1. Introduction

The precipitation radar (PR) at 13.8 GHz on the Tropical Rainfall Measuring Mission (TRMM) satellite has played a very significant role in the characterization of clouds and precipitation particles aloft (Kummerow et al. 1998; Kozu et al. 2001). Future spaceborne radars are envisaged to operate at higher frequencies such as 35.5 GHz for the radar planned for the Global Precipitation Measurement (GPM) satellite and its European version (EGPM; Mugnai 2003). The “CloudSat” radar at 94 GHz is already deployed to investigate the vertical structure of clouds (Stephens et al. 2002).

The interpretation of radar scans is always made on the assumption of negligible multiple-scattering (MS) effects. However, as the frequency of operations of the PR is raised, some factors may come into play that have the potential to enhance the significance of higher order of scattering in the radar echo. Among such factors are the increase in the optical thickness of the medium, the decrease of the mean free path of radiation (defined as the inverse of the extinction coefficient) toward values comparable to or smaller than the radar footprint dimension, the increase of scattering versus absorption cross sections [e.g., for raindrops see Fig. 3.6 in Gasiewski (1993)], and the modification of hydrometeor scattering-phase functions, which become more peaked in the forward direction. This factor is highlighted by the increase of the asymmetry parameter g [e.g., Fig. 3.7 in Gasiewski (1993) and center-left panels in Figs. 5 and 7–9 of Battaglia et al. (2006), hereinafter referred to as Part II]. Because previous studies such as Ito et al. (1995), Marzano et al. (2003), Kobayashi et al. (2004, 2005), and Battaglia et al. (2005a) showed that the borderline of relevance for MS effects lies between the new-generation spaceborne radars (at K_a and K_u band) and the TRMM PR at X band (see Kozu et al. 2001), a detailed investigation of the problem of MS at frequencies in the millimeter region is a timely endeavor.

The MS phenomenon in active remote sensing has been well studied within the lidar community (see Bissonnette et al. 1995; Bissonnette 1996, and references therein). In typical lidar configuration, this phenomenon is enhanced by the higher optical thickness and higher single-scattering (SS) albedos typical for visible and near-infrared regions. Because of the small horizontal footprint of lidar systems (fields of view are within 0.1–5 mrad), the MS effect is enhanced by the strongly peaked phase function that is characteristic of ice/water particles in the optical-scattering regime. This property allows the analysis to focus on scattering in the forward direction only and to derive a closed form of the MS lidar equation (Bissonnette 1996).

When radar systems are used for active remote sensing of precipitation, it is observed that very few studies on MS effects have been carried out (Hubbert and Bringi 2000; Oguchi et al. 1994; Ito et al. 1995; Marzano et al. 2003; Kobayashi et al. 2005). Hubbert and Bringi (2000) developed a three-body scatter model to explain flare echoes or “hail spikes,” which are special signatures in reflectivity and differential reflectivity observed at S-band and C-band signals in hailstorms. The numerical model is restricted to three-body scatter, and it cannot be extended to general MS theory. A similar model aimed at validating the second order of scattering has been developed in our general MS model (section 4b). Note also that the mirror-image return, frequently observed in most of the TRMM PR profiles over ocean (e.g., Olson et al. 2001), is based on an MS mechanism that involves surface, rain, and then surface.

