029531-1-F MILLIMETER-WAVE BISTATIC RADAR SCATTERING FROM TERRAIN FINAL REPORT U.S. Army Research Office Box 12211 Research Triangle Park, NC 27709 Contract DAAL03-92-G-0190 July, 1995 APPROVED FOR PUBLIC RELEASE, DISTRIBUTION UNLIMITED Fawwaz T. Ulaby 29531-1-F = RL-2405

MILLIMETER-WAVE BISTATIC RADAR SCATTERING FROM TERRAIN FINAL REPORT U.S. ARMY RESEARCH OFFICE CONTRACT DAAL03-92-G-0190 July, 1995 THE VIEW, OPINIONS, AND/OR FINDINGS CONTAINED IN THIS REPORT ARE THOSE OF THE AUTHOR(S) AND SHOULD NOT BE CONSTRUED AS AN OFFICIAL DEPARTMENT OF THE ARMY POSITION, POLICY, OR DECISION, UNLESS SO DESIGNATED BY OTHER DOCUMENTATION.

MA37n L OP' KEEP THIS COPY FOR RFPRnT)11U-V"i'(-,,N PITPPric' c, REPORT DOCUMENTATION PAGE Form A~_ov PutliC reoorting CurBen tor tns collection of information is estimated to average: mour oer rfoorse. inctuoing the time for reviewing instructions. searcninq existing oata sources gathering and maintaining the data needed. and compreting and reviewing the collection of information. Send comments regarding this burden estimate or any otner asoect of tnis collection of information. including suggestions for reducing this ourden to ashing:on Heaouarters Services. Directorate for information Operations and Reports. 1215 Jeferson Davis Highway. Suite 1204. Arlington. VA 22202-4302. and to the Otfice of Management and Budget. Paoerwork Reduction Project (0704-0188). Wastngton. DC 20503. 1. AGENCY USE ONLY (Leave blank) 12. REPORT DATE 3. REPORT TYPE AND DATES COVERED | ]995, July 24 Final 6-]-92 - 5-31-95 4. TITLE AND SUBTTIE 5. FUNDING NUMBERS MILLIMETER-WAVE BISTATIC RADAR SCATTERING FROM TERRAIN 6. AUTHOR(S) Fawwaz T. Ulaby 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION Radiation Laboratory, University of Michigan REPORT NUMBER Ann Arbor, Michigan 48]09 02953]-]-F 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITORING AGENCY REPORT NUMBER U. S. Army Research Office P. 0. Box 12211 DAAL03-92-G-0190 Research Triangle Park, NC 27709-2211 11. SUPPLEMENTARY NOTES The view, opinions and/or findings contained in this report are those of the author(s) and should not be construed as an official Department of the Army position, policy, or decision, unless so designated by other documentation, 12a. DISTRIBUTION/ AVAILABIUTY STATEMENT 12b. DISTRIBUTION CODE Approved for public release; distribution unlimited. 13. ABSTRACT (Maximum 200 words) This final report provides a summary of the results realized over the past 3 years with regard to the characterization of bistatic radar scattering from terrain. The papers and report published under this program describe hardware and techniques developed for bistatic measurements, and an enhancement to the Physical Optics model for surface scattering. 14. SUBJECT TERMS 15. NUMBER OF PAGES radar scattering, terrain clutter 16.- RICE CODE 16. PRICE CODE 17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT OF REPORT OF ABSTRACT UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL i I i, KI~ 7AI ---~-CA I Nl^5N / )4-Q 1 -itOU-s5UU 8 Standard Form 298 (Rev. 2-39) Prescrtod by ANSI Std Z39-.6 298-102

1 Introduction A program for measuring the bistatic scattering of microwaves from rough surfaces has been conducted for the purpose of understanding the nature of the role of bistatic scattering in the radar backscatter response of complex scenes such as forests and crops. A Bistatic Measurement Facility, an automated radar system operating with a center frequency of 9.25 GHz, has been designed and built for measuring the bistatic scattering response of rough surfaces in a laboratory setting. In addition, an enhancement to the Physical Optics model for surface scattering has been developed. This report provides a summary of the results realized under this program, with the details provided in Appendix A in the form of reprints of articles published in scientific journals and symposia proceedings. Not included in the Appendix is a long report [1] documenting the operation and maintenance of the Bistatic Measurement Facility, created for use by the U.S. Army Waterways Experiment Station. These reports have been requested by and provided to numerous U.S. Army and other DOD groups and laboratories. The references cited in the body of this report refer to the publication list given in Section 3. 1

2 Summary of Results Previous research, which resulted in the development of the Michigan Microwave Canopy Scattering (MIMICS) model, demonstrated that an important mechanism for describing the radar scattering from complex natural targets, such as forests and crops, involves a double bounce of energy involving both the ground and overstructure components, such as stems, branches, trunks and/or the canopy. In order to understand this double bounce mechanism, the individual bistatic scattering mechanisms at the ground and the overstructure components can be explored individually. This report contains the results of our efforts to understand the nature of bistatic scattering from the ground. 2.1 Bistatic Measurement Facility While extensive theoretical developments exist regarding bistatic scattering, experimental confirmations are very scarce, and therefore an emphasis was placed on developing an accurate means of measuring the bistatic scattering from the ground. From our experience with previous efforts to make such measurements, a Bistatic Measurement Facility has been designed from scratch and is now in operation. Critical design considerations include main2

taining polarization purity, polarimetric calibration, positional and angular accuracy, large numbers of independent samples, automation for acquiring large data sets, and long term stability. A second Bistatic Measurement Facility has been designed and built for the U.S. Army Waterways Experiment Station and is awaiting delivery pending the preparation of appropriate laboratory space at WES. 2.2 An extension to Physical Optics One of the oldest and most popular rough surface scattering models is the Physical Optics model. An extension of the model, involving higher order slope dependence, has been developed and confirming measurements in the specular direction have been made on the Bistatic Measurement Facility. These results are published in [2, 3]. The first experimental evidence of Brewster angle migration toward nadir as the surface roughness increases, predicted by the Small Perturbation Method, is also contained in this paper. 3

3 List of Publications References [1] R. De Roo, R. Hartikka, N. Peplinski, and A. Zambetti, "Bistatic measurement facility user's manual," Technical Report 031163-F, The University of Michigan, Ann Arbor, MI, August 1994. [2] R. D. De Roo and F. T. Ulaby, "Bistatic specular scattering from rough dielectric surfaces," IEEE Transactions on Antennas and Propagation, vol. 42, no. 2, pp. 220-231, February 1994. [3] R. D. De Roo and F. T. Ulaby, "Authors' reply to comments on bistatic specular scattering from rough dielectric surfaces," IEEE Transactions on Antennas and Propagation, vol. 43, no. 2, pp. 225-226, February 1995. 4 Participating Scientific Personnel The following people participated in the bistatic scattering program: Faculty and Research Scientists Professor Fawwaz T. Ulaby Dr. Kamal Sarabandi 4

Graduate Students Mr. Roger De Roo, received MS (1990), expected PhD completion in 1995. Mr. Neil Peplinski, received MS (1993). Mr. Andrew Zambetti, received MS (1995). Mr. Bryan Hauck, expected MS completion in 1996. Research Engineer Mr. Ronald Hartikka 5 Conclusions The bistatic scattering program has led to the development of hardware and techniques for the accurate measurement of bistatic scattering from rough surfaces. A number of important measurements has already altered our understanding of the nature of bistatic scattering, particularly in the specular direction. By combining these measurement techniques with rough surface manufacturing techniques developed under a separate research project, a set of high quality bistatic measurements of deterministic rough surfaces is under 5

way. These measurements are expected to be of great value to theoreticians studying surface scattering. A Reprints of Selected Papers 6

