INTENSITY MODULATIONS IN SYNTHETIC APERTURE RADAR IMAGES OF OCEAN SURFACE CURRENTS AND THE WAVE/CURRENT INTERACTION PROCESS

Transcription

1 DONALD R. THOMPSON INTENSITY MODULATIONS IN SYNTHETIC APERTURE RADAR IMAGES OF OCEAN SURFACE CURRENTS AND THE WAVE/CURRENT INTERACTION PROCESS The physis that determines the perturbatin f the ean surfae-wave spetrum due t the presene f a slwly varying surfae urrent is examined. In the limit f Bragg sattering, this perturbatin is related t the intensity mdulatin in radar baksatter frm the ean surfae. Using grund-truth data btained frm reent experiments as input, we mpute the expeted mdulatin at L and X band and mpare the results with the bserved mdulatin frm nurrent syntheti aperture radar images. At L band, the mparisn is fund t shw gd agreement; at X band, a mre mpliated tw-step mehanism must be invked. INTRODUCTION During the past deade r s, rapid advanes in the field f remte sensing have enabled us t examine many interesting prperties f the ean surfae. In partiular, the develpment f syntheti aperture radar (SAR) has prvided an extensive and rih database fr studying a wide variety f eangraphi phenmena, an exellent desriptin f whih is given in Ref. 1, alng with images shwing suh messale features as majr urrent bundaries, warm and ld water eddies assiated with the Gulf Stream, and surfae manifestatins f the bathymetry. Beause f the large number f unntrllable and unmeasured parameters during a SAR verflight, it is usually nt pssible t determine whih f many pssible mehanisms is respnsible fr prduing the features that are visible n the SAR images. Reently, hwever, APL has been invlved in experiments designed t btain relevant grund-truth data nurrently with the SAR verflight in rder t understand sme f the pssible imaging mehanisms. In partiular, the SAR Signature Experiment (SARSEX) (whih is desribed in the artile by Gasparvi et al. elsewhere in this issue) is nerned with understanding the SAR imaging mehanism fr internal waves. In Fig. 1, we shw X-band (fr = 9.35 X 1 9 hertz) and L-band (fr = 1.25 X 1 9 hertz) SAR images f internal waves taken in the late summer f 1984 in the New Yrk Bight area ff the eastern United States ast. The waves are generated by tidal flw ver the ntinental shelf break abut 15 kilmeters sutheast f Lng Island. The researh vessels that lleted grund-truth data an be seen as bright spts near the bttm f the images. Sine the 1 9 hertz mirwave radiatin frm the SAR penetrates nly a few entimeters int seawater, it is lear that the bright 346 and dark streaks in these images are aused by sme surfae manifestatin f internal waves. Furthermre, the mean surfae displaement arising frm the passage f the internal wave is muh t small t be reslved by the SAR. It is, in fat, the hrizntal surfae velity field f the internal wave interating with the surfae waves that is respnsible fr the SAR baksatter mdulatin seen in Fig. 1. This interatin results in a mdulatin in the surfae-wave displaement spetrum. Beause the SAR is sensitive t surfae rughness n sales f the rder f the radar wavelength, mdulatins f the surfae spetrum at these sales by the internal wave surfae urrents prdue the intensity mdulatins seen n the SAR images. It is the purpse f this artile t develp a mre preise desriptin f the wave/urrent interatin press. In the next setin, it will be shwn hw the degree f mdulatin in the wave-height spetrum f the surfae waves is related t the prperties f the surfae urrent field; hw this mdulatin is related t the bserved intensity mdulatins in the SAR image will als be disussed. Finally, we will return t ur disussin f Fig. 1 and attempt t desribe quantitatively the magnitude f the bserved mdulatins n the basis f ur understanding f the wave/urrent interatin press. WAVB/CURRENT INTERACTION THEORY W ave/ urrent interatin thery has been disussed in many bks and artiles ver the past years. One shuld see, fr example, Refs. 2 thrugh 6 and the referenes ntained therein fr a thrugh treatment f the prblem. We present here a brief develpment based n the gemetrial tehniques f ray thery and the statinary phase apprximatin, whih is an asymptti representatin f dispersive wave fields. fhns Hpkins APL Tehnial Digest, Vlume 6, Number 4