Oguchi et al. (1994) and Marzano et al. (2003) evaluated MS effects and their impact on rainfall-rate estimates. These studies concluded that MS signals are significant, especially in heavy-rain cases. In particular, Marzano et al. (2003) showed that, in the presence of strongly attenuating media, MS contributions will enhance the detection of rain by partly overcoming the apparent path attenuation. This effect has an immediate impact on rainfall retrieval algorithms (e.g., Testud et al. 1992; Marzoug and Amayenc 1994; Iguchi et al. 2000). In fact, the correction of attenuation is important in spaceborne monitoring of rainfall for which the frequencies used are higher but the path through the rain is fairly short. The studies by Oguchi et al. (1994) and Marzano et al. (2003), however, did not explicitly account for the effect of the antenna pattern and of the backscattering enhancement on radiation undergoing MS. In contrast to these theoretical studies that considered infinitely extended plane waves (which assumption overestimates the MS effect, as will be shown later in this study), more recent studies have taken into account the antenna pattern effect. In the study by Kobayashi et al. (2004, 2005), reductions in MS effects were derived by considering spherical waves with a Gaussian antenna pattern, including depolarizing effects. Their conclusion can be summarized as follows. The differences from the plane-wave results are not significant when the optical thickness is small. However, when the latter increases, the differences become significant and essentially depend on the ratio of radar footprint radius to the mean free path of hydrometeors. In this regime, for a radar footprint that is smaller than the mean free path, the MS (inclusive of the backscattering enhancement) in reflectivity corresponding to spherical waves is significantly less pronounced than in the case of the plane-wave theory. Whereas the theory of Kobayashi et al. (2005) treats a uniform single-layered cloud illuminated by continuous waves, Battaglia et al. (2005a) simulated the MS for random media in multiple cloud layers with pulsed radars. The findings by Kobayashi et al. (2004, 2005) were confirmed in Battaglia et al. (2005a) in which the reduction in MS due to illumination by a finite beamwidth is demonstrated for a PR down-looking configuration for heavy convective rain. In particular, it was shown how the MS effect vanishes when passing from a typical PR footprint diameter (on the order of 4–5 km) to very high resolution airborne systems (on the order of hundreds of meters) at 35 GHz. As a conclusion from these results, infinitely extended plane-wave computations can be used only to pinpoint when MS is negligible; when MS effects are relevant, this kind of computation is accurate only when the radar footprint radius is much larger than the mean free path.

In this two-part series of papers, attention is focused on the two frequencies to be used in the planned GPM core satellite: 13.8 and 35.5 GHz. First, a general insight into MS effects is given by discussing the impacts of the different factors involved in the problem (e.g., scattering properties, radar configuration, or antenna pattern). Then, in Part II estimates of the MS effects at the two GPM frequencies are provided when using both the configurations of the envisaged spaceborne radars and those set up in preliminary airborne campaigns.

In this first part, a numerical Monte Carlo (MC) model suitable for a pulsed active-sensor MS evaluation is described and validated. In section 2, the radar equation, which is extended to account for MS contributions, is briefly reviewed. Section 3 describes the MC code developed to evaluate the MS effect in radar echoes. Using the model, the results obtained for the first and second order of scattering are validated in section 4. In section 5, the model is applied to a uniform homogeneous layer to explain the general features of the MS signal. Conclusions are drawn in section 6.

2. The radar equation in the presence of multiple scattering

The classical radar equation (e.g., Doviak and Zrnić 1984; Bringi and Chandrasekar 2001) usually provides the apparent reflectivity as a function of SS contributions only. Taking an approach similar to that adopted by Marzano et al. (2003), how to generalize the radar equation to include MS components is briefly summarized. Here emphasis is on antenna-related quantities because of their importance in the following developments.

The apparent (or effectively measured) received power due to the range gate at distance r in the presence of MS (higher than the SS contributions) can be generally expressed (Tsang et al. 1985) as

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with 〈I_a(r, Ω̂)〉 being the mean value of the apparent received specific intensity. Here, F_n(Ω̂) is the normalized (in the sense that the maximum is equal to 1) antenna pattern (equal to the square of the modulus of the antenna field pattern); G(Ω̂) = G₀ f_n(Ω̂) is the antenna gain, with G₀ = (4π)/Ω_p being the maximum directive gain and Ω_p = ∫F_n(Ω̂) dΩ being the antenna pattern solid angle. The maximum antenna equivalent area is A_e0 = λ²/(4π)G₀. When only the first order of scattering is retained in the apparent specific intensity, the single- (back-) scattering power is obtained as

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The integral over the solid angle on the right-hand side of Eq. (2) can be simplified under the hypothesis that the SS specific intensity 〈I^SS_a(r, Ω̂)〉 is constant within the antenna main lobe. This condition is true in particular when dealing with a pencil beam, hereinafter referred to as the SS pencil-beam (SSpb) approximation. When carrying out the integral, Eq. (2) reduces to (Bringi and Chandrasekar 2001)

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where λ is the wavelength, P_T is the transmitted power, Δr is the range resolution, and Z^SSpb_a is the SS apparent radar reflectivity at range r and is related to the equivalent reflectivity factor Z_e by

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with |K|², σ_back, and N(D) being the dielectric factor, the backscattering cross section, and the size distribution of the scatterers contained inside the radar backscattering volume, respectively. In the simulations, the dielectric factor is always normalized to that of water at 0°C (assumed to be equal to 0.93 for all frequencies considered hereinafter). Equation (4) explicitly defines also the two-way attenuation A_2-way(r) at range r.

