Comparison of σo Obtained from the Conventional Definition with σo Appearing in the Radar Equation for Randomly Rough Surfaces

A comparison is made of the radar cross section of rough surfaces calculated in one case from the conventional definition and obtained in the second case directly from the radar equation. The objective of the analysis is to determine how well the conventional definition represents the cross section appearing in the radar equation. The analysis is executed in the special case of perfectly conducting, randomly corrugated surfaces in the physical optics limit. The radar equation is obtained by solving for the radiation scattered from an arbitrary source back to a colocated antenna. The signal out of the receiving antenna is computed from this solution and the result put into a form recognizeable as the radar equation. The conventional definition is obtained by solving a similar problem but for backscatter from an incident plane wave. It is shown that these two forms for σ° are the same if the observer is far enough from the surface; However, the usual far-field criteria are not sufficient. For the two cross sections to be the same, the observer must be far from the surface compared to the radii of curvature of the surface at the reflection (specular) points. Numerical comparison of the two cross sections has been made for normally distributed surfaces and the difference can be significant.


R-oo e1e
where is is the scattered field and ji is the incident electric field. In this limit, all radiation has approximately plane phase structure. Of course, in an actual radar measurement the distance between the scattering element and the observer/transformer is finite and the incident radiation does not have a truly plane phase structure. The measurement is governed not by (1) but by the radar equation, and in the radar equation the scatterer/observer geometry and the phase structure of the incident radiation must be taken into account. The validity of the conventional definition for radar cross section and any simplifying assumptions such as the use of incident plane waves to calculate the scattered fields lies in their ability to yield the same cross section as appears in the radar equation. The objective of this paper is to seek insight into conditions under which the conventional definition yields the same cross section as appears in the radar equation. This will be done by Manuscript  examining a special case in which both the conventional definition and the radar equation can be obtained analytically under an identical set of assumptions. The special case to be treated is that of a randomly rough conducting surface in the case of two dimensions (line sources and corrugated surfaces). This is an idealized model relevant in a first order to scattering from long crested ocean waves and perhaps to other rough surfaces such as plowed fields. A solution will be obtained in the physical optics limit, adopting a Kirchoff (i.e., tangent plane) approximation to obtain the fields on the surface and then evaluating the Helmholtz integral (Green's theorem) asymptotically in the high-frequency limit. This procedure has proven to be reasonable for microwave scattering from ocean surfaces [5], [6]. Two problems are to be solved using this approach (see Fig. 1). In the first case, a plane wave will be assumed to be incident on the surface and the fields scattered to an arbitrarily located observer will be computed. This result will then be used in the conventional definition to determine the radar cross section for the case of backscatter (monostatic cross section). In the second problem, the source of the radiation will be an antenna at an arbitrary position above the surface. The fields scattered from the surface back to a co-located receiving antenna will be obtained and will be used to compute the available power. It will be shown that this expression for power has the form of the radar equation and the term corresponding to radar cross section will be identified. It will be shown that the two forms for radar cross section are identical if the observer is far from the surface compared to the radii of U.S. Government work not protected by U.S. copyright curvature of the surface at the points of reflection (station points). This restriction is not a far field (e.g., Fraunhof requirement. Rather it is a consequence of focusing of scattered rays (caustics) which can occur in the physical op approximation [14]. This focusing does not appear in conventional definition because the observer is at infinity.
To obtain an indication of the significance of the differe between the two cross sections, numerical examples have b computed for a normally distributed random process. Thi an example surface which has been used in studies of the s tering from fully developed ocean waves [3], [17]. The ferences between the two cross sections can be significant,; values for representative situations are presented.

SCATTERED FIELDS
Solutions are desired for the electric field scattered from irregularly corrugated, perfectly conducting surface in 1 cases: 1) the incident radiation is a plane wave, and 2) the i dent radiation is produced by an antenna located at an a trary point above the surface. These two problems are ill trated in Fig. 1. The solutions will be specialized to the c in which there are no variations perpendicular to the plan the figure (i.e., two dimensions) and results will be obtail explicitly for the case of backscatter.
