New Derivation for the Rough Surface :c cc: i Coefficient and for the Distribution of Sea Wave Elevations

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1 4; "00' V NRL Report 814,' New Derivation for the Rough Surface :c cc: i Coefficient and for the Distribution of Sea Wave Elevations A. R. MILLER Software Systems and Support Branch Research Computation Division R. M. BROWN ENEWS Program Tactical Electronic Warfare Division E. Vicii Identification Systems Branch Radar Division September 30, 1983 UELECTEa Noy 1OV NAVAL RESEARCH LABORATORY 'Washington, D.C. E Approve! for public releasee; distribution unlimited, S

2 SECURITY CLASSIFICATIOh OF T.IS PA(,E f"lo.n D0ta Lnlered) REPORT DOCUMENTATION PAGE READ INSTRUCTIONS IEBEFORE COM.PL"'TING FORM I REPO)RT NUMBER ;2 G T C-SIO O P:PtENT'S CATALOC NUM13ER NRL Report CA 4 TITLE (and Subtitlh) ITYPE OF REPORT & PERIOO COVERED A NEW DERIVATION FOR THE ROUGH SURFACE REFLECTION COEFFICIENT AND FOR THE DISTRIBUTION OF SEA WAVE ELEVATIONS Final report A. PERVORM:NG ORG. REPORT NURSER?. AUTHOR(e) I CONTRACT OR t.rant NUMBERfe) A. R. Miller, R. M. Brown, and E. Vegh S. PERFORMING ORGANIZATION NAME AND ADDRESS 10 PROGRAM ELEMENT. PROJECT. TASK Naval Research Laboratorr' Washington, DC AREA & WORK, NIT NUMUERS PE 63267N, W and A-3 II. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Naval Air Systems Command September 30, 1983 Washington, DC NUMBER 0o PAGES *7ORING AGENCY NAME & AODDRESS($ different from Controlling Office) is. 1 SECUlITY CLASS. (of thls report) UNCLASSIFIED ISO. DECLASSIFICATION/DOWNGRADING SCHEUULE 1,. OISTRItUTION STATEMENT (of Wleg Roseat) Approved for public release; distribution unlimited. 17. OiSTRIIuTION STATEMENT (of the abstract efterod In Stock 20. I dltflenept from R port) 16. O.JPPLEMIENTARtY NOTES is. KEY woims (Ce."*wu1 m otea,ete *ldo If nc*o"w). and lnff r by bloch k Rough surface reflection coefficient Sea wave elevation distribution Coherent reflected field m"bot) Fourier transform Bessel functions t AGSTRACT (C*hou. sat rt o.. old Id it neessay and 1dtlte Iy bpbloch owmbwj. In this report the rough surface reflection coeffkient for the coherent reflected field is derived. The result obtained assumes a plane wavefront incident on a Gaussian collection of sinusoidal surface waves with uniform phase distribution. The theoretical result obtained is compared with experimental data. In addition, the probability density function of sea wave elevations is obtained. Do, ' EO,TION OF I NoV 6S5It OBSOLET[E - S/N SECURITY CLASSIFICATION OF THIS PAGE (1.Mto Dee0 So-ftd) I - -

3 A NEW DERIVATION FOR THE ROUGH SURFACE REFLECTION COEFFICIENT AND FOR THE DISTRIBUTION OF SEA WAVE ELEVATIONS In 1961 C. I. Beard [1 found that experimental values of the coherent,eflected field, R = IE/EgrI were larger that, the values given by the generally accepted theoretical formula -T = exp (-2(27rg) 2 ) (1) for values of g - (a-h sin X)/X greater than 0.1 radian. Here E is the average electric field due to the "sea surface" SA, E& is the field due to the direct wave, r is the smooth sea refl.ction coefficient, 0 -h is the standard deviation of the sea-surface elevation, X is the grazing angle, and X is the electromagnetic wavelength (Fig. 1). 1.o-0 1 *O Accession For NTIS GRA&I DTIC TAB Unannounced 0 Justification u0.5 - THEORY _.\%,,#-MILLER Et BRTOWN) By EXPERIMENT Distribution/ ".N 7 W.ARD) Availability Codes ".,.,EA.D) --- Avail and/or THEORY0I EN Dist Special 0 o._o (AMENT) S(oh sin X)/A Fig. I - Comparison of theoretical and ey.perimental results for coherea. forward scattering This expression for IE,/EgrIsA was first publishrd by W.S. Ament [21 in 1953 and was obtained from the equation R * exp 4,y 5in X Dy y 2 ID essentially also derived in Ref. 2. Here D(Y) "ds the normalized probability density function for sea wave elevations, y. Equation (1) was obtairi ";nm Eq. (2), by assuming that sea waves were flat surfaces with sea height H, distributed normaiiy,.t:., y - H fo- "sea surfac-e Sý 4, and D~y - -exp H--3 f~w a If 2 T2 I- H1 One might note W!J that Eq. (2) is the Fourier transform of the density function, D(y). Manuscript approved May J