2 X band Wave 1 The k value that satisfies Eq. 3, ks = kax,t), defines the pint f statinary phase. Fr slwly varying G(k) and w" (k), HX,t) :::::: a(x,t) exp [ix(x,t)j, (4) Wave 2 Wave 3 Wave 4 where the phase funtin x(x,t) has the frm x(x,t) = ks(x,t) x - w(ks)t. (5) Nte that Eq. 4 has the frm f a plane wave but with amplitude, wave number, and frequeny that are slwly varying funtins f x and t. By analgy with the plane wave ase, we may define the lal wave number and frequeny assiated with Eq. 4 as I I 1 meters USNS Bartlett R/ V Cape L band Wave 1 and k w = These equatins imply that ax at (6a) (6b) Wave 2 k + Vw =, at (7) Wave 3 Wave 4 whih is knwn as the nservatin f wave rests. This kinemati nditin states that when n lal sures are present, the rate f hange f the wave number is balaned by the flux f wave rests past a given pint; thus, rests are nserved. If we write the dispersin relatin as! USNS Bartlett Azimuth Figure 1-Cnurrent X- and L-band SAR images f internal waves rerded during the SARSEX. where the additinal spae-time dependene speified by A(x,t) allws fr nnunifrmity in the medium, Eq. 7 may be written as w = W [k(x,t), A(X,t)], (8) A general desriptin f a linear dispersive wave field, HX,t), an be written as where (x,t) 1 G(k) exp Uk x - w(k)t] dk, (\) G(k) = (2.y' 1 Hx,O) exp [-ik x] dx, (2) and w(k) is the dispersin relatin between wand k. In the limit f large x and t, Eq. 1 may be slved apprximately by regnizing that fr slwly varying G(k), the main ntributin t the integral mes frm the regin in k spae that satisfies V k [k x - w(k)t] = O. (3) Jhns Hpkins APL Tehnial Digest, Vlume 6, Number 4 (9a) (where repeated indies are summed). We nw ntie frm Eq. 3 that the pint f statinary phase prpagates with a velity C g given by (9b) whih is knwn as the grup velity and, in fat, determines the spatial psitin at whih Eq. 4 is valid fr a partiular k s Using the grup velity and the fat that V x k = (see Eq. 6a), we may rewrite Eq. 9a as 347

3 D. R. Thmpsn - Intensity Mdulatins in SAR Images f Oean Currents and the Wave/Current Interatin ak; ak + C _I at g} ax J aw a'j a'j... ax;, (loa) and, by differentiating Eq. 8 with respet t time, we btain Nte that alng the spatial urves defined by dx- _J = C. dt g} (lob) (11 ) the left-hand sides f Eqs. 1 may be written as ttal time derivatives. These urves are alled rays, and frm Eqs. 1 ne an see that fr a hmgeneus, timeindependent medium ('J... = nstant), k and ware nstant alng a ray. Fr waves prpagating in a mving medium fr whih the urrent velity U(x,t) varies ver sales that are large mpared t the wavelengths and perids f the waves, the frequeny measured by a statinary bserver is Dppler shifted t w = W + k U, (12) and the pint f statinary phase nw prpagates alng the urve defined by and dk; dt au -k. _J J ax; (15b) Equatins 15 define ray trajetries r harateristis in fur-dimensinal phase spae; it will be shwn in the fllwing paragraphs that the nature f the trajetries determines hw energy is transprted ver the fluid surfae. We turn ur attentin nw t the dynamial nditins that gvern energy prpagatin. Let 8(x,t) be the average energy density in the wave field desribed by Eq. 4 (averaged ver time and spae sales O(I/w) and O(1/k)). Then the ttal wave energy between pints XI and X2 is given by (16) where we nsider the ne-dimensinal ase fr simpliity. Ray trajetries with assiated grup velities C gi and C g2 may be fund that pass thrugh XI and X2' respetively. Then, sine XI. = Cit and g X2 = C g2 t, we may wnte de 1" a8 dt at dx + C g C gl 8 1 XI dx _J = C. + U. dt g} J Equatins 1 then take the frm (13) =1" [ a8 + :x (C.8)] dx. (17) XI at Fr a unifrm medium with n urrent, the wave energy is nserved, i.e., de/dt =, and we may write and aw aw at + (C gj + U) ax. J (14a) (14b) Nte that as in the n-urrent ase, Eqs. 14 may be written as ttal time derivatives alng the rays defined by Eq. 13. Fr the ase f a unifrm medium ('J... = nstant) with a slwly varying urrent U(x,t), the kinemati nditins n k and w (within the ntext f the statinary-phase apprximatin) are determined by the knwn dispersin relatin (e.g., Eq. 8) and the upled system f equatins 348 dx; dt (15a) a8 at + v (Cg 8) =, (18) where we have reverted bak t vetr ntatin. Frm this equatin we see that the energy is transprted at the grup velity. Als, fr a unifrm medium with n urrents, V C g =, and the energy density remains nstant alng a ray trajetry. Equatin 18 may be generalized t the ase f a nnunifrm medium-in partiular, t the ase where a slwly varying urrent U(x,t) is present. This is mst easily amplished using variatinal methds 3,4 and, in fat, leads t a new nservatin equatin fr the quantity 8/ w, alled the wave atin, f the frm () + V [(C g + U) ] at W W O. (19) In this equatin, W is the intrinsi frequeny in a lal rest frame and C g = V k W. Equatin 19 hlds Jhns Hpkins APL Tehnial Digest. Vlume 6, Number 4

4 D. R. Thmpsn - Intensity Mdulatins in SAR Images f Oean Currents and the Wave/Current Interatin fr waves prpagating in a medium whse prperties are slwly varying mpared t the waves, when n sures r sinks f energy are present. By separating ut the W variatins, we may rewrite it (assuming a unifrm medium, A = ) as an energy balane equatin f the frm af, at + v [(Cg + U)f,] W k (C g v) U. (2) Cmparing Eq. 2 with Eq. 18, we see that when a surfae urrent field U is present, the surfae waves interat with the urrent via the term n the right side f Eq. 2. It is this term that generates the wave/urrent interatin. Nte that when U is nstant, there is n interatin and Eq. 2 takes the frm f Eq. 18 with f, transprted alng the ray trajetries with velity C g + U. The result given by Eq. 19 may be mdified t desribe a general wave field by writing the wave atin in terms f its spetral dempsitin. If we define the atin spetral density N(k;x,t) by A = I N(k;x,t) dk, (21) where A is the ttal atin, it an be shwn (see e.g., Ref. 5) that when n sures r sinks f energy are present, dn dt an at + (Cg + U) V N =, (22) where N(k;x,t) may vary slwly in spae and time beause f the presene f the urrent field U. Equatin 22 states that (when n sures r sinks are present) the atin spetral density N is nstant alng a ray trajetry, even when a surfae urrent field is present. Nte that this is nt true, in general, fr the ttal atin A, whih beys Eq. 19. Equatin 22 expresses nservatin f wave atin. The ttal wave atin A is related t the surfaewave-height pwer spetral density S(k;x,t) by A(x,t) = pg I [1 + (k/km )2] S(k'x t) dk (23) w(k),,, where the (k/ k m )2 term is due t the effets f surfae tensin and W is given by the gravity-apillary dispersin relatin In these equatins, g is the aeleratin f gravity; the fluid density, p, is assumed t be nstant; and k m = 363 radians per meter. Frm ur previus defi- Jhns Hpkins APL Tehnial Digest, Vlume 6, Number 4 nitin f the atin spetral density in Eq. 21, we btain W N(k;x,t) = p S(k'x t) (25) fki ", whih relates the atin spetrum t the wave-height pwer spetrum. Thus, the nservatin f atin equatin als desribes the behavir f the surfaewave spetrum. In partiular, when n urrents are present, k (and hene w) is nstant and, therefre, arding t Eq. 22, s is the pwer spetrum. When urrents are present, the waves and urrent interat, and the surfae-wave pwer spetrum is mdified by the urrent field. Up t this pint, we have negleted sures r sinks f wave energy in ur disussin. We knw, hwever, that when a nstant wind begins t blw n the smth sea surfae, surfae waves develp-shrt waves at first, then lnger and lnger-until an equilibrium wave-height spetrum is reahed and n further wave grwth urs. The equilibrium urs when the energy input is balaned by dissipatin effets. If