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The assumptions in and reliance on effective parameters of turbid medium models means that they are not ideal for understanding the measurement of forests through simulations. Therefore more detailed models that do not need effective parameters are needed. Models exist that explicitly define the position, orientation and spectral properties of all elements in the scene then calculate the radiation field by a form of numeric integration, though these are far more computationally expensive than the more abstract turbid medium methods. A complete description of a vegetation scene contains far too many variables to allow direct inversion (attempts have been made using fitting to look up tables of simulated waveforms, but with slightly abstracted forests (Koetz et al.

2006)) but it also minimises assumptions, allowing physically realistic simulations of remote sensing signals.

In the beginning the computational expense of explicit methods limited their use (Myneni et al. 1992), users tending towards more abstract volumetric methods (Kimes and Kirchner 1982). The simulation of optical remote sensing signals has much in common with computer graphics; both produce light signals (or images if a 2D array) from given objects, also known as rendering. This synergy with computer graphics and the explosive growth in computer power, graphics hardware and the increasing sophistication of rendering algorithms in the late 1980s and 1990s made explicit simulations of vegetation feasible (Borel et al. 1991). Care must be taken when employing com- puter graphics acceleration methods as these tend to be designed to produce a “psycho-physical” result, one that looks right to a human eye rather that a physically accurate result obeying energy conservation required for remote sensing images (Koenderink and van Doorn 1996).

2.2.1 Radiosity

One popular computer graphics method is known as the “radiosity” technique (Cohen and Wallace 1993). These calculate how much light reflects between every surface in the scene, producing a set of “form factors” that can be used to calculate the radiance measured by a detector from an illumination source placed anywhere within the scene. This is ideal for films since each frame can be quickly rendered from the form factors, though the initial calculation is computationally expensive and the set of form factors can be large if there are many elements in the scene (which there invariably are in forests).

They were first used for remote sensing of vegetation in the early 1990s (Borel et al. 1991) and thus far only scenes with relatively sparse canopies have been modelled with radiosity methods (for example scrub in semi-desert regions with a ground cover of <25% (Qin and Gerstl 2000) and, more recently, short crops (Liu et al. 2007)) and the scaling of computational and memory expense with scene complexity and size has been cited as a major disadvantage of the technique (Disney

et al. 2000, Liang 2004). The computer requirements can be reduced by splitting the scene into

repeatable subsections, but these units cannot be smaller than the scale of heterogeneity (Qin and Gerstl 2000), limiting its applicability to forests.

It is questionable whether it would be practical to use radiosity methods with scenes as complex as a coniferous forest (they being one of the most structurally complex biomes). Also radiosity models do not typically record the path length of light and so it has been suggested than they are not suitable for simulating the range resolved signals required for lidar (Disney et al. 2000).

2.2.2 Monte Carlo ray tracing

Monte Carlo techniques are a form of numeric solution whereby a finite number of samples are made of a complex system, building up a picture of the system’s behaviour (Metropolis and Ulam 1949). They are very popular for problems that have no analytic solution (such as radiative transfer), without making many assumptions and whose complexity makes alternate methods slow. Previous uses include electron microscopy (Shimizu and Ze-Jun 1992), neutron diffusion for nuclear fission

(Metropolis and Ulam 1949) and CO2flux (Hollinger et al. 2004) and they are an obvious choice

for solving radiative transfer.

2.2.3 Details of operation

For radiative transfer the process is called Monte Carlo ray tracing and has been used for many years, including optical instrument design on the very first electronic and mechanical computers (Comrie 1940). It can work in two modes, the most physically intuitive of which is forwards ray tracing. In this mode rays of light are traced from an illumination source (whether it be the sun, a laser or a diffuse sky) towards the target. At the first object it reaches it is either absorbed, reflected or transmitted depending on a randomly drawn number, the probability and direction of each is set by the bidirectional reflectance and transmittance spectra of the object struck. Using a probability distribution to choose either reflection or transmission prevents the ray from splitting into two, both of which would need to be traced, doubling the computational load with each interaction. This process is repeated until the ray is either absorbed, leaves the scene, undergoes a maximum number of interactions or reaches the detector, in which case its intensity is added to the signal. By this process a “ray tree” is generated, a set of intensities throughout the target. This is repeated for many (typically tens of thousands to millions) rays until a representative signal from the target is reached.

