Two-scale models for rough surface scattering: Comparison between the boundary perturbation method and the integral equation method

Authors


Abstract

[1] This paper presents a comparison between two models for rough surface scattering computations: the boundary perturbation method (BPM) and the integral equation method (IEM). They differ in two fundamental aspects: the method used to compute the electric and magnetic fields on the surface and the surface information required for the computation. The two approaches lead analytically to the same solution in the two asymptotic cases of very large and very small vertical displacements, with no intermediate scales. For a composite surface the solution of the BPM is expressed as the sum of two terms, while a series development up to higher orders can be formulated with the IEM. In this paper, the comparison is restricted to composite surfaces and, particularly, to the ocean surface. After presenting a method for the two-scale decomposition of a rough surface, which satisfies the constraints of the electromagnetic model for each scale, we compute the scattering by the ocean surface using both models for various instrumental configurations and surface conditions. We show that considering the accuracy of usual radar measurements and under the assumptions made for the development of the models, both methods give very similar results. Since the BPM is based on a simple physical argument and appears to be more efficient than the two-scale IEM with regard to the computation time, the BPM should be preferred for ocean-like rough surfaces.

1. Introduction

[2] Radar probing of the Earth's surface has now been in use for a long time. Several Earth observation missions have already proven radar potentialities, with numerous scientific and operational products being distributed. Microwave and millimeter wave instruments present the major advantage of being able to sense the Earth's surface through cloud cover. Their main drawback compared to optical or infrared sensors is their weak spatial resolution due to the longer electromagnetic wavelength they use. This problem can be overcome using synthetic aperture techniques, leading, however, to particular disturbing effects such as speckle, target displacements, etc…

[3] Radar echoes are directly related to the scattering properties of the observed scenes. Therefore a number of methods have been developed for scattering simulations and for the retrieval of geophysical parameters from microwave measurements. Among these models, empirical and theoretical ones should be distinguished. Though the first ones are usually very efficient considering their computation time, they do not allow a full understanding of the physical processes implied. Physically based theoretical models, on the other hand, allow the analysis of the sensitivity of the scattering characteristics to both target parameters and radar technical specifications. Moreover, being tabulated for particular instrumental configurations or implemented on high-performing recent computers, they can also be used as the basis for inversion algorithms for operational systems. Finally, when proven efficient, theoretical models could also be used as references for the intercalibration of multichannel active and passive instruments.

[4] Since the basic work of Rice [1951], an abundant literature has covered the subject of random rough surface scattering, both for the scalar case (acoustics) and for the vector case (electromagnetics), under either extreme boundary conditions (soft and hard surfaces, or perfectly conducting surfaces) or more general conditions. Extensive lists of references are to be found in some recent textbooks, e.g., those by Fung [1994] or Voronovitch [1994]. Among all proposed methods, two-scale methods work rather well for the case of the ocean surface, because the separation between the large-scale component and the small-scale component is somehow related to gravity and capillary waves. We have developed a two-scale approach [Guissard and Sobieski, 1987; Guissard et al., 1992], based on the boundary perturbation method (BPM) [Burrows, 1973]. It leads to numerical results that compare rather well with experimental data [Lemaire et al., 1999]. Another method that has encountered a large audience in recent years is the integral equation method (IEM) proposed by Fung et al. [1992] and Fung [1994], which can be applied for both ocean and soil surfaces.

[5] We present in this paper a comparison between these two theoretical models for rough-surface-scattering computation. Since both methods lead to the same asymptotic solution in the cases of either very small vertical displacements of the surface with small slopes or very large surface displacements with large curvature radii, we shall focus on the case of a composite surface where different scales of roughness may be present. In section 2, the two models are presented following a scheme of increasing complexity, so that it remains as clear as possible and facilitates the following discussions. The presentation is restricted to the main steps that highlight the differences between the methods, and the reader is referred to the literature whenever the mathematical developments are not relevant to the purpose of this paper. Next, the results obtained with both methods are compared in the case of the ocean surface and for various radar configurations. Since the solution appears as a series development for both methods, we analyze the results starting with the zeroth-order term and then continuing with higher-order terms.

2. Rough-Surface-Scattering Models

[6] A usual quantity derived from the radar echo measurements is the normalized radar cross section (NRCS) or scattering coefficient σo. Consider a rough surface z = ζ(x, y), horizontal in the mean. A portion of the surface with horizontal area A is illuminated by a plane wave Epi = E0pe(−jki·r) from direction i and polarization p. It induces a scattered field Eqs with polarization q into the direction s. The bistatic scattering coefficient is then defined as

equation image

where R stands for the observation distance.

