EM models for evaluating rain perturbation on the NRCS of the sea surface observed near nadir-Radar, Sonar and Navigation, IEE Proceedings -

The authors address the problem of evaluating the normalised radar cross section gURCS) of the sea surface perturbed by the joint effect of rain and wind, when observed close to nadir. They present a model, based on the full wave theory, for evaluating such an NRCS when varying polarisation, frequency and incidence angle (not far from nadir) for different values of wind velocity and of the root mean square height of the corrugation induced by rainfall. Some comparisons are made with the integral equation model results in the case of rain-induced corrugation alone. The two models are found to be in good agreement. In addition, partial comparisons made with experimental data suggest that the proposed model is well grounded and exploitable for application. It is indeed expected that the model can be exploited to improve precipitation measurements over the sea through spaceborne rain radar and to improve wind measurements using scatterometers in the presence of rain.


Introduction
Signals backscattered by the sea depend on several physical phenomena, but mainly on wind-and rainfallinduced corrugation. While the influence of wind on sea normalised radar cross section (NRCS) has been investigated in depth, the effects of rainfall are not yet well known. However, it has been shown they are not negligible [l]. Therefore, in principle, given a suitable analytical model (possibly taking advantage of specific experimental results) to represent changes of the NRCS of the water surface under the effect of rainfall, rainfall intensity could just be retrieved by measuring the sea NRCS. Furthermore, as shown in the companion paper [2], such a model can be exploited to improve the estimate of vertical profiles of rainfall rate through spaceborne weather radar. Raindrops falling on a water surface generate cavities with a crown, that collapse forming a vertical stalk of water, which subsides to spawn rings of gravity-capillary waves that propagate outward [3,41. Analysis of radar data provides evidence that the ring-waves are the dominant feature contributing to the backscattered power for incidence angles not far from nadir [l, 41, while at grazing angles stalks are the dominant feature [3]. Though the case of nadir incidence is of particular interest for spaceborne weather radars, very few results are available in the literature about the influence of rainfall rate on the power backscattered by the water surface; in particular, some results have been published on the relationship between rainfall rate and the statistic parameters of the water surface [1, 51. In this paper, we present the application of an electromagnetic (EM) model for computing the sea surface NRCS, with the main purpose of pointing out the effects of rainfall when such a surface is jointly corrugated by wind. We limit our analysis to incidence angles not too far from nadir, accounting only for the distributed ring-waves phenomenon. Due to the total lack of experimental data referring to the third case, we proceed by analysing separately the three different cases of: (a) rainfall only, (b) wind only, (c) wind plus rainfall. Thus, we first refer to experimental results obtained through artificial rain [l] and derive a plausible relationship between rainfall rate and surface roughness variance. Next, to predict the NRCS in case (a), we use the full wave model (FWM) [6, 71 and the experimentally derived sea roughness frequency spectrum reported in [5]. Then, we check consistency of FWM predictions with experimental data in case (b). Finally, theoretical FWM results are presented for case (c) at K, band.
Though the FWM has attracted some controversy in the past , the formulation adopted here [6, 71 was found in agreement with the integral equation model (IEM) theory [12-141 and with some experimental results. The purpose of this paper is not a validation of the FWM, but the formulation of a model able to predict satisfactorily the sea NRCS for a wide range of wind velocities and rainfall rates. In Sections 4 and 5.1 FWM is shown to provide accurate results when these two phenomena act separately. Section 4 deals with case (a); for cross-validation purposes we compare the FWM NRCS with the IEM NRCS at both C and K, bands when varying incidence angle, polarisation and root mean square (rms) height of the sea surface, obtaining excellent agreement, in particular for low incidence angles. Incidentally, a comparison between FWM and IEM has never before been presented for a two-dimensional azimuth symmetric rough surface. In Section 5.1, good agreement between FWM results and experimental data by Schroeder et al. [15] is shown for case (b). As mentioned above, in case (c), no comparison was possible with experimental data reported in the literature, however the FWM was found particularly suitable for sea NRCS prediction under the effect of these two statistically independent effects. Indeed, as shown in Section 5, the conditions that corrugated surfaces must meet to be implemented in a unified manner through the FWM according to [7], are largely satisfied.

