IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 3, MARCH 2006 597
A Nonlinear Optimization Algorithm for
WindSat Wind Vector Retrievals
Michael H. Bettenhausen, Member, IEEE, Craig K. Smith, Member, IEEE, Richard M. Bevilacqua, Nai-Yu Wang,
Peter W. Gaiser, Senior Member, IEEE, and Stephen Cox
Abstract—WindSat is a space-based polarimetric microwave ra-
diometer designed to demonstrate the capability to measure the
ocean surface wind vector using a radiometer. We describe a non-
linear iterative algorithm for simultaneous retrieval of sea surface
temperature, columnar water vapor, columnar cloud liquid water,
and the ocean surface wind vector from WindSat measurements.
The algorithm uses a physically based forward model function for
physically based model are discussed. We present evaluations of
initial retrieval performance using a six-month dataset of WindSat
measurements and collocated data from other satellites and a nu-
merical weather model. We focus primarily on the application to
wind vector retrievals.
Index Terms—Microwave radiometer, ocean surface winds,
polarimetric, retrieval, WindSat.
WindSat’s primary mission is to provide measurements for the
evaluation of polarimetric microwave radiometry in retrieving
the ocean surface wind vector. WindSat also provides measure-
ments for retrieving sea surface temperature
atmospheric water vapor
, and columnar atmospheric cloud
The polarization properties of an electromagnetic wave
can be fully characterized by measuring the modified Stokes
vector. The modified Stokes vector includes the vertical and
horizontal polarizations and the third and fourth Stokes param-
.1Modeling and aircraft measurements
have shown that
andare even periodic functions of
, andandare odd periodic functions of
HE first space-based fully polarimetric microwave ra-
diometer, WindSat , was launched in January 2003.
Manuscript received June 27, 2005; revised October 12, 2005. This work was
supported by the U.S. Navy under Grant N000WX0573003.
M. H. Bettenhausen, R. M. Bevilacqua, and P. W. Gaiser are with the Naval
Research Laboratory, Remote Sensing Division, Washington, DC 20375 USA
C. K. Smith was with Computational Physics, Inc., Springfield, VA 22151
USA. He is now with The Aerospace Corporation, Los Angeles, CA 90009
N.-Y. Wang was with the Office of Research and Applications, National En-
vironmental Satellite, Data, and Information Service, National Oceanic and At-
Park, MD 20742 USA.
S. Cox is with Computational Physics, Inc., Springfield, VA 22151 USA.
Digital Object Identifier 10.1109/TGRS.2005.862504
1The symbols ? and ? are often used to denote the third and fourth Stokes
parameters. We use ? and ? here to avoid confusion with other notation used
in this paper.
wind direction minus the radiometer look direction. Therefore,
dual-polarization radiometers, which measure only
do not provide enough information to unambiguously retrieve
wind direction. However, a fully polarimetric radiometer such
as WindSat, which also measures
information to, at least in principle, retrieve the ocean surface
Dual-polarization observations from radiometers such as
the Special Sensor Microwave/Imager (SSM/I)  and the
Advanced Microwave Scanning Radiometer–EOS (AMSR-E)
 have been used to retrieve ocean surface wind speed2
andwith both statistical and physically based
methods. Most statistical regression algorithms empirically
derive regression coefficients for the retrieved parameters
using collocated in situ measurements or retrievals from other
satellites (see e.g., –). Wentz and Meissner  used a
multiple linear regression algorithm where the coefficients are
determined using brightness temperatures
a physically based model function. Wentz  used physically
based model functions for the SSM/I
equations in four unknowns which are solved using an iterative
procedure. Wentz and Meissner  also outlined a nonlinear
iterative retrieval algorithm, but they do not discuss retrieval
results obtained with the algorithm.
Algorithms for retrieving wind direction from a polarimetric
microwave radiometer have previously been investigated using
measurements are limited in scope because only a relatively
small amount of data is available. Piepmeier and Gasiewski 
Their algorithm used maximum–likelihood estimation (MLE)
for separate retrievals of wind direction and wind speed and it-
erated between the two retrievals to arrive at a final wind vector
solution. Liu and Weng  used simulated polarimetric data
to demonstrate wind vector retrievals using a physical inversion
method. Their retrieval algorithm used the polarimetric mea-
surements for only one frequency.
WindSat provides the first opportunity to evaluate the
wind vector retrievals from polarimetric radiometer data on a
global scale. We previously described an empirically derived
combined statistical and MLE algorithm for retrieving ocean
surface wind vectors from WindSat measurements . Here
, the relative wind direction, is defined as the compass
and, provides sufficient
s simulated with
s, to obtain a set of four
2Throughout this paper wind speed refers to the equivalent neutral-stability
wind speed at a 10-m reference height.
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A Nonlinear Optimization Algorithm for WindSat Wind Vector
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598IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 3, MARCH 2006
NOMINAL NEDT VALUES FOR RAIN-FREE OCEAN
RETRIEVAL CELLS (IN KELVIN)
we present a nonlinear optimization algorithm which simulta-
algorithm is designed to produce four solutions (ambiguities)
for each set of WindSat
s using a physically based forward
in our study. In Section III we describe our retrieval algorithm.
Section IV describes the first generation geophysical forward
model function used in our retrieval algorithm. We then present
retrieval performance results and a discussion of the results in
the final two sections.
andfrom WindSat s. The
II. WINDSAT DATA DESCRIPTION
A description of the WindSat sensor and data processing
system is provided in . The data processing system produces
sensor data records (SDRs) which contain
formation and data quality information for 16 separate channels
at five different frequencies.
vector are provided at 10.7, 18.7, and 37 GHz. Dual-polariza-
tion measurements, vertical and horizontal, are provided at 6.8
and 23.8 GHz. The antenna temperature measurements, which
have a different beamwidth and relative pointing angle for each
frequency, are resampled and averaged to provide collocated
s for the SDRs at a common resolution. The SDRs used for
this paper have an effective field of view (EFOV) or footprint
of approximately 40 km
60 km. The nominal effective noise
equivalent differential tempertures (NEDT) for ocean scenes
after resampling and beam averaging were given in  and
are repeated here in Table I.
