# A nonlinear optimization algorithm for WindSat wind vector retrievals

**ABSTRACT** WindSat is a space-based polarimetric microwave radiometer designed to demonstrate the capability to measure the ocean surface wind vector using a radiometer. We describe a nonlinear 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 the WindSat brightness temperatures. Empirical corrections to the 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 numerical weather model. We focus primarily on the application to wind vector retrievals.

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**ABSTRACT:**In this paper, a wind vector retrieval algorithm has been developed for the first satellite borne polarimetric radiometer WindSat based on a simplified forward model. Sea surface wind speed retrieval algorithm was developed using multiple linear regression. Multiple ambiguous wind directions were estimated with Maximum Likelihood, and the median filter was used to remove the ambiguity of wind direction. Using the buoy wind vector from U.S. National Data Buoy Center (NDBC) as ground truth, the RMS error of the retrieved wind speed is 1.17 m/s with a bias of 0.13 m/s. The wind direction retrieval results are agree with the results given in published research.Remote Sensing, Environment and Transportation Engineering (RSETE), 2012 2nd International Conference on; 01/2012 -
##### Conference Paper: WindSat ocean surface wind vector and sea surface temperature retrievals

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**ABSTRACT:**WindSat is the first spaceborne fully polarimetric microwave radiometer. We describe a nonlinear optimization algorithm for WindSat wind vector and sea surface temperature retrievals. The algorithm simultaneously retrieves the atmospheric water vapor and cloud liquid water, sea surface temperature and the ocean surface wind vector that matches our forward model to the measured brightness temperatures. The forward model is based on parameterizations of a radiative transfer model with empirical corrections. We describe our retrieval algorithm and present analysis of retrieval performanceOCEANS, 2005. Proceedings of MTS/IEEE; 01/2005 -
##### Conference Paper: A digital correlation full-polarimetric microwave radiometer design and calibration

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**ABSTRACT:**Since the first polarimetric radiometer satellite WindSat was launched on January 6, 2003 [1] and has been working normally until now, microwave polarimetry has demonstrated its ability on retrieving global wind vector [2].The first digital correlation microwave polarimeter was designed in 2001 [3], and digital technology has made a great progress during the last decade. We describe design of a digital-correlation full-polarimetric microwave radiometer system (DPMR), which operates at the same frequencies as WindSat, i.e. 6.8, 10.7, 18.7, 23.8 and 37.0 GHz. A direct digital cross-cerrelation technique is used in the system to produce four Stokes parameters at each frequency simultaneously. The main specifications are list in TABLE 1.Geoscience and Remote Sensing Symposium (IGARSS), 2012 IEEE International; 01/2012

Page 1

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 3, MARCH 2006597

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

theWindSatbrightnesstemperatures.Empiricalcorrectionstothe

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.

I. INTRODUCTION

T

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

liquid water

.

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-

eters

.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 [1], was launched in January 2003.

, columnar

[2]–[4]

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

(e-mail: michael.bettenhausen@nrl.navy.mil).

C. K. Smith was with Computational Physics, Inc., Springfield, VA 22151

USA. He is now with The Aerospace Corporation, Los Angeles, CA 90009

USA.

N.-Y. Wang was with the Office of Research and Applications, National En-

vironmental Satellite, Data, and Information Service, National Oceanic and At-

mosphericAdministration,CampSprings,MD20746USA.Sheisnowwiththe

EarthSystemScienceInterdisciplinaryCenter,UniversityofMaryland,College

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.

where

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

wind vector.

Dual-polarization observations from radiometers such as

the Special Sensor Microwave/Imager (SSM/I) [5] and the

Advanced Microwave Scanning Radiometer–EOS (AMSR-E)

[6] 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., [7]–[9]). Wentz and Meissner [10] used a

multiple linear regression algorithm where the coefficients are

determined using brightness temperatures

a physically based model function. Wentz [11] used physically

based model functions for the SSM/I

equations in four unknowns which are solved using an iterative

procedure. Wentz and Meissner [10] 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

aircraftmeasurementsandsimulateddata.Studiesusingaircraft

measurements are limited in scope because only a relatively

small amount of data is available. Piepmeier and Gasiewski [4]

usedaircraftdata toretrievewinddirectionusingmeasurements

of

andat10.7and37GHzand

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 [12] 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 [13]. Here

, the relative wind direction, is defined as the compass

and,

and, provides sufficient

s simulated with

s, to obtain a set of four

andat18.7GHz.