4 MILLER, BROWN, AND VEGH Brown and Miller [31 derived Eqs. (1) and (2) as well, using geometric optics. Moreover, they showed that the coherent reflected field was given by R = V exp (-2 (2irg) 2 )Io(2(2irg) 2 ) (4) where Io(x) is the modified Be~sel function J0(ix). They derived this result by using geometric optics, assuming a planar wavefront incident on a Gaussian collection S of sinusoidal surface waves with uniform phase distribution. Further, IE/E 1I.s agreed with Beard's experimental curve (Fig. 1), with a systematic difference in g of 10%. Beard had estimated that wave elevations were within 10% of their correct values. After. reading Ref. 3, Ament suggested to the authors that a short proof of Eq. (4) might be obtained along the lines that he produced Eq. (1) from Eq. (2). In this report we present such a proof. We assume that the "sea surface," S, has sea wave elevation that is a random variable, y, with y = H sin -x where A is water wavelength, H is distributed as in Eq. (3), and x is distributed uniformly on the interval (-A/4, A/4). It is a straightforward computation to show that the probability density function of y is given by H2 exp DMy)- " dh (5) Tr _3/2 'H VHy y where (see appendix). We see that D(y) 0/ ay - adl (6) _I 1 exp -_ - T i3/24,. fiý 1Hj 2 _ di, exp _12] 1-2o'h and using Ref. 4, page 319, , D(,)-r(I/2) /-Ae CXP 1-1 Y JJY~ K' o 2 J I80,2 2w 312 crh 8a2J 2 i 4--- I I.

5 NRL REPORT 8744 where Ko(x) is a Bessel function of imaginary argument defined by Substituting Eq. (7) into Eq. (2) we have Ko(x) cos (x sinh t)dt, x > 0. R- I Cos -- y sin X exp - K I-Y. 2 d &Ih Jo- S j- crh sincr X 0 = 7/0 2cos ( N 4 (x) dx. Now using Ref. 4, rage 765, we obtain Eq. (4). Since Eq. (4) agrees with the experimental data and Eq. (2) is the Fourier transform of D(y), Eq. (7) must be a good approximation for the actual distribution of sea, wave elevations (s-.e Fig. 2). 1.0 Oh K 0.5 1" Fig. 2 - Probability density function for sea wave elevations, y (0 in feet and the standard deviatiorn of y. ag. also in feet) CONCLUSIONS In this report we derive the rough surfac,- reflection coefficient for the coherent reflected field with the assumption of a plane wave front incident on a Gaussian collection of sinusoidal surface waves with uniform distribution. The theoretical result agrees with the experimertal data, when the systematic difference in the experimental data is taken into account. Furthermore, D(y) as given here is a good approximation for the actual distribution of sea wave elevations. REFERENCES 1. C.I. Beard, "Coherent and Incoherent Scattering of Microwaves from the Ocean," IRE Ttans. Antennas & Propag. AP-9(S), 470 (1961). 2. W.'S. Ament, "Toward a Theory of Reflection by a Rough Surface," Proc. IRE 41(1), 142 (1953). 3. R.M. Brown and A.R. Miller, "Geometric-Optics Theory for Coherent Scattering of Microwaves from the Ocean Stirfa.-e," NRL Report 7705 (1974). 4. i.s. Gradshieyn and I.M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, 1980). 3

6 Appendix A Consider the random variable y = H sin 0, where H is a random variable distributed normally with mean 0 and standard deviation o-h and 0 in a random variable, independent of H, distributed uniformly on the interval 101 < ir/2. Let K, F, and D be the densities of H, 0, and y respectively. Then and K(H) - exp (H) (-H/' 2 /2o"]/), IHIr<o SF(O) = r-' Io1 101> < 7r/2 ir/2. Since 0 and H are independent, the joint density, g (0,H), of 0 and H is given by K (H)F(O) or exp (-H 2 f2o') g(0,h) /2ir 3 / 2 'rh Let y - H sin 9, v - H so that 8 - Sind (Y/v) and let f(y, v) be the joint density of y and v. 1 Then Then f0yv) - AJi g(sin-t(y/v),v) where ay O v OH OH A straightforward computation yields and I.I f1h exly, f-h2/2,,h) I(,)"21/2,tr3/2 oq/ vltl. _ yi Finally, integrate f(yh) over H to obtain D (y)-f f(yh)df +,f(yh) dh. Since f(yh) - f(y,-h) is it clear that D(y) is given by Eq. (5). Note that the result ah - - a'lh/%r of Eq. (6) follows easily from the definitions of variance, expected value, and the hypothesis that 0 and Hare independent. *-i 4