorem

Though the inverse solutions to a diffusion problem are infinite, we may wish to exclude certain classes of solutions based on our prior knowledge. Using a Bayesian approach, we can restrict the solution space by applying our prior knowledge of the problem. Bayes’ theorem states that
e7
where m is a vector of input parameters and dobs denotes the observations. Vector m is the set of unknowns we wish to solve for, in our case the ocean bottom water history of salinity or δ18O and the unknown diffusion parameter , and dobs is the sediment pore fluid depth profile of measured concentrations. The probability that a given set of input parameters is correct, given the measured data, is proportional to the product of the prior probability of those input parameters and the likelihood of the data produced by the model using those input parameters. The prior probability P(m) encompasses our knowledge of the input parameters independent of the measurements dobs. The likelihood P(dobs | m) is computed using the differences between the measured data and output of the forward model using a given set of input parameters m. The posterior probability distribution, or posterior, is P(m | dobs).

d. Bayesian estimation for a linear inverse problem with Gaussian prior parameters

Assuming a linear model with Gaussian prior probabilities on the input parameters and Gaussian distribution of data error, there exists an analytical solution to the posterior probability of a set of input parameters P(m | dobs), equal to
e8
where
e9
The vector dobs denotes the observations, D is the error covariance matrix for the data in dobs, and M is the prior error covariance matrix for the parameters in m (Tarantola 2005).

e. Bayesian estimation for nonlinear problem or non-Gaussian prior parameters: Bayesian MCMC parameter estimation

For more general cases, either nonlinear problems or when we consider parameters with arbitrary prior probability distributions, we could determine the posterior probability distribution with a grid-search approach. In such an approach, we would construct an n-dimensional grid of parameter values (n is the dimension of m, the number of unknown parameters), with the ranges of each grid dimension determined by the prior probability distribution P(m) of each parameter. Each point on the grid would represent a vector set of choices for m. The initial spacing of the grid would be arbitrarily coarse and we would run the forward model at every grid point and then refine the grid spacing until the estimate of P(m | dobs) converges. Our inverse problem is high dimensional (22–23 parameters), so that a grid-search approach is intractable [because of the curse of dimensionality (Bellman 1957)]. In a Bayesian MCMC approach, instead of directly computing the multidimensional integral of the probability distribution for each parameter, we build an ensemble of inverse solutions whose density in model space is proportional to the Bayesian posterior P(m | dobs). For this search, we use CATMIP (Minson et al. 2013), which behaves efficiently for high-dimensional problems. CATMIP is initialized with the prior probability distribution and evolves from the prior to the posterior in a series of “tempering” or “cooling” steps. At each cooling step the algorithm recovers an intermediate probability distribution between the prior and posterior distributions. For a given set of parameter values m, the likelihood P(dobs | m) is computed by evaluating the forward model and computing a misfit cost function between the model prediction and the measured data.

f. Parameterization of inverse problem

In practice, we must solve for a discrete rather than continuous set of variables. We therefore frame our inverse problem, the solution for a time series from a vertical concentration profile, as a set of discrete parameters. We are primarily interested in recovering the bottom water history, the time series of [Cl] or δ18O at the ocean floor, but its solution is not independent of the other unknowns. As we show later (see section 7), the solution is particularly sensitive to ; therefore, we seek to simultaneously recover the bottom water history and the coefficient . The bottom water history is parameterized as salinity at 22 unevenly spaced, fixed time points, which we henceforth refer to as nodes. The nodes are spaced 2000 years apart near the present-day, with coarser resolution further back in time to 125 000 years BP. The number of nodes is consistent with the observation that only O(10) nodes can be resolved by the data [nodes arranged at fixed points in time, determined by examination of the singular value decomposition (SVD) of the discrete Green’s functions generated from the 1D advection–diffusion equation (Miller 2014)]. Higher temporal resolution of the parameterization increases the computational expense of the fully nonlinear inverse problem without a qualitative change in the inverse solution set. To retain accuracy in the forward finite difference solver, we linearly interpolate between the parameters at fixed times to generate the input to the forward model.

g. Likelihood function

We assume our error has a Laplace (or double exponential) distribution to compute the likelihood function
e10
The data covariance matrix D has only diagonal terms σ2, where σ is one standard deviation of the measurement error.

We use a Laplace error distribution rather than the commonly used Gaussian error distribution in order to reduce the effect of data outliers on our solutions. Note that in general the form of the Laplace distribution is ∝ , while the Gaussian distribution is ∝ . To maintain roughly the same nominal probability within two standard deviations of the mean, we assign b = 2σ.

