Flujo del proyecto

The functions described in section 2.1.1 go some way to recreating the angular reflectance of a vegetation canopy, however it does not fully account for all the effects. The bidirectional reflectance curve in figure 4 shows that there is a sharp peak in reflectance when the viewer looks in the same direction as the illumination (at a zenith of 60o_{). This is known as the opposition or hotspot effect.}

It was first observed in the rings of Saturn (Seeliger (1895), cited in Hapke et al. (1996)) and has been well documented in laboratory conditions.

A lidar detector (described in section 3.5) always looks in the hotspot direction, so fully ex- plaining the bidirectional reflectance is not necessary to understand the signal; in fact the signal is easier to understand when views are limited to the hotspot (Strahler and Jupp 1990). Whilst it is not essential for inversion, if the lidar data is to be fused with passive optical measurements or is to be used to create a canopy model to predict off-hotspot reflectance, an appreciation of the issue is needed.

The hotspot is primarily caused by shadow hiding (Liang 2004, Hapke et al. 1996), when the viewer is looking along the the same vector as the illumination, all shadows are cast behind objects and so hidden from the viewer. As the vectors move apart shadows appear, reducing the measured reflectance.

A ray of light that passes any distance through a canopy will always be able to reflect back along the same path as there must be a gap for it to have got in. In a turbid medium model the path length is used to calculate the probability of a light ray interacting with the canopy (with equation 11) and so gaps are not remembered. There is a chance that a returning ray will interact

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 30 40 50 60 70 80 90 Reflectance

Viewer zenith (degrees)

Figure 4: Sample bidirectional reflectance of a sparse Sitka spruce canopy from Monte Carlo ray tracer simulation

with elements that were not there on the way in. This forgetting of gaps is unrealistic and prevents the hotspot effect from being modelled.

In a typical forest canopy there are two different scales of shadow casting, leaves and crowns. Efforts have been made to account for both within radiative transfer models.

Crown scale shadows Except for very dense canopies a horizontally homogeneous model is not

sufficient to capture all the details (Ross 1981). Within a vegetation canopy scattering elements are arranged or clumped into crowns. Shadows cast by these crowns will contribute to the hotspot effect as well as altering the visible area of foliage (also known as effective LAI, more on this later). One method to account for the hotspot is to clump the turbid medium into geometric primitives with free space between and a layer of ground below. From a defined crown size and density the proportion of directly sunlit crown and ground, shaded crown and ground for a given viewing and illumination direction can be determined. From this the reflectance can be calculated, treating the sunlit and shaded areas of crown as 1-D turbid media. This technique was pioneered by Li and Strahler (1985) and has been refined over the years (Li and Strahler 1988, Li and Strahler 1992, Li et al. 1995) leading to the “GORT” model (Ni et al. 1999) which has been applied to a wide range of problems, including lidar (Ni-Meister et al. 2001).

This approach is known as a hybrid geometric model as it combines geometric optics for the discrete crowns with turbid media for canopy. With GORT type models the locations of individual

crowns is never explicitly defined, only the density and distribution through stochastic means. This level of abstraction reduces the number of model parameters, greatly easing inversion (Woodcock

et al. 1997) but may cause problems when the detector’s field of view approaches the scale of

heterogeneity. The “FOREST” model (Cescatti 1997a) was created to extend the hybrid geometric approach to irregular crowns, greatly increasing the complexity of the model. The model has been shown to perform well when inverting above canopy measurements (Cescatti 1997b) but the additional complexity was not justified by comparison of inverted parameter accuracies with simpler models (those with fewer unkowns). Certain models exist that do explicitly define the locations of geometric primitives (North 1996) however these tend to use the Monte Carlo method and so will be covered later.

Kimes and Kirchner (1982) included a further level of detail where instead of geometric primi- tives of turbid media with a bounding plane, the scene was split into voxels (volumetric pixels or cubes). Each was filled by either a plane (for ground, trunks or buildings) or some turbid medium (for crowns). This is also known as the “discrete ordinate” method. The approach is very similar to geometric primitives with explicit locations except that the crowns are not constrained to fit into simple shapes. The light reaching and reflected by each voxel can be calculated, leading to a complete picture of the radiation regime within the canopy. The method has been refined to create the DART model (Gastellu-Etchegorry et al. 1996, Gastellu-Etchegorry et al. 2004).

Splitting the scene up into voxels increases the computational load compared to geometric primitives, but it is questionable whether real tree crowns can be represented as such simple, hard edged objects as cones and ellipses (Parker and Brown 2000).

