# Product Tutorial on Sea Surface Temperature (SST)

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## Chapter IV: Physical concepts

### Table of Contents

- Physical concepts

### Physical concepts

As seen in the chapter "SST physical meaning", temperature is an ambiguous term. It depends on distinct physical meanings and also on the rendering instrument. A temperature measured by a thermometer in good contact with the object corresponds to what is physically defined as a thermodynamic temperature. Generally thermodynamic temperature indicates the direction of the heat flow. It is defined by the second law of thermodynamics in which the theoretically lowest temperature is absolute zero, the point at which the particle constituents of matter have minimal motion and can become no colder. On the other side, the faster these particles are moving, the higher the reading on the measuring instrument. Thermodynamic and kinetic temperatures are often considered as equivalent, but to be acurate, kinetic temperature is a macroscopic quantity defined in terms of the mean kinetic energy of the particles (Eq. 1):

(Eq. 1)

here m is the mass of a particle, v2 is the mean square velocity of particles, K is Boltzmann's constant and T is the kinetic temperature.

SST measured from a satellite corresponds to a distinct physical meaning, and it is called radiometric temperature. What quantity is this?

According to the Prevost law, all bodies at a temperature greater than absolute zero radiate energy and the quantity of energy emitted depend on the properties of the body and do not depend on the properties or presence of neighbouring bodies.

When radiative energy strikes a body, generally a part of the energy, Er, is reflected, a part Ea is absorbed and a part Et is transmitted through the body (Figure 4):

Fig. 4: Schematic representation of the basic interactions of electromagnetic energy when striking a body

Since energy is conserved we have:

E = Er + Ea +Et (Eq. 2)

Defining reflectivity ρ, absorptivity α and transmissivity τ respectively as the ratios of Er, Ea and Et to the total incident energy, E, we obtain:

1 = ρ + α + τ (Eq. 3)

Kirchhoff created a thought experiment involving a radiative equilibrium between two totally opaque (i.e. τ ≡ 0 and therefore ρ=1-α) infinite parallel plates of different materials facing each other. The two plates, say A and B, will be emitting and absorbing radiation. Let αA(λ,T) and αB(λ,T) denote the absorbtivities of plates A and B at wavelength λ, and the two plates to reach thermodynamic equilibrium at common thermodynamic temperature T. Let MA(λ,T) and MB(λ,T) denote the emitted energy per unit time at wavelength λ by plates A and B at equilibrium temperature T. Kirchhoff demonstrated that the ratios of emitted energy to absorptivity have to be the same for all materials, i.e.

MA(λ,T) / αA(λ,T) = MB(λ,T) / αB(λ,T) = Bλ(T) (Eq. 4)

where Bλ(T) is a universal function. It may be noted that this function would be given by the emitted energy per unit time per unit wavelength of a totally opaque body with unit absorptivity (and therefore zero reflectivity) for all wavelengths. Kirchhoff envisioned a material with zero reflectivity and coined the term black body to label it.

Radiation emitted by a blackbody in thermal equilibrium per unit time at a specific wavelength λ (μm) and temperature T (K) is given by Planck's law:

(Eq. 5)

The spectral blackbody radiance is given in units of W m-2 sr-1 μm-1, λ is the wavelength in μm, c1 and c2 are the Planck constants (c1=1.19104x108 W m-2 sr-1 μm4; c2=1.43877 x104 μm K).

As the temperature in Eq. 5 is related to emitted radiation, it is referred to as radiometric temperature.

So radiometric temperature should be obtained by inverting Planck´s function given a certain radiance measured at a certain wavelength. Yet, radiation is not monochromatic, instead it comprises a range of wavelengths. F↑(Eq. 5) is given by the integration of Planck's function for a gray body, i.e. a body where emissivity is the same for all wavelengths and directions,

F↑ = εσTskin4 (Eq. 6)

This equation represents the so called Stefan-Boltzmann law, where σ = 5.67 x 10-8 W m2 K-4, and Tskin is called skin temperature.

Radiometric temperature is not measured with thermometers, but with radiometers. Moreover, there is no direct measuring method, as radiometric temperature is not directly measured, but the corresponding emitted radiance is measured instead.

#### Temperature from infrared sensors

If no atmosphere existed between the surface and the sensor, and assuming the emissivity was known, the surface temperature could be obtained from the measured spectral radiance by inverting Planck's function.