the equilibrium is disturbed, fr example, by a lal surfae urrent, the wind-dependent sure/sink term will tend t fre the spetrum bak tward equilibrium. Therefre, in rder t generalize the nservatin f atin as develped abve, a sure/sink term is required in rder t aunt fr the wind effets. The frm f this term is a tpi f urrent researh and remains ntrversial (see e.g., Refs. 5, 7, and 8 and the referenes ited therein). In the present study, we adpt the sure term desribed by Hughes 8 and reast Eq. 22 as an atin balane equatin in the frm dn(k;x,t) _. [ N(k;x,t)] - - {3(k)N(k,x,t) 1 -, dt Neq(k) (26) where the time dependene f k and x is given by Eqs. 15, and Neq(k) represents the equilibrium (n-urrent) atin spetrum that is independent f spae and time. Nte that the sure/sink term n the right side f Eq. 26 is zer when N = Neq and psitive (sure) r negative (sink) depending n whether N is less than r greater than N eq, respetively. Cnsequently, the sure/sink term always tends t fre N tward its equilibrium value N eq. The strength f the fring is determined by the funtin,b(k), the s-alled relaxatin rate. Sine shrter surfae waves respnd mre quikly t hanges in the wind, {3(k) is an inreasing funtin f Ik I, and fr fixed k, {3(k) inreases with wind speed. There is, hwever, nsiderable unertainty as t its exat funtinal frm (see e.g., Refs. 8 and 9). With the time dependene f k and x given by Eqs. 15, Eq. 26 is a first-rder nnlinear differential equatin fr N. Fr N, we may use the transfrmatin Q = 1/ N t write Eq. 26 in the frm 349

5 D. R. Thmpsn - Intensity Mdulatins in SAR Images f Oean Currents and the Wave/Current Interatin dq dt + {3Q = (3Qeq, (27) where Qeq = 11 N eq. The slutin t this equatin may be written as Q(k;x,t) exp [-L J3(k N ) dt J N dt', (28) : : u... a. <:Q..1.1 v w = 3 meters per send = 6 degrees where the primes and duble primes n the k and x vetrs rrespnd t the primes and duble primes n the rrespnding integratin variables. Nte that fr (3(k) =, Eq. 28 yields the nservatin f atin result Q[k(t);x(t),t] = Qeq[k(-)] as it shuld. We may nw use this equatin t determine the hange in wave-height spetral density relative t the equilibrium value. In partiular, using Eq. 25 t relate S t N (and ur transfrmatin Q = lin), we btain where S(k;X,t) 1 = [1 + P(k;x,t)r, (29a) Seq(k) exp [-r" J3(k N) dt N J dt', (29b) Equatins 29 enable us t express the perturbatin f the surfae-wave-height spetrum fr a partiular k value at a speified spatial latin as a funtin f the lal surfae urrent field, a speified equilibrium spetrum, and the relaxatin rate (3(k). In the fllwing setin, we disuss the results f ur attempts t predit the intensity mdulatins bserved in the SAR images f Fig. 1 using the wave/urrent interatin thery disussed abve, with the measured internal-wave surfae urrents as input. In btaining the results, we assume that the equilibrium wave-height spetrum Seq (k) is given by the mdified Piersn spetrum desribed in Ref. 1, with a s 4 [(8 k - 8 w )/2] angular dependene, where 8 k and 8 w speify the k and wind diretins, respetively. This empirial spetrum prvides a reasnable representatin f spetral measurements ver a wide range f Ik I inluding the gravity-apillary and the apillary ranges. Fr the k values f interest in ur study, the spetrum falls rughly like k- 4 Als, we have hsen fr the relaxatin rate (3(k), the parameterizatin given by Hughes. 8 The k dependene f (3(k) is shwn in Fig. 2 fr wind speeds (V w) f 3 and 6 meters per 35.1l.L..- -L-... -L- -..J- --' I kl (radians per meter) Figure 2-k-dependene f the surfae-wave relaxatin rate, {3. The {3 urves are fr wind speeds, V w, f 3 and 6 meters per send and wind diretins, </J, with respet t k f and 6 degrees. The parameterizatin f (3(k) is that f Hughes. 