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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598 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 3, MARCH 2006

TABLE I

NOMINAL NEDT VALUES FOR RAIN-FREE OCEAN

RETRIEVAL CELLS (IN KELVIN)

we present a nonlinear optimization algorithm which simulta-

neously retrieves

algorithm is designed to produce four solutions (ambiguities)

for each set of WindSat

s using a physically based forward

model.

WebeginbydescribinginSectionIItheWindSatdatasetused

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.

and from WindSats. The

II. WINDSAT DATA DESCRIPTION

A description of the WindSat sensor and data processing

system is provided in [1]. 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 [13] 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

positionontheedgeoftheswath).Thereare80pixelsinthefor-

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

in the

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

forward swath.

TheSDRsusedfortheretrievalstudiesdescribedinthispaper

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-

denceanglesaremorethan0.5 fromtheirnominalvalues.Data

are also excluded for rain, ice, radio-frequency interference at

10.7 GHz [14], 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 [15], was used for the training set.

Rain was considered to be present if any of the following con-

ditions were satisfied:

s are

III. RETRIEVAL ALGORITHM

A. Optimal Estimation

Our retrieval algorithm uses an optimal estimator [16] which

is a Gauss–Newton iterative method with a priori constraints.

The method is equivalent to minimization of the cost function

(1)

where the superscript

the state vector of quantities to be retrieved. The state vector

is comprised of

aprioriconstraintsaregivenbytheaprioristatevector,

the a priori error covariance matrix,

of the goodness of fit of the forward model, evaluated using

the retrieved state vector, to the measurements [16], [17]. An

estimate of the

is obtained from

indicates the matrix transpose andis

andor a subset thereof. The

,and

. Theis a measure

(2)

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

theforwardmodel,

.Theforwardmodelisdescribed

in Section IV.

The iteration used can be written

.

s used for the

(3)

The error covariance of the solution is approximated by

(4)

where

is the final iteration.

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BETTENHAUSEN et al.: NONLINEAR OPTIMIZATION ALGORITHM FOR WINDSAT 599

The matrix

the derivative of the forward model with respect to the state

parameters

, called the weighting function or kernel, is

(5)

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

covariance [16]

(6)

where

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

cept

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

andare chosen to be near the center of their respective

ranges:

K and

and are 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

rangeoftheretrievedparameters.Thesquarerootoftheapriori

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

states.

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

m/s and

, is chosen to

and biases toward

and low.

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

s. WindSat

ters are used. The first stage retrievals for

as a priori values for the second stage. The square root of the a

priorierrorcovariancevaluesare6K,4m/s,5mm,0.5mm,and

45 for

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

below.

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

tangentofthefirstharmonicupwindandcrosswindcomponents

of the wind vector from the two-stage regression algorithm de-

scribed in [13]. 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

matrixis oftenusedtosimplifytheretrievalcalculationsandthe

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.

Therearemanypossiblecontributionstothesecorrelations—we

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

in

improves the weighting of the channels for the retrievals.

s for all Stokes parame-

are used

will have

with

.

.

does not account

and) and the ob-

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600IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 3, MARCH 2006

TABLE II

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

estimates of

which, in turn, improves ambiguity selection.

is estimated using the differences between the measured

s from theWindSatSDRsand

model

s simulated withtheforward

where

The

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

usedforempiricalcorrectionstotheseasurfaceemissivity(Sec-

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

ranges:

m/s, m/s

m/s m/s, and

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

rootsofthediagonalelementsof

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 WindSats from the SDRs and

. Here the

m/s, m/s m/s,

m/s. The resulting error

s and the

areshowninTableIIwhere

for the

at 6.8 GHz are

TABLE III

? 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

frequenciesandpolarizations.ThevaluesshowninTablesIIand

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

for the

those for the

channels at the same frequency because

is more sensivite to changes in the wind vector and the atmo-

spheric parameters.

andchannels between

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

on the

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 [18]. 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 [13]. 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

Ouroverallapproachtodevelopmentofaforwardmodelisto

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-

denceoftheforwardmodelonempiricallyderivedrelationships

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

s.

