Fieldwise Wind Retrieval
In pointwise wind retrieval, wind
estimation at each wind vector cell (wvc) was performed independently.
However, by making use of wind field models,
improved optimization may be achieved.
The Fieldwise Objective Function
The wind field model assumes a correlation between the wind vectors
at adjoining wvc's. Specifically, we may write a wind field as:
W = FX,
where W is a column scanned vector of a wind field region, F is
the model matrix, and X is a vector of the model parameters. This
representation reduces the dimension of the problem. (For example,
if we are examining a 24 x 24 region, then the W vector has 1028
elements. If we truncate a model at only six parameters, the dimensionality
of the problem is greatly reduced.) As in the case with the pointwise objective function, we may write the
density of the measurements given a wind field p_{z}(Z 
W) by multiplying the densities of each measurement at each cell
in the region. To take advantage of the wind field model, however,
we write p_{z}(Z  FX) or simply p_{z}(Z  X). Once
again the joint probability density function is just a product of
independent Gaussian random variables:
This is a function of the same dimension as X. As with pointwise
estimation, we may form a maximum likelihood estimate for the wind
field by maximizing the pdf with respect to W. For convenience,
we take the negative log of the pdf to form the fieldwise objective
function. Like the pointwise objective function, the fieldwise
objective function has multiple minima representing ambiguous solutions.
For this reason, we generally optimize to all of the local minimaa
by a QuasiNewton gradient descent algorithm. (See Numerical
Methods for Unconstrained Optimization and Nonlinear Equations
by J.E. Dennis, Jr. and Robert B. Schnabel, 1983) In this manner,
there are multiple solutions for each wind region.
The Fieldwise Median Filter
In order to select between the ambiguous fieldwise solutions,
a fieldwise median filter was developed. The swath is divided into
overlapping regions, and each region is optimized to generate the
ambiguities. Each region is initialized to the first ambiguity and
the entire swath estimate is created by averaging the overlapping
regions.
For each region, all of the neighboring regions are averaged together
and lowpass filtered. This is comparable to averaging surrounding
cells in the pointwise median filter. All of the ambiguities
are compared to this field, and the closest field (in a least squares
sense) is selected. The filter iterates until either a maximum number
of iterations is reached, or there are no more ambiguity changes
across the swath.
Back to Wind Scatterometry