Physical characterisation of the water surface corrugated by rainfall only
In this Section we describe the solution adopted for water surface modelling in the presence of rainfall only, allowing us to relate both the rms height and the spatial wave spectrum of the sea surface to rainfall rate.

I Surface roughness height distribution
Water surface roughness is modelled through a zero mean Gaussian height distribution, with variance (h2). An approximate relationship between (A2) and rainfall rate is obtained referring to experimental results carried out with artificial rain [I]. In such experiments, a fixed raindrop size (2.8" diameter) was used with drops falling from l m above the water surface, thus hitting the water surface with a velocity (4.4m/s) lower than the terminal velocity of real rain. Through linear regression applied to those results (see Table 1 in [I]), we derived the following relationship between rainfall intensity (mm/h) and height variance (mm2): ( P ) = 11.6 x 10-38' (1) where R' is the rainfall rate of that experiment.
Our goal is to extrapolate such results to derive an analogous relationship for drops falling at their terminal velocity. As a consequence of the precipitation process, ring-waves are generated by transfer of some fraction q of the kinetic energy of the falling raindrops. We assumed that the same fraction q is transferred to generate the ring-wave phenomenon also when raindrops fall with their terminal velocity, and have different diameters. Thus, the same energy transfer for the generation of ring-waves (and the same (h2)) corresponds to different equivalent rainfall rates, depending on the kind of rainfall. We used this assumption to determine expressions, similar to eqn. 1, for the two 'natural' rainfall cases (a) and (b) that follow.
In the experiment of [ 11, each raindrop contributed to the total energy with a kinetic energy Ek = (1/2)mw2 = 1.113 x lo4 J, where m is the mass of the drop (kg) and w is its impact velocity. The water volume fallen over a lm2 area was V = IO6R' mm3/h. Hence, the number of drops fallen per hour was Nd2 Accordingly, the total kinetic energy reaching the unit area per hour was Etot = Nd2.8) . (112) . m . w2 = 9.679R'. Using eqn. 1 we get and, as implicitly suggested in [l], eqn. 2 is assumed independent of the type of rainfall and drop size distribution (DSD). Let us consider now two types of DSD:  [16]); its kinetic energy is the same as that of a 2.8" diameter raindrop, but carries a different water content. Repeating the above computations, we get Nd2.l) = V/Vd2.1) = 2.06 x 105R, with R in mmh. The total kinetic energy per unit area and per hour is now 1 From eqns. 2 and 3 we get the relationship between the rms height h(rms) in mm and R, plotted in Fig. 1 ( [16]. The total kinetic energy for drops at terminal velocity is where pw is the water density (kg/m3). The second curve of Fig. 1 is obtained by computing eqn. 5 against R, and using eqn. 2. It is reasonably believed that the above simplifying assumptions hold for not too high rainfall rates; in any case, in the following we suppose eqn. 4 to be approximately valid up to 100mdh. The two curves of Fig. 1 do not differ much from each other, indicating that the relationship obtained is not very sensitive to the adopted DSD. This feature is desirable, since it makes the modelling approach robust with respect to the DSD model.

