## 4 Observation Simulations of GMR, GeoSTAR, and GIMS
### Simulation of the Observed Brightness Temperature TA of GMR, GeoSTAR, and GIMS
#### 4.2.1 Geostationary Microwave Radiometer (GMR)  
A real aperture radiometer is mainly composed of antennas and receiver channels. The measured antenna temperature of the real aperture radiometer GMR at location \((\theta_{0},\varphi_{0})\) (TA(\(\theta_{0},\varphi_{0})\)) can be formulated as the integration of the products of the upwelling brightness temperature \(TB\) image and the normalized antenna gain pattern \(G_{n}\) plus the system noise \(\Delta T_{rms}\)[44]:  
\[T_{A}(\theta_{0},\varphi_{0})=\int TB(\theta,\varphi)G_{n}(\theta_{0},\varphi_ {0};\theta,\varphi)d\Omega+\Delta T_{rms} \tag{2}\]  
where \((\theta,\varphi)\) denotes the elevation and azimuth angles of the point on the Earth's surface in the spherical coordinate system of the antenna, respectively, \(T_{A}(\theta_{0},\varphi_{0})\) represents observed brightness temperature at \((\theta_{0},\varphi_{0})\), \(G_{n}(\theta_{0},\varphi_{0};\theta,\varphi)\) represents gain in the field of view \((\theta,\varphi)\) when the antenna is aligned with \((\theta_{0},\varphi_{0})\), \(\Omega\) is unit solid angle, \(\Delta T_{rms}\) is random  
Figure 4: Simulated upwelling brightness temperature TB (**A–H**) and observation TA of GMR (**I–P**), GeoSTAR (**a–h**), and GIMS (**i–p**). (**A,I,a,i**) represents channel-1 50.3 GHz, (**B,J,b,j**) represents channel-2 51.76 GHz, (**C,K,c,k**) represents channel-3 52.8 GHz, (**D,L,d**),represents channel-4 53.596 GHz, (**E,M,e,m**) represents channel-5 54.4 GHz, (**F,N,f,n**) represents channel-6 54.94 GHz, (**G,O,g,o**) represents channel-7 55.5 GHz, and (**H,P,h,p**) represents channel-8 57.29 GHz.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Instrument** & **BW (MHz)** & **NF (dB)** & **\(\mathbf{\tau}\) (s)** & **N** & **d (cm)** & **D (m)** & **T (s)** \\ \hline GMR & 180–400 & 5 & 0.04 & / & / & 5 & \(\sim\)600 \\ GeoSTAR & 200 & 5 & 300 & 312 & 2.1 & 3.7 & 300 \\ GIMS & 200 & 5 & 1 & 70 & 1.87 & 3.7 & \(\sim\)300 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The simulation configurations of the three GEO-MW sounders.  
noise. Equation (2) is a standard signal model for a real aperture microwave radiometer, which presents the intrinsic degradation quality in the GMR observation process. The main influence on the observed brightness temperature TA data is the smoothing effect introduced by the limited antenna beam main lobe width and the system noise. In the model, it is usual practice that the system noise \(\Delta T_{rms}\) is simulated by white Gaussian noise, whose standard deviation is approximately equal to the noise-equivalent brightness temperature (NEAT) of each frequency channel of the GMR. The simulated GMR uses a circular aperture parabolic antenna with a 5 m diameter, whose antenna gain pattern \(G_{n}\) is calculated using Equation (3).  
\[G_{n}(\theta,\varphi)=2J_{1}\left(\frac{\pi D}{\lambda}sin\theta\right)/\left( \frac{\pi D}{\lambda}sin\theta\right) \tag{3}\]  
where \(J_{1}\) represents the first-order Bessel function, \(D\) is the antenna diameter, and \(\lambda\) is the wavelength. Because the circular aperture antenna is symmetrical in the azimuth angle \(\varphi\), its pattern \(G_{n}\) is only a functsion of the elevation angle \(\theta\). Figure 5a,f shows the simulated 2-D and E-plane 1-D antenna patterns of GMR at 50.3 GHz, and the corresponding 3-dB beamwidths are 0.087\({}^{\circ}\).  
To generate the simulated observation of GMR, first, the coordinates of each pixel of the upwelling brightness temperature TB are converted from the latitude and longitude coordinate system to the antenna spherical coordinate system of GMR, and then, \(TB(\theta,\varphi)\), \(G_{n}(\theta,\varphi)\), and \(\Delta T_{rms}\) are substituted into Equation (2) to calculate the observed antenna brightness temperature TA. The simulated GMR antenna brightness temperature TA images of Typhoon Hagibis for all 8 channels are shown in Figure 4I-P.