WindSat was designed with both one-look and two-look ca-
pability with measurements taken in both the forward and aft
viewing directions. The width of the forward swath is about
950 km and the width of the aft swath is about 350 km where
the swath width is defined to be the arc length on the Earth’s
surface where there are common measurements available for all
WindSat frequencies (except 6.8 GHz due to the 6.8-GHz horn
ward scan with an approximate spacing of 12.5 km along scan
and along track. Retrievals are performed for all 80 pixels in the
forward scan. The common swath with 6.8-GHz measurements
contains 63 pixels but due to rolloffs at the edge of the swath
s at 6.8 GHz we only use the 6.8-GHz measurements
from 55 pixels in the retrievals. Due to the narrowness of the
aft swath our initial forward modeling and retrieval efforts have
focussed on the forward swath. The discussion in the remainder
of this paper applies only to the SDRs and retrievals from the
were produced with version 1.8.1 of the WindSat ground data
processing system. We use six months (September 1, 2003 to
s, geolocation in-
s for the full modified Stokes
February 28, 2004) of WindSat SDRs using every third day for
retrieval analysis with the remainder reserved as training data to
develop empirical corrections to the geophysical forward model
(as explained in Section IV). Data are excluded if the
outside of physical bounds for ocean scenes or the Earth inci-
are also excluded for rain, ice, radio-frequency interference at
10.7 GHz , land contamination, inland lakes, for satellite
attitude anomalies and if less than 60% of the measurements
nominally used for beam averaging are available. Rain is as-
sumed to be present if the retrieved cloud liquid water is greater
than 0.2 mm. A more conservative rain flag, which is based on
a flag developed for SSM/I , was used for the training set.
Rain was considered to be present if any of the following con-
ditions were satisfied:
III. RETRIEVAL ALGORITHM
A. Optimal Estimation
Our retrieval algorithm uses an optimal estimator  which
is a Gauss–Newton iterative method with a priori constraints.
The method is equivalent to minimization of the cost function
where the superscript
the state vector of quantities to be retrieved. The state vector
is comprised of
the a priori error covariance matrix,
of the goodness of fit of the forward model, evaluated using
the retrieved state vector, to the measurements , . An
estimate of the
is obtained from
indicates the matrix transpose and is
andor a subset thereof. The
. Theis a measure
where the subscript denotes the th iteration. In this equation,
is the measurement vector, with error covariance matrix
The measurement vector is the set of WindSat
retrieval. The state and measurement spaces are related through
in Section IV.
The iteration used can be written
s used for the
The error covariance of the solution is approximated by
is the final iteration.
BETTENHAUSEN et al.: NONLINEAR OPTIMIZATION ALGORITHM FOR WINDSAT599
the derivative of the forward model with respect to the state
, called the weighting function or kernel, is
The kernel is calculated numerically at each stage of the itera-
tion using a centered finite difference scheme.
The convergence criteria we use compares the change in the
state vector at each iteration to an estimate of the retrieval error
This retrieval method requires an accurate forward model as
a function of the retrieval parameters, a priori estimates for the
retrieval parameters, an error covariance matrix for the a priori
and an estimate of the measurement error covariances. At this
time, we are using all of the WindSat
at 6.8 GHz andat 37 GHz.
excluded for this version due to a wind speed bias problem but
we expect to include it in the future following improvements to
our forward model.
at 37 GHz has been excluded because
the signal is too small (
K) to improve the retrievals.
We use constant a priori for
and are chosen to be near the center of their respective
andare highly skewed so the a priori constants are chosen
near the median for each distribution:
mm. The a priori error covariance matrix,
be diagonal with values that allow the retrievals to cover the full
error covariance values are 12 K, 6 m/s, 50 mm, and 1 mm for
and, respectively. To a large degree, the a priori
error covariance values can be considered tuning parameters.
Initial analysis of the retrievals from a single-stage algorithm
with constant a priori values showed
the respective a priori values due to the large ranges of values
and reduced forward model sensitivity at high
Therefore, we use a two-stage retrieval algorithm where the
first-stage retrieval is performed to provide more accurate a
priori values for the second stage. In the first stage,
and are retrieved using only the WindSat
and horizontal polarizations. The forward model for the first
stage is not a function of the wind direction. Both retrieval
stages employ the optimal estimation method descibed above.
A diagram of the retrieval process is shown in Fig. 1. This
two-stage process could be used in the future with a priori for
the first stage derived from climatologies or numerical weather
models without biasing our final retrievals. The primary ad-
vangtage would likely be faster convergence of the retrieval
The second stage retrieval solves for all five retrieval param-
eters and includes the azimuthal wind direction dependence in
the forward model. Simultaneous retrieval allows the algorithm
is the dimension of the state vector.
s for our retrievals ex-
at 6.8 GHz has been
. The a priori for
mm. The distributions for
, is chosen to
s for the vertical
Fig. 1.Simplified flowchart of the retrieval algorithm.
to adjust all five parameters to optimally match the forward
model to the measured
ters are used. The first stage retrievals for
as a priori values for the second stage. The square root of the a
and, respectively. These values have
been chosen to be roughly two to three times larger than the ex-
pected root mean squre (RMS) errors in the first stage retrievals
(the a priori values). As with the first stage a priori error co-
variance, these values can be treated as tuning parameters. The
method used to obtain the wind direction a priori is described
The procedure outlined by (3)–(6) is equivalent to minimiza-
tion of the cost function (1), as noted above. The
multiple local minima primarily due to the dependence of the
forward model on
; so, the choice of a priori may effectively
choose a local minima. Multiple solutions or “ambiguities” can
therefore be obtained by performing separate retrievals for mul-
tiple a priori state vectors. We use four a priori state vectors,
where only the wind direction differs, to obtain four ambigu-
ities. The first a priori wind direction is obtained from the arc
of the wind vector from the two-stage regression algorithm de-
scribed in . The three additional a priori wind directions are
chosentobe90 ,180 ,and270 fromtheregressionresult.The
retrieved ambiguities are ranked by the corresponding
the first rank ambiguity having the lowest
The remaining information needed by the retrieval algorithm
is an estimate of the measurement error covariance matrix,
We include both the effects of measurement noise and forward
model errors in
. A diagonal measurement error covariance
task of estimating
. However, a diagonal
for correlations in the errors for different channels. While the
measurement noise in different channels is uncorrelated, there
are significant correlations between the forward model errors.
provide two examples here. The atmospheric absorption and
emission are largely unpolarized so that errors in estimating
them will be common to all polarizations at a given frequency.
