3.1.1  Vibration and Wave

To mathematically describe a wave, we must start with its vibration. A vibration is a back-and-forth movement that is caused by a force, f = −κz, in the opposite direction of the displacement z (κ is the restoring coefficient). Examples of vibration include a spring and a pendulum, as shown in Figure 3.1.

Based on Newton’s second law, the equation for the displacement is written as

where m is the mass. We can rewrite Equation 3.1 to yield

where ω² = κ/m.

Solving Equation 3.2 yields

where A is the amplitude, which is defined as the maximal displacement of vibration from its equilibrium position, ω = 2πf is the angular frequency ( f = 1/T is the frequency, with T as the period), and ϕ₀ in Equation 3.3 is the initial phase at t = 0. The phase is the argument of the cosine function, which is zero in Figure 3.2.

As a wave is the propagation of a vibration, its mathematical expression can be obtained by changing the time t to t − x/v in Equation 3.3, yielding

where v is the phase velocity of the wave, and the wavelength is λ = vT (as shown from peak to peak in Figure 3.3).

3.1.2  Phasor Representation of Time-Harmonic Waves

Although a time-harmonic wave can be simply represented by a sinusoidal function like Equation 3.4, it can be more conveniently represented by a complex quantity. For example, Equation 3.4 can be rewritten as

where z(x) = Ae^j(ϕ0-εx/v) is a complex quantity, which is equivalent to the real quantity z(x,t) in representing the wave, except for the omission of [e^jωt] in writing. The complex quantity z(x) is called a phasor because it is a complex quantity with phase information, but omits the time dependence term. The advantage of introducing the phasor in the complex representation is the convenience it allows in mathematical manipulation. For example, a derivative with respect to time t is simply a multiplication of jω: $\frac{\partial z\left(x,t\right)}{\partial t}\leftrightarrow j\mathrm{\omega }z\left(x,t\right)$

.

3.1.3  Em Wave

As mentioned earlier, an EM wave is a transverse wave that needs a vector in the form of an electric field $\overrightarrow{E}\left(\overrightarrow{r},t\right)$

to represent it. An EM wave propagates in space or a medium such that a changing electric field causes a magnetic field. This magnetic field is also changing, which in turn yields a varying electric field, and so on. Using the analogy of the phasor for a scalar wave, the phasor representation of the electric field is $\overrightarrow{E}\left(\overrightarrow{r},t\right)=\text{Re[}\overrightarrow{E}\left(\overrightarrow{r}\right)e^{j\mathrm{\omega }t}\text{]}$ . In the case of the time-harmonic wave, the Maxwell equations (Equation 2.14) for the phasors of the electric and magnetic fields become

Taking the curl of Equation 3.6 and using the vector identity of $\nabla \times \nabla \times \overrightarrow{E}=\nabla \left(\nabla •\overrightarrow{E}\right)-{\nabla }^2\overrightarrow{E}$

, and using Equation 3.7, we have

with the wave number

In the uniform medium where the permittivity and permeability are constant, a plane wave is the solution of Equation 3.10, specifically:

and

where $\eta =\sqrt{\frac{\mathrm{\mu }}{\mathrm{\epsilon }}}$

is the intrinsic impedance of the medium, $\widehat{e}$ is the unit vector for polarization, $\overrightarrow{k}=k\widehat{k}$ is the wave vector, and $\widehat{k}$ is the unit vector of wave propagation (Figure 3.4).

In the case of a spherical wave, the electric field can be expressed by

and it is illustrated as in Figure 3.5, where the equal phase constitutes spherical surfaces.

3.1.4  Wave Polarization and Representation

An EM wave is a transverse wave, and the electric field $\overrightarrow{E}$

varies sinusoidally with time (vibration) at a given location. The wave polarization is a description of the vibration direction, which is the locus of the tip of the $\overrightarrow{E}$ vector as time progresses.

Figure 3.6 shows examples of the polarization of an electric wave field from a coherent source or sources. If the locus of the $\overrightarrow{E}$

vector is a straight line, as shown in Figure 3.6a, the wave is said to be linearly polarized (or in linear polarization); if the loci form a circle as in Figure 3.6b, it is circularly polarized (circular polarization); and if the locus is an ellipse as in Figure 3.6c, it is elliptically polarized (elliptical polarization). If the locus is random, however, the wave is unpolarized. Sunlight or light reflected from a Lambertian surface (Born and Wolf 1999) is an example of unpolarized waves. A wave can also be partially polarized if it is from both coherent and incoherent sources.