[7] The time dependence ejωt being understood, the scattered field Es, at large distance from the target, can be evaluated from the knowledge of the induced electric and magnetic surface fields E and H, by

equation image

where r represents a point of the surface S, n is the normal to the surface at that point, and k is the electromagnetic wave number.

[8] For a rough surface, r, n, E, and H are stochastic functions of time and position. Since the measurement of radar echoes from a rough surface proceeds by averaging a large number of independent return pulses, each of them being the combination of echoes produced by a large amount of individual scatterers, one defines the total scattering coefficient associated with the total field as follows (the angle brackets denoting an ensemble average):

equation image

[9] We follow hereafter the scheme presented by Guissard [1996]. We normalize the surface fields with respect to the incident electric field,

equation image

with η the intrinsic impedance of the propagation medium, and we extract the component of the scattered field with polarization q, i.e., Eqs = q · Es. The total bistatic scattering coefficient σpqo can be evaluated by

equation image

where r′ = (x′, y′, ζ′) and r″ = (x″, y″, ζ″) are two points on the surface, v = k(si), and Spq is given by

equation image

with

equation image

and the asterisk indicating the complex conjugate. In expression (6), z is the unit vertical vector, and n′ and n″ are the unit vectors normal to the surface at r′ and r″, respectively. In (7) the field vectors ep and hp and the unit vector n normal to the surface must be evaluated at the location r on the surface.

[10] In the above relations, ζ′ = ζ(x′, y′) and ζ″ = ζ(x″, y″) represent the elevation of the surface above the horizontal plane of reference. They are stochastic functions of the horizontal coordinates of r′ and r″. Under the assumption of spatial stationarity the averaged quantity in (5) only depends on the difference in position r = (r′ − r″). Further, if the correlation length of this random quantity is small compared to the dimensions of the area A, (5) may be simplified as follows [Ulaby et al., 1982, volume 2, chapter 12]:

equation image

where v · r = vz (ζ′ − ζ″) + vxx + vyy has been used.

[11] In the expression above, the surface field term Spq and the phase term equation image are both difficult to evaluate. Existing methods are based on different sets of assumptions. First, the ensemble average involves the product of two factors. The exponential term only depends on the vertical displacement, while Spq is a function of the shape of the surface, i.e., of the slopes, curvatures, etc…. It can be shown, furthermore, that under the assumption of spatial stationarity the displacement at one point of the surface is not correlated with the slopes at that point. In the particular case of Gaussian surface statistics this implies that these variables are independent. Although the same argument does not apply to two separate points, we shall assume that the ensemble average may be factorized as

equation image

[12] It must be stressed that in certain derivations this assumption is not necessary or is simply induced, because of the set of initial assumptions (for instance, under the small-slopes assumption, the Spq term is usually slope-independent, hence position-independent too). The product factorization in (9) is based on a heuristic argument and does not ensure a correct solution. The proof of its validity can only be obtained a posteriori, showing that it leads to a good approximation of the reality. This factorization is made explicitely in the BPM and implicitly in the IEM.

[13] The second factor of (9) can be identified as the joint characteristic function χ2 of the displacements at two points, evaluated at (vz, −vz). If the surface dimensions are large compared to the correlation length of the variables over which the ensemble averages are taken, the limits of integration may be extended to infinity, and we obtain

equation image

[14] For the particular case of a normal surface the joint probability density function (pdf) of the vertical displacements at two points is a Gaussian function, and we have

equation image

where C(ρ) is the autocorrelation function of surface heights and ρ = (x, y) denotes a position in the horizontal plane of reference. When the joint pdf of the vertical displacements is not Gaussian, the expression of χ2 becomes more complicated. The analysis of this case is given by Guissard [1996]. It yields σpqo components whose evaluation requires high-order statistics (bicovariance, bispectrum, etc.) of the surface displacements. In the following analysis, we assume that relation (11) is valid.

[15] To make the following discussion as clear as possible, we shall recall hereafter the results in the two asymptotic cases of very large and very small surface displacements compared to the electromagnetic wavelength.

