Laroche & Zawadzki (1994) developed a constraining model for the purpose of re- trieving all three components of the wind{vector using monostatic single{Doppler radar observations. Protat & Zawadzki (1999) extended this constraining model for the use of multiple{Doppler data and bistatic Doppler radar data.
5.2 Methodology 65 The principle in the constraining model is to combine observations with certain physical constraints, as explained by Laroche & Zawadzki (1994, 1995). If the con- straint is applied as a strong one, it must satisfy exactly the dened constraining model. Otherwise, if observational errors, assumptions or approximations are al- lowed, the constraint is set as weak one. In constraining model discussed here, the continuity equation is applied as a strong constraint, while the velocity measurements are used as weak constraints. Observational errors are allowed. The sum of the weak constraints is minimized in the model using the method of conjugate gradients.
5.2.1 The minimization procedure
Here, a sequence for the minimization procedure is given. A detailed description of the constraining model can be found in Laroche & Zawadzki (1994) and Protat & Zawadzki (1999). Those variablesu;v;wto be retrieved are called 'control variables'. 1. The control variables u;v are initialized. They can be either set to zero or the translation velocity obtained by the retrieval at the previous time step can be applied as the rst guess.
2. The anelastic form of the continuity equation is used to calculate the vertical wind{componentw from the retrieved 'control variables' u;v
@u
@x + @v@y + 1@z@ (w) = 0 (5.2)
with as the mean air density as a function ofz. The retrieved variablew has to satisfy the equation of continuity exactly due to its application as a strong constraint.
3. The dierences between the measured Doppler velocities (with a monostatic and bistatic receiver) and the respective Doppler velocities calculated from the control variables are computed by the cost functionJv as
Jv = (
V
t;V
0 t)TWt(V
t;V
0 t) + n X p=1(V
ep ;V
0 ep)TWb(p)(V
ep;V
0 ep) : (5.3)Here
V
t is a vector containing all the radial components of the wind{eld,dened at a given grid{point i by vti, and
V
0t contains the corresponding ob-
servations. The vector containing all the Doppler velocities, measured by the
p0th bistatic receiver, is
V
ep, dened at a given grid{point i by veip, and
V
0ep
contains the corresponding observations. Herein, n is the number of bistatic receivers and Wt,Wb(p) are the weighting matrices for monostatic and bistatic
data, respectively. If all the quality criteria are fullled according to Sec. 4.3, the weights for the monostatic Doppler velocity observations are set to one. The Doppler velocities measured by the bistatic receiver are weighted according to the scattering angle [Wb(p) = W(i) in Eq. (3.3)]. The transposed matrix is
66 A variational analysis method to determine the vertical velocity denoted by an upper{caseT. Moreover, a horizontal smoothness penalty func- tion is applied to prevent the retrieved wind{eld from being too noisy. The smoothness constraint, suggested by Wahba & Wendelberger (1980)21, is given
by Js= X X xyz " @2 @x2x2 2 + @2 @y2y2 2 + 2 @2 @x@yxy 2 # ; (5.4)
where presents the control variable (u;v) and is the weighting factor. Smoothing is applied as a weak constraint.
Retrieval of the three wind components is then performed by minimizing the following cost function:
J =Jv+Js (5.5)
4. The minimization procedure is iterated until the convergence criterion is met. For setting the convergence criterion, balance has to be found between computa- tion time (number of iterations) and a reasonably converged solution. The opti- mal number of iterations is case dependent. An appropriate number is estimated by examining the behavior of the cost function. To obtain a reasonably con- verged solution, 200 iterations are used in the following tests on the vertical inte- gration using simulated radar data. The minimization procedure is terminated when the change in the cost function becomes small ([(Ji;1
;Ji)=Ji]<110 ;20)
or the number of iterations is reached. If these predened convergence criteria have not been met, a new estimate of the control variables u;v is calculated with the conjugate{gradient method for a new iteration (Powell, 1977). The gradient of the cost function is used to determine the search direction.
5.2.2 Vertical integration of the equation of continuity
On a non-staggered grid, the vertical component can be estimated by discretizing the continuity equation with a trapezoidal scheme starting at ground level:
wx;y;z = X xyz h ; 1 + Hz wx;y;z;1 ; z 4x(ux+1;y;z;ux ;1;y;z+ux+1;y;z;1 ;ux ;1;y;z;1) ; z 4y(vx;y+1;z;vx;y ;1;z+vx;y+1;z;1 ;vx;y ;1;z;1) i ; (5.6)
where H;1 = (ln)=z. The lower boundary condition is given by
wx;y;1 =; z 2x(ux+1;y;1;ux ;1;y;1) ; z 2y(vx;y+1;1;vx;y ;1;1) : (5.7)
5.3 Testing vertical integration with simulated radar data 67