Microwave Specular Measurements and Ocean Surface Wave Properties

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Article Microwave Specular Measurements and Ocean Surface Wave Properties

Paul A. Hwang * , Thomas L. Ainsworth and Jeffrey D. Ouellette

Remote Sensing Division, U.S. Naval Research Laboratory, Washington, DC 20375, USA; [email protected] (T.L.A.); [email protected] (J.D.O.) * Correspondence: [email protected]

Abstract: Microwave reflectometers provide spectrally integrated information of ocean surface waves several times longer than the incident electromagnetic (EM) wavelengths. For high wind condition, it is necessary to consider the modification of relative permittivity by air in foam and whitecaps pro- duced by wave breaking. This paper describes the application of these considerations to microwave specular returns from the ocean surface. Measurements from Ku and Ka band altimeters and L band reflectometers are used for illustration. The modeling yields a straightforward integration of a closed- form expression connecting the observed specular normalized radar cross section (NRCS) to the surface wave statistical and geometric properties. It remains a challenge to acquire sufficient number of high-wind collocated and simultaneous reference measurements for algorithm development or validation and verification effort. Solutions from accurate forward computation can supplement the sparse high wind databases. Modeled specular NRCSs are provided for L, C, X, Ku, and Ka bands with wind speeds up to 99 m/s.

Keywords: ocean surface roughness; normalized radar cross section; specular reﬂection; relative permittivity; whitecaps Citation: Hwang, P.A.; Ainsworth, T.L.; Ouellette, J.D. Microwave Specular Measurements and Ocean Surface Wave Properties. Sensors 2021, 1. Introduction 21, 1486. https://doi.org/10.3390/ The range of ocean surface wavelengths important to microwave remote sensing s21041486 extends several orders of magnitude. Crombie [1] reports the Doppler frequency spectrum of 13.56 MHz radar sea echo at low grazing angle. A distinct spectral peak at 0.38 Hz Academic Editor: Ali Khenchaf is illustrated, corresponding to the resonance ocean surface wavelength of about 10 m (wavenumber k about 0.6 rad/m). He goes on to suggest that variable frequency equipment Received: 29 December 2020 can be used to measure the ocean surface wave spectrum. Depending on frequency and Accepted: 17 February 2021 Published: 20 February 2021 incidence angle, the range of resonance surface wavenumbers spans from about 20 rad/m (L band) to about 500 rad/m (Ku band) for microwave sensors operating at moderate and

Publisher’s Note: MDPI stays neutral high incidence angles [2,3]. with regard to jurisdictional claims in For altimeters and reflectometers, the specular reflection mechanism dominates. The published maps and institutional affil- NRCS is proportional to the number of specular points and the average radii of curvature of iations. the specular reflection facets [4–6]. With the Gaussian distribution describing the elevation and slope of the moving ocean surface [7], a simple inverse relationship between NRCS and surface mean square slope (MSS) is established [5,6]. Further analysis [8–10] indicates that the responsible MSS is contributed by surface waves longer than the EM wavelengths. The frequently cited ratio between the EM wavenumber k and the upper cutoff wavenumber Copyright: © 2021 by the authors. r k Licensee MDPI, Basel, Switzerland. u of Low-Pass MSS (LPMSS) integration is between 3 and 6 [9,10]. Thus, for Ku band This article is an open access article (~14 GHz) altimeter, ku is about 50 to 100 rad/m, and for L band (~1.6 GHz) reflectometer distributed under the terms and it is about 6 to 11 rad/m. conditions of the Creative Commons These theoretical and empirical analyses provide useful guidelines for quantitative Attribution (CC BY) license (https:// investigation of the connection between specular NRCS and ocean surface roughness. creativecommons.org/licenses/by/ Ku band altimeters have been in operation for many decades, and there is a rich trove 4.0/). of well-calibrated Ku band altimeter NRCS data for a close examination of the specular

Sensors 2021, 21, 1486. https://doi.org/10.3390/s21041486 https://www.mdpi.com/journal/sensors Sensors 2021, 21, 1486 2 of 21

2 2 point theory (SPT) applied to nadir-looking altimeters: σ0 = |R(0)| /s f , where σ0 is 2 NRCS, R(0) is the Fresnel reflection coefficient for normal incidence, and s f is the Ku band 2 LPMSS [8–10]. One peculiar outcome is that the resulting s f calculated from measured Ku band NRCS is larger than the optical total MSS [11,12]. The difference is especially obvious in low and moderate wind speeds (U10 less than about 10 m/s). To address this paradox, an effective reflectivity ranging from 0.34 to 0.5 has been suggested [8–10]; those numbers are much smaller than the nominal relative permittivity of 0.62 computed for the Ku band frequency. An alternative explanation is that the peculiar result can be reconciled if the tilting effect of the reflecting specular facets is considered when applying the SPT [13,14]. It has been about two decades since the study presented in [13,14] and our understanding of the ocean surface wave spectrum has advanced considerably with incorporation of remote sensing data into the relatively sparse databases of short-scale ocean surface waves accumulated from direct observations [15–17]. Here we revisit the Ku band altimeter NRCS analyses. The results are applied to other frequencies including the observations of L band LPMSS [18–21] and recent reports of NRCS results derived from the CYclone Global Navigation Satellite System (CYGNSS) mission [22–24], and Ka band altimeter NRCS from the Satellite ARgos and ALtiKa (SARAL) mission [25,26]. The overall objectives are (a) to use specular microwave returns to understand the mean square slopes of surface waves several times longer than the radar wavelengths; spaceborne altimeter and reflectometer measurements coupled with the SPT are employed for this study; and (b) to supplement the high wind in situ data with accurate forward computation of the altimeter and reflectometer solutions. Section2 gives a brief review of the SPT [ 4–6,8]. Section3 describes its application to spaceborne microwave observations. Section4 discusses issues such as wind speed and wave age relationship on LPMSS, whitecap effects caused by wave breaking on surface reflectivity in high winds, and extending the analysis to various frequencies. The NRCS dependence on incidence and scattering angles are presented and discussed with computed examples; the limitations and range of application are described. Section5 is summary.

2. Review of Specular Point Theory Kodis [4] presents a theoretical analysis of backscattering from a perfectly conducting 1D irregular surface at very short EM wavelengths (Kirchoff approximation), with the application of the stationary phase principle to the Kirchoff integral for the complex scattered field. The integral formulas are derived directly from the vector field theory. He shows that to the first order approximation, the backscattering cross section is proportional to the product of the average number of specular points illuminated by the EM waves nA, and the geometric mean of the two principal radii of curvature of those specular points r1 and r2, i.e., σ(ki, −ki) ∼ πh|r1r2|inA (1)

where ki is the incidence EM wavenumber. This analysis elicits the close connection between EM scattering and statistical and geometrical properties of the rough surface. In order to carry out the calculation further, it is necessary to specify the statistics of the rough surface, in particular in regard to the average number of illuminated specular points and their average curvature. Barrick [5,6] extends the analysis to 2D horizontal plane, full scattering geometry configuration, polarization states, and finite surface conductivity. Following his notations as defined by the scattering geometry depicted in his Figure1, which is simplified and reproduced here as Figure1, the NRCS for arbitrary incident and scattered polarization states (η and ξ, respectively), incidence angles (θi, φi = 0), and scattering angles (θs, φs) is

2 σ0ξη = π Rξη(ι) h|r1r2|inA (2)

where Rξη(ι) is the reflection coefficient from infinite plane tangent to the surface at the specular points for incidence and scattered states, and ι is the local (effective) incidence Sensors 2021, 21, 1486 3 of 21

2 σπιξη= Rrrn ξη () (2) Sensors 2021, 21, 1486 012A 3 of 21

where Rξη(ι) is the reflection coefficient from infinite plane tangent to the surface at the specular points for incidence and scattered states, and ι is the local (effective) incidence angleangle at at the the specular specular points. points. From From Kodis’s Kodis’s stationary stationary phase phase analysis analysis [4], [it4 ],is itshown is shown that thethat effective the effective incidence incidence angle angle ι is halfι is the half angle the anglebetween between the incidence the incidence and scattering and scattering prop- agationpropagation directions directions [5], and [5], andit can it can be beexpressed expressed as asa function a function of of incidence incidence and and scattering scattering anglesangles (after correcting a couple of typographic errors):

1/2 cosιθθφθθ=−() 1 sin sin cos + cos cos / 2 1/2 (3) cos ι = [(1− sin θi sinisθs cos φs s+ cos θ ii cos θ ss)/2] (3)

FigureFigure 1. TheThe scattering scattering geometry geometry from from a surface facet.

TheThe average number of specularspecular pointspoints perper unitunit areaarea for for a a rough rough surface surface and and the the aver- av- erageage radii radii of of curvatures curvatures are are then then derived derived in in terms terms of theof the surface surface statistics. statistics.Sections Sections2 and 2 and3 of 3 ofBarrick Barrick [5 ][5] and and Appendix Appendix B ofB Barrickof Barrick [6] [6] give give the the detailed detailed mathematical mathematical derivation derivation of nofA nandA and <| <|r1rr21r|>.2|>. The The ﬁnal final results results are are copied copied here: here: ! −− 2 γ 7.255= 7.255tan tan γ nA =nA expexp (4) π2πl222s2 2 lsff

π 2 =0.1380.138πl 44γ h|r rrr12|i = 2 sec γ (5) 1 2 s2 s ff where ll isis the the correlation correlation length length between between surface surface points points ζ(ζx(,x y, )y )and and ζ(ζx’(x, 0y’, y) 0separated) separated by by a horizontala horizontal distance distance r =r =[(x [(−xx’−)x2 0+)2 (+y− (y’y−)2]y1/20) 2and]1/2 andassuming assuming a Gaussian a Gaussian distribution distribution of the of correlationthe correlation coefficient coefﬁcient as r as →r →0. The0. The numerical numerical constants constants in in Equations Equations (4) (4) and and (5) (5) result fromfrom carrying carrying out out triple triple integral integral functions functions of of (ζ (xxζ,xx ζxy, ,ζ ζxyyy,);ζ yythe); details the details are provided are provided in Bar- in rickBarrick [5,6]. [5 ,6]. SubstitutingSubstituting Equations Equations (4) and (5) into Equation (2), then

4 42γγ − 2 ! σι = () 2 sec2 secγ − tan γ σ =0ξη R R( ξηι) expexp (6) 0ξη ξη 2ss222 s f ffs f

where s22 is the ocean surface LPMSS, and tanγ is the surface slope at the specular point, where s f f is the ocean surface LPMSS, and tanγ is the surface slope at the specular point, which can be expressed as a function of incidence and scattering angles.

