Atmospheric Sounding using
Passive Microwave Sensing

Passive Microwave Sensing of the Ocean

Microwave radiometers are commonly used for remote sensing of the ocean surface. As illustrated in Figure 1, the signal from a microwave radiometer viewing the ocean surface consists of contributions from downwelling radiation reflected from the sea surface, radiation emitted from the surface, and atmospheric emissions.

The microwave brightness temperature observed by a satellite radiometer $T_B$ is given by: $$\begin{equation} T_B(f,\theta,p) = \epsilon_s T_s \, t_{atm} + r_s\left[ T_{sky} + T_{\cos} \, t_{atm} \right] t_{atm} + T_u \end{equation}$$ for frequency $f$, sensor look angle $\theta$ and polarization $p$, where:

• $T_s$ = surface brightness temperature
• $\epsilon_s$ = surface emissivity
• $t_{atm}$ = atmospheric transmittance = $\exp(-\tau_a)$
• $\tau_a$ = atmospheric opacity (optical depth)
• $T_{sky}$ = sky brightness temperature
• $T_{\cos}$ = cosmic radiation (2.7 K)
• $r_s$ = surface reflectivity (from Fresnel equations, note $r_s = 1 - \epsilon_s$)
• $T_u$ = upwelling atmospheric temperature.

The first term is the surface brightness temperature times the atmospheric transmittance. This is the one term that contains the signal of interest for ocean measurement.

The second term is the brightness temperature reflected from the surface. The downwelling brightness temperature is the sum of the sky brightness temperature plus cosmic radiation attenuated by the downward trip through the atmosphere. This downwelling brightness temperature is then multiplied by the surface reflectivity and the attenuation from the trip back up through the atmosphere.

The last term is the brightness temperature of upwelling from the atmosphere.

Atmospheric Sounding

PMR is commonly used to estimate vertical profiles of atmospheric parameters, a measurement technique referred to as atmospheric sounding. Figure 2 plots two examples of vertical temperature and humidity profiles derived from PMR (solid curves) and measurements made from a radiosonde (dotted curves). The skew-T log-P diagrams show that while the PMR soundings do not have the vertical resolution of a radiosonde, they can be quite accurate in the mean.

These soundings are derived from PMR measurements made at multiple wavelengths on the edges of an absorption band. For example, the temperature soundings for the MIRS instrument on NOAA-12 are largely derived from measurements made at 12 channels on the edge of the oxygen absorption band with center frequencies ranging from 50 to 57.6 GHz. The humidity profiles from the same instrument are largely derived from measurements made at 6 channels on the edge of water vapor absorption band at 180 GHz.

While the theory of passive microwave radiometry is developed in the next section, the basic idea of atmospheric sounding is simple enough. Like humans who can only see a few meters in front of us in a heavy fog, a microwave radiometer operating at a frequency where the atmosphere is highly absorptive will only respond to the temperature of the atmosphere relatively near the sensor. Photons originating further away from the sensor are absorbed before they can reach the sensor. So such a radiometer looking down from space will detect emissions coming from the upper portions of the atmosphere and not the lower.

A similar measurement, made at a frequency where the atmosphere is slightly more transparent, will originate further away from the sensor and hence slightly lower in the atmosphere. If the measurement is instead made at a frequency where the atmosphere is mostly transparent, then some portion of the measured radiance will come from the surface and some will come from the atmosphere. But because the atmosphere is denser and more absorptive at lower altitudes, most of the atmospheric radiance will originate in the lowest portions of the atmosphere. Therefore measurements made at different frequencies correspond to emissions coming from different altitudes in the atmosphere, leading to information on the vertical profile of temperature.

Theory of Passive Microwave Radiometry in the Atmosphere

The basic physics of passive microwave radiometry in the atmosphere is relatively simple. In the absence of rain, there is negligible atmospheric scattering at microwave frequencies, and so the atmosphere can be modeled as purely absorptive. Figure 3 illustrates the spectral radiance $L_{\lambda}(h)$ seen by a sensor at height $h$ pointed down at a surface with spectral radiance of $L_{\lambda}(0)$. The tube in this figure indicates the absorbing path between the surface and the sensor. Absorption along this path is characterized by a total absorption coefficient $\kappa(z)$, which consists mostly of contributions from oxygen and water vapor.

