where
V
f
=(
x
,
y
,
z
) is the location of the footprint in terrestrial
reference system,
V
g
= (
x
g
,
y
g
,
z
g
) is the location of the phase
center of
PS
in terrestrial reference system,
R
t
is the rotation
from the celestial reference frame to the terrestrial reference
system,
R
c
is the rotation from the attitude reference system
to the celestial reference frame,
R
a
is the rotation from the
positioning reference system to the attitude reference system,
R
s
is the rotation from the laser altimeter reference system to
the positioning reference system,
R
l
is the rotation from the
laser pointing reference system to the laser altimeter reference
system,
r
l
= (
x
l
,
y
l
,
z
l
) is the vector from the phase center of
PS
to the reference point of laser altimeter in the positioning
reference system,
P
m
= (0, 0,
ρ
) is the laser range vector from
the reference point of laser altimeter to the laser footprint in
the laser pointing reference system,
ρ
is the laser range value.
The 3 × 3 transformation matrix
R
t
related with precession,
nutation and changes in Earth’s rotation can be resolved by
UT1 and Earth spin axis components
[24]
.
If we define that
M
x
(
θ
x
),
M
y
(
θ
y
) and
M
z
(
θ
z
) are respectively
the rotation about x axis, y axis and z axis in rectangular co-
ordinate system by
θ
x
,
θ
y
and
θ
z
angle, the rotation matrices in
Equation 2 can be rewritten as:
R
t
=
M
y
(
ϕ
),
R
s
=
M
z
(
α
z
)
M
y
(
α
y
)
M
x
(
α
x
),
(3)
R
a
=
M
z
(
β
z
)
M
y
(
β
y
)
M
x
(
β
x
),
R
c
=
M
z
(
θ
)
M
y
(
φ
)
M
x
(
ω
)
Here, the laser pointing angle
Φ
is the acute angle between
the nadir direction and laser beam center line,
ω
,
φ
, and
θ
are
the roll, pitch and heading angles of the platform. The angular
components of (
α
x
,
α
y
,
α
z
) and (
β
x
,
β
y
,
β
z
) respectively denote
the mounting angle biases of laser altimeter and OS relative to
the positioning system, which are usually confined at insig-
nificant level by pre-launch alignments to keep their coordi-
nate axes parallel approximately with each other.
Because of the restriction of calibration and performance
level for sensors, several kinds of errors including the mount-
ing errors and collected data errors may be introduced into
LFG
in Eq. (2). When adding these error components, we can
obtain actual
LFG
V
f
*
by the following equation
V
*
f
,
V
g
+
Δ
V
g
+
R
t
Δ
R
c
R
c
Δ
R
a
R
a
[(
r
l
+
Δ
r
l
)+
Δ
R
s
R
s
Δ
R
l
R
l
(
P
m
+
Δ
P
m
)] (4)
where
Δ
V
g
= (
Δ
x
g
,
Δ
y
g
,
Δ
z
g
) is the location error of the phase
center of
PS
,
Δ
r
l
= (
Δ
x
l
,
Δ
y
l
,
Δ
z
l
) is the displacement vector
error of laser altimeter reference point, and
Δ
P
m
= (0, 0,
ρ
+
Δ
ρ
)
is the range vector error of laser altimeter,
Δ
ρ
is the laser
range error. The angle error matrices of
Δ
R
c
,
Δ
R
a
,
Δ
R
s
and
Δ
R
l
corresponded respectively to attitude angle error, mounting
angle errors and laser pointing angle error can be simplified
since these angle errors are extremely slight. If we define f
(
δ
x
,
δ
y
,
δ
z
) as the error matrix function about
δ
x
,
δ
y
, and
δ
z
with the
following expression:
δ
δ
, ,
f
M M M
z
y
x
δ δ δ
δ
δ δ
δ
δ
δ δ
x y z
z
y
x
z
y
z
x
y
x
(
)
=
( )
( )
( )
=
−
−
−
1
1
1
, (5)
and then we can describe the angle error matrices by intro-
ducing the angle errors corresponding to the angles in Equa-
tion 3 as:
(
)
θ
β
,
,
∆
∆ ∆ ∆ ∆
∆ ∆ ∆
R f
R f
c
a
=
=
(
)
ω φ
β β
, ,
,
x
y
z
R f
s
l
z
x
α
ϕ
,
,
∆
∆ ∆ ∆ ∆
∆ ∆ ∆
R f
=
(
)
=
(
)
α α
ϕ ϕ
x
y
y
z
,
,
,
.
(6)
Therefore, the absolute error of
LFG
is directly calculated
by subtracting equation 2 from Equation 4, which is given by:
(
)
R R
(
)
+
+
∆
∆ ∆
∆ ∆ ∆
∆ ∆ ∆
V R R R R R R R R R R R
R R R R R
f
c c a c a l
c c a a l
t
c c a a s s
=
r +
r
+
t
t
−
∆
∆ ∆ ∆ ∆ ∆ ∆
R R R R R
R R R R R R R R R
l
c a s l
t
c c a a s s
l l
m
−
P
P
V
m
g
.
(7)
Actually, the displacement vector error
Δ
r
l
can be achieved
at millimeter level by pre-launch calibration, which is much
less than
Δ
V
g
and
Δ
P
m
components. Meanwhile, the range vec-
tor
P
m
is much greater than the displacement vector
r
l
due to
the orbital altitude of several hundred kilometers. Then, the
contribution of displacement vector and its error vector to the
geolocation of laser footprint might be neglected. Consequent-
ly, we rewrite the Equation 7 with the Cartesian coordinate
components (
Δ
x
f
,
Δ
y
f
,
Δ
z
f
) by merging all matrix elements as:
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
x m x m y m z
y m x m y m z
z m x
f
fc
fc
fc
f
fc
fc
fc
f
fc
=
+
+
=
+
+
=
1
2
3
4
5
6
7
+
+
m y m z
fc
fc
8
9
∆
∆
,
(8)
where
m
i
(
I
= 1,2,3...9) is the matrix element of
R
t
, and the
LFGerror of (
Δ
x
fc
,
Δ
y
fc
,
Δ
z
fc
) in the celestial reference frame
can be described by:
3
2
n
s
a
a
∆
∆
+
+
1
3
(
)
a
a
ϕ
ϕ
n c
(
)
s
c
∆
∆
∆
∆
x
a
a
a
a
a
fc
x
y
z
= −
(
)
+
−
(
)
+
(
)
{
+ −
ρ
ϕ γ
ϕ
ϕ γ
ϕ γ
ρ
2
1
co
os
si
in
4
6
7
9
sin cos
sin cos
si
os
ϕ
ϕ θ
ϕ
ϕ φ
ρ
−
+
+
(
)
a
∆
+ ∆
x
gc
,(9)
n
s
a
a
+
+
(
)
a
a
∆
∆
(
)
s
c
∆
∆
∆
∆
z
a
a
a
a
a
fc
x
y
z
= −
(
)
+
−
(
)
+
(
)
{
+ −
ρ
ϕ γ
ϕ
ϕ γ
ϕ γ
ρ
8
7
9
8
co
os
si
in
1
3
4
6
7
9
sin cos
sin cos
sin cos
ϕ
ϕ φ
ϕ
ϕ ω
ϕ
ϕ ρ
−
+
+
(
)
a
∆
+ ∆
z
gc
.
,(10)
n
s
a
a
+
+
(
)
a
a
∆
∆
(
)
s
c
∆
∆
∆
∆
z
a
a
a
a
a
fc
x
y
z
= −
(
)
+
−
(
)
+
(
)
{
+ −
ρ
ϕ γ
ϕ
ϕ γ
ϕ γ
ρ
8
7
9
8
co
os
si
in
1
3
4
6
7
9
sin cos
sin cos
sin cos
ϕ
ϕ φ
ϕ
ϕ ω
ϕ
ϕ ρ
−
+
+
(
)
a
∆
+ ∆
z
gc
.
.(11)
Here
a
i
(
I
= 1,2,3...9) is the matrix element of
R
c
,
Δ
γ
j
(
j
=
x
,
y
,
z
) is the cumulative angle error,
Δ
γ
j
=
Δ
ϕ
j
+
Δ
α
j
+
Δ
β
j
, and
(
Δ
x
gc
,
Δ
y
gc
,
Δ
z
gc
) is the location error of
PS
in the celestial refer-
ence frame.
The
LFG
error is determined mainly by the errors of the
laser pointing angle, the mounting angle and the attitude
angle, the range error, and the
PS
location error. Considering
that the influence of these error factors on
LFG
are mutually
independent, we can again express
LFG
error (d
x
f
, d
y
f
, d
z
f
) in
the terrestrial reference system by using root mean square er-
ror (
RMSE
) and the law of error propagation [25] as
dx m dx m dy m dz
dy m dx m dy m d
f
fc
fc
fc
f
fc
fc
2
1
2 2
2
2 2
3
2 2
2
4
2 2
5
2 2
6
2
=
+
+
=
+
+
z
dz m dx m dy m dz
fc
f
fc
fc
fc
2
2
7
2 2
8
2 2
9
2 2
=
+
+
,
(12)
where the
RMSE
of the
LFG
(
dx
fc
,
dy
fc
,
dz
fc
) in the celestial refer-
ence frame can be given according to Equations 9, 10, and 11
by the following:
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
October 2018
649