220 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 42, NO. 2, FEBRUARY 1994 Bistatic Specular Scattering from Rough Dielectric Surfaces Roger D. De Roo, Student Member, IEEE, and Fawwaz T. Ulaby, Fellow, IEEE Abstract- An experimental investigation was conducted to determine the nature of bistatic scattering from rough dielectric surfaces at 10 GHz. This paper focusses specifically on the dependence of coherent and incoherent scattered fields on surface roughness for the specular direction. The measurements, which were conducted for a smooth surface with ks < 0.2 (where k = 27r/A and s is the rms surface height) and for three rough surfaces with ks = 0.5, 139, and 1.94, included observations over the range of incidence angles from 20~ to 65~ for both horizontal and vertical polarizations. For the coherent component, the reflectivity was found to behave in accordance with the prediction of the Physical Optics model, although it was observed that the Brewster angle exhibited a small negative shift with increasing roughness. The first-order solution of Physical Optics also provided good agreement with observations for hh-polarized incoherent scattering coefficient, but it failed to predict the behavior of the vv-polarized scattering coefficient in the angular range around the Brewster angle. A second-order solution is proposed which appears to partially address the deficiency of the Physical Optics model. I. INTRODUCTION S CATTERING of electromagnetic fields by random rough surfaces in the backscatter direction has many uses and has been investigated extensively over the past few decades. By comparison, very few experimental investigations have been attempted to evaluate forward scattering in the specular direction. This is in part because the applications for specular scattering are not as straightforward as for backscattering. The many theoretical developments for scattering from random rough surfaces, while developed for the general bistatic case, have only been extensively used and tested for backscattering. Therefore, the usefulness and validity of these theories for specular scattering is largely unknown. Recent developments in the modeling of terrain for radar backscattering indicate that specular scattering from a rough ground surface combined with scattering by an overstructure (such as trees or crops) can contribute significantly to the backscattering from the terrain as a whole [1]-[5]. Therefore an understanding of the nature of specular radar scattering and knowledge of the behavior of specular scattering theories are needed. Several experimental investigations were conducted at centimeter wavelengths in the 1946-1960 period to evaluate the variation of the coherent and incoherent components of the -specularly reflected energy as a function of surface Manuscript received Aprn30, 1993; revised September 17, 1993. This work was supported by the Army Research Office, contract DAAL03-92-G-0269. The authors are with The Radiation Laboratory, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, 48109-2122. IEEE Log Number 9215659. 0. I.. co -10. -20. -30. -40. -50. -60. -70. -80. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. Incidence Angle 9 (degrees) Fig. 1. Calculated coherent reflectivity for v polarization using the Physical Optics model for a Gaussian surface with ks vaired from 0 to 2.5, and 6 = 3. roughness. The results for the coherent component, which is represented by the reflection coefficient, are summarized in Beckmann and Spizzichino [6]. According to these results, the overall variation of the reflection coefficient with ks, where k = 27r/A, and s is the rms height, may be explained by the coherent scattering term of the Physical Optics surface scattering model [6], [7]. The data, however, are rather lacking in several respects: (1) marginal accuracy with regard to both the measured reflected signal and the surface rms height, (2) limited dynamic range (10 dB relative to the level of the signal reflected from a perfectly smooth surface), and (3) no examination of the behavior in the angular region around the Brewster angle. Additional bistatic measurements were reported by Cosgriff et al [8] in 1960, but the data were not calibrated, nor were the surfaces characterized. More recently, Ulaby et al. [9] measured bistatic scattering from sand and gravel surfaces at 35 GHz, and while the data were calibrated and the surfaces were characterized, no comparison to a theoretical prediction was given. In the optical regime, Saillard and Maystre [10] have measured the bistatic scattering of light from dielectric surfaces, and have observed a change in the Brewster angle as the roughness of the surface increased. Greffet [ 11] explained their observations using the Small Perturbation Method [12]. However, the Small Perturbation Method does not explain the observations reported in our present study: the first-order Small Perturbation Method predicts that the Brewster angle will move toward grazing as the roughness increases, while our observations indicate the opposite. t 0018-926X/94$04.00 ~ 1994 IEEE

E ROO AND ULABY: SPECULAR SCATTERING FROM ROUGH SURFACES 221 I ball screw Fig. 2. Diagram of Laser Profiler. There are three major rough surface scattering approachIs which have long held the acceptance of the scientific:ommunity as valid for some ranges of surface roughness. the Geometric Optics (GO) model, the Physical Optics (PO) nodel, and the Small Perturbation Method (SPM) are the three heoretical approaches most commonly used at microwave 'requencies for characterizing scattering from random rough;urfaces [7]. While both GO and PO models rely on the (irchhoff approach of using the tangent-plane approximation, hey yield very different predictions for the scattering of waves 'rom a rough surface. In particular, the GO model does not predict any coherent reflectivity from rough surfaces. This may ippear to be an inadequacy of the GO approach; however, he GO approach is valid only for surfaces so rough that my coherent scattering would be very small anyway. The PO nodel, on the other hand, has a simple expression for the:oherent scattering component, but the complete expression.or incoherent scattering, even for single scattering, has not let been formulated. SPM is a different approach, but its range Af validity is restricted to surfaces with small rms heights and;lopes. Because the ranges of validity for the GO and SPM heories are outside the range of rms heights described in this?aper, comparisons to GO and SPM will not be discussed. In this paper we will examine experimental measurements 3f the coherent and incoherent scattering components in the specular directions at 10 GHz for several dry sand surfaces:overing a wide range of rms heights (0.5 mm to 1.4 cm). The measurements were conducted over an incidence angle range Extending from 20~ to 65~ for both horizontal and vertical polarizations. The sand, with a relative dielectric constant f 3.0 and negligible loss factor, exhibits a null for vertical polarization at the Brewster angle of 60~. The experimental data are compared with predictions based on a revision of Lhe Physical Optics solution that is slightly different from and more accurate than the standard form available in the literature [7]. II. PHYSICAL OPTICS MODEL-A VECTOR SOLUTION The Physical Optics approach involves integration of the Kirchhoff scattered field over the rough surface. The coherent field reflection coefficient from a surface with a Gaussian height distribution is given by [7]: Rq(O) = Rqoe-2k22 CO2 e6pq (1) where the polarization subscripts p, q are either h or v, and cos = -i * ki. The angle 0 is both the angle of incidence and reflection; coherent scattering occurs only in the specular direction from the mean surface. The reflection coefficient Rqo for a plane surface is given by (15) and (16) below. Fig. 1 shows Fv(0) = JR,(0)\2 for several values of ks. The Brewster angle does not change with surface roughness, but the coherent scattered power decreases very rapidly with increasing roughness. The power in the incoherent reflected field may be given by a Taylor series in surface slope distributions. In Ulaby et al. [7] the Physical Optics solution is called the Scalar Approximation because slopes are ignored in the surface local coordinate system, leading to a decoupling of polarizations in the vector scattering equations. As a result, co-polarized scattering in the plane of incidence is fairly accurate, but cross-polarized scattering is zero. With the inclusion of surface slopes transverse to the plane of incidence in the vectorial solution to the Physical Optics approximation, depolarization in the plane of incidence is predicted when the Taylor series is expanded to the second order in surface slopes. The derivation is given in Appendix A. In the specular scattering direction, the first order terms are zero. The Physical Optics expression for incoherent scattering in the specular direction, which includes the zeroth and the important part of the second order terms, is given by (see last paragraph of Appendix A): 0 2k2 COS2 0a 2 2 l I + 4 cos40(la + Iatql2)I20 pVq Iopqu + \lpq 1.tpq.44o (2) where I = e -4k22 co2 (e4k22 cos2 ep() - )id (3) 0 ' ( p() ) 2 e4k2s2 cs2 (1-p()),d (4),20 9o

222 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 42, NO. 2, FEBRUARY 1994 and aOhh = Rho alhh = Rhl avv = Rvo aIvv = Rvl athh = ate = 0 athh = tvv = 0 I i I iI a aOvh = al = aOhv = alhv = 0 atvh = (Rho cos2 + Rvo)/ sin 0 athV = -(Rvo cos2 + Rho)/ sin (5) IV (6) (7) i (8). (9) I (10) (11) | (12) The a'pq terms are the zeroth-order terms (scalar approximation); the apq and a'q terms are the second-order terms due to slopes longitudinal and transverse to the plane of incidence. The function p({) is the normalized correlation function of the surface, and the parameters Rv,Rho, Rvl,Rhl are the coefficients of the (field) reflectivity local to the surface when expanded in terms of surface slopes: Rh(x, y) = Rho + RhlZI(x, y) + * * (13) R,(x,y) = Rvo + Rv,1Z(x,y) + * (14) where Zi(x, y) is the surface slope longitudinal to the direction of the incident wave at the (x, y) lateral coordinates of the surface. The zeroth-order terms are identical to the reflection coefficients for a smooth surface. In particular, R 72 cos 0 - 1 cos (15) r72 COS 0 + 7/1 COS Ot r71 COS 0 - 7r2 COSt t Rvo= (16) 771 COS 0 + r72 cos Ot R 72 sin (1 - Rho) - 71 k cos sin Ot (1 + Rho) Rhl =2 CS O 772 COS 0 + 71 cos Ot (17) 71 sin 0(1 - Rvo) - 2 k cos sin t(1 + Rvo) 7r1 cos 0 + 72 cos Ot (18) where Ot is related to 0 by Snell's Law: kl sin 0 = k2 sin Ot, and r7 = /f/e. These first-order coefficients are different from those found in Ulaby et al [7] and Ulaby and Elachi [13] due to the incorporation of a more precise method for expanding the local angle of transmission. The zeroth-order terms dominate co-polarized scattering except near the Brewster angle, where the zeroth-order terms tend toward zero. For cross-polarization, the zeroth-order terms predict no scattering for all specular angles, and therefore cross-polarized scatteing is determined by the second-order terms. Unfortunately, this is still a single-scattering theory, and depolarization is very small compared to other possible sources, such as multiple scattering or volume scattering. Thus it provides a simple estimate of the minimum expected crosspolarized scattering in the specular scattering direction. The surface height above mean surface z (mm) (a) aE 3 Q. V 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 0. 100. 200. 300. 400. 500. 600. 700. Displacment 4 (mm) (b) Fig. 3. Typical results of surface characterization of one of the surfaces measured in this study: (a) Histogram of measured heights for a slightly rough surface and the Gaussian probability distribution used to model it; a total of 4353 height measurements were made, from which the rms height was calculated to be s = 6.9 mm. (b) Measured autocorrelation function of the same surface. co-polarized second-order terms are all negligible except in the vicinity of the Brewster angle, where the zeroth-order terms vanish for vv polarization. The second-order terms all tend toward zero at grazing, and the cross polarized terms tend toward zero at nadir. The fact that the Kirchhoff approximation is capable of predicting any cross-polarized single scattering in the plane of incidence is somewhat surprising. However, this is possible because the Physical Optics approximation is used in the derivation; use of the Geometric Optics approximation neglects diffraction and is incapable of predicting cross-polarized scattering. While expressions similar to, but not identical with, those described here for the higher order terms of the Physical Optics model have been presented In the past [14], and have been unsuccessful in matching experimental measurements of