Myneni proposed a rigorous and spatially explicit solution of the radiative transfer equations (Myneni et al. 1992). In practise computational limits would cause this to be similar to the voxel based methods of (Gastellu-Etchegorry et al. 1996).

Leaf scale shadows Turbid medium models were first developed for diffuse gases common in

astrophysics (Chandrasekhar 1960). The scattering elements are treated as point particles with no size and so cast no shadows. This model is not appropriate for vegetation canopies where elements have finite size and cast shadows (Ross 1981, Knyazikhin et al. 1992).

There are far more leaves in a crown than there are crowns in a canopy (for a single pixel), therefore it is much more computationally expensive to explicitly define leaf scale gaps and elements than it is for crown scale shadows (Ni et al. 1999, Gastellu-Etchegorry et al. 2004). This approach of remembering all gaps has been used (Knyazikhin et al. 1992, Disney et al. 2006) but more abstract approaches are more common as they require fewer parameters, easing inversion.

One conceptually simple approach is to treat gaps and leaves as circles of a set diameter, so that a ray penetrating into a canopy can be described by a cylinder (Verstraete et al. 1990). The increased probability of a ray returning along the same path is accounted for by modifying the optical thickness by the fraction of overlap of the incoming and outgoing cylinders (see figure 5).

Figure 5: Illustration of the modelling of hotspot by overlapping cylinders from Verstraete et al (1990)

Jupp and Strahler (1991) used statistics to describe the probability of returning rays passing through existing gaps; the optical thickness either being zero (gap) or the same as for the turbid medium (canopy). This was an extension of earlier work to describe crown scale shadows (Strahler and Jupp 1990). It was later generalised to to use rectangular rather than circular gaps (Qin and Xiang 1994), although it was found that the hotspot is dominated by leaf size and distribution and the effect of leaf shape could be accounted for by modifying leaf size.

Kuusk (1995) used a similar approach, but with Markov chain theory rather than Boolean statistics. The probability of an interaction in a canopy layer is modified from the standard turbid medium case by whether or not there have been interactions in higher layers and a correlation fac-

tor. These three methods are mathematically equivalent (Liang 2004), requiring only an additional leaf size factor and can adequately recreate the hotspot.

Knyazikhin et al. (1992) gave a more rigorous, but more complicated, solution in which the canopy is taken as a random collection of leaves (turbid medium) and gaps. Interactions are remembered and influence the probability of a ray returning in the hotspot direction.

Coherent backscatter Another potential contribution to increased reflectance in the hotspot

direction is a phenomenon known as “coherent backscatter”. Some believe it to be significant factor in lidar reflectance (Harding, DJ, 2008, pers comm.).

Coherent backscatter, as its name suggests, results from light scattered from the target inter- fering with itself constructively, increasing the intensity (Stephen and Cwilich 1986). Ordinarily scattered light will be completely incoherent, so no enhanced reflectance is observed. In the hotspot direction, when the illumination source and viewer are co-aligned, an interesting effect is observed. For every photon path from the illumination source to the viewer, a reversed path can be traced from the viewer to the illumination. As these two are co-aligned, photons from both of these paths will contribute to the measured signal and as they have travelled exactly the same distance they will constructively interfere. This was first noticed by Kuga and Ishimaru (1984) and for a disordered medium, such as vegetation, is the only significant interference effect (Stephen and Cwilich 1986).

The scale of the target’s roughness controls the magnitude of the effect (partially from shad- ows). Scenes with many elements around the size of the light’s wavelength show more coherent backscatter than scenes with larger elements. Experiments found that for a typical forest canopy, where objects are generally larger than the light’s wavelength, the contribution to the hotspot from coherent backscatter is insignificant compared to that from shadow hiding (Hapke et al. 1996). It was found to be important for dry soils and some fine structure vegetation, such as mosses.

Hapke’s study focused on the visible region, where foliage reflectance is low. As coherent backscatter depends upon multiple scattering it is especially sensitive to element reflectance. There- fore it may be slightly more important to the hotspot in the near infra-red, with higher foliage reflectance. However, for the visible it was found to be such a small effect that increasing the

reflectance (from around 0.1 to 0.6) is not likely to make coherent backscatter more important than shadow hiding. It would seem to be an unnecessary level of detail although a quantitative analysis has not, as far as he author is aware aware, been performed in the near infra-red.