In Eq. (3), Ω_2A is the two-way main-lobe solid angle given by

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For instance, a Gaussian antenna has an antenna pattern of the form

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where η and ξ are the angles in the vertical and horizontal antenna planes and (η_3dB, ξ_3dB) are the respective beamwidths at 3 dB. When the Gaussian antenna is narrow, it follows that

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By comparing Eqs. (1) and (3) it is therefore possible to define an apparent MS reflectivity as

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In a similar way, by making use of Eq. (2) Z^SS_a (≤Z^MS_a) can be computed by Eq. (8) with the single-scattering contribution I^SS_a instead of I_a.

Last, note that Eq. (8) further simplifies to G₀Ω_2A = 2π for a Gaussian antenna pattern and to G₀Ω_2A = 4π for pencil-beam-like antennas.

3. A Monte Carlo approach to simulate the multiple-scattering reflectivity

To compute the apparent MS reflectivity defined by Eq. (8), it is necessary to evaluate the received specific intensity in terms of all relevant scattering orders, as depicted in Fig. 1. To compute Z^MS_a, the vector radiative transfer equation has been solved by using a sophisticated MC solution technique (Battaglia et al. 2005b). When reverting to radiative transfer theory, the present MC approach cannot account for the backscattering enhancement phenomenon (see Tsang et al. 1985, chapter 5) because it can only be described adequately by wave theory. For instance, at the second order of scattering the model accounts for the second-order ladder term only [i.e., the conventional MS term as illustrated in diagram 1b in Kobayashi et al. (2004)] but does not for the second-order cross term [i.e., the contribution from interference of two ray paths mutually satisfying the condition of time-reversal paths as illustrated in diagram 2b in Kobayashi et al. (2004)]. This condition will lead to an underestimation of the MS effect (see Kobayashi et al. 2004, their Fig. 5). However, when considering spaceborne systems, the high-speed platform motion will generally break the right backscattering condition. Conversely, the treatment of scattering order higher than 2 is much easier in our frame.

Instead of using the backward MC technique of Roberti and Kummerow (1999), as done by Marzano et al. (2003), we have modified the forward MC by Battaglia and Mantovani (2005). In the case of an active sensor response simulation, the backward and the forward approaches give identical statistics when the same number of photons is launched. The forward code has, however, the advantage of a complete polarimetric treatment and the capability of accommodating nonisotropic media. Both aspects are crucial for the simulation of radar polarimetric variables. The MC technique was adopted in this study because of its flexibility in simulating both 3D-structured clouds and arbitrary antenna patterns. Because of the 3D nature of the problem as introduced by the presence of the antenna pattern, it is not possible to simplify the radiative transfer equation to a 1D approximation (which has the possibility of analytic and fast codes) as was done by Ferrauto et al. (2004).

The forward MC code by Battaglia and Mantovani (2005) (which was available online at http://www.meteo.uni-bonn.de/mitarbeiter/battaglia/) was developed for passive remote sensing applications. The algorithm has been modified to simulate the active sensor response by keeping track of the distance traveled by each photon and by allowing the radar to be the only source of radiation. As illustrated in Fig. 1, the radar is characterized by

1. a locally transmitting r_tr and receiving position r_rec (to allow for bistatic antenna configurations)—the radar antennas can be positioned at any altitude above, below, or within the cloud, thus allowing for ground-based, spaceborne, and airborne configurations;

2. two pointing angles Ω̂^tr/rec_rad or, equivalent, two angles (Θ^tr/rec_rad, Φ^tr/rec_rad) that correspond to the direction of the main lobe of the transmitting and of the receiving antenna, whose unit vector is expressed as

   

   

   View Expanded

   

   with the vertical and horizontal polarization directions being

   

   

   View Expanded

   

3. two antenna pattern functions F^tr/rec_n(Ω̂) (that can be different in the transmitting and in the receiving segment), typically identified by the two beamwidths defined in Eq. (6);

4. a transmission and a receiving polarization state (like vertical, horizontal, left/right circular polarized); and

5. a range resolution Δr = cΔt/2, with Δt being the pulse duration.

