NSCAT Probability Distribution
NSCAT makes only indirect measurements of wind. The direct
measurement is of the backscattered radar power. This signal power
is contaminated by radiometric noise so a separate measurement of
the noise power is subtracted from the signal+noise measurement to
estimate the backscattered power. Using the radar equation, sigma0
is computed from the measured signal power. From multiple sigma0
measurements made at different azimuth angles, the wind is
estimated. In wind retrieval, the NSCAT sigma0 measurements are
assumed to have a Gaussian probability distribution with a variance
which depends on the mean. Given this distribution model, the
maximumlikelihood estimator is formed and optimized to estimate
the wind.
Because of the onboard signal processing used by NSCAT, the
model of a Gaussian distribution for sigma0 is only an
approximation to the actual distribution. Working from first
principles and the design of the NSCAT signal processor we derive
the distribution of the NSCAT measurements as a function of the
surface sigma0, the signal to noise ratio and the cell number. The
resulting hypergeometric distribution is skewed relative to the
traditional Gaussian model. This PDF has been derived by Paul E.
Johnson, Uncertainties in Oceanic Microwave Remote Sensing: the
Radar Footprint, the WindBackscatter Relationship, and the
Measurement Probability Density Function, Ph.D. Dissertation,
Brigham Young University, Provo, Utah, 1999. (A related paper is
P.J. Johnson and D.G. Long, "The Probability Density of Spectral
Estimates Based on Modified Periodogram Averages," IEEE
Transactions on Signal Processing, Vol. 47, No. 5, pp.
12551261, 1999.)
The observed NSCAT probability distribution (pdf) will be the
combination (convolution) of the pdf described above and the pdf of
the sigma0 of the surface. The latter will be a function of the
pdf of the nearsurface wind field and the geophysical model
function.
From wavetank and lake experiments there is strong evidence
that sigma0 falls off rapidly at low winds. While this effect is
clear in such data, it is not as clear in spaceborne data where the
radar footprint covers 25² km or more. However, if sigma0
does drop to low values at low wind speeds, the probability of
seeing negative sigma0 measurements (which occur because of noise
in the power measurements) increases. As evidence that negative
sigma0 measurements are correlated with low wind speeds consider
the following figures. Figure 1 shows the spatial distribution of
negative sigma0 measurements over a three day period. Figure 2
shows a map of 3 days of NSCATderived wind speeds. Figure 3 shows
shows a map of the standard deviation of NSCATderived wind speeds
over the three day period. The later image is useful in evaluting
the temporal variation over the three day period.
Figure 1: Map of the count of the number of negative sigma0
measurements in 3 days of NSCAT data. This is a shunk down version
of the original
image (58 K) which is a 1/4 deg by 1/4 deg map. In this
image, the number of measurements in each grid element are
indicated by a gray value with white as the most (thresholded to
5).
Figure 2: Map of the average NSCATderived wind speed over a
three day period. This is a shunk down version of the original image (605 K)
which is a 1/4 deg by 1/4 deg map. Whiter values indicate higher
wind speeds with pure white 18 m/s and higher.
Figure 3: Map of the average NSCATderived wind speed over
a three day period. This is a shunk down version of the original image (608 K)
which is a 1/4 deg by 1/4 deg map. Whiter values indicate
higher wind speeds with pure white 18 m/s and higher.
Figure 4: Map of the count of the number of sigma0
smaller (but positive) than the square of root of Gamma in the
quadratic Kpc equation in 3 days of NSCAT data. This is a shunk
down version of the
original image (58 K) which is a 1/4 deg by 1/4 deg map. In
this image, the number of measurements in each grid element are
indicated by a gray value with white as the most (thresholded to
5).
Figure 5: Map of the weekly NCEP analyzed sea surface
temperture (SST). This is a shunk down version of the original image (136 K)
which is a 1/4 deg by 1/4 deg map. In this image, the
temperature is represented as a gray value from 10 to 40 deg C.
The images presented above are for JDs 810 in 1997. Full size
images for JDs 1921 are available below.
Spatial distribution
negative sigma0 image (55 K)
Wind speed image (619 K)
Wind speed standard
deviation image (621 K)
Spatial distribution
of low sigma0 values (58 K)
SST map (135 K)