2 2 1/2 sin θi − 2 sin θi sin θs cos φs + sin θs tan γ = (7) cos θi + cos θs

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A case of special interest is backscattering: φs = π, θs = θi, γ = θi, and the NRCS is ! sec4 θ − tan2 θ = ( ) 2 i i σ0ξη Rξη ι 2 exp 2 (8) s f s f

3. Application 3.1. Ku Band Altimeter Analysis

For a nadir-looking altimeter (θi = θs = ι = 0) with linear polarization (h or v for horizontal or vertical), Equation (8) becomes

|R(0)|2 = = = σ0 2 σ0hh σ0vv (9) s f

where R(0), shorthand for Rηη(0) with η = h or v, is the Fresnel reflection coefficient for normal incidence, and the NRCS is independent on the polarization state. Apply- ing Equation (9) to Ku band altimeter measurements, a rather peculiar result is discov- ered [8–10,13,14]: the computed Ku band LPMSS is larger than the total optical MSS in 2 clean water s∞ [11,12]. 2 The LPMSS s f is an integrated surface wave property, which is defined as

Z ku 2 2 s f = k S(k)dk (10) 0

where S is the surface wave elevation spectrum, k is surface wavenumber, and ku is the upper limit of lowpass filter, which is in turn proportional to the EM wavenumber kr. The ratio kr/ku is generally given as between 3 and 6 [9,10]. When distinction of EM 2 2 frequency is desired, s f is also given as sku in this paper. For example, for Ku band EM 2 2 frequency fr = 14 GHz and ku = kr/3 = 293/3 = 98 rad/m, the corresponding s f is s98 for clarification. The optical EM frequency is many orders of magnitude higher than those of 2 the microwave sensors used in ocean remote sensing, so s∞ is expected to be the upper 2 bound of s f observed by microwave equipment. Figure2a shows the Ku band altimeter NRCS with collocated buoy winds in the Gulf of Alaska and Bering Sea [14], and the calculated NRCSs based on the optical MSS from sun glitter analyses in clean and slick waters with wind speed measured onboard a research vessel in the region [11] or collocated spaceborne scatterometer wind product [12]. 2 Figure2b shows the optical MSS and s f derived from the Ku band altimeter NRCS using Equation (9). Two sets of MSS reported in Cox and Munk (C54) are also from sun glitter analysis but from a spaceborne optical sensor (wind speed up to 15 m/s); the results are essentially identical to those of the clean water condition in Cox and Munk [11]. For the slick waters, surface waves shorter than about 30 cm, corresponding to ku = 21 rad/m, are 2 2 suppressed [11]; therefore, the two sets of optical MSS are s∞ and s21. The Ku band (14 GHz) EM wavelength is about 2.1 cm, with the factor kr/ku = 3 to 6 applied, the observed LPMSS 2 2 is between s98 and s49. The surface roughness sensed by the Ku band altimeter is expected to be between the optical data in clean and slick waters. This, however, is not the case for the result in low and moderate winds (U10 ≤ ~10 m/s), as illustrated in Figure2b. Especially intriguing is that the discrepancy increases toward lower wind condition for which the ocean surface is less nonlinear and simplifications made in the EM theoretical development are better justified. Sensors 2021, 21, 1486 5 of 21

Sensors 2021, 21, 1486 5 of 21 ocean surface is less nonlinear and simplifications made in the EM theoretical development are better justified.

FigureFigure 2. 2.( (aa)) Ku Ku band band altimeter altimeter NRCS NRCS and and comparison comparison with with computational computational results results using using Equation Equation (9) with(9) with the opticalthe optical MSS MSS in clean in clean and slickand slick waters: waters: C54 andC54 B06.and B06. (b) The (b) MSSThe MSS computed computed using using Equation Equation (9) with the Ku band altimeter NRCS and comparison with the optical MSS in clean and (9) with the Ku band altimeter NRCS and comparison with the optical MSS in clean and slick waters. slick waters. The smooth curves are linear fitting to the three sets of MSS data. The smooth curves The smooth curves are linear fitting to the three sets of MSS data. The smooth curves in (a) are the in (a) are the NRCS computation using Equation (9) with the linear fitted curves of the MSS data. NRCS computation using Equation (9) with the linear fitted curves of the MSS data. As mentioned in the Introduction, many researchers resort to using an effective re- As mentioned in the Introduction, many researchers resort to using an effective flectivity much smaller than that computed from the relative permittivity [8–10]; more reflectivity much smaller than that computed from the relative permittivity [8–10]; more discussion on reflectivity is deferred to Section 4.2. In Hwang et al. [13,14], the authors discussion on reflectivity is deferred to Section 4.2. In Hwang et al. [13,14], the authors stressstress thatthat Equation (8) (8) carries carries the the physical physical meaning meaning of ofan anexponentially exponentially attenuating attenuating contribution with respect to the incidence angle θi. Borrowing the two-scale concept of scat- contribution with respect to the incidence angle θi. Borrowing the two-scale concept of scatteringtering at moderate at moderate incidence incidence angles angles [8] [that8] that the the local local incidence incidence angle angle can can be bemodified modified by bythe the background background waves, waves, and and reexamining reexamining Equation Equation (6), (6), the the left-hand left-hand side of EquationEquation (6)(6) can be written as σθξη (),γ , and Equation (8) is explicitly written as can be written as σ0ξη0(θs, γs), and Equation (8) is explicitly written as

42 2 secγγ − tan ! σθθγ()==() ι4 γ − 2 γ 0ξηsi,exp R ξη 2 sec tan (11) σ0ξη(θs = θi, γ) = Rξη(ι) 22exp  (11) 2 ssff2 s f s f The alternative explanation offered in [13,14] is that Equation (8) can be interpreted The alternative explanation offered in [13,14] is that Equation (8) can be interpreted as as the specular scattering pattern with respect to the incidence angle θi. This is illustrated the specular scattering pattern with respect to the incidence angle θ . This is illustrated in a in a conceptual sketch (Figure 7, [13]), which is reproduced as Figurei 3 here. In the Figure conceptual sketch (Figure 7, [13]), which is reproduced as Figure3 here. In the Figure 7 of Ref7 of [ 13Ref], the[13], parallel the parallel horizontal horizontal lines representlines represent the far-field the far-field EM wave EM frontswave fronts emitted emitted from zenithfrom zenith and impinge and impinge on the on ocean the ocean surface. surface. Fivescattering Five scattering patterns patterns are illustrated. are illustrated. Patterns Pat- 1,terns 4, and 1, 4, 5 areand from 5 are threefrom incrementallythree incrementally rougher rougher patches patches located located on background on background surfaces sur- thatfaces are that locally are locally parallel parallel to the incoming to the incoming wave fronts wave such fronts that such the localthat the incidence local incidence angle is notangle changed is not changed from the from nominal the nominal incidence incidence angle (0 inangle this (0 case). in this The case). backscattering The backscattering returns fromreturns the from three the patches three patches are inversely are inversely proportional proportional to the surface to the surface roughness roughness as expected as ex- frompected Equation from Equation (9). Patterns (9). Patterns 2, 3, and 2, 3, 4 and are from4 are from three three statistically statistically identical identical roughness rough- patchesness patches and located and onlocated background on background surfaces with su differentrfaces with orientations. differentThe orientations. backscattering The strengthsbackscattering from thestrengths three patches from the observed three patches by the observed antenna atby zenith the antenna are different, at zenith although are dif- theferent, reflecting although patches the reflecting have identical patches statistical have identical roughness. statistical The roughness. difference ofThe the difference returns towardof the returns the nominal toward incidence the nominal direction incidence (from zenith)direction reflects (from thezenith) tilting reflects effect asthe described tilting ef- byfect the asexponential described by term the inexponential Equation (8)term or in Equation Equation (11). (8) or Equation (11).

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FigureFigure 3.3. AA conceptualconceptual sketchsketch depictingdepicting thethe scatteringscattering ofof radarradar waveswaves byby surfacesurface patchespatches ofof variousvari- roughnessous roughness (1, 4, (1, and 4, and 5) and 5) and the effectthe effect of tilting of tilt backgrounding background surface surface on theon the backscattering backscattering intensity inten- fromsity from patches patches of identical of identical statistical statistical roughness roughness (2, 3, (2, and 3, 4);and reproducing 4); reproducing (Figure (Figure 7, [13 7,]). [13]).

Consequently,Consequently, there there is is one one more more step step to to obtaining obtaining the the backscattering backscattering NRCS NRCS accounting account- foring modiﬁcation for modification of local of local incidence incidence angle, angle, that that is, is,