8 send blwing alng the diretin P f degrees, and 3 meters per send fr P = 6 degrees. Nte that fr all ases shwn, (3(k) inreases relatively slwly fr k ;;::: 6 radians per meter but begins t derease rapidly fr k :5 6 radians per meter. INTENSITY MODULATION IN SAR IMAGES OF INTERNAL WAVES Figure 1 shws L- and X-band SAR images f the surfae manifestatin f internal waves prpagating tward the tp (range diretin) f the image. Eah image spans abut 5 kilmeters in azimuth and abut 6 kilmeters in range, with the spaing between the fur brightest waves at the tp f the sene measuring abut.5 t 1.5 kilmeters. The SAR inidene angles t these waves range frm arund 25 t 47 degrees, whih rrespnds t Bragg wave numbers, k8 (equal t twie the prjetin f the radar wave number n the hrizntal surfae), f abut 22 t 38 radians per meter fr L band and abut 166 t 287 radians per meter fr X band. The researh vessel USNS Bartlett an be seen learly near the bttm enter f eah image, and the smaller vessel R/V Cape is the dim pint seen mst easily in the X-band image t the right and belw Bartlett. The vessels lleted grund-truth data by traversing the waves n parallel traks tward the tp f the sene abut 15 minutes after the images shwn in Fig. 1 were taken. The wind velity was relatively nstant at 6 meters per send tward an angle f 145 degrees with respet t the internal wave prpagatin diretin (tward the tp f the sene). One an easily see bright and dark streaks assiated with eah f the large waves in the L-band image but nly bright streaks fr the X-band ase. Jhns Hpkins APL Tehnial Digest, Vlume 6, Number 4

6 D. R. Thmpsn - Intensity Mdulatins in SAR Images f Oean Currents and the Wave/Current Interatin The relative mdulatin in the SAR intensity alng the trak f Bartlett is shwn fr X band in the upper plt f Fig. 3 and fr L band in the lwer plt. Bth urves were btained by averaging the SAR intensity ver abut 14 meters in the hrizntal diretin (azimuth), smthing ver abut 3 meters alng trak (range), and nrmalizing s that the bakgrund intensity is unity. The peaks labeled waves 1 thrugh 4 n the plts rrespnd t the first thrugh furth waves frm the tp f the apprpriate image in Fig. 1. One an see frm Fig. 3 that the enhanement in refletivity (mdulatin greater than 1) fr these fur waves is nearly the same at L and X bands, while nly the L-band return shws a definite regin f redued refletivity (mdulatin less than 1) assiated with eah wave. Als, nte that the psitin f the maximum mdulatin fr eah wave urs at rughly the same distane frm the ship at bth L and X bands. An aeptable thery f the SAR imaging mehanism shuld aunt fr eah f these bservatins. Fr the radar frequenies and inidene angles suh as thse in SARSEX, the radar baksatter frm the ean surfae is given by the small perturbatin r Bragg sattering limit, 11,12 where the radar rss setin is prprtinal t the sum f the surfae-wavedisplaement pwer spetral densities S(k,x) at the Bragg wave numbers ± k B Thus, fr range-traveling waves suh as thse shwn in Fig. 1, the SAR intensity mdulatin M(k B,x) (r relative intensity) is given by M(kB,x) S(kB,x) + S(-kB'X) Seq (k B) + (3) Seq (-kb) 3. where bth M and S are funtins f psitin due t the presene f the surfae-urrent field as disussed abve. We nw use the measured surfae urrents and winds frm SARSEX as inputs t the mdel desribed previusly and mpare the alulated intensity mdulatin with that measured frm the SAR image. The surfae urrents were determined by using shipbard eh-sunder measurements f the internal-wave amplitude and twed-hain measurements f the thermline displaement in njuntin with internalwave thery 13 t nstrut analyti representatins fr the surfae urrent assiated with the waves. These urrents were fund t agree with diret measurements frm the shipbard urrent meters and an be represented by the funtin U seh2 [K(x - CIt)]. Typial values fr the urrent U and peak urrent gradients fr the waves shwn in Fig. 1 are abut.3 t.6 meter per send and.2 t.8 inverse send, respetively. In Fig. 4, we shw a mparisn f the measured L-band intensity mdulatin ver waves 2, 3, and 4, as desribed earlier, with the preditins frm the wave/ urrent interatin mdel. The mdel results shwn