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The wind estimate is determined from the maximum correlation peak in three steps. First, the region with greatest weight is selected as a global maximum correlation region. Then, the highest peak of the global maximum region is determined. Finally, the wind speed and direction are calculated.

The maximum correlation peak was determined
from a spatially low pass filtered correlation function
in an earlier study
[45].
The optimal filter length, however, depends on the noise level and
aerosol structure formations, which both vary with atmospheric conditions.
To reduce the number of user-adjustable parameters in the analysis,
this study uses the following approach to locate the maximum
correlation peak.
First, the global maximum correlation amplitude is found.
Then, all regions with an amplitude greater than of the global maximum
correlation amplitude are located.
The mass **m** of each region is calculated by summing all
correlation amplitudes inside the region above the level.
The region with greatest mass
is assumed to contain the maximum correlation peak.
The masses of the regions are also used in the
reliability analysis (see section 4.2.1).

To use information from more than one point to reduce noise, the top of the global maximum correlation region is determined by least-squares-fitting a two-dimensional polynomial function to a 55 domain around the highest point of the global maximum correlation region. The maximum point of the fitted function is used as the maximum peak position to gain sub-pixel resolution. Sub-pixel resolution is achievable since aerosol structure formations undergoing approximate Gaussian turbulence [65] lead to smooth-peaked cross correlation functions. The fitted function is expressed as

where **x** and **y** denote coordinates in correlation plane and
s are model parameters.

The model parameters are solved from a linear equation group

where are the fit point coordinates
and is the cross correlation function amplitude.
The least-squares solution of the parameter vector ** a**
in a matrix form is
[66]

where denotes the transpose of matrix
** G** and the inverse of a square matrix.
After solving the model parameters,
the position of the maximum point
of function is determined by solving equations

leading to

Figure 26 shows an example of the fit to the maximum CCF region. The top of the region is wide and the points near the local maximum have almost the same intensity. The maximum point of the fitted function interpolates between pixels to gain sub-pixel resolution. The polynomial fit is less sensitive to noise than the bicubic spline fit used in the earlier studies [45]. The least-squares fit slightly filters the top, while the bicubic spline may have noise-induced oscillations near the maximum peak.

**Figure 26:**
The cross correlation function maximum (wire frame)
on July 28, 1989, from 12:00 to 12:03 CDT at 500 meters altitude,
fit points (diamonds), and the peak determined from the fit (rectangle).
The horizontal axes represent the distance from the maximum point in pixels
(one pixel equals 47 m).
The correlation coefficient values are plotted on the vertical axis.

The wind speed and direction are determined from the
maximum point of the fitted polynomial
representing mean movements of the aerosol structures
between the CAPPI scans.
The wind speed **u** and direction are given by

where is the time separation between subsequent volume scans.

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Tue Mar 26 20:53:05 CST 1996