2.1. Very Large Displacements

[16] In the first case, kσ is assumed very large. Ulaby et al., [1982, volume 2, chapter 12] express the limit of validity as

equation image

[17] The characteristic function (11) has then negligible values except in the vicinity of origin where C(0,0) = 1. Assuming, in addition, that the curvature radii are large compared to the electromagnetic (EM) wavelength, the Kirchhoff's approximation can be used for the evaluation of Spq. Expanding the correlation function C(x, y) around the origin, the integration can be performed, and the well-known physical optics solution is found [Guissard et al., 1992]:

equation image

where the Fresnel coefficient Rpq(nsp) is evaluated on a plane oriented so that s and i correspond to a specular reflection, while the slope probability density function T is evaluated for the values (−vx/vz, −vy/vz) leading to a tangent plane with such an orientation. Here, nsp is the normal to the surface at these specular points, and γsp is the angle between this normal and the vertical, i.e., cos γsp = nsp · z.

2.2. Very Small Displacements

[18] For very small surface displacements the small perturbation method (SPM) [Rice, 1951] applies. The result can be directly obtained from (10) using a much simpler derivation. With the assumption of small slopes, the normal n to the surface is close to the vertical z, and we can use the approximation n · z ≈ 1. To the first order, we have

equation image

Assuming a Gaussian joint pdf of the surface displacements at two points, χ2 can be written as (11), and using a series expansion of the exponential, it can also be expressed as

equation image

With the small displacement assumption, kσ ≪ 1, χ2 simplifies to the first order as

equation image

and we get

equation image

where δ represents the Dirac function. Here, γ(K) is the surface vertical displacement spectrum, i.e., the Fourier transform of the autocovariance function Γ(x, y) = σ2C(x, y),

equation image

taken at the Fourier components Kx = vx and Ky = vy. The function ∣Fpqn = z2 in (17) can be evaluated with the boundary perturbation method (BPM) [Burrows, 1967], which is equivalent to the SPM for the first-order term as proved by Craeye et al. [2000], with the result that

equation image

where θi and θs stand for the incidence and scattering off-nadir angles and an expression of the αpq is given, for instance, by Ruck et al. [1970]. (The same coefficients are also given by Ulaby et al., [1982, volume 2], with sign errors, however.) In the specular direction, i.e., for vx = vy = 0, relation (19) reduces to

equation image

Finally, the total bistatic scattering coefficient is given by

equation image

where ∣Reff,pq2 = (1 − σ2vz2) ∣Rpq2 denotes an effective Fresnel coefficient, taking into account the decrease of the power in the specular direction due to the scattering of the incident wave in all directions by the surface ripples.

[19] In (21) the total scattering coefficient appears as the sum of two terms. The first term corresponds to the coherent scattering coefficient, defined as

equation image

and becomes negligible as the ratio between the surface roughness and the electromagnetic wavelength increases. The second term corresponds to the incoherent scattering coefficient. It is commonly referred to as the Bragg scattering mechanism, since it shows that the scattering results from the components of the spectrum vx and vy that produce a constructive interference in the scattering direction.

[20] The same result can be obtained using the integral equation method (IEM) to obtain an expression of the field coefficient Spq in (10) [Fung, 1994, chapter 4]. An advantage of this method is that it provides also an expression of the scattering coefficient when the development of χ2 in (15) is not limited to the first order, yielding then an improved accuracy when kσ is not negligible compared to unity. A complete expression in this case is given by Fung [1994].

2.3. Composite Surface

[21] For natural targets such as the ocean, for instance, the surface is composed of many scales of roughness. In this case, the physical optics and the Bragg solutions apply strictly to millimeter waves and HF waves, respectively. The IEM can yield a better solution over a larger range of wavelengths when enough terms of the series expansion (15) are used; however, it becomes practically inapplicable when too many terms must be computed. We present hereafter two methods for the computation of the scattering by a composite surface. One is based on a boundary perturbation approach and is developed by Guissard et al. [1992]; the other is an extension of the integral equation approach for the case of a composite surface. We briefly recall the main assumptions of the BPM and present the developments for the two-scale extension of the IEM. These two methods are numerically compared in sections 3 and 4.

