An earlier analysis of pore-water salinity (chlorinity) in two deep-sea
cores, using terminal constraint methods of control theory, concluded that
although a salinity amplification in the abyss was possible during the LGM,
it was not required by the data. Here the same methodology is applied to

Based upon the work of McDuff (1985), Schrag and DePaolo (1993), Schrag et al. (1996, 2002), Adkins and Schrag (2001, 2003), Insua et al. (2014), and several others, on the properties of pore waters in abyssal cores, inferences have been made about the salinity and temperature of the regional and global abyssal oceans during the Last Glacial Maximum (LGM). A summary of the central conclusion (e.g. Adkins et al., 2002) would be that the deep ocean was almost everywhere near freezing temperatures, with an abyssal salinity much above the global volume average, particularly in the Southern Ocean.

Those inferences have become a part of the ongoing discussion of climate physics, including the whereabouts of global carbon during the LGM, and are a standard against which models are being tested: e.g. Otto-Bliesner et al. (2006), Intergovernmental Panel on Climate Change (2013), and Kobayashi et al. (2015). Recently Miller (2014) and Miller et al. (2015) have challenged this interpretation showing, using a Monte Carlo method, that the uncertainties of the inferences were too great to assert that the LGM abyssal stratification could be determined with useful accuracy.

Their conclusion was tested by Wunsch (2016; hereafter W16) using salinity
(chlorinity) data obtained from the pore waters of two of the cores used by
Adkins and Schrag (2003), Miller (2014), and Miller et al. (2015; hereafter
M15). In contrast to the latter authors, the analysis was carried out in the
physically more direct context of standard control theory: pore fluid data
were treated as a “terminal constraint” on the time-evolving pore-water
properties.

Miller et al. (2015) used a Markov chain–Monte Carlo (MCMC) approach. Whether this stochastic method is intuitively more accessible than the one used here is a matter of taste.

Using highly optimistic assumptions (a known one-dimensional advection–diffusion model with perfect parameters, known initial conditions, etc.), the uncertainties in the estimated abyssal salinity through time supported the M15 inference. In general, the very high local values of abyssal salinity,The purpose of this present paper is to extend the W16 salinity analysis to
the pore-water measurements of the oxygen isotope ratio,

As in W16, the focus here is on the upper 100 m of the cores, where the
observed

The general procedure here for both salinity and

If

Notation used for initial, final, and boundary conditions. In the discrete form, two time steps of the concentration

Because the core properties are measured only at time

Prior estimate of the ocean–sediment interface boundary condition on

Modern

What of the boundary condition

The bottom boundary
condition is problematic. With finite diffusion and/or an upward directed

Following W16, the problem is written in discrete numerical form using a
DuFort–Frankel method (Roache, 1976) with the exception that now

Anyone interested in the rigorous mathematics of such problems in continuous space and time is urged to consult Lions (1971).

Again, with the goal of finding the most optimistic estimates of uncertainty,The notation is that bold lower-case letters indicate column
vectors; bold upper-case letters (Latin or Greek) are matrices, and the
superscript

An equation governing the observations,

The estimation equations are described more fully in W16. Full specification of the system includes these equations plus all of the a priori estimates of uncertainty in parameters, initial conditions, measurement noise, etc.

Standard control theory and state-space methods (e.g. Goodwin and Sin, 1984;
Franklin et al., 1998; Wunsch, 2006) commonly distinguish between two
problems associated with Eqs. (

Identification must be done, at least in part, before estimation. In the
present situation, the identification step is simply assumed to have been
properly carried out by previous authors, who used one-dimensional
structures, various orders of magnitude for

If the identification problem has been solved, producing a useful model (or
“plant” in the engineering control literature), available data can be used
instead to determine the adjusted boundary condition

Control methods introduce the concepts of observability and controllability (Wunsch, 2006; Marchal, 2014) as well as a series of related ideas such as “reachability” (see Goodwin and Sin, 1984). Here, “observability” means that the observations are adequate to perfectly reconstruct the initial conditions. “Controllability” implies that the system can be driven from any initial condition to an arbitrary terminal value.

