## 4 Retrieval of Atmospheric Water Vapor  
In the MiRS system, all state variables, for example, atmospheric profiles of temperature, water vapor, and cloud water contents, are retrieved simultaneously by utilizing brightness temperatures from all of the channels. In the following, we focus on the ATMS applications. The same as the MiRS operational setting, the latitude/longitude-dependent climatological background is used as the a priori background as well as the first guess in the retrieval. ATMS channel 1, 2, and 16, 17 are often called window channels that contain information about total precipitable water (TPW), clouds, and surface parameters, including surface emissivity. ATMS channels 3 to 15 are used for retrieving the atmospheric profile of temperatures. Together with ATMS channel 1, ATMS channels 18 to 22 are used for acquiring the atmospheric profile of water vapor. To avoid saturation for a very high water vapor amount, ATMS channel 1 is centered at 23.8 GHz instead of at the water vapor absorption line at 22.235 GHz. ATMS channel 2 at 31.4 GHz is less affected by water vapor but more sensitive to cloud liquid water. A simple radiative transfer equation may be helpful to understand what signals the passive microwave sensor receives from space. In the absence of precipitation (or insignificant scatterings in microwave), a radiative transfer equation is composed of upward surface radiation and upward atmospheric radiation as well as downward atmospheric radiation reflected upward from the surface. The equation may be expressed as:  
\[R = \varepsilon\ B\Big{(}T_{surface}\Big{)}\tau(atm)+R_{up}(atm)+(1- \varepsilon\ )\tau(atm)R_{down}(atm) \tag{5}\]  
where \(\varepsilon\) is the surface emissivity; \(B\Big{(}T_{surface}\Big{)}\) is the Planck radiation at the surface temperature, \(\tau(atm)\) is the atmospheric transmittance at the satellite height, and \(R_{up}(atm)\) and \(R_{down}(atm)\) are the upward atmospheric radiation at the satellite height and downwelling atmospheric radiation at Earth's surface, respectively. The radiance can be converted into the brightness temperature [14], which is commonly used in satellite data assimilation and in the retrieval of geophysical parameters. We also use the brightness temperature in the following equations. Table 1 shows the sensitivity of the ATMS channel 1 brightness temperature (\(T_{b}\)) on TPW (\(W\)) at a sensor scan angle of 30 degrees for standard atmospheres. The surface emissivity values are given in the second row. The ocean surface emissivity value is the CRTM model calculated for a sea surface wind speed of 7 m/s at 10 m above water. We choose U.S. standard atmosphere (low TPW), a mid-latitude atmosphere (close to global mean TPW), and a tropic atmosphere (high TPW). Over water, the sensitivity is very high because of low surface emissivity and high atmospheric emission. Over land, surface emission is dominant and the sensitivity to atmospheric TPW becomes weak. For high land surface emissivity, the sensitivity can become zero or negative. In general, the higher the sensitivity to a parameter, the better it is for acquiring the parameter. If only channel 1 is used, the perturbation of \(T_{b}\) may be expressed as:  
\[\Delta T_{b}(1) = \frac{\partial T_{b}(1)}{\partial\log(W)}\Delta\log(W) \tag{6}\]  
where \(W\) denotes the TPW. Thus, the TPW increment may be calculated as:  
\[\Delta\log(W) = \Delta T_{b}(1)/\bigg{(}\frac{\partial T_{b}(1)}{\partial\log(W)} \bigg{)} \tag{7}\]In the retrieval, the water vapor Jacobian is often expressed as the sensitivity of the brightness temperature to a logarithm of water vapor because water vapor exponentially decreases as height increase in the troposphere in general. Obviously, Equation 7 is singular when the water vapor Jacobian equals zero. The solution is unstable or has a large error for a very weak sensitivity. Therefore, using single-channel information, the microwave measurement at 23.8 GHz is not very helpful for estimating TPW over land.