[22] In order to develop a simple and efficient method, Guissard et al. [1992] extended the boundary perturbation method initially proposed by Burrows [1967, 1973] and Brown [1978] to bistatic scattering by a dielectric rough interface. In this method, it is assumed that the surface is composed of the superposition of small ripples with RMS value σR such that kσR ≪ 1 on large waves with RMS vertical displacements σL. The two surface components are assumed to be independent stochastic processes, and their corresponding spectrum is computed by splitting the spectrum of the composite surface γ(K) at a given wave number Klim, such that the large-scale spectrum γL and the small-scale spectrum γR satisfy

equation image

where a cylindrical coordinate system such that (Kx, Ky) = (K cos α,K sin α) has been assumed for the last two relations.

[23] The total scattered field is then expressed as an expansion with respect to σR,

equation image

The zeroth-order term is the solution for the field scattered by the large-scale component without ripples. Assuming large curvature radii ρL for the large scale, Kirchhoff's approximation can be used to determine it. Next, assuming constant values of the zeroth-order electromagnetic fields inside the volumes determined by the small waves, the perturbation field at large distance from the surface Eq1s due to the ripples is evaluated. Finally, removing the cross term 2 ReEq0s · Eq1s*〉 in the evaluation of 〈∣Eqs2〉, thanks to the assumption of independence between large and small roughness scales, the scattering coefficient is expressed as the sum of two terms:

equation image

The zeroth-order term is given by the Kirchhoff solution for the large-scale surface (13) with the Fresnel coefficient Rpq replaced by an effective Fresnel coefficient Reff,pq similarly to (21). This latter takes into account the effect of the ripples present on the “tangent planes” on the large waves as

equation image

On the other hand, the expression of the first-order term is

equation image

where

equation image

[24] Tsl(α, β) corresponds to the large-scale slope pdf, and the expression of the surface field function HPQ is given by Guissard et al. [1992]. Analyzing this expression closely, we note that as for the incoherent term of (21), the scattering results from the Fourier components of the surface spectrum that produce a constructive interference in the scattering direction. The difference with (21) is that, since in this case the ripples lie on large tilting undulations (with a certain probability density Tsl(α, β)), a certain range of Fourier components may have the correct wavelength and orientation to yield a constructive interference. This results in the convolution between the ripple spectrum γR and the large-scale slope probability density function Tsl. Moreover, the HPQ(α, β) function has also been shown by Craeye et al. [2000] to be identical to the αpq in (21), and therefore we sometimes refer to expression (27) as “tilted Bragg” solution hereafter. In particular, when only ripples are present on a flat surface, Tsl(α, β) reduces to a product of Dirac functions and the solution degenerates into the SPM or Bragg scattering solution [Guissard et al., 1989].

[25] Another approach can be followed in the case of scattering by a composite surface. It is based on a two-scale expansion of the IEM. The detailed developments are given by Lemaire [1998]. We summarize them hereafter. Using the IEM, the surface fields appear as the sum of a Kirchhoff and a complementary term. The scattering coefficient appears as the sum of several terms, and each term can be factorized under a form similar to (10). If the spectrum γ of the composite surface is expressed as the sum of a large-scale spectrum γL and a small-scale spectrum γR using (23), the covariance function is equal to the sum of two functions corresponding to the inverse Fourier transform of the respective spectra, i.e.,

equation image

Consequently, the characteristic function (11) can be written as

equation image

[26] Next, as done above when only one scale of roughness is present, the covariance function of the large scale can be expanded around the origin (see section 2.1), and the joint characteristic function of the small scale can be developed in series (see section 2.2). The surface fields being evaluated using the integral equation, the integration along the horizontal coordinates, can be performed. Since, similarly to (10), all terms of the scattering coefficient appear as the Fourier transform of a product of two functions χ2L and χ2R, two partial solutions can be found in a first step by integration on the large and the small scales separately. Then, using the properties of the Fourier transform with regard to the product of two functions, the complete solution is the convolution of these partial solutions. We obtain, finally, the series expansion [Lemaire, 1998]

equation image

[27] As with the BPM, the zeroth-order term is given by Kirchhoff's solution for the large-scale surface (equation (13)) with an effective Fresnel coefficient Reff,pq taking into account the effect of the ripples:

equation image

Jpq denotes the difference with BPM and is given by

equation image

where

equation image
equation image

[28] Similarly to (27), the single scattering terms σpqno are given by

equation image

In this expression, the γR(n) functions are defined as the spatial Fourier transform of the nth power of the ripples autocovariance function, i.e.,

equation image

The surface field functions Ipqn are given by

equation image

where the expressions of the fpq and Fpq factors are given by Fung [1994] and the wave numbers

equation image

geometrically correspond to the local tangential wave number components on large waves, if the large-scale slopes are small compared to unity.