3. The prior probability of the bottom water history P(m)

We represent our prior knowledge of the salinity or δ18O bottom water histories as multidimensional Gaussian distributions. The value of salinity or δ18O at each fixed time point in the discrete bottom water history is a parameter in our problem. We assign a prior mean and variance to each node. The mean represents our estimate of the global deep ocean mean, while the variance represents our best guess of the global spread of deep ocean salinities or δ18O values around the mean at that time. To ensure smoothly varying solutions, we further impose temporal covariance between node values.

a. Prior information from sea level records

There is little information about spatial or temporal variations in past deep ocean salinity from paleo proxies. However, sea level reconstructions give us a useful prior estimate of the mean ocean salinity. Assuming the major ions in seawater are conservative over our time period of interest, the mean ocean salinity is proportional to the volume of water in the oceans, itself proportional to sea level. We treat the ocean as a rectangular basin that is 3800 m deep today and 131.5 m lower at the LGM (Bard et al. 1990a,b) so that the concentrations scale directly to changes in sea level as
e11
Here mod indicates the modern mean ocean value and h is the mean ocean depth. For consistency with the assumptions made in Adkins et al. (2002), we take the modern mean ocean salinity to be 34.68 g kg−1 and the LGM mean ocean salinity to be μS|LGM = hmodμS|mod/hLGM = 35.9 g kg−1. Our sea level curve is interpolated from a new compilation of published values (Fig. 2, Table 1).
Fig. 2.
Fig. 2.

Reconstructions of past sea level relative to present (black circles) and the points we use for sea level in computing the prior mean salinity and δ18O (red triangles). See also Table 1.

Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0203.1

Table 1.

Sea level compilation.

Table 1.

Conversion from total ice mass to the mean δ18Ow of the ocean requires knowledge of the spatial and temporal variability of δ18O of mountain glaciers and continental ice sheets. Estimates of these variations of δ18O in the past are relatively unconstrained by measurements, making it impossible to quantify their associated errors. There are many existing estimates of the past contribution of δ18O|w (δ18O of water) to the δ18O|c (δ18O of carbonate) of benthic foraminifera at the LGM (e.g., Emiliani 1966; Shackleton 1967; Dansgaard and Tauber 1969; Chappell and Shackleton 1986; Fairbanks 1989; Mix 1987; Schrag et al. 1996). Duplessy et al. (2002) reviews the traditional ice-mass-based estimates of δ18O at the LGM and suggests a lower bound of ocean mean δ18O|w of 0.85‰ based on the freezing point of seawater and an upper bound based on ice mass approximations of 1.47‰. We take the best current guess of the LGM mean ocean δ18O to be the midpoint of the Duplessy et al. (2002) ice mass bounds, 1.16‰. The equation for deep ocean δ18O as a function of time (sea level) is similar to that of μS(t) [Eq. (11)], with modern mean δ18O = 0.

b. Prior information from modern ocean property spatial spreads

Today, salinities and δ18O everywhere in the deep ocean are close to their mean ocean value. Distributions of δ18O and salinity in the past are unknown. Lacking other constraints, we might guess that past spatial distributions of deep δ18O and salinity around their mean ocean values were the same as their modern spatial distributions. The full ocean modern distributions of salinity and δ18O deeper than 2000 m (LeGrande and Schmidt 2006) are not simple Gaussians (Fig. 3); at best they are a mixture of Gaussians. The bimodal nature of the modern distributions comes from the dominance in the modern deep ocean of North Atlantic Deep Water and Antarctic Bottom Water, two water masses with distinct properties. However, the modern ranges of salinity and δ18O at a given site are narrow. The bimodality of the global distributions of salinity and δ18O are due primarily to spatial rather than temporal (e.g., seasonal, interannual) variability.

Fig. 3.
Fig. 3.

Modern deep ocean measurements below 2000 m, excluding the Mediterranean Sea from the GISS Global Seawater Oxygen-18 gridded database (Schmidt et al. 1999; LeGrande and Schmidt 2006). Red curves are Gaussian distribution fits to data. (a) Modern S below 2000 m and (b) modern δ18O below 2000 m.

Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0203.1

In the vicinity of each site on the ocean floor, the modern distribution of salinity and δ18O could be described by a Gaussian. However, we wish to avoid imposing the modern ocean state on the glacial ocean state by assuming, for example, that salinity is high in the North Atlantic and low in the Southern Ocean. A less stringent prior assumption is that the feasible end-to-end ranges of salinity and δ18O at any site in the past match the modern deep ocean end-to-end ranges. As a conservative estimate for the modern spread of salinity and δ18O, we fit a Gaussian probability distribution envelope around each histogram, as shown by the red curves in Fig. 3. Applying this method gives a modern deep ocean standard deviation of salinity of σS = 0.12 g kg−1 and a standard deviation of δ18O of = 0.25‰. In addition to the possible values for deep ocean properties, the prior variances must also account for error in the sea level curve. An error of 10 m in height propagated through the scaling for sea level gives an error of 0.093 g kg−1 in salinity, assuming the LGM height and salinity are known. The error propagation formula is , with δh = 10 m and . The error in δ18O is not directly related to the sea level; however, Duplessy et al. (2002) suggest an error in their estimate for mean δ18Ow should be 0.2‰. Summing together the modern variance and the estimated error in sea level yields σS = 0.213 g kg−1 ( = 0.05 g2 kg−2) and = 0.45‰ [ = 0.20(‰)2].