The scattered electric field J,(r, v) for both problems car expressed in terms of the Helmholtz integral which in the c of perfectly conducting boundaries has the following form: eS(Fr v)= -i a g(r/r')ds an surface where g(F/r') is the two-dimensional Green's function j/4 H (k IF -I), and n is a unit vector normal to the surface Z( The kernel aM/an in this integral will be evaluated for b problems by making a Kirchoff approximation: That is, fields at a particular point on the surface are assumed to be same as would exist on an infinite plane tangent to the surf at that point. Assuming perpendicular polarization (E EOX), the Kirchoff approximation yields the following kei for incident plane waves [14]: =2j (k * n)Eoxeik * r and in the case of radiation from the finite source the Kirch approximation yields the form [ 15] a = 2 i y k3 cos a) -HO') (M)k) F(y,v x . ( an 2 Y )Î n (3b), the function F(y, v) is the Fourier transforn of source current density J(z, y), evaluated at spatial frequenc vy = v/c sin ¢(y) and v, = v/c cos ¢(y). R(y) is the dista: from the source to the surface, a(y) is the slope of the s face, ¢(y) is the angle R(y) makes with the vertical (z-ax and k = 2rrv/c. The frequency of the incident radiation i and 4o')(kR) 7 V2TK exp [j (kRir/4)] is the asyr totic form for large kR of the Hankel function of first k [1]. Equation (3b) was obtained by representing the source terms of an equivalent current distribution, xJ(z,y) (e.g., equivalent aperature illumination of the antenna). Then the Kirchoff approximation was used to find the fields on the surface taking into account radiation from the image of the source below the tangent plane. This results in an integral over the equivalent current distribution and its image which has been evaluated here using a Fraunhoffer approximation [15, appendix A and B]. This is an appropriate procedure when the antenna is small relative to the distance to the surface, a restriction which is to be distinguished from requiring that the antenna be in the far field of the scattering elements (surface). The latter restriction implies limitations on the relative size of the object while the former implies a restriction on the size of the antenna (for given distance and wavelength).
Substituting (3) into (2) and performing the integration in the limit kR -oo by means of a saddle point approximation [8] yields the physical optics solution for the scattered fields. Assuming in the plane-wave case that the wave is incident at angle 0 and that scattering is to an observer at (y = 0, z = H), d. and R,(y) is the radius of curvature of the surface at y. The (Y)' Yn are the stationary points defined by 34./ay = 0. To obtain soth the radar cross section for backscatter, (4a) will be evaluated the in the limit that the observer recedes to infinity (R(y) oo) the along a ray parallel to the direction of incidence 0 of the plane *ace w'ave. In this limit, scattering occurs at points where a = ,B = 0 o = and the result for very large R(y) is rnel s(r,v, ' ). iV°a 2R(yn) kD (5) :3a) These are cylindrical waves propagating from the stationary off points back in the direction of the incident plane wave. The equivalent result in the case of radiation from the antenna is obtained by substituting (3b) into (2). Assuming a 3b) source and observer co-located at (y = 0, z = H) one obtains the following result for backscatter: tihee5 The amplitude in (6a) has been normalized so that the electric field radiated by the antenna in the direction 0, the angle of incidence of the plane wave, is equal to the amplitude of the plane wave. In this normalization F(y, v) is the far-field radiation pattern of the antenna. F(O, v) is a constant; and if F(O, v) is the maximum value of F(y^, V), then F(yn, v) is the relative field pattern of the source [7] .
Prior to using the scattered fields given by (4) and (6) to obtain expressions for radar cross section, some comments are in order. First, the solutions for js(F, v) are applicable to surfaces which are large compared with the distance R(y), to the observer. In particular, although the assumption kR > 1 was used to justify an asymptotic expansion of the Helmholtz integral, it was not necessary to make a sagital approximation for R(y) in the evaluation of (2). That is, it has not been necessary to expand R(y) in a power series in y and Z(y) keeping only terms up to first order iny and Z(y). As a result, (4) and (6) are applicable as long as the observation point is many wavelengths above the surface (kR large) regardless of the size of the surface. This is in contrast to making a Fraunhoffer approximation in which it is necessary to impose restrictions on the size of the surface relative to the distance to the observer.
(Typically LIR < 1 and kL2/R < 27r where L is a dimension characteristic of the surface.) Secondly, in the limit R(y,) -+ oo backscatter from the plane wave and finite source ((5) and (6), respectively) become very similar. If the size of the surface is restricted so that one can also do a binomial expansion of R(y) about Ro, the distance from the observer to the center of the illuminated surface, then one can show that ¢(y) = 2R(y)-Ro and in this case, the phase factors in (5) and (6) Hence, in this far-field (Fraunhoffer) approximation, except for the arbitrary phase, the two solutions differ only by the factors F(yn, 'I)/I/R(yj) which are the antenna pattern (F) and the cylindrical spreading (1/ /R(yn)) present in the case of radiation from a finite source. If this ratio is kept constant as R -+ oo, the two solutions are directly proportional.
As a final comment consider the singular case of a flat surface, Z(y) = 0. In this case there is only one stationary point in both (4) and (6) and this corresponds to the ray normally incident on the surface. Of course R,(yn) --oo in this case. Setting a = 0 in (4) and letting RC°°yields a reflected plane wave S = -EOX exp (jkR(O)) as expected. In (6) set-ting Rc -o°also yields one solution, which in this case is just radiation from the image of the antenna below the surface.