[29] The single-scattering terms are thus given by a convolution between the ripple spectral functions γR(n) and the large-scale slope probability density function. Hence, as for (27), this expression can be physically interpreted as resulting from the interaction between small waves propagating on large tilting waves. Finally, the higher-order terms σpqmno take multiple scattering into account. They represent the influence of the electromagnetic fields at one point of the surface on the neighboring points. Their expression is given in Appendix A since they are quite long and complicated. They will not be considered in the following analysis.

[30] For a slightly rough surface, Tsl(α, β) degenerates into a product of Dirac functions. In this case, the σpqno can be shown to reduce to the expressions given by Fung [1994, section 5.5.1], which are themselves equivalent to the SPM solution when the development is limited to the first order for small kσR compared to 1. However, contrary to BPM, it is important to note that relation (36) for n = 1 is not equivalent to the tilted Bragg solution. Indeed, as already mentioned, when no large scale is present and when the exponential terms exp {−[σR2vz2]} in (36) and exp {−[σR2kizksz]} in (38) can be approximated to unity, the solution reduces to the SPM solution (21), as does (27). Thus, for a composite surface, not considering the exponential factors in (36) and (38), if all the wave number components in expression (38) of Ipqn were evaluated as local wave numbers with reference to the tilting large scale, (36) would correspond like (27) to the tilted Bragg solution. However, although (39) shows that the Fpq functions in (38) are indeed calculated for tangential wave number components (kix, kiy, ksx, and ksy) evaluated locally, the vertical components kiz and ksz in (38) are taken with reference to the global incidence angle. Hence expression (36) of σpqno with n = 1 does not correspond exactly to the tilted Bragg solution.

[31] This important difference is due to the fact that the integration of the various terms (similar to (10)) of the scattering coefficient is performed along the horizontal coordinates. Therefore, when a two-scale decomposition is used, the convolution is performed only on the horizontal wave number components; whereas, since vz or kiz and ksz appear only as multiplicative factors of the covariance function in the series development of χ2, no convolution is performed on the vertical components of the vectors, and they are thus evaluated in the global coordinate system. This might result from initial assumptions done to compute the surface fields using the IEM, such as the small-slopes assumption. Although it would seem more intuitive to use also vertical wave number components evaluated locally in the expression of the Ipqn functions, we use expression (38) obtained analytically, knowing that its validity is limited to the assumptions made for the developments.

3. Limit Wave Number Problem

[32] In section 2, we briefly reviewed electromagnetic models for rough surface scattering. We ended with the BPM and the two-scale development of the IEM that will be compared in section 4. Different conditions were imposed on each scale for the application of the scattering models. These constraints depend on the electromagnetic wavelength (and thus on the instrumental configuration) and on the surface roughness, which is related to a number of environmental conditions for natural rough surfaces such as the ocean. Both the surface and the instrumental characteristics should therefore be taken into account to decompose the surface so that the electromagnetic constraints are satisfied. This is the purpose of this section.

[33] For the application of the two-scale models in the case of a composite surface, we split the surface into small and large waves following (23). For the large-scale component, besides the condition (23) on its vertical displacements, the curvature radius should be much larger than the electromagnetic wavelength everywhere on the surface for the Kirchhoff's approximation to be valid. For the small-scale component the surface vertical displacements need to be small compared to the electromagnetic wavelength. This was required for the BPM to assume constant fields inside the volumes determined by the small roughness; while for the two-scale expansion of the IEM, this assumption allows the use of a series expansion of the joint characteristic function of the small-scale vertical displacements.

[34] For the small scale, the condition on the displacements is given by

equation image

with σR given by (23). From its definition the surface spectrum γ is a symmetric function with respect to the origin. It is usually factorized as [Lemaire et al., 1999]

equation image

where S(K) is the radial surface spectrum and F(K, α) represents the azimuthal variation of the spectrum and is normalized so that ∫0F(K, α)dα = 1. Using this factorization, σR is given by

equation image

On the other hand, the constraint on the very rough surface is that the mean curvature radius is large compared to the wavelength. With equation image being the standard deviation of the curvature radius, we write it as

equation image

The curvature radius at one point of a surface in a given direction is expressed as

