9-11 Nov. 1999 
.0, -0.2}. The correlation 
nted in table 5. 
| 0.190221 
| 0.229318 
1.000000 
for opposing slopes 
indicates a well-balanced 
ng 30% slope we obtain a 
in 20%, indicating that the 
ident of one another. In 
re reliable. Introducing a 
ariances for the parameters 
alent to +5m on the ground 
ce of dr = +0.69m for the 
n (SD) is on the order of 
> error of £5m we obtain a 
e effect of the position bias 
roduced a bias of 500m in 
ged only few millimeters. 
ive parameter model, the 
on diverged for a good set 
| is just another indication 
rameters. We also checked 
g error (50m). Using the 
| still converged to the true 
ation both by reducing the 
tude of the slopes. For a 
d following slopes {-0.2, 
elation was reduced to the 
0.070234 
-0.192688 
1.000000 
rfaces with opposing, steep 
ut this is still a reasonable 
y = 12.8, and the variances 
nd dr = 10.83. Remaining 
inimal configuration) and 
st - {-0.15, 0:151, 10.15, 
wing figures. 
0.095204 
-0.036995 
1.000000 
surfaces with opposing, 
3s 
   
International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 3W14, La Jolla, CA, 9-11 Nov. 1999 
The correlation, presented in table 7, was at most 10% and the 
condition number rose to 91.8. This is to be expected because 
of the decrease in the slope. Having almost an uncorrelated 
solution implies that each parameter is being solved 
independently. The variances are the following: dg, dw = 
12.6”, dr = +0.86m. 
Reducing the slopes even further {-0.1, 0.1}, {0.1, 0.1], (0., - 
0.07} still yields a good solution. The condition number is 200, 
but the correlation does not exceed 15%. Although the 
significance of the parameters decreases, their independence is 
still maintained. The variances for the mounting biases rise to 
de, dw = £4” while the ranging bias variance did not change. 
The rise of the mounting bias variances is a direct effect of 
reducing the surfaces slope. This makes sense because flat 
surfaces do not provide good support for these values. The rise 
in the condition number presents another implication of the 
slope decrease, at the limit when the slopes are 0 the condition 
number will approach infinity. 
The decrease in the correlation between the parameters due to 
the opposing surface trends shows that the surface topography 
is indeed the dominant factor affecting the robustness of the 
solution. The increase in the mounting bias variances as the 
slope decreases shows that steep slopes provide a better 
accuracy estimate for these parameters. However, steep slopes 
raise practical problems. Opposing trends are realistic 
requirements since any rise in elevation is followed by a 
descend. Even gentle slopes provide good solutions and it 
should not be too difficult to find suitable calibration sites. As 
for the size of the calibration site, the presented results were 
obtained with 12-18 laser shots, i.e., surface segments that do 
not exceed the order of hundreds of meters: (this is since the 
spacing between two consecutive shots for GLAS is ~160m). 
This would imply a 2.4km calibration site. We conclude that 
the size of the calibration site has no direct effect on the 
robustness of the solution but more on the algorithm and on the 
reliability of the estimated parameters. 
5. CONCLUSIONS 
The two prevailing problems encountered in calibrating 
airborne and spaceborne laser ranging systems are the unknown 
correspondence between laser surface and control surface, and 
the non-redundant determination of laser points. The latter 
circumstance causes some of the calibration parameters to 
become highly correlated; and the solution is very sensitive to 
the surface shape of the calibration site. In this paper we 
proposed an algorithm that utilizes natural terrain to resolve the 
calibration parameters. The method solves the unknown 
correspondence between laser and control surface by an 
adaptive coarse-to-fine segmentation of the terrain and by 
sequential refinement of the calibration parameters. Since not 
all parameters can be resolved simultaneously (due to their 
correlation), we derived an appropriate model and analyzed the 
parameter dependencies. Although the derived model is 
general, we have applied it in this paper for a spaceborne 
profiler. 
The experiments included planar surfaces but it is simple to 
extend the algorithm to include other surfaces, such as 
quadratic or higher order surfaces. We have computed 
correlation coefficients among calibration parameters to 
express parameter dependencies more quantitatively. It is well 
known that GPS timing errors and correction to the GPS 
position have similar effects and are highly correlated, but it is 
important to understand, how this correlation changes as a 
function of the surface topography of the calibration site. The 
results reported in this paper are in close agreement with a 
study by [Schenk, T. 1999]. 
Experiments with the proposed calibration method 
demonstrated that natural surfaces with moderate slopes but 
oriented in different directions are perfectly adequate to solve 
the calibration parameters. The key is that such surfaces reduce 
the correlation between parameters to negligible values. The 
compelling conclusion is that natural terrain with slopes in 
different direction is suitable for in-flight calibration, yielding 
results that are accurate and robust. 
The derived model can be extended without much effort to 
solve calibration of other laser altimeter configurations, such as 
laser scanners. We intend to extend the model to include other 
scanning systems. In addition we plan to incorporate the 
returned waveform signal into the calibration scheme. For this 
purpose we have developed a 3-D waveform simulator that 
copes wit any type of terrain. We are also analyzing further the 
nature of the deformation caused by systematic errors in order 
to refine the mathematical model for the transformation. 
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