n
c
+
+
(
)
a
a
n c
+
+
(
)
(
)
n
c
n
s
s
c
dx a
d a
a
d a
d
fc
x
y
z
2
2
2 2
1
3
2 2
2
2
=
(
)
+
−
(
)
+
(
)
+
ρ φ γ
ρ ϕ ρ ϕ γ
ρ ϕ γ
co
os
si
in
a
a
d a
a
d
4
6
2 2
7
9
2 2
1
3
ρ ϕ ρ ϕ θ
ρ ϕ ρ ϕ φ
ϕ
ϕ
si
os
si
os
si
os
+
+
2 2
2
d d
gc
ρ x
,(13)
n c
(
)
n
c
+
+
(
)
1
3
a
a
ρ ϕ
n
c
+
−
n
s
d a
x
y
(
)
ϕ γ
s
c
dy a
d a
a
d
fc
z
2
5
2 2
4
6
2 2
5
2 2
=
(
)
+
(
)
ρ ϕ γ
ρ ϕ ρ
ρ ϕ γ
co
os
si
in
+
+
(
)
+
+
d a
a
d
a
a
2 2
7
9
2 2
4
6
ρ ϕ θ
ρ ϕ ρ ϕ ω
ϕ
si
os
si
os
si
os
ϕ ρ
+
2 2
2
d dy
gc
,(14)
n c
(
)
n
c
n
c
n
s
s
c
dz a
d a
a
d a
d
fc
x
y
z
2
8
2 2
7
9
2 2
8
2 2
=
(
)
+
−
(
)
+
(
)
ρ ϕ γ
ρ ϕ ρ ϕ γ
ρ ϕ γ
co
os
si
in
+
+
(
)
+
+
(
)
+
+
a
a
d a
a
d
a
a
1
3
2 2
4
6
2 2
7
9
ρ ϕ ρ ϕ ϕ
ρ ϕ ρ ϕ ω
ϕ
si
os
si
os
si
os
ϕ ρ
+
2 2
2
d dz
gc
.
.(15)
Here,
d
γ
j
(
j
=
x
,
y
,
z
) is the
RMSE
of the cumulative angle
error,
d
γ
2
j
=
d
ϕ
2
j
+
d
α
2
j
+
d
β
2
j
, and
d
ρ
is the
RMSE
of laser range.
When the surface slope and roughness at the original geoloca-
tion and actual geolocation are comparable, if we only take
into account the effects of dominating photon noise and laser
pointing error on the range error, and then the
RMSE
of laser
range can be expressed simply [26].
(
)
+
+
2
2
(
)
2
2
d
c F
N
t
c
s d
s
s
p
x
ρ
σ
ρ
ϕ
ϕ
ϕ
=
+
+
⊥
2
12
4
2
2
2
2
2
∆
tan
tan cos cos
co
//
//
s
//
2
2
ϕ
ϕ
+
s
d
y
,(16)
where
F
is the detector excess noise factor,
Δ
t
is the sampling
interval of A/D converter,
c
is the speed of light in vacuum,
s
and
s
^
are the surface slope angles parallel and normal to
nadir directions.
N
and
σ
p
are the detected photon count and
the
RMS
pulse width [27]:
4
4
N
E T A
E
c
s
s
r d t a r
p
t
f
h
=
= + +
+
(
)
+
η η β
πρ
σ σ σ
σ
ϕ
2
0
2
2
2
2
2
2
2
2
,
cos
cos
//
//
ρ
θ
2
2
2
2
tan
tan
t
c
s
,(17)
where
η
r
and
η
d
are the efficiency of the receiver optics and
detector respectively,
T
a
is the one-way atmospheric transmit-
tance,
β
is the diffuse reflectivity of target,
A
r
is the receiver
aperture area,
E
t
and
E
0
are the transmitted laser energy of
single pulse and photon,
σ
t
and
σ
f
are the
RMS
pulse width of
the transmitted laser pulse and electrical filter, and
σ
h
is the
standard deviation of surface roughness within laser foot-
print,
θ
t
is the
RMS
of laser divergence angle,
s
corresponding
to a composite slope angle has the following expression:
s
s
s
s
s
=
+
(
)
+
+
(
)
⊥
arctan tan
tan cos
cos
//
//
//
2
2
2
2
ϕ
ϕ
.
(18)
In general, the
RMSE
of
LFG
would be described by the com-
ponents of horizontal error (
dx
f
,
dy
f
) and a vertical error
dh
.