42 R ∞ ∞∞R ∞ 2 sec 2 secθθ4 θ − tan− tan2 θ ( σθθ= ) === R ι( ) ilil il il σ0ξη θs ξη()θi −∞ −∞R ξη ξη()ι 2 expexp2 . . 0 si−∞ −∞ 22s f s f (12) ssff (12) p tan γx, tan γy d tan γxd tan γy γγ γ γ pdd()tanxy , tan tan x tan y where θil is the local incidence angle satisfying the specular reflection condition (i.e., θil = θsl on thewhere tilting θil surface,is the localθsl isincidence the local angle scattering satisfying angle), theγ is specular the global reflection tilting angle, condition tanγ =(i.e., (tan θγilx =, tanθsl γony) the = ( ζtiltingx, ζy) surface, is the corresponding θsl is the local surfacescattering slope, angle), and γp (tanis theγx global, tanγy )tilting = p(ζx angle,, ζy) is tan theγ probability= (tanγx, tan densityγy) = (ζx function, ζy) is the (pdf) corresponding of the background surface (global)slope, and surfaces p(tanγ thatx, tan tiltγy the) = p specular(ζx, ζy) is patches.the probability The term density inside function the double (pdf) integrals of the background and before the (global) pdf function, surfaces that that is tilt essentially the spec- Equationular patches. (8), isThe now term interpreted inside the as double the scattering integrals pattern, and before and Equation the pdf (12)function, is equivalent that is es- to thesentially tilting Equation modulation (8), of is local now incidence interpreted angle as inthe the scattering discussion pattern, of scatterometer and Equation returns (12) [8 is] withequivalent the pdf to function the tilting accounting modulation for theof local off-specular incidence contribution. angle in the For discussion the Gaussian of scatterom- distribu- tioneter ofreturns sea surface [8] with slopes the pdf with function equal up-downwind accounting for and the crosswindoff-specular slope contribution. components, For i.e.,the 2 2 2 sGaussianf x = s f y =distributions f /2, of sea surface slopes with equal up-downwind and crosswind slope 222== 2 ! components, i.e., sssfx fy f /2, 1 − tan γ = p tan γx, tan γy 2 exp 2 (13) πs f s f 1tan− 2 γ p()tanγγ , tan= exp For the altimeter application, wexy have π 22 (13) ssff 4 − 2 For the altimeterσ application,(0) = R ∞ weR ∞ have|R( 0)|2 sec γ exp tan γ · 0 −∞ −∞ s2 s2 f f (14) ∞∞ − tan2 γ 42γγ− 1 exp 2 secd tan γ d tantan γ σ ()00exp=⋅πs2 R ()s2 x y 0 −∞f −∞ f 22 ssff (14) Figure4a shows the NRCS results1tan computed− with2 γ Equation (14) and their comparison expdd tanγγ tan with altimeter observations. For reference,π ss22 the results computedx withy Equation (9) are also ff2 shown. Figure4b shows the two sets of s f used in the computation. They are based on 2 the H18Figure spectrum 4a shows model the (NRCSsH18)[ results17,27] integratedcomputed towithku =Equationkr/3 and (14)kr/5. and The their agreement compari- betweenson with measurementaltimeter observations. and theoretical For refere computationnce, the results using computed Equation with (14) Equation with H18 (9) MSS are 2 integratedalso shown. to Figureku = k r4b/3 shows is improved the two considerably. sets of sf used More in discussion the computation. on LPMSS They analysis are based and 2 reﬂectivity |R(0)| is deferred to2 Section4. on the H18 spectrum model ( sH18 ) [17,27] integrated to ku = kr/3 and kr/5. The agreement between measurement and theoretical computation using Equation (14) with H18 MSS integrated to ku = kr/3 is improved considerably. More discussion on LPMSS analysis and reflectivity |R(0)|2 is deferred to Section 4.

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FigureFigure 4. 4. ((aa)) KuKu band NRCS NRCS and and comparison comparison with with model model computation computation with with Equation Equation (14), (14), for for comparison;comparison; computation computation results results withwith EquationEquation (9)(9) are also shown (labeled (labeled GO). GO). ( (bb)) The The LPMSS LPMSS used for the NRCS computation shown in (a), and comparison with the optical MSS in clean and used for the NRCS computation shown in (a), and comparison with the optical MSS in clean and slick waters. The LPMSS is computed with the H18 spectrum model integrated to kr/3 and kr/5. slick waters. The LPMSS is computed with the H18 spectrum model integrated to kr/3 and kr/5.

ForFor specular specular returns, returns, thethe wind direction enters enters into into the the mean mean square square slopes slopes in inthe the up- up-downwinddownwind and and crosswind crosswind components. components. The The ratio ratio of the of thecrosswind crosswind to up-downwind to up-downwind com- componentsponents is close is close to toone one based based on on Cox Cox and and Munk’s Munk’s optical sun glitterglitter datadata [11[11],], and and our our analysisanalysis assumes assumes equal equal up-downwind up-downwind and and crosswind crosswind slope slope components. components. More More discussion discussion onon the the up-downwind up-downwind and and crosswind crosswind surface surface roughness roughness components components is also is also given given in Section in 4Section of [17], 4 see of their[17], Figuresee their 10 Figure and related 10 and discussion. related discussion.

3.2.3.2. L L Band Band CYGNSS CYGNSS Analysis Analysis TiltingTilting modiﬁcation modification is is expected expected to to impact impact specular specular reﬂections reflections at at oblique oblique angles. angles. Ap- Ap- σθ()γ plyingplying the the same same procedure procedure discusseddiscussed inin thethe altimeteraltimeter case to Equation (6) (6) for for σ00ξηξη(θs,sγ, ), the, the NRCS NRCS for for 2D 2D Gaussian Gaussian pdf pdf of tiltingof tilting surfaces surfaces is: is:

42 ∞ ∞∞∞ 22 sec4 γγγ−− tan2 γ ( ) = R R R ( ) sec tan · σ0 θs σθ()−=⋅∞ −∞ Rξη () ιι 2 expexp2 0 s −∞ −∞ s 22f s f ssff (15) 1 − tan2 γ (15) 2 exp 2 2 d tan γxd tan γy πs f 1tan−s f γ expdd tanγγ tan π ss22xy It turns out that for global positioningff system reflectometry (GPSR) with right-hand- 2 circularIt transmitturns out and that left-hand-circular for global positioning receive, system the | Rreflectometrylr(ι)| is almost (GPSR) independent with right-hand- on local ◦ 2 incidence angle, ι, up to about 50 [27], so |R (ι)| plays a rather2 minor role in computing circular transmit and left-hand-circular receive,lr the |Rlr(ι)| is almost independent on local circular polarization NRCS except at low grazing angles. The reflectivity in Equation (6) or incidence angle, ι, up to about 50° [27], so |Rlr(ι)|2 plays a rather minor role in computing 2 Equationcircular polarization (15) can be approximated NRCS exceptby at |lowRlr(0)| graziforng GPSRangles. from The thereflectivity ocean surface. in Equation (6) Through GPSR delay Doppler waveform analyses, there are now several sets of or Equation (15) can be approximated by |Rlr(0)|2 for GPSR from the ocean surface. 2 L bandThrough LPMSS GPSR collected delay in Doppler TC conditions waveform [18 analys–21].es, These there data are now are identifiedseveral sets as of sLGPSR band and illustrated in Figure5 (labeled K0913 and G1318 in the legend); the2 least-squares LPMSS collected in TC conditions [18–21]. These data are identified2 as sGPSR and illustrated fitted curve is given by the solid black line (labeled GPSR). The sGPSR data have served toin address Figure 5 one (labeled of the K0913 most and unsettled G1318 issuesin the lege in thend); ocean the least-squares surface wind fitted wave curve spectrum is given 2 functionby the solidS(f ), i.e.,black the line spectral (labeled slope GPSR). in the The high sGPSR frequency data have region served [17,28 to]. address The clarification one of the ofmost spectral unsettled slope issues is especially in the ocean critical surface to the wind LPMSS wave determination spectrum function using S an(f), ocean i.e., the wave spec- spectrumtral slope model.in the high The frequency refinement region of the[17,28].S(f ) Th function,e clarification in turn, of spectral offers the slope feasibility is especially to derivecritical LPMSS to the LPMSS given U determin10 and windseaation using dominant an ocean wave wave period spectrumTp from model. operational-system The refinement measurements [29] or to create synthetic high-wind LPMSS data set with a small number of the S(f) function, in turn, offers the feasibility to derive LPMSS given U10 and windsea of critical tropical cyclone (TC) parameters [17,27]. Also shown in Figure6 are the LPMSS p dominant wave period T 2 from operational-system2 measurements [29] or to create synthetic computed from the G18 (sG18) and H18 (sH18) spectrum models [17,27] with ku = kr/3 and high-wind LPMSS data set with a small number2 of critical tropical cyclone (TC) parame- kr/5. In addition, the results based on E97 (sE97)[30] are also displayed; the E972 is used in theters CYGNSS [17,27]. Also project shown [31,32 in]. Figure For comparison, 6 are the LPMSS the optical computed data obtainedfrom the G18 in clean ( sG18 and) and slick H18 2 ( sH18 ) spectrum models [17,27] with ku = kr/3 and kr/5. In addition, the results based on

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2 E97 ( sE97 ) [30] are also displayed; the E97 is used in the CYGNSS project [31,32]. For com- Sensors 2021, 21, 1486 s2 8 of 21 E97parison, ( E97 the) [30] optical are also data displayed; obtained inthe clean E97 andis us slicked in sea the surfaces CYGNSS [11,12] project are [31,32]. illustrated For com- with parison,black markers the optical in the data Figure. obtained The GPSR in clean data and have slick expanded sea surfaces the [11,12]wind speed are illustrated coverage withcon- black markers in the Figure. The GPSR data have expanded the wind speed coverage con- seasiderably surfaces [(from11,12] are 15 illustrated to 59 m/s), with and black they markers are incritical the Figure. for Therefining GPSR datathe haveocean surface wind expandedsiderablywave spectrum the (from wind speedmodels15 to coverage 59 in m/s), high considerably and wind they conditions. (from are 15critical to 59More m/s),for detailrefining and they on arethederiving critical ocean LPMSS surface from wind a forwave reﬁning spectrum the ocean ismodels surface deferred windin high to wave Section wind spectrum 4.1.conditions. models in More high wind detail conditions. on deriving More LPMSS from a detailwave on spectrum deriving LPMSS is deferred from a waveto Section spectrum 4.1. is deferred to Section 4.1.

Figure 5. L band LPMSS derived from GPSR delay Doppler analyses (labeled K0913 and G1318), FigureFigurethe least-squares 5. L5. bandL band LPMSS LPMSScurve derived fitted derived from through GPSR from delay allGPSR the Doppler GPSRdelay analyses Dodatappler (labeledis given analyses K0913 by the and(labeled solid G1318), black K0913 the line and (labeled G1318), least-squares curve ﬁtted through all the GPSR data is given by the solid black line (labeled GPSR). theGPSR). least-squares Three sets curve of computation fitted through from all sp theectrum GPSR models data is (E97, given G18, by the and solid H18) black with line ku = (labeledkr/3 and Three sets of computation from spectrum models (E97, G18, and H18) with ku = kr/3 and kr/5 are GPSR).kr/5 are Threealso illustrated. sets of computation For comparison, from sp opticalectrum MS modelsS in clean (E97, and G18, slick and waters H18) withis shown ku = kwithr/3 and black also illustrated. For comparison, optical MSS in clean and slick waters is shown with black markers. kmarkers.r/5 are also illustrated. For comparison, optical MSS in clean and slick waters is shown with black markers.