RADAR CROSS SECTION
With minor modifications to specialize the results to two dimensions, the conventional fonnula for radar cross section may be written [2], [101, [11] (O) = him 2rR | s ) eSQ(F )| (8) where e v) is the scattered field and e(J, v) is the incident electric field. For distributed targets such as land and ocean surfaces in which there are many randomly located scattering elements per radar footprint, it is convenient to define a normalized cross section (u0(O)) [16], [181 as follows: where the brackets ( ) denote an ensemble (statistical) average. Alternatively, the -radar equation itself may also be regarded as the definition of radar cross section. In the case of distributed targets, one may write the radar equation for the received power, Pr(v), in the following form in two dimensions [18]: (Pr(v)) = f4 Pt (s) Gt(s) Gr() d sfaek (2irR)2 (0()d (10) where Pt is the transmitted power and Gt and G, are the gains of the transmitting and receiving antennas, respectively. The objective of this paper is to compare the conventional definition for (a°) (see (9)) with the normalized cross section appearing in the radar equation (equation (10)). This will be done in the special case of rough, corrugated conducting surfaces using the solutions for the scattered fields obtained in the previous section. To obtain the cross section from the definition, denoted here by (a0(O))pw, (4a) will be substituted into (9). To obtain the form for the normalized cross section as it appears in the radar equation, to be denoted here by (a (G))RE, first, the expression for the scattered fields (see (6)) will be used to determine the power available from the receiving antenna and then this solution will be cast into the form of a radar equation and the terms corresponding to (a°) identified. Finally, to make the required averages tractable, it will be assumed that the fields scatter incoherently and that the stationary points (specular points) are homogeneously distributed over the surface (i.e., spatially stationary). These latter assumptions have been used in the past to analyze scattering from rough surfaces [3], [13]. Ideally, (a0))pw will be identical to (a0(O))RE. However, as will be shown, this is only true if the receiving antenna is far from the illuminated surface compared to the radii of curvature of the surface at the stationary points: R(yn) > R,(y,).
To begin, consider the radar cross section obtained from the definition (see (9)). Substituting (4a) into (9) and assuming that the scattered fields j,(F, v) are incoherent, one obtains the following: 2-2 /COS' (0-a) Cos aRc(Y) (U0(j))PW = L cos (0) +cos(3) IRC'oYfl/ (11) In the case of backscatter, the incident and reflected rays are locally normally incident on the surface and so ax = ,3=0. In this case, (11) simplifies to (00(0)),W = L (IRC(Yn)I) (12), L allYn and when the surface is homogeneous, this result may be written (O(0)) =lrn (IRc(Yn)I) (13) where n is the number of stationary points per unit length. This is the two-dimensional equivalent of results which have appeared in the literature on scattering from rough surfaces [31, [13]. It is a well-known result which in three dimensions has shown agreement with data on ocean surfaces. The dependence of this cross section on the radii of curvature of the surface is a characteristic of the physical optics solution [2].
To obtain an expression for the radar cross section from the radar equation, it is necessary to first compute the power available from the receiving antenna (using (6) to express the fields incident on the antenna) and then to put the result in the form of (10). To do this note that the time average power, Pr(V) available from the receiving antenna can be written in terms of the scattered fields, kS(F, i.), incident on the antenna in the form Pr(V)) = \/l74 [kS(, v)~*eS (r v)] k4Gr(,P) (14) where 4Gr(F, v)/k is the equivalent area of the receiving antenna in two dimensions (a line source) and Gr(F, *) is its gain [10] . Now substituting (6a) for es(F, v) and assuming that the scattered fields are incoherent, one obtains (Pr(0)) = E 2Ry)| 1 -R(y ) -k4 ) (15) Examination of the definition of Eo and F (see (6b), (6c)) and comparison with the form taken by radiation from a twodimensional source in the far field yields [141 jEOF 12=v hE -PtGt (16) 27T where Pt is the transmitted power. Now using (16) and assuming that the stationary points are homogeneous with density per unit length, n, and replacing the summation in (15) formally by an integral, one obtains (P( =, suJfae 4 Pt {r G 1-R(s) |-1 R(s) dsX Equation (17) is the desired form for the scattered power seen by the observer in the case of backscatter. It is to be compared with the radar equation (equation (10)) to obtain an expression for the normalized radar cross section. Comparing (17) and (10) Clearly the two forms of the radar cross section are not the same. However, when R RC the denominator in (20) approaches unity and the two results agree. That is, the cross sections are equivalent if the observer is far from the surface compared to the radii of curvature of the surface at the specular points. Notice the singularity appearing in (20) (and also in (4a) and (6a)). This singularity is a manifestation of focusing of the scattered rays which can occur in the physical optics solution (e.g., [14]). In the physical optics solution the reflected rays emerge from the surface as if reflected from a small mirror with focal length proportional to the radius of curvature of the surface at that point. If the surface is concave in the direction of the observer, the rays will converge at a point above the surface. When the normalized cross section is computed from the conventional definition, the limit is taken as R -* o. Consequently, the observer is removed beyond any potential caustic (focusing) and as a result there is no singularity appearing in the solution (see (19)). However, when the radar equation is used to determine the cross section, the observer is maintained at a finite distance from the surface, and in this case the influence of focused rays appears explicitly in the solution (see (20)).