A mathematical representation of a polarized wave can be made by describing the wave field $\overrightarrow{E}$

vector in the polarization plane. Assuming that a plane wave propagates along the x-axis, the $\overrightarrow{E}$ vector will be in the y–z plane:

and each component follows the time-harmonic solution (Equation 3.5), which is rewritten as follows:

where (ωt − kx) is the time–space variable phase term; A_y and A_z are the amplitudes for the y and z components, respectively and ϕ₀₁ and ϕ₀₂ are their corresponding initial phases. Because the variable phase (ωt − kx) in Equations 3.16 and 3.17 is a common term, the different types of polarization, such as those illustrated in Figure 3.6, are then represented by Equations 3.15 through 3.17 with different amplitude ratios and phase differences as follows:

3.2.1  Scattering Amplitude, Scattering Matrix, and Scattering Cross Sections

When the wave is incident on the particle, a part of the wave power is absorbed by the particle, and the other part is scattered out to all directions. It is expected that the scattering and absorption properties depend on the wave properties and the physical and EM characteristics of the particle, as described in Chapter 2.

Let the particle have an arbitrary shape that will be specified later. Its EM properties are represented by permittivity ε and permeability μ as

where ε₀ and μ₀ are free space permittivity and permeability, respectively; ε_r is the relative dielectric constant with ε′ as its real part and ε″ as its imaginary part.

Assume that the incident wave is a linearly polarized plane wave propagating in free space with a permittivity and permeability of (ε₀,μ₀). Hence, the phasor representation of the plane wave is Equation 3.12, which is

where ${\widehat{e}}_{\text{i}}$

is the unit vector for the incident wave polarization; E₀ is the amplitude; and ${\overrightarrow{k}}_{\text{i}}=k{\widehat{k}}_{\text{i}}$ is the incident wave vector, with k as the wave number for the background medium and the unit vector ${\widehat{k}}_{\text{i}}$ for the propagation direction.

With the incident wave impinging on the particle, charges inside the particle are excited and vibrate, yielding a reradiated wave or waves in all directions. If the observed point is far away from the particle, it is called the far field if it meets the condition: r > 2D²/λ. In this case, the scattered field behaves as a spherical wave and can be represented by an equation similar to Equation 3.14:

where $\overrightarrow{S}\left({\widehat{k}}_{\text{s}},{\widehat{k}}_{\text{i}}\right)=S\left({\widehat{k}}_{\text{s}},{\widehat{k}}_{\text{i}}\right){\widehat{e}}_{\text{s}}$

is the scattering amplitude, representing the amplitude, phase, and polarization of the scattered wave field for a unit plane wave incidence on the particle. The scattering amplitude is, in general, complex and contains both magnitude and phase information.

As discussed in Section 3.1.4, wave polarization is fully described by two orthogonal components. Assuming that $\left({\widehat{e}}_{\text{i}1},{\widehat{e}}_{\text{i2}}\right)$

are the reference unit vectors for the incident wave field and $\left({\widehat{e}}_{\text{s}1},{\widehat{e}}_{\text{s2}}\right)$ are the reference unit vectors for the scattered wave field, as shown in Figure 3.9, we have

The scattering equation (Equation 3.33) can then be written in matrix form as

The matrix $\left[S\right]=\left[\begin{array}{cc}{S}_{11}& S_{12}\\ S_{21}& S_{22}\end{array}\right]$

is called the scattering matrix, with its diagonal terms s₁₁ and s₂₂ representing the wave scattering for co-polar components and the off-diagonal terms s₁₂ and s₂₁ representing cross-polar scattering. There are alternate ways to choose the orthogonal unit vectors $\left({\widehat{e}}_1,{\widehat{e}}_2\right)$ to define the scattering matrix— two commonly used methods are based on (i) the scattering plane and (ii) the horizontal and vertical directions.

The scattering plane is the plane that contains the incident and scattered wave vectors. The vector perpendicular to both the wave vector and the scattering plane is called the perpendicular vector, defined as ${\widehat{e}}_1={\widehat{e}}_{\perp }=\frac{{\widehat{k}}_{\text{s}}\times {\widehat{k}}_{\text{i}}}{\left|{\widehat{k}}_{\text{s}}\times {\widehat{k}}_{\text{i}}\right|}$

. The vectors on the scattering plane are defined as ${\widehat{e}}_{\text{i}\left|\right|}={\widehat{k}}_{\text{i}}\times {\widehat{e}}_{\text{i}\perp }$ and ${\widehat{e}}_{\text{s}\left|\right|}={\widehat{k}}_{\text{s}}\times {\widehat{e}}_{\text{s}\perp }$ for the incident and scattered waves, respectively. In this case, Equation 3.36 becomes

The other coordinate system to define the scattering matrix is based on the horizontal and vertical (more precisely described as longitudinