The Weight Matrix Determination of
Systematic Bias Calibration for a Laser Altimeter
Ma Yue, Li Song, Lu Xiushan, Yi Hong, Zhou Hui, and Cui Tingwei
Abstract
The geolocation accuracy of satellite laser altimeters is sig-
nificantly influenced by on-orbit misalignment and ranging
biases. Few researchers have investigated the weight matrix
determination method, which plays a critical role in bias
estimation. In this article, a systematic misalignment and
ranging bias model was deduced. Based on the least squares
criterion, a bias calibration method was designed for use with
solid natural surfaces; and the weight matrix was defined
according to the ranging uncertainty theory. Referring to the
Geoscience Laser Altimeter System (
GLAS
) parameters, the es-
tablished model and method were verified using programming
simulations, which indicated with a misalignment of tens of
arc-seconds in the pitch and roll directions and a ranging bias
of several centimeters, by using the weight matrix, the estima-
tion accuracies of the misalignment and ranging bias increased
by 0.22″ and 2 cm, respectively. Consequently, the geolocation
accuracy increased by approximately 0.64 m horizontally and
3 cm vertically for a 1° sloping surface.
Introduction
The surface topography of Earth and other planets can be ac-
quired by satellite laser altimeter systems. The range between
a satellite and target is calculated by measuring the round-trip
propagation time of laser pulses, which are reflected from
the surface and combined with the position, attitude, and
pointing information; the elevation of a footprint illuminated
by a spaceborne laser can be determined from the sum of the
ranging and position vectors (Gardner, 1992; Schutz, 2002).
Compared with radar altimeters, laser altimeters have a small-
er beam divergence angle, which gives a smaller instantaneous
field of view (
IFOV
) and more accurate surface elevation measure-
ments. Moreover, the parameters of received waveforms, such as
pulse widths and peaks, are influenced by the slope, roughness,
and reflectivity of a surface. Therefore, the target information can
be inversed through waveform processing (Brenner
et al
., 2011).
Laser altimeters represent an important development direction
in remote sensing and Earth observation at high precision.
For a laser altimeter at an altitude of several hundred
kilometers, boresight misalignment will induce significant
horizontal geolocation errors and large elevation errors for
surfaces with moderate slopes, and it is typically the dominant
error source. The Geoscience Laser Altimeter System (
GLAS
) on
the Ice, Cloud and land Elevation Satellite (
ICESat
) is the first
polar-orbiting satellite laser altimeter used for Earth observa-
tion, and its derived elevation accuracy is less than 13 cm on
ice-sheet surfaces with small slopes (Schutz
et al
., 2002). How-
ever, a 30 arc-second misalignment in attitude for an
ICESat
altitude (600 km) instrument with a 1° incident angle on the
sloping surface will produce significant geolocation errors on
the order of 87 m horizontally and 1.5 m vertically (Luthcke
et
al
., 2002).
Therefore, various calibration/validation experiments
involving misalignment and ranging bias were performed
when
GLAS
was in orbit. One approach was based on the
GLAS
range-residual method, using commanded spacecraft attitude
maneuvers and ocean range scans (Luthcke
et al
., 2000 and
2005). One
in situ
technique used multiple ground-based
detectors and corner cubes to capture the laser footprints (Ma-
gruder
et al
., 2005, 2007, and 2010). Other techniques used
natural solid surfaces, such as salt flats (Fricker
et al
., 2005),
deserts (Martin
et al
., 2005), and vegetation (Harding
et al
.,
2005), as calibration sites. A calibration/validation method
based on natural surfaces is highly cost effective, and many
researchers have established calibration models for systematic
misalignment and ranging bias. Moreover, accurate Digital
Elevation Models (
DEM
s) of the Lunar and Mercury cannot be
surveyed on the ground as
a priori
to calibrate the misalign-
ment; thus,
LOLA
and
MESSENGER
altimeters use other methods
(Sun
et al
., 2014 and 2015). However, fewer researchers have
employed the weight matrix determination method, which
plays a critical role in bias estimation.
The purpose of this research is to determine whether a
weight matrix derived from the ranging uncertainty improves
the estimation accuracy of systematic bias parameters and
causes specific improvement in the geolocation accuracy
of a laser altimeter. The calibration model of the systematic
bias should be given first because the weight matrix cannot
be used without the calibration model. The weight matrix
will be derived for use in the calibration. Before a simulated
calibration process is performed to evaluate the effect of the
weight matrix, the normal coefficient matrix will be expressed
to provide guidance regarding choosing calibration surfaces
for the simulated calibration process.
Methods
Calibration Model of Systematic Bias
In this subsection, the derivation of the systematic bias cali-
bration model of misalignment and ranging is outlined based
on the footprint geolocation theory. The geolocation of laser
footprints is calculated and converted from the laser beam
reference frame to the geocentric reference frame. Based on the
analysis of Filin (2003 and 2006), the footprint can be modeled
in the local vertical reference frame for simplicity because the
rotation matrix from the local vertical reference to the ellipsoi-
Ma Yue, Li Song, Yi Hong, and Zhou Hui are with the School
of Electronic Information, Wuhan University, Wuhan 430079,
China (
).
Lu Xiushan is with the Institute of Ocean Engineering, Shan-
dong University of Science and Technology, Qingdao 266590,
China.
Cui Tingwei is with the First Institute of Oceanography, State
Oceanic Administration, Qingdao 266061, China.
Photogrammetric Engineering & Remote Sensing
Vol. 82, No. 11, November 2016, pp. 847–852.
0099-1112/16/847–852
© 2016 American Society for Photogrammetry
and Remote Sensing
doi: 10.14358/PERS.82.11.847
