lidar data at the reference
LFG
, we compute the
RMSE
of
LFG
(
dx
f
,
dy
f
,
dh
) by the Equation3 12 and 21. According to three-
standard-deviations principle of error, the
RMSE
models are
rational if all the differences would be confined within three
times
RMSE
[34].
Validation Results
1. Validation of Horizontal error models
By using the aforementioned determination method on
LFG
, we can obtain 20 reference
LFGs
and compare them
with original
LFGs
in GLA14. The differences between
them in horizontal components and offset (
Δ
x
,
Δ
y
,
Δ
d
) are
shown in Table 4. The original
LFGs
in Table 4 are offered
with the formation of (latitude, longitude, elevation).
The offset
Δ
d
in Table 4 can be described as
Δ
d
s
±
Δ
d
n
,
where
Δ
d
s
and
Δ
d
n
are the systematic offset and random
offset, respectively. The systematic offset is the average of
all offsets, and the differences of
LFG
caused by random
offset (
Δ
x
n
,
Δ
y
n
) are calculated by the following expressions:
∆
∆
∆
∆
∆
∆
∆
∆
x
d
d
x
y
d
d
y
n
s
n
s
= −
= −
1
1
and
.
(25)
Certainly, the difference of
LFG
due to systematic offset (
Δ
x
s
,
Δ
y
s
) can be expressed as (
Δ
x
s
,
Δ
y
s
) = (
Δ
x
,
Δ
y
) – (
Δ
x
n
,
Δ
y
n
).
According to the results in Table 4, we obtain the random
differences in x-axis and y-axis components and present
them in Figure 7, which are compared with theoretical
horizontal errors computed by the
RMSE
models of
LFG
in
Equation 12.
We observe that all differences in x-axis and y-axis com-
ponents are restricted within three times horizontal errors
(±3
dx
f
and ±3
dy
f
) in Figure 7. Because the horizontal dif-
ferences are approximately independent on surface slope
and roughness, then we compute the standard derivations
of
Δ
x
n
and
Δ
y
n
from 20 measurements, which respec-
tively are 7.56 m and 7.49 m against the theoretical value
5.86vm. The causes of the difference
between the theoretical value and field
test value are assumed to be (1) the
lidar spatial resolution of 2 m; (2) lim-
ited
GLAS
measurement numbers; and
(3) the accuracy of waveform matching
method.
2. Validation of vertical error model
For deriving the random difference
of elevation and theoretical vertical
error, we need to extract the terres-
trial features including surface slope,
roughnes,s and elevation within laser
footprint at the systematic offset posi-
tions and reference
LFGs
, by implement-
ing the plane-fitting for the correspond-
ing airborne LiDAR data [20]. The 20
terrestrial features are listed in Table 5.
Depending on the results in Table 5, we
calculate the random difference of ele-
vation (
Δ
h
n
) by subtracting the elevation
at reference
LFG
from that at systematic
offset position, and obtain the vertical
error using Equation 21 and surface
slope and roughness at reference
LFG
.
The distribution of elevation differences and three times
vertical error are shown in Figure 8.
We observe that 17 of elevation differences (85%) are
confined within three times vertical error (±3dh) and
only three of them are out of range. By comparing surface
slope and roughness at systematic offset positions with
those at reference
LFGs
, we discover the major cause for
the outliers is that the differences of slope and roughness
are much greater than others. It also implies that Equation
16 is applicable only if that surface slope and roughness
parameter at the systematic offset positions and reference
LFGs
are close.
Table 4. The differences of
LFG
in horizontal components.
Index
Correlation
coefficient
Original LFG in GLA14
Difference Offset
Latitude (°) Longitude (°) Elevation (m)
Δ
x (m)
Δ
y (m)
Δ
d (m)
1
0.94
34.49
112.95
842.65
-18
16 24.08
2
0.95
34.50
112.95
788.03
32
-16 35.78
3
0.98
34.50
112.95
793.98
-14
-4 14.56
4
0.94
34.50
112.95
747.48
40
40 56.57
5
0.95
34.50
112.95
723.92
-14
2 14.14
6
0.93
34.50
112.95
720.20
-40 -14 42.38
7
0.97
34.50
112.95
717.96
6
-26 26.68
8
0.98
34.51
112.94
662.90
20
28 34.41
9
0.97
34.51
112.94
618.08
4
34 34.23
10
0.97
34.51
112.94
586.45
-38
22 43.91
11
0.65
34.51
112.94
553.49
34
6 34.53
12
0.89
34.51
112.94
535.22
18
40 43.86
13
0.96
34.51
112.94
541.20
0
24 24.00
14
0.86
34.51
112.94
541.29
-16 -22 27.20
15
0.87
34.52
112.94
542.54
-22
28 35.61
16
0.93
34.52
112.94
565.74
4
-22 22.36
17
0.93
34.52
112.94
573.36
12
-16 20.00
18
0.87
34.52
112.94
600.60
30
-40 50.00
19
0.79
34.52
112.94
637.07
2
-26 26.08
20
0.93
34.52
112.94
703.61
38
40 55.17
Figure 7. The random differences in x-axis and y-axis
components.
654
October 2018
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