We have developed a parameterized forward model similar to

that described in [10]. 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

s.

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BETTENHAUSEN et al.: NONLINEAR OPTIMIZATION ALGORITHM FOR WINDSAT 601

radiation and the direct emission are attenuated by the atmos-

phere. The

s at each WindSat frequency can be expressed as

(7)

(8)

where

larized

Stokes parameters. The sea surface emissivity for polarization

is ,andthecorrespondingreflectivityis

upwelling atmospheric brightness temperature at the top of the

atmosphere,

is the downwelling atmospheric brightness

temperature at the surface, and

sivity.

isthecosmicbackgroundradiationtemperature,which

is approximately 2.7 K. The

term is a correction factor to ac-

countfornonspecularreflectionoftheatmosphericdownwelling

radiation from the rough sea surface [19]. We are currently ne-

glectingtheazimuthaldependence of thereflected downwelling

radiationbecauseitisdifficulttomodelempirically,andtheoret-

ical work in this area has been done only recently [19]. We have

also neglected the effect of nonspecular reflection of the cosmic

background radiation since the effect is small.

refersto the

s, and

s for theverticallyor horizontallypo-

refers to the s for the third and fourth

. isthe

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

(9)

where

theverticalatmosphericabsorptionsduetooxygen,watervapor,

and cloud liquid water, respectively.

Theupwellinganddownwellingatmosphericbrightnesstem-

peratures are parameterized in terms of effective upwelling and

downwelling atmospheric temperatures,

is the Earth incidence angle, and andare

and

(10)

(11)

Values for

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 [20]. 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 [21]. We use

a double Debye model for the dielectric constant of water [22].

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

1 longitude/latitude grid. These data were filtered to only

include grid points that are in the ocean between

andare computed at each

and

TABLE IV

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.

(12)

(13)

(14)

(15)

(16)

where the

fits,

These functional forms are similar to those used by Wentz

and Meissner [10] with the following differences. Wentz and

Meissnerfit

toonlysecondorderin

order in

and include an additional term which includes a

dependence. We also have chosen to use the

thefit for

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, andand andare in Kelvin.

.Theyfit tofourth

dependence in

to limit coupling of the

B. Sea Surface Emissivity

Itfollowsfromreflectionsymmetrypropertiesofpolarimetric

scattering and emission from the sea surface that

even periodic functions of

, and

functions of

[2]. A Fourier cosine series for

Fourier sine series for

and

monic in

, can be used to accurately represent the sea surface

emission [3], [23], [24].

We consider the emissivity and reflectivity of the sea sur-

face as determined by a combination of the effects from large-

scalegravitywaves,smallscalecapillarywaves,andseasurface

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

and are

andare odd periodic

andand

, expanded to the second har-

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602 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 3, MARCH 2006

radiometer brightness temperature measurements from aircraft

[24],[25].Weuseatwo-scalemodelimplementation[19]toob-

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 [26]. The presence

of foam increases the measured

emissivity than water [27], [28]. We have chosen to account for

sea surface foam emission using only empirical estimates be-

causethereislargeuncertaintyincurrentfoamcoveragemodels

[29]. 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.

WehaveusedtheDurden–Veseckymodeloftheseaspectrum

[25],[30] withmodificationstothecutoffwavenumberbetween

the large- and small-scale waves, the hydrodynamic modulation

and

, 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

of

andwhere the magnitudes increase with increasing

. We have used

used in [30] and used in [24] and [25]. 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 [30], used in [25], and

[24]. The

term is the electromagnetic wavenumber for the

individualWindSatfrequencies.Thehydrodynamicmodulation

term is defined in terms of the upwind slopes as in [25] but

modified to take on minimum and maximum values of 0 and

2, respectively, as opposed to the 0.5–1.5 range of [25]. 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 [30] 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

[22], 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 [31], 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

s,

, affects the mag-

as opposed to the

versus

used in

s, which is indicated by

and the cutoff

retrievals resulting from

values

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 [32]. 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-

sivity,

, 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

and

and from the matchup

. These values for

s from the corresponding

. We then calculate

and in 2-m/s-wide

channels

(17)

