In this chapter, we proposed two methods in the multiple Doppler format to integrate the data from two or more radars to recover the full velocity from radial velocity data using Optical Flow techniques. The first one uses the least squares approach [48] and the other uses the global regularization method [37]. The least squares method is based on the local smoothness assumption, while the second regularization method works on a global energy function to minimize all the derivatives. Besides, the regularization method uses not only the radial velocity data but also implements the pre-calculated results from the least squares method to generate reasonable smooth results. In the next chapter we will design a synthetic experiment to simulate the 3D motion of wind, and use the generated artificial data to evaluate the performance of the dual radar retrieval methods compared with the single ones. Real data results will also be discussed.
Synthetic and Real Data Experiment
4.1
Synthetic Experiment Design
In order to evaluate the claimed improvements to the calculation of full velocity provided by the Dual Doppler methods, a quantitative evaluation of the accuracy and robustness of the retrieved full velocity flow is necessary. Here we design a synthetic experiment where the ground truth of the 3D full velocities is already known. By comparing the estimated velocities from single and dual methods with the correct velocities, we can measure the performances of our recovery methods quantitatively and claim which one works the best. Since the wind motion in meteorology can be very complex, we adopt different groups of synthetic data and furthermore add several levels of variation and noise to the synthetic data. It allows the performance of our methods to be evaluated exhaustively under various experimental scenarios. By variation, we mean changes to the preset magnitude and direc- tion of the synthetic data. By noise, we mean the errors introduced by the radar detection procedure.
As in Chen’s paper [13], this artificial experiment can be divided into the following steps:
1. Set the synthetic full velocity at each position in the dataset of each Doppler radar. The synthetic data is determined by two factors: the constantbase velocityand the
changeablevariation velocity.
2. Decide the radial velocity for each voxel based on its synthetic full velocity and the relative radial direction. This radial velocity calculated here is then “polluted” by noise in order to simulate the inevitable errors during radar detection.
3. Apply the least squares method or regularization method to the synthetic data and obtain the estimated full velocity flow. Both the single and dual radar methods will be tested.
4. Compare the estimated results with the correct velocity. Several quantitative analyses for the error estimation will be performed to evaluate their performances.
4.1.1
The Generation of Synthetic Velocity
Our synthetic data is determined by two factors: the constantbase velocityand the change- ablevariation velocity:
~
V =V~base+V~var, (4.1)
where V~ = (U,V,W), each element representing one component of velocity flow along one of the three axes: U is the positive west-east direction on the xaxis,V is the positive north-south direction on theyaxis and theW is the positive bottom-top direction vertically on the zaxis. V~base = (Ubase,Vbase,Wbase) represents the constant values assigned on each
component. V~var= (Uvar,Vvar,Wvar) represents the variation values added later.
To test the performance under various weather conditions, we choose 6 different values for base velocity:
1. Group 1: Base velocity is set asV~base =(20.0,20.0,20.0), where all three components
of full velocity are assigned the same values. This is the simplest case of our synthetic velocity model.
2. Group 2: Base velocity is set as~Vbase = (20.0,10.0,5.0). According to the observa-
tions on the real data, this is close to the real wind motion in the Great Lakes area. 3. Group 3: Base velocity is set asV~base = (5.0,5.0,5.0). This one is similar to group 1
but has smaller base velocity values. We examine how the retrieval is influenced by the changing of the overall velocity magnitude.
4. Group 4: Base velocity is set to~Vbase = (20.0,20.0,5.0). This one has very limited
motion on thezaxis compared to that on thexandyaxes. Here we test the influence of a relatively small motion in the z direction on the overall retrieval and how the retrieved vertical velocityW is affected.
5. Group 5: Base velocity is set asV~base= (20.0,5.0,20.0). This is for comparison with
the previous group 4. We try to examine what the difference is between the retrievals on different axes.
