An algorithm for estimating Absolute Salinity in the global ocean

To date, density and other thermodynamic properties of seawater have been calculated from Practical Salinity, SP. It is more accurate however to use Absolute Salinity, SA (the mass fraction of dissolved material in seawater). Absolute Salinity SA can be expressed in terms of Practical Salinity SP as 5 SA=(35.165 04 g kg /35)SP + δSA(φ, λ, p) where δSA is the Absolute Salinity Anomaly as a function of longitude φ, latitude λ and pressure. When a seawater sample has standard composition (i.e. the ratios of the constituents of sea salt are the same as those of surface water of the North Atlantic), the Absolute Salinity Anomaly is zero. When seawater is not of standard composi10 tion, the Absolute Salinity Anomaly needs to be estimated; this anomaly is as large as 0.025 g kg−1 in the northernmost North Pacific. Here we provide an algorithm for estimating Absolute Salinity Anomaly for any location (φ, λ, p) in the world ocean. To develop this algorithm we use the Absolute Salinity Anomaly that is found by comparing the density calculated from Practical Salinity to the density measured in 15 the laboratory. These estimates of Absolute Salinity Anomaly however are limited to the number of available observations (namely 811). To expand our data set we take advantage of approximate relationships between Absolute Salinity Anomaly and silicate concentrations (which are available globally). We approximate the laboratorydetermined values of δSA of the 811 seawater samples as a series of simple functions 20 of the silicate concentration of the seawater sample and latitude; one function for each ocean basin. We use these basin-specific correlations and a digital atlas of silicate in the world ocean to deduce the Absolute Salinity Anomaly globally and this is stored as an atlas, δSA(φ, λ, p). This atlas can be interpolated to the latitude, longitude and pressure of a seawater sample to estimate its Absolute Salinity Anomaly. 25 For the 811 samples studied, ignoring the Absolute Salinity Anomaly results in a standard error in SA of 0.0107 g kg −1. Using our algorithm for δSA reduces the error to 0.0048 g kg−1, reducing the mean square error by a factor of five. The number of sea 216 water samples used to develop the correlation relationship is limited, and we hope that the algorithm and error can be improved as further data becomes available.

To date, density and other thermodynamic properties of seawater have been calculated from Practical Salinity, S P .It is more accurate however to use Absolute Salinity, S A (the mass fraction of dissolved material in seawater).Absolute Salinity S A can be expressed in terms of Practical Salinity S P as S A =(35.165 04 g kg −1 /35)S P + δS A (ϕ, λ, p) where δS A is the Absolute Salinity Anomaly as a function of longitude ϕ, latitude λ and pressure.When a seawater sample has standard composition (i.e. the ratios of the constituents of sea salt are the same as those of surface water of the North Atlantic), the Absolute Salinity Anomaly is zero.When seawater is not of standard composition, the Absolute Salinity Anomaly needs to be estimated; this anomaly is as large as 0.025 g kg −1 in the northernmost North Pacific.Here we provide an algorithm for estimating Absolute Salinity Anomaly for any location (ϕ, λ, p) in the world ocean.
To develop this algorithm we use the Absolute Salinity Anomaly that is found by comparing the density calculated from Practical Salinity to the density measured in the laboratory.These estimates of Absolute Salinity Anomaly however are limited to the number of available observations (namely 811).To expand our data set we take advantage of approximate relationships between Absolute Salinity Anomaly and silicate concentrations (which are available globally).We approximate the laboratorydetermined values of δS A of the 811 seawater samples as a series of simple functions of the silicate concentration of the seawater sample and latitude; one function for each ocean basin.We use these basin-specific correlations and a digital atlas of silicate in the world ocean to deduce the Absolute Salinity Anomaly globally and this is stored as an atlas, δS A (ϕ, λ, p).This atlas can be interpolated to the latitude, longitude and pressure of a seawater sample to estimate its Absolute Salinity Anomaly.
For the 811 samples studied, ignoring the Absolute Salinity Anomaly results in a standard error in S A of 0.0107 g kg −1 .Using our algorithm for δS A reduces the error to 0.0048 g kg −1 , reducing the mean square error by a factor of five.The number of sea water samples used to develop the correlation relationship is limited, and we hope that the algorithm and error can be improved as further data becomes available.