c. Prior information about ocean property temporal variations

As neighboring points in a time series at a given location are not independent, we assign covariances to the points in the time series. We adopt covariances with the form
e12
where T is a time scale of covariance and i and j are matrix indices denoting discrete values of the parameterized time series. When titj = T, the correlation between the time series values at those times is ~0.6, while if titj = 2T, the correlation between the values is ~0.1.
The covariance time scale T represents roughly half of an ocean site’s memory of its previous salinity or δ18O value. Time scale T is a spatially variable function of ocean mixing, which will also depend on the global distribution of temperature and salinity (through density). Assuming that large-scale steady-state ocean tracer mixing is a balance between advection and eddy diffusion, that is, as in Ferrari and Wunsch (2009),
e13
where v is the fluid velocity; C is the tracer concentration field; and K is the (eddy) diffusivity of a tracer, in general a spatially variable tensor. We can scale Eq. (13) to relate the time scale T to an eddy diffusivity κ using a characteristic length scale L. That is,
e14
where 2T is the time needed for an ocean site to forget its previous value. Assuming that mixing predominantly proceeds along isopycnals, recent estimates suggest κ is O(100) m2 s−1 (Zika et al. 2009, 2010). For deep ocean sites under the influence of spatially homogeneous deep water masses, we assume L is at least half the meridional extent of the Atlantic Ocean, roughly 8000 km. With these values for L and κ, T is ~10 000 years.

Our scaling argument assumes a steady-state ocean circulation and is simplistic. Paleoceanographic records suggest that the ocean does not remain in steady state over O(104) years (e.g., Charles et al. 1996; Hodell et al. 2003; Lamy et al. 2004; Lynch-Stieglitz 2004; Pahnke and Zahn 2005; Robinson et al. 2005). Using radiocarbon ages to estimate the reservoir age of the deep ocean, some authors conclude that the ocean can reach a new steady state after 2000 years (Duplessy et al. 1991; Skinner and Shackleton 2005), implying that T = 1000 years. However, Wunsch and Heimbach (2008) demonstrate that radiocarbon ages can be misleading; the ocean may need up to 10 000 years to reach a new equilibrium in response to regional changes in surface forcing. In our framework this is equivalent to T = 5000 years or more.

d. Accounting for different-than-modern past ocean property spreads

Adkins et al. (2002) conclude that the spread in LGM deep ocean salinity was larger than it is today. Their results demonstrate that the measurements are consistent with a larger-than-modern spread in salinity, but they do not require that the salinity spread was so large. It is possible that the ocean had a wider salinity spread in the past, but we do not have any a priori information on what the magnitude of the spread was.

Bayesian methods allow us to shrink but not expand the distribution on any given parameter. With an assumed Gaussian prior, posterior estimates more than two standard deviations from the mean of the prior are unlikely, as any parameters with 0 prior probability will, by definition, have 0 posterior probability [recall that P(m | dobs) ∝ P(dobs | m)P(m), so P(m) = 0 implies that P(m | dobs) = 0]. To allow for the possibility of the LGM salinities estimated by Adkins et al. (2002), we consider the sensitivity of our solution to wider-than-modern prior variances, in particular a prior variance equivalent to that implied by the spread in Adkins et al. (2002), = 0.5 g2 kg−2.

e. Diffusion coefficient prior

We assign a lognormal prior to the diffusion coefficient because is a Jeffreys parameter. Jeffreys parameters are physical parameters that are always positive but could vary by orders of magnitude (Tarantola 2005). Laboratory studies find values of D0 for chloride between 1 × 10−6 and 20 × 10−6 cm2 s−1, varying as a function of temperature, pressure, and ionic content (Li and Gregory 1974). The D0 of δ18O is the relative diffusivity of water oxygen isotopes, which mirrors the diffusivity of O in water, O(1 × 10−5) cm2 s−1, also a function of temperature (Wang et al. 1953). In our nonlinear inversions in which we estimate , we choose a prior mean diffusion coefficient of 5.05 × 10−5 cm2 s−1 and the standard deviation of the ln equal to 1.5, as shown in Fig. 4.

Fig. 4.
Fig. 4.

Prior distribution for is lognormal centered on 5.05 × 10−5 cm2 s−1, with standard deviation of the natural logarithm equal to 1.5. (a) Prior probability density plotted as a function of the natural logarithm of . (b) Prior probability density plotted as a function of .

Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0203.1

4. Synthetic example framework

We consider several synthetic problems to examine the extent to which we can recover bottom water salinity histories from pore fluid profiles. The details of each synthetic example vary, but all follow the same basic approach. We choose hypothetical bottom water salinity histories, which we refer to as SBW(t). Using a known SBW(t) as the top (ocean–sediment interface) boundary condition in the forward problem (see section 2a), we generate synthetic data: an ocean sediment pore fluid salinity profile. Random synthetic Gaussian noise of 0.05% is added to the synthetic data to represent measurement uncertainties. We then use CATMIP to invert the noisy synthetic data and recover a posterior distribution of SBW(t), which we compare to the known SBW(t) to determine how well we can recover the true solution by inverting the data. The difference between the posterior (inverse) estimate and the true SBW(t) is summarized by the root-mean-square difference in each figure caption. The experiments are summarized in Table 2.