The radar outgoing radiation is simulated by a number N_T of photons per unit time, with N_T high enough to represent the stochastic variability of all processes involved in photon propagation and interactions when released from the transmitting radar antenna with an appropriate polarization state I⁽⁰⁾(r_tr) [the Stokes basis is the one defined in Tsang et al. (1985), e.g., for a vertical-polarization transmitting radar I⁽⁰⁾(r_tr) = (1.0, 0.0, 0.0, 0.0)]. Note that the overbar indicates the four-dimensional Stokes vector. Each photon is considered to carry a unitary energy (so that N_T corresponds to the total emitted power P_T), and the angular distribution function is that provided by the antenna pattern. For instance, to reproduce the Gaussian pattern of Eq. (6) it is sufficient (House and Avery 1969) to select two uniform random numbers R₁ and R₂ and to define

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with σ_η = η_3DB/8 log2 and σ_ξ = ξ_3DB/8 log2, so that ξ and η are Gaussian-distributed random numbers with zero mean and standard deviations equal to σ_η and σ_ξ. The outgoing photon direction can be identified by

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where a₁ = x₁/(1 + x²₁ + x²₂)^1/2, a₂ = x₂/(1 + x²₁ + x²₂)^1/2, and a₃ = 1/(1 + x²₁ + x²₂)^1/2, with x₁ = tanη and x₂ = tanξ.

By so doing, all radiance field properties can be computed from statistical properties of the photon density. For instance, when the radiance in a particular direction Ω̂ at a certain point r is looked for, it is sufficient to take a surface ΔΣ centered in r and normal to Ω̂, and to count the number of photons n that cross the surface at Ω̂ within a small solid angle ΔΩ. The MC stochastic estimation of the radiance will be

View Expanded

In the code, each photon is traced inside the medium: all stochastic processes (distance to collision, probability of scattering, direction of scattered photons, etc.) are simulated as in the MC version for passive sensors and the associated Stokes vectors are properly modified. Biasing techniques are applied to prevent the photons from escaping from the medium and to avoid absorption events (see section 5.2 in Battaglia and Mantovani 2005).

Each time a k-order scattering event occurs at point r^(k)_scat, the algorithm computes the contribution to the received intensity. Because we are dealing with an active sensor, this k-order scattering contribution has to be assigned to a range bin. Therefore, the apparent range is computed by adding the total distance traveled by the photon from the transmitting antenna to the interaction point and the distance between the interaction point and the receiver antenna divided by two:

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This distance will correspond to a particular range bin encompassing all ranges between r − Δr/2 and r + Δr/2. The k-order-of-scattering contribution to the Stokes vector received by the radar for a particular photon (j) will be

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𝗦_scat represents the normalized phase matrix (see definition in Mishchenko et al. 2000) for the medium at the point r^(k)_scat relative to the incoming direction of the photon before the k-order scattering Ω̂^(k)_i and to the scattered direction toward the receiving antenna Ω̂^(k)_υ (unit vector corresponding to the vector difference r_rec − r^(k)_scat). It is obvious that, at the first scattering order for monostatic radar, Ω̂⁽¹⁾_υ = −Ω̂⁽¹⁾_i. In Eq. (10), the F^rec_n[Ω̂^(k)_υ] scalar term takes into account the antenna pattern in the receiving segment (this term suppresses all photons scattered to the radar receiver from outside the field of view). Term 𝗣_prop[r^(k)_scat → r_rec] is the 4 × 4 propagation matrix that accounts for the extinction of the signal scattered back to the radar receiver. When propagating in nonisotropic media, this propagation matrix will induce polarization effects as well (see section 5.3 in Battaglia and Mantovani 2005). When the photon intensity become smaller than a certain threshold, the tracing is terminated and a new photon is launched. By summing up over N_T photons and over the different orders of scattering, the algorithm provides an estimate of the mean apparent intensity received by the radar from range r as

View Expanded

with N_s being the maximum scattering order. When using N_s = 1, only SS effects are taken into account.