Introduction
The composition of the dissolved material in seawater is not totally constant, but varies a little from one ocean basin to another.Brewer and Bradshaw (1975) and Millero (2000) point out that these spatial variations in the relative composition of seawater impact the relationship between Practical Salinity (which is essentially a measure of the conductivity of seawater at a fixed temperature and pressure) and density.
The thermodynamic properties of seawater are more accurately written as functions of Absolute Salinity (as well as of temperature and pressure) rather than as functions of Practical Salinity (Millero, 1974;Millero et al., 1976b).One can make reasonable estimates of the thermodynamic properties of seawater from the concentration and known properties of the components of the solution (this simple additivity is known as Young's rule).All the physical properties of seawater as well as other multicomponent electrolyte solutions are directly related to the concentrations of the major components not the salinity determined by conductivity.Some of the variable nonelectrolytes (e.g., SiO 2 , CO 2 and dissolved organic material) do not have a conductivity signal but they do contribute to the thermodynamic properties such as density, enthalpy, entropy etc.It is for this reason that the new thermodynamic definition of seawater (IAPWS-2008, Feistel, 2008) has the Gibbs function g of seawater expressed as a function of Absolute Salinity as g(S A , t, p) rather than as a function of Practical Salinity S P or of Reference Salinity, S R .Consider for example exchanging a small amount of pure water with the same mass of silicate in an otherwise isolated seawater sample at constant temperature and pressure.The conductivity is almost unchanged but the Absolute Salinity is increased and Young's rule indicates that the density, enthalpy etc. are changed in proportion to the change in Absolute Salinity.Similarly, if a small mass of say NaCl is added and the same mass of silicate is taken out of the sample, the Absolute Salin-217 ity will not have changed (and by Young's rule the density, enthalpy etc will be almost unchanged) but the Practical Salinity will have increased.
Ocean models treat their prognostic variables as possessing the "conservative" property, and the interaction of the ocean with the ice and the atmosphere already proceeds in a manner consistent with the ocean model's salinity variable being Absolute Salinity (see Jackett et al., 2006).In order to make ocean models totally consistent with TEOS-10 the models need to be initialized with Absolute Salinity and the salinity output of the models need to be compared with Absolute Salinity values derived from observations.As a first step towards incorporating the difference between Practical Salinity S P and Absolute Salinity S A in oceanographic practice, Millero et al. (2008a) defined a reference composition of seawater.This reference composition defines exact mole fractions of the major components of seawater (see Table 4 of Millero et al., 2008a).Up to the accuracy of measurements to date, this reference composition is identical to that of Standard Seawater (which is surface water from a specific region of the North Atlantic).Using the most recent atomic weights, Millero et al. (2008a) calculated the Absolute Salinity of seawater of reference composition, and this salinity they called Reference-Composition Salinity, S R .For the range of salinities where Practical Salinities are defined (that is, in the range 2<S P <42) it was shown that S R ≈ u PS S P where u PS ≡ (35.16504/35) g kg −1 . (1) For practical purposes, this relationship can be taken as an equality since the approximation is dominated by the extent to which estimates of Practical Salinity, as determined from measurements of conductivity ratio, temperature and pressure, may vary depending on the temperature and pressure at which the measurements are made.(Although Practical Salinity was formulated so as to not depend on temperature or pressure, the algorithms used to estimate Practical Salinity at conditions other than 15  Millero et al. (2008a) list six reasons for introducing Reference Salinity, the last of which was to be able to use Reference Salinity as a stepping stone to Absolute Salinity, thereby being able to calculate density more accurately.Heuristically this can be thought of as reflecting the fact that some non-ionic species (such as silicate) affect the density of a seawater sample without significantly affecting its conductivity or its Practical Salinity.
The fundamental measurements required to provide a method for estimating Absolute Salinity in terms of values of Practical Salinity have been reported in Millero et al. (1976aMillero et al. ( , 1978Millero et al. ( , 2000Millero et al. ( , 2008bMillero et al. ( , 2009)).The data for samples from the Southern Ocean on the CASO SR3 AA0806 voyage south of Tasmania are given in Table 2.These papers describe measurements of 811 seawater samples from around the globe at the locations shown in Fig. 1.The Practical Salinity S P and the density ρ lab of each seawater sample are measured in the laboratory at 25 • C and at atmospheric pressure (assumed to be p=0 dbar, or an absolute pressure P of exactly 101 325 Pa) using a vibrating tube densimeter (Kremling, 1971).The Absolute Salinity of the seawater sample is estimated from the laboratory density measurement and the equation of state, essentially by solving the equation ρ lab =ρ(S A , 25 • C, 0 dbar) for S A .In practice the laboratory data were used to determine the density difference δρ=ρ lab −ρ(S R , 25 • C, 0 dbar) and this density difference was used with the partial derivative of density with respect to Absolute Salinity at 25 • C and 0 dbar, )].This is the method for estimating δS A suggested by Millero et al. (2008a) (their Eq. 7.2).The difference δS A is plotted in Fig. 2a using all the data published in Millero et al. (1976aMillero et al. ( , 1978Millero et al. ( , 2000Millero et al. ( , 2008bMillero et al. ( , 2009, the Southern Ocean data of this paper).These papers have considered the correlation of various measured properties of seawater with δS A (such as silicate, total alkalinity, total carbon dioxide and nitrate) and found that silicate correlates the best.This is fortunate as there are more measurements of silicate in the ocean data bases than either total alkalinity or total carbon dioxide.