Spatial wave spectrum
As shown in [l] and [4], the short-term transient (crowns and stalks) that follows the impulsive inputs due to falling drops does not affect the radar cross section at incidence angles close to nadir, but only the long-term transient (ring-waves) is relevant. Thus, we adopted the ring-wave frequency spectrum described in [5], converting it to the ring-wave wavenumber spectrum. Under the hypothesis that the ring-wave phenomenon is independent of the angular direction (azimuth-symmetric surface), such spectra can be expressed in terms of the radial wavenumber spectrum 221 (RWS), as shown in the following Section. As suggested in [SI, the dependence of the RWS on rainfall intensity can indeed be approximated as follows: where S(K) is the RWS normalised with respect to (h2). S(K) is defined by S(K)dK = @ (co)dco and by the appropriate capillary-gravity wave dispersion formula [ 17, 181, where @(U) is the normalised folded frequency spectrum given in [SI (defined for o > 0) (6) where& -6Hz, Af -SHz and qo depends on R. Fig. 2 (after [5]) shows how such a log-Gaussian model fits the aforementioned experimental data relative to a normalised frequency spectrum. Here we report, for ease of reference, the basic expressions of the adopted EM model, based on the FWM for the backscatter case, described in detail in [6, 71, utilising the same original notation. The two-dimensional surface spectrum for an azimuth-symmetric surface, disregarding a normalisation factor, can be expressed as [17, 181

SR(K) = ( h 2 ) S ( K )
where W,(K) is the two-dimensional spectrum to be used in the FWM. If the mean square slope CT?~ of the rough surface is small, say o?l < 0.2, the effects of height-slope correlation can be neglected [19]; 02, is then calculated as in [6,191, and for ring-waves corrugation with spectrum W,(K) we obtained os? = 3.34 x 10-2(h2). Thus, condition c& < 0.2 is met for all values of (h2) considered in the following Section. Under this hypothesis, the unified FWM expressions of the likeand cross-polarised backscattering cross sections &' become [6,19,201 a P q = F ( i i f , iiZ)Q(iif, ii') (9) where p and q stand for the scattered and incident 228 wave polarisations (V or H), respectively. Furthermore  (12) 1"' are defined as in [6]. Expressions for the coefficients DPq in eqn. 11 are reported in the Appendix. In the backscattering case, ' i~ = v, 9, where k is the free space wavenumber of the EM wave [21], and ii' = -Ef, where iif = -sin6 . ax + cos6 . XY is the unit vector pointing from the surface towards the radar; the unit vectors si,, si, are, respectively, parallel and perpendicular to the plane of the surface and 6 is the incidence angle. Eqn. 12 reports the Q( iif, iiq func- where (h2) is the mean square surface height, and (hh') = (h2)p(rd) is the azimuth-symmetric surface autocorrelation function.
In our analysis, we disregard the shadow function considered in [6], since we refer only to incidence angles not too far from nadir and to a small rms surface slope. The correlation coefficient p(rd) is related to the normalised spectrum through [6,191 where W,(v,) is the normalised wavenumber spectrum, defined as in [6].

Evaluation of the NRCS of the sea surface
We report here the NRCS computed through the FWM in the case of a sea surface perturbed by rain only, for different values of frequency, incidence angle and rms height. Correspondingly, we also report the NRCS computed through the IEM. The purpose is to support and strengthen the cited experimental results in [l] and [SI with some EM modelling effort. A good agreement between FWM and IEM, as that shown below, can indeed be interpreted as a cross-validation of their capability of predicting the radar backscatter, accounting for rain effects. Concerning the IEM, since a complete study about its range of validity has not been carried out for dielectric surfaces, we considered one of the most restrictive conditions, namely a Gaussian correlation function [12]. Thus, if L is the correlation length, we required (kh(ums)) . (kL) < l.2dsr, which is certainly met by the adopted spectrum.
The results refer to 5.6 and 13.75GHz. The dielectric constant against frequency of the EM waves is in [22, 231; we have sr = 65 -j36 and =43 -j40 for 5.6 and 13.75GHz, respectively. In Figs. 3 and 4, the NRCS against incidence angle is plotted for both the copolar horizontal (HH) and vertical (VV) returns at 13.75GHz. Dots and lines refer to FWM and IEM, respectively. At all incidence angles the agreement is fairly good both for lmm rms height (Fig. 3), and for 2mm rms height (Fig. 4). Near nadir (incidence angles close to 5-10") the NRCS is lower, in agreement with the shape of the spatial wavenumber spectrum, namely decreasing for decreasing wavenumbers [5]. However, at K, band the NRCS decreases when the incidence angle exceeds 5-10", owing to the absence of high wavenumber components in the spectrum (see Fig. 2). Note also that, when sufficiently far from nadir, the VV response is larger than the HH.  In Figs. 5 and 6 the NRCS is plotted against rms height for both frequencies and for both HH and VV returns. The incidence angle is 10" in Fig. 5 and 30" in Fig. 6. h(rms) ranges from 0.1-2.5mm, and the NRCS increases with it. The corresponding rainfall intensity can be obtained from Fig. 1. Similar results were obtained in the cited K, band experiment at 30" incidence angle [l]. At both bands the same trend is observed, but at K, band the NRCS is greater, being the EM wavelength closer to the rms height of the surface.