Equation (11) represents the statistical MC equivalent of the nondimensional term inside the brackets on the right side of Eq. (8), with the only difference being that it includes the four Stokes components. When the radar emits a horizontal wave and the horizontal (vertical) component is considered in the receiving segment, the copolar reflectivity Z^MS_hh (the cross-polar reflectivity Z^MS_hυ) will be computed by inserting Eq. (10) in Eq. (8). Therefore, parameters such as Z^MS_dr and linear depolarization ratio LDR^MS_υh are easily derived in the MC simulation as well. Although in this study we consider only isotropic media, because of MS effects cross-polar reflectivity signals can also of interest, as will be shown in section 5d.

4. Validation of the code

a. Single-scattering results

To validate the MC procedure, a very simple five-layer raining atmosphere is used. The five layers are each 1 km thick and contain raindrops with an exponential hydrometeor size distribution with fixed intercept as in the Marshall–Palmer size distribution (Marshall and Palmer 1948) and equivalent water contents equal to 0.3, 1.0, 1.8, 2.0, and 2.5 g m⁻³, with the intensity increasing downward. At 35.5 GHz, this configuration leads to equivalent reflectivities equal to 33.5, 40.5, 43.4, 43.9, and 45.0 dBZ and to attenuations equal to 1.2, 5.1, 9.8, 11.0, and 13.9 dB km⁻¹, respectively. From these values it is straightforward to reconstruct the SS pencil-beam apparent reflectivity profile by accounting for the two-way path attenuation. One detail worth mentioning concerns the attenuation contribution from the same range bin where the reflectivity is computed. In fact, although the reflectivity value is attributed to the center of a fixed range bin at a distance r, the attenuation is not that corresponding to the same distance: Eq. (4) is not exact. In contrast, suppose that the equivalent reflectivity factor is constant and equal to Z_e(r) in the radar bin of thickness Δr; then we can write

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with k_ext and τ_r→r+Δr = k_extΔr being the extinction coefficient and the optical thickness of the bin, respectively. When passing to logarithmic values, Eq. (12) provides an attenuation factor (dB) equal to

View Expanded

Note that only in the limit of small τ_r→r+Δr, [1 − exp(−2τ_r→r+Δr)]/2τ_r→r+Δr → exp(−τ_r→r+Δr) so that A_2-way(r) → 2 × 4.343(τ_{0→r−Δr/2} + 0.5τ_r→r+Δr) and the attenuation can be computed as the two-way attenuation relative to the center of the bin 2 × 4.343τ_0→r.

Figure 2 shows the departure from the exact analytical solution for Z^SS_a in the case of a spaceborne radar with a vertical resolution of 500 m and a footprint diameter of 1 km. To give an idea of the accuracy of the computations performed with the MC technique, in Fig. 2 the MC simulations have been performed for three different numbers of photons: 10², 10⁴, and 10⁶. Even in the presence of a strong attenuating profile one million photons are enough to reach an accuracy of better than 0.2 dB in the whole profile. The accuracy obviously deteriorates going deeper inside the cloud because of the reduced number of photons that successfully reach the deepest layers. In the extreme case when N_T = 100, no photon is able to reach the layer below 2 km. In fact, the optical thickness above this layer is approximately equal to 5 so that, on average, only N_T exp(−5) ≈ 0.6% of N_T photons reach that layer. This is the reason why the dashed-diamond line stops at 2.25-km altitude in Fig. 2. Based on the MC statistics of each range bin (which will depend on the selected Δr), the code internally computes the variance of 〈I_a(r)〉 defined in Eq. (11) for each range bin. This can be converted to an error (related to the MC noise) for each of the analyzed radar quantities. In Fig. 2, results for each simulation are plotted with a corresponding error bar that represents the 3-σ confidence level based on MC statistical noise.