All interactions occur in the geometric domain (Houstoun 1938) so that no diffraction or re- fraction effects have to be calculated. Any features smaller than the radiation’s wavelength are taken into account by the element bidirectional reflectance and transmittance.

The alternative mode is reverse ray tracing, where rays are traced from the detector back to the scene. This ensures that every ray has a chance to contribute to the signal (Lewis 1999). For cases when the field of view of the detector is smaller than the field of illumination (which is generally the case for passive optical sensors with slightly diffuse sky conditions) reverse ray tracing is more efficient than forwards as with forwards it would be possible for a ray traced from the illumination to never enter the detector’s field of view, wasting computer time (Disney et al. 2000). However, if the field of view is smaller than the field of illumination the opposite is true and a ray traced from the detector might never enter the field of illumination. Therefore in these cases, which include most lidar detectors (Schutz et al. 2005), forwards ray tracing would be more efficient.

How many rays to sample the scene with and what to limit the maximum number of interactions to depend upon the scene. Often the maximum number of iterations is set as the number of interactions after which the ray’s contribution to the total intensity is negligible (found from test runs) and the number of rays is set as the number after which the fractional change in recorded radiance (from one new ray to another) drops below a certain level. Limiting the maximum number of interactions can cause a small truncation in energy but the effect is small and can be corrected by a small factor (Disney et al. 2000). Obviously if only a psycho-physical result is needed, as is the case for computer graphics, less physically stringent limits can be set, using enough rays and interactions to ensure a human observer will not notice the difference.

Finding which object a ray strikes through intersection tests is the most computationally ex- pensive process in ray tracing, especially if the scene contains many elements. Methods to increase the efficiency have been introduced such as using hierarchical “bounding boxes” around objects (Glassner 1989). This way many millions of elements can be grouped into relatively few boxes which can be tested for intersection, then only test the elements within intersected boxes.

One obvious short coming of the individual photon tracing method described above is that if a photon is absorbed, it makes no contribution to the signal and all the computational effort has

been in vain. A more computationally efficient method is to attenuate a ray at each interaction rather than completely absorb it. Rays are then “bundles of photon paths” rather than individual photons (Lewis 1999).

Another acceleration method is known as “fictitious flight”. With this, rather than waiting for a ray to happen to scatter back to the detector, at each interaction a line is traced to the detector and if the point is visible a contribution is added (Ross and Marshak 1988). If reverse ray tracing is to be used exactly the same process is carried out but with lines back to the illumination at each interaction, a slightly more complex process (Disney et al. 2000).

This concept was further developed into the “ray spreading” method to speed calculations of bidirectional reflectance (Thompson and Goel (1998) cited in Liang (2004)). Periodically an interaction is spread out into a wavefront travelling in all directions. This contributes to the reflectance across the whole hemisphere making calculation of the brdf faster than with traditional ray tracers. It is not clear whether this technique would be of any benefit to simulations of lidar, where the illumination and detector are highly directional.

2.2.4 Ray tracing of plants

Monte Carlo methods were first suggested for solving radiative transfer in vegetation canopies in the late 1960s and early 1970s, seemingly independently by Smith and Oliver (1974) and Tanaka (1969) (cited in Ross and Marshak (1988)). Due to the computational expense of the method the early work represented plant canopies with turbid media, the Monte Carlo sampling dealing with multiple scattering.

As computers became more powerful, simulations with explicit geometric models models be- came possible (Ross and Marshak 1988). One major advantage Monte Carlo ray tracing has over other explicit 3-D methods is that the number of rays to be traced and the maximum number of interactions are set independently of the scene. This means that computation time does not scale directly with complexity as it does for radiosity (Liang 2004) making it preferable for complex or dense canopies.