Table 2.

Summary of experiments.

Table 2.

5. Linear problem: Inversion for bottom water salinity history, SBW(t) only

a. Inversion of synthetic stretched sea level example with varying prior assumptions

Our first example is modeled after Adkins et al.’s (2002) reconstructed bottom water salinity history in the Southern Ocean (ODP site 1093), the location with highest inferred salinity at the LGM. We suppose that this was the true bottom water history in order to determine what information about such a history would be retained by the modern measured pore fluid profile. To construct SBW(t), we start with a presumed mean ocean salinity history, μS(t) (see section 3a). We multiply μS(t) by a constant factor, determined by assuming that the LGM salinity in the Southern Ocean was 37.1 g kg−1 [as inferred by Adkins et al. (2002)]. We call this synthetic SBW(t) as the “stretched sea level” example. The diffusivity parameter is 2.5 × 10−5 cm2 s−1, and r and dr/dz are computed from the temperature profile at ODP site 1093 (Gersonde et al. 1999). The variable uH is 0.01 cm yr−1, and the model domain (sedimentary column) is 350-m long. The initial salinity profile is interpolated from the measured chlorinity profile (converted to salinity) at ODP site 1093, as in Adkins et al. (2002). Here we assume that the cyclical nature of glaciations and deglaciations would have caused the depth profile of [Cl] to be the same at the last interglacial (125 000 yr BP) as it is today. The bottom boundary condition is .

For the purpose of our present work, we assume that the mean ocean salinity history μS(t) is known up to an error of 0.093 g kg−1 (see section 3b). Deep ocean salinities should be near the mean ocean value; however, we do not have any prior information on the variance of salinity at the LGM. The range in deep ocean salinities at the LGM reported in Adkins et al. (2002) corresponds to a variance of approximately = 0.5 g2 kg−2, while a generous estimate of deep ocean salinity variability from existing modern compiled measurements combined with LGM sea level error is ≈ 0.05 g2 kg−2. The full range of modern ocean salinity is encompassed by a variance = 100 g2 kg−2 (one standard deviation range is ~25–45 g kg−1). Similarly, we expect temporal correlation in SBW(t), but we also have few constraints on the value of the correlation time scale T. As discussed in section 3, T could be as short as a few hundred years or it could be greater than 5000 years.

With these points in mind, we invert the synthetic data assuming prior variances = 0.05, 0.5, and 100 g2 kg−2. As shorthand, we refer to these as “modern,” “intermediate,” and “uninformative” variances, respectively. We also vary the prior assumed covariance time scale T and consider its effect on the posterior distribution of SBW(t).

The results of the uninformative prior variance example, = 100 g2 kg−2, illustrate that the measured data constrain the absolute values of salinity between 6 ka BP and present when T is assumed to be 0 years (Fig. 5). Note that even when T is assumed to be 0 years, there is some temporal correlation implicit in μS(t). By increasing the value of T while holding constant, we find that the posterior variance of SBW(t) is a strong function of T. Increasing T decreases the posterior variance of SBW(t), particularly between ~26 ka BP and the present (Fig. 6). However, a reduction of the posterior variance relative to the prior variance of SBW(t) does not guarantee that the mean posterior estimate aligns with the true solution, as demonstrated in Fig. 5.

Fig. 5.
Fig. 5.

Linear stretched sea level synthetic example with uninformative prior variance = 100 g2 kg−2. Red line with triangular markers is the known synthetic SBW(t). The black open circles are the mean posterior values of SBW(t) (i.e., the salinity values estimated by inversion). Gray shading denotes the standard deviation of the posterior distribution. Black dashed line with star markers is the prior mean. (a) Case where no temporal covariance in bottom water salinity is assumed (T = 0 years), ϵ = 0.66. (b) Case with a covariance time scale T = 6000 years, ϵ = 0.92.

Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0203.1

Fig. 6.
Fig. 6.

Histograms of the prior (blue) and posterior (red) distributions of bottom water salinity. Histograms are directly proportional to the marginal distributions of each variable (i.e., the absolute probability of the variable not conditioned on any other variable). The prior distribution is characterized by a variance = 100 g2 kg−2. (a) Case where no temporal covariance in bottom water salinity is assumed (T = 0 years), ϵ = 0.66 (b) Case with a covariance time scale T = 6000 years, ϵ = 0.92.

Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0203.1

As shown in Fig. 5, the mean posterior estimate of SBW(t) (here equivalent to the mode or highest-probability inverse solution) oscillates around the true synthetic SBW(t) at times that have a wide posterior variance. This oscillation suggests that while the data do not constrain the value of salinity at these times, they do constrain temporal correlations in SBW(t), that is, the relative value of salinities at different times in the bottom water history. For this reason we compare the frequency content of the prior and posterior estimates (Fig. 7). The frequency content of the prior μS(t) is set equal to the frequency content of the true synthetic stretched sea level SBW(t), so it is unsurprising that the frequency content of the posterior estimates are similar to the frequency content of the true SBW(t). Imposing nonzero values of T does filter the frequency content of the posterior estimate, removing power at the upper and lower ends of the frequency range. In this particular example, the data do not constrain the frequency content of the bottom water salinity history separately from the prior assumptions.

Fig. 7.
Fig. 7.

Lomb–Scargle periodogram of the posterior mean SBW(t) for the stretched sea level example using a Gaussian prior with variance 100 g2 kg−2. Spectral power units are variance of the time series (g2 kg−2). The p value is the probability (at the 99% significance level) that the power detected at a given frequency is generated by N independent random Gaussian variables (noise), where N is the number of frequencies sampled in computing the periodogram. The black dashed line is the periodogram of the prior mean that is equivalent to the periodogram of the true synthetic SBW(t). Each color is the periodogram of the posterior mean with a different prior covariance time scale T, from 0 to 6000 years. The vertical lines overlain show the peak frequencies for the prior and those of the posterior for the example T = 6000 years.

Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0203.1

Assuming = 0.05 g2 kg−2 and T = 0 years in the prior distribution, the synthetic SBW(t) falls outside one standard deviation of the posterior distribution (Fig. 8). Recall that the synthetic SBW(t) in this example is inconsistent with the modern prior variance assumption, because it is modeled after the high-salinity Southern Ocean case in Adkins et al. (2002). Increasing the prior variance assumption from 0.05 to 0.5 g2 kg−2, holding T constant at 0 years, widens the posterior variance but does not significantly alter the mean posterior time series (cf. Fig. 9a to Fig. 8a).

Fig. 8.
Fig. 8.

Linear stretched sea level synthetic example with modern prior variance ( = 0.05 g2 kg−2). Red line with triangular markers is the known synthetic SBW(t). The black open circles are the mean posterior values of SBW(t) (i.e., the salinity values estimated by inversion). Gray shading denotes the standard deviation of the posterior distribution. Black dashed line with star markers is the prior mean. (a) Case where no temporal covariance in bottom water salinity is assumed (T = 0 years), ϵ = 0.28. (b) Case with a covariance time scale T = 4000 years, ϵ = 0.13.

Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0203.1

Fig. 9.
Fig. 9.

As in Fig. 8, but with intermediate prior variance. (a) Case where no temporal covariance in bottom water salinity is assumed (T = 0 years), ϵ = 0.23. (b) Case with a covariance time scale T = 4000 years, ϵ = 0.16.

Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0203.1

Increasing the prior covariance time scale T from 0 to 4000 years improves the match of the posterior mean estimate SBW(t) to the true synthetic SBW(t) over the time period between 40 ka BP and present (Figs. 8b, 9b). The improved match of the posterior mean estimate of SBW(t) to the true synthetic SBW(t) when T = 4000 years results from the underlying temporal correlation of the synthetic SBW(t) (see Fig. 7). As expected, a more accurate prior guess leads to a more accurate posterior estimate.

b. Inversion for random time series consistent with the Gaussian prior assumption

In section 5a we found that our ability to recover SBW(t) from the modern pore fluid profile decreases with time in the past. We also showed that prior assumptions more consistent with the true synthetic SBW(t) led to more accurate posterior estimates. However, the prior assumptions in section 5a were not strictly consistent with the known SBW(t). Specifically, some values of the synthetic SBW(t) were outside the range prescribed by the prior variances. The term SBW(t) had the same frequency content as the prior μS(t), but the prior distribution had an additional constraint of a covariance time scale T. As shown in section 5a, the prior constraint T reduces the frequency content of the posterior mean estimate relative to the frequency content of the prior mean guess and the true SBW(t). For this reason, the posterior estimate may be dominated by frequencies or harmonics consistent with the prior covariance time scale, and an accurate estimate of tracer variability at one frequency may trade off with the fit to another, such that parts of the inverse estimate (e.g., millennial variability, centennial variability) are closer to the true synthetic than others.

Because the prior constraints are parameters of a multivariate Gaussian distribution, complete consistency between prior assumptions and a synthetic example requires considering a distribution of synthetic examples. We therefore construct examples in which we randomly draw an ensemble of 10 synthetic SBW(t) from a multivariable Gaussian distribution consistent with (equivalent to) the prior distribution. In these examples, the mean of the Gaussian is equal to μS(t), and we show representative cases for two different variances, = 0.05 and 0.5 g2 kg−2, permuted with two different covariance time scales, T = 0 and 4000 years. We show that even when the synthetic time series are completely consistent with the prior distribution, the pore fluid data still only constrain salinity a few thousand years before present day.