b. Relative importance of the contribution from the second order of scattering

As a test, the importance of the second order of scattering (relative to the first order) has been evaluated in a homogeneous layer having constant extinction and SS albedo and a generic phase function p(Θ) [normalized by the condition ∫¹₋₁ p(Θ) d cos(Θ) = 2]. Note that Θ is the scattering angle, that is, the angle between the incident and the scattered direction. We expect that the second order of scattering becomes more relevant with deeper penetration into the layer; this contribution should also increase linearly with the SS albedo. For illustration, consider Fig. 3 in which the nadir down-looking radar is positioned at an altitude H_radar above the homogeneous layer (at the point F₁). Here we consider a monostatic radar, the antenna pattern of which is pencil-beam-like in the emitting segment and Gaussian in the receiving segment. At depths between z and z + Δz, the second-order-of-scattering effect results from an integral over all possible second order-of-scattering contributions by photons interacting the first time at P₁ and the second time at P₂. Note that P₂ has to be located between the two spheroids with the same foci F₁ and F₂ (located at the radar position and in P₁) and with the rotational symmetry axis length equal to 2a₁ = 2z − z₁ + H_rad and 2a₂ = 2(z + Δz) − z₁ + H_rad. In fact, by using the ellipse definition property, the path undergone by the photons emitted by the radar, scattered in P₁ and then scattered back to the radar in P₂, is between H_rad + z₁ + 2a₁ and H_rad + z₁ +2a₂, corresponding to an apparent range between H_rad + z and H_rad + z + Δz. The probability of a photon being scattered at a penetration depth between z₁ and z₁ + dz₁, dP(z₁), is proportional to the SS albedo ϖ and, according to Beer–Lambert–Bouguer law, to the factor k_ext exp(−k_extz₁)dz₁, that is,

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The fraction of photons being scattered in the ellipsoidal segment with polar angles between ψ and ψ + dψ will be

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with r(ψ) = and Δr(ψ) = MN (see Fig. 3 for definition of points M and N). This fraction of photons, when going back to the radar, undergoes a path attenuation equal to exp(−k_extΔξ), with Δξ = + and an antenna attenuation corresponding to the angle θ; the scattering angle is given by Θ_sca = θ + ψ so that a correction factor equal to p(Θ_sca)/(4π) also has to be included. Note that the photon scattered both at P₁ and P₂ is always less attenuated than the SS photons with the same apparent range gate. The medium path difference is equal to − , and it increases with θ.

Therefore, using azimuthal symmetry, the total fraction of photons first scattered at P₁ and then at any point will be

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Thus, the fraction of photons backscattered twice with an apparent range between z and z + Δz will be

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On the other hand, the fraction of photons being backscattered only once and with apparent range between z and z + Δz will undergo an additional z-long path inside the attenuating medium (but no antenna suppression), so that

View Expanded

where p denotes the phase function. The fraction of photons backscattered twice divided by the fraction backscattered once, that is, the ratio obtained by dividing Eq. (17) by Eq. (18), provides a way to assess the magnitude of the second order of scattering relative to the first order. As a first obvious consideration, this ratio has a dependence on ϖ, and the higher the SS albedo is, the higher is the effect.

The analytical model has been exploited to validate the second-order-of-scattering results as computed with the MC method. Figure 4 shows the ratio F[z, (2)]/{ϖF[z, (1)]} evaluated for different phase functions when considering an infinite-aperture antenna (so no antenna suppression factor is present). Figure 4 shows an almost perfect agreement between the analytical (line style) and the MC (symbol style) results. The disagreement at optical thickness higher than 5 is due to MC noise that becomes relevant at high optical thicknesses. In this case, the impact of the second order of scattering depends only on the optical thickness τ traveled inside the medium: the higher the optical thickness is, the higher is the MS effect. As an example (see Fig. 4), in an isotropic medium with an albedo equal to 0.1 the second order of scattering has almost the same magnitude as the SS at an optical thickness of 7 whereas in a Rayleigh medium with the same albedo it is almost one-half as much at an optical thickness of 6. To evaluate the effect of the increased forward scattering for higher microwave frequencies, different phase functions have been used in Fig. 4. In particular, we have included a Rayleigh (for unpolarized radiation), an isotropic, and a Heyney–Greenstein (H–G) phase function with different g parameters. For comparison purposes, and because of the difference between this and realistic phase functions in the backscattering region, we have included a phase function for a distribution of spherical ice crystals with the same g = 0.4. The simulations show that an increase in g causes first an increase in the MS contribution and then a decrease with the turning point for g around 0.5. For instance, at an optical thickness of 3 and a phase function with g = 0.2, 0.4, and 0.95, the corresponding magnitude of the second order of scattering is 1.57, 1.92, and 1.25 times the value of the corresponding isotropic phase function. In fact, as g increases, more photons contribute to the integration in Eq. (16) because photons that are scattered with scattering angle higher than π − ψ₁ (with ψ₁ = ) do not contribute to the integral in Eq. (16). At a certain point at which practically all radiation is scattered in P₁ with scattering angles lower than π − ψ₁, a further increase in g actually has a negative effect in terms of the magnitude of the second order of scattering. In effect, the scattering phase function p(Θ_sca) [which also enters the integral Eq. (16)] decreases monotonically with Θ_sca (and in particular has a minimum at Θ_sca = π). When the scattering at P₁ is mainly in the forward direction, the scattering angles in P₂ will be primarily close to π. The opposite behavior occurs when the scattering at P₁ distributes photons in all the shaded area in Fig. 3, when scattering angles as large as π/2 are allowed to happen in P₂. The balance between the two effects decreases the ratio F[z, (2)]/{ϖ F[z, (1)]} for high g (g ≥ 0.5) for the H–G-like phase functions. This turning point for the g value will move to higher values when the antenna pattern is considered. In this case, the effective area to be considered will be the intersection between the shaded area of Fig. 3 and the antenna pattern. In the hypothetical case of a phase function that scatters radiation only in the forward direction (and just allowing for a small fraction to be backscattered), we can actually forecast the ratio F[z, (2)]/{ϖ F[z, (1)]} to be equal to the optical thickness τ itself. In this case, the photons scattered twice are simply those that make the same exact path as the photons scattered once. Another factor should be taken into consideration: with more forward scattering, the differential attenuation between the first-order-scattered and second-order-scattered photons (which favors the second-order contribution) is less pronounced. However, this effect can be neglected when considering spaceborne configurations (for which the difference − is typically negligible).