Forward model errors for the same polarization but different
frequencies may be correlated, for example, due to differences
in the ocean wave spectrum or sea surface foam coverage be-
tween the global means (for a given
served conditions. Therefore, including the off-diagonal terms
improves the weighting of the channels for the retrievals.
s for all Stokes parame-
does not account
and) and the ob-
600 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 3, MARCH 2006
SQUARE ROOT OF THE DIAGONAL OF ? USED FOR THE WIND SPEED
RANGE OF 7–13 m/s. THE SUPERSCRIPTS IN THE CHANNEL
COLUMN ARE THE FREQUENCIES IN GIGAHERTZ
This improves the accuracy of the state vector estimates and the
which, in turn, improves ambiguity selection.
is estimated using the differences between the measured
s from theWindSatSDRsand
s simulated withtheforward
is the total number of SDRs used to estimate
are state vectors consisting of collocated data from a numer-
ical weather model and other satellite retrievals. The method
for simulating the
s and the data used are the same as those
tion IV). The measurement error covariance values vary signif-
icantly with wind speed. For example, the error covariance for
the third Stokes
s is much smaller at low wind speeds, where
the magnitude of the signal is small, than it is at high wind
speeds. Therefore, to account for these variations we calculate
separate error covariance matrices for five different wind speed
covariance matrices account for modeling error, measurement
noise, and calibration error. The estimated covariances also in-
clude “matchup noise” due to spatial and temporal differences
between the collocated data used for simulating
values actually measured in the WindSat footprint. The square
the superscripts in the channel column are the frequencies in gi-
gahertz. Table III shows the lower triangular part of
6.8- and 10.7-GHz channels. The values for
are vectors of the WindSat s from the SDRs and
. Here the
m/s. The resulting error
s and the
at 6.8 GHz are
? TERMS FOR THE 6.8- AND 10.7-GHz CHANNELS USED FOR THE WIND
SPEED RANGE OF 7–13 m/s. ? IS SYMMETRIC, SO ONLY THE LOWER
TRIANGULAR PART IS SHOWN. THE SUPERSCRIPTS IN THE CHANNEL
COLUMN ARE THE FREQUENCIES IN GIGAHERTZ
included for information even though that channel is not cur-
rently in the retrievals. The values in Table III illustrate the cor-
relationsbetween the errorsfor the
III are used for the wind speed range of 7–13 m/s. The values
in Table II decrease up to 50% for the lowest wind speed range
and increase up to 100% for the highest wind speed range. The
diagonal elements of
those for the
channels at the same frequency because
is more sensivite to changes in the wind vector and the atmo-
channels are larger than
B. Median Filtering
We apply a spatial vector median filter (MF) to the retrieval
cells to correct isolated errors in the ambiguity selection based
ranking. The MF cost function for a given retrieval
cell is computed on a 7
7 cell box (in scan-based coordinates)
centered on and including that cell . The median filter can
be initialized using the first rank retrieval from the optimal esti-
mation results. Alternatively, the median filter can be initialized
with a “nudged” windfield where a backgroundwind field from
an external data source is used to select the first or second rank
ambiguity closest to the background field. Further detail on the
the median filter is given in . We use spatially interpolated
wind fields from the National Centers for Environmental Pre-
diction Final Analysis (NCEP) as the background wind field for
the results presented in this paper.
IV. FORWARD MODEL
begin with a physically based radiative transfer model and then
apply empirical corrections to better match the measured
The empirical corrections are then used as a guide for improve-
ments to the radiative transfer model. In addition, the depen-
is reduced relative to a completely empirically derived model,
so that the dependence of the retrievals on the training data used
is limited. Here we present an overview of our forward model
for the WindSat
We have developed a parameterized forward model similar to
that described in . The
s measured by the satellite are the
sum oftheupwellingatmospheric radiation,thereflected down-
welling atmospheric and cosmic background radiation, and the
direct emission of the sea surface. The reflected downwelling
BETTENHAUSEN et al.: NONLINEAR OPTIMIZATION ALGORITHM FOR WINDSAT601
radiation and the direct emission are attenuated by the atmos-
s at each WindSat frequency can be expressed as
Stokes parameters. The sea surface emissivity for polarization
upwelling atmospheric brightness temperature at the top of the
is the downwelling atmospheric brightness
temperature at the surface, and
is approximately 2.7 K. The
term is a correction factor to ac-
radiation from the rough sea surface . We are currently ne-
glectingtheazimuthaldependence of thereflected downwelling
ical work in this area has been done only recently . We have
also neglected the effect of nonspecular reflection of the cosmic
background radiation since the effect is small.
s for theverticallyor horizontallypo-
refers to thes for the third and fourth
is the atmospheric transmis-
A. Atmospheric Parameterization
We use a one-layer isotropic atmosphere approximation be-
cause the WindSat frequency band set does not provide the in-
formation necessary to estimate atmospheric profiles. In addi-
tion, the one-layer atmosphere approximation facilitates rapid
evaluation of the forward model for the retrieval algorithm. The
atmospheric transmissivity is taken to be
and cloud liquid water, respectively.
peratures are parameterized in terms of effective upwelling and
downwelling atmospheric temperatures,
is the Earth incidence angle, andand are
frequency so that the parameterized forward model matches
a plane-parallel atmospheric radiative transfer model. The ra-
diative transfer calculation uses the dry-air (primarily oxygen)
and water vapor absorption models given in . Our current
model excludes precipitating clouds. For nonprecipitating
clouds at WindSat frequencies, scattering from cloud liquid
water is negligible because the drop size is small relative to
the radiation wavelength. The cloud liquid water absorption
coefficient is therefore proportional to the cloud liquid water
content and given by the Rayleigh approximation . We use
a double Debye model for the dielectric constant of water .