FigureFigure 6. 6.NRCS NRCS computed computed with LPMSSswith LPMSSs derived derived from E97, from G18, andE97, H18 G18, spectrum and H18 models, spectrum and models, and comparisonFigurecomparison 6. withNRCS with the computed CYGNSS the CYGNSS NRCS with observations:NR LPMSSsCS observations: derived R18 and from B20. R18 E97, and G18, B20. and H18 spectrum models, and comparison with the CYGNSS NRCS observations: R18 and B20. Retrieving NRCS from GPSR, or global navigation satellite system reﬂectometry (GNSSR)Retrieving in general, remainsNRCS afrom challenging GPSR, task or [22 global–24]. Here na wevigation investigate satellite the properties system reflectometry of(GNSSR) L-bandRetrieving NRCS in general, through NRCS forwardremains from computation GPSR,a challenging or usingglobal task Equation na [22–24].vigation (15). TheHere satellite results we investigate aresystem then reflectometry the proper- compared(GNSSR)ties of L-band with in general, the NRCS most recentremains through publications a forward challenging of Lcomputation band task NRCS [22–24]. results using Here from Equation thewe CYGNSSinvestigate (15). The the results proper- are missiontiesthen of compared [L-band23,24]. Figure NRCS with6 shows throughthe themost results forward recent computed computationpublications with Equation usingof (15)L bandEquation and the NRCS LPMSS (15). results The results from arethe derivedthen compared from the three with spectrum the most models recent shown publications in Figure5. Superimposedof L band NRCS with the results from the computationCYGNSS mission curves are [23,24]. two sets Figure of CYGNSS 6 shows data: the the results black crosses computed and pluses with labeledEquation (15) and the CYGNSSLPMSS derived mission from [23,24]. the Figurethree spectrum 6 shows themodels results shown computed in Figure with 5. Equation Superimposed (15) and with the LPMSSthe computation derived from curves the are three two spectrum sets of CYGNSS models data: shown the inblack Figure crosses 5. Superimposed and pluses labeled with theR18 computation [23] are the GMFcurves at are 10° two and sets 50°, of and CYGNSS the black data: squares the black labeled crosses B20 and [24] pluses are the labeled inci- R18dence-angle-averaged [23] are the GMF atresult. 10° and It is 50°, clear and that the the black CYGNSS squares GMF labeled is still B20 evolving. [24] are theInterest- incidence-angle-averagedingly, there is a good agreement result. It isbetween clear that the the mo CYGNSSst recent version GMF is [24] still (B20: evolving. black Interest-squares) ingly, there is a good agreement between the most recent version [24] (B20: black squares)

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R18 [23] are the GMF at 10◦ and 50◦, and the black squares labeled B20 [24] are the incidence- angle-averaged result. It is clear that the CYGNSS GMF is still evolving. Interestingly, there is a good agreement between the most recent version [24] (B20: black squares) and the 2 2 computed NRCS with sG18 and sH18 integrated to ku = kr/3; the model-data difference is 2 less than about 1 dB to wind speed up to about 60 m/s. The computed NRCS with sE97 is also in good agreement with data for low to moderately high winds (U10 ≤ ~14 m/s) but fares far worse in higher winds, because E97 underestimates the LPMSS for U10 greater than about 14 m/s (Figure5). It illustrates the importance of choosing an accurate surface wave spectrum model for ocean remote sensing analysis. More discussion on LPMSS and 2 reﬂectivity |Rζη(ι)| is deferred to Section4.

4. Discussion 4.1. LPMSS, Wind Speed, and Wave Development Stage The functional form of the ocean surface wind wave spectrum remains one of the most uncertain quantities in ocean remote sensing problems. For specular return, the relevant property is the LPMSS with the upper integration wavenumber ku determined by the EM wavenumber kr. From altimeter analyses [9,10] the range of kr/ku is generally determined to be between 3 and 6. In this paper, we have shown results of specular computation obtained with kr/ku = 3 and 5, which corresponds to ku = 11 and 6.6 rad/m for L band (1.575 GHz), and 98 and 59 rad/m for Ku band (14 GHz). The contribution of long surface waves in the energetic dominant wave region becomes more important as wind speed increases and EM frequency decreases. The value of the wave spectral slope –s in the high frequency region of the wind wave frequency spectral function is one of the most uncertain spectral properties critical to the determination of LPMSS. Traditional wind wave spectrum models assume s to be either 4 or 5 [33–36], although ﬁeld observations have shown a wide range between about 2 and 7 [17,28]. The wave height spectral level drops sharply toward both high and low frequencies from the spectral peak; therefore, wave height measurements are not very sensitive for addressing the spectral slope issue. Wave slope data are much more useful for this task [27,29]. For many decades, the airborne sun glitter analyses in clean and slick waters reported in 1954 [11] have remained the most comprehensive ocean surface MSS dataset; the wind speed range is between 0.7 and 13.5 m/s for the clean water condition, and between 1.6 and 10.6 m/s for the slick water condition. The spaceborne sun glitter analysis reported in 2006 [12] expands slightly the wind speed range of clean water condition to about 15 m/s, and the results are essentially identical to those of the clean water condition reported in 2 1954 [11]. The recent results of sGPSR further extend the LPMSS wind speed coverage to 59 m/s [18–21]. With the EM frequency of 1.575 GHz, the ku is between about 5 and 2 11 rad/m, and sGPSR data are most useful for investigating the wind wave spectrum slope. The study leads to establishing a general wind wave spectrum function G18 [17], with the applicable upper limit (kmax) of the G18 spectrum function estimated to be about the upper range of L band ku (11 rad/m). For Ku band application, the hybrid model H18 is more suitable. The H18 model uses G18 for long waves and H15 [16] for short waves, with linear interpolation between k = 1 and 4 rad/m [27]. Wind speed U10 and windsea dominant wave period Tp are the only required input for computing the G18 and H18 spectrum (and many other spectrum models such as the E97 discussed in Section3). The combination of U10 and Tp can be expressed as the dimensionless spectral peak frequency ω# = U10/cp = U10/(gTp/2π), where cp is the wave phase speed of the spectral peak component, and g is gravitational acceleration. The inverse of ω# is wave age, which represents the stage of wave development. Determining the wave spectrum requires consideration of both wind speed and wave development stage. With a 2 wave spectrum function, the sku can be pre-calculated for a range of U10, ω#, and ku. For 2 2 example, Figure7a,b show the contour maps of H18 s98 and s11, respectively. They are illustrated for U10 between 0 and 70 m/s, and ω# between 0.8 and 5.2. These pre-calculated Sensors 2021, 21, 1486 10 of 21

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2 results serve as design curves or lookup tables for quickly obtaining the desired s f through through interpolation.interpolation. Superimposed Superimposed in the in theFigures Figures are are the the observed observed ω#(ωU#10(U) in10 )TC in TCand and non-TC non-TC conditions.conditions. Because Because the wave the wave age age1/ω# 1/ is ωdefined# is deﬁned as the as ratio the ratioof dominant of dominant wave wave phase phase speedspeed cp andc windp and speed wind speedU10, U10U, and10, U ω10# ,= and U10/ωcp# are= U not10/ independentcp are not independent variables. The variables. The observed ω#observed(U10) in TCω #and(U10 non-TC) in TC andconditions non-TC show conditions the general show thelinear general relationship linear relationship with with a a narrow rangenarrow of ω range# variation of ω# forvariation a particular for a wind particular speed wind U10 as speed illustratedU10 as by illustrated the color by the color markers in Figuremarkers 7a,b. in Figure7a,b.

Figure 7. Pre-calculated H18 LPMSS contour maps for (a) ku = 98 rad/m and (b) ku = 11 rad/m. Su- Figure 7. Pre-calculated H18 LPMSS contour maps for (a) ku = 98 rad/m and (b) ku = 11 rad/m. Superimposed in the Figures perimposed in the Figures are the observed ω#(U10) in TC and non-TC conditions. (c) Color mark- are the observed ω#(U10) in TC and non-TC conditions. (c) Color markers show the interpolated LPMSS for ku = 98 and ers show the interpolated LPMSS for ku = 98 and 11 rad/m (upper and lower sets, respectively) 11 rad/m (upper and lower sets, respectively) using the observed ω#(U10) in TC and non-TC conditions. For comparison, using the observed ω#(U10) in TC and non-TC conditions. For comparison, illustrated with black illustrated withmarkers black markers are the optically are the optically sensed MSS sensed in clean MSS and in clean slick andwaters. slick Al waters.so shown Also in shownthe Figure in the are Figurethe are the interpolated LPMSSinterpolated assuming LPMSS constant assumingω# (1 constant and 2), and ω# (1ω# andapproximated 2), and ω# approximated by Equation (16). by Equation (16).

2 2 The interpolateds 2 H18s2 s98 and s11 are presented with color markers in Figure8. For The interpolated H18 98 and 11 are presented2 with color markers in Figure2 8. For comparison, the optically2 sensed s∞ in clean water [11,122 ] and s21 in slick water [11] are comparison,illustrated the optically with sensed black markerss∞ in clean in the water Figure. [11,12] Also and shown s21 in in Figure slick 8water are the [11] interpolated 2 2 are illustratedH18 withs98 andblacks 11markersassuming in the constant Figure.ω Also# (1 and shown 2 are in used Figure for 8 illustration). are the interpo- As wind speed 2 2 2 lated H18 sincreases,98 and s11 the assuming difference constant increases ω# (1 between and 2 ares f computedused for illustration). with constant As andwind observed ω#. Interestingly, if the approximation 2 speed increases, the difference increases between sf computed with constant and ob-

served ω#. Interestingly, if the approximation ω# = max(0.8, 0.065U10) (16) ω = max() 0.8,0.065U #102 (16) is employed, the resulting s f is very close to that obtained with observed ω#. For the 2 2 is employed,purpose the resulting of obtaining sf is verys f fromclose ato wave that obtained spectrum with function, observed approximation ω#. For the pur- Equation (16) simpliﬁes2 the procedure in practical applications, since it requires only the U10 input (with pose of obtaining s f from a wave spectrum function, approximation Equation (16) sim- ku speciﬁed). plifies the procedure in practical applications, since it requires only the U10 input (with ku specified).

