9-71 Nov. 1999 
(cluded from processing 
istics. 
es for peak positions, 
based on the first and 
signal, with exception 
s. 
on fitting to obtain first 
tudes, with exception 
1s. Editing peaks based 
)ximity (zero amplitude 
exception handling for 
icance if necessary, and 
ocation, and half-width. 
> start of the waveform 
peaks, 2) centroid of the 
fined as follows: 
(2) 
(3) 
ting the model, and the 
ude 
Y; is the observations 
the 1-sigma uncertainties 
; are calculated in the 
(4) 
in Y, then the total chi- 
(5) 
fit Gaussian distributions 
dded, user-supplied IDL 
ardt, NASA/GSFC Code 
Aarquardt technique as a 
h for the best fit in the 
International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 3W14, La Jolla, CA, 9-11 Nov. 1999 
parameter space (Bevington and Robinson, 1992). A Gaussian 
distribution is used because SLA does not digitize the shape of 
the transmitted pulse. Lacking information on the shape of the 
transmitted pulse on a per shot basis, a Gaussian distribution is 
used as a reasonable approximation. A non-Gaussian, user- 
supplied fitting function can be input to MPFIT. Updated 
versions can be found on  http://astrog.physics.wisc.edu/- 
-craigm/idl.html. ^ MPFIT uses the Levenberg-Marquardt 
technique to solve the least-squares problem. Within certain 
  
constraints, these routine will find the set of parameters which 
best fits the data in the least-squares sense; that is, the sum of 
the weighted squared differences between the model and data is 
minimized by minimizing the Chi-square value, calculating 
derivatives numerically via a finite difference approximation. A 
set of starting parameters are specified, and the user can apply 
constraints to individual parameters by setting boundaries on 
the lower and/or upper side. Otherwise, it is assumed that all 
parameters are free and unconstrained. The step size to be used 
in calculating the numerical derivatives of the model with 
respect to the parameters can be defined by the user or it is 
computed automatically, and the covariance matrix can also be 
computed. 
The following constraints are applied during the fitting search: 
1) lower bounding constraint that all return position, amplitude 
and width parameters be non-negative; 2) upper and lower 
bounding constraints on each peak position such that its final 
position can not be outside the estimated position bounded by 
its half-width (stopping peaks from migrating away into larger 
adjacent peaks); 3) if detector saturation occurs, only that part 
of the signal up to the saturation point is fit, the estimated peak 
position is seeded earlier in time to compensate for the falsely 
broadened return, and the area under the saturated peak is 
preserved in the Gaussian fit; 4) if the digitizer 8-bit dynamic 
range is exceeded causing clipping of the signal, then the signal 
before and after clipping is fit making up for the clipped part of 
the signal with reasonable accuracy; 5) shots with more than 10 
peaks were excluded as they were difficult to fit, usually lacking 
convergence after 40 iterations (these are probably low-lying 
cloud returns mis-classified as ocean or land surface returns). 
4.2. Deriving Elevations from the Waveform 
The SLA ranging electronics provide the range from the start of 
the transmit pulse to the start of the backscatter return. It is this 
range that is used in the geolocation processing and, thus, the 
location of the geolocated bounce point (latitude, longitude, 
and elevation) refers to the highest detected feature within the 
100 m diameter laser footprint. The distances from the start of 
the waveform signal to the centroid of all the peaks, the 
centroid of the last peak, and the end of signal, all provided in 
the SLA-02 data structure, can be used to correct the bounce 
point elevation depending on the character of the return signal 
(e.g., single or multi-peaked), the assumed nature of the surface 
type, and the intent of the end user. 
For a measure of the mean elevation of illuminated surfaces 
within the laser footprint (assuming uniform reflectance of all 
the surfaces at the 1064 nm laser wavelength) the appropriate 
range would be from the centroid of the transmit pulse to the 
centroid of the backscatter return. An approximation of this 
mean elevation is obtained by: 
[geolocated bounce point elevation] + [transmit pulse 
centroid] — [centroid distance for all return peaks] (6) 
The transmit pulse centroid corresponds to the distance from 
the start to the centroid of the transmit pulse. SLA does not 
provide a digitized record of the transmit pulse. However, the 
transmit pulse impulse response can be obtained from returns 
from flat, smooth surfaces such as water, defining the narrowest 
possible returns which have not been broadened in time by 
surface relief. The impulse response pulse width is a function 
of peak amplitude. Examination of a plot of waveform centroid 
versus peak amplitude for single-peak returns defines an 
envelope of data points, with the minimum boundary defining 
the impulse response centroid which varies from 2 meters for 
low amplitude returns to 4 meters for high amplitude returns. 
For multiple-peaked waveforms where it is assumed that the 
last waveform peak is due to laser energy backscattered from 
the ground and that preceding peaks are energy returned from 
higher surfaces such as vegetation or buildings, the mean 
elevation of the ground surface can be approximated by: 
[geolocated bounce point elevation] + [transmit pulse 
centroid] — [centroid distance to last peak] (7) 
Inference that the last peak corresponds to the ground surface 
within a 100 m diameter footprint requires that the height 
distribution is simple, as for example due to an open vegetation 
canopy or building above a flat ground surface. Ground slope 
across the footprint can cause convolution of the ground return 
with returns from overlying surfaces. 
For a measure of the lowest detected surface within a footprint, 
the appropriate range is the distance from the start of the 
transmit-pulse to the end of the waveform signal, with a 
correction for the full width of the transmit pulse impulse 
response. This elevation can be approximated by: 
[geolocated bounce point elevation] + 2%*[transmit pulse 
centroid]— [distance to end of waveform signal] (8) 
These calculations assume the laser vector is at nadir. For off- 
nadir pulses, the distances along the laser vector to the 
centroids and end of signal should be modified by multiplying 
by the cosine of the off-nadir pointing angle, although this 
modification is very small for the near-nadir SLA observations. 
The waveform processing and resulting products thus provide a 
means to correct the SLA first return elevation to a mean 
elevation for illuminated surfaces, a mean elevation of the last 
return, and the elevation of the lowest return. The appropriate 
use of these elevations will depend on the assumed character of 
the surface within the laser footprint and the intent of the user.