Atmospheric profiles from NCEP were used for the radiative
transfer calculations. The profiles were taken from the 1st and
15th of each month between July 2001 and June 2002 on a
1 longitude/latitude grid. These data were filtered to only
include grid points that are in the ocean between
andare computed at each
COEFFICIENTS FOR THE ATMOSPHERIC PARAMETERS
AT EACH WINDSAT FREQUENCY
latitudeand at least 75 km from land. In addition,we only
included points where
We use least squares fits to the radiative transfer modeling
results tocalculate theatmospheric parameters for eachforward
model evaluation. The form of the fits are
mm to exclude possible rain.
These functional forms are similar to those used by Wentz
and Meissner  with the following differences. Wentz and
and include an additional term which includes a
dependence. We also have chosen to use the
as a proxyfor cloudtemperature, whileWentzand
Meissner used. We have intentionally avoided connecting
the atmospheric parameterization to
atmospheric and surface parameters during our initial forward
model development. A separate set of coefficients is needed
for each WindSat frequency. The full set of coefficients used is
given in Table IV.
are the coefficients derived from the least squares
are in millimeters, andandand are in Kelvin.
to limit coupling of the
B. Sea Surface Emissivity
scattering and emission from the sea surface that
even periodic functions of
. A Fourier cosine series for
Fourier sine series for
, can be used to accurately represent the sea surface
emission , , .
We consider the emissivity and reflectivity of the sea sur-
face as determined by a combination of the effects from large-
foam. The two-scale model approximation of dividing the wave
spectrum into large-scale gravity waves and small-scale capil-
lary waves has been shown to provide general agreement with
andare odd periodic
, expanded to the second har-
602 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 3, MARCH 2006
radiometer brightness temperature measurements from aircraft
tain an initial approximation of the sea surface emissivity and
reflectivity and then use empirically derived corrections to ac-
count for modeling errors and sea surface foam. Emission from
sea surface foam increases with increasing wind speed due to
increasing foam coverage of the sea surface . The presence
of foam increases the measured
emissivity than water , . We have chosen to account for
sea surface foam emission using only empirical estimates be-
. However, this is a subject of our current work and we an-
ticipate that future versions of our forward model will include
an explicit foam formulation.
the large- and small-scale waves, the hydrodynamic modulation
, the scalar multiplier of the wave spectra. We continue
to investigate improvements to the assumed wave spectrum
model based on analysis of WindSat
to predictions from the two-scale model. The wave spectrum
parameters we are currently using for the Durden–Vesecky
model, which are based on our initial analysis of WindSat
are described here. The scalar multiplier,
nitude of all the harmonics including the isotropic components
and where the magnitudes increase with increasing
. We have used
used in  andused in  and . The cutoff
wavenumber primarily affects the magnitude of the second
harmonic terms where the magnitudes increase with increasing
cutoff wavenumber. We have chosen a cutoff wavenumber
between the large- and small-scale waves of
used in ,used in , and
term is the electromagnetic wavenumber for the
term is defined in terms of the upwind slopes as in  but
modified to take on minimum and maximum values of 0 and
2, respectively, as opposed to the 0.5–1.5 range of . This
range of values for the hydrodynamic modulation increases the
magnitude of the first harmonic of the azimuthal wind direction
dependence of the third Stokes
the WindSat measurements. Our values for
wavenumber are close to those given in  and within the
ranges of values in the previous studies.
We use a Gaussian model of the long wave slope probability
distribution function in the two-scale model calculations. The
“modified Stogryn” model of the sea water permittivity is used
, with a fixed sea surface salinity of 34 psu. Variations in
salinity have a small impact on
small changes in the
s at 6.8 and 10.7 GHz. We plan to use a
sea water salinity climatology to improve sea surface tempera-
ture retrievals in future work.
To develop empirical corrections to the sea surface emis-
sivitywe usevaluesfor thegeophysicalparametersfrom NCEP,
QuikSCAT , SSM/I, and TMI collocated to the training set
of WindSat SDRs described in Section II. We use
from NCEP analysis closest in time and spatially interpolated
to the location of the WindSat SDRs. We use wind speed
s because foam has a higher
s and comparisons
, affects the mag-
as opposed to the
s, which is indicated by
and the cutoff
retrievals resulting from
and direction from QuikSCAT retrievals within 25 km and
60 min of the WindSat measurement when available. Before
collocation, the eight retrieval cells along both edges of the
QuikSCAT swath were removed, because they contain less than
the optimal four beam combinations, and have degraded wind
vectors . When a QuikSCAT matchup is not available we
use NCEP wind speed and direction within 1 h of the NCEP
analysis time and spatially interpolated to the location of the
WindSat SDRs. Finally, we use SSM/I and TMI retrievals that
are averaged into 0.25
0.25 longitude-latitude cells for
and(see geophysical data at http://www.remss.com). The
SSM/I and TMI observations are collocated to within 25 km
and 40 min of the WindSat observations.
We use the values for
dataset and the two-scale model to calculate values for emis-
, atmospheric parameters, and
andalong with the
SDRsandare thenusedin(7)and (8)tosolvefor the“measured
emissivity” of the ocean surface,
empirical corrections to the emissivity using least squares fits
to the difference between
wind speed bins. Our analysis showed that the following form
worked well for the
andfrom the matchup
. These values for
s from the corresponding
. We then calculate
We neglect variations of
within a wind speed bin and calculate empirical fits of the form
anddue to changes inand
A different set of coefficients is calculated for each 2-m/s wide
wind speed bin and each WindSat channel where
channel. The corrections for
andchannels the correction is applied as a ratio to
the result from the two-scale model, e.g.,
and where the model subscript denotes the
harmonic term calculated from the two-scale model using the
andfrom the wind speed bin and the nominal Earth
as a radiometer calibration offset.