Consider a small absorbing volume in the path at height $z$ along the path with infinitesimal length $dz$. The radiance $L$ leaving this volume will be attenuated by an amount $dL$ equal to the absorption coefficient times the radiance entering the volume times the length of the volume $dz$:

$$\begin{equation} dL_{\lambda}^a(z) = -\kappa(z) L_{\lambda}(z) dz \end{equation}$$

By Kirchoff’s law, the emissivity of the material in the infinitesimal volume element is $\varepsilon = \kappa(z) \, dz$. Hence the emitted radiance from the volume element is given by:

$$\begin{equation} dL_{\lambda}^e(z) = \kappa(z) L_{BB}(T(z)) dz \end{equation}$$

where $T(z)$ is the temperature of the gas at height $z$ above the surface, and $L_{BB}$ indicates the radiance of a blackbody at the specified temperature.

Then the net change in radiance from the volume element is the emissions minus the attenuation:

$$\begin{equation} dL_{\lambda} = \kappa(z) L_{BB}(T(z)) dz - \kappa(z) L_{\lambda}(z) dz \end{equation}$$

This is a differential equation with the solution:

$$\begin{equation} L_{\lambda}(h) = L_{\lambda}(0) \exp \left[ -\tau(0,h) \right] + \int_0^h \kappa(z) L_{BB}(T(z)) \exp \left[ -\tau(z,h) \right] dz \end{equation}$$

where $\tau(z_1,z_2) = \int_{z_1}^{z_2} \kappa(z') dz'$ is the optical thickness or optical depth of the layer between $z_1$ and $z_2$. The first term is the radiance of the surface attenuated by absorption along the total path. The second term is the integral of the emissions from each element in the path, attenuated by the path from the emission to the sensor.

At microwave frequencies the brightness temperature is proportional to radiance, so the previous equation can alternatively be expressed in terms of the brightness temperatures: $$\begin{equation} T(h) = T(0) \exp \left[ -\tau(0,h) \right] + \int_0^h \kappa(z) \, T(z) \exp \left[ -\tau(z,h) \right] dz \end{equation}$$

Now T(0) in turn consists of the sum of two components: the thermal emissions from the surface $\varepsilon_s T_s$ and the downwelling thermal emissions from the sky reflected off the surface $(1-\varepsilon_s ) T_{sky}$, where $\varepsilon_s$ is the emissivity of the surface, and $T_{sky}$ is the downwelling sky temperature as observed at the surface. The sky temperature is then given by: $$\begin{equation} T_{sky} = T_{cosmos} \exp \left[ -\tau(0,\infty) \right] + \int_0^\infty \kappa(z) \, T(z) \exp \left[ -\tau(0,z) \right] dz \end{equation}$$ where $T_{cosmos}$ is the background temperature of space, which is generally just a few degrees K.

The next few sections take the following steps to build on this basic physics to illustrate and evaluate atmospheric sounding:

• A notional sensor is selected whose primary attribute is a set of channels with fixed center frequencies.

• The attenuation and refractive delay characteristics of moist air at a given pressure, temperature, and humidity is evaluated using the MPM93 model described in Liebe and Hufford (1993).

• Profiles of atmospheric pressure, temperature, and humidity are computed based on the International Telecommunication Union (ITU) standard atmosphere. The differential attenuation and refractive delay characteristics of the air is evaluated as function of altitude for these profiles.

• The total attenuation and refractive delay for a vertical path through the entire atmosphere is evaluated.

• Radiative transfer theory is extended to compute temperature weighting functions. These are used to illustrate the altitudes that contribute to the brightness temperature in the specified channels observed from both space and an aircraft flying at altitudes of 1 or 5 km.

• The response of a passive microwave radiometer to variations in surface temperature and roughness is computed.

AMSU-A and MHS Sensors

The AMSU-A passive microwave sensor was chosen to illustrate the concepts in this memo. This sensor orbited at an altitude of 830 km, pointing at an angle of $53.3^{\circ}$ relative to nadir, which results in an incident angle of $65^{\circ}$. The primary characteristics of the channels for AMSU-A are listed in Table 1.

The channel frequencies in this table follow a standard notation. A single value $x$ indicates the center frequency of a single band. A pair of values $x\pm y$ indicates a pair of bands, one centered at a frequency of $x+y$, and the other at a frequency of $x-y$. Similarly, three values $x\pm y \pm z$ indicates four bands. These patterns of sidebands are easily implemented with radio frequency receivers, and effectively double or quadruple the signal (2 or 4 pass bands instead of one). The bandwidth specified in the table is the total bandwidth across 1, 2 or 4 sidebands in each channel.