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The reason for the good performance of silicate alone is thought to be that (a) it is itself substantially correlated with the other variables responsible for errors in using Practical Salinity to determine Absolute Salinity, (b) it accounts for a substantial fraction (of around 0.63) of the typical variations in concentrations (g kg −1 ) of the above species and (c) being essentially non-ionic, its presence has little effect on conductivity while having a direct effect on density (Millero et al., 1976c(Millero et al., , 2000)).
The thermodynamic properties of seawater are naturally functions of Absolute Salinity rather than of Practical Salinity, and the new Thermodynamic Equation Of Seawater -2010 (TEOS-10 for short, see McDougall et al., 2009) has algorithms for density, potential temperature, Conservative Temperature, enthalpy, entropy etc, all of which need to be called with Absolute Salinity S A , not Practical Salinity.The algorithm of the present paper is intended to be used to estimate S A given the "measured" values of a seawater sample's Practical Salinity, longitude, latitude and pressure.

The global regressions of δS
A with silicate The data in Fig. 2a, representing seawater samples from throughout the world ocean can be fitted by the simple proportional relationship with silicate (as indicated by the straight line in the figure) The standard error in this fit on Fig.
. When the data in Fig. 2a are coloured by ocean basin it becomes clear that the data from different ocean basins lie either predominantly above or below the straight line fit of Eq. ( 2) as a function of silicate concentration.For example, the data from the North Pacific and North Indian basins clearly were on average above the straight line of Fig. 2a while the data from the Southern Ocean were clustered below the line.This is not unexpected since the spatially variable relative concentrations of different constituents of seawater will not exactly co-vary with silicate.
In order to incorporate this spatially distinct information we decided to perform different fits for the different ocean basins.Because of the dominant role of the Antarctic Circumpolar Current in transporting seawater zonally in the high southern latitudes, we posit that the zonal variation in the relative constituents of seawater may be weak and so we grouped all the data south of 30 • S together and these data were fitted in a separate linear fit with silicate, as shown in Fig. 3.This fit (for latitudes south of 30 The dots on Fig. 3a  The data north of 30 • S in each of the Pacific, Indian and Atlantic Oceans was treated separately.In each of these three regions we constrained the fit to match Eq. ( 3) at 30 • S and allowed the slope of the fit to vary linearly with latitude.The resulting fits were (for latitudes north of 30 • S, that is for λ≥−30 These fits in the Pacific, Indian, and Atlantic Oceans north of 30 • S are shown in Figs.4-6.These fits are intended to be used from 30 • S through the equator and up to the northernmost extent of these ocean basins.In the absence of density data from the Arctic Ocean, our present recommendation is that the Arctic Ocean be characterized by the same equation as the Atlantic, namely Eq. ( 6).

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The fitted circles in panels (a) of Figs. 4, 5 and 6 do not fall on a straight line on these plots because the fit depends on both latitude and silicate.The reason why part of a straight line is visible for the Pacific data is because much of the Pacific data is from a single latitude (see Fig. 1).It is not known why the standard deviation of the data for the Pacific and Indian Oceans are significantly larger than for the Southern Ocean.
It may be that the laboratory technique for determining the conductivity and density of the samples has improved, since the Southern Ocean data was the most recent data to be measured; but this is only a conjecture at this stage.As is well-known, the silicate concentrations in the North Atlantic are quite low and it is comforting to see in Fig. 6 that the laboratory-determined values of δS A =S A −S R are also rather small there.
For each of the Southern, Pacific and Indian data sets we also performed fits that allowed an offset of δS A at zero silicate.In no case did this significantly improve the fits.We have also plotted the residuals as functions of pressure and of latitude and detect no obvious trend in either plot.
We now have a "model" for estimating the Absolute Salinity for data from the major ocean basins.One needs to know the Practical Salinity, the location of the sample (its pressure, its latitude and which ocean basin it is from) and the silicate concentration of the sample.Having these pieces of information, one can use the appropriate equation from Eqs. (3) to (6) to calculate δS A =S A −S R for the seawater sample.This we have done for the 811 samples for which we also have the laboratory-determined values of The error between the laboratory-determined values of δS A =S A −S R and the "model"-based values is shown as the scatter plot of errors in Fig. 2b.The standard error of these data is 0.0048 g kg −1 which is a little less than that from the straightline fit of Eq. ( 2) and Fig. 2a, namely 0.0054 g kg −1 . This improvement amounts to a reduction in error variance of 21% ((0.0048/0.0054) 2 =0.79) and has been obtained by having different fits in the different ocean basins.