We assume that variations in the Earth incidence angles that
are within the nominal ranges (about 0.6 ) have a negligible ef-
fect on the corrections. This is valid for the
because the corrections account for less than 5% of the total
except at very high wind speeds. The effects of Earth inci-
dence angle and sea surface temperature variations on the
andchannels are small but these effects, as predicted by the
two-scale model, are included since the empirical corrections
are applied as a scaling factor. The corrections do not account
for modeling errors in the azimuthal harmonics of the wind di-
rection dependence for the
wind direction dependence in the forward model for
is just the result from the two-scale model. We have developed
empirical corrections to the sea emissivity for wind speeds up
to 20 m/s; therefore, we limit the results shown in this paper to
wind speeds less than 20 m/s. The model function for high wind
speeds will be improved in future work.
andare added to. For
channels; therefore, the
BETTENHAUSEN et al.: NONLINEAR OPTIMIZATION ALGORITHM FOR WINDSAT603
The two-scale model is computationally expensive and the
full calculation cannot practically be done during the retrieval
process. Therefore,we calculateemissivity valueswith thetwo-
scale model and store the results in a three-dimensional lookup
and Earth incidence angle. The emissivity used
for the retrievals is calculated by linearly interpolating between
values in this table. The maximum
interpolation method is less than 0.1 K.
error introduced by this
V. RETRIEVAL PERFORMANCE
Wind vector retrievals are the primary focus of this paper.
However, analysis of the
tional information about how well the retrieval algorithm is per-
to the wind vector retrievals because our algorithm simultane-
ously retrieves all five geophysical parameters. In this section,
we present measures of retrieval performance by comparing our
WindSat retrievals with NCEP for
wind speed and direction, and SSM/I retrievals for
using the WindSat dataset described in Section II. The results
shown here are intended to demonstrate the efficacy of the re-
trieval algorithm rather than to verify accurate calibration.
Ambiguity selection has only a small impact on our sea sur-
face temperature retrievals, and the effect on the water vapor
and cloud liquid water retrievals is negligible. Therefore, the re-
for the ambiguity selected after median filtering with nudging.
For the wind vector results the ambiguity selection method is
more important, and we consider the differences that are related
to ambiguity selection. A 25-km collocation distance window
is used for all of the collocated (matchup) datasets. This dis-
tance is roughly half the diameter of the WindSat SDR foot-
print. For datasets where there are multiple measurements that
satisfy both the temporal and spatial thresholds, the measure-
ment closest to the WindSat location is used. The retrieval re-
from the six-month dataset.
andretrievals provides addi-
, QuikSCAT retrievals for
A. Water Vapor and Cloud Liquid Water
Wecompare ourwatervaporandcloudliquidwater retrievals
to SSM/I retrievals (http://www.remss.com). It is desirable to
use the smallest collocation time window that is feasible due to
the high temporal variability of cloud liquid water. A 40-min
window is the smallest window that can be used while still al-
yields more than 30 million matchups for the six-month dataset.
The estimated bias and RMS errors for the SSM/I water vapor
retrieval algorithm are 0.6 and 1.0 mm, respectively . While
no in situ measurements for cloud liquid water over the ocean
are available for validation Wentz  used an analysis of the
distribution of retrieved cloud liquid water to estimate a RMS
retrieval accuracy of 0.025 mm.
The overall differences between our WindSat water vapor re-
and a RMS difference of 1.05 mm. Fig. 2 shows the difference
in millimeters between the WindSat and SSM/I water vapor re-
trievals versus the SSM/I water vapor. The differences were cal-
versus SSM/I water vapor.
Difference between the WindSat and SSM/I water vapor retrievals
than 33 million collocations.
Histograms of the WindSat and SSM/I water vapor retrievals for more
retrievals versus SSM/I retrievals.
Difference between the WindSat and SSM/I cloud liquid water
culated for measurements in 5-mm water vapor bins. The max-
imum RMS difference is about 2 mm at 65 mm, or about 3%,
and the maximum bias difference is slightly larger than 1 mm.
Fig. 3 shows the corresponding histograms for the water vapor
retrievals. The histograms agree well with only small differ-
ences above 60-mm water vapor. The differences we show here
between our WindSat retrievals and SSM/I retrievals are also
on the order of (or better than) the differences between various
SSM/I algorithms noted in  and .
Fig. 4 shows the difference in millimeters between the
WindSat and SSM/I cloud liquid water retrievals versus the
604IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 3, MARCH 2006
more than 33 million collocations.
Histograms of the WindSat and SSM/I cloud liquid water retrievals for
versus NCEP ? .
Difference between the WindSat ?
retrievals and NCEP ?
retrieved SSM/I cloud liquid water. The differences were cal-
culated for measurements within 0.02-mm cloud liquid water
bins. Fig. 5 shows the corresponding histograms for the cloud
liquid water retrievals. The high spatial and temporal variability
of cloud liquid water makes quantitative evaluation of these
results difficult since a substantial portion of the differences
may be due to collocation differences. However, the WindSat
and SSM/I retrievals are in good qualitative agreement.
B. Sea Surface Temperature
We have used NCEP sea surface temperature data with an
analysis time within one hour of the WindSat retrievals to
evaluate our sea surface temperature retrievals. The one hour
window is small enough such that temporal changes in
should not affect our results. The NCEP
interpolated in space to the WindSat measurement location.
The dataset includes about 25 million matchups. We expect the
accuracy of the NCEP
data used here to be similar to the
accuracy of the Reynolds optimum interpolated (OI)
 which is produced using similar methods. The Reynolds
data have an overall standard deviation error of about
0.5 K and bias errors less than or about 0.1 K , .
and the interpolated NCEP
dard deviation is 0.98 K. Fig. 6 shows the difference in degrees
Celsius between the WindSat retrievals for
results versus the NCEP
in 2 bins. Fig. 7 shows the same
results are linearly
K, and the stan-
and the NCEP
versus NCEP wind speed.
Difference between the WindSat ?
retrievals and NCEP ?
using the forward model.
Derivative of ? with respect to ?
at 6.8 and 10.7 GHz calculated
differences plotted versus NCEP wind speed in 2-m/s wind
speed bins. There is generally good agreement for
to 30 C range and for
and higher wind speeds the standard deviation of the
ences increases significantly. There are two reasons for the ob-
served increases in the standard deviation of the
at lower temperatures and higher wind speeds. One is primarily
a low-temperature effect, and the other is primarily a high wind
speed effect. It is difficult to separate the contribution of these
effects because the global mean
retrieval accuracy decreases as
sensitivity to changes in
and 10.7 GHz decreases. This behavior is shown in Fig. 8
, as calculated from our foward model for zero
wind speed and cloud liquid water and 20-mm water vapor, is
. Theretrievals primarily rely upon the
have the highest sensivity to
to atmospheric variations. As discussed in Section II,
6.8 GHz is only used for 55 of the 80 retrievals in the WindSat
forward scan. The
channels also have a significant role in
retrievals because they are needed to separate the wind speed
and atmospheric contributions to the
contribution. The variations in
in the permittivity of sea water with
bias nearoccurs because, even with the two-stage
in the 20
m/s. At lower temperatures
decreases with increasing
and are relatively insensitive
measurements from the
are due to changes
. The large positive
BETTENHAUSEN et al.: NONLINEAR OPTIMIZATION ALGORITHM FOR WINDSAT605
versus NCEP wind direction relative to the WindSat look direction for three
wind speed ranges.