equation image

Here, ζ denotes, as mentioned in section 2, the elevation of the surface with respect to a horizontal plane of reference, and ζ′ and ζ″ are the first and second derivatives of ζ, respectively, with respect to horizontal coordinates following the chosen direction. For small surface slopes we have

equation image

and (43) is transformed as

equation image

where σc stands for the standard deviation of the surface curvature ζ″. It may be estimated from the surface spectrum by

equation image

Then, letting (40) and (46) have the same limit value to obtain a unique limit wave number, we have

equation image

[35] We applied this relation to the case of the ocean surface. We used the surface spectrum presented by Lemaire et al. [1999], with 0.7% significant slope, a fetch of 500 km, and the peak wave number given by Pierson's formula for fully developed state. The ratio δ = Klim/k is plotted in Figure 1a for C, Ku, and Ka Bands. We also plot the kσR product in Figure 1b. We see that kσR values lie below unity as required for the small scale. As the conditions on the small and large scales have the same limit value, a small kσR product also means that the curvature radius to EM wavelength ratio is large, as necessary for the large scale. We also verified that (12) is satisfied for the large scale. Finally, several authors use a constant value δ to split the spectrum into two scales. The range of values usually recommended is 0.3 < δ < 0.5 [Kim and Rodriguez, 1992]. We note from Figure 1 that the δ values computed using the proposed method lie in this recommended range.

Figure 1.

Two-scale decomposition for various frequencies. (a) Ratio δ = Klim/k and (b) kσR product.

[36] Besides the constraints on the height of the small scale and the curvature of the large scale, conditions on the correlation length of each scale have also been proposed in the literature. These conditions are usually expressed for the case of a Gaussian surface and for the asymptotic cases of the physical optics solution or the SPM solution. We verify hereafter that these conditions are satisfied by the above decomposition of the surface.

[37] Let us denote as lL and lR the correlation lengths of the large and the small scale, respectively. In the case of a Gaussian surface, imposing the constraint that the curvature radius of the large scale should be much larger than the electromagnetic wavelength, Ulaby et al., [1982, volume 2, chapter 12] state that the expression of the physical optics solution, i.e., the zeroth-order term without correction of the Fresnel coefficient, should be valid if

equation image

[38] For the first-order term, Thorsos and Jackson [1989] expressed the limit of validity of the small-perturbation method (i.e., the first-order term of the two-scale model, without taking the tilting of the small-scale surface by the large waves into account) as

equation image

where B is a number close to unity and corresponds to the intersection of both expressions. This condition was established for the case of a one-dimensional surface with a Gaussian spectrum. Since it is not easy to conclude that this condition would remain exactly the same in the case of a two-dimensional non-Gaussian spectrum, we use it hereafter merely as an indication.

[39] To check if our two-scale decomposition of the surface satisfies these constraints, we have computed the correlation length of the large and small scales. For a non-Gaussian autocovariance function we define the correlation length as the maximum distance beyond which the absolute value of the autocovariance function never exceeds 1/e = 1/2.718 times its maximum value. The large- and small-scale autocovariance functions ΓL(x, y) and ΓR(x, y) have been computed as the inverse Fourier transform of their respective spectrum. Moreover, to avoid defining a correlation length for each direction in the horizontal plane, the spectrum has been assumed isotropic, i.e., F(K, α) = 1/2π in (41). This does not modify our conclusions. We used the spectrum presented by Lemaire et al. [1999] with the same values for the significant slope and the fetch as above. We considered the three wind friction velocity values 20 cm/s, 60 cm/s, and 100 cm/s and three frequencies 5.3 GHz, 13.6 GHz, and 36 GHz, for which we perform the comparisons between the BPM and the two-scale IEM in the section 4.

[40] The results confirm that condition (49) for the zeroth-order term is satisfied. For the first-order term, Figure 2 presents the computed couples (kσR, klR) together with the limit of validity (50) expressed by Thorsos and Jackson [1989]. We note that in most configurations the couples (kσR, klR) are close to Thorsos and Jackson's limit. Since condition (50) was established for a rough surface with a one-dimensional (1-D) Gaussian spectrum and not for a natural surface such as the ocean (whose radial spectrum is close to a power law), we may not conclude that the proximity of (kσR, klR) to this limit implies that the contribution of terms higher than the first order to the total σo will be negligible. Nevertheless, it does still indicate that their contribution should be small, as confirmed by the calculations (see section 4). Similarly, at high frequency, large wind friction velocity, and large incidence angle, the computed couple (kσR, klR) slightly departs from the limit expressed by Thorsos and Jackson, indicating that terms higher than the first order might be necessary for an accurate computation of the total scattering coefficient. Let us also mention that the computed values of the couples (kσR, klR) compare very well with the validity range computed by Chen and Fung [1988]. Finally, we also tested other spectra from the literature [Bjerkaas and Riedel, 1979; Elfouhaily et al., 1997] with the same conclusions.