DE ROO AND ULABY: SPECULAR SCATTERING FROM ROUGH SURFACES 223 Dual Pol Horn Receiver.. -center o.. —..-. e i s:ct*:e: *inth Pav of the rews-e*r angl Illuminated....'. Fig. 4. Diagram o the cross-polarized scattering coefficient [15], the authors are not aware of any attempt to use the higher order terms to explain scattering in the vicinity of the Brewster angle. III. MEASUREMENT DESCRIPTION A. Laser Profiler The measurements shown in this paper are for random surfaces with varying roughness. The surfaces were characterized by a Laser Profiler, a device engineered at the University of Michigan to measure 2 meter linear or 1 meter by 1 meter square sections of surface profiles. The Profiler is shown in 2 meter linear mode in Fig. 2. Using a Pulsar 50 Electronic Distancemeter manufactured by GEO Fennel, it can measure profiles of surfaces without direct contact. The profiler has a horizontal resolution of 1 mm and a vertical resolution of 2 mm. Fig. 3(a) is an example of the height histogram generated from the profile measured for one of the surfaces, and Fig. 3(b) shows the corresponding correlation function. B. Bistatic Facility The configuration shown in Fig. 4 depicts the indoor bistatic radar system used for making the measurements reported in this paper. It is a stepped-frequency (8.5-10 GHz) measurement system capable of measuring the scattering matrix S of the target contained in the area or volume formed by the intersection of the transmit and receive antenna beams. Using an HP8720 vector network analyzer with an amplifier on the transmitting antenna, the system measures a complex voltage ntable f the Bistatic Facility. for any pair of v or h receive and transmit polarization states. With proper calibration, it is capable of measuring all four complex elements of the scattering matrix of the target surface. The hardware allows the transmitter and receiver to be located independently at any point on a hemispherical shell 2.1 m from the center of the target. In practice, however, measurements are accurate only when both antennas are within 70~ of nadir. The receive antenna is a dual-polarized horn antenna with a beamwidth of 12~, and the transmit antenna is a dualpolarized parabolic dish whose feed was designed such that the main beam of the parabolic dish is focused at a range equal to the distance to the target surface, which is held constant for all measurements. Because of the larger aperture (30 cm diameter), the transmit antenna has a narrow beam of 5~, which dictates the extent of the surface area responsible for the scattered energy. By using a focused beam antenna, we achieve a narrow-beam configuration without having to satisfy the usual far-field criterion. A baffle made of radar absorbing material was placed in the direct path between the transmitter and reciever to insure proper isolation of the two antennas. To separate the measured signal into its coherent and incoherent components, it is necessary to measure many statistically independent samples of the random surface characterizing the target surface. This is achieved by rotating the sample holder in increments of 10~. thereby realizing 36 spatial samples per full rotation. The spatial correlation of the measured incoherent power indicates that measurements decorrelated every 15", resulting in 24 independent sample per surface. Measurements of smooth surfaces indicate that phase coherence is maintained between independent samples.

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 42, NO. 2, FEBRUARY 1994 224 go 10 40 0. -5. -15. -IS. -20. -25. -30. -35. -40. -45. -50. boresight. The system is extremely stable; while the calibration procedure is performed for each day of measurements, calibrations have been good for up to 5 days. The bistatic facility measures Epq, the p polarized field due to a q polarized transmitted field. The power in this measured field Ppq, is composed of a coherent and an incoherent component: p _ pcoh + pincoh q- p q Pq (21) 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. Incidence Angle 0 (degrees) It KpqI2 pqh q (471)2(ror + rot)2 ~pq (22) pncoh - t Iq 4I q o0A2 gP(x, y)gt(x, y) dxdy (23) Pp q q(47r)3 pq M r2x ~y^ xy) -= pq 1Pqt T qIA pq (24) q (47r)30'pqZ in where the coherent power reflection coefficient, and therefore the coherent power, exists only for co-polarized scattering: coh= =pq 0 if p ~ q. The co-polarized coherent power reflection coefficient is calculated by comparing the coherent power from a target to that from a large flat conducting plate, for which Fcal = 1 and =. Thuspp orp0 - 0. Thus, Erpq Fig. 5. Measured coherent reflectivity of a smooth surface with e = 3.0 and ks < 0.2: squares denote horizontal polarization and circles denote vertical polarization. Continuous curves are predictions based on Physical Optics. The angle of minimum reflectivity for vertical polarization is 60~. The total path length, from transmitter to target to receiver, has a standard deviation less than 4 mm (7~ at 9.25 GHz) for the set of independent samples. C. Separation of Coherent and Incoherent Power The Bistatic Facility measures a complex voltage Vm which is proportional to the scattered electric field for each polarization state at each spatially independent sample of the surface. Because the scattered electric field is composed of a coherent component from the mean surface and an incoherent component from the rough surface and/or volume, the measured voltage will also have a coherent and incoherent component: Vm = Vcoh + Vincoh. These two components can be separated because the incoherent component has a zero mean: (Vincoh) = 0. Provided a large number of independent samples are measured, the coherent power pcoh is proportional to the square of the complex average of the measured voltages: p coh = I =oh12 I= V12 (19) The incoherent power pincoh is then proportional to the variance of the fluctuating component of the measured voltage: p h incoh 12) IVm V2 (20) D. Calibration The bistatic measurement system is calibrated using a bistatic adaptation of the polarimetric backscattering calibration technique developed by Sarabandi and Ulaby [16]. For measurements corresponding to the bistatic specular configuration, a large, flat condunting plate is used as a calibration target. Further verification is obtained by measuring a conducting hemisphere placed on the calibration target. The radar cross section of the hemispherical target was calculated via Physical Optics. Calibration to date has been accurate to within 0.5 dB in magnitude and 10~ in co-polarized phase difference at pcoh r - PP.PP pcal pp (25) The co-polarized differential scattering coefficient is calculated by comparing the incoherent power from a target to that of the calibration power. IA4,pq was calculated from extensive measurements of the normalized antenna patterns for bo.th antennas, for each of the principal polarization states, over the entire main lobe of the antennas, at the boresight ranges ror and rot. Thus, 0 47r pncoh PP (rot + ror)2IAill pp pcal (26) IV. RESULTS A. Surface Characterizations The shape of a random rough surface is described by the surface height distribution function and the surface height correlation function. For a surface whose height is given by z = f(x, y), the surface-height probability density function is given by pf(z), and is assumed to be Gaussian: i 1 z2 Pf(Z) - e- 7 V'Us (27) Measurements by this and other experimenters [17], [18] indicate that this assumption is appropriate. Fig. 3(a) shows the fit between a histogram of measured surface heights and equation (27). The surface height characteristics can be specified by a single parameter, s, which is the root-meansquared surface deviation from the mean planar surface located at z = 0. The other statistical descriptor offrandom rough surfaces is the normalized correlation function, denoted by p. It describes