Bias difference between the WindSat ?
retrievals and NCEP ?
retrieval algorithm, the sensitivity to
enough at low
a priori value of 287 K.
increases due to the increasing dependence of the
wind direction. Small forward model errors for the directional
dependence can produce substantial errors in the
in the0–5-m/s range thereis no
there is a large bias for wind speeds in the range of 10–15 m/s.
directional signals for
and). In Figs. 6 and 7, the bias in
increases the standard deviation of the differences but the mean
is approximately zero.
variations is not strong
from the constantto pull the retrieved
retrieval accuracy decreases as wind speed
s on the
bias withwind direction,but
are largest there (since they
C. Wind Vectors
We use QuikSCAT wind vector retrievals within one hour of
the WindSat measurements to evaluate WindSat wind vector
mise between minimizing the time difference and maximizing
global coverage. The resulting dataset contains more than
29 million matched retrievals. For comparison to the results
presented here, analysis of the differences between QuikSCAT
wind speed and direction and in situ measurements from bouys
,  show RMS wind speed differences of 1.2 m/s. The
directional differences between QuikSCAT and bouys are about
20 at 5-m/s wind speed and rapidly decrease to values in the
range of 10 to 15 for wind speeds above about 8 m/s.
The histograms of wind speed and direction retrievals in
Figs. 10 and 11 show good agreement between WindSat and
QuikSCAT. These plots are based on the selected ambiguity
after median filtering with nudging. The difference between
the WindSat and QuikSCAT wind direction histograms near
90 is due to differences at wind speeds below 5 m/s where
the directional signal is small and, as a result, the WindSat
retrievals are less accurate.
Fig. 12 shows the difference between WindSat and
QuikSCAT retrieved wind speed versus the QuikSCAT wind
more than 29 million collocations.
Wind speed histograms for WindSat and QuikSCAT in 1-m/s bins for
more than 29 million collocations.
Wind direction histograms for WindSat and QuikSCAT in 5 bins for
in 2-m/s wind speed bins.
Difference between WindSat and QuikSCAT retrieved wind speeds
speeds for the selected ambiguity after median filtering with
nudging. The bias of the difference is less than about 0.2 m/s
for wind speeds less than 20 m/s. The standard deviation of
the difference is below 1 m/s for wind speeds below about
12 m/s. At higher wind speeds the standard deviation increases
but remains well below 2 m/s. The overall RMS difference for
all wind speeds is 0.89 m/s. The corresponding overall RMS
differences for the ambiguities based on
1.04, and 1.28 m/s for the first through fourth rank ambiguities,
respectively. This shows that there are small but significant
differences between the wind speeds retrieved for the four
rank are 0.91, 0.97,
606 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 3, MARCH 2006
DIFFERENCE BETWEEN WINDSAT AND QUIKSCAT WIND
DIRECTION RETRIEVALS VERSUS QUIKSCAT WIND SPEED
retrievals in 2-m/s wind speed bins.
RMS difference between WindSat and QuikSCAT wind direction
The RMS differences between WindSat and QuikSCAT re-
trieved wind directions versus the QuikSCAT wind speed are
shown in Table V and Fig. 13. The biases, which are not shown
in the table or the figure, are all small (less than 3 ) for all wind
speeds less than 20 m/s. In the table, “First” refers to the first
ranked ambiguity; “MF” refers to results with median filtering;
“MF/NG” refers to results with median filtering and nudging
with the NCEP background wind field; and “Closest” refers to
direction. The overall RMS wind direction difference is 30.0
for the selected ambiguity (MF/NG) and 60.2 for the first rank
Fig. 14 shows an example of a retrieved wind field from
WindSat data for September 12, 2003 with the wind speed scale
shown in the colorbar. For clarity, the vectors are plotted for a
subset of the retrieval cells, but no averaging is done. Vectors
are not plotted for wind speeds less than 3 m/s. This figure
gives qualitative verification that the wind vector retrieval is
producing realistic wind fields. The apparent noise in the wind
convergence zones is not unexpected because the WindSat
footprint used here (40 km
fully resolve the fronts and many of the abrupt changes in wind
direction are occurring at low wind speeds.
60 km) is likely too large to
wind speed is indicated by the color. These retrievals are based on the selected
ambiguity after median filtering with nudging (MF/NG).
Retrieved wind field from WindSat data for September 12, 2003. The
D. Ambiguity Selection
The difference between the wind direction retrieval perfor-
mance for the first rank and the closest ambiguities, as shown
in Fig. 13, demonstrates the importance of ambiguity selec-
tion. Fig. 15 shows the ambiguity selection skill for the se-
lected ambiguity after median filtering with nudging and for
each of the four ranked ambiguities. Skill is defined as the per-
centage of retrievals where the ranked (selected) ambiguity was
the closest to the QuikSCAT wind direction. The selected am-
biguity (MF/NG) is the closest over 80% of the time above
m/s. The skill for the third and fourth ranked ambi-
guities is small above
m/s so that the closest ambiguity
is usually the firstor second rankambiguity. The combinedskill
for the first rank and second rank ambiguities is always greater
than 50% and is greater than 90% above about
We use two steps for ambiguity selection as discussed in Sec-
tion III. The discussion of ambiguity selection presented here
focuses on the first step of ranking each ambiguity based on
Detailed evaluation of the effects of the second step of median
filtering, and in particular median filtering with nudging, should
include the study of local phenomena such as at weather fronts
and the impact of nudging on the retrieved wind fields. Such a
study is beyond the scope of this paper.
Fig. 15 shows that skill varies significantly with wind speed.
