Pointwise Wind Retrieval
The most common method for scatterometric wind estimation is known
as pointwise wind retrieval. In this method, the measured swath
is divided into resolution elements known as wind vector cells (wvc's)
and the wind is estimated independently at each wvc. The independent
estimates are then processed to remove ambiguous solutions.
The Geophysical Model Function
To estimate the wind from scatterometer measurements, we must
establish the relationships between the normalized backscatter and
the wind speed and direction. This relationship is described mathematically
by the Geophysical Model Function (GMF). Much research has been
and continues to be done to accurately characterize the GMF. (See
YSCAT and JPL). The GMF can be generically represented by the following
equation:
The forward model function accepts as inputs: wind speed, relative
azimuth angle (or the angle between the instrument azimuth and the
wind direction), incidence angle, instrument frequency, and instrument
polarization. The output is the normalized backscatter, sigma0,
or the measurement the instrument would return under such conditions.
In general, the wind speed and the relative azimuth (which will
be referred to hereafter simply as the wind direction) are unknowns,
while the incidence, frequency and polarization are known for each
measurement.
It is important to recognize that the GMF accepts two wind inputs
(wind speed and direction) to generate sigma0. Unfortunately, the
inverse model function is not unique. This implies that the inverse
problem (i.e. deriving the wind vector from the measurements) is
not trivial. Indeed, in the absence of other information, a single
measurement generates a continuum of potential wind vectors, all
of which would have given rise to the observed backscatter.
This figure demonstrates some of the characteristics of the GMF.
For a fixed incidence, frequency, and polarization, sigma0 is shown
as a function of wind direction (x axis) and wind speed (labeled
on each curve.) Note that in general, higher wind speeds generate
more surface roughness and therefore higher sigma0 values. We note
that the model function has a cos(2X) symmetry.
Although the problem has been studied for years, an accurate analytic
derivation of the GMF remains beyond the current state of the art.
Thus the GMF is typically empirically derived.
The Pointwise Objective Function
In order to resolve the inverse problem and solve for both wind
speed and direction, multiple backscatter measurements are taken
at each wvc. The following figure illustrates four different measurements
(with different azimuth and incidence angles) taken with the same
wind speed and direction. For each measurement, the figure shows
a curve of the wind speeds and directions that could have given
rise to the observed backscatter. Where the curves intersect is
the wind vector consistent with the measurements.
In the absence of noise, the different measurements allow the
wind speed and direction to be resolved as 11 m/s and 150 degrees.
Note however that there are near intersections at approximately
330 degrees, 40 degrees, and 235 degrees. The intersection shown
is the idealized case where there is no noise in the measurements
or the GMF. Unfortunately, the measurements are not noise free,
so the problem becomes more complicated. In the presence of noise,
the alternate intersections might appear to be more acceptable options.
Noise in the backscatter measurements is due to radiometric and
thermal noise in the radar receiver and has an approximately Gaussian
distribution. To generate a statistical model for use in estimation,
each measurement is assumed to be a realization of a Gaussian random
variable dependent on the wind speed and wind direction. Each measurement
is assumed to be independent, thus the joint density function of
the measurements given the wind vector is given by:
A maximum likelihood estimator for the wind vector can be formulated
by maximizing this expression for W. For convenience, we take the
log of this expression to create the maximum likelihood objective
function.
Ambiguities in the Pointwise Objective Function
As mentioned above, there are often several directions near intersections
when resolving the sigma0 measurements into a single estimate.
These alternate intersections show up as peaks in the objective
function. A true maximum likelihood estimate would ignore them as
inconsequential, but for pointwise wind retrieval, the maximum
likelihood estimate is not sufficient to generate a realistic swath
of wind estimates. A common practice is to keep several ambiguous
solutions at each wind vector cell (wvc), and attempt to resolve
the ambiguity thereafter.
This figure shows a typical region with each of the wind ambiguities
displayed at each cell. Blue is the first (most likely) ambiguity,
red is the second, green the third and aqua the fourth. In order
to determine an estimate of the wind across the entire swath, an
ambiguity selection algorithm is needed.
The Pointwise Median Filter
In order to select the proper ambiguity, we assume that the estimates
are correlated from one cell to the next. In other words, we assume
that the wind is unlikely to shift radically from one cell to the
next. Using this assumption, we may use a variety of techniques
to select a single ambiguity at each wvc.
A common technique used to correlate the estimates is based on
the pointwise median filter. This method does not alter any of
the wind estimates generated; rather is selects between the ambiguities
at each cell. Like median filters used in image processing, the
pointwise median filter attempts to preserve edges and avoid blurring.
The entire swath is initialized to the first ambiguity. For each
cell in the swath, a 7cell window is generated around the cell
in question, and an average of all of the wind vectors is taken.
The ambiguity closest to this windowed average is selected. This
process is repeated until there are no more changes (or until a
maximum number of iterations is achieved).
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