Revisiting the vegetation hot spot modeling: Case of Poisson/Binomial leaf distributions Abdelaziz Kallel a, ⁎, Tiit Nilson b a Institut Supperieur d'Electronique et de Communication de Sfax, 3000 Sfax BP 868, Tunisia b Tartu Observatory, 61602, Travere, Estonia abstract article info Article history: Received 1 January 2012 Received in revised form 15 November 2012 Accepted 17 November 2012 Available online xxxx Keywords: Hot spot effect Bidirectional gap distribution Poisson/Binomial distributions Homogeneous discrete vegetation The accuracy of spaceborne/airborne sensor measurements in the solar domain keeps increasing over time. High resolution, multi-directional and hyperspectral image acquisitions start to be abundant. With regard to the multi-angular remote sensing data, the hot spot, i.e. the exact backscattering direction of direct sunlight together with its neighboring directions, is of special interest. Accurate hot spot models have to be used to adequately simulate the hot spot signature and to allow reliable inversion of multi-angular data. In this paper, we propose a physical hot spot model (Leaf Spatial Distribution based Model, LSDM) assuming that for a given point inside the vegetation to be sunlit (respectively, observed) it should be located within a cylinder free from leaf centers. The cylinder is oriented to the sun (respectively, sensor) direction. Assuming a leaf random, regular or clumped spatial distributions, the gap probabilities in the sun and sensor directions are expressed as a function of these cylinder volumes. Based on the same hypothesis, the bidirectional gap probability is estimated as a function of the total volume of the two cylinders. The evaluation of the needed common volume of two cylinders having different radii is reduced to calculation of some elliptic integrals. Finally, the hot spot signature is estimated based on the bidirectional gap probability distribution. Different model versions with different leaf spatial distri- bution functions are compared. Particularly, it is shown that compared to the random distribution, the regular (the clumped, respectively) distribution increases (decreases, respectively) the reflectance due to single scat- tering contribution from foliage. The proposed model is validated using the ROMC web-based tool and its better performance compared to the Semi-Discrete Model and Kuusk's model is confirmed. © 2012 Elsevier Inc. All rights reserved. 1. Introduction The results of spaceborne multi-angular remote sensing mea- surements have been available at least during the last 15 years. New sensors such as CHRIS, which is an imaging spectrometer car- ried on board the space platform PROBA, allow high resolution multi-angular and hyperspectral acquisitions. In terms of multi- angular observations, this sensor can be pointed off-nadir in both along-track and across-track directions. The sophistication of such instruments keeps increasing over time, particularly the sensor agil- ity is a hot topic. It allows to rapidly acquire off-nadir targets, in order to sequence images of the same area in different observation angles leading to sampling of the directional reflectance factor of the canopy [e.g., Pleiades-HR constellations (Lebegue et al., 2010)]. Although the hot spot region, corresponding to the bright area close to the exact backscattering direction, has been recognized as a potentially informative angular region, a majority of existing instruments with multi-angular capability do not currently measure in the hot spot di- rection. However, a considerable amount of images showing the hot spot region as well as its angular signature have been recorded in the last two decades. For instance, the airborne version of the Multi- angle Imaging SpectroRadiometer (AirMISR) (Gerstl et al., 1999), and the spaceborne Polarization and Directionality of the Earth's Re- flectances (POLDER) (Grant et al., 2003) instruments provided the Bidirectional Reflectance Distribution Function (BRDF) signatures that included the hot spot region. Recent studies have shown that a number of biophysical features can be retrieved from a sampled BRDF (just two or three parameters per inversion). For instance, one can cite the canopy architecture (Schlerf & Atzberger, 2006) (i.e., the tree spatial distribution, canopy cover, leaf area index), the tree macro structure (Mõttus et al., 2006) (e.g., tree height, the size and shape of the crowns and leaves), the understory reflectance (Canisius & Chen, 2007) and the clumping index (He et al., 2012). As the main aim of canopy remote sensing is to derive canopy prop- erties (Combal et al., 2002) from the BRDF, it is important to ade- quately model it as a function of canopy features and scene geometry. For that a lot of theoretical works are trying to increase the accuracy of the BRDF modeling (Widlowski et al., 2006b), there- fore the proposed approach complexities keep increasing and the models are becoming time consuming, particularly those based on Monte Carlo (MC) ray-tracing (for which the inversion is not Remote Sensing of Environment 130 (2013) 188–204 ⁎ Corresponding author. E-mail addresses: abdelaziz_kallel@yahoo.fr (A. Kallel), nilson@aai.ee (T. Nilson). 0034-4257/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.rse.2012.11.018 Contents lists available at SciVerse ScienceDirect Remote Sensing of Environment journal homepage: www.elsevier.com/locate/rse