4.2. Surface Reflectivity Considering Wave Breaking

The reflectivity |Rξη(ι)|2 is a function of relative permittivity and generally treated as a constant for a given EM frequency (with the assumption of some representative sea surface temperature and sea surface salinity: 293 K and 35 psu are used throughout this paper; the relative permittivity of sea water is computed with the formula given by Klein and Swift [37]). In high winds when air is entrained by wave breaking into the water surface layer and foam covers the water surface, the modification of relative permittivity by the mixed air needs to be considered. Figure 8 shows the Fresnel reflectivity at 0°, 10°, 30°, and 50° incidence angles for Ku (14 GHz) and L (1.575 GHz) frequencies. The Ku band altimeters discussed in this paper operate with linear polarizations (h and v), so |Rhh(θ)|2

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and |Rvv(θ)|2 are illustrated. The GPSR signals are right-hand-circular transmit and left- hand-circular receive, so for L band |Rlr(ι)|2 is given; its dependence on incidence angle is very weak up to about 50° incidence angle [27]. For altimeter reflection from the ocean surface, |Rξη(0)|2 is the quantity of interest (Section 3.2), and it is independent of polarization states hh, vv, and lr, but can vary considerably with wind speed as a result of air entrainment by wave breaking. The foam modification is more severe toward higher frequency as expected; this can be seen from comparing the black solid line of L band and red/green (overlapped) solid lines of Ku band. The procedure to account for the foam effect is briefly described below. Through analyses of microwave radiometer measurements collected in TCs that cover a wide range of frequencies, incidence angles, and both horizontal and vertical polarizations [38–46], the effects of surface foam generated by wave breaking are expressed as a function of wind speed, microwave frequency, and incidence angle [47,48]. The effective air fraction Fa is related to the whitecap coverage Wc as described in Appendices A and B of [48]. A brief summary is given here. The proposed Fa/Wc function is

β F f α a = max 1, cos θ (17) Wf cref From empirical fitting the computed brightness temperature with observations [38–46], the following values are recommended for the three parameters in Equation (17): fref = 14 GHz, α = 1.3, and Sensors 2021, 21, 1486 11 of 21 β = max{ 0,0.5 - min{ 0.5,0.5 exp() 1.1ff / -1.5 }} ref 

FigureFigure 8.8.The The seasea surfacesurface reﬂectivity reflectivity at at 0 ◦0°,, 10 10°,◦, 30 30°◦,, and and 50 50°◦ incidence incidence angles angles (solid, (solid, dashed, dashed, dashed– dotted,dashed–dotted, and dotted and curves, dotted respectively) curves, respectively) for Ku (14 for GHz, Ku (14 green GHz, and green red and curves) red andcurves) L (1.575 and L GHz, black(1.575 curves) GHz, black bands. curves) The GPSR bands. signals The areGPSR right-hand-circular signals are right-hand-circular transmit and left-hand-circular transmit and left-hand- receive, circular receive, so for2 L band |Rlr(ι)|2 is given (black curves). For Ku band, the reflectivity for hh so for L band |Rlr(ι)| is given (black curves). For Ku band, the reﬂectivity for hh and vv polarizations and vv polarizations is shown (green and red, respectively). is shown (green and red, respectively).

4.2. SurfaceThe whitecap Reflectivity fraction Considering Wc(U10 Wave) is defined Breaking by 2 The reflectivity |Rξη(ι)| is a function of relative permittivity≤ and generally treated  0,u* 0.11m/s as a constant for a given EM frequency (with the assumption of some representative sea Wu=−0.30() 0.113 , 0.11 <≤ u 0.40m/s surface temperature and seac surface salinity:** 293 K and 35 psu are used throughout this(18)  0.07uu2.5 ,> 0.40m/s paper; the relative permittivity of sea water** is computed with the formula given by Klein and Swift [37]). In high winds when air is entrained by wave breaking into the water surface layer and foam covers the water surface, the modification of relative permittivity by the mixed air needs to be considered. Figure8 shows the Fresnel reflectivity at 0 ◦, ◦ ◦ ◦ 10 , 30 , and 50 incidence angles for Ku (14 GHz) and L (1.575 GHz) frequencies. The Ku band altimeters discussed in this paper operate with linear polarizations (h and v), so 2 2 |Rhh(θ)| and |Rvv(θ)| are illustrated. The GPSR signals are right-hand-circular transmit 2 and left-hand-circular receive, so for L band |Rlr(ι)| is given; its dependence on incidence angle is very weak up to about 50◦ incidence angle [27]. For altimeter reflection from the 2 ocean surface, |Rξη(0)| is the quantity of interest (Section 3.2), and it is independent of polarization states hh, vv, and lr, but can vary considerably with wind speed as a result of air entrainment by wave breaking. The foam modification is more severe toward higher frequency as expected; this can be seen from comparing the black solid line of L band and red/green (overlapped) solid lines of Ku band. The procedure to account for the foam effect is briefly described below. Through analyses of microwave radiometer measurements collected in TCs that cover a wide range of frequencies, incidence angles, and both horizontal and vertical polarizations [38–46], the effects of surface foam generated by wave breaking are expressed as a function of wind speed, microwave frequency, and incidence angle [47,48]. The effective air fraction Fa is related to the whitecap coverage Wc as described in Appendices A and B of [48]. A brief summary is given here. The proposed Fa/Wc function is

 !β Fa f = max1, cosα θ  (17) Wc fre f

From empirical ﬁtting the computed brightness temperature with observations [38–46], the following values are recommended for the three parameters in Equation (17): fref = 14 GHz, Sensors 2021, 21, 1486 12 of 21

α = 1.3, and n n h ioo β = max 0, 0.5 − min 0.5, 0.5 exp 1.1 f / fre f − 1.5

The whitecap fraction Wc(U10) is deﬁned by Sensors 2021, 21, 1486  12 of 21 0, u∗ ≤ 0.11 m/s  3 Wc = 0.30(u∗ − 0.11) , 0.11 < u∗ ≤ 0.40 m/s (18)  2.5 0.07u∗ , u∗ > 0.40 m/s where u* is the wind friction velocity; the drag coefficient connecting u* and U10 input ( uCU= 0.5 ) is given by where*1010u∗ is the wind friction velocity; the drag coefﬁcient connecting u∗ and U10 input 0.5 (u∗ = C U ) is given by 10 10 10−42() -0.0160UU++ 0.967 8.058 , U ≤ 35m/s =  10 10 10 (C10  (19) −4 2 -3 −1 10 −0.01602.23U ×> 10+ 0.967()UUU / 35+ ,8.058 , U10 ≤ 35m/s35m/s C =  10 1010 10 10 −3 −1 (19) 2.23 × 10 (U10/35) , U10 > 35m/s With the effective air fraction Fa determined from the whitecap coverage Wc, the ef- fectiveWith relative the effective permittivity air fraction εe is computedFa determined with the from refractive the whitecap mixing rule coverage [49–51]:W c, the

effective relative permittivity εe is computed with the refractive2 mixing rule [49–51]: εε=+−1/2() ε 1/2 eaaFF1 asw (20) 2 h 1/2 1/2i εe = Faε + (1 − Fa)ε (20) where εa and εsw are the relative permittivitiesa of air andsw (foamless) sea water, respectively. The effective relative permittivity εe is then used to compute the Fresnel reflection coeffi- where ε and ε are the relative permittivities of air and (foamless) sea water, respectively. cient witha windsw speed dependence as a result of foam effects caused by surface wave The effective relative permittivity ε is then used to compute the Fresnel reﬂection coefﬁcient breaking. e with wind speed dependence as a result of foam effects caused by surface wave breaking.

4.3.4.3. FrequencyFrequency DependenceDependence andand VeriﬁcationVerification WithWith aa surfacesurface wavewave spectrum,spectrum, thethe LPMSSLPMSS cancan bebe computedcomputed forfor applicationapplication toto anyany 2 EM frequencies. For example, Figure 9 shows 2s H 18 integrated to ku = kr/3 (black curves) EM frequencies. For example, Figure9 shows sH18 integrated to ku = kr/3 (black curves) andandk kuu == kr/5 (red (red curves) curves) for for several several microwave microwave fr frequencyequency bands usedused frequentlyfrequently inin oceanocean remoteremote sensingsensing (L,(L, C,C, X, X, Ku, Ku, and and Ka) Ka) with with wind wind speeds speeds up up to to 99 99 m/s. m/s. AlsoAlso illustratedillustratedfor for comparisoncomparison is is the the optical optical MSS, MSS, which which is is restricted restricted to to wind wind speed speed below below about about 15 15 m/s. m/s. TheThe specularspecular NRCSNRCS cancan bebe computedcomputed withwith thethe LPMSSLPMSS input,input, asas illustratedillustrated inin FigureFigure 10 10.. ForFor 2 2 clarity,clarity, only only results results based based on onsH 18s Hintegrated18 integrated to ku to= kkru/3 = arekr/3 shown. are shown. It is of It interest is of interest to verify to theverify model the computationsmodel computations with available with available data. data.

FigureFigure 9.9. LPMSS for for L L (1.575 (1.575 GHz, GHz, solid solid lines), lines), C C(6 (6GH GHz,z, dashed), dashed), X (10 X (10GHz, GHz, dashed-dotted), dashed-dotted), Ku Ku(13.575 (13.575 GHz, GHz, dotted), dotted), and and Ka (35.75 Ka (35.75 GHz, GHz, dots) dots) band bandss based based on the on H18 the H18spectrum spectrum (for normal (for normal incidence application), black curves are integration to kr/3 and red curves are integration to kr/5. incidence application), black curves are integration to kr/3 and red curves are integration to kr/5.

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FigureFigure 10. 10. SpecularSpecular NRCS (normal (normal incidence) incidence) dependence dependence on on wind wind speed speed for for L (1.575 L (1.575 GHz), GHz), C (6 C (6GHz), GHz), X X (10 (10 GHz), GHz), Ku Ku (14 (14 GHz), GHz), and and Ka Ka (35.75 (35.75 GH GHz)z) bands. bands. All All computed computed with with the the H18 H18 LPMSS LPMSS integrated to kr/3. integrated to kr/3.

