Geoscience Reference
In-Depth Information
J oJd sJb
(1)
J
o
represents the difference between the analysed radial velocity and the observed radial
velocity,
J
d
is the mass continuity equation constraint term,
J
s
is the smoothness constraint
term, and
J
b
is the background constraint term. The definition of
J
o
used here differs from
that used by G99.
J
o
is defined as
1
ijkm
,, ,
2
(2)
J
C (
Vr
Vr
_
obs
)
o
o
N
ijkm
,, ,
Vr
is the analysed radial velocity on a specified Cartesian grid, where
i
,
j
, and
k
indicate
spatial location in the x, y, and z directions, respectively, and
m
indicates the
m
th
radar in the
network.
N
is the total number of observations, which is equal to the product of the number
of grid cells and the number of radars in the network. C
o
is the reciprocal of the mean
squared error in the observations.
Vr_obs
is the radial velocity interpolated to the regular
Cartesian grid. The cost function is evaluated at each grid point in the Cartesian coordinates,
rather than in spherical coordinates.
Each constraint is weighted by a factor that accounts for its respective proportion of the
reciprocal of the mean squared error. As noted by G99, it is usually difficult to obtain
appropriate values for the weighting coefficients. In particular, the value of the weighting
coefficient for the anelastic mass conservation constraint plays an important role in
determining the vertical wind component. This study uses the coefficient value used by G99,
although G04 introduced a more objective method for estimating this coefficient.
The variational method uses the derivative of J with respect to the analysis variables to
obtain an optimal solution. The gradient of the cost function is derived with respect to the
control variables, namely the two horizontal wind components (u, v) and the vertical wind
component (w). The form of the gradient used here differs slightly from that used by G99
because the form of the observational constraint differs. The gradient of the observational
constraint with respect to u is given by
J
x
ijkm
,, ,
(3)
o
C (
Vr
Vr
_
obs
)
o
u
r
where
r
is the distance between the radar and the grid point and
x
is the component of r in
the x direction.
After the cost function is evaluated and its gradients are obtained, a quasi-Newton-type
optimization scheme is used to update the control variables. This analysis uses a limited-
memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method (Liu & Nocedal, 1989). For
most meteorological applications, the L-BFGS method is more efficient than the conjugate
gradient method (Navon & Legler, 1987). L-BFGS uses an approximation of the second-
order derivative, so that an iteration of the L-BFGS method typically requires less
computation than an iteration of the CG method. L-BFGS is therefore a better choice for
optimizing a computationally expensive cost function.
3. Observational system simulation experiment and model description
The performance of the variational technique is evaluated in the context of an observational
system simulation experiment (OSSE). This OSSE is conducted using numerical simulations
