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I recently became interested in calculating water salinity from conductivity and temperature. I failed to find any theoretical model to how this is done, only empirical models derived from fitting to experimental data.

One such model (first Google hit) I found was this, of which if you look in the page source you find:

$r = \frac{C/42914}{c_0+T*(c_1+T*(c_2+T*(c_3+T*c_4)))}$

$ds_1 = b_0+r^2*(b_1+r^2*(b_2+r^2*(b_3+r^2*(b_4+r^2*b_5))))$

$ds_2 = ((T-15.0)/(1.0+0.0162*(T-15.0)))$

$d_s = ds_1 ds_2$

And finally:

$salinity=a_0+r^2*(a_1+r^2*(a_2+r^2*(a_3+r^2*(a_4+r^2*a_5))))+d_s$

where $a_i, b_i, c_i$ are fitted coefficients.

Without going into depth about the fysics of salinity, doesn't this model seem overly complicated? I mean whats the highest order of T? Alot.

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You'd think they'd have tried simpler models first, no? In any event, I'm not sure what the mathematical question here is... –  J. M. Sep 14 '11 at 14:40
    
@J.M. - My point is that does it really add that much to have that many degrees of freedom? –  Theodor Sep 14 '11 at 14:53
    
Hard to tell. Were any physical justifications given for the polynomials in the expression? –  J. M. Sep 14 '11 at 14:55
    
No, I might try plotting salinity as a function of conductivity and temperature and see weather the function looks overfitted. –  Theodor Sep 14 '11 at 15:00

1 Answer 1

In 1978, the United Nations Educational, Scientific and Cultural Organization (UNESCO), International Council for the Exploration of the Sea (ICES), Scientific Committee on Oceanic Research (SCOR) and International Association for the Physical Sciences of the Oceans (IAPSO) adopted a practical method for calculating salinity of seawater based on electrical conductivity measurements compensated for temperature.

The Practical Salinity Scale of 1978 (PSS-78) defines a scale for 'Practical Salinity', $S$ (a dimensionless quantity) which is calculated from electrical conductivity and temperature measurements, in the range $2 \leq S \leq 42$, by the following formula:

$S = 0.0080 - 0.1692K_{15}^{1/2} + 25.3851K_{15} + 14.0941K_{15}^{3/2} - 7.0261K_{15}^{2} + 2.7081K_{15} ^{5/2}$

Where $K_{15}$ is the ratio of conductivity of the sea water sample to that of potassium chloride (KCl) solution of mass fraction 32.4356x10$^{-3}$, at 15$^{o}C$ and atmospheric pressure.

This formula is extended to compensate for the variation in conductivity with temperature in the range $-2^{o}C \leq t \leq 35^{o}C$ (at atmospheric pressure) by incorporating temperature coefficient (see http://unesdoc.unesco.org/images/0004/000479/047932eb.pdf).

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