MPM93

The Microwave Propagation Model 1993 (MPM93) models the major refractive components in air including oxygen and water vapor. The inputs to MPM are microwave frequency, and the pressure, temperature, and absolute humidity of a volume of air. MPM then returns the total extinction coefficient in units of dB/km through that air, and the total refractivity (change in the speed of light) in parts per million.

MPM also includes models for rain, fog, and snow, but those effects are ignored here. Figure 4 plots the absorption computed from MPM93 at sea level for oxygen and water vapor, assuming 75% relative humidity. Oxygen has major absorption lines at 60 GHz and 118 GHz, while water vapor has absorption lines at 22, 183.5 and 325.5 GHz.

The vertical bars indicate the notional sensor channels. The coloring of the channels indicates the primary use of that channel. Note that a number of the channels are closely spaced along the edges of the 60 GHz band – these are predominantly used for producing temperature profiles. Similarly there are a number of closely spaced channels near 183 GHz that are primarily used for humidity profiling.

The remainder of this memo will concentrate on the channels below 60 GHz. Figure 5 plots the attenuation as a function of frequencies near the oxygen absorption line, but for four different pressures, corresponding to different altitudes in the atmosphere. A number of distinct oxygen absorption lines are apparent higher in the atmosphere where the pressure is lower. The higher pressures found lower in the atmosphere causes the individual absorption lines to broaden into a single broad line.

Figure 6 further expands the horizontal axis to illustrate the nature of the overlapping bands that precisely sample a pair of the absorption lines at 56.9681 and 57.6125 GHz. These bands have the same center frequencies but use different bandwidths to obtain different average absorption levels.

While all of the plots in this section illustrate the refractive effects of moist air, passive radiometric measurements are made along a path in the atmosphere. So profiles of the key parameters in the atmosphere are needed to model the response of a passive microwave radiometer.

ITU Atmosphere

The International Telecommunication Union (ITU) promulgates a standard atmosphere for use in telecommunication calculations. The first three panels of Figure 7 illustrate the temperature, humidity and temperature profiles of the ITU standard atmosphere for mid-latitude locations in summer and winter. The bottom right panel shows how the absorption varies in the summer as a function of altitude for 3 different frequencies. At 23.8 GHz, the absorption is low throughout the atmosphere, so a radiometer should be expected to be able to detect at least some surface radiance. At 54.94 GHz, the absorption may exceed 1 dB/km in the lower 10 kilometers of the atmosphere, making the measurement sensitive to the temperature of the lower atmosphere. And at 56.9681 GHz, the peak of the oxygen absorption line, the absorption is so significant at altitudes of 40-50 km, that any surface radiance will be absorbed in the atmosphere, and the radiometer will be sensitive to atmospheric temperature high in the atmosphere.

In the ITU standard atmosphere the pressure varies nearly, but not quite, like an exponential with atmosphere. the hydrostatic equation states that the change in pressure $(dP)$ over an infinitesimal change in height $dz$ is equal to the weight of the air in that small layer: $$\begin{equation} \frac{dP}{dz} = -\rho g \end{equation}$$ where $\rho$ is density and $g$ is the gravitational constant. Water is nearly incompressible, so underwater pressure increases linearly with depth. But air is very compressible. The ideal gas law $PV=nRT$, can be manipulated to get an equation for density that varies with pressure: $$\begin{equation} \rho = \frac{PM}{RT} \end{equation}$$ where $P$ is pressure, $M$ is the molar mass of air, $R$ is the ideal gas constant, and $T$ is temperature. Thus $$\begin{equation} \frac{dP}{dz} = -\frac{PMg}{RT} \quad \rightarrow \quad \frac{dP}{P} = -\frac{Mg}{RT} dz \end{equation}$$ By inspection, this equation has a solution $$\begin{equation} P(z) = P_0 \; e^{-\frac{Mg}{RT}z} \end{equation}$$ So theoretically, if the air in the atmosphere was all the same temperature and humidity, then the pressure would be distributed exponentially.

The actual mid-latitude, summertime ITU atmospheric pressure model is given by: $$\begin{equation} P(z) = \begin{cases} 1012.8186 - 111.5569 z + 3.8646 z^2 & \text{if } z \leq 10 \\ 283.71 \exp(-0.147 (z-10)) & \text{if } z > 10 \mbox{ and } z \leq 72 \\ 0.03124 \exp(-0.165 (z-72)) & \text{if } z > 72 \end{cases} \end{equation}$$ where $z$ is measured in kilometers. The main reasons that this is not a pure exponential is that the temperature and humidity in the atmosphere vary with altitude, which impacts air density and hence pressure. The three layer model reasonable accurately captures the common characteristics of the atmosphere. It is the type of model that is never 100% accurate, but captures the major effects in the mean.

