10.2 In-Situ Measurements

Step 1: Conductivity Ratio Definition

The PSS-78 algorithm defines practical salinity in terms of a conductivity ratio $R$ rather than absolute conductivity. The ratio is taken relative to Standard Seawater at $S = 35$, $T = 15°C$, $p = 0$:

$$R = \frac{C(S, T, p)}{C(35, 15, 0)}, \quad C(35, 15, 0) = 4.2914 \;\text{S/m}$$

Step 2: Factor the Ratio into T and p Corrections

The conductivity ratio is factored as $R = R_p \cdot r_t \cdot R_t$, separating pressure ($R_p$), temperature at $S = 35$ ($r_t$), and the salinity-dependent part ($R_t$). The pressure correction:

$$R_p = 1 + \frac{p(e_1 + e_2 p + e_3 p^2)}{1 + d_1 T + d_2 T^2 + (d_3 + d_4 T)R}$$

Step 3: Polynomial Expansion for Salinity

With the corrected ratio $R_t$ isolated, practical salinity is computed from a polynomial in $R_t^{1/2}$ (half-integer powers provide better fit across the salinity range 2--42):

$$S_P = \sum_{i=0}^{5} a_i R_t^{i/2} + \Delta S(T, R_t)$$

Step 4: Sensor Calibration via Bottle Samples

CTD conductivity sensors drift over time. Post-cruise calibration fits a correction polynomial by comparing CTD-derived salinity against discrete water samples analysed on a laboratory salinometer. The correction is:

$$C_{\text{corrected}} = C_{\text{raw}} \cdot (1 + \alpha_0 + \alpha_1 p + \alpha_2 t_{\text{station}})$$

Step 1: General Error Propagation Formula

For a derived quantity $f(x_1, x_2, \ldots, x_n)$ computed from measured variables $x_i$ with uncertainties $\sigma_{x_i}$, the combined uncertainty (assuming uncorrelated errors) is:

$$\sigma_f^2 = \sum_{i=1}^{n} \left(\frac{\partial f}{\partial x_i}\right)^2 \sigma_{x_i}^2$$

Step 2: Apply to Salinity from Conductivity

Practical salinity $S_P = f(C, T, p)$ depends on three measured quantities. The salinity uncertainty is dominated by conductivity and temperature sensor accuracy:

$$\sigma_{S_P}^2 = \left(\frac{\partial S_P}{\partial C}\right)^2 \sigma_C^2 + \left(\frac{\partial S_P}{\partial T}\right)^2 \sigma_T^2 + \left(\frac{\partial S_P}{\partial p}\right)^2 \sigma_p^2$$

Step 3: Evaluate Partial Derivatives

At typical ocean conditions ($S \approx 35$, $T \approx 10°C$), the sensitivities are approximately: $\partial S/\partial C \approx 10$ PSU/(S/m), $\partial S/\partial T \approx -0.02$ PSU/°C. For an SBE 911plus with $\sigma_C = 3 \times 10^{-4}$ S/m and $\sigma_T = 10^{-3}$ °C:

$$\sigma_{S_P} \approx \sqrt{(10 \times 3\!\times\!10^{-4})^2 + (0.02 \times 10^{-3})^2} \approx 0.003 \;\text{PSU}$$

Step 4: Propagation to Derived Quantities

The salinity and temperature uncertainties propagate further into density ($\rho$) and geostrophic velocity. For density via the equation of state, the thermal expansion coefficient $\alpha$ and haline contraction coefficient $\beta$ give:

$$\sigma_\rho = \rho_0\sqrt{\alpha^2 \sigma_T^2 + \beta^2 \sigma_S^2} \approx 1025 \sqrt{(2\!\times\!10^{-4} \times 10^{-3})^2 + (7.5\!\times\!10^{-4} \times 0.003)^2}$$

Step 5: Impact on Geostrophic Transport

Geostrophic velocity from the thermal wind equation is proportional to horizontal density gradients. Over a station pair separated by $\Delta x$, the velocity uncertainty is $\sigma_v \approx g/(f \rho_0) \cdot \sqrt{2}\sigma_\rho / \Delta x$. This sets the minimum station spacing needed to resolve currents above measurement noise.