E ROO AND ULABY: SPECULAR SCATTERING FROM ROUGH SURFACES 225 -a '3t Ct 6) 94 0. -5. -10. -15. -20. -25. -30. -35. -40. -45. -50. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. Incidence Angle 6 (degrees) (a) iES 4) 0. -5. -10. -15. -20. -25. -30. -35. -40. -45. -50. given by z' = f(x'. y'). For surfaces described by a stationary random process, p can be expressed in terms of the lateral separations u = x - x' and v = y - y' between the two locations on the surface. Moreover, if the surface statistics are symmetric under azimuthal rotations, the-correlation function can be described by a single variable l = u2 + v2, which specifies the absolute value of the lateral separation. Unlike the surface height distribution function, the correlation function may take on several forms for naturally occurring randomly rough surfaces. The vast majority of the literature on rough surface scattering assumes that the surface statistics are azimuthally symmetric and Gaussian, while many measurements of commonly occurring surfaces in microwave remote sensing situations indicate that an exponential correlation function may be more appropriate [19]. A correlation function for a surface with a correlation length of 52.5 mm is shown in Fig. 3(b). It was generated by averaging the individual autocorrelations of 3 linear profiles of the surface. Experimentation has shown that only 3 profile measurements averaged together are neccessary to accurately determine the correlation length and shape of this and other surfaces, but many more are needed to demonstrate that the correlation function tends toward zero beyond a few correlation lengths. For the purposes of this paper, the correlation length of a surface is that length at which the normalized correlation function is e-1. As a result of the negative values of the correlation function, several of the integrals used to predict scattering characteristics ((3) and (4)) may yield values which are obviously incorrect. However, only a few surface profiles are needed to determine the shape of the correlation function within one correlation length, and if the rest of the correlation function tends towards zero, this portion of the correlation function dominates the integrals. The effect of the shape of the correlation function within one correlation length can be explored by considering several analytical forms for the correlation function. B. Coherent Scattering 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. Incidence Angle 0 (degrees) (b) 0. -5. -10. -15. -25. -30. -35. -40. -45. -50. L- r...: / At the Brewster angle, the reflectivity for the vertical o' / polarization is identically zero for a smooth interface between.. two lossless dielectric media. Whether or not it remains identically zero for a slightly rough surface is not clear. / The Physical Optics approach clearly predicts that this is so; / moreover, it predicts that the minimum reflectivity remains at / ~ the same incidence angle as for a smooth surface. This can? | be seen in Fig. 1. However, the Small Perturbation method..predicts that the angle of minimum reflection for vertical o. 10. 20. 30. 40. 50. 60. 70. 80. 90. polarization increases slightly with increasing roughness of the surface [11]. The fact that Physical Optics does not predict Incidence Angle O (degrees) Ina change in angle while the Small Perturbation does is a consequence of the fact that the correction to the Fresnel easured coherent reflectivity of three rough surfaces. Continuous coefficient is multiplicative for Physical Optics while it is predictions based on Physical Optics. In all cases, the surfaces ii or rrio A ii i:ive dielectric constant = 3.0. (a) = 0.15. k = 5.4. (b) additive for Small Perturbation. Additional terms in the Small tve dielectric constant e = 3.0. (a) ks = 0.515. k1 = 5.4, (b) kl = 10.6. (c) ks = 1.94.kl = 11.8. Perturbation expansion may move this minimum angle back toward the smooth-surface Brewster angle.:e to which the height at one location given by a Fig. 5 shows measurements of the reflection coefficient,y), is correlated to the height at another location, for a smooth dry surface with ks < 0.2 (the rms height s was Fig. 6. Me curves are 1 have a relat ks = 1.39, the degre z = f(x

226 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 42, NO. 2, FEBRUARY 1994 10. l~. -IO. n_ -5. I -1.- K f o -15. -- frmn PhysicalOpadc -20. - ~ mtaeund -25. -30., I, 0.0 02 0.4 0.6 0.8 1.0 12 y = ks cosO Fig. 7. The reduction of coherent scattering from a surface due to roughness. Shown is the measured coherent reflectivity of several surfaces (all with with e' = 3) but differing roughness parameters W, normalized to the reflection coefficient of a smooth surface. The angles of incidence range from 20~ to 70~ and the roughness ks ranges from 0 to 2. The continuous curve is the Physical Optics prediction for surfaces with Gaussian-height probability densities. smaller than 1 mm, the measurement precision of the laser profileometer). The curves in Fig. 5 were calculated using the Fresnel reflection coefficient formulas given by (15) and (16) for a surface with a relative dielectric constant e = 3.0 + jO. The dielectric constant for the sand medium was measured by a dielectric probe, which gave a value of C' = 3.0 for the real part and a value of c" < 0.03 for the imaginary part. Because e"/e' << 1 and the inclusion of c" as high as 0.05 does not significantly change the results of any of the calculations in this paper, it was ignored. The excellent agreement between the measured data and the calculated curves presented in Fig. 5 provides testimony to the measurement accuracy of the system. Fig. 6 compares measured values of the power reflection coefficient F with curves calculated using Physical Optics (equation (1)) for surfaces with ks = 0.515,1.39, and 1.94. Although good overall agreement is observed between theory and experimental observations, it should be noted that the location of the Brewster angle exhibits a slight shift towards decreasing angle of incidence; the Brewster angle shifts from 60~ for the smooth surface shown in Fig. 5 to 58~ for the surface with ks = 1.39 (Fig. 6(b)) and to about 56~ for the surface with ks = 1.94 (Fig. 6(c)). The shift is toward decreasing angle of incidence, which is opposite to the direction predicted by the Small Perturbation Method. By way of summary, Fig. 7 shows the dependence of the q-polarized normalized power reflection coefficient Yq on the roughness parameter V/, where.. = no}2 (28) ) qo(E, a IR (28) = e (29) I 0 00 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. Incidence Angle 0 (degrees) (a) 10. 0 Ct I 5. 0. -5. -10. -15. -20. -25. -30. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. Incidence Angle 0 (degrees) (b) 10. 0 -2 ~o -E a rA S. 0. -5. -10. -15. -20. -25. -30. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. Incidence Angle 0 (degrees) (c) Fig. 8. Measured co-polarized specular scattering coefficient for three rough surfaces. Continuous curves are based on Physical Optics: the dashed curve corresponds to the zeroth-order term for oao, the solid curve to a(To, with both the zeroth-order and second-order terms included; the dotted curve corresponds to roh. The second-order term is negligible for hh polarization. In all cases, the surfaces have a relative dielectric constant e = 3.0 and an exponential correlation function was used. (a' ks = 0.515,kl = 5.4. (b) ks = 1.39,kl = 10.6, (c) ks = 1.94,kl = 11.8. I and o = kscos9.