Even when the prior distribution is consistent with the parent (source) distribution of the synthetic SBW(t), larger values of T and smaller values of lead to more accurate posterior estimates (Fig. 10). When T = 0 years and = 0.05 g2 kg−2, at best the posterior mean estimate resembles the synthetic salinity history in the most recent 8 or 6 ka BP (Fig. 10b). Further back in time, the posterior mean estimate decreasingly resembles the synthetic salinity history, more so the more the synthetic SBW(t) deviates from the prior mean. In the recent past, if the true synthetic SBW(t) is very different from the prior mean, it is easier to recover through the inversion. In the past, if the true synthetic SBW(t) is very different from the prior mean, it is harder to recover through inversion of the measured pore fluid profile data.

Fig. 10.
Fig. 10.

Synthetic, prior mean, and posterior mean bottom water salinity from representative linear synthetic random ensembles. (a) Variance 0.05 g2 kg−2 and covariance T = 4000 years, ϵ = 0.10. (b) Variance 0.5 g2 kg−2 and covariance T = 0 years, ϵ = 0.60. Red is the target or true model from which the data were generated. Black circles are the mean of the posterior distribution. Black stars and dashed lines show the mean of the prior distribution.

Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0203.1

The difference between the posterior estimate and an individual random synthetic SBW(t) varies. However, the ensemble difference between the posterior mean and the synthetic time series is reduced relative to the ensemble difference between the prior mean and the synthetic time series, and the degree of reduction is a strong function of T (Fig. 11). With prior assumption T = 0 years, the posterior mean estimate for SBW(t) is more accurate than the prior estimate between 10 ka BP and the present. With T = 4000 years, the posterior mean estimate is more accurate than the prior mean estimate between 20 ka BP and present.

Fig. 11.
Fig. 11.

Difference between the ensemble of random synthetic examples and the prior (black) compared to the difference between the ensemble of random synthetic examples and posterior (blue) distribution means. Values closer to zero indicate better estimates. Shading and dotted lines represent one standard deviation of the ensemble difference: (a) = 0.05 g2 kg−2, T = 4000 years, = 0.13 and (b) = 0.5 g2 kg−2, T = 0 years, = 0.58.

Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0203.1

6. Nonlinear problem: Simultaneous inversion of bottom water salinity and diffusivity

The diffusivity in ocean sediment pore waters is difficult to measure in situ or to estimate from ocean sediment cores. The values for [Cl] can vary up to an order of magnitude from site to site on the ocean floor. Because the value of is uncertain, we examine our ability to invert modern measured pore fluid profile data for the value of simultaneous to SBW(t). We repeat all of the synthetic examples from section 5 in which is known, now inverting for as an unknown parameter with a lognormal prior assumption. The synthetic data are identical to the data inverted for the linear synthetic examples, such that the true value of is 2.5 × 10−5 cm2 s−1.

a. Nonlinear inversion of synthetic stretched sea level example with varying prior assumptions

A larger range of (here larger than a single, fixed value) in the prior allows for a wider possible range of bottom water histories. With the uninformative prior = 100 g2 kg−2 and T ranging from 0 to 6000 years, the posterior distributions of SBW(t) are wider than those of the corresponding linear examples. The true value of is at the low end of the posterior estimate and does not coincide with the posterior mean (Fig. 12). The posterior mean is also not a good descriptor of the posterior distribution, as the posterior distribution is skewed.

Fig. 12.
Fig. 12.

Prior and posterior distributions of ln for the nonlinear stretched sea level inverse problem with uninformative prior variance ( = 100 g2 kg−2). Gaussian noise of 0.05% is added to the synthetic salinity data before inversion. The black dot in each panel shows the true (target) value of that is used to generate the data. (a) Case where no temporal covariance in bottom water salinity is assumed (T = 0 years), ϵ = 1.3. (b) Case with T = 6000 years, ϵ = 0.90.

Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0203.1

The posterior estimate of and the posterior estimates of salinities at individual times are correlated (Fig. 13). Near the present, higher-than-modern salinities coincide generally with high , while high salinities further in the past co-occur generally with low . The pore fluid profile data contain the information that salinity was higher-than-modern at some time in the past. The higher it was recently, the higher must have been to diffuse away the signal in the sediment (the peak at some depth) to what we measure today. Conversely, if SBW(t) was higher only in the distant past, must have been smaller to retain the observed peak in the sediments. However, without independent knowledge of the value of , the data tell us only that salinity was either high recently or high very far in the past. The data cannot distinguish between these two cases. Higher assumed T increases the correlation between and SBW(t).

Fig. 13.
Fig. 13.

Correlation coefficients of vs salinity time series for the nonlinear stretched sea level synthetic example with different assumptions in the prior distribution: (a) = 0.05 g2 kg−2 and T = 4000 years covariance time scale, ϵ = 0.23 and (b) = 0.5 g2 kg−2 and T = 0 years, ϵ = 0.28.

Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0203.1

When we solve for as a parameter in addition to the values of SBW(t), the frequency content of the posterior mean estimate differs from that of the prior mean and true synthetic SBW(t). Different values of enable significantly different SBW(t) (as judged by their frequency content) to create similar pore fluid profile data. Though the minimum and maximum posterior estimates of span less than half an order of magnitude, the frequency content of their associated time series are distinct (Fig. 14). That is, the estimated time series of bottom water salinity compensates for the inaccurate estimate of . With the modern prior assumption ( = 0.05 g2 kg−2) and T = 0 or 4000 years, SBW(t) associated with the minimum posterior estimate has less power at high frequencies than the mean posterior estimate of SBW(t), and SBW(t) associated with the maximum posterior estimate has more power at high frequencies than the mean posterior estimate of SBW(t). With the intermediate prior variance assumption ( = 0.05 g2 kg−2), the most significant frequencies in the posterior distribution of SBW(t) break into different harmonic groups at the extreme values of (Figs. 14b,d).

Fig. 14.
Fig. 14.

Lomb–Scargle periodograms for the nonlinear stretched sea level synthetic example of bottom water salinity estimated by inversion (nonlinear problem) for different assumptions about the prior distribution. (a) Salinity variance = 0.05 g2 kg−2 and covariance time scale T = 0 years, ϵ = 0.34; (b) = 0.5 g2 kg−2 and T = 0 years, ϵ = 0.28; (c) = 0.05 g2 kg−2 and T = 4000 years, ϵ = 0.23; and (d) = 0.5 g2 kg−2 and T = 4000 years, ϵ = 0.26.

Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0203.1

b. Nonlinear inversion for random time series consistent with the Gaussian prior assumption

We use the identical ensembles of synthetic SBW(t) and corresponding synthetic data described in section 5b, but now we solve simultaneously for as an unknown, with the lognormal prior assumption described in sections 3e and 6a.

In these nonlinear inversions, the posterior mean frequency content is a worse estimate of the true frequency content of the synthetic SBW(t) than in the linear inversions (Fig. 15). The inverse problem is significantly nonlinear in , which means that it is essential to simultaneously solve for when inverting for SBW(t). In all of these cases the (peak) power in the inverse solution is more broadly distributed at the maximum and for least restrictive prior distributions.

Fig. 15.
Fig. 15.

Lomb–Scargle periodograms for representative examples from the nonlinear random ensemble synthetic examples of bottom water salinity estimated by inversion (nonlinear problem) for different assumptions about the prior distribution. (a) Salinity variance = 0.05 g2 kg−2 and covariance time scale T = 0 years, ϵ = 0.23; (b) = 0.5 g2 kg−2 and T = 0 years, ϵ = 0.68; (c) = 0.05 g2 kg−2 and T = 4000 years, ϵ = 0.13; and (d) = 0.5 g2 kg−2 and T = 4000 years, ϵ = 0.47.

Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0203.1

As in the stretched sea level examples, the posterior distributions of the estimates of are wider in the cases when the prior assumed distributions are less informative (Fig. 16). Just as increasing the prior range of increases the variance of the posterior distribution of SBW(t), increased prior variance of SBW(t) leads to increased variance in the posterior estimate of . However, the posterior distribution of is better resolved by addition of information in the data than the imposed input prior. The posterior variance of is smaller by one order of magnitude than its prior variance in all of the cases we considered (Fig. 16). However, the mean posterior estimate of is not equal to the true value of used to generate the synthetic data. A greater spread in allowed salinities as imposed through the prior ( = 0.5 g2 kg−2 rather than 0.05 g2 kg−2) allows a greater spread in possible in the posterior (or vice versa), demonstrating the fundamental covariance between and the time series of salinities that produce the same data.

Fig. 16.
Fig. 16.

Prior and posterior distributions of ln from representative examples from the nonlinear synthetic random ensembles. (a) Salinity variance = 0.05 g2 kg−2 and covariance time scale T = 4000 years, ϵ = 0.13; (b) = 0.5 g2 kg−2 and T = 0 years, ϵ = 0.68.

Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0203.1

7. Discussion and summary

Synthetic examples allow us to determine the maximum information about the past ocean that can be extracted from pore fluid profiles of δ18O and [Cl]. Without assuming a prior covariance time scale or a narrow range of salinity, the measured pore fluid data constrain the bottom water history between 6 ka BP and present. When the assumed range of possible salinity is narrower ( = 0.05 or 0.5 g2 kg−2) the measured pore fluid data constrain the bottom water history between 10 ka BP and present. The difference between the distribution of possible inverse solutions and the true bottom water salinity history is a strong function of the assumed time scale of covariance T. Our results demonstrate that modern profiles of pore fluid δ18O and [Cl] can constrain the range of salinity and δ18O at the LGM only if T is known. When there is no prior assumption of T, the spread of possible salinities at the LGM is equivalent to the initial guess or the variance of the prior distribution. While the absolute values of salinities prior to 10 ka BP are unconstrained by the pore fluid data, the data do constrain the correlation between the salinities at different times in the past. This information about correlation is useful because, combined with other independent data, it could narrow the range of past salinity.

