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### Noise in the Cross Correlation Function

The wind speed and direction errors are estimated by calculating the change in the maximum correlation peak position when the CCF has additional noise. In this section, the error of the fit parameters is determined and then used to estimate the position error of the maximum of the fitted function. The expected root-mean-square errors in the wind speed and direction are derived from the position error.

Most of the signal noise in the daytime VIL measurements originates from the photon noise due to background light. This leads to spatially uncorrelated noise in the CAPPI scans. However, since the histogram equalization of a CAPPI scan is a highly nonlinear operation, a CCF does not provide enough information about the spatial distribution of noise. To make analytical formulation of the noise effects possible, the noise is assumed uncorrelated with a constant variance. However, the accuracy of this approximation remains undetermined. If the noise in the CCF plane is spatially uncorrelated and the variance of the noise is , the covariance matrix of the estimated model parameters solved from equation (14) is given by

The noise variance of the CCF is estimated by the following. First, an average of three vertically closest CCF planes are subtracted from the CCF

Then, the CCF planes are masked by rejecting all pixels inside the global maximum correlation region

Finally, noise variance is calculated from the masked CCF

where denotes the mean value of the masked correlation function and the number of nonzero elements in the masked correlation function.

The mean-squared determination error of the maximum peak position is given by

where s are error variances of model parameters estimated from the model parameter covariance matrix as

Finally, the root-mean-square errors and for wind speed and direction estimates, respectively, are determined from

The direction error is expressed in radians. These equations represent root-mean-square errors of the mean wind estimate due to a position error of the maximum correlation peak from a noisy CCF. Since the previous error analysis does not take into account errors due to possible deformation of aerosol structures between scans, the errors may be underestimated. Furthermore, the error variance may be poorly estimated due to nonlinear effects of the histogram equalization of the CAPPI scans.

Aerosol deformation is expected to become a significant error source in the presence of gravity waves or strong vertical wind shear. The gravity waves affect the formation of aerosol structures by providing an additional energy source. The aerosol structures tend to follow the wave structure. Therefore, the aerosol structures move by the group velocity of the gravity waves, which differs from the velocity of the air mass. A strong, vertical wind shear may systematically deform aerosol structures, which may tilt the maximum peak of the cross correlation function. The effects of both error sources on the VIL wind profiles are difficult to examine. A comprehensive study would require insitu measurements with an instrument which has similar averaging properties and accuracy as the VIL, and which also can directly measure the air mass movements. Therefore, this problem is not examined further in this study.

Next: Results and Discussion Up: Error Analysis Previous: Random Cross Correlation