rE ROO AND ULABY: SPECULAR SCATIERING FROM ROUGH SURFACES. Incoherent Scattering As was discussed previously in Section II, the expression;iven by (2) for the bistatic specular scattering coefficient,onsists of two terms, a zeroth-order term and a second-order erm, with the latter being the result of a new derivation of he Physical Optics model given in Appendix A. Fig. 8(a) shows the measured values of or% and ahh for a slightly rough surface with ks = 0.515, plotted as a function of incidence mgle, as well as plots for the same quantities calculated in Lccordance with (2). The calculated curves include a pair or the zeroth-order term alone, and a pair for the sum of he zeroth-order and second-order terms. For hh polarization, he second-order term is much smaller than the zeroth-order erm, and therefore its contribution is insignificant. The same )bservation applies to vv polarization for angles more than 5~ away from the Brewster angle, but in the vicinity of the Brewster angle, the second order term becomes the domiiant contribution and it correctly predicts the level of the experimental observations. Unfortunately, for the rougher surfaces shown in Figs. 8(b) md (c) with ks = 1.39 and 1.94, the model overestimates the level of a~, particularly for vv polarization in the vicinity of the Brewster angle. The measured data fall in between the curves calculated on the basis of the zeroth-order term alone and the curves based on both terms. Thus, despite the improvement that the proposed model provides for the slightlyrough case shown in Fig. 8(a), it is inadequate for very rough surfaces. V. CONCLUSION Several measurements of specular scattering from rough surfaces at 10 GHz are presented. They indicate that the Physical Optics predictions for coherent scattering are very good for surface roughness as large as ks = 2.0, with the exception that the theory does not predict a shifting of the Brewster angle by a few degrees towards nadir. A theoretical explanation of this phenomenon is unresolved. For co-polarized incoherent scattering, Physical Optics has been shown to be an adequate descriptor for a surface with ks = 0.515, but rapidly loses its quantitative predictive value for ks > 1. APPENDIX A. A VECTOR PHYSICAL OPTICS DEVELOPMENT In Ulaby et al. [7], the Physical Optics solution for scattering from a dielectric rough surface is presented under the scalar approximation. This approximation leaves out many terms, some of which change the results of the calculations significantly. In particular, cross-polarized scattering is neglected in the plane of incidence under the scalar approximation. What follows is a full vector solution to the Physical Optics problem, including vector terms which are neglected via the scalar approximation and some higher order terms in the expansion of the solution with respect to surface slope, with the rest of the assumptions remaining the same as in Ulaby et al. [7]. The solution starts with the exact Stratton-Chu integral equation for the p-polarized scattered far field due to a qpolarized wave (qEoe-'jk' r) incident upon a rough surface: 227 E -q = nkJk h x J [(h x E) - rs x (n x H)] x ejk'u sFdS (A.1) - jke-ksR Eo UdS (A.2) where the singly scattered fields on the surface are calculated via the tangent plane approximation and are given by: ni x E = [Rh( t)(hi x t) + Rv(ni i)( - d)t]Eo 77ini x H = [Rh(-n(. )(i x (q t)t - R,.(q4 d)(nl x t)]Eo (A.3) (A.4) where ni is the incident wave direction, q is the incident wave polarization direction, fn is the unit normal to the surface, t = ni x ni/jni x nil, and d = hi x t. Also, R/ and Rh are the v- and h- polarized Fresnel Reflection coefficients local to a point on the surface. These quantities, and the appropriate vector products, are defined in Ulaby et al. [7]; however, the exact expressions for Upq under the single scattering tangent plane approximation do not appear there and hence are given here for reference: Uhh = DD2 [RvZt((sin O sin AO + Z' cos 0 + ZtZj sin 0) + (cos 0 + Zi sin 0)(sin 0 cos 0s sin Aq + Zt cos 0 cos 0 - Zt sin 0 sin 09)) - Rh(sin 0 - Z cos 9)((cos 0 + Zi sin 0) x (sin 0 cos AO - Z' cos 0) + (sin 0 cos 9, cos AO - Z' cos 0 cos 09 + ZtZ' sin 0 cos 0 - Zt sin 0 sin s, + (Z + Zt ) cos 0 sin 0))] 1 Uvh = D2D [-Rh(sin 6- Zicos6)((sin0sinA(b (A.5) + Zf cos 6 + ZtZ[ sin 6) + (cos 0 + ZI sin 0) x (sin 0 cos 09 sin AO> + Zt' cos 0 cos 0s - Zt sin 0 sin 09)) - RZt((sin 0 cos 0, cos A< - Zj' cos 0 cos 0s + ZtZt' sin 0 cos 0, - Zl sin 0 sin 0s + (Z2 + Z2) cos 0 sin 0s) + (cos 0 + Zt sin 0) x (sin 0 cos AO - Z' cos 0))] (A.6) Uv = D2D[RhZt((cos + Zi sin 6) x (sin 0 cos AO - Z/' cos 0) + (sin 0 cos 09 cos AO - Zl cos 9 cos s, + Zt Z sin 0 cos 9, - ZI sin 0 sin 09 + (Z2 + Z2) cos sin 08)) + Rv(sin 0 - Z cos 0) x ((sin 0 sin AO + Zt' cos 0 + ZtZr sin 0) + (cos 0 + Z1 sin ) (sin 0 cos 0, sin AOp + Zt cos 0 cos 63 - Zt sin 0 sin 6,))] (A.7)

228 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 42, NO. 2, FEBRUARY 1994 Uvv = D2 [RhZt((cos + Zl sin 0) (sin 0 cos 0s sin AO + Zt9 cos 0 cos 0 - Zt sin sin s) + (sin 0 sin Aq + Zt' cos a + ZtZf sin 0)) - Rv(sin 0 - Zi cos 0)((sin 0 cos 0s cos AO - Z' cos 0 cos 09 + Zt Zt sin 0 cos 0s - Z sin 0 sin 0s + (Z4 + t2) cos 0 sin 0s) + (cos 0 + Zi sin 9)(sin 0 cos AL - Zf cos ))] (A.8) where 0 and 0, describe the incident and scattered elevation angles measured from nadir, respectively, AO describes the angular change in azimuthal direction between the incident and scattered waves, Zt and Zt represent the surface slopes within (longitudinal to) and transverse to the plane of incidence, respectively, Zf and Zt represent the surface slopes within (longitudinal to) and transverse to the plane of the scattered wave, respectively, D1 = \i x nif = /(sin0- Zos)2+ Z2, and D2 =/1 + Z2 + Z2. Unfortunately, the exact expressions for U are not mathematically tractable in the Stratton-Chu integral, as the surface slopes are random functions of the location on the surface. An approximate solution can be obtained by expanding U in a Taylor series in slopes and retaining only the first two terms: aovv = -Rvo(cos 0 + cos 08) cos AO (A. 18) alihh = Rho(sin 0 sin 09 - (1 + cos 0 cos 08) cos A^) - Rhl sin 0 cosq A(cos + cos 0s) (A. 19) alshh = RhO cos 9(cos 0 + cos 08) (A.20) atihh = Rvo sin A (1 + cos 0 cos 8 ) (A.21) atshh = 0 (A.22) alivh = -Rho sin A/(cos 0 + cos 0,) - Rhl sin 0 sin Ao(1 + cos 0 cos 0s) (A.23) alsvh = 0 ativh = Rho sin 0 cos 0 sin 0, - Ro cos A4(cos 0 + cos 0,) atsvh = -RhO cos 0(1 + cos0 cos 0s) alihv = Rvo sin A<q(cos 0 + cos 0s) + R/1 sin 0sin Ao(1 + cos0 cos 08) (A.24) (A.25) (A.26) (A.27) alshu = 0 (A.28) atihv = -Rvo sin 0 cos 0 sin 0s + Rho cos Aqb(cos 0 + cos 0s) (A.29) atshv = Rvo cos 0(1 + cos 0 cos 0s) (A.30) alivv = Rvo(sin 0 sin 8s - (1 + cos 0 cos 0s) cos A4) - R,1 sin 0(cos 0 + cos 0s) cos A\ (A.31) alsvv = Rvo cos 9(cos 0 + cos 0s) (A.32) ativ = Rho sin A0((1 + cos 0 cos 0s) (A.33) atsvv = 0 (A.34) A. Differential Radar Cross Section The elements of the covariance matrix [13] are given by: (SmnS;q) = /(unuUp*qeJk(hs -i)(-f'))dSdS' (A.35) from which the differential radar cross section can be derived: Upq = U(o) + U ) +... 1 (A.9) = D (aopq + (alipqZI + atipqZt + alspqZp + atspqZt)/ sin 0 + -. - ) (A. 10) Note that D2 need not be expanded as it will be cancelled upon integration over dS = D2dxdy, but D1 is Taylor expanded as follows: 1 1 D2 (sin 0 - Zt cos 0)2 + Z2 - ( Ztfl 2 2 sin s2 tan 0 tan 2 ) (A. 11) k2 0 - Spq Srq) pq - 4irAo (A.36) Using (A.9): (A. 12) Similarly, the reflection coefficients local to the surface are also expanded in terms of slopes:.U*, u(~) uv)* +'U( ) U(0l) +u *u()*+uWu(l)* Umn p u mnpg mnpg *mn )pq (mn) U 1pq) q n pqq UmnU q (A.37) and evaluating these separately, we can express a~ as: k2 PTq 4rA ((spqspq)so + (SpqSpq)s +(SpqS;q)s2) (A.38) (Smnpq)o = /(U(U(o *ejk(se -i) ))dSdS' Rv Rvo + RIlZI 4-+.. Rh = RhO + RhlZl + * (A. 13) (A.14) The reflection coefficients are not Jependent on Zf or Zt, and depend on even powers only of Zt. The a coefficients in the expansion of U -i, for all principal linear polarization combinations, given by: (Smn;Sq)s, = Jf((UnUp(q) + U(i Uq)0) ejk(,'-'i)'(r-r'))dSdS' (A.39) (A.40) aOhh = - RhO (cos 0 + cos 0,)csOS A aOvh = -RhO(1 + cos0 cosS 0) sin AO aohv = Ro(l + cos 0 cos 0s) sin AO (A 1 \ (A.16) (SnnSq)2 = (U(l))*e -i(-))dSdS' (A. 17) (A.41)