However, water vapour, clouds, trace gases and aerosols all disturb the propagation of radiation from the surface towards the Top of the Atmosphere (TOA) by attenuating and re-emitting thermal infrared radiation. Figure 4 illustrates the several contributions to the OLR (outgoing longwave radiation), which is the total energy emitted from Earth and its atmosphere out to space in the form of thermal radiation. The major contribution to the OLR is from the surface emission (Planck's law) followed by the upward radiance emitted by the atmospheric constituents (mainly water vapor) at several layers of the atmosphere, L↑atm. This term varies strongly with the vertical structure of the atmosphere-warm/moist layers increase its contribution. The atmosphere emits also longwave radiation in a downward direction that is then reflected back by the surface to the TOA. The actual longwave radiation reaching the Top Of Atmosphere (TOA) corresponds to the sum of all of these terms multiplied by the atmospheric transmittance (τλ):

Lλ = ελBλ(SST) τλ + Lλ↑atm τλ + (1-ελ) Lλ↓atm τλ (Eq. 7)

Equation 7 is known as the Radiative Transfer Equation (RTE). Note that aerosol absorption and scattering are considered to be negligible and are generally ignored (Prata et al 1995).

Fig. 5: Contributions to the total outgoing longwave radiance reaching the top of the atmosphere (TOA) from a water surface.

If, as previously mentioned, emissivity of the water surface is assumed to be ~1, the RTE becomes:

Lλ = Bλ(SST) τλ + Lλ↑atm τλ (Eq. 8)

Summarising, Lλ is the total radiance we can measure by a satellite sensor at TOA while the surface temperature is what we want to retrieve. So in principle if we had the necessary information about the atmospheric composition we could solve the RTE to get the SST. This also assumes that cloud masking is solved, since surface temperature from thermal infrared measurements can only be achieved for clear sky pixels.

#### Surface sensing - which infrared channels?

Computing Planck's radiation emission for a representative Earth temperature of 300 K in respect to wavelength, as represented in Figure 6 leads to the conclusion that the radiation emission from surface peaks at around 10 μm (with 90% of total emission at wavelengths >4 μm).

Fig. 6: Terrestrial radiation emission curve for a representative temperature of 290 K.
Source: http://gis.humboldt.edu/OLM/Courses/GSP_216_Online/lesson1-2/blackbody.html

As previousy mentioned, some atmospheric constituents (N2, O2, H2O, CO2, N2O, O3, CH4) interact with the propagating radiation in its path from the surface towards the satellite, with a dependance on the concentration of these components but also on the wavelength of the propagating radiation. Fortunately, there are some regions of the electromagnetic spectrum where these effects are less pronounced, which are called atmospheric windows. Satellite sensors designed to sense the surface are provided with spectral channels ideally located within these regions.

Fig. 7:Radiation emission curves for distinct surface temperatures ranging between 200-300 K computed from Planck's function superimposed with a real observation of radiation emitted from a representative surface at 295 K (line in bold), revealing the attenuation effects of some atmospheric components (H2O, O3 and CO2).

Figure 7 shows the surface emission spectra for different temperatures (200–300K), computed by Planck's function between 2.0 - 18.0 μm. The line in bold represents a real measurement by a satellite of earth-emitted radiation from a surface at 295 K. As observed, the curve varies with wavelength due to absorption and emission by atmospheric gases at different pressure levels (O3, CO2 and mostly water vapour), revealing the presence of absorption bands but also atmospheric windows (where measured radiance is closer to the theoretical 300 K curve). As concluded from Fig. 7, some atmospheric windows for thermal infrared radiation coincide with the spectral band where radiation emission from the surface is at a maximum. Yet, despite the atmosphere being almost transparent to thermal infrared radiation in the window regions, the effect of the trace gases is not negligible in the SST retrieval, and therefore some correction is still required.

Figure 8 shows the locations of some spectral channels that can potentially be used for SST retrieval. The atmospheric transmittance (gray line) is plotted between 4.0–14.0 μm. On the top of Fig. 8, infrared channel locations for 3 geostationary meteorological satellites: MSG, GOESR and HIMAWARI-8 are indicated.

Fig. 8: Atmospheric transmittance with respect to wavelength. Spectral range/location of the thermal infrared channels for 5 geostationary satellites are indicated by the coloured bars at the top of the graph.

For this group of satellites, 4-5 channels are located within thermal infrared windows.

There are however some issues with channel IR3.9 that hamper its use for SST retrieval. Shown by Fig. 8, transmissivity is close to 1 around the 3.9 μm wavelengths, making it a good candidate since there is weak water vapor absorption in this band. However, this channel is not only sensitive to infrared emissions but is also partially sensitive to the reflected solar radiation, as illustrated in Fig. 9, which is problematic for SST daytime retrievals.

Fig. 9: This figure shows the solar and thermal emission curves as a function of wavelength. It is clear from the curves that the 3.9 μm channel is sensitive both to solar and thermal radiation, whereas the 8.7 μm, 10.8 μm and 12.0 μm spectral channels are only sensitive to thermal radiation.

The channels of IR10.8 and IR12.0 are the most common choice since the atmospheric effect may be estimated by the difference between the radiance between these two channels.