6. Group 6: Base velocity is set as V~base = (5.0,10.0,20.0). This one, which is the
reverse of group 2, furthers our investigation on the differences of retrievals in thez direction compared to that onxandyaxes.
From observing historical data, we believe the simulation using group 2 is the most sim- ilar to the actual wind motion around the Great Lakes area. The other groups of synthetic data are used to evaluate how the performance would change under potentially extreme conditions. In particular, it helps us understand better how the performance changes as the components on the x, y, and zaxes vary. Comparisons between these six groups of data, from which we can explore some characteristics from our 3D velocity recovery algorithm, are given in detail in the next section.
~
Vvar is introduced to add variation to the synthetic data. The variation is proportional
to the base velocityV~base and can be adjusted with different variation levels, K. Here we
use 6 levels of variation in total. As K increases, more variation is added to the data. The highest one is 50% of the base velocity whenK= 5. The distribution of variation added to
each voxel is determined by a sinedistribution function. The synthetic velocity in the Kth variation level can be calculated as in Equation (4.4):
Uvar = Ubase· K 10.0 ·sin (x−ωx)· PI 180.0 , (4.2) Vvar = Vbase· K 10.0 · sin ((y−ωy)· PI 180.0 , (4.3) Wvar =Wbase· K 10.0 ·sin (z−ωz)· PI 180.0 , (4.4)
where the three phase parameters ωx, ωy andωz add changes according to the precis po-
sitions, while variation level K and the base velocity determine the highest variation that could be implemented. It should be noted that variation level K = 0 will generate no vari- ation at all, so the synthetic velocity will be constant everywhere. As K increases from 0 to 5, the possible maximum value of variation will change from 0% to 50% of the base velocityV~base, each increment in theK-value representing a 10% increment.
4.1.2
Radial Velocity And Noise Generation
Radial velocity can be considered as the projection of full velocity on the direction of radial beams. The calculation of radial velocity is:
Vr = ~V·rˆ =(U,V,W)·(rx,ry,rz)=(Urx+Vry+Wrz),
whereV~ represents the full velocity and ˆris the unit vector of the radial direction.
Noise is also added to the experimental procedure as a simulation of the potential pol- lution to the real data during the radar detection procedure. We add noise to the radial velocity directly. The distribution of random noise is set according to a Gaussian normal distribution. The density function can be described as:
f(x)= √1
2πσe
whereµis the mean value of the distribution andσis the standard deviation. In our exper- imentµis always set to be 0 to get an unbiased noise around 0.0. The value of σis used to control the distribution shape. Here we useσ = 1.0. The Gaussian distribution in this experiment is generated using the Teichroew method [50]. (It produces a standard normal distribution N(0,1), i.e. with µ = 0 and σ = 1.0.) To estimate the error quantitatively, we also define the Input Error, EI, as a ratio of the input noise magnitude to the radial
velocity magnitude. In our experiment we have 5 noise levels,L, where the input error EI
is set as 0%, 5%, 10%, 15%, and 20% of the overall radial velocity.
Here we present how to adjust the generated noise to the required noise level. Assume the total number of data points involved in the calculation to beN. All the radial velocities can be treated as components of one big vector, (V~r)= (Vr0,Vr1,Vr2, . . . ,VrN). We can cal-
culate the magnitude of this vector askV~rk2in Equation (4.6). Similarly, the magnitude of all the noise can be calculated ask~nk2 in Equation (4.7). Divide all the generated Gaussian noise byk~nk2. We have the~n normalized, and the new noisen0i at each data point (x,y,z) can be calculated as:
kV~rk2= q V2 r1 +V 2 r2 +. . .+V 2 ri +. . .+V 2 rN, (4.6) k~nk2= q n21+n22+. . .+n2i +. . .+n2N, (4.7) n0i = ni k~nk2 ·EI · kV~rk2, (4.8)
whereni is the old noise at any data point (x,y,z), andn
0
i is the modified one. EI is set as
0%, 5%, 10%, 15%, or 20% according to the current noise levelL.