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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