TheThe KuKu band altimeter altimeter systems systems have have many many more more well-calibrated well-calibrated NRCS NRCS datasets datasets com- comparedpared to other to other frequencies frequencies due due to the to thelong long history history of using of using Ku band Ku band for ocean for ocean wind wind sens- sensinging [9,10,13,14,52–64]. [9,10,13,14,52–64 Three]. Three examples examples are areexamined examined here: here: 1.1. TOPEX/POSEIDONTOPEX/POSEIDON (T/P) altimeter NRCS (13.575(13.575 GHz)GHz) andand collocatedcollocated National National OceanographicOceanographic and and Atmospheric Atmospheric Administration Administration (NOAA) (NOAA) National Nation Dataal Data Buoy Buoy Center Cen- (NDBC)ter (NDBC) buoy buoy datasets datasets in threein three geographic geographic regions regions (Gulf (Gulf of of Alaska Alaska and and Bering Bering Sea, Sea, GulfGulf ofof Mexico,Mexico, andand HawaiiHawaii islands) islands) have have been been reported reported in in [ 13[13,14],14] with with up up to to seven seven yearsyears of of simultaneous simultaneous measurements measurements and and a totala total of of 2174 2174 (U (10U,10σ, 0σ)0) pairs. pairs. The The maximum maximum windwind speedspeed inin thethe datasets is about about 20 20 m/s. m/s. The The maximum maximum temporal temporal and and spatial spatial dif- differencesferences between between buoy buoy and and altimeter altimeter data data are are 0.5 0.5 h and 100100 km,km, respectively.respectively. These These datadata areare referredreferred to to as as H02 H02 from from here here on. on. 2.2. AA one-yearone-year TropicalTropical Rainfall Rainfall Mapping Mapping Mission Mission (TRMM) (TRMM) Precipitation Precipitation Radar Radar (TPR, (TPR, 7 13.813.8 GHz)GHz) dataset dataset [ 10[10]] with with more more than than 1.13 1.13× ×10 107( U(U1010,,σ σ00)) pairs.pairs. TheThe TPR TPR dataset dataset is is quitequite unusualunusual becausebecause thethe windwind sensor sensor and and altimeter altimeter are are on on the the same same satellite. satellite. The The nadirnadir footprint footprint of of the the TRMM TRMM Microwave Microwave Imager Imager (TMI) (TMI) wind wind sensor sensor is is collocated collocated with with thethe footprintfootprint of of the the TPR, TPR, and and there there is is no no need need for for temporal temporal interpolation. interpolation. The The spatial spatial resolutionresolution of of TPR TPR altimeter altimeter is is about about 4.3 4.3 km km and and that that of of the the TMI TMI is is about about 25 25 km, km, so so the the spatialspatial separationseparation betweenbetween TPR,TPR, NRCS,NRCS, and TMI wind speed data data is is no no more more than than ± ±12.512.5 km. km The. The maximum maximum wind wind speed speed in in the the da datasettaset as presented inin theirtheir FigureFigure4 4is is aboutabout 2929 m/s.m/s. These data are referred to as F03 from herehere on.on. 3.3. AnAn extensiveextensive collectioncollection ofof 3333 years years wind wind speed, speed, wave wave height, height, and and altimeter altimeter NRCS NRCS fromfrom 1313 satellite satellite missions missions ranging ranging from from Geosat Geosat to to Sentinel-3A Sentinel-3A and and Jason-3 Jason-3 [ 64[64].]. The The altimeteraltimeter NRCSNRCS isis mergedmerged withwith EuropeanEuropeanCentre Centrefor forMedium-Range Medium-RangeWeather Weather Fore- Fore- ◦ × ◦ castscasts (ECMWF)(ECMWF) model model wind wind speed speed in in 1 1° ×1 1° grids;grids; theirtheir FigureFigure3 3shows shows an an example example ofof thethe Jason-3Jason-3 (J3)(J3) NRCS NRCS dependence dependence on on wind wind speed. speed. The The maximum maximum wind wind speed speed is is aboutabout 2020 m/s.m/s. These data are referred to as R19 from herehere on.on. FigureFigure 11 11 shows shows the the bin-averaged bin-averaged Ku Ku band band altimeter altimeter data data from from the the three three examples examples described above, the data are labeled H02, F03, and R19. The H02 T/P results (red pluses) described above, the data are labeled H02, F03, and R19. The H02 T/P results (red pluses) are processed from our in-house data sets. The R19 J3 results (red circles) are digitized are processed from our in-house data sets. The R19 J3 results (red circles) are digitized from their Figure 3 [64]. The F03 TPR results (blue diamonds) are digitized from their from their Figure 3 [64]. The F03 TPR results (blue diamonds) are digitized from their Figure4[ 10]. Compared to the T/P and J3 results, there is a 1-dB systematic bias in the Figure 4 [10]. Compared to the T/P and J3 results, there is a 1-dB systematic bias in the TPR data, which is subtracted in the Figure. A 1.92 dB bias is reported [10] from comparing TPR data, which is subtracted in the Figure. A 1.92 dB bias is reported [10] from comparing with the modiﬁed Chelton and Wentz (MCW) GMF (Table 1, [59]). The MCW is designed with the modified Chelton and Wentz (MCW) GMF (Table 1, [59]). The MCW is designed for the Geosat altimeter. With improved algorithm and including atmospheric correction, for the Geosat altimeter. With improved algorithm and including atmospheric correction, the T/P NRCS differs from the Geosat NRCS by 0.7 dB [54]. The MCW GMF with 0.7-dB the T/P NRCS differs from the Geosat NRCS by 0.7 dB [54]. The MCW GMF with 0.7-dB

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Sensors 2021 21 adjustment is shown with the red solid line, which goes through the center of T/P, J3, and Sensors 2021, , 21, 1486, 1486 1414 of of 21 21 adjusted TPR data. The blue dashed line is the TPR GMF with 1 dB adjustment applied. 2 The black line is the model computation with sH18 integrated to ku = kr/3. The agreement adjustmentisadjustment very good is is shown betweenshown with with model the the red red computation solid solid line, line, which wh andich goes allgoes three through through Ku the bandthe center center data of of sets. T/P, T/P, J3,In J3, andhigh and winds, adjustedtheadjusted model TPR TPR computation data. data. TheThe blueblue even dasheddashed outperforms lineline isis the the the TPR TPR empirically GMF GMF with with 1 established1 dB dB adjustment adjustment GMFs. applied. applied. Also shown 2 Thefor reference black line isis thethe model GO solution computation Equation with s(9)2 withintegrated the black to ku dashed= kr/3. Theline, agreement which is about 1 The black line is the model computation with Hs18H18 integrated to ku = kr/3. The agreement is very good between model computation and all three Ku band data sets. In high winds, dBis very too highgood comparedbetween model to the computation data. and all three Ku band data sets. In high winds, the model computation even outperforms the empirically established GMFs. Also shown the model computation even outperforms the empirically established GMFs. Also shown for reference is the GO solution Equation (9) with the black dashed line, which is about for reference is the GO solution Equation (9) with the black dashed line, which is about 1 1 dB too high compared to the data. dB too high compared to the data.

Figure 11. Ku band altimeter NRCS from T/P, J3, and TPR missions (normal incidence) and com-

parison with NRCS computations using the H18 LPMSS integrated to kr/3 (black solid line). The FigureMCWFigure 11.and 11.Ku Ku TPR band band GMFs altimeter altimeter (with NRCS NRCS adjustment) from from T/P, T/P, are J3, J3, andshown and TPR TPR with missions missions red (normalsolid (normal and incidence) incidence) blue dashed and and compari- lines;com- more sondetailparison with is NRCSwithgiven NRCS computationsin the computations text. using us theing H18 the LPMSSH18 LPMSS integrated integrated to kr/3 to (blackkr/3 (black solid solid line). line). The MCWThe andMCW TPR and GMFs TPR (withGMFs adjustment) (with adjustment) are shown are shown with red with solid red and solid blue and dashed blue dashed lines; morelines; more detail is detail is given in the text. given inFigure the text. 12 shows the data from all three EM frequencies together and the comparison 2 withFigure Figuremodel 12 12 computation shows shows the the data data using from from all(14) all three three with EM EM s frequenciesH frequencies18 integrated together together to k andu and= k ther /3.the comparison Thecomparison number of re- with model computation using (14) with s22 integrated to k = k /3. The number of portedwith model data computationsets is considerably using (14) less with in othersHH1818 integrated EM frequencies. to kuu = k rr/3.As The mentioned number ofin re-Section 3, reported data sets is considerably less in other EM frequencies. As mentioned in Section3 , thereported are data two sets L bandis considerably NRCS data less sets in otherreported EM frequencies.from the CYGNSS As mentioned mission in [23,24].Section 3,A couple there are two L band NRCS data sets reported from the CYGNSS mission [23,24]. A ofthere Ka areband two altimeter L band NRCS (ALtiKa) data setsdata reported sets have from also the been CYGNSS reported mission from [23,24]. the SARAL A couple mission couple of Ka band altimeter (ALtiKa) data sets have also been reported from the SARAL of Ka band altimeter (ALtiKa) data sets have also been reported from the SARAL mission mission[25,26]. [For25,26 the]. For Ka the band, Ka band, the model-data the model-data agreement agreement (red (red markers markers and and solid solid line) is close [25,26]. For the Ka band, the model-data agreement (red markers and solid line) is close isto closethat toof thatthe Ku of the band Ku comparison band comparison (black (black markers markers and andsolid solid line). line). For ForL band, L band, the model to that of the Ku2 band comparison2 (black markers and solid line). For L band, the model the model results with s integrated to ku = kr/3 are shown (magenta marker and solid results with s2H18 integratedH18 to ku = kr/3 are shown (magenta marker and solid line). Less line).results Less with than s 1H18 dB integrated difference isto found ku = kr/3 between are shown model (magenta and measurements marker and solid to wind line). speed Less than 1 dB difference is found between model and measurements to wind speed about 60 aboutthan 1 60 dB m/s. difference Also shown is found for between reference model are the and GO measurements solutions Equation to wind (9) speed for the about three 60 frequenciesm/s.m/s. AlsoAlso shownshown illustrated for for reference reference with dashed are are the lines,the GO GO solutions which solutions are Equation about Equation 1 (9) dB for too (9) the high for three the compared frequencies three frequencies to theillustratedillustrated data. withwith dashed dashed lines, lines, which which are are about about 1 dB 1 too dB high too highcompared compared to the data.to the data.

Figure 12. Model–measurement comparison for L, Ku, and Ka microwave frequencies; more detail FigureFigureis given 12. 12. inModel–measurement Model–measurementthe text. comparison comparison for L, for Ku, L, and Ku, Ka an microwaved Ka microwave frequencies; frequencies; more detail more is detail givenis given inthe in the text. text.