Interpolation of the silicate atlas
We now use these correlations between δS A =S A −S R and the silicate concentration to develop a practical algorithm that can be used by oceanographers to estimate Absolute Salinity, given the Practical Salinity and the location of a seawater sample.We do this by utilizing the global atlas of Gouretski and Koltermann (2004) for (among other properties) silicate.We first use the above four Eqs.(3-6) to replace all the silicate data in the world ocean with values of Absolute Salinity Anomaly δS A =S A −S R .Given the location of an observation in space, that is, the latitude λ, longitude ϕ, and pressure p, we interpolate the global atlas values of δS A =S A −S R to this location and then add this value of δS A to the Reference Salinity S R (evaluated from Practical Salinity using Eq. 1) to find the estimated Absolute Salinity S A .We now describe in detail how the global atlas of δS A was formed.The Gouretski and Koltermann (2004) atlas has its silicate field (SiO 2 ) at a 1/2 degree by 1/2 degree horizontal resolution at 45 pressure levels ranging from the sea surface to 6131 dbar.Unfortunately this silicate field does not cover the entire global ocean, but only 99.75% of the ocean for which other hydrographic data is defined.These missing values are here filled in by averaging over the silicate values found at the four locations in the east/north/west/south directions at a distance of 1/2 degree in latitude or longitude from the point in question.In the first instance this was done along isopycnals using precise calculations of the four neutral tangent planes in all four directions.This was performed iteratively until no further missing values needed filling, when only 0.05% of the data remained without silicate values.Apart from the Caspian Sea where silicate values are not available from the Gouretski and Koltermann atlas (the Caspian Sea is excluded from the present data set; see Millero et al., 2008c for an equation of state for these waters), the locations of the remaining missing values are all coastal and so were filled in by averaging along geopotentials.This still left 0.03% of the ocean without a silicate value.All of these were against continental boundaries and were very shallow, so these missing values were set to zero, consistent with the 223 surrounding near-zero silicate data at these shallow depths.
We then sub-sampled this 1/2 degree resolution ocean at 4 degrees in both latitude and longitude in the domain [0 . The latitude numbers have been chosen to exactly capture the northern boundary, making the computational scheme in the latitudinal direction straightforward.Since the southernmost data is located at 78.5 • S, the lower boundary at 78 • S captures silicate values down to 82 • S.
The east/west boundary condition at the Greenwich meridian is accommodated by replicating the data along the 0 • E meridian at 360 • E. The final step in building the three-dimensional look-up table is to calculate from Eqs. (3-6) the Absolute Salinity Anomaly δS A corresponding to the silicate values in the 4-degree global ocean.This is straight-forward and provides the additive adjustment to Reference Salinity that is required to complete the conversion from Practical Salinity to Absolute Salinity.The same offset can be subtracted from Absolute Salinity to yield Reference Salinity and thus obtain Practical Salinity for the inverse of the Practical to Absolute Salinity function.
To find the value of the Absolute Salinity Anomaly δS A of a seawater sample, the global atlas of δS A (ϕ, λ, p) is interpolated as follows.The "cube" containing the data point requiring the salinity adjustment can be identified with the simplest of arithmetic operations in x − y space since the longitude and latitude grids are regular.Finding the pressure index in the vertical is made using an efficient bi-section technique that can be found in, for example, Press et al. (1992).All these operations can be vectorized.When values on the upper or lower faces of the cube are missing, these values are replaced with the mean of the valid δS A values on these same faces.When the seawater sample is deeper than the deepest non-zero δS A (ϕ, λ, p) data in the global atlas at this (ϕ, λ) location, the pressure of the seawater sample is artificially deemed to be the maximum pressure of non-zero δS A values directly above the sample and the interpolation then proceeds as normal.
In Fig. 7a we have plotted a map of the silicate data (in µmol kg −1 ) from the Gouretski and Koltermann ( 2004) atlas at a pressure of 2000 dbar (20 Mpa), while in Fig. 7b is shown the Absolute Salinity Anomaly δS A at the same pressure.The maximum difference between the North Atlantic and North Pacific oceans at 2000 dbar is in excess of 0.025 g kg −1 .