E ROO AND ULABY: SPECULAR SCATTERING FROM ROUGH SURFACES To obtain explicit expressions for or, the following relaions will be used [7]: - _,_ _ -/\X -2 -2 t. -l \\ 229 (e3q z -Z )) (Zzejqz(z-z')) (Zyeiqz(Z -)) (ZxZeJqz(z- z')) {ZyZ eJq(z-z')) = e-q< -wp) (A.42) = (ze- (z-z')) = -jqzs 2 C P( q/2 —q —p(I ()) e- s(-)) (A.43) ap(~) = (ZyeJ (z-z')) =-jqS2 sin a e z (A.44) 2 2 2 — _cos2a(qzs2 2 e-q2s2(1-p()) (A.45) 2 2 2A 2 =- sin2a (qzSP())) e-q2(1-pf)) (A.46) tangent plane approximation: k2 -k2 (Ssmn ) 1s i [(aomnapq + alimn a*)qli 47rAo 47rpq 4Tq sin O lipq OP + (aOmna*spq +,~lsmna;pq)qls + (aOmnatpq + attmnapgq)qti + (aomnatspq + atsmnapgq)qts]IO (A.52) where qli = k(sin 8s cos AO - sin 0) qti = k sin 0s sin AOq ql, = k(sin 0s - sin 0 cos AO) qts = k sin 0 sin Aq (A.53) (A.54) (A.55) (A.56) (ZZejqz(z-z))- = (ZyZ'eiq(-Z' )) = -sin a cosa x (qzs2P )2e- '(1-P(0)) (A.47) as well as the following Bessel function integral identity: j cos(na + l)eixco~ da = 27rjn cos fJn(x) (A.48) 7r where qz = k(cos0 + cos 0,), s is the rms height, (J,a) are the polar coordinates of the difference between the unprimed and primed surface locations, and p(Q) is the normalized surface height correlation function. B. Zeroth Order Term The analysis of the zeroth order term is straightforward and yields the traditional coefficients for Physical Optics: k2 1ao 1k2 47rA( (SmnSpq)3s0 k= mapqIo4 k2amnaopqI (A.49) This term represents the expected powqr in a particular direction due to the correlation of height of the surface at one point to the slope at another point. This term is negligible for scattering in the plane of incidence. D. Second Order Term The cross-slope term does not appear in Ulaby et al [7], but is nonetheless an analytic term. It is given by: k2 4 (SmnSpq)S2 4i7Ao 4 sin2 {[alimnaipq + alsmnaspq + atimnatipq + atsmnatspq + ((aIsmnatpq + atimna*lpq) - (alimnat*pq + atsmnaqip)) sin zA + ((alimnalspq + alsmnaipq) + (atimn atspq + atsmnaipq)) cos A]2ho - [(alimn aipq - atimn atipq)(qi - t2i) + (alsmna*spq - atsmn atspq)(q - q s) - (alimnatipq + atimnalipq)qtiqli - (alsmnatspq + atsmnalpq)qtsqls + ((aimn atspq + atsmnapq) + (alsmnaipq + atimna*spq))(qtiqis + qiQts) where Io = 27re-2 [eq P() - l]Jo(qtO)>d~ r (A.50) + ((alimnalpq + alsmnaipq) - (atimnatspq + atsmnaipq)) x (qliql + qtiqts)]I22} (A.57) qt = k sin2 0 + sin2s - 2 sin s sin cos A (A.51) This term represents the expected power in a particular direction due to the correlation of the height of the surface at one point to the height at another point. This term is the largest contribution to a~. C. First Order Term The first order term in Ulaby et al. [7] is that of the scalar approximation. Below is the full vector solution under the where 1o 2 2 I20 = Jo(qt)e-qs (1-P())d (A.58) __ _2 J2(qt~) 2 2 I22 = j e-q (1-P(d))d (A.59) This term represents the expected power in a particular direction due to the correlation of slope of the surface at one point to the slope at another point. This term is significant for cross polarization in the plane of incidence, and when the angle of incidence is near the Brewster angle for the mean surface.

230 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 42, NO. 2, FEBRUARY 1994 E. Evaluation of the I Integrals for Common Correlation Functions The remaining integrals can be further simplified if we assume a form for the correlation function p(s). In particular, if it is Gaussian, i.e., p(- ) = e-~ 2/, then o = 27re-q82 [eqs2 P( - l] (A.78) r 0 I20= / IJo e Ie-t s2(1-P(A)).d 7 (A.79) then Io - 27rl2e-q22 e (qzs )2i e e 4, l (i!i 2 =1 i 2- 2 i i(qzs)2(i- 1 12 -. = (i + 1)!(i 1)e-4(+) 1 1 2 2 -Jo = 27r12e-gfz8 2 00(i - 1)q(i2 + q2) i (i - 1)!(i2 +l 2)3 + )2 + q - (i F. Special Case: Forward Scattering in the Spec i=l-1)!(i - 1)21e then For forward scattering in the specular dir 0, AO -+ 0, and qt -- 0, and the general expi simplify considerably: For the principal linear polarizations pq = hh, hv, vh, vv, (A.60) the incoherent specular scattering coefficient can be obtained by setting mn = pq in (A.66) and (A.68) and the resultant l 2 'z expressions in (A.38) This process leads to the expression 4(i + 1) given in 920. (A.61) (A.62) REFERENCES [1] K. C. McDonald, M. C. Dobson and F. T. Ulaby, "Using MIMICS to model L-band multi-angle and multi-temporal backscatter from a walnut Pp() = e-, orchard," IEEE Transactions on Geoscience and Remote Sensing, vol. 28, no. 3, pp. 477-491, May 1990. [2] K. C. McDonald, M. C. Dobson and F. T. Ulaby, "Modeling multifrequency diurnal backscatter from a walnut orchard," IEEE Transac(A.63) tions on Geoscience and Remote Sensing, vol. 29, no. 6, pp. 852-863, 1991. [3] K. C. McDonald, Modeling Microwave Backscatter from Tree Canopies, PhD dissertation, University of Michigan, Ann Arbor, 1991. (A.64) [4] F. T. Ulaby, K. Sarabandi, K. C. McDonald, M. Whitt and M. C. Dobson, "Michigan Microwave Conopy Scattering Model," International 2 - + Journal of Remote Sensing, vol. 11, no. 7, pp. 1223-1253, 1990. It2 - (i - 1)) [5] M. Whitt, Microwave Scattering from Periodic Row-Structured Vegeta1~2\ 3 212 tion, PhD dissertation, University of Michigan, Ann Arbor, 1991. qt [6] P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic \ Waves from Rough Surfaces, Pergamon Press, Oxford, 1963. (A.65) [7] F. T. Ulaby, R. K. Moore and A. K. Fung, Microwave Remote Sensing: ) Active and Passive, volume 2, Addison-Wesley, Reading, Massachusetts, 1982. [8] R. L. Cosgriff, W. H. Peake and R. C. Taylor, 'Terrain scattering properties for sensor system design," Technical Report 181, Ohio State:ular Direction University, 1960. [9] F. T. Ulaby, T. F. Haddock and M. E. Coluzzi, "Millimeter-wave bistatic ection, 8 - radar measurements of sand and gravel," in Digest, IEEE International ressions above Geoscience and Remote Sensing Symposium (IGARSS'87), pp. 281-286, Ann Arbor, MI, May 1987. [10] M. Saillard and D. Maystre, "Scattering from metallic and dielectric rough surfaces," Journal of the Optical Society of America A, vol. 7, no. (A.66) 6, pp. 982-990, June 1990. [11] J.-J. Greffet, "Theoretical model of the shift of the Brewster angle on a rough surface," Optics Letters, vol. 17, no. 4, pp. 238-240, February 1992. (A.67) 1992 ( ) [12] S. 0. Rice, "Reflection of electromagnetic waves from slightly rough surfaces," Communications in Pure and Applied Mathematics, vol. 4, * + a pp. 351-378, 1951. lipq aIspq) [13] F. T. Ulaby and E. C. Elachi, Radar Polarimetry for Geoscience +* Applications, Artech House, Norwood, Massachusetts, 1990. oq atspq)] 20 [14] J. C. Leader, "Bidirectional scattering of electromagnetic waves from (A.68) rough surfaces," Journal of Applied Physics, vol. 42, no. 12, pp. 4808-4816, November 1971. [15] J. C. Leader and W. A. J. Dalton, "Bidirectional scattering of electromagnetic waves from the volume of dielectric materials," Journal of Applied Physics, vol. 43, no. 7, pp. 3080-3090, July 1972. (A.69) [16] K. Sarabandi and F. T. Ulaby, "A convenient technique for polarimetric calibration of single-antenna radar systems," IEEE Transactions on Geo(A.70) science and Remote Sensing, vol. 28, no. 6, pp. 1022-1033, November 1990. (A.71) [17] K. A. O'Donnell and E. R. Mendez, "Experimental study of scattering (A.72) from characterized random surfaces," Journal of the Optical Society of America, vol. 4, no. 7, pp. 1194-1205, July 1987. (A.73) [18] E. R. Mendez and K. A. O'Donnell, "Observation of depolarization and backscattering enhancement in light scattering from Gaussian rendom (A.74) surfaces," Optics Communications, vol. 61, no. 2, pp. 91-95, January 1987. 0 (A.75) [19] Y. Oh, K. Sarabandi and F. T. Ulaby, "An empirical model and an (A.76) inversion technique for radar scattering rom bare soil surfaces," IEEE Transactions on Geoscience and Remote Sensing, vol. 30, no. 2, pp. (A.77) 370-381, March 1992. k2 4rA (SomnSpq)so ks rA (SmnSpjq)s k2 A7)Ao "5 1 = -k2aomna, pqIo =0 tan2 [(alimn + alsmn)(6 (atimn tsmn + (atimn + atsmn)(ati where aohh = -2Rho cos aovh = aohv = 0 aovv = -2Rvo cos 0 aliv- + atl.v = -2Rv sin 0 cos 0' alivh + alsvh = r1it, + alshv = 0 alihh + alshh = -2Rhl sin 0 cos 0 ativh + atsvh = -2(Rho COS2 0 + Rvo) COS atihv + atshv = 2(Rvo cos2 0 + Rho) cos 0 ativv + atsvv = atihh + atshh = 0

