10.1 Satellite Oceanography

Step 1: Top-of-Atmosphere Radiance Budget

The total radiance measured by a satellite sensor at wavelength $\lambda$ is the sum of atmospheric path radiance, surface-reflected radiance, and water-leaving radiance, each modified by atmospheric transmittance:

$$L_t(\lambda) = L_{\text{path}}(\lambda) + t_d(\lambda)\,L_r(\lambda) + t_d(\lambda)\,t_u(\lambda)\,L_w(\lambda)$$

Step 2: Isolate Water-Leaving Radiance

Atmospheric correction removes $L_{\text{path}}$ (Rayleigh scattering + aerosols) and $L_r$ (sun and sky glint). The water-leaving radiance is:

$$L_w(\lambda) = \frac{L_t(\lambda) - L_{\text{path}}(\lambda) - t_d(\lambda)\,L_r(\lambda)}{t_d(\lambda)\,t_u(\lambda)}$$

Step 3: Normalize to Remote Sensing Reflectance

Divide the water-leaving radiance by the downwelling irradiance $E_d(\lambda)$ just above the surface to obtain the dimensionless remote sensing reflectance:

$$R_{rs}(\lambda) = \frac{L_w(\lambda)}{E_d(\lambda)} \quad [\text{sr}^{-1}]$$

Step 4: Relate $R_{rs}$ to Inherent Optical Properties

The remote sensing reflectance is related to the ratio of backscattering $b_b$ to absorption $a$ coefficients through the Gordon approximation:

$$R_{rs}(\lambda) \approx \frac{f}{Q} \cdot \frac{b_b(\lambda)}{a(\lambda) + b_b(\lambda)}$$

where $f/Q \approx 0.09$ sr$^{-1}$ depends on the underwater light field geometry.

Step 5: Band-Ratio Chlorophyll Algorithm

Chlorophyll-a absorbs strongly in the blue (443 nm) and weakly in the green (555 nm). The OC4 algorithm exploits this spectral contrast:

$$\log_{10}[\text{Chl-}a] = a_0 + \sum_{i=1}^{4} a_i \left(\log_{10}\frac{R_{rs}(\lambda_{\text{blue}})}{R_{rs}(\lambda_{\text{green}})}\right)^i$$

The maximum band ratio (MBR) approach selects the largest of $R_{rs}(443)/R_{rs}(555)$, $R_{rs}(490)/R_{rs}(555)$, or $R_{rs}(510)/R_{rs}(555)$ to extend the dynamic range.

Step 1: Radar Range Measurement

The altimeter transmits a short radar pulse (Ku-band, ~13.6 GHz) and records the two-way travel time $\Delta t$. The measured range is:

$$R_{\text{meas}} = \frac{c \cdot \Delta t}{2}$$

where $c \approx 3 \times 10^8$ m/s is the speed of light in vacuum.

Step 2: Ionospheric Path Delay Correction

Free electrons in the ionosphere slow the radar signal. The range error depends on the total electron content (TEC) and varies as $f^{-2}$. Using dual-frequency measurements at $f_1$ (Ku) and $f_2$ (C-band):

$$\Delta R_{\text{iono}} = \frac{f_2^2}{f_1^2 - f_2^2}(R_1 - R_2) = \frac{40.3 \cdot \text{TEC}}{f^2}$$

Step 3: Tropospheric Corrections

The dry troposphere delay is modeled from surface atmospheric pressure $P_s$ using the hydrostatic approximation:

$$\Delta R_{\text{dry}} = -\frac{0.2277 \, P_s}{1 - 0.00266 \cos 2\phi - 0.00028 \, h_s} \quad (\text{cm})$$

The wet troposphere delay (0–40 cm) is measured by the onboard microwave radiometer from brightness temperatures at 18, 21, and 37 GHz.