by the lred urve in the upper plt in Fig. 4 were btained using the analyti representatin fr the internal-wave surfae urrents, while thse in the lwer plt were mputed diretly frm the shipmeasured urrents. In bth ases, we have used the measured wind velity (6 meters per send at p = 145 degrees relative t the internal-wave prpagatin diretin), Hughes relaxatin rate,8 and the mdified Piersn.equilibrium height spetrum 1 disussed previusly. Fr the wind velity and Bragg k values f interest here, nly waves with k = -kb ntribute signifiantly t Eq. 3. One an see frm Fig. USNS Bartlett 'en "en Wave 4 r ai 2. '';::; 1. r "';::; 1. ai 3. USNS L band Bartlett 'en 2. +-' 'en +-' Wave 4 '';::; r 1. ai 2. '';::; Distane (meters) 7 2 Distane (kilmeters) Figure 3-Relative intensity traes alng the Bartlett trak. Nte that the large signal frm Bartlett has been lipped at a relative intensity f 3. Jhns Hpkins APL Tehnial Digest, Vlume 6, Number 4 Figure 4-Cmparisn f the measured (blak urves) and mputed (lred urves) relative L-band intensity mdulatin fr waves 2 thrugh 4. The lred urve in the upper plt shws the alulated results btained using an analyti representatin f the internal-wave surfae urrent in the viinity f eah wave. The lred urve in the lwer plt shws the alulated results using the urrents measured by the shipbard urrent meter. 351

7 D. R. Thmpsn - Intensity Mdulatins in SAR Images f Oean Currents and the Wave/Current Interatin 4 that althugh the alulatin smewhat underestimates the measured L-band mdulatin, the general agreement is quite enuraging. Nte in partiular that the alulated mdulatin reprdues the enhanement and redutin with nearly the same spatial variatin as was bserved in the measured data. We have als fund that the mputed mdulatin is rughly prprtinal t the hrizntal urrent gradient, in agreement with the thery f Alpers. 14 Thus, within the experimental unertainties in the exat value f the relaxatin rates, we feel that the wave/urrent interatin thery utlined abve an adequately explain the L-band SAR imaging f internal waves. The situatin at X band is nt as lear. Appliatin f the thery t mpute the mdulatin f the Xband Bragg waves (A 3 entimeters) fr the internal waves in Fig. 1 yields peak mdulatins f nly a few perent f the bakgrund intensity as ppsed t 2 t 5 perent at L band. Qualitatively, this urs beause the shrter X-band waves respnd muh mre quikly t the wind than d the L-band waves. Thus, the relaxatin rate in the X-band regin is signifiantly larger and the sure term in Eq. 26 is mre effetive in fring the spetrum tward equilibrium than at L band. Hwever, the measured X-band data, as shwn by the upper plt in Fig. 3, shw enhanements in the SAR intensity f the same rder as thse bserved at L band and little redutin in intensity. These bservatins seem t indiate that althugh the presene f the internal-wave surfae urrent des indeed prdue enhaned rughening f the surfae at X-band sales, the rughening is nt due t diret mdulatin f X-band Bragg waves as desribed by Eq. 26. It is well knwn, hwever, that as gravity waves steepen t near the breaking pint, they begin t lse energy by generating pathes f small-sale rughness and by wave breaking. 5 We are therefre led t the hypthesis that the bserved X-band mdulatin results frm a tw-step press in whih inreased small-sale rughening is generated as a result f the perturbatin f the dminant wind waves by the internal-wave surfae-urrent field. In rder t substantiate ur hypthesis, we have mputed the perturbatin f the lnger wavelength mpnents f the surfae spetrum by the measured internal-wave urrents. Results fr the internal wave 3 (as named abve) surfae urrent are presented in Fig. 5 fr wavelengths ranging frm.21 (L band) t 271" meters; the family f urves between A = 271" and A =.271" rrespnds t wave numbers between 1 and 1 radians per meter in steps f 1 radian per meter. The prpagatin diretin fr eah wave number was taken t be ppsite the internal-wave prpagatin diretin at