by both wind and rain
The FWM was applied assuming that the surface roughness is the superposition of two statistically independent random processes h, and h induced by rainfall and wind, respectively. This assumption is mainly based on the different scale of the two perturbations: (e) The wind-and rainfall-induced perturbations have quite different correlation lengths. These are some metres for wind [6, 241 and a few centimetres for rainfall [5].
(4 Statistically averaging the above effects over the sea surface illuminated by radar (in a stationary field), as implicit for cross-section evaluation, should improve conditions for the application of the followed approach. Therefore, besides independence, the following hypotheses were exploited: (i) The large-scale process hw is ridden by the smallscale process hR, defined through a local reference system jointly with h, . At a generic point of the surface, h, is represented by a set of three orthogonal unit vectors that depend upon the local slope of the wind roughness h (ii) The correlation length L , of h , is much larger than the correlation length L R of h, . Exploiting such hypotheses, since os? < 0.2 for both surfaces, the CF and JCF are properly decomposed [7], and the NRCS expression changes from eqn. 9 to [25, 261 (16) Qw and QR depend on the statistics of the phase of the EM wave, as determined by the height distribution of the rough surface relevant to wind and rain, respectively. Qw is given by eqn. 12, substituting the wind surface height hw in eqn. 13. We state in advance that, as specified in Section 5.1, we considered an azimuthsymmetric approximation of the sea surface wind corrugation spectrum. In fact, examining the NRCS data reported by Schroeder et al. [15], for incidence angles ranging from 0-60" and different wind velocities, it can be noticed that the difference between up-, down-and cross-wind measurements becomes remarkable only for angles larger than 20". Thus, for incidence angles close to nadir the natural asymmetry of the wind-induced corrugation can be neglected. For Q R we have instead x Jo ('UwzzTwd)TwddTwd (17) where the CF and JCF are now calculated by means of the height distribution h~ with respect to the local vector ii normal to h , (and not with respect to ay as in eqn. 12). The integration variable Y, d is also the argument of the correlation function appearing in the JCF of eqn. 17; the subscript w recalls that it rides h, . The term v , , , ] is the projection of v on the local plane tangent to hw [7]. The result is averaged over the slopes of the large-scale process. Note that Q R ( i i f , ii', ii) is weighted by the slope of the largescale surface while Qw is not, as a consequence of the fact that h, rides h, . This corresponds to computing 230 the ring-waves contribution through a statistical average over the slopes of the wind-roughened surface. Also in this case, shadowing effects were not accounted for. Finally, notice that the EM model requires the height standard deviation hR(?") of the small-scale process h,, that can be related to the rain rate R through the curves of Fig. 1

