## 3 Measurement Theory  
### Reflectance and NDVI  
The geometry of the downward-looking reflectance sensors is shown in Figure 3. The average footprint reflectance [44] was computed by (see Appendix 1 for deductions):  
\[\bar{\rho}=\frac{\pi\bar{L}_{r}}{E}=\frac{\int_{\Omega_{A}}\pi f(\theta_{r}) \cdot d\omega^{\prime}}{\int_{\Omega_{A}}d\omega^{\prime}}=\frac{\int_{A}\ \pi f(\theta_{r})\cdot\frac{\cos\ (\theta_{r})}{r^{2}}dxdy}{\int_{A}\ \frac{\cos\ (\theta_{r})}{r^{2}}dxdy}, \tag{1}\]  
where \(\bar{L}_{r}\) is the mean radiance of the area (sensor footprint) \(A\), \(E\) is the hemispherical irradiance, \(\pi f(\theta_{r})\) is the hemispherical-directional reflectance factor (HDRF), \(d\omega^{\prime}\) is the projected solid angle subtended by the differential area \(dA\) (_i.e._, \(dxdy\)), and \(\Omega_{A}\) is the integration range of the projected solid angle subtended by the footprint. Thus, the measured average footprint hemispherical-conical reflectance is the average of \(\pi f(\theta_{r})\) over the projected solid angle subtended by footprint \(A\). \(\pi f(\theta_{r})\) is written as [36]:  
\[\pi f(\theta_{r})=\int_{2\pi}d\phi\int_{0}^{\frac{\pi}{2}}f(\theta_{i},\theta_ {r})\cdot\sin(\theta_{l})\cdot\cos(\theta_{l})\cdot d\,\theta_{l}, \tag{2}\]  
where \(f(\theta_{i},\theta_{r})\) is the bidirectional reflectance distribution function (BRDF) at the differential area \(dA\) with the incoming light angle \(\theta_{l}\) and reflected light angle \(\theta_{r}\). The spatial contribution from a Lambertian surface to the total footprint reflection decreases away from nadir towards the far end of the footprint according to a Cauchy distribution. It shows that the radiance captured by our sensors is dominated by flux originating from the closest half of the footprint area. Figure 3: Geometry diagram of the oblique-looking sensor. The downward-looking sensor, mounted at height \(H\) with an oblique off-nadir angle \(a\) or around \(55^{\circ}\), has an ellipse footprint with semi-major axis \(a\), and semi-minor axis \(b\). A reflection beam, with an angle \(\theta_{r}\) to the normal of the small area \(dA\), reaches the sensor at a distance \(r\). The yellow arrow shows the direct incoming light, which is added to the diffuse light from the upper hemisphere. NDVI [38] is computed as:  
\[NDVI=\frac{\bar{\rho}_{NIR}-\bar{\rho}_{red}}{\bar{\rho}_{NIR+}\bar{\rho}_{red}}, \tag{3}\]  
where \(\bar{\rho}_{NIR}\) and \(\bar{\rho}_{red}\) are average footprint HCRF's in NIR and red bands, respectively. Modelling Anisotropic Effects on the Footprint HCRF  
In order to investigate the feasibility of sampling anisotropic reflectance using a wide-angle sensor we used the Kuusk and Nilson forest reflectance and transmittance (FRT) model [39] to simulate reflectance measured by our tower sensors under hemispherical-conical geometry. The model was driven with field measurements on average stand characteristics (Table 5), and ground vegetation properties. In the model, PROSPECT [40] is used for leaf optics, and 6S [41] is used for atmosphere correction, simulating both direct and diffuse sun light. The sun was fixed at a zenith angle of 40\({}^{\circ}\), and simulations were made over all azimuth angles, and zenith angles varying between 0 and 80\({}^{\circ}\). Figure 4 displays simulated HDRF for red and NIR bands, and for NDVI. The HDRF properties of the ground vegetation can also be given by the FRT model. The modeling results were further integrated over the footprint area to simulate HCRF, reflectance and NDVI, as measured by our tower sensors under hemispherical-conical geometry. These computations were made for one forest canopy simulation and one ground vegetation simulation separately. Figure 4: HCRF and sensor footprint at the Abisko forest stand as modeled with FRT [39]. The sensor is looking at 261\({}^{\circ}\) direction with an off-nadir angle of 55\({}^{\circ}\) and an FOV of 60\({}^{\circ}\). The sun zenith angle is 40\({}^{\circ}\) at 180\({}^{\circ}\) relative azimuthal direction, shown as an asterisk.