In Monte Carlo ray tracing the number of rays, and so samples of the scene, are defined indepen- dently of the scene’s complexity; this is its main advantage over other realistic simulation methods (Disney et al. 2000). To ensure that the Monte Carlo results are representative of reality there must be a sufficient number of rays to adequately sample the scene and this number will be related to scene complexity. The number of rays required depends upon the variability of the scene at the scale of the field of view, if the scene is a single plane all rays will see exactly the same surface and return the same reflectance, so a single ray will suffice. For a scene as complex as a conifer forest many hundreds of rays will be needed to sample all the objects and curved surfaces.
In addition to the sampling of surfaces by directly reflected light (rays that have undergone a single interaction) multiple scattered light must also be adequately sampled. The intensity of multiple scattered light will depend upon the element reflectance and the probability of a multiple scattered ray returning to the detector (which in turn will depend upon the field of view, scattering element density and orientation of scatterers relative to each other). For computational efficiency the ray tree is truncated after a certain number of interactions (once the contribution from scattered light has become negligible).
To ensure an even areal sampling and because a single lidar beam’s measurement is not spatially explicit within the footprint (unlike a camera), the laser footprint was divided into segments and a set number of rays traced per segment. The segments were made to be of constant solid angle, or as near as possible without leaving gaps. The field of illumination was first divided into zenith annuli of a given angular resolution αres. These were then divided into azimuthal segments, each of
as close to the sine of αres as would fit an integer number of times into a full circle. This gives the
sampling pattern seen in figure 18. Rays were traced from the lidar scan centre through a random point within a segment. This random jittering allows different scans with the same resolution which may give different results if not enough rays are used. These segments could also be used as pixels to create a two dimensional picture, although the full three dimensional distribution of intensity was not recorded to reduce the memory requirements.
Figure 18: Starat’s sampling pattern
using so many rays as to make the simulations impractically slow.
Above canopy lidars The above canopy lidars used in this investigation (in section 4.3) will have 30m footprints, about the size of LVIS (Hofton et al. 2000) and the proposed DesDyni (Dubayah et al. 2008) instruments. These will be simulated over Sitka spruce and birch forests of different densities, ages and topographies (the creation of which is describes in section 4.2). The denser the forest the greater the scene complexity, also the larger the trees in the forest the greater their complexity. The coniferous trees used contained more elements than the birch and so were more complex. A sparse young Sitka spruce forest was used as an example of a simple forest whilst a dense old Sitka spruce forest was used to find the number of rays needed for the most complex scenes. The dense forest was used to set the number of rays; the sparse forest was tested to see how the required number of rays changed with complexity. Topography should not affect the number of rays needed to sample the scene. It does slightly spread out scattering elements which may have an impact on multiple scattering though this should cause a reduction in the contribution if at all.
Maximum interactions The number of rays needed to characterise a scene cannot be deter-
mined until it has been decided how many interactions to limit the ray tree to. Multiple scattered light is attenuated by the element reflectance to the power of the order of scattering, therefore to ensure that the simulation’s accuracy is never limited by ray tree truncation the wavelength with the brightest reflectance were used to choose the truncation point. Within the canopy (where the majority of multiple scattering will occur) there is a mixture of leaves and bark, therefore the
spectrum of interest will be a mixture of these two spectra, depending upon the probability of light scattering between the various elements. As leaves are by far the densest scattering elements in conifers (whilst bark can have as large a surface area as leaves this is made up of a few large elements) and so the majority of multiple scattering is likely to come from them. For the particular spectra used in this investigation the maximum leaf reflectance was at a wavelength of 920nm.
0 0.05 0.1 0.15 0.2 0.25 0.3 0 5 10 15 20 25 30 35 40
Cumulative energy from higher scattering orders
Scattering order
Figure 19: Fraction of energy at higher orders of scattering for a dense (99.9% canopy cover) forest
A set of twenty five simulations were run over different locations in a dense old and a sparse young Sitka spruce forest. Ray trees were allowed to include up to one hundred interactions (by which time there should be no contribution). The fraction of total intensity that would be truncated if the ray tree were limited to a maximum number of interactions (adding up over all ranges) was calculated for each footprint. The mean and standard deviation of the fractional contribution was calculated from these twenty five footprints.
No contributions were recorded above sixty interactions, so limiting to one hundred interactions has not affected the results’ realism in any way. From figure 19 it can be seen that for the maximum multiple scattering conditions (dense canopy, high reflectance) around 84% of the signal comes from singly scattered light. Closer examination reveals that limiting the ray tree to 30 interactions
would truncate only 6.5 × 10−3% of the total energy with a standard deviation of 5.3 × 10−3%, a
negligible fraction on the edge of a computer’s floating point precision. After fifty eight interactions this fraction dropped below the precision of computer doubles and cannot be measured by the ray tracer (and so is effectively zero).
it varies. For this case 86% of the signal comes from single scattered light and rays stopped contributing after 42 interactions (after thirty interactions the contribution was 1.5 × 10−3% with
a standard deviation of 5.2 × 10−3%). This shows that the number of interactions necessary to
sample a scene is related to tree density, but the fractional contribution of multiple scattered light is not. The majority of the multiple scattering contribution was from low orders of scattering, possibly within individual tree crowns so that tree density does not affect it.
Therefore limiting the ray tree to thirty interactions should not limit simulation accuracy and will not take too long to compute (a few hours for the most complex scenes on a 2GHz processor).
Number of rays Again the densest, oldest and so most complex forest was used to determine
the number of rays required to accurately characterise a 30m diameter footprint. With starat the number of rays was controlled by the number of segments in a footprint and the number of rays per sector. Care must be taken not to make the solid angle of the sectors too small, to avoid any danger of rounding issues in the jittering of rays within segments.
