The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
assumption that ignores the heterogeneity of real trees. It is not 
clear how such an assumption would affect derived results, 
especially when parameters are derived using the same 
assumptions used to create the forests (Widlowski et al, 2005). 
Figure 2 shows a simulated waveform over a single thirty year 
old Sitka spruce tree model. The heterogeneity and subterranean 
echoes caused by multiple scattering are apparent. The ground 
is at a range of 1,200m. 
Figure 2. Simulated waveform from a single Sitka spruce tree. 
Simulations were run with a resolution of 25cm (which can be 
coarsened for analysis), a range of wavelengths (including 
532nm, 850nm, 1064nm, 1650nm and 2060nm), a 30m ground 
footprint (the optimum for forestry, Zwally et al, 2002) and with 
and without a temporal laser pulse (100ns is proposed for A- 
scope). Gaussian noise can be added before analysis. The width 
of the Gaussian is defined as a percentage of the maximum 
signal return. The methods will be developed for an 
infinitesimally short pulse before the extra complication of 
deconvolution is added. Systems such as GLAS have short 
pulses (around 2ns) which should not need any deconvolution 
unless the range sampling is significantly finer. 
Derivation of parameters 
Estimation of forest parameters from lidar relies on the ground 
returns being distinguishable from the canopy returns (Hofton et 
al, 2002). This can either be achieved with multiple first return 
scans (Koetz et al, 2007) or a single full waveform measurement 
(Zwally et al, 2002) to get a distribution of returns from 
throughout the canopy. Due to the speed of spacebome 
platforms and the subsequent sparsity of sampling only full 
waveform lidar is suitable for measuring vegetation from space. 
The standard method is to decompose the waveform into a set 
of Gaussians by non-linear regression (Hofton et al, 2000). The 
distribution of Gaussians can be used to classify cover type 
(Wagner et al, 2008, Reitberger et al, 2008) and (taking into 
account relative cross sections) can be used to derive vegetation 
height (Blair et al, 1999), estimate canopy cover (Lefsky et al, 
2005) and through metrics derive other biophysical parameters 
such as leaf biomass and leaf area index (Lefsky et al, 1999). 
Often there is no clear separation of ground and canopy returns, 
either due to dense understory, small separation of canopy and 
ground or topography. Attempts have been made to improve 
height estimates in these situations by using another data source 
to estimate the ground position (Rosette et al, 2008). Care must 
be taken that these ground elevation datasets give the true 
height (for example SRTM saturates over forests). Accurate 
datasets are not available globally. 
Method 
This investigation explores methods that use only the waveform 
to estimate tree height (which can be linked to biomass through 
allometric relationships and stand counts). Other characteristics 
would need to be inferred with additional information and will 
not be investigated in this paper. Fusing lidar with hyper- 
spectral and multi-angular data would greatly help in the 
derivation of these biophysical parameters however the lidar 
waveform alone should provide the best height profile. 
Before Gaussians are fitted to the simulated signal it is pre- 
processed in the following order; 
• 5% Gaussian noise was added, as described above. 
• The signal was pre-smoothed by convolution with a 
3m Gaussian. 
• Noise statistics are calculated from a known empty 
portion of signal (above canopy to avoid echoes). 
• The signal was de-noised by subtracting a threshold of 
the mean noise plus (an arbitrary) three standard 
deviations 
• The signal was post-smoothed with a lm Gaussian. 
The empty tails are cropped from the signal to constrain the 
Gaussian decomposition. The positions and amplitudes of all 
turning points are recorded along with the width of peaks. If 
more features than the number of Gaussians to be fitted are 
found (due to heterogeneous or noisy signals) the Gaussians 
with the largest cross sections are used first. If too few are 
found (skewed Gaussians for example) the extra Gaussians are 
evenly spaced in the gaps. An implementation of the 
Levenberg-Marquardt method was used to minimise the root 
mean square difference between the fitted Gaussians and 
original signal (Press et al, 1994). It has been found that the 
best fits are achieved when the x and y axes are rescaled to 
between 0 and 100. 
The fitted Gaussians and the pre-processed signal were analysed 
to derive biophysical parameters. The centre of the last 
Gaussian is taken as the ground position if the energy contained 
within is more than an arbitrary percentage (1%) of the total 
energy. This should avoid any Gaussians fitted to noise or 
subterranean echoes caused by multiple scattering. If the 
Levenberg-Marquardt method fails to find a solution or the 
derived parameters are unrealistic the fitting is repeated with 
one less Gaussian. 
The tree top is calculated form the pre-processed signal. Taking 
it as the point at which the signal rises above the noise threshold 
will always lead to an underestimate of height. Data 
assimilation schemes such as the Kalman filter rely on unbiased 
observations (Williams et al, 2005). For this reason it may be 
better to try to estimate a point that could be an over or under 
estimate. The first point at which the signal drops to the mean 
noise level before it rises above the noise threshold would seem 
to be a sensible, unbiased estimate of tree top position. Figure 3 
shows a histogram of the signal start position error with and 
without tracking back from the noise threshold to the mean 
noise value. One hundred simulated waveforms were used with 
ten thousand separate sets of random noise added to give one 
million estimates. A negative error means a premature signal 
start; this was common in both methods.