# Simplify water salinity equations

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

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).