Note that in Fig. 4, because of the different slopes, the Rayleigh phase function has less MS than the isotropic one—this is mainly due to the fact that the backscattering ratio p(Θ_sca)/p(π) is generally lower and because more photons are “lost” when scattered at scattering angles higher than π − ψ₁. The result labeled as “Rayleigh unpol” (“Rayleigh pol”) (see Fig. 7, described in section 5b), represents the scalar (vector) solution of the radiative transfer equation (but in both cases with unpolarized impinging radiation). When polarization is considered, the Rayleigh results are closer to the isotropic results. In a similar way, the real phase function gives less MS effect relative to that of the H–G with the same g; because of the backscattering peak of the real phase function, the backscattering ratio is typically lower in this case.

For an infinite antenna beamwidth (see Fig. 4), the MS tends to increase almost linearly with the optical thickness: in fact, no antenna factor suppresses the contributions from regions far away from the radar zenith foot. When the beamwidth is considered (Fig. 5), the increase with the optical thickness is strongly damped. When k_ext becomes sufficiently high, the antenna pattern is no longer important and the results tend to be equal to those for an infinite-beamwidth-antenna pattern (reproduced in Fig. 5 and indicated as k_ext = ∞ in the legend). Moreover, at the same τ, the effect is larger for media with higher k_ext because the antenna factor suppresses all contributions from regions far from nadir. The difference between the dotted lines and the other lines practically accounts for the departures between taking into account the antenna pattern or working in the 1D assumption like Marzano et al. (2003) (which generally overestimates the effect). As before, very good agreement between the MC results and the analytical model can be seen by comparing the symbols and line-style curves in Fig. 5.

5. Evaluation of MS effects for a homogeneous layer

The MC code has been used to understand which of the scattering properties are most relevant when considering the MS signal. For this purpose, the MS effect is evaluated for a single-layer atmosphere characterized by fixing an SS albedo ϖ and a phase function and varying the extinction coefficient k_ext independently.

It is useful to introduce the “scattering optical thickness, ” defined as the integral over distance of the scattering coefficient (i.e., particle density times scattering cross sections):

View Expanded

where H_top indicates the altitude of the top of the medium.