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4.4. Incidence and Scattering Angle Dependence 4.4. Incidence and Scattering Angle Dependence The NRCS dependence on the incidence and scattering angles hinges on ι and γ as The NRCS dependence on the incidence and scattering angles hinges on ι and γ as prescribed by Equation (3) and Equation (7). Figure 13 a,b show examples of in-plane (ϕs prescribed by Equation (3) and Equation (7). Figure 13a,b show examples of in-plane = 0°) vv◦ and hh scattering of Ku band (EM frequency f is 14 GHz) at 5 m/s wind speed. For (φs = 0 ) vv and hh scattering of Ku band (EM frequency f is 14 GHz) at 5 m/s wind clarity, the contour lines are 2 dB apart for NRCS greater than 5 dB, 5 dB apart between -30 speed. For clarity, the contour lines are 2 dB apart for NRCS greater than 5 dB, 5 dB apart and 5 dB, and 10 dB apart for NRCS less than −30 dB. The NRCS contours are symmetric between −30 and 5 dB, and 10 dB apart for NRCS less than −30 dB. The NRCS contours are to the θi = θs line. This is a consequence of reciprocity, since we get the same set of (ι, γ) symmetric to the θi = θs line. This is a consequence of reciprocity, since we get the same set from Equation (3) and Equation (7) when θi and θs are swapped. The dropoff along the θi of (ι, γ) from Equation (3) and Equation (7) when θi and θs are swapped. The dropoff along = θs line is relatively mild: about 2.1 dB between θi = θs = 0° and θ◦ i = θs = 60°. The ◦NRCS the θi = θs line is relatively mild: about 2.1 dB between θi = θs = 0 and θi = θs = 60 . The dropoff with respect to Δθ = θi − θs is dependent on incidence angle. For example, with θi NRCS dropoff with respect to ∆θ = θi − θs is dependent on incidence angle. For example, = 0°, the NRCS◦ difference is about 4 dB for θs from 0° to 20°, ◦and much◦ steeper at large θs, with θi = 0 , the NRCS difference is about 4 dB for θs from 0 to 20 , and much steeper at about 14 dB for the same Δθ with θs from 20° to 40°. Figure◦ 13c,d◦ show variation of vv and large θs, about 14 dB for the same ∆θ with θs from 20 to 40 . Figure 13c,d show variation hh NRCS along θi = θs for scattering azimuth angles ϕs = 0°, 45°, 90°, 135°,◦ ◦ and◦ 180°.◦ Inter- of vv and hh NRCS along θi = θs for scattering azimuth angles φs = 0 , 45 , 90 , 135 , and ◦ s ◦ 180estingly,. Interestingly, in the in-plane in the condition in-plane (ϕ condition = 0°, solid (φ sblack= 0 ,lines solid in black Figure lines 13c,d), in Figure the SPT 13 c,d),predicts strong NRCS even at low grazing angles. For this configuration (θi = θs, ϕs = 0°), γ = 0°, ι the SPT predicts strong NRCS even at low grazing angles. For this configuration (θi = θs, i ◦s ◦ =φ sθ= =0 θ), (Figureγ = 0 ,14a),ι = θ ithe= θ in-planes (Figure specular14a), the forward in-plane scatte specularring forward is just like scattering altimeter is justthat like re- flectsaltimeter from that flat reflects facets from(γ = 0°), flat facetsexcept ( γthat= 0 ◦now), except the effective that now incidence the effective angle incidence is ι = θ anglei = θs, σι= ()2 2 2 2 andis ι = theθi = NRCSθs, and is the0 NRCSξηRs is ξη σ0ξη =/ fR.ξη For(ι) the/s fsame. For theocean same surface ocean roughness, surface roughness, it varies onlyit varies with only the surface with the reflectivity surface reflectivity (Figure 14b). (Figure Bistatic 14 b).radars Bistatic have radarsbeen used have in been establish- used ingin establishing forward-scatter forward-scatter fences for air fences and sea for craft air and detection sea craft based detection on the strong based onforward the strong scat- forward scattering principle [65–69]. In the out-of-plane conditions (φ 6= 0◦), the dropoff tering principle [65–69]. In the out-of-plane conditions (ϕs ≠ 0°), the dropoffs toward graz- toward grazing is rather rapid at large θ = θ . More discussion on the dependence on ing is rather rapid at large θi = θs. More discussioni s on the dependence on scattering azi- muthscattering angle azimuth is given angle later. is given later.

Figure 13. ExamplesExamples of of NRCS NRCS dependence dependence on on incidenc incidencee and and scattering scattering angles. angles. In-plane In-plane forward forward ◦ ◦ ◦ ◦ ◦ ◦ scattering ((φϕss = 0°):0 ): ( (aa)) vvvv,,( (b)) hh. Variation Variation ofof NRCSNRCS alongalongθ θi i= = θθss forfor φϕss == 00°,, 4545°,, 90°, 90 , 135°, 135 , and 180° 180 scattering azimuth azimuth angles: angles: (c (c) )vvvv, ,((dd) )hhhh. .The The microwave microwave frequency frequency is is14 14 G GHz Hz and and wind wind speed speed is 5 is m/s.5 m/s.

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◦ FigureFigureFigure 14. 14. 14.RelevantRelevant Relevant parameters parameters parameters governing governing governing in-plane in-plane specular in-plane specular reflection reflectionspecular (θi =(reflectionθθis,= ϕθs ,=φ 0°):s = (0aθ) ):i local= (θas), localϕs = 0°): (a) local incidenceincidenceincidence angle angle angle ιι andand ι facet facetand angle anglefacet γ γangle, ,((bb)) surface surface γ, (b )reflectivity. reflectivity. surface reflectivity. ◦ ◦ FigureFigure 15a15a shows shows the the in-plane in-plane ( ϕφs = 0°) 0 ) NRCS NRCS dependence dependence on on θs for θii = 50°, 50 , U10 == 5, 5, Figure 15a shows the in-plane (ϕs = 0°) NRCS dependence on θs for θi = 50°, U10 = 5, andand 15 15 m/s. m/s. In In this this figure, figure, positive positive θθs sisis in in the the forward forward scattering scattering direction direction and and negative negative θθss ◦ isisand in in the the 15 backward backwardm/s. In this scattering scattering figure, direction. direction. positive As As θ expected,s expected, is in the the the forward maximum maximum scattering NRCS NRCS is is atdirection at θθss == 50°, 50 ,and negative θs andandis inthe the the magnitude backward isis largerlarger scattering forfor the the lower lower direction. wind wind condition condition As expected, with with smoother smoother the maximum surface surface roughness. rough- NRCS is at θs = 50°, ness.Theand scatteringThe the scattering magnitude beamwidth beamwidth is islarger narrower is narrower for the for theforlower the smoother smoother wind surface condition surface roughness, roughness, with sosmoother so when whenθs surface rough- is different from θ and with large |∆θ|, the NRCS is larger in higher wind than that in θsness. is different The scattering from iθi and beamwidthwith large |Δ θis|, narrowerthe NRCS is for larger the in smoother higher wind surface than that roughness, in so when lowerlower wind. wind. The The hhhh NRCSNRCS is is larger larger than than the the vvvv NRCSNRCS as as a a result result of of the the larger larger hhhh reflectivityreflectivity comparecompareθs is different to to vvvv (Figure(Figure from 88). ).θ Thei and SPT with isis basicallybasically large | followingΔfollowingθ|, the aaNRCS geometricgeometric is larger opticsoptics in approach.approach. higher Awind than that in comparisoncomparisonlower wind. of of GO, GO, The small small hh NRCSslope slope approxim approximation is largeration than (SSA), (SSA), the and andvv NRCS two two scale scale as model modela result (TSM) (TSM) of thesolutions solutions larger hh reflectivity isiscompare given given in in [70], [ 70to], here vv here (Figurereferred referred to 8). to as as TheA06. A06. FiguSPT Figurere is 3 of3basically ofA06 A06 is reproduc is reproducedfollowinged here herea as geometric Figure as Figure 15b,c 15 optics b,cfor approach. A 5for comparisonand 5 and15 m/s, 15 m/s, respectively. of respectively. GO, small They Theyslope give give theapproxim the results results ofation ofSSA SSA (SSA),and and GO GO and comparison comparison two scale for for modelthe the EM EM (TSM) solutions and wind conditions used in Figure 15a; the E97 spectrum [30] is used in the A06 analysis. andis givenwind conditions in [70], here used referred in Figure to 15a; as theA06. E97 Figu spectrumre 3 of [30] A06 is usedis reproduc in the A06ed analysis. here as Figure 15b,c for FigureFigure 15c15c can can be be compared compared with with Figure Figure 15a15a 5 m/s results results given given with with solid solid line line ( (hhhh)) and and circlecircle5 and marker marker 15 m/s, ( (vvvv),), respectively.and and Figure Figure 15d15d Theycan can be givecompared the re with sults Figure of SSA 1515aa 15and m/s GO results results comparison given given for the EM withwithand dashed dashed wind line lineconditions ( (hhhh)) and and square squareused markerin marker Figure ( (vvvv). ).15a; There There the is is aE97 a good good spectrum quantitative quantitative [30] and and is qualitativeused qualitative in the A06 analysis. agreementagreementFigure 15c between between can bethe the comparedSPT SPT and and GO GO with results. results. Figure 15a 5 m/s results given with solid line (hh) and circle marker (vv), and Figure 15d can be compared with Figure 15a 15 m/s results given with dashed line (hh) and square marker (vv). There is a good quantitative and qualitative agreement between the SPT and GO results.

◦ Figure 15. (Top row) Examples of in-plane NRCS dependence on θs, f = 14 GHz, θi = 50 .(a) Specular Figure 15. (Top row) Examples of in-plane NRCS dependence on θs, f = 14 GHz, θi = 50°. (a) Specu- point computation, U = 5 and 15 m/s, (b) 5 m/s, and (c) 15 m/s SSA and GO solutions reproduced lar point computation,10 U10 = 5 and 15 m/s, (b) 5 m/s, and (c) 15 m/s SSA and GO solutions repro- ◦ ducedfrom Figurefrom Figure3 of A06. 3 of (Bottom A06. (Bottom row) Examplesrow) Examples of out-of-plane of out-of-plane forward forward scattering scattering ( φs = (45ϕs =) 45°) NRCS ◦ NRCSdependence dependence on θs, onf = θ 14s, f GHz,= 14 GHz,θi = 40θi =.( 40°.d) ( Speculard) Specular point point computation, computation,U10 U10= = 5 5 and and 15 15 m/s, m/s, (e) (5e) m/s, 5 m/s, and and (f) ( 15f) 15 m/s m/s SSA SSA and and GO GO solutions solutions reproduced reproduced from from Figures Figure6a and6a and7a, respectively,Figure 7a, respec- of A06. tively, of A06.