Skill for a radiometer also depends on
and second ranked ambiguities is shown in Fig. 16 versus the
the QuikSCAT wind direction for
skillhas two maxima near
0 and 180 . The second rank skill varies inversely with
the directional dependence of the
. The skill for the first
– m/s. The first rank
90 and 270 and minima near
BETTENHAUSEN et al.: NONLINEAR OPTIMIZATION ALGORITHM FOR WINDSAT607
Fig. 15. Ambiguity selection skill versus QuikSCAT wind speed.
ambiguities versus the QuikSCAT wind direction for ? ? ?–? m/s.
model) at 8-m/s wind speed and ?
Directional dependence of the ? s (as calculated by our forward
? ??? K for 10.7 GHz.
Fig. 17 shows the directional dependence of the 10.7-GHz
s (as calculated by our forward model) at
K. We have chosen
m/s for illustration
purposes because the wind direction difference for the closest
ambiguity, as shown Fig. 13, is near the minumum value while
first rank and closest ambiguities than there is at higher wind
signals are nearly pure second harmonic for all
frequencies and wind speeds and are, therefore, approximately
have significant first and second harmonic terms; so, while
is the same at
and are distinct. Also, the ratio of the first and second har-
varies with frequency and wind speed, and there-
values at which
maximum also vary.
as zero mean with a standard deviation of
m/s, the peak-to-peak amplitude of the
directional signals is less than the corresponding
while the opposite is true for the
Therefore, the retrieval of
for a specified a priori depends
primarily on the
andmeasurements. However, for some
, ambiguity selection is dependent on
or. Then both
and contributions to the
and a second ambiguity at
Therefore, ambiguity selection must rely on the
and. Conversely, near
is negative near
lection, and first rank skill is high. The second rank skill shown
in Fig. 16 is higher than the first rank skill near
, which indicates a likely forward model error. Also, note
that for the lowest wind speeds the peak-to-peak signal for the
all polarizations is less than the corresponding
which leads to poor first rank skill for all
We have chosen to use four separate a priori wind directions
to retrieve four ambiguities. This choice is supported by the di-
rectional dependence of the
shows the retrieved wind direction from WindSat versus the
QuikSCAT wind direction for all four of the ambiguities for
– m/s (left panel) and
contours have been normalized so that the sum of all points in
each 5 QuikSCAT wind direction bin is 100. This was done so
that the shape of the overall wind direction distribution, shown
in Fig. 11, would not obscure the changes due to the directional
dependence of the
s. There are four ambiguities for each
as can be seen by noting four local maxima along any vertical
line in the plots. The four ambiguities are spaced at nearly 90
intervalswhenisnear0 ,90 ,180 ,or270 andaregrouped
in two pairs when
is near 45 , 135 , 225 , and 315 . This
pairing of ambiguities near the local amplitude maxima in
andresults from the uncertainty in the measurements and
forward model (
, as discussed above). Our discus-
sion here of the ambiguity selection features has been simpli-
fied for the sake of illustration. We have explained the primary
and . Thesignals
or has a local minimum or
accounts for measurement
(see Table II).
are near zero, and
will be the same for an
and positive near.
andmeasurements. Fig. 18
–m/s(right panel). The
608IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 3, MARCH 2006
Fig. 18.Two-dimensional histogram of the ambiguities in 5 ? 5 bins of WindSat versus QuikSCAT wind direction relative to the WindSat look direction.
features using only the directional dependence of the
retrieval algorithm is actually considering the ambiguities in a
five-dimensional space. However, variations between ambigui-
ties intheretrievedvaluesfortheotherparametersare relatively
We have developed a nonlinear iterative retrieval algorithm
for wind vector retrievals from WindSat data. The algorithm
can easily be adapted to use different subsets of measured
s, and therefore, it can easily be adapted for use with future
polarimetric microwave radiometers. The comparisons of our
WindSat retrievals to QuikSCAT retrievals verify that the
retrieval algorithm is performing well. The accuracy of the
retrievals is limited by measurement noise and the accuracy of
the forward model. The differences between the forward model
and the measurements are currently dominated by modeling
errors as can be seen by comparing the values given in Tables I
and II. The results presented here are for the lowest resolution
footprint of about 40 km
60 km. Work is ongoing to produce
s and wind vector retrievals at a higher resolution
of about 25 km
40 km. For this higher resolution, beam
averaging will provide less noise reduction and measurement
noise will contribute significantly to the differences between
the forward model and the
clear that minimizing the measurement noise in the third and
fourth Stokes measurements is important for measuring wind
direction with a polarimetric microwave radiometer. This will
be even more important as our forward model is improved.
The RMS difference between WindSat and QuikSCAT wind
speed retrievals is less than 1.0 m/s for wind speeds below 10
andmeasurements. It is
m/s. The increase in the standard deviation of the wind speed
difference at higher wind speeds is likely due to several factors.
The largest concentration of WindSat-QuikSCAT matchups
is in the midlatitudes with relatively few matchups at high
latitudes where sustained high winds are more prevalent. As
a result, many of the high wind speed cases are from storms
where the spatial and temporal variability of the wind speeds is
high. The storm-related localized variabilities would increase
the wind speed differences between WindSat and QuikSCAT
at high wind speeds. Another possible cause of the increasing
variability at high wind speeds is errors in the forward model
function for the directional dependence of the
channels. This is similar to the effect of these forward model
errors on the
retrieval performance that are shown in Fig. 9.
Finally, for high wind speeds less training data are available for
the empirical corrections to the sea surface emissivity which
results in greater uncertainty in the forward model.
The wind direction performance results are primarily wind
comes greater than the noise in the measurements around 4 m/s.