We neglect variations of

within a wind speed bin and calculate empirical fits of the form

anddue to changes inand

(18)

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

the

andchannels the correction is applied as a ratio to

the result from the two-scale model, e.g.,

andwhere the model subscript denotes the

harmonic term calculated from the two-scale model using the

mean

andfrom the wind speed bin and the nominal Earth

incidenceangle.The

termfor the

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

and

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.

denotes the

andare added to . For

and channelsis used

and channels

channels; therefore, the

and

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BETTENHAUSEN et al.: NONLINEAR OPTIMIZATION ALGORITHM FOR WINDSAT 603

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

table in

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-

forming.The

and retrievalperformanceisalsorelevant

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-

sultswepresentinthissectionforthoseretrievalsareonlygiven

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-

sultspresentedareforeverythirddayofWindSatmeasurements

from the six-month dataset.

andretrievals provides addi-

, QuikSCAT retrievals for

and

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-

lowingforasufficientlylargedatasetwithglobalcoverage.This

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 [11]. While

no in situ measurements for cloud liquid water over the ocean

are available for validation Wentz [11] 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-

trievalsandtheSSM/Iretrievalsareabiasdifferenceof0.43mm

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-

Fig. 2.

versus SSM/I water vapor.

Difference between the WindSat and SSM/I water vapor retrievals

Fig. 3.

than 33 million collocations.

Histograms of the WindSat and SSM/I water vapor retrievals for more

Fig. 4.

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 [33] and [34].

Fig. 4 shows the difference in millimeters between the

WindSat and SSM/I cloud liquid water retrievals versus the

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604IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 3, MARCH 2006

Fig. 5.

more than 33 million collocations.

Histograms of the WindSat and SSM/I cloud liquid water retrievals for

Fig. 6.

versus NCEP ? .

Difference between the WindSat ?

retrievals and NCEP ?

results

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)

[35] which is produced using similar methods. The Reynolds

OI

data have an overall standard deviation error of about

0.5 K and bias errors less than or about 0.1 K [36], [37].

TheoverallbiasdifferencebetweentheWindSat

and the interpolated NCEP

values is

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

data

retrievals

K, and the stan-

and the NCEP

Fig. 7.

versus NCEP wind speed.

Difference between the WindSat ?

retrievals and NCEP ?

results

Fig. 8.

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

wind speed.

First, the

retrieval accuracy decreases as

because the

sensitivity to changes in

and 10.7 GHz decreases. This behavior is shown in Fig. 8

where

, as calculated from our foward model for zero

wind speed and cloud liquid water and 20-mm water vapor, is

plotted versus

. The retrievals primarily rely upon the

measurementsof

at6.8and10.7GHzbecausethosechannels

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

differ-

differences

decreases with increasing

decreases

at 6.8

and are relatively insensitive

at

measurements from the

are due to changes

[22]. The large positive

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BETTENHAUSEN et al.: NONLINEAR OPTIMIZATION ALGORITHM FOR WINDSAT605

Fig. 9.

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.

Second, the

increases due to the increasing dependence of the

wind direction. Small forward model errors for the directional

dependence can produce substantial errors in the

Fig.9showsthemean

differencesplottedversusNCEPwind

directionrelativetotheWindSatlookdirection.Forwindspeeds

in the0–5-m/s range thereis no

there is a large bias for wind speeds in the range of 10–15 m/s.

Thedifferenceislargestnear180 becausethemagnitudeofthe

directional signals for

and

vary as

and). In Figs. 6 and 7, the bias in

versuswinddirectionisaveragedoverallwinddirectionswhich

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

retrievals.

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

retrievalperformance.Thistimewindowischosenasacompro-

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

[38], [39] 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

Fig. 10.

more than 29 million collocations.

Wind speed histograms for WindSat and QuikSCAT in 1-m/s bins for

Fig. 11.

more than 29 million collocations.

Wind direction histograms for WindSat and QuikSCAT in 5 bins for

Fig. 12.

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

ambiguities.

rank are 0.91, 0.97,

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606 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 3, MARCH 2006

TABLE V

DIFFERENCE BETWEEN WINDSAT AND QUIKSCAT WIND

DIRECTION RETRIEVALS VERSUS QUIKSCAT WIND SPEED

Fig. 13.

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

theretrievedWindSatambiguitythatisclosesttotheQuikSCAT

direction. The overall RMS wind direction difference is 30.0

for the selected ambiguity (MF/NG) and 60.2 for the first rank

ambiguity.

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

Fig. 14.

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

thefirstrankskill.Thesefeaturescanbeexplainedbyexamining

the directional dependence of the

m/s.

.

. The skill for the first

– m/s. The first rank

90 and 270 and minima near

s.

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BETTENHAUSEN et al.: NONLINEAR OPTIMIZATION ALGORITHM FOR WINDSAT607

Fig. 15. Ambiguity selection skill versus QuikSCAT wind speed.

Fig.16.

ambiguities versus the QuikSCAT wind direction for ? ? ?–? m/s.

Ambiguityselectionskillfor(solid)thefirstand(dashed)secondrank

Fig. 17.

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 and

m/s for illustration

purposes because the wind direction difference for the closest

ambiguity, as shown Fig. 13, is near the minumum value while

thereisalargerdifferencebetweenretrievalperformanceforthe

first rank and closest ambiguities than there is at higher wind

speeds. The

signals are nearly pure second harmonic for all

frequencies and wind speeds and are, therefore, approximately

equal

at

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-

monics for

varies with frequency and wind speed, and there-

fore, the

values at which

maximum also vary.

Theoptimalestimationmethodeffectivelyweightseachmea-

sured

based onwhere

noiseandmodelingerror.Ifweneglecttheoff-diagonaltermsin

,themeasurementandmodelingnoisearecollectivelytreated

as zero mean with a standard deviation of

For

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

and measurements. However, for some

, ambiguity selection is dependent on

when

or. Then both

the

andcontributions to the

ambiguity at

and a second ambiguity at

Therefore, ambiguity selection must rely on the

surementswhichresultsinlowfirstrankskillnearboth

and. Conversely, near

is negative near

Then

and providesufficientinformationforambiguity se-

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

intervalswhen isnear0 ,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

and results 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. The signals

values at

or has a local minimum or

accounts for measurement

(see Table II).

and

values

and directional signals.

and

are near zero, and

will be the same for an

. Consider

and

.

and mea-

and, while

and positive near.

and

value,

.

andmeasurements. Fig. 18

– m/s(right panel). The

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608 IEEE 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

small.

s. The

VI. CONCLUSION

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

WindSat

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

speeddependent.Themagnitudeofthewinddirectionsignalbe-

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-

fectsmeansthatthereispoorwinddirectionperformancebelow

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

for

and 1.0 K for at 10.7 GHz and higher at the higher

frequencies. Therefore, the

and

a greater impact on the wind direction retrievals above about

10 m/s.

The retrieval error covariance matrix,

(4) provides an estimate of the expected variance of the re-

and

andchannels

and channels are

directional signals have

, calculated from

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BETTENHAUSEN et al.: NONLINEAR OPTIMIZATION ALGORITHM FOR WINDSAT609

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

forthesquarerootofthediagonalof

m/s for

, 1.0 mm for , and 0.032 mm for

are similar to the overall standard deviation of the differences

betweenourWindSatretrievalsandcollocateddatafromNCEP,

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

windspeedbinsthatweusedforthecomparisonstoQuikSCAT.

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.

should approxi-

give0.86Kfor ,0.78

. These values

, 0.95 mm for , and 0.045

for the same 2-m/s wide

s. We continue to

ACKNOWLEDGMENT

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

Earth Science REASoN DISCOVER Project. Data are available

at www.remss.com.

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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

microwave radiometer.

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

spiral galaxies.

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

antenna performance.

Dr. Smith is a member of Sigma Pi Sigma and the AGU.

Richard M. Bevilacqua photograph and biography not available at the time of

publication.

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-

terometer data.

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

radiometry.

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

project.

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,

VA,supporting seasurfacetemperatureandaerosolretrievalsfromAVHRRand

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.

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