Special treatment of ocean boundaries
The North Atlantic and North Pacific Oceans are closer than four degrees of latitude or longitude apart in the vicinity of the Panama Canal and if this region was not treated in a special way, the interpolation procedure described above would interpolate the silicate data of the atlas across this boundary, whereas in fact, it should be a hard boundary across which there should be no such interpolation.The contrast in silicate values is significant between the two different oceans as can be seen in the plot of silicate shown in Fig. 7a; at a pressure of 2000 dbar the difference of silicate on either side of the Panama Canal is of the order 150 µmol kg −1 .Thus Pacific waters should be treated as no data for interpolations in the Atlantic region and viceversa.An expanded view of the Panama region is shown in Fig. 8 where we also show the simple 6-point piecewise linear function of five straight lines in longitude and latitude (in magenta) that separates the two oceans.These six points yield an efficient test to decide if a user's location is in the North Pacific or the North Atlantic.This test is only performed when the location is near this small region of the global ocean.
The other water mass barrier which might potentially need special treatment is the Indonesian archipelago, but in this case there is no problem since water in the Pacific and Indian oceans are well mixed above 1200 m.At great depths in this region, where the water properties do become significantly different, the two oceans are separated by more than four degrees of latitude and so the issue does not arise.225 5 The Baltic Sea Millero and Kremling (1976) have made density measurements using the vibrating tube densimeter technique on samples from the Baltic Sea.In that paper the Absolute Salinity of water from the surface North Atlantic at S P =35 was thought to be 35.171g kg −1 .Using our updated estimate from Millero et al. (2008a) This equation has the reasonable property that there is no compositional correction to S A when the conductivity of a sample is such that S P =35 which indicates that any Baltic Sea water has been substantially diluted with North Atlantic seawater.Equation ( 7) is used in our algorithm to estimate δS A for any seawater sample from all depths inside the Baltic Sea.

Summary
Thermodynamic properties of seawater are naturally functions of Absolute Salinity rather than of Practical Salinity, and because the new algorithms for evaluating the thermodynamic properties of seawater are functions of Absolute Salinity (McDougall et al., 2009), a method is needed to estimate this type of salinity in terms of properties that are measured at sea.Here we have described an algorithm for estimating the Absolute Salinity (g kg −1 ) of seawater from its Practical Salinity S P as well as the latitude, longitude and pressure of the seawater sample.The estimated standard error in the resulting value of Absolute Salinity is 0.0048 g kg −1 which is considerably less than the standard error involved in present oceanographic practice of effectively equating Absolute Salinity to Reference Salinity (0.0107 g kg −1 ).
The algorithm exploits the correlation between the difference between Absolute and Reference Salinities and the silicate concentration.The global atlas of silicate values of Gouretski and Koltermann (2004) has been used together with our Eqs.(3-6) to obtain a global atlas of S A .To estimate the Absolute Salinity of a particular seawater sample, our algorithm uses interpolations in space over the data in this atlas of Absolute Salinity Anomaly to find the value of δS A =S A −S R appropriate for the location of the seawater sample.In the Baltic Sea the approach of Millero and Kremling (1976) based on the Practical Salinity of the seawater sample has been used (Eq.7).An example of the difference between Absolute Salinity and Reference Salinity, namely the Absolute Salinity Anomaly δS A =S A −S R , is shown for a meridional vertical section through the Pacific Ocean in Fig. 9.
The algorithm described in the present paper should be regarded as a first attempt at providing a practical means of estimate Absolute Salinity.Many more measurements of density and Practical Salinity on samples collected from around the globe would probably enable the residual error to be reduced in a future algorithm.
Of the 811 samples of seawater from the world ocean that have to date been analyzed for density and hence for δS A =S A −S R , the standard error of δS A is 0.0107 g kg −1 .This is the standard error that is incurred if one approximates the Absolute Salinity as Reference Salinity.The maximum value of δS A =S A −S R in the open ocean occurs in the Northernmost North Pacific and is approximately 0.025 g kg −1 .The algorithm of this paper has been used to give estimates of Absolute Salinity for the 811 seawater samples from the world ocean.These estimates of δS A differ from the measured density-derived values of δS A with a standard error of 0.0048 g kg −1 .That is, the mean square error in evaluating Absolute Salinity by the algorithm of this paper is a factor of five less than the mean square error of equating Absolute Salinity with Reference Salinity (i.e.(0.0107/0.0048) 2 ≈5).Some of the remaining error of 0.0048 g kg −1 is due to the error in measuring density in the laboratory (perhaps a standard error of 0.0020 g kg −1 ) and the remaining error is due to the fact that deviations from the standard relative concentrations of the constituents of seawater are not perfectly correlated with the silicate concentration.
The computer software, in both Fortran and Matlab, which evaluates Absolute Salin-
] Table 2. Practical Salinity S P and silicate SiO 2 measured in the Southern Ocean on the CASO SR3 AA0806 voyage south of Tasmania at the longitudes, latitudes and pressures shown.The densities of seawater samples collected at these locations were measured in the laboratory at t=25 • C and p=0 dbar.The values of δρ in the table are differences between the measured densities in the laboratory and those evaluated via the equation of state using the Practical Salinity at t=25 • C and p=0 dbar.The Absolute Salinity Anomaly δS A is calculated from δρ as δρ/[0.75179kg m −3 /(g kg −1 )].
• 55 −65    2) and fits these data with a standard error of 0.0054 g kg −1 .This straight-line fit is not the model that is adopted in this paper.The mean square of these values of δS A is the square of 0.
are the individual data points and the open circles are the values of the straight line fit Eq. (3) to the data, evaluated at the same silicate values as the data points.The error in the fit between the laboratory-determined values of δS A =S A −S R and the value from the linear fit Eq. (3) is shown in Fig.3b.The associated standard error is 0.0026 g kg −1 .

