Geoscience Reference
In-Depth Information
k
represents the wavenumber vector of the focused direction with the zenith and azimuth
2
angle of
θ
and , respectively (
k
[sin
sin
, sin
cos
, cos
]
), and the vectors which
represent the center of each receiving receiver are denoted by
D
for the
m
th receiver.
e
is
referred to as the steering vector. The constrained minimization problem can be solved
using the Lagrange method, and Palmer et al. (1998) describe details of solving the
constrained minimization problem (see their appendix). As the solution of the constrained
minimization problem, the optimal weight
wk
is given by
()
1
H1
Re
wk
eRe
.
()
(7)
C
Using Equations (1) and (7), scalar output of the filter
y t
is given by
()
H
y t
()
wks
.
( )()
t
(8)
By calculating the Doppler spectrum of
y t
, brightness
B
(i.e., power density), radial
Doppler velocity, and spectral width are able to be computed with improved angular
resolution.
()
B
is able to be obtained without calculating
w
and given by
1
B
()
k
eRe
.
(9)
C
H1
When the brightness at arbitrary Doppler velocity needs to be calculated,
R
is replaced by
the cross-spectral matrix of the
N
receiver signals. Palmer et al. (1998) showed a clear
difference in angular distribution of brightness between positive and negative Doppler
velocities (see their Plate 2). Brightness at the arbitrary Doppler velocity of received data is
also able to be calculated by applying band-pass filtering to
s
.
In the Fourier-based method, in which all the signals from receivers were synthesized with
equal weight, a weight vector
wk
which steers the beam in the direction
k
is given by
F
()
j
kD kD
j
j
kD
T
wk
F
() (
e
,
e
, ,
e
)
.
(10)
1
2
N
Scalar output of the filter is calculated by replacing
wk
in Equation (8) with
C
()
wk
.
F
()
Other methods are able to be used for CRI. Details of multiple signal classification (MUSIC)
method and maximum entropy method (MEM) are explained by Hélal et al. (2001) and Yu
et al. (2000), respectively. For more general review of CRI, see Woodman (1997).
There are factors that affect the performance of CRI. Using numerical simulation, effects of
receiver noise and turbulence distribution were evaluated by Yu et al. (2000) and Cheong et
al. (2004). Further, Yu et al. (2000) evaluated relation between CRI performance and receiver
arrangement. Effects of uncertainty of receiver gain and phase were evaluated for the case of
turbulent Eddy Profiler (TEP), which was developed by the University of Massachusetts in
order to carry out CRI measurement in the boundary layer (Mead et al. 1998).