Simulations were run with different numbers of rays, at first with one ray per sector until the solid angle became small, at which point the sector size was limited and the number of rays per segment increased.
The literature reports a range of methods to ensure adequate sampling, however these are mainly for psycho-physical results (Koenderink and van Doorn 1996). Whether an image would look correct to a person is not of interest, only the reflectance with range. The method of (Hofton
et al. 2000) uses a cumulative threshold of 1% of the total energy to remove noise, therefore it
would be sensible to ensure that the measured intensity is correct to this level. For safety the accuracy should be higher than this, if computational expense allows.
From figure 20 it can be seen that the fractional variation was relatively small compared to the overall intensity, even down to a hundred rays. The variance began to flatten off at around five hundred rays reaching a plateau at around two thousand rays, suggesting that there is little point in using more than this many rays. For these forests one thousand rays would seem to be a sensible number. Not as many samples were available for the dense forest due to the extra computational expense but one thousand rays would seem to be sufficient.
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Standard deviation divided by mean
Number of rays (a) Sparse forest
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 100 200 300 400 500 600 700 800 900 1000 1100
Standard deviation divided by mean
Number of rays (b) Dense forest
Figure 20: Standard deviation divided by mean against number of rays for a 30m footprint over Sitka
spruce forests
Below canopy lidar The thesis will also include simulations with CSIRO’s Echidna terres-R
trial lidar (described in section 5.14). This has a much smaller footprint than the above canopy instruments, a beam divergence of between 2mrad and 15mrad from a starting diameter of 29mm (Jupp et al. 2009) giving a footprint of between 49mm and 179mm at a range of 5m and between 149mm and 929mm at a range of 30m and so fewer elements will contribute and fewer samples should be needed per beam.
Forests Echidna simulations take a considerable amount of time and computer resources to
complete. Even at the coarsest resolution (15mrad beam divergence) a full scan between -100o
and 100o zenith contains 48,740 individual beams, therefore only segments of the scans will be
investigated. The total canopy density is less of an issue than it is for large footprint lidar as the beam footprint will always be completely filled by a single tree and it is unlikely that light will be multiply scattered into an adjacent crown and back into the field of view. For this reason sparse canopies will suffice, using only small sectors covering trees.
The small footprint of Echidna means that some returns may come entirely from hard targets, such as trunks, which are unlikely to have any contribution from multiple scattering so the search for convergence was limited to diffuse targets (targets which have returns in multiple range bins). Comparing figure 21 to figure 19 it can be seen that the smaller footprint of Echidna leads to much smaller contributions from multiple scattering. In this case light makes no contribution after twenty interactions whilst limiting the ray tree to only five orders of scattering would truncate
(a) Hemispherical image -0.005 0 0.005 0.01 0.015 0.02 0.025 0 5 10 15 20 25 30 35 40
Cumulative energy from higher scattering orders
Scattering order
(b) Contribution split by scattering order
Figure 21: Intensity of contribution against scattering order for Echidna simulations of a sparse birch canopy at 920nm
only 6.4−3% of the energy, around the precision of a floating point.
As footprint size is proportional to range it is reasonable to assume that more distant targets may have more multiple scattering than closer targets. Examining a different sector with more distant trees (at 8.5m rather 7m) gave similar results, with multiple scattering contributions com-
pletely disappearing after twenty orders of scattering, although the fraction stayed above 10−3%
until seven orders of scattering, adding weight to the argument that that larger footprint sizes include more multiple scattering.
0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0 200 400 600 800 1000 1200 1400 1600
Maximum standard deviation as fraction of intensity
Number of rays
Figure 22: Maximum standard deviation between ten different samples against number of samples for an
Echidna simulation of a birch forest
Sitka spruce, with its needle shoots, should suffer much more multiple scattering than birch and on a much smaller scale. To save computations with the more complex scenes only a single sector was examined, containing distant trees.
(a) Whole scene -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 5 10 15 20 25 30 35 40
Cumulative energy from higher scattering orders
Scattering order
(b) Contribution split by scattering order
Figure 23: Intensity of contribution against scattering order for a Sitka spruce canopy at 920nm
the total energy. The contribution dropped below the precision of a double variable after thirty
six interactions and down to 4.8 × 10−3% with a standard deviation of 1.4 × 10−2% after thirteen
interactions. Therefore a limit of thirteen interactions will not affect simulation accuracy. In fact all orders of scattering beyond four contributed only 1% of total energy, so a much lower ray tree limit may be acceptable if computation time becomes an issue.
0 0.005 0.01 0.015 0.02 0.025 0 500 1000 1500 2000 2500 3000
Maximum standard deviation as fraction of intensity
Number of rays
Figure 24: Maximum standard deviation between ten different samples against number of samples for an
Echidna simulation of a Sitka spruce forest at 920nm
From figure 24 it can be seen that using one thousand samples would have a maximum error of around 0.5%. For a beam divergence of 15mrad this corresponds to segments of solid angle
4.8×10−8strad and 11 rays per segment. This sampling number will be used for all 15mrad beam
divergence simulations with Sitka spruce forests. At this sampling rate each beam took a maximum of 31 minutes and a mean of 8 minutes on a 2GHz processor. This is for the most complex portion of the forest and so it should be possible to complete a number of complete scans within a reasonable
time.
With this level of sampling and the lack of assumptions of ray tracing described in section 2.2.3 the ray traced signals can be taken as accurate representations of reality. Inversion algorithms can be tested as if the signals were real data and compare the results against a known truth, a feat impossible in reality (Pinty et al. 2001). Of course more intelligent sampling could be employed, with denser sampling around curved and complex areas than flat planes or empty sky, but this thesis is about understanding the physical processes that make up lidar signals, not about ray tracer development so the extra effort will not be expended, even though it means wasting some computer time.