Previous work suggested that the diffusivity coefficient could be determined from a scaling argument separately from the bottom water salinity history. Here we show that the inverse estimate of strongly covaries with the inverse estimate of bottom water salinity. Over a narrow, physically realistic range of , the correlation between salinities at different times in the inverse estimate changes dramatically. For this reason, when we estimate bottom water salinity histories from measured pore fluid profiles, must be included as an unknown parameter.

In all the synthetic cases described here, the bottom boundary condition (vertical gradient of S at z = 350 m) and the initial condition (vertical profile of S from the sediment–water interface at z = 0 m to z = 350 m at 125 000 years ago) are fixed and assumed to be known. We assumed that the cyclical nature of glaciations and deglaciations caused the data profile at 125 ka BP (our initial condition) to be equivalent to the modern measurement profile. We also assumed that the gradient boundary condition at the bottom of the sediment column is constant over the time period of integration and that it can be estimated from the slope of the measured pore fluid data over the final (bottom) tens of meters in the sediment column.

Our inverse framework allows us to relax these assumptions and instead solve for the initial condition and bottom boundary condition as additional unknown parameters. Our preliminary work in this area (see, e.g., Miller 2014) confirms our intuition that, in general, an increased number of free parameters in an ill-posed problem such as this one always increases the uncertainty of the posterior parameter distributions. We make many other assumptions in our advection–diffusion model [Eq. (4)], such as the assumption of steady-state compaction and the assumed relationship between sediment tortuosity and porosity. If these assumptions were relaxed, it is likely that the uncertainty in bottom water salinity and δ18O deduced from pore water profiles would further inflate.

Adkins et al. (2002) relied on stretching and shrinking the salinity history scaled to a composite benthic δ18Oc and mean sea level history, rather than allowing the salinities at each time to vary independently, and this is equivalent to assuming that the salinity covariance time scale T is 4000 years or longer. We have shown here that the variance of the inverse estimate of salinity at the LGM (assuming the LGM occurred around 20 ka BP) is entirely determined by the prior assumption of and T. Therefore, the error reported in Schrag and DePaolo (1993), Schrag et al. (1996), Adkins and Schrag (2001), Paul et al. (2001), Adkins et al. (2002), Schrag et al. (2002), Malone et al. (2004), and Lado Insua et al. (2014) may underestimate the true uncertainty of estimated LGM deep ocean salinity and δ18O.

Adkins and Schrag (2003) investigated the sensitivity of estimating LGM salinity to various assumptions about the bottom water salinity history, the diffusivity of [Cl], the effective advective velocity of [Cl], and the initial condition pore fluid profile. They concluded that the overall uncertainty in the relative difference between LGM and present salinity was 0.1%–0.5% (out of 3.5%). Their approach involved small perturbations around a fixed value, varying only one parameter at a time. Our approach allows us to vary the parameters simultaneously and therefore explore the inverse solution space more completely. We find that salinities, diffusivity, and initial conditions can compensate for each other. This ability to compensate increases the uncertainty on any one parameter.

Previous authors (Lado Insua et al. 2014; Malone et al. 2004; Adkins et al. 2002; Schrag et al. 2002; Adkins and Schrag 2001; Paul et al. 2001; Schrag et al. 1996; Schrag and DePaolo 1993; McDuff 1985) who have reconstructed LGM salinity and δ18O from pore fluid profiles have all assumed that the bottom water salinity history was a scaled version of the benthic δ18Oc history or the sea level curve. Such a scaling imposes a specific set of temporal correlations on the bottom water history. It is difficult to justify assuming that the temporal correlation of salinity at a deep ocean site exactly matches that of the changing sea level or of δ18Oc. The temporal correlation in δ18O is only equivalent to that in a benthic δ18Oc record if the proportion of changes in δ18O to temperature remains constant over time. Without this assumed temporal correlation, the uncertainty of the LGM salinity and δ18O inferred from pore fluids is greater. If we can narrow the likely range of temporal correlation further, as is done through assumption in previous work, we can use the pore fluid profiles to help constrain the bottom water salinity and δ18O histories from the LGM through the deglaciation to the present. Increased prior information on T could be gained from analyzing temporal correlations in benthic foraminiferal records of bottom water properties in the same cores.

Acknowledgments

M.D.M. thanks William Jenkins for introduction to the Lomb–Scargle periodogram and Eli Tziperman for many helpful discussions and constructive reviews of drafts of this manuscript.

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1

While there is significant two- or three-dimensional advective fluid flow in many places on the ocean floor associated with spreading and converging plate boundaries, submarine groundwater discharge from continental aquifers, gas seeps (Judd and Hovland 2007), and bathymetric pressure perturbations due to current–obstruction interactions (Huettel and Webster 2001), the evolution of pore fluid concentration profiles in impermeable muddy sediments in abyssal plains is dominated by one-dimensional diffusion (Spinelli et al. 2004; Huettel and Webster 2001; Boudreau 1997; Berner 1980).

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