## 3 Methods and Results
### Electro-Optical and Infrared
#### 3.1.3 Height from Shadow  
In sunny and cloud-free conditions, all features protruding from the level ice surface produce shadow. The method for height estimation using a single high-resolution image is to analyze the ice-feature shadow, which is dependent on the orientation of the feature.  
Figure 4: MYHF on a WorldView-2 image (**a**) and DEM (**b**); rubble field shown in WorldView-2 image (**c**) and in DEM (**d**). Images © DigitalGlobe, 2014.  
The developed algorithm [27] facilitates the extraction of ice-feature height from shadow and the deriving of statistical information on ice deformation parameters.  
In the work [49], a vertically downward-looking camera mounted on a helicopter was used to collect photographs with ground resolutions of 0.5-0.8 m. The ice-feature height \(H\) was calculated from the measured shadow length \(L_{BS}\) (Figure 5) and from knowing the sun elevation angle \(\epsilon_{sun}\) as  
\[H=L_{BS}\ \text{tan}(\epsilon_{sun}) \tag{1}\]  
However, the approach described by Equation (1) cannot be applied to satellite imagery, which is collected from different angles (e.g., elevation angle for our GeoEye-1 image is 74.57\({}^{\circ}\)).  
The height can be derived from the shadow considering image-acquisition geometry without utilizing the rational polynomial coefficient (RPC) and the sensor (camera) model [50]. In this method, the height, \(H\), was calculated as  
\[H=\frac{L_{PS}}{\sqrt{\frac{1}{\text{tan}^{2}(\epsilon_{sun})}+\frac{1}{\text{tan} ^{2}(\epsilon_{sat})}-\frac{2\text{cos}(\epsilon_{sat}-\epsilon_{sun})}{\text {tan}(\epsilon_{sat})\text{tan}(\epsilon_{sun})}}} \tag{2}\]  
where \(\alpha_{sun}\) is the azimuth angle of the sun; \(\epsilon_{sun}\) is the elevation angle of the sun; \(\alpha_{sat}\) and \(\epsilon_{sat}\) are the azimuth and the elevation angles of the satellite, respectively (Figure 5a); and \(L_{PS}\) is the measured distance between the image of the top of the ridge and the shadow (Figure 5b).  
Equation (2) was applied to analyze the height of different ice features using height-from-shadow techniques and to compare the height to DEM. Figure 6 shows a WorldView-2 image containing ice-island fragments. The red line over the edge of the fragment was  
Figure 5: Acquisition geometry for image of the ridge: (**a**) the azimuth and the elevation angles for the sun and the satellite, (**b**) explanatory diagram of the ridge and shadow.  
used for height estimation. The stereo DEM of these ice-island fragments was compared to the height profile in Figure 6 (bottom) extracted using Equation (2) along the shadow base. It can be seen that the height from shadow provides a height estimate of about 3.5 to 4 m. Similar height values can be found in the DEM for this ice-island fragment. The quantitative comparison between these two techniques can be found in paper [27].  
More EIF (ice-island fragments and forming hummocks) shown in Figure 7 were also investigated with the height-from-shadow method, and the results were compared to stereo DEM (Table 1). The comparison shows good agreement between the two methods. The error of height estimation for ice-island fragments is about 5-10% but about 15% for the hummock peak.  
\begin{table}
\begin{tabular}{c c c} \hline
**EIF \#** & **Height from Shadow, m** & **Max Height from Stereo DEM, m** \\ \hline
1 (Figure 7a) & 4.4 & 4.7 \\
2 (Figure 7b) & 4.1 & 4.5 \\
3 (Figure 7c) & 25.6 & 29.3 \\
4 (Figure 7d) & 3.0 & 3.3 \\ \hline \end{tabular}
\end{table}
Table 1: Heights extracted from shadow-based methods and stereo.  
Figure 6: WorldView-2 image with ice-island fragments with red line indicating edge used for height estimation (**a**); stereo DEM of these ice-island fragments (**b**); and height profile extracted using Equation (2) along the shadow base (**c**). Images © DigitalGlobe, 2014.