Step 4: Sea State Bias (Electromagnetic Bias)

Wave troughs reflect radar more effectively than crests, biasing the measured range toward the troughs. The sea state bias is empirically modeled as a fraction of significant wave height:

$$\Delta R_{\text{SSB}} \approx -(0.02\text{--}0.05) \cdot H_s$$

Step 5: Corrected SSH and Dynamic Topography

The fully corrected sea surface height is the orbit altitude minus the corrected range:

$$\text{SSH} = h_{\text{orbit}} - R_{\text{meas}} - \Delta R_{\text{iono}} - \Delta R_{\text{dry}} - \Delta R_{\text{wet}} - \Delta R_{\text{SSB}} - \Delta R_{\text{tide}} - \Delta R_{\text{IB}}$$

Step 6: Geostrophic Currents from SSH Gradients

The absolute dynamic topography (ADT) is $\eta = \text{SSH} - N$, where $N$ is the geoid. The geostrophic balance on a rotating Earth gives surface velocities:

$$u_g = -\frac{g}{f}\frac{\partial \eta}{\partial y}, \quad v_g = \frac{g}{f}\frac{\partial \eta}{\partial x}, \quad f = 2\Omega\sin\phi$$

A 10-cm SSH gradient over 100 km at 35$^\circ$N yields $v_g \approx (9.81 \times 0.1)/(8.4 \times 10^{-5} \times 10^5) \approx 0.12$ m/s.

Step 1: Planck's Radiation Law

The spectral radiance emitted by a blackbody at temperature $T$ is given by the Planck function:

$$B(\lambda, T) = \frac{2hc^2}{\lambda^5} \cdot \frac{1}{e^{hc/(\lambda k_B T)} - 1}$$

where $h = 6.626 \times 10^{-34}$ J s is Planck's constant, $c = 3 \times 10^8$ m/s, and $k_B = 1.381 \times 10^{-23}$ J/K.

Step 2: Define Brightness Temperature

The brightness temperature $T_B$ is the temperature a blackbody would need to emit the same radiance as measured by the sensor. For a surface with emissivity $\varepsilon(\lambda)$ at physical temperature $T_s$:

$$L_{\text{measured}}(\lambda) = \varepsilon(\lambda) \, B(\lambda, T_s) + (1 - \varepsilon(\lambda)) \, L_{\text{reflected}}(\lambda)$$

Step 3: Atmospheric Radiative Transfer

The radiance reaching the satellite passes through the atmosphere, which attenuates the surface signal and adds its own emission:

$$L_{\text{TOA}}(\lambda) = \tau(\lambda)\,\varepsilon(\lambda)\,B(\lambda, T_s) + L_{\text{atm}}^{\uparrow}(\lambda) + \tau(\lambda)(1-\varepsilon)\,L_{\text{atm}}^{\downarrow}(\lambda)$$

where $\tau(\lambda)$ is the atmospheric transmittance.

Step 4: Multi-Channel SST (MCSST) Algorithm

Rather than inverting the full radiative transfer equation, operational SST is retrieved using a statistical regression of brightness temperatures in multiple IR channels (e.g., 11 and 12 $\mu$m split-window):

$$\text{SST} = a_0 + a_1 T_{B,11} + a_2 (T_{B,11} - T_{B,12}) + a_3 (T_{B,11} - T_{B,12})(\sec\theta - 1)$$

Step 5: Physical Basis of the Split-Window Technique

Water vapor absorbs differently at 11 and 12 $\mu$m. The brightness temperature difference $\Delta T_B = T_{B,11} - T_{B,12}$ is proportional to the atmospheric water vapor correction needed. The view-angle term $(\sec\theta - 1)$ accounts for the longer atmospheric path at oblique viewing:

$$T_{B,i} \approx T_s - \frac{(1-\tau_i)\,(T_s - T_{\text{atm}})}{\tau_i}$$

Step 6: Microwave SST Retrieval

In the microwave (Rayleigh-Jeans regime, $hc/\lambda k_B T \ll 1$), the Planck function simplifies and $T_B$ is linearly related to surface temperature:

$$T_B \approx \varepsilon(\theta, S, T_s, f) \cdot T_s, \quad \varepsilon = 1 - |R(\varepsilon_r)|^2$$

The Fresnel reflectivity $R$ depends on the complex dielectric constant $\varepsilon_r(S, T_s, f)$ of seawater, which varies with salinity, temperature, and frequency. This enables SST retrieval through clouds at ~25 km resolution.