In addition, forward modeling errors for the
are currently on theorder of 0.1 K. The combination of these ef-
5 m/s. The directional signals for the
superposed on much larger isotropic signals. The forward mod-
eling errors for the isotropic signals are on the order of 0.7 K
and 1.0 K for at 10.7 GHz and higher at the higher
frequencies. Therefore, the
a greater impact on the wind direction retrievals above about
The retrieval error covariance matrix,
(4) provides an estimate of the expected variance of the re-
directional signals have
, calculated from
BETTENHAUSEN et al.: NONLINEAR OPTIMIZATION ALGORITHM FOR WINDSAT 609
trievals. The square root of the diagonal of
mate the standard deviation of the retrieval errors provided the
foward model, the measurement error covariance matrix, and
the a priori are good approximations. The overall mean values
, 1.0 mm for, and 0.032 mm for
are similar to the overall standard deviation of the differences
QuikSCAT, and SSM/I as presented in Section V. Those values
are 0.98 K for
, 0.89 m/s for
mmfor .Forwinddirection,we computedthemeanvaluesfor
the square root of the diagonal of
The results are within 2 of the values for differences between
the closest WindSat ambiguity and the QuikSCAT wind direc-
tion as give in Table V. These results confirm that our retrieval
algorithm is performing well.
The retrieval results described in this paper provide a base-
line for WindSat performance and show that our retrieval algo-
rithm effectively retrieves all five geophysical retrieval param-
eters. These results have been obtained using a first-generation
forward model function for the WindSat
improveour forward modelfunction and WindSat calibrationto
improve the retrievals. We anticipate that retrieval performance
will improve significantly as our forward model is refined.
. These values
, 0.95 mm for , and 0.045
for the same 2-m/s wide
s. We continue to
The authors thank J. Johnson for providing his implementa-
tion of the two-scale sea surface emission and reflection model.
The authors also thank B. Johnston and L. Connor for their as-
sistance in assembling the collocated datasets and E. Twarog
for comments on the manuscript. SSM/I and TMI data are pro-
duced byRemote SensingSystemsand sponsoredbytheNASA
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Michael H. Bettenhausen (S’88–M’95) received
the B.S., M.S., and Ph.D. degrees in electrical engi-
neering from the University of Wisconsin, Madison,
in 1983, 1990, and 1995, respectively. His graduate
research focussed on theoretical and computational
studies of radio-frequency heating in plasmas.
While employed by Mission Research Corpora-
tion, Santa Barbara, CA, from 1997 to 2000, he did
software development and algorithm research for
particle simulation. In 2000, he joined Integrated
Management Services, Inc., Arlington, VA, where he
worked on projects for analysis and processing of hyperspectral remote sensing
data. He joined the Remote Sensing Division, Naval Research Laboratory,
Washington, DC, in 2002. His current research interests include development
of forward models and retrieval algorithms for the WindSat polarimetric
Craig K. Smith (M’03) received B.A. degrees in
physics and mathematics in 1983, and the Ph.D.
degree in physics in 1995, all from the University
of California at Berkeley. His doctoral thesis was on
the determination of supernova production rates in
From 1995 through 1996, he was a Post Doctoral
Fellow in the Energy and Environment Division,
Lawrence Berkeley Laboratory, where he examined
methods of measuring and mitigating urban heat
islands. From 1997 to 2001, he worked for Remote
Sensing Systems, Santa Rosa, CA, where he developed sea surface tempera-
ture, wind speed, and wind direction retrieval algorithms, a sensor model and
sensor requirements, and fully polarimetric expressions for microwave antenna
cross-polarization for the Conical Scanning Microwave Imager Sounder
(CMIS). From 2001 through 2004, he was with Computational Physics, Inc.,
Springfield VA, where he developed statistical and physical retrieval algorithms
and operational retrieval code for WindSat. In November 2004, he joined
The Aerospace Corporation, Los Angeles, where he conducts analyses of
operational microwave radiometer requirements, retrieval algorithms, and
Dr. Smith is a member of Sigma Pi Sigma and the AGU.
Richard M. Bevilacqua photograph and biography not available at the time of
Nai-Yu Wang received the B.S. degree in meteorology from the Chinese Cul-
ture University, Taipei, Taiwan, R.O.C., in 1987, and the M.S. degree in atmo-
spheric sciences and the Ph.D. degree from the University of Michigan, Ann
Arbor, in 1993 and 1998, respectively. Her doctoral thesis focussed on esti-
mating sea surface temperature using satellite microwave radiometer and scat-
After graduation, she was a Postdoctoral Fellow at the University of
Michigan, working on microwave polarimetric ocean surface emission
modeling. From 2001 to 2005, she was with the University Corporation for At-
mospheric Research (UCAR), Boulder, CO, working with the NOAA/NESDIS
Office of Research and Applications and the Naval Research Laboratory devel-
oping algorithms to retrieve ocean surface wind vectors and atmospheric water
vapor and cloud liquid water from WindSat measurements. She is currently
at the Earth Science System Interdisciplinary Center, University of Maryland,
College Park, working on the precipitation applications of microwave satellite
remote sensing measurements over land and ocean.
Peter W. Gaiser (S’91–M’93–SM’04) received
the B.S. degree in electrical engineering from
Virginia Polytechnic Institute and State University,
Blacksburg, in 1987, and the Ph.D. degree from
the University of Massachusetts, Amherst, in 1993,
where he studied microwave remote sensing, with
emphasis on synthetic aperture interferometric
He has been with the Naval Research Laboratory
(NRL), Washington, DC, since 1993, and currently
Acting Head of the Remote Sensing Physics Branch,
Remote Sensing Division at NRL. While at NRL, he has been involved in po-
larimetric radiometry research. His research interests also include instrument
design, data collection, and model development specifically for the purpose of
ocean wind vector measurements from space. He is the Principal Investigator
for the WindSat spaceborne polarimetric microwave radiometer demonstration
Stephen Cox received the Ph.D. degree in applied mathematics from the Uni-
versity of Maryland, College Park, in 1988. His graduate research focussed on
the analysis and computation of nonlinear waves with applications to fluid dy-
namics, continuum mechanics, and meteorology.
From 1989 to 1994, he worked for Hughes STX Corporation, Lanham, MD,
providing postprocessing data analysis and algorithm development for ozone
profile retrieval from SBUV for NASA’s Goddard Space Flight Center, Green-
belt, MD. In 1996, he joined Science and Technology Corporation, Hampton,
HIRS for NOAA/NESDIS’s Environmental Product Systems, Suitland, MD,
with software development, analysis, and operational monitoring. He currently
works for Computational Physics Incorporated, Springfield, VA, providing data
analysis and software development for WindSat ocean wind retrievalsat the Re-
mote Sensing Division, Naval Research Laboratory, Washington, DC.