an angle f 35 degrees with the wind (i.e., alng -kb as fr the L-band wave). One an see frm the plt that the psitin f maximum perturbatin fr the lnger waves is near the peak f the internalwave surfae urrent (shwn in the upper plt f Fig. 5). As the wavelength dereases, the psitin f this maximum shifts tward the psitin f the maximum (negative) gradient (shwn in the middle plt in Fig ,_-----,----, , ,----., u Sl.4 CIJ... :J U... <l.l Cl. 2. CIJ O-----==-----L ,_-_ ' C. 8 -CIJ 15r '" Q; Cl L- L_ ,----,------, ,-- ' ;:::; CIJ E--3 Spetral perturbatin 3 A= 2 2 1T meters CIJ"U E O_ L - L 1 2 _ 3 4 Distane (meters) Figure 5-Cmparisn f the spatial behavir f the urrent and urrent gradient determined fr wave 3 with that f the mputed spetral mdulatin at several different wavelengths. 5). There are, mrever, peak mdulatins f between tw and three times the equilibrium value thrughut this entire regin f the internal-wave phase. Althugh we annt yet estimate quantitatively the degree t whih the surfae rughness is affeted by the mdulatin f the lnger meter-sale waves, it seems likely that it is large enugh t ause a signifiant inrease, fr example, in the density f parasiti apillary waves r small-sale lally breaking waves t aunt fr the bserved inrease in the X-band SAR return. The regin f large mdulatin urs between the peak internal-wave urrent and the peak (negative) urrent gradient, rughly the same latin as fr the maximum L-band perturbatin. This is nsistent with the bservatin frm the data that the maximum L- and X-band mdulatin urs at rughly the same psitin. In fat, the mdulatin f the lnger waves shuld als prdue additinal L-band satterers, perhaps explaining the slight underpreditin f the maxima in the L-band SAR signatures disussed earlier. Finally, we see frm Fig. 5 that there is an extended regin where the meter-sale waves shw a mdulatin less than unity. In that regin, there shuld be n additinal prdutin f small-sale rughness via the tw-step press disussed abve, and the magnitude f the X-band Bragg mpnents shuld therefre be mparable t their equilibrium value, thereby explaining why n ntieable dark bands, suh as thse bserved at L band, are visible n the X-band images. CONCLUSION We have presented a brief disussin f the press by whih surfae waves are affeted by the presene f lal surfae urrents. In partiular, we have develped a wave/urrent interatin mdel that an be Jhns Hpkins APL Tehnial Digest, Vlume 6, Number 4

8 D. R. Thmpsn - Intensity Mdulatins in SAR Images f Oean Currents and the Wave/Current Interatin used t predit the mdulatin f the surfae-waveheight spetrum as a funtin f psitin in a variable urrent field. With the assumptin f Bragg sattering, the mdulatin evaluated at the Bragg wave number shuld be prprtinal t the intensity mdulatin in a SAR image f the urrent feature. These ideas have been tested by using grund-truth measurements frm SARSEX as inputs t the wavel urrent interatin mdel in rder t mpute the mdulatin f the surfae-wave spetrum. With the assumptin f Bragg sattering, the mdulatin at the Bragg wave number is prprtinal t the SAR intensity mdulatin. We have fund, using the relaxatin rate (3(k) f Hughes, that the predited mdulatin at L band shws gd verall agreement with measured data, althugh the magnitude f the predited mdulatin is generally lwer. Als, we have fund that the predited L-band mdulatin is rughly equal t -4.5 n Ulx[l/{3(k B )]. This means that the L-band mdulatin measurements and a knwledge f the hrizntal urrent gradient uld, in priniple, be used t determine (3(k B ). Alternatively, knwledge f the L-band mdulatin and (3(k B ) uld be used t determine the urrent gradient. The measured SAR mdulatin at X band shwed regins f enhaned intensity f the same rder and at nearly the same psitins as thse bserved at Lband, an enhanement that was almst an rder f magnitude larger than that predited by diret appliatin f the wave/urrent interatin mdel t the X-band Bragg waves. Hwever, the mdel