IE ROO AND ULABY: SPECULAR SCATTERING FROM ROUGH SURFACES 231 Roger D. De Roo (S'88) was born in Ridgewood, NJ, on February 29, 1964. He received the B.S. degree in letters and engineering from Calvin College, Grand Rapids, MI, in 1986, and the B.S.E. and M.S.E. degrees from the University of Michigan, Ann Arbor, both in electrical engineering, in 1986 and 1989, respectively. He is currently a candidate for the Ph.D. degree at the University of Michigan. His research interests include modeling and measurement of bistatic scattering of electromagnetic waves from rough surfaces md modeling electromagnetic scattering from vegetation. Fawwaz T. Ulaby (M'68-SM'74-F'80) received the B.S. degree in physics from the American University of Beirut, Lebanon, in 1964 and the M.S.E.E. and Ph.D. degrees in electrical engineering from the University of Texas, Austin, TX, in 1966 and 1968, respectively., Dr. Ulaby is the Williais Professor of Electrical Engineering and Computer Science and the Director of theNASA Center for Space Terahertz Technology at the University of Michigan. Ann Arbor. His current interests include microwave and millimeterwave remote sensing, radar systems, and radio wave propagation. He is the recipient of numerous awards, including the Eta Kappa Nu Association C. Holmes MacDonald Award as "An Outstanding Electrical Engineering Professor in the United States of America for 1975," the IEEE Geoscience and Remote Sensing Distinguished Achievement Award (1983), the IEEE Centennial Medal (1984), the American Society of Photogrammetry's Presidential Citation for Meritorious Service (1984), the Kuwait Prize in applied science (1986), the NASA Group Achievement Award (1990). and the University of Michigan Distinguished Faculty Achievement Award (1991). Professor Ulaby served as president of the IEEE Geoscience and Remote Sensing Society (1980-1982), as Executive Editor of its Transactions (1983-1985), and as General Chairman of several international symposia. he is a member of URSI Commission F and served on several scientific boards and professional committees.

224 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 43, NO. 2, FEBRUARY 1995 Comments on "Bistatic Specular Scattering from Rough Dielectric Surfaces" Jean-Jacques Greffet Abstract-In a recent paper', De Roo and Ulaby have reported experimental results for the coherent bistatic scattering by a random rough surface. They have found a negative shift of the Brewster angle. We show that their results are in agreement with all the previously reported theoretical and numerical calculations. In a recent paper,' De Roo and Ulaby have presented a comparison between experimental data and the Kirchhoff specular bistatic scattering. They have shown that the simple Kirchhoff result agrees with the data, except near the Brewster angle OB. In the vicinity of,B. the authors have found that the zero becomes a minimum and that there is a shift A0 of this minimum. If we define the shift as A\0 = 0,ni -B., the shift is negative, which means that the minimum moves toward the normal of the surface. I would like to correct two statements that appear in the paper by De Roo and Ulaby in the discussion of this point. The shift was first observed on numerical simulations (but not experimentally, as stated in the paper) for one-dimensional random rough surfaces by Saillard and Maystre [1]. Hence. the paper by De Roo and Ulaby is the first experimental report on this effect for random rough surfaces. The second point deals with the sign of the shift of the Brewster angle NEB. De Roo and Ulaby have observed that the shift is always negative. This result was reported in [1]. In a later paper [2], I have proposed a simple model based on a perturbation model to explain the physical mechanism responsible of this shift. It predicts a negative shift. Saillard has published a slightly different approach which also yields the same result [3]. The above references deal with one-dimensional surfaces. In a recent paper [4], we have derived a perturbative expression for the coherent reflected field for a twodimensional surface. This new expression also yields a negative shift. Therefore, the observed shifts reported by De Roo and Ulaby agree with all the previous works in contrast with their statement in their Introduction and their Conclusion. Let us briefly describe a simple mechanism in order to get some physical insight on this effect. Using a perturbative approach, the amplitude.4 (0) of the coherent field can be written as the sum of two contributions: the zero-order field (amplitude A(~) which is given by Fresnel reflection factor, and the second-order contribution (amplitude 4(2)) due to the roughness A.4(0) = A(~)(0) + 42)(). Manuscript received June 6. 1994. The author is with the Laboratoire d'Energetique Moleculaire, Macroscopique Combustion, CNRS, Ecole Centrale lParis F-92295 ChatenayMalabry Cedex, France. IEEE Log Number 9408253' ^ I R. D. De Roo and F. T. Ulaby, IEEE Trans. Antennas Propagat., vol. 42, pp. 220-231, 1994. It can be shown that the imaginary part of.4(2) is much smaller than its real part. Thus, we may assume as a first approximation that both.4(~)(#) and A(2) (9) are real. The minima are produced when the two contributions have the same modulus and opposite sign. This happens for 0 smaller than OB. Finally, I would like to mention that the same mechanism holds for the shift of the total absorption observed in the numerical calculations reported by Garcia and Stoll [5]. In this case, the reflection minima are due to the total absorption of light by a sinusoidal grating when a surface plasmon is resonantly excited. In their calculations, Garcia and Stoll have added a small random component to the sinusoidal surface. They have found that the minima for the reflected field are shifted. Again, this effect can be understood on the basis of an interference between the specularly reflected field A(~0(O) by the nonperturbed grating and the coherent amplitude A(2) () due to the roughness. REFERENCES [1] M. Saillard and D. Maystre, "Scattering from metallic and dielectric surfaces," J. Opt. Soc. Amer., A vol. 7. pp. 982-990, 1990. [2] J. J. Greffet, "Theoretical model of the shift of the Brewster angle on a rough surface," Opt. Lett., vol. 17, pp. 238-240, 1992. [3] M. Saillard, "A characterization tool for dielectric rough surfaces: Brewster's phenomenon," Waves Random Media, vol. 2, pp. 67-79, 1992. [4] C. Baylard. J.-J. Greffet, and A. A. Maradudin. "Coherent reflection factor of a random rough surface. Applications," J. Opt. Soc. Amer., A, vol. 10, pp. 2637-2647, 1993. [5] N. Garcia and E. Stoll, "Monte-Carlo calculation for electromagnetic wave scattering from random rough surfaces," Phys. Rev. Lett., vol. 52, pp. 1798-1801, 1984. Comments on "Bistatic Specular Scattering from Rough Dielectric Surfaces" Marc Saillard Recently, the above paper1 has been published describing an experimental work on scattering by dielectric rough surfaces, and investigating the shift of the Brewster angle when a plane interface becomes randomly modulated. I have some comments about this aspect of the paper. As mentioned by the authors,1 the numerical evidence of the phenomenon has been given in [1], where rigorous computations have shown that the minimum of the reflected intensity is shifted toward lower incidence angles. This conclusion, concerning one-dimensional surfaces under p polarization, is in agreement with the aforementioned experimental work.t Later, with the help of the perturbation methods, a more detailed study has been achieved by several authors for both Manuscript received June 13, 1994. M. Saillard is with the Faculte des Science de St. Jerome, Optique Electromagnetique, 13397 Marseille Cedex 20, France. IEEE Log Number 9408251. I R. D. De Roo and F. T. Ulaby, IEEE Trans. Antennas Propagat., vol. 42, pp. 220-231, 1994. I 0018-926X/95$04.00 ~ 1995 IEEE

