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for the radial velocity component. It is to ensure that the retrieved velocity field
.
u
;
could observe the conservation of momentum approximately.
The weighting coefficients are taken as:
v
/
m
2
s
2
/
m
2
s
2
/
W
1
D
0:1.1=
,
W
2
D
1.1=
,
m
2
s
2
/
W
6
D
10
4
.1=
m
2
s
4
/
W
3
D
W
4
D
W
5
D
0:1.1=
. They are chosen empiri-
cally in this paper to ensure that the constraints have proper orders of magnitude.
Following
Qiu et al.
(
2006
), the background velocity field is determined by
expanding it in terms of second-order Legendre polynomials:
and
2
X
2
X
u
B
.x;y/
D
a
nx;ny
P
nx
.x/Q
ny
.y/;
nx
D
0
ny
D
0
2
X
2
X
v
B
.x;y/
D
b
nx;ny
P
nx
.x/Q
ny
.y/:
(22.13)
nx
D
0
ny
D
0
P
nx
.x/
are the orthonormal functions (Legendre polynomials). The
background field is then fully determined by the expansion coefficients
and
Q
ny
.y/
a
nx;ny
and
b
nx;ny
, which are the retrieved variables in this step. The cost function for retrieval
is similar to (
22.8
), except that the first term vanishes (i.e. setting
).
Before performing the retrieval, the radial velocity data are quality-controlled
to remove the outliers due to, for instance, reflection from clutters (
Shun and
Chan 2008
). The main source of clutter is the moving aircraft in the sky and
the clutter does not occur very frequently (in the order of a few per day). Such
outliers could be detected by mimicking visual inspection to compare each piece
of radial velocity with the data points around, and replaced by a median-filtered
value if the difference between them is larger than a pre-defined threshold. The
threshold is determined from the frequency distribution of velocity difference
between adjacent range/azimuthal gates of the LIDAR over a long period of time.
The quality-controlled radial velocity in the range-azimuth coordinate system is
then interpolated to a Cartesian grid with resolution of 100 m using Barnes scheme.
According to
Chan and Shao
(
2007
), the root-mean-square errors of the retrieved
wind components (
u
and
v
) were about 2 m/s when compared with the anemometer
measurements (Table 2 and Fig. 2 in
Chan and Shao
(
2007
)).
One application of the 3DVAR retrieved 2D wind field is the identification of
coherent structure in the airflow at HKIA, which may be related to the low-level
windshear and turbulence to be encountered by the aircraft. The monitoring of
airflow near HKIA using the LIDAR's radial velocity data is a kind of Eulerian
descriptions of the flow field. It has recently been established that, such descriptions,
inefficient and somewhat arbitrary at best, could lead to serious flaws as instanta-
neous streamline sketches is not an objective representation of actual particle motion
in an unsteady flow. Lagrangian analyses, however, provide frame-independent
description when the flow field is not evolving too quickly, and certain trajectories
of an unsteady flow persist with coherent motion over some period of time. The
method analyzes the relative motion of fluid particles in the Lagrangian frame.
In this framework, the Lagrangian coherent structures (LCSs) are identified as
distinguished sets of fluid particle trajectories that attract or repel nearby trajectories
W
1
D
0
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