Figure 15d shows the out-of-plane NRCS dependence on θs for θi = 40° and ϕs = 45°. TheFigure high 15.wind (Top NRCS row) is Exampleslarger than of that in-plane of the NRCSlow wind dependence because of on the θ sbroader, f = 14 GHz, beam- θi = 50°. (a) Specu- lar point computation, U10 = 5 and 15 m/s, (b) 5 m/s, and (c) 15 m/s SSA and GO solutions reproduced from Figure 3 of A06. (Bottom row) Examples of out-of-plane forward scattering (ϕs = 45°) NRCS dependence on θs, f = 14 GHz, θi = 40°. (d) Specular point computation, U10 = 5 and 15 m/s, (e) 5 m/s, and (f) 15 m/s SSA and GO solutions reproduced from Figure 6a and Figure 7a, respectively, of A06.

Figure 15d shows the out-of-plane NRCS dependence on θs for θi = 40° and ϕs = 45°. The high wind NRCS is larger than that of the low wind because of the broader beam-

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◦ ◦ Figure 15d shows the out-of-plane NRCS dependence on θs for θi = 40 and φs = 45 . widthThe high of scattering wind NRCS from is larger the rougher than that surfac of thee. lowThe wind SSA becauseand TSM of solutions the broader for beamwidththe condi- tionsof scattering given in from Figure the rougher15d are given surface. in TheFigure SSA 6a and and TSM Figure solutions 7a of forA06, the which conditions are repro- given ducedin Figure as Figure15d are 15e,f, given respectively, in Figures6a andfor 57 aand of A06, 15 m/s. which The are agreement reproduced between as Figure SPT 15 ande,f, TSMrespectively, or SSA is for fair. 5 and The 15 difference m/s. The between agreement SPT between and TSM SPT or and SSA TSM is comparable or SSA is fair. to Thethe differencedifference between SPTTSM and and TSM SSA. or All SSA three is comparable produce similar to the results difference near between the specular TSM andan- gle.SSA. All three produce similar results near the specular angle. Figure 16a shows the NRCS dependence on the scattering azimuth angle φs; the Figure 16a shows the NRCS dependence on the scattering azimuth◦ angle ϕs; the inci- incidence and scattering elevation angles are ﬁxed at θi = θs = 40 . The SPT gives a dence and scattering elevation angles are fixed at θi = θs = 40°. The SPT gives a monoton- monotonically decreasing NRCS with respect to φs, because the computed ι and γ vary ically decreasing NRCS with respect to ϕs, because the computed ι and γ vary with ϕs with φ monotonically. Figure 16b,c show the SSA and TSM solutions reproduced from monotonically.s Figure 16b,c show the SSA and TSM solutions reproduced from Figure 8 of Figure8 of A06 for U10 = 5 and 15 m/s, respectively. There is a lobe structure especially A06 for U10 = 5 and 15 m/s, respectively. There is a lobe structure especially pronounced pronounced in the SSA solution. The SPT is not able to produce such a lobe variation in the in the SSA solution. The SPT is not able to produce such a lobe variation in the scattering scattering azimuthal dependence. azimuthal dependence.

◦ Figure 16. (a) Examples of NRCSNRCS dependencedependence onon φϕss,, ff = 14 GHz, θii == θss == 4040°,, U10 == 5 5 and and 15 15 m/s; (b (b) )and and ( (cc)) are are the the SSA SSA and TSM solutions reproduced from Figure8 8 of of A06A06 forfor U 10 == 5 5 and and 15 15 m/s, m/s, respectively. respectively.

TheThe GO GO approach approach (SPT (SPT included) included) is is applied applied to to very rough surface accounts for the large-scalelarge-scale roughness roughness [71]. [71]. For For nadir-looking nadir-looking (altimeter), (altimeter), it it works works well well for for small small incidence incidence ◦ angles,angles, up toto aboutabout 15 15°.. As As incidence incidence angle angle increases, increases, the the Bragg Bragg scattering scattering from from small-scale small- scaleroughness roughness becomes becomes important, important, and and methods methods such such as TSM, as TSM, SSA, SSA, and and small small perturbation perturba- tionmethod method (SPM) (SPM) have have been been developed developed [8,70 [8,7,72–0,72–74].74]. These These models models are used are used for forward for forward com- computationputation of the of scatteringthe scattering coefficients coefficients of the of sea the surface sea surface as a function as a function of various of various parameters, parameters,such as the such angles as the ofincidence angles of forincidence different fo frequencies,r different frequencies, polarizations, polarizations, wind speeds, wind and speeds,wind directions. and wind Others directions. are dedicated Others are to thededicated inversion to the problems inversion where problems empirical where methods em- piricalare usually methods used are to usually estimate used the to wind estimate speed the and wind sea speed state and from sea altimeters, state from scatterome-altimeters, scatterometersters (including (including SAR images), SAR im andages), more and recently more recently the bistatic the bistatic reflectometers reflectometers such as such the asGNSSR the GNSSR [23,24 ,[23,24,75–79].75–79]. FromFrom the computed examples presentedpresented here,here, the the SPT SPT has has been been based based on on the the geometric geomet- ricoptics optics approach, approach, has has limited limited validity. validity. Comparison Comparison with with the the GO, GO, TSM, TSM, and and SSA SSA solutions solu- suggests that the SPT works well for small |∆θ| = |θ − θ | and within a narrow azimuthal tions suggests that the SPT works well for small |Δi θ| s= |θi − θs| and within a narrow angle range in the forward specular scattering direction, estimated to be within about ±10◦ azimuthal◦ angle range in the forward◦ specular scattering direction, estimated to be within to ±20 in |∆θ| and |φs| (φ = 0 ). The applicable incidence angle range is unclear, but about ±10° to ±20° in |Δθ| andi |ϕs| (ϕi = 0°). The applicable incidence angle range is unclear,comparison but comparison with the SSAwith and the TSMSSA and solutions TSM so [70lutions] as shown [70] as in shown Figures in 15 Figures and 16 15 suggests and 16 that the SPT gives reasonable results near the specular reflection direction for incidence suggests that the SPT gives reasonable results near the specular reflection direction for elevation angle up to about 50◦ but maybe beyond, even to low grazing angle for the incidence elevation angle up to about 50° but maybe beyond, even◦ to low grazing angle in-plane specular forward scattering condition (θi = θs, φs = 0 ). Despite its limitation, for the in-plane specular forward scattering condition (θi = θs, ϕs = 0°). Despite its limita- the computation of SPT is relatively simple, and it elicits a clear connection between EM tion, the computation of SPT is relatively simple, and it elicits a clear connection between scattering and the ocean surface properties. EM scattering and the ocean surface properties. 5. Conclusions 5. Conclusions The specular point theory [4–6,8] establishes a firm relationship between specular NRCSThe and specular surface point wave statisticaltheory [4–6,8] and geometric establishes properties a firm relationship in a closed-form between expression specular for NRCS and surface wave statistical and geometric properties in a closed-form expression

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a given γ (7). Specifically, it states that the NRCS is linearly proportional to the reflectivity 2 2 |Rξη(ι)| , inversely proportional to the LPMSS s f , and multiplied with a term dominated by the exponential attenuation with respect to the surface slope at the specular point (tanγ). To consider the local incidence angle modification, the left-hand side of Equation (6) is

expressed as σ0ξη(θs, γ). For small incidence and reflection angles, the modification of the local incidence angle can be significant as a result of the exponential term. To account for the local angle modification by the background tilting, a straightforward 2D Gaussian integration of Equation (6) is performed, which leads to Equation (14) for altimeters and Equation (15) for oblique reflectometers. The model results are in very good agreement with a broad collection of specular NRCS data from L, Ku, and Ka band instruments (Figures 11 and 12). We emphasize that remote sensing measurements are indisputably among the most important data sources of ocean surface roughness. The satellite platforms allow data collection in extreme wind conditions from global oceans. Such capability is unimaginable for traditional in-situ ocean sensors. Since ocean remote sensing is interdisciplinary, it requires coherent consideration from both remote sensing and oceanographic perspec- tives in order to access this precious ocean data source for improving the ocean surface wave spectrum that is important to remote sensing analysis. In return, the improved ocean surface wave spectrum provides more accurate forward computations of microwave altimeters, reflectometers, scatterometers, and radiometers covering a wide range of frequencies, incidence angles, and polarizations. The study presented here offers a method to use specular microwave returns for understanding the mean square slopes of ocean surface roughness several times longer than the radar wavelengths. There are plenty of spaceborne altimeter and reflectometer measurements in existence. These measurements, coupled with the SPT, represent a valuable data source of the ocean surface wave properties covering a wide range of wind conditions. There are limitations associated with the general geometric optics approach. The applicable incidence angle range based on our comparison with SSA and TSM solutions suggests that the SPT gives good results near the specular ◦ ◦ ◦ reflection direction, within about ±10 to ±20 in |∆θ| = |θi − θs| and |φs| (φi = 0 ), for incidence elevation angle up to about 50◦ but maybe beyond, even to low grazing angle ◦ for the in-plane specular forward scattering condition (θi = θs, φs = 0 ). The SPT provides a clear physical interpretation between the microwave scattering from a rough surface and the statistic and geometric properties of the rough surface. The computation of SPT is relatively simple and the results presented here suggest its usefulness for the interpretation of satellite altimeter and reflectometer data for the purpose of retrieving the surface wave properties, particularly the LPMSS.

Author Contributions: Conceptualization: P.A.H., T.L.A., and J.D.O.; methodology: P.A.H., T.L.A., and J.D.O.; software: P.A.H.; validation: P.A.H., T.L.A., and J.D.O.; formal analysis: P.A.H., T.L.A., and J.D.O.; investigation: P.A.H., T.L.A., and J.D.O.; resources, P.A.H., T.L.A., and J.D.O.; data curation: P.A.H.; writing—original draft preparation: P.A.H.; writing—review and editing: P.A.H., T.L.A., and J.D.O.; visualization: P.A.H.; supervision: P.A.H., T.L.A., and J.D.O.; project administration: P.A.H., T.L.A., and J.D.O.; funding acquisition: T.L.A. and J.D.O. All authors have read and agreed to the published version of the manuscript. Funding: This work is sponsored by the Ofﬁce of Naval Research (Funding Doc. No. N0001416WX00044). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Data sets used in this analysis are given in the references cited. The processing codes and data segments can also be obtained by contacting the corresponding author. Conﬂicts of Interest: The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results. Sensors 2021, 21, 1486 19 of 21

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