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 43, NO. 2, FEBRUARY 1995 225 one-dimensional surfaces (2]-[4] and two dimensional surfaces [5]. The main point is that although the theories are not the same, they all lead to the same conclusion and confirm the previous results. Therefore, I am very surprised to read in the introduction1 that a shift toward grazing angles was predicted. Section B' also requires some comments. It is suggested that not enough terms of the expansion are taken into account in the perturbation method. I recall that in [3] the reflection coefficient is expanded up to second order (the first order does not concern coherent scattering) and that, within the Rayleigh hypothesis, the coefficients are rigorously calculated. Even though we do not know accurately what is the radius of convergence of such an expansion, a range of rms heights exists where the theory gives accurate results. In addition, a confrontation with rigorous computations is also achieved in [3] and the results agree quite well when the rms height is less than a tenth of the incident wavelength, thus for significan: shifts of the Brewster angle. The perturbation theory, as well as rigorous computations, also predicts that the minimum of the reflectivity is no longer rigorously zero. Fig. 6(a)' deals with parameters within the domain of validity of perturbation methods, thus the experimental results could be compared with theoretical ones, while in Fig. 6(b) the rms height probably exceeds the possibilities of these theories. I would like to add one more comment. On p. 1, the authors write that Greffet and Saillard "state that the small perturbation method predicts just the opposite..." but (later) they refer to [1], which only contains rigorous results from an integral theory. The results from the.perturbation theory are given in [3]. This remark permits me to recall that the shift of the Brewster angle has also been investigated with the help of a rigorous theory which suffers no limitation with respect to the geometrical parameters of the surface. In [3] a comparison of rigorous computations with data from the perturbation method is given, and shows a very good agreement for shallow surfaces. In the Author's Reply, it seems that two kinds of perturbation theories exist; those which give a shift toward nadir and those which lead to the opposite. According to me, it is more important to notice that the perturbation theories which, in the frame of the Rayleigh hypothesis, calculate rigorously the coefficients of the expansion of the reflection coefficient, lead to a good quantitative agreement with both rigorous computations and measurements (as shown in Fig. 1 of the reply). As a conclusion, I suggest modifying a few sentences,' related to the shift of the Brewster angle: the last sentence of p. 220, the end of the first paragraph of section B, p. 225, the last comment on Fig. 6, p. 226, and the concerned sentence of the conclusion. REFERENCES [11 M. Saillard and D. Maystre, "Scattering from metallic and dielectric rough surfaces," JOSA A, vol. 7, no. 6, pp. 982-990, 1990. [21 J. J. Greffet, "Theoretical model of the shift of the Brewster angle on a rough surface," Optics Letters, vol. 17, pp. 238-240, 1992. [3] M. Saillard, "A characterization tool for dielectric random rough surfaces: Brewster's phenomenon." Waves in Random Media, vol. 2, pp. 67-79. 1992. 141 A. A. Maradudin, R. E. Luna. and E. R. Mendez, "The Brewster effect for a one-dimensional random surface." Waves in Random Media, vol. 3. pp. 51-60. 1993. 151 C. Baylard. J. J. Greffet. and A. A. Maradudin, "Coherent reflection factor of a random rough surface: applications." JOSA A. vol. 10. pp. 2637-2647, 1993. Authors' Reply Roger D. De Roo and Fawwaz T. Ulaby The comments contributed independently by Greffet and Saillard are basically the same and may be distilled down to two issues: a) Their theoretical and numerical calculations predict that the Brewster angle shifts in the direction of normal incidence (negative shift) as the surface roughness increases, and our experimental results indeed support their prediction. We clearly have no disagreement with this comment and are in fact pleased that our data confirms their theory. b) In our paper [I], we stated that "the small perturbation method does not explain the observations reported in our present study: the first-order small perturbation method predicts that the Brewster angle will move toward grazing as the roughness increases, while our observations indicate the opposite." This statement is incorrect and both Greffet and Saillard take exception to it. The small perturbation method (SPM) predicts that the Brewster angle is toward the nadir. Our statements regarding SPM were based on the expressions given in Tsang et al. [2]. Unfortunately, the published expression for the SPM vertically polarized reflection coefficient contains a sign error [3], which leads to the incorrect conclusion. Equation (51) of Chapter 2 in [2] as it apparently should appear is given by RI\'(SP-A = 2 2 2 2212 f-(k2+k2 )12/4 (k. 2, + k2k, )2 )) { k.I1 9~~:1+ It() k - k2 kX k, - k2 k I 2II (X)+ k2 piko2 - kl2: k2 k: + ki: k2 klkl- k2 +21 A2 A2~kkj"- _ + k --- —k -- k" k1:1 k Io (. ~ - k A,, k:- + kl:: A.2 A. 2 + A-z i:.11 p (1) where x = kpkp,12, Rx'(.spM) is the vertical small perturbation Fresnel reflection coefficient, R0o is the vertical Fresnel reflection coefficient for a plane interface, a is the rms surface height, I is the surface correlation length, k. ki are the wavenumbers in the incident and other medium, respectively, k-, = k cosO,, kli = V/2' -.2 sin2,, kp = Asin,, k2 = k2 - k2, kA = k2 - k2, i is the angle of incidence and I,, (.) is the modified Bessel function of order n. A Gaussian surface height distribution and a Gaussian correlation function have been assumed. The change in sign is on the second to last term in the integrand (which is the last term in [2]). It is interesting to note that while the SPM expressions given by Tsang et al. [2] and Baylard et al. [4] for the H-polarization reflection coefficient agree to order '2, our translation of Baylard's SPM expression for the V-polarized reflection coefficient (to order Manuscript received July 18, 1994; revised October 3, 1994. The authors are with the The Radiation Laboratory. Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109-2122 USA. IEEE Log Number 9408265. 0018-926X/95$04.00 ~ 1995 IEEE

226 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. VOL. 43. NO. 2. FEBRUARY 1995.o 0 U I u c ~3 w; 0. -5. -10. -15. -20. -25. -30. -35. -40. -45. -50. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. Incidence Angle 0 (degrees) Fig. 1. Measured rough surface reflectivities compared to various theoretical expressions. ka = 0.515. l = 5.4.~ = 3.0 + jO, r,,, = \R,1112. 02) into the above notation is as follows R\'(sP.\I) = kk i 2 - k.k2 1 T.2 2 (k2+k2i)12/4 RiAo - k- k A- zla I ~ - (k2k:, + k2k1:)2 - -2.2, k2 k.2 k2 _ k2 {:1X (j tP I~o (I)(- 2A, ) k.- k. + k:k, k +, A + /, A'2 k' - k'2 k. k2 - k'2 k'2 k. k: + kt: 2 k + k __ k2+ kk_- k2 k+ - k2 k:, +k (o(r)- - )) }kdk. (2) The expressions are very similar, but not identical. Fortunately, the differences are numerically insignificant. Fig. I shows the reflectivity of one of the surfaces we measured compared to the correct expression for SPM, as well as the predictions of physical optics (PO). It is clear that our data agrees with the prediction of SPM. including the shift of the Brewster angle toward nadir, for this slightly rough case. To the extent that we may have misrepresented the works of Greffet or Saillard [5], [6], we apologize. We are pleased that our observations, at least for slightly rough surfaces, have a rigorous explanation. Unfortunately, a number of the rough surfaces that we have measured have roughnesses sufficiently large that they fall outside of the accepted range of validity for the SPM. The region of validity for the PO approach does encompass the roughnesses of the surfaces in our paper, and for this reason we emphasized that approach in the analysis of the entire body of data which we presented. Nonetheless, PO does not predict any shift of the Brewster angle, and the SPM is perhaps more appropriate for a detailed analysis of this phenomenon. REFERENCES [1] R. D. De Roo and F. T. Ulaby, "Bistatic specular scattering from rough dielectric surfaces," IEEE Trans. Antennas Propagat., vol. 42, no. 2, pp. 220-231, Feb. 1994. (21 L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing. New York: Wiley-Interscience. 1985. [31 L. Tsang, private communication, Sept. 1994. 141 C. Baylard, J.-J. Greffet, and A. A. Maradudin, "Coherent reflection factor of a random rough surface: Applications," J. Opt. Soc. Amer. A, vol. 10, pp. 2637-2647, Dec. 1993. 151 M. Saillard and D. Maystre, "Scattering from metallic and dielectric rough surfaces," J. Opt. Soc. Amer. A, vol. 7, no. 6, pp. 982-990, June 1990. [6] J.-J. Greffet, "Theoretical model of the shift of the Brewster angle on a rough surface," Opt. Lett., vol. 17, no. 4, pp. 238-240, Feb. 1992. Correction to "A Study of Transient Radiation from the Wu-King Resistive Monopole-FDTD Analysis and Experimental Measurements" James G. Maloney and Glenn S. Smith The above paper' contained the following errors. Equation (4) should read (TO la(lb/a,,,) o' 0 -. d(O t 'o and in the first paragraph of Section III. crod(o should equal 8.45, not 1.34. Equation (10) should read 1 2;ra,,, d7T The correct expressions were used in all calculations so all of the numerical results, figures and tables in the paper are correct. Manuscript received September 16, 1994. The authors are with the Georgia Institute of Technology, Atlanta, GA 30332 USA. IEEE Log Number 9408252. 'J. G. Maloney and G. S. Smith. IEEE Trans. Antennas Propagat., vol. 41, no. 5, pp. 668-676, May 1993. 0018-926X/95$04.00 ~ 1995 IEEE