I Evaluation of the NRCS of the sea surface corrugated by wind and rainfall at nadir incidence
As mentioned in Section 1, nadir incidence experimental NRCS data for a sea surface perturbed by both wind and rain are not available in the literature. However, a partial evaluation of the applicability of the described FWM-based model was possible, by comparing its predictions with experimental data obtained by Schroeder et al. at 13.9GHz in the case of a sea surface perturbed by wind only [15]. Fig. 7 reports such a comparison. Measurement values (dots) were taken from the regression line in [ 151 and were made in two distinct experiments; the two curves represent two predictions based on the approximated Pierson-Moskowitz spectrum [6, 271 and on the spectrum reported by Apel in [24]. In particular, when using the latter, the azimuth dependence was averaged to deal with an azimuth-symmetric spectrum consistent with the assumptions made in this paper. As expected, the Apel spectrum fits the actual NRCS behaviour better than the approximated Pierson-Moskowitz spectrum, even if the predicted NRCS is slightly larger. Schroeder et al. [15] indeed noted a similar discrepancy by comparing their measurements with the SASS I model. However, the difference is of the order of the uncertainty of the measurements, and may also be attributed to the Apel spectrum. The approximated Pierson-Moskowitz spectrum is not accurate for nadir incidence, but we found that its predictions off nadir are in rather good agreement with the experimental results.
Our general theoretical results are finally summarised in Fig. 8, that reports the sea surface NRCS against rainfall rate R at 13.9GHz, for some wind velocities and nadir incidence obtained using the FWM-based model and the Apel spectrum [24]. We point out that the NRCS variations due to rainfall rate are of the same order of magnitude as those due to wind velocity. This justifies the inclusion of rainfall-induced corrugation in the EM model. In fact, trying to predict sea NRCS without accounting for rainfall corrugation may easily lead to bias errors comparable with those that may be caused by an incorrect choice of the wind spectrum.

Conclusions
The characterisation of the NRCS of the sea surface, when perturbed by rainfall only and by the joint action of wind and rainfall was the main objective of this paper. An EM model has been considered, based on the FWM theory.
To analyse phenomena related to rainfall-induced variations of NRCS, it was necessary to derive a relationship between the rainfall intensity and the rms height of the sea surface, this result is not available in the literature for natural rain. By exploiting results referring to an artificial rainfall experiment and energetic considerations, we obtained an approximate theoretical relationship for two different DSD models.
Considering rainfall-induced corrugation only, the FWM NRCS was compared with the IEM NRCS, and a very good agreement was shown at 5.6 and 13.75GHz. Incidentally, this can be considered as a confirmation of the consistency of both models with respect to the problem examined. Typically expected NRCS behaviour has been highlighted or confirmed; near nadir, the predicted echo is smaller than at greater incidence angles, and it increases as the surface rms height increases. It has been shown that in the case of sea surface perturbed by rainfall-induced ring-waves, the VV response is larger than the HH, at both C and K, bands, and that such a difference increases with increasing incidence angle. Moreover, the NRCS is always larger at K, band than at C band. As a complementary analysis, the predicted NRCS by the FWM for the sea surface corrugated by only wind has been compared with experimental data, with rather good results. Finally, the case of induced corrugation by the joint action of wind and rain has been considered, showing that additional roughness induced by rainfall cannot be considered as a side issue in the backscatter mechanisms at the K, band. Actually, experiments would be IEE Proc-Radar, Sonar Navig., Vol. 145, No. 4, August 1998 needed to determine errors due to the EM model assumptions made in Section 5, as well as possible limitations associated with the use of high-frequency bands. Indeed, major model limitations are expected to be related to strong andlor nonstationary winds.
All the results summarised above were obtained disregarding the surface damping effects, typically induced by heavy rainfall, that cause an increase of NRCS at nadir incidence [28,291. This problem, requiring a quite complex physical analysis and EM characterisation, is worth further investigation, necessarily supported by experimental measurements. At the Ku band it is however expected that, even when damping effects occur, ring-waves should play a dominant role since their size is comparable with the EM wavelength. This seems to be confirmed by the analysis of TOPEX/Poseidon altimeter data [30].
The proposed EM model is exploitable by algorithms devoted to rainfall retrieval over the sea surface, as discussed in the companion paper [2]. Its use for revealing features (also polarimetric) and for separating sea surface radar return from volumetric rainfall return is foreseeable. Also, it could be exploited in scatterometry to improveievaluate performance of algorithms for wind speed computation, to highlight the bias effects due to rain roughness, or in altimetry to predict effects of rainfall (e.g. to estimate K, band attenuation by atmospheric liquid water, as suggested in [30]).