The extent to which the terminal data are determined by the initial conditions
is an important issue here. Thus (e.g. Wunsch, 2006) with a single
observation at the end time and in the absence of any external disturbance,
the observability matrix, a special case, is

Suppose that the initial condition were zero. Then controllability would answer the question of whether any
choice of control in

Neither of these concepts depends on the actual data. The formalisms can be
used to find explicit descriptions of the terminal data structures
determinable from the initial conditions and controls. Here we proceed instead
by direct construction of the solutions, having inferred that there will be a
strong dependence on both initial conditions and controls, with some
inevitable residuals (the null spaces) to be regarded as “noise.” A fuller discussion of controllability and
observability depends upon understanding whether the smaller, but non-zero,
eigenvalues of

Cores from which salinity (chlorinity) data were used, along with a reference to their initial description in the Ocean Drilling Program (ODP) and with a geographical label. A nominal water depth of the core top is also listed.

The four

Figure 2 displays the positions of the five cores for
which

Measured terminal porosity in each of the cores is displayed in W16. Figure 3
shows the

The visible fluctuations in all cores exceed the estimated analytical accuracy of 0.03 ‰ (Adkins and Schrag, 2001), but the extent to which they represent real changes in boundary conditions through time, their initial conditions, and fluxes from below the measured core depth, as opposed to a variety of noise processes in the formation of a core undergoing active sedimentation, remains obscure. One of the major issues is whether structures other than the visible overall maximum, presumably at the LGM, are signals to be understood or mere noise to be suppressed.

Differences among the core

Measured salinity minus 35 g kg

Variations among the cores imply that there need not be any overall, that is
global, control on their time histories. (See the cautionary statements in
Schrag et al., 2002.) Dynamics and modern oceanographic structures (as in Fig. 2) instead support the accepted inference of different
time histories of the values of

The mean of four core

Figure 5 shows normalized versions of the oxygen
isotope and salinity data in the cores. If these two properties satisfy the
same advection–diffusion Eq. (

Begin as in W16, in which the observed core provides the terminal constraint, and the initial condition is assumed to be the same as the terminal one, but with a larger error estimate. In the absence of any more compelling possibility, the same sea level curve is used, but scaled as in Fig. 1.

Purely diffusive solution for the mean core, to 100 m depth, with

Knowledge of oceanic dynamics and the modern distribution as well as the core

As a simplified context for later discussion of the individual cores, a start is made by averaging the four cores displayed in Fig. 3, with the result also shown there, and limited to the depth of the shallowest record (138 m). An average core does not exist in nature but provides a generic data set to discuss the methodology and results. In any core, one can guess at the structures to be treated as a noise process rather than as signal. Averaging is a data-based noise reduction process, in which incoherent small vertical-scale features will tend to be suppressed. With four examples, the standard error of the result, shown in Fig. 6, is very large, having at most only 3 degrees of freedom. Nonetheless, we proceed. No attention has been paid to differences in sedimentation rate or other depth controlling processes. Results will be used as a framework for later discussion of the individual cores. Because of the linearity of the problem, the final estimation uncertainties do not depend upon the data themselves. In addition, the control solution for the average core will be the same as averaging the controls of the individual cores – if the same statistics are used for them.

The analysis follows much of the earlier literature in setting

The fit to the terminal state is statistically acceptable, with an isotopic maximum at 60–70 m. On the other hand, no significant LGM maximum appears in the control – instead, the smoother places most of the structure into the initial conditions – which, consistent with the observability discussion, persists as a local maximum through the 100 ky time interval. This result emphasizes the ambiguity of initial conditions and control.

Examples such as this one render concrete a number of interlocking elements of
the problem. (1) Noise or uncertainty covariances for the initial conditions,
the terminal data, and the prior

In contrast to the quasi-periodic initial and final conditions, Fig. 8 shows the result when the initial condition was taken to be zero: the system responds by reconstructing the near-periodic initial condition. Again, the residual is acceptable. Initial conditions are very important with these diffusivity values and the time interval.

Now consider what happens when the prior is taken to be the scaled sea level
curve of Fig. 1, with zero initial conditions, and as shown in Fig. 9. The
terminal fit is once again acceptable, and the control adjustments are very
small. The standard inference of enhanced

The strong dependence upon the initial condition is striking. It can be
suppressed as shown in Fig. 10 where the initial
condition was set to zero, with a minute uncertainty, and the terminal
uncertainty was strongly downweighted in the vicinity of the depth of the
local maximum. Then as shown in the figure, a solution reproducing the
terminal maximum is found, with a time-varying control over almost the entire
record whose changes are interesting but not readily interpretable.
Uncertainty in

Same as Fig. 7 except that the initial condition was zero with a large uncertainty, rendering the Kalman filter solution zero until the very end when the terminal data are encountered so that the prediction is zero. The smoothed solution is very similar to that with a near-periodic initial condition, and the initial condition is roughly reconstructed.