a. Magnitude of the successive orders of scattering

The MC code records the contribution of each order of scattering I^(k)_a(r) to the total signal I_a(r) [see Eq. (11)]. In particular, it is possible to evaluate the magnitude of the successive orders of scattering by computing the ratio f (k) = I^(k)_a(r)/I_a(r). Figure 6 shows the magnitude of different orders of scattering when going inside the medium for an unpolarized Rayleigh phase function. The incoming radiation is collimated at nadir, and no antenna pattern suppression factor is considered in the receiving segment. The radar vertical resolution is selected to be so small (Δr = 50 m) that k_extΔr ≪ 1; in other words, the radar pulse (multiplied by 0.5 times the speed of light) is much shorter than the mean free path. If this condition is not fulfilled, the scaling properties with the scattering optical thickness no longer hold. At ranges within which the scattering optical thickness is equal to 1, the contributions of the first and second orders of scattering to the radar signal are approximately 30%, whereas the third and the fourth orders of scattering contribute approximately 20% and 10%, respectively. It is obvious that the contributions from higher scattering orders increase with the scattering optical thickness. Figure 6 indicates that the peak contribution of each scattering order decreases with scattering order while the scattering optical depth associated with the peak contribution of each scattering order increases with scattering order. Note that the results plotted in Fig. 6 correspond to an albedo ϖ = 0.5, but identical results have been found for other albedos. Therefore, at the same scattering optical depth, the result is independent of the values assumed for the albedo and the extinction coefficient of the medium but depends only on the product of the two. For the second order of scattering, this can be understood by looking at the integral in Eq. (17): when spaceborne configurations are considered, the product of the three terms exp[−k_extΔξ(ψ)], exp[−k_extr(ψ)], and exp(−k_extz₁) can be taken out of the integral because it is practically equal to exp(−2k_extz). Therefore, Eq. (17) becomes

View Expanded

where H⁽²⁾ is related to the phase function of the medium and the radar bin width Δz only. The same kind of argument can essentially be used for the successive order of scattering, except that the geometry is not as simple. Because at order (n) it is expected that

View Expanded

in the relative contributions of the different orders of scattering f (k) the term exp(−2k_extz) will cancel out (and with it the dependence on k_ext). The same results are therefore found for equal ϖk_ext values. Slightly different curves are found when different phase functions are used. This invariance property fails when the antenna suppression factor exp[−4 log2(θ_υ/η_3dB)²] is taken into account.

6. Discussion and conclusions

A numerical code based on an MC procedure that is capable of simulating the polarized signal of a (monostatic/bistatic) radar as sensed from nonisotropic media has been described and validated. The method is not able to account for backscattering enhancement. However, this effect has been shown to be negligible (Kobayashi et al. 2005) in the spaceborne systems that are the foci of this work. The code has been exploited to have insight into the MS phenomenon by using a uniform layer as a case study. The magnitude of the SS with respect to the total radar return (see Fig. 7) is primarily a function of the scattering optical depth [defined in Eq. (19)], with the MS effect increasing with the traveled scattering optical depth. When the antenna suppression factor is omitted, results in spaceborne configuration are found to depend only on the product ϖk_ext. The asymmetry parameter of the phase function also plays an important role, with MS contributions increasing when passing from the Rayleigh phase function to the phase function with g values around 0.4 (see Fig. 4). Realistic asymmetry factors do not exceed this value at GPM frequencies (see Part II).

Similar to the results of Kobayashi et al. (2004, 2005), the Battaglia et al. (2005a) study also shows that, at scattering optical depth greater than 2, antenna pattern plays a key role in reducing the MS contributions. In essence, what really matters is the interplay among the radiation mean free path, the radar footprint (and thus the product of the satellite altitude and the tangent of the beamwidth), the scattering-layer SS albedo coefficient, and the distance traveled inside the medium. The importance of MS effects increases with the depth traveled, with the SS albedo of the scattering medium (which enhances the probability of scattering events vs absorption events), and for mean free paths that are much lower than the horizontal resolution. When GPM footprints are considered, the MS effects are certainly reduced to those values obtained for an infinite-beamwidth-like antenna for extinction coefficients on the order of 5 km⁻¹. When less extinguishing media are considered, however, the suppression by the antenna pattern can be substantial (see Fig. 8). Therefore, the approach used by Oguchi et al. (1994) and Marzano et al. (2003) generally overestimates the MS effects when typical spaceborne radar conditions are met, because they effectively assume plane-wave incidence. Their solution can a fortiori only exclude the presence of MS when absent but cannot be used to assess the effective amount of the MS contributions. This factor is even more critical when airborne configurations are examined. In agreement with results already present by Ito et al. (1995), section 5c reveals that MS effects can be responsible for overestimation in the detection of cloud bottom boundaries, and section 5d shows that MS enhances the cross-polarized component. This is believed to be a key signature of MS effects.

In Part II, the MC numerical model presented here serves as the basis for simulating the SS and MS radar return at the GPM frequencies for different inhomogeneous hydrometeor profiles extracted from some cloud-resolving-model simulations.