Figure 2.

Limit of validity of the first-order term expressed by Thorsos and Jackson [1989] (lines) and values of the correlation length and height variance of the small scale given by our two-scale decomposition of the surface (symbols). The condition expressed by Thorsos and Jackson can be considered as satisfied.

[41] These computations show that the proposed method for the two-scale decomposition of the surface provides a good compromise between the constraints expressed for each scale. Moreover, this method takes the roughness of the surface and the instrumental configuration into account for the computation of the limiting wave number and constitutes therefore an improvement compared to models in which δ = Klim/k is given an arbitrary fixed value.

4. Results

[42] Simulations have been performed for the case of the ocean surface using the BPM and the two-scale IEM. We considered a large range of incidence angles for three different frequencies (5.3 GHz, 13.6 GHz, and 36.0 GHz) and three incident-scattered polarization combinations (VV, HH, and VH). We used the ocean surface spectrum presented by Lemaire et al. [1999] with a fetch of 500 km and a significant slope equal to 0.70% as previously. Three different wind friction velocities have been considered: 20 cm/s, 60 cm/s, and 100 cm/s. For the large scale we used the Cox and Munk [1954a, 1954b] slope PDF, but with the slope variance computed from the surface spectrum. Finally, all simulations have been performed for a view angle aligned with the wind direction.

[43] First, we analyzed the difference between zeroth-order correcting factors for the Fresnel coefficient as expressed by (26) for the BPM and (32) for the IEM. The analysis has been performed for both HH and VV polarizations and for low incidence angles at which σpq0o is the dominant contribution to σpqo. The results are presented in Figures 3 and 4, where we plotted the correction factor αcorr to the Fresnel coefficient, defined as

equation image

We note that there is no important difference between the two corrective coefficients. The maximum difference reaches a few tenths of a decibel for 30 deg. Since, for such incidence angle, σpq1o has the same order of magnitude as σpq0o (see Figures 8 and 9), this means an even lower difference on the total scattering coefficient. We conclude therefore that the exponential correction (26) is an excellent approximation of (32) and presents an advantage of computational efficiency since it avoids a two-dimensional integration.

Figure 3.

Correction factor αcorr to the Fresnel coefficient for HH polarization: BPM (solid lines) and IEM (dashed lines). Numbers 1, 2, and 3 stand for C, Ku, and Ka bands, respectively.

Figure 4.

Correction factor αcorr to the Fresnel coefficient for VV polarization: BPM (solid lines) and IEM (dashed lines). Numbers 1, 2, and 3 stand for C, Ku, and Ka bands, respectively.

[44] Next, we computed the total scattering coefficient using the BPM and the two-scale IEM (up to third order). The results are displayed in Figures 5, 6, and 7, where angular points in the curves are due to the fixed 10 deg increment in the incidence angle. The results of the simulations using the SPM are also displayed to show the importance of the tilting by large waves, i.e., the difference between the BPM and SPM curves.

Figure 5.

Comparison between BPM, IEM, and SPM models for backscattering case and VV polarization.

Figure 6.

Comparison between BPM, IEM, and SPM models for backscattering case and HH polarization.

Figure 7.

Comparison between BPM and IEM models for backscattering case and VH polarization. SPM reduces to σo = 0 in this configuration.

[45] In most configurations the difference between the BPM and the two-scale IEM results is small, if not negligible, with regard to the accuracy of radar measurements. The largest differences occur at large incidence angle, large wind (i.e., important roughness), and high frequency, where it is of the order of 1 dB. The difference is also slightly larger in HH than in the other polarization configurations.

[46] As underlined at the end of section 2.3, the two-scale expansion of the IEM does not correspond exactly to the tilted Bragg solution, as the BPM does. The differences are (1) the use of global incidence angles for the vertical components of the incident and scattered wave number vectors and (2) the exponential factors exp{−[σ2vz2]} and exp{−[σ2kizksz]} in the expressions (36) and (38) of σpqno and Ipqn, which do not appear in the tilted Bragg solution. It is therefore interesting to analyze the contribution of each term of the two-scale IEM.

