## 4 Understanding the Physical Principles of a Meteorological System  
The principles underlying the design and operation of meteorological radar are essential for effective weather forecasting [20]. In this section, we introduce some general concepts related to this subject. It begins with defining the electromagnetic interaction of EM waves with individual molecules. The total internal energy of a molecule can be categorized into three types of energy states--electronic, rotational, and vibrational energy:  
\[\varepsilon_{molecule\ energy}=\varepsilon_{e}+\ \varepsilon_{v}+\ \varepsilon_{r} \tag{1}\]  
These energy states define quantum states and, as a consequence, dictate how an atom responds to electromagnetic (EM) waves with energy values corresponding to the differences between energy levels, \(\Delta\)E. This interaction leads to the creation of absorption and emission spectrum due to transitions between these energy states. Each atom has unique energy differences between states; therefore, when an EM wave with energy at specific frequencies interacts with an atom, absorption and scattering spectral lines are produced. From these spectra, one can infer which types of atoms constitute the molecules in question. From a spectral graph, an absorption coefficient can be derived. Since spectral lines have tails that extend over a certain bandwidth, it is necessary to sum the contributions of all frequency lines. Based on this principle, a microwave propagation model was developed in the 1980s. This model computes the gaseous absorption coefficient, (\(K_{g}\)) as the sum of the oxygen absorption coefficient and the water vapor absorption coefficient for a clear-sky atmosphere. That is,  
\[k_{g}=k_{O_{2}}+k_{H_{2}O}\left[\frac{\mathrm{dB}}{\mathrm{km}}\right] \tag{2}\]  
where \(k_{O_{2}}=0.182f\sum_{j=1}^{44}S_{i}F_{i}\left[\frac{\mathrm{dB}}{\mathrm{km} }\right]\), and \(k_{H_{2}O}=0.182f\sum_{j=1}^{34}(S_{i}F_{i}+S_{c}F_{c})\left[\frac{\mathrm{dB }}{\mathrm{km}}\right]\). The summation is performed over 44 oxygen spectral lines and 34 water vapor spectral lines, calculated as a function of line strength, \(S_{i}\), and shape function, \(F_{i}\). Both parameters are influenced by atmospheric conditions, specifically temperature (T), barometric pressure (P), and the partial pressure of water vapor (P\({}_{\mathrm{H_{2}O}}\)) at different altitudes (Z). In light of these principles, it becomes evident that the choice of operating frequency is critically important and depends on the specific atmospheric parameters one seeks to measure remotely using spectral absorption or emission lines. While the discussion thus far has focused on clear-sky atmospheric conditions, the presence of extinction hydrometeors--such as rain, clouds, snow, or fog particles--introduces both absorption and scattering effects. In such cases, estimating the volume extinction coefficient becomes essential. This coefficient depends on the density, shape, size distribution, and dielectric properties of the particles within the volume interacting with the propagating electromagnetic (EM) waves. To model this interaction, it is commonly assumed that the EM waves interact with spherical particles within the volume. This simplification allows for the analysis of the power density, \(S_{i}\), of an incident EM wave upon a particle with cross-sectional area A and radius r (see Figure 3). A portion of the energy is absorbed, while the remainder is scattered in all directions within a radius R from the center of the particle outward (see Figure 3). As a consequence, we can define an absorption cross-section  
\[Q_{a}=\frac{P_{a}}{\mathcal{S}_{i}}\left[\mathrm{m}^{2}\right] \tag{3}\]  
where \(\mathrm{P_{a}}\) is the absorbed power.