233Fig. 1 .
Fig. 1.Map showing the locations where the 811 seawater samples were collected whose density measurements form the basis of this paper.The first number in the brackets indicates the number of casts from which the samples were collected in each region and the second number is the number of seawater samples.
Fig. 2. (a)The laboratory-determined values of δS A =S A −S R for all 811 samples from the world ocean plotted against the silicate value of each sample.The straight-line fit to the data is given in Eq. (2) and fits these data with a standard error of 0.0054 g kg −1 .This straight-line fit is not the model that is adopted in this paper.The mean square of these values of δS A is the square of 0.0107 g kg −1 .(b) The difference between the laboratory-determined value of δS A and the model for δS A developed in this paper represented by Eqs.(3-6).The standard error of these residuals is 0.0048 g kg −1 .

Fig. 3 .
Fig. 2. (a)The laboratory-determined values of δS A =S A −S R for all 811 samples from the world ocean plotted against the silicate value of each sample.The straight-line fit to the data is given in Eq. (2) and fits these data with a standard error of 0.0054 g kg −1 .This straight-line fit is not the model that is adopted in this paper.The mean square of these values of δS A is the square of 0.0107 g kg −1 .(b) The difference between the laboratory-determined value of δS A and the model for δS A developed in this paper represented by Eqs.(3-6).The standard error of these residuals is 0.0048 g kg −1 .

Fig. 4 .
Fig. 4. (a) Laboratory-determined values of δS A =S A −S R for seawater samples from both the North and South Pacific Ocean basins north of 30 • S. The data are plotted against the silicate value of each seawater sample.The data are the small dots and the open circles are the values obtained from the fit Eq. (4) to this data.(b) The residuals between the laboratory-determined values of δS A and the values found from the fit Eq. (4).

Fig. 5 .
Fig. 5. (a) Laboratory-determined values of δS A =S A −S R for seawater samples from both the North and South Indian Ocean north of 30 • S. The data are plotted against the silicate value of each seawater sample.The data are the small dots and the circles are the values obtained from the fit Eq. (5) to this data.(b) The residuals between the laboratory-determined values of δS A and the values found from the fit Eq. (5).

Fig. 6 .
Fig. 6.(a) Laboratory-determined values of δS A =S A −S R for seawater samples from the North Atlantic Ocean.The data are plotted against the silicate value of each seawater sample.The data are the small dots and the open circles are the values obtained from the fit Eq. (6) to this data.(b) The residuals between the laboratory-determined values of δS A and the values found from the fit Eq. (6). 239

Fig. 8 .
Fig. 8.An expanded view of Fig. 7b showing the series of six straight lines (in magenta) that serve to demark the North Pacific form the North Atlantic in this region.