The mean core, upper 100 m, and zero initial conditions but with the sea
level prior in Fig. 1. The small adjustment,

Using the full 140 m of the average core, with the zero initial
conditions prevented from changing significantly with very small uncertainty
and a flat prior, a solution reproducing the

Core 1063 with the sea level prior, periodic initial conditions, and a
solution forced to reproduce the local maximum through the terminal
uncertainty estimate (small

When the sea level prior is used in this situation (not shown), the final total control is visually very similar to that shown in Fig. 9. This result suggests that suppression of the initial conditions as unknowns brings the system closer to producing a unique control, but a zero initial state is not easily justified.

In turning to the individual cores, all of the problems arising in the discussion of the
mean core remain, including the sensitivity to the assumptions about the initial conditions,
in turn depending upon the values, structures, and signs of

None of the five cores is obviously “typical”, but core 1063 on the Bermuda Rise, a focus of the study of Adkins and Schrag (2001), has the characteristic maximum at depth with a quasilinear decrease with depth. Again, only the upper 100 m are considered.

Core 1063 with a flat prior, quasi-periodic initial–final conditions,
and with the terminal data uncertainty matrix

Core 1093 with a quasi-periodic initial condition, a flat prior, and
terminal uncertainties forcing solution to the

Results for Core 981 in the northeast Atlantic Ocean using a flat prior and quasi-periodic boundary conditions for a solution forced to produce the maximum at depth by uncertainty variance weighting.

Figure 11 shows the Core 1063 solution with quasi-periodic boundary conditions when it is forced to the water column maximum by greatly reducing the estimated error in its vicinity, with the sea level prior. The fit near the maximum is, as forced and expected, good, but the smaller-scale structures are not reproduced. The same situation but with the flat zero prior is shown in Fig. 12. This solution is marginally better than for the sea level prior, but no LGM maximum appears in the control. In terms of the residuals, this solution effectively treats all structures in the top 100 m of the core, except for the maximum excursion, as a noise process. If that inference is accepted, then a posteriori, an estimate of the variance of the noise structure in the core data has been made.

Core 1093, in the Southern Ocean on the Southwest Indian Ridge, was the main
basis of the inference of a strongly salinity-stratified abyssal ocean during
the LGM. For the top 100 m of

Similar results emerge from the remaining two cores, and so only representative
solutions are shown in Figs. 14 and 15, both for the case of
quasi-periodic initial conditions, the flat prior, and forcing to the terminal time

Results for Core 1123, east of New Zealand, for a flat prior,
quasi-periodic initial conditions, and forcing to the

Comparison, for a 100 m deep core of the numerical solution with

To a very great extent, the results of analysing these cores depend very
directly upon a long list of assumptions of which a rough summary would
include the following:

Physics and chemistry are one-dimensional.

Sedimentation rates are constant.

Rules for diffusivity, porosity, and tortuosity are accurate.

Advection and diffusion without chemical reaction processes is adequate.

Initial conditions are similar to the terminal measurements but with a larger uncertainty.

Structures in the terminal values are (are not) signals or are (are not) noise, and accuracy is dominated by analytical accuracy (or not).

Lower boundary condition at

The scaled sea level curve is a useful prior boundary condition estimate
of the order of

Variance estimates for the uncertainties in the terminal data and in the initial conditions are approximately correct.

The complex boundary layers at the sediment–water interface are adequately replaced by a simple concentration boundary condition.

The overall inference here, consistent with both M15 and W16, is that the
conventional picture of a very cold, highly saline abyssal ocean during the
LGM remains possible but is not a requirement of the existing data. If LGM

Consider a purely diffusive system, with

Supported in part by the National Science Foundation under Grant OCE096713 to MIT. I am again grateful to M. Miller for making the data available to me and for discussions of their use. D. Schrag was very helpful in skeptical discussions of the background and of the details of this problem. I had very useful comments from the general to the very detailed by G. Gebbie, O. Marchal, and the anonymous referee. Edited by: L. Skinner