[47] Figures 8 and 9 present the results in VV and HH polarizations at 36 GHz and large wind friction velocity, i.e., for the configurations where the difference between the BPM and the two-scale IEM is the largest. Let us concentrate on the VV configuration first. We note that the difference between two σpqno in successive orders (higher than the first) is roughly about 10 dB. This difference is smaller around 45° than at lower or higher incidence angles. It is also smaller as the wind speed, hence the roughness, increases. Thus in general, for the VV polarizations combination, two σpqno in successive orders (higher than the zero order), are in a ratio of about one decade. The contribution of the second-order σo is therefore small, while the contribution of the third-order term can be considered as negligible. Nevertheless, the first-order term of the two-scale IEM being slightly lower than with the BPM, the addition of higher-order terms reduces the difference between the two methods.

Figure 8.

Successive terms of σo using the two-scale IEM (36 GHz, VV polarization).

Figure 9.

Successive terms of σo using the two-scale IEM (36 GHz, HH polarization).

[48] This tendency appears more clearly when considering the HH combination. As can be observed in Figure 9, the difference between the BPM and the first order of the two-scale IEM is rather large at large incidence angle (where the contribution of the zeroth-order term is negligible). However, in this case, the second-order term is not negligible compared to the first order, and as successive terms of the two-scale IEM are added, the total σo converges to the BPM solution. This tendency was found for the two other simulated frequencies as well.

[49] Finally, to conclude this section, an important remark must be made concerning the implementation of the two-scale IEM. Indeed, to compute the second- and higher-order terms of σo, it is necessary to evaluate the second- and higher-order surface spectral functions defined by (37). This was performed by (1) evaluation of the 2–D inverse Fourier transform of the small scale spectrum to get the autocovariance function ΓR of the small-scale displacements; (2) elevation of ΓR at the nth power; and (3) computation of the 2-D Fourier transform of ΓRn. These computations have to be done carefully to avoid aliasing and windowing problems. They are, furthermore, time-consuming operations. Therefore, from the computation time point of view, the BPM appears to be more efficient than the two-scale IEM, as was the case for the evaluation of the effective Fresnel coefficient for the zeroth-order term.

5. Conclusion

[50] We have compared two models for scattering of electromagnetic waves by rough surfaces, the BPM and the IEM, applicable to the case of a composite surface such as the ocean. We first reviewed the models and derived a two-scale extension of the IEM. The total σpqo appears as the sum of a zeroth-order and a first-order term in the BPM, while a series expression up to any order can be formulated in the two-scale IEM. An expression of multiple-scattering contributions to σpqo can also be found in the two-scale IEM but has not been implemented. Next, we proposed a method that takes the instrumental configuration and the roughness of the surface into account for the two-scale decomposition of the surface, in order to satisfy the constraints on each scale for the application of the electromagnetic model. Finally, we compared the BPM and the two-scale IEM for various instrumental configurations and roughness conditions in the case of the ocean surface.

[51] In most configurations (frequency, polarization, view angle, roughness conditions) the two models give very close results. Slight differences appear at large incidence angle, high frequency, and strong wind. For the zeroth-order term of σpqo we showed that the difference between the effective Fresnel coefficients given by the BPM and the two-scale IEM can be considered as negligible compared to the usual accuracy of radar measurements. For the higher-order terms we showed that the contribution of the second and higher orders in the two-scale IEM is small in the VV and VH polarization combinations, while it becomes important in HH at large incidence angle as the surface roughness or the frequency increases.

[52] We also showed that when a sufficient number of terms are taken into account in the two-scale IEM, the difference between the BPM and the IEM solutions tends to be small, and even negligible in most configurations. From the computation time point of view, using the two-scale IEM, the evaluation of the effective Fresnel coefficient for the zeroth-order term is complicated, as well as the computation of the spectra of the nth power of the displacements correlation function for the second- and higher-order terms. Therefore we conclude that the BPM appears more efficient than the two-scale IEM and is the best candidate for operational algorithms that have to provide scattering coefficients.

Appendix A: Expression of Multiple-Scattering σpqmno Terms

[53] Using the integral equation method to determine the electromagnetic fields on the surface, and the separation of the surface characteristic function into large and small scales, it can be shown that the multiple-scattering terms in (31) are given by

equation image

with

equation image

Ancillary