Vertical Path Integrals

Figure 8 plots the total attenuation through the atmosphere as a function of frequency. It was produced by integrating the attenuation profile for the mid-latitude summer ITU atmosphere along a vertical path from the surface to space.

A simple integration with a vertical step size of 100 m was initially used for this calculation: $$\begin{equation} I = \int_{0}^{100 \; km} \alpha(z) \, dz \end{equation}$$ Unfortunately the resulting evaluation was a bit too slow, so a faster implementation was developed based on using logarithmically based steps. This is a common trick that allows fine resolution in the lower portion of the atmosphere where it is needed, and coarser resolution at higher altitudes. Define a new coordinate system $u = \ln(z)$, so $z = \exp(u)$ and $du = 1/z dz$. Then $$\begin{equation} I = \int_{ln(0.001)}^{ln(100)} \alpha[\exp(u)] \, z \, du \end{equation}$$ where the bottom of the atmosphere has been arbitrarily set to an altitude of 1 m (= 0.001 km). This trick speeds up the calculation by allowing it to be performed with a factor of 10 fewer points.

The code for all the key mathematical calculations are called from the doAllBasicCalculations() function in this page's javascript code. This function also calls key functions in the mpm93.js and itu_atm.js files.

Figure 9 plots the attenuation over a vertical path as a function of frequency for varying levels of relative humidity. This plot indicates that the 23.8 GHz channel responds predominantly to water vapor at high and low humidity, while the higher frequency channels respond more equally to temperature and water vapor, with a balance that depends on the relative humidity.

Temperature Weighting Functions

The previous equation for brightness temperature can be decomposed to determine what part of the atmosphere contributes to the measured brightness temperature. Begin by defining a weighting function $W(z)$ corresponding to two of the three terms in the integrand for the path radiance: $$\begin{equation} W(z) = \kappa(z) \exp \left[ -\tau(z,h) \right] = \kappa(z) \exp \left[ -\int_z^h \kappa(z') dz' \right] \end{equation}$$ with the last step arising from the definition of the optical depth in terms of an integral over altitude of the absorption coefficient $\kappa(z)$. The absorption coefficient $\kappa$ is related to the attenuation rate $\alpha$ by: $$\begin{equation} \kappa(z) = 1 - 10^{\alpha(z)/10000} \end{equation}$$ where $\kappa$ is measured in units of m$^{-1}$ and $\alpha$ is measured in units of dB/km. Note that dividing the attenuation rate $\alpha$ by 10 and taking the power converts from dB to natural units.

With this definition of the weighting function, the atmospheric brightness temperature then becomes a simple integral over altitude of the weighting function times the temperature. For a downward-looking sensor at altitude $h$, the expression is: $$\begin{equation} T_a(h) = \int_0^h W(z) \, T(z) \, dz \end{equation}$$ Note that when the attenuation rate is measured in dB/km, the units for the integral also has to be kilometers.

The shape of the weighting function tells us the relative contribution of various altitudes in the atmosphere to the measured brightness temperature. A change of variables is all that is needed to to compute the weighting function for a slanted path through the atmosphere.

Figure 10a plots the temperature weighting functions for the sensor channels viewing a mid-latitude summer profile. The vertical (altitude) axis is linear on the left panel and logarithmic on the right panel. The logarithmic scale is used to illustrate the minor differences in the weighting functions in the lowest portions of the atmosphere. The curves are computed assuming that the sensor is looking at a 65° incidence angle through a plane-parallel atmosphere. The extra slant path increases absorption and moves the peak of the weighting functions higher in the atmosphere.

Figure 10b also plots the temperature weighting function, but allows you to set the frequency and incident angle.

References

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Liebe, H. J., & Layton, D. H. (1987). Millimeter-wave properties of the atmosphere: Laboratory studies and propagation modeling. NASA STI/Recon Technical Report N, 88, 21387.

Liebe, H., & Hufford, G. (1993). MPM93–propagation model of moist air at frequencies below 1000 GHz. In Agard conf. proceedings.

NOAA. (2019). NOAA Products Validation System. (accessed December 20, 2019). Retrieved from https://www.star.nesdis.noaa.gov/smcd/opdb/nprovs/

Reale, T., Sun, B., Tilley, F. H., & Pettey, M. (2012). The NOAA Products Validation System (NPROVS). Journal of Atmospheric and Oceanic Technology, 29(5), 629–645.

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