predits that the dminant wind waves shw a maximum enhaned mdulatin tw r three times the bakgrund level at rughly the same psitin in the internal-wave phase as the L-band peak. This finding has led t the speulatin that the X-band mdulatin is diretly nneted t the small-sale rughness prdued as a result f the mdulatin by the internal-wave urrent f the lnger (mre than a meter) wind waves, rather than by diret mdulatin f the X-band Bragg waves. We hpe t strengthen this speulatin by further analysis f additinal SAR imagery under different envirnmental and imaging nditins where the harater f these lnger wave mdulatins an be quite different. The results f these analyses shuld add signifiantly t ur grwing understanding f the physis that gverns SAR imaging f ean surfae urrents. REFERENCES R. C. Beal, P. S. DeLenibus, and I. Katz, eds., Spaebrne Syntheti Aperture Radar fr Oeangraphy, The Jhns Hpkins University Press, Baltimre and Lndn (1981). 2 M. S. Lnguet-Higgins and R. W. Stewart, "The Changes in Amplitude f Shrt Gravity Waves n Steady Nnunifrm Currents," 1. Fluid Meh. I 1, 529 (1961). 3 F. P. Brethertn and C. J. R. Garrett, "Wavetrains in Inhmgeneus Mving Media," in Pr. R. S. Lndn S32, p. 529 (1969). 4 G. B. Whitham, Linear and Nnlinear Waves, Jhn Wiley and Sns, New Yrk (1974). 5 O. M. Phillips, The Dynamis f the Upper Oean, Cambridge University Press, New Yrk (1977). 6p. H. LeBlnd and L. A. Mysak, Waves in the Oean, Elsevier Sientifi Publishing C., New Yrk (1978). 7 K. Hasselmann, "Weak Interatin Thery f Oean Waves," in Basi Develpment in Fluid Dynamis, M. Hld, ed., Aademi Press, New Yrk (1968). 8 B. A. Hughes, "The Effet f Internal Waves n Surfae Wind Waves, Theretial Analysis," 1. Gephys. Res. 83C, 455 (1978). 9W. J. Plant, "A Relatinship between Wind Stress and Wave Slpe," 1. Gephys. Res. 87, (1982). t A. W. Bjerkaas and F. W. Riedel, Prpsed Mdel fr the Elevatin Spetrum f a Wind-Rughened Sea Surfae, JHU/ APL TG 1328 (1979). II G. R. Valenzuela, "Sattering f Eletrmagneti Waves frm a Tilted Slightly Rugh Surfae," Radi Si. 3, (1968). 12J. W. Wright, "A New Mdel fr Sea Clutter," IEEE Trans. Antennas Prpag. AP-14, (1966). 13 J. R. Apel and F. I. Gnzales, "Nnlinear Features f Internal Waves ff Baja Califrnia as Observed frm the Seas at Imaging Radar," 1. Gephys. Res. 88, (1983). 14W. Alpers, "Thery f Radar Imaging f Internal Waves," Nature 314, (1985). ACKNOWLEDGMENTS-Aknwledgment is due t E. Kasishke at the Envirnmental Researh Institute f Mihigan fr supplying us with the SAR data and fr disussins nerning alibratin and grund-range transfrmatin. I als wish t aknwledge the help f my lleagues at APL: J. Apel, R. Gasparvi, B. Gtwls, R. Sterner, and T. Taylr, fr useful disussins during the urse f this wrk; J. Kerr and B. Raff, fr pressing the SAR images; and K. MMakin and M. Wilfrd, fr preparatin f the manusript. The wrk was spnsred by the Offie f Naval Researh. THE AUTHOR DONALD R. THOMPSON was brn in Sidney, Ohi, in He reeived a B.S. in physis frm Case Western Reserve University in 1964 and a Ph.D. in physis frm the University f Minnesta in 1968, where his thesis dealt with the quantum-mehanial desriptin f few-nulen sattering prblems. After tw years at the Califrnia Institute f Tehnlgy where he applied these methds t the study f reatins that gvern stellar nulesynthesis, Dr. Thmpsn returned t the University f Minnesta in 197. In 1976, he was awarded an Alexander vn Humbldt Fundatin dent fellwship, whih he used t ntinue his wrk n nulear sattering thery at the Institute fr Theretial Physis at the University f Tilbingen in West Germany. Dr. Thmpsn ame t APL in 198, where he has been wrking n the physis f ean surfae waves and remte sensing in the Systems Grup f the Submarine Tehnlgy Department. He has published extensively and is a member f Tau Beta Pi, the Amerian Physial Siety, the Amerian Assiatin fr the Advanement f Siene, and the New Yrk Aademy f Sienes. 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