The vertical error can be defined as the elevation error along
the nadir direction of satellite when laser pointing angle is
small [5]. Therefore, we may utilize the abovementioned
method to derive the expression of the vertical error based
on the approximation that the height can be expressed as the
z-coordinate in satellite positioning reference system, which
can be calculated by
s
s
dh d
d d dz
y
y
s
2
2
2
2 2
2
2
2
=
+
+
(
)
+
ρ
ϕ ρ
ϕ ϕ α
co
in
,
(19)
here
dz
s
is orbital determination error of platform in radical
direction. According to the mathematical expressions of the
range error, if all components of laser pointing angle error are
equal, we define a new angular variable
d
Φ
as:
d
ϕ
=
d
ϕ
x
=
d
ϕ
y
=
d
ϕ
z
.
(20)
Thus, we can rewrite the vertical error based on the clas-
sification of impact factors, which is described as
n t
+
+
d
s
n c
(
)
dh
c
F
N
t
F
N
s
t
f
h
2
2
2
2
2
2
2
2
2
4
12
=
+
(
)
+
+
cos
cos
cos
cos
//
ϕ σ σ
ϕ σ
∆
2
2
2
2
2 2
2
2
2
ϕ
ρ
θ
ρ ϕ
ϕ ϕ
ϕ
+
+
+
(
s
s
t
//
//
ta an
si
os tan
)
+
+
(
)
+
⊥
tan cos cos
cos
sin
//
//
2
2
2
2
2 2
s
s
s
ϕ
ϕ
ρ
ϕ
d dz
y
s
α
2
2
+
(21)
The vertical error is made up of five terms orderly caused
by the laser altimeter system error, the pulsewidth broadening
effect, the laser pointing uncertainty, the instrument misalign-
ment, and the position error.
Data Set
Based on the error models of
LFG
, we find that many factors
including the specifications of the
GLAS
, various sensor errors
and terrestrial features would affect the horizontal errors and
vertical error of
LFG
. To simulate the distributions of the
LFG
error versus the surface slope and roughness and verify the
error models, we describe the data set as the following.
The performance of the
GLAS
depends primarily on the
specifications [28] of its transmitter and receiver. The
GLAS
system emits a laser pulse with 1064 nm wavelength, 6 ns
pulsewidth, 75 mJ average pulse energy, and 110 µrad diver-
gence angle at a 40 Hz repetition rate to measure diverse in-
formation of the Earth’s surface. The returned echo is received
by a Cassegrain telescope with aperture area of 0.638 m
2
and
optics transmission of 0.55, and then converted to the elec-
tronic signal by a silicon avalanche photodiode with quantum
efficiency of 0.35 and excess noise factor of 3.24. Finally, the
analog waveforms are mainly fed to a low pass filter with
impulse response width of 4 ns and sampled through a 1-
GHz
rate digitizer. All waveforms are recorded by onboard memory
and downlinked to the ground section for the postprocessing.
The sensor errors induce extra errors for locating the laser
footprint. According to the simulation results from the orbit
and attitude determination by
NASA
research teams, we can
assume that the errors about angle and position relevant to
LFG
may be assigned as the following [29; 30; and 31]: (1) the three-
axes error components are supposed to be respectively equiva-
lent , which includes the mounting errors (0.5 arcsecond) and
laser pointing error (1 arcsecond); (2) the attitude errors in the
roll, pitch and heading axes are individually allocated to be
1.6 arcseconds, 1.6 arcseconds and 3 arcseconds; (3) the posi-
tion of the
GLAS
instrument can be achieved at an accuracy of
5 cm and 20 cm in radial and horizontal components.
The surface slope and roughness depicting the geomor-
phic features of terrain have a dominant effect on the shape
of waveforms and the accuracy of
LFG
. Harding used the
DEM
data to classify nine terrains that can be adapted to the
various Earth science applications [20]. The terrain contains
different landforms, ranging from low-relief glaciated con-
tinental shield to high-relief convergent mountain front, as
shown in Table 1.
To verify the proposed error models of
LFG
,
GLAS
data are
utilized as the referenced data source to evaluate the model.
In the product data derived from the
GLAS
,
GLA01
contains raw
waveforms of received signal and
GLA14
gathers original
LFG
on land surface [32]. We identify the potential reference
LFG
as the best-match location with maximal correlation coeffi-
cient between raw
GLAS
waveforms and simulated waveforms
by using a waveform matching method. The simulated wave-
forms are generated with a waveform simulator that requires
the specifications of
GLAS
and
DEMs
coincident with
GLAS
650
October 2018
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