The magnitude and phase of the reflected rays depend on the focal length of the equivalent mirror and on the phase of the incident wave which illuminates the mirror. Thus, for a given mirror, incident plane waves and cylindrical waves are reflected differently. This accounts for differences in the radical in (4) and (6). Also, the location of the stationary points and therefore the radii of curvature are different in (4) and (6), and as a result the averages in (19) and (20) are not over the same points. This is so because the stationary points occur where the local angle of incidence equals the angle of reflection, and the incidence angles are different in Q Q he two cases. When R > RC the observer is removed beyond any possible singularity and in this case the incident radiation is nearly a plane wave for surfaces of finite extent in both cases and the two solutions for the radar cross section are identical.
The formulas for radar cross section (see (19) and (20)) have been derived for one special example-corrugated, statistically homogeneous, conducting surfaces. However, the differences which appear in the cross sections are the result of features inherent in the method of analysis (the physical optics approximation) and not to the specific scattering object (surface) chosen. It would seem reasonable, therefore, to expect such differences in real three-dimensional configurations when the scattering is predominately specular (situations amenable to analysis by physical optics). An example is scattering at angles near nadir from the ocean surface or from rough soil [5] .
Granted differences between the two cross sections, the question remains as to the significance of these differences.
It is possible that R > RC is typical of measurements of real surfaces, in which case the differences would be of no practical consequence. To obtain an estimate of the possible order of magnitude of the difference between (U0)pw and (U0)RE the averages in (19) and (20) have been carried out in the case of a surface Z(y) which is a Gaussian random process with a Gaussian correlation function. This is a model which has been used for the analysis of scattering from ocean waves [3], [17] and might apply to plowed fields. Since RC [1 + (dZ/dy)2] 3/2/[d2Z/dy2] in two dimensions, the joint densities for Z, dZ/dy and d 2Z/dy2 are required to evaluate the averages.
The calculations of these densities is straightforward for this surface [17]  . This is an ad hoc assumption made to simplify the mathematics which at least is consistent with the approximations made in arriving at (20) [3] In particular, the asymptotic evaluation of the Helmholtz integral (equation (2)) employed here does not apply at the singular points R = RC. At stationary points for which R = RC it is necessary to modify this asymptotic form by taking higher order terms [81, [9] . Doing so, one obtains finite fields at the singular points but the focused character  of the radiation when R is not close to Rc remains as described b) above. The assumption (1/A) = 1/(A) avoids this subtelty and makes it possible to do the average analytically. It is imposed here in a purely ad hoc way without attempt to bound the error. Making this approximation one obtains in a straight forward manner A (&0(0))Pw ll(IRIRcI) (a0(0))RE = 1/(I -R/Rcj) It is clear from (23) that the ratio of the two cross sections decreases from unity at t = 0 to zero at = oo. The behavior of F(Q) has been plotted in Fig. 2. In addition, values of t are listed in Table I for a range of the parameters 1 and R representative of the extremes one might encounter in observations of natural surfaces. Consider, as an example, a situation representative of observations from a low, earth orbiting satellite (e.g., R = 1000 km). In this case, it is clear from  Fig. 2, one finds r _ 1. Consequently for observations from space it is unlikely that the differences between the cross section calculated from the conventional definition and that actually measured by a radar will be significant. In the case of observations from aircraft (e.g., 1 km < R < 20 km) the situation is less clear. In this case, one sees from Table I that t can be large for large values of the correlation length (e.g., t = 1.7 at R = 1 km and 1 = 100 m) in which case differences between (a°)pw and (W0)RE could be important, or t can be quite small (e.g., an observation at R = 10 km over plowed fields with 1= 1 m) in which case F_ 1. For towerbased operations, on the other hand, R is much smaller and differences between (a0)Pw and (U0)RE are much more likely.
For example, at R = 10 m and I = 5 m, one finds from Table I that, t = 0.4. In this case using Fig. 2 one finds F = 0.45 which is about 6 dB. Clearly, significant differences between the two cross sections are to be expected in such cases.

CONCLUSIONS
The results of the preceding section, and of this paper in general, are not reflections on measurement techniques. Rather, they are an indication to those concerned with the calculation of radar cross sections that the classical definition may not apply equally well in all cases. Specifically, in the case of remote sensing of rough surfaces where the scattering is predominately specular, it would appear that more specific attention needs to be given to the geometry under which an actual experiment might be performed before routinely using the classical definition to compute radar cross section.