## 5 Properties of Shape Groups
### Relationships between Microphysical Properties  
#### 5.2.1 Cross-Sectional Area and Area Ratio  
The cross-sectional area versus particle size (\(A\) vs. \(D_{\text{max}}\)) and area ratio versus particle size (\(A_{\text{r}}\) vs. \(D_{\text{max}}\)) relationships for the shape groups are analyzed in this section. The \(A\) vs. \(D_{\text{max}}\) relationship is fitted to a power law given by [PERSON] and [PERSON] [26]:  
\[A=C_{1}\cdot\left(\frac{D_{\text{max}}}{1\upmu\text{m}}\right)^{C_{2}}. \tag{3}\]  
Equation (3) yields straight lines on a logarithmic plot; therefore, the parameters \(C_{1}\) and \(C_{2}\) are determined from linear fits to the data expressed as \(\log(A)\) versus \(\log(D_{\text{max}})\), so that if all particles were spherical (i.e., they would appear circular in the images), \(C_{1}\) and \(C_{2}\) would be equal to \(\frac{\uppi}{4}\times 10^{-12}\) m\({}^{2}\) and 2, respectively. The area ratio is calculated according to Equation (1) but can also be expressed as a power law function:  
\[A_{\text{r}}=C_{3}\cdot\left(\frac{D_{\text{max}}}{1\upmu\text{m}}\right)^{C_ {4}}, \tag{4}\]  
where parameters \(C_{3}\) and \(C_{4}\) can be obtained from Equations (3) and (1) so that \(C_{3}=C_{1}\times\frac{4}{\uppi}\times 10^{12}\) m\({}^{-2}\) and \(C_{4}=C_{2}-2\). Figure 5 shows the fitted \(A\) vs. \(D_{\text{max}}\) (_left_) and \(A_{\text{r}}\) vs. \(D_{\text{max}}\) (_right_) relationships as lines for each shape group. The point on each line represents the median of the distribution of \(D_{\text{max}}\), the lines' endpoints correspond to the percentiles 15.9% and 84.1%, and hence, the spread of data is illustrated by the length of the line. The fit parameters \(C_{1}\) and \(C_{2}\) as well as the corresponding correlation coefficients, \(R^{2}\), which are used to analyze correlation, are shown in Table 9. In general, judging from the correlation coefficient, \(R_{a}^{2}\), in all shape groups, there is a good correlation (\(R_{a}^{2}\geq 0.73\)) in the \(A\) vs. \(D_{\text{max}}\) relationship. The shape groups (_5) Plates_, (_11) _Spatial stellar crystals_, (_12) _Grappel_, (_13) _Ice and melting or sublimating particles_, (_14) _Irregulars and aggregates_, and (_15) _Spherical particles_ have \(R_{a}^{2}>0.9\). For most shape groups, area ratio decreases with increasing particle size, as can be seen in Figure 5. Clearly, group (_15) _Spherical particles_ is an exception, and its area ratio is almost constant, i.e., size independent. A nearly constant \(A_{\text{r}}\) means that \(C_{4}\simeq 0\) and \(C_{2}\simeq 2\) (see Equations (3) and (4)), which is the case for this group. Two more groups have values of \(C_{2}\) close to 2. In group (_12) Graupel_, many particles have a close to spheroidal shape, and this appears to be similar over all sizes encountered, so that the area ratio is again size independent. The other group with \(C_{2}\) fairly close to 2 and an almost constant \(A_{\text{r}}\) is (_6) Stellar crystals_. Apparently, the stellar particles in this group also remain similar at different sizes, so that \(A_{\text{r}}\) hardly varies. While \(A_{\text{r}}\) for group (_15) Spherical particles_ is close to 1, \(A_{\text{r}}\) for (_6) Stellar crystals_ is much smaller, approximately 0.4, as one would expect. Particles in shape groups (_1) Needles and thin or long columns_, (_2) _Crossed needles and crossed columns_, and (_3) _Thick columns and bullets_, which are highly nonspherical, have \(C_{2}\) values furthest from 2. As a consequence, \(C_{4}\) values are furthest from zero and these groups have the steepest size dependence of \(A_{\text{r}}\), as can be seen in Figure 5.  
Tables 8 and 9 as well as Figures 4 and 5 show, in addition to the results for each shape group described above, also the results of \(A\) vs. \(D_{\rm max}\) and \(A_{\rm r}\) vs. \(D_{\rm max}\) fits to all data (regardless of shape). These are provided as references and for situations where one might need average relationships instead of shape-dependent ones.