vertical error produced by
d
φ
is given by:
(
)
σ
φ φ
α
φ
h
H
B
H d d
cos
=
−
2
1
2
(4)
Since
d
φ
1
and
d
φ
1
are independent from each other, Equa-
tion 5 can be obtained as the
RMSE
of the formula above:
σ
σ
α
φ
φ
h
H
B
H
cos
=
2
2
.
.
.
(5)
where
σ
φ
is the error of the satellite attitude measurements in
the pitch direction.
The vertical error induced by attitude measurement errors
in the yawing direction (Wang, 2006) is given by:
σ
h
κ
=
H
B
· Y
·
(
d
k
2
–
d
k
1
)
(6)
Since
d
φ
1
and
d
φ
1
are independent from each other, Equation
7 can be obtained as the
RMSE
of the formula given above as:
σ
h
κ
=
2
·
H
B
· Y
·
σ
κ
(7)
where
d
κ
1
is the attitude yawing angle error of camera sta-
tion-1,
d
κ
2
is the attitude yawing angle error of camera sta-
tion-2;
σ
κ
is the error of the satellite attitude measurements
in the yawing direction, and
Y
represents half of the satellite
image coverage width
.
The vertical error generated by the orbit measurements er-
ror is given by:
σ
h
GPS
=
d
gps
z
(8)
where
d
gps
z
is the positioning error of the satellite orbit mea-
surements in the vertical direction.
After the geometric calibration, the influence on the verti-
cal error of the calibration errors for the camera inner orienta-
tion elements is similar to the image matching error on the
vertical accuracy expressed as:
σ
h
calib_M
=
H
B
· d
calib_M
· pixel
(9)
where
d
calib_M
is the calibration error of the camera inner orien-
tation elements after the geometric calibration and less than
0.25 pixels (Jiang
et al.
2014; Zhang
et al.
2014)
.
The effect of the calibration errors of the installation angle
for on-board equipment on the vertical accuracy is similar to
the random error of the attitude measurements on the vertical
accuracy, with the vertical error in the pitch direction given by:
σ
σ
α
φ
φ
h
calib
calib
H
B
H
cos
_
_
=
2
2
.
.
.
(10)
Similarly, the vertical error in the yawing direction is given by:
σ
h
calib_k
= 2
·
H
B
· Y ·
σ
calib_
k
(11)
where
σ
calib_
φ
and
σ
calib_
k
are the calibration errors of the instal-
lation angle for on-board equipment in the pitch and yawing
directions, respectively
.
The integrated vertical
RMSE
is given by:
M
c
_
_
+
+
+ +
h
h
+ +
h
h
=
+
σ σ σ
σ σ σ
σ
σ
σ
ϕ
κ
φ
h
h
h
h
h
r
M
GPS
calib
alib
cali
2
2
2
2
2
2
2
b
_
κ
2
(12)
Planar Accuracy Estimation Model
The planar accuracy caused by the time synchronization error
is given by:
σ
p
T
=
d
T
· V
.
(13)
Similarly, the orbit and attitude measurements errors
induced by the time synchronization error are low and can be
neglected
.
The attitude measurement errors in the pitch, rolling, and
yawing directions affect the planar accuracy of the satellite
images. However, the error in the yawing direction is negli-
gible considering its minor effect on the planar accuracy. The
influence of the attitude measurements error on the planar
accuracy in the pitch direction is calculated as:
σ
p
φ
=
H ·
(tan(
φ
+
d
φ
) – tan(
φ
))
(14)
where
φ
is the pitch angle of the camera, and
d
φ
is the error of
the attitude measurements in the pitch direction. The influ-
ence of the attitude measurement error in the rolling direction
is calculated as:
σ
p
ω
=
H ·
(tan(
ω
+
d
ω
) – tan(
ω
))
(15)
where
ω
is the rolling angle of the camera and
d
ω
is the error
of the attitude measurements in the rolling direction
.
The effect on the planar error by the orbit measurement
error is given by:
σ
p
gps
gps
gps
X
Y
d d
=
+
2
2
(16)
where
d
gps
X
and
d
gps
Y
are the positioning accuracies of the orbit
measurements on the along track and vertical track directions
.
After the geometric calibration, the planar error induced by
the calibration error of the camera inner orientation elements
is given by:
σ
p
calib_M
=
d
calib_M
· pixel
.
(17)
The planar error caused by the calibration error of the in-
stallation angle for on-board equipment in the pitch direction
is given by:
σ
p
calib_
φ
=
H ·
(tan(
φ
+
d
calib_
φ
))
– tan
(
φ
)).
(18)
where
d
calib_
φ
denotes the calibration error of the installation
angle for on-board equipment in the pitch direction. Similar-
ly, the calibration error of the installation angle for on-board
equipment in the rolling direction is given by:
σ
p
calib_
ω
=
H ·
(tan(
ω
+
d
calib_
ω
))
– tan
(
ω
)).
(19)
where
d
calib_
ω
is the calibration error of the installation angle
for on-board equipment in the rolling direction
.
Thus, the comprehensive planar
RMSE
is given by:
M
c
_
_
+
+
+ +
2
2
=
+
σ σ σ
σ σ σ
σ
σ
φ
ω
ϕ
ω
p
p
p
p
p
p
p
p
t
gps
calib
alib
calib
+
2
2
2
2
2
_
(20)
Geometric Accuracy Estimation without
GCP
s
The characteristics and parameters of the
ZY-3
satellite orbit
and cameras, which are related with the geometric position-
ing accuracy, are presented in Table 1
.
Moreover, after the geometric calibration, the calibration
errors of the camera inner orientation elements and installa-
tion angle for on-board equipment were less than 0.25 pixels
and 0.5” (tri-axis, 1
σ
), respectively (Zhang
et al.,
2014). Table
2 presents the theoretical geometric accuracies for
ZY-3
images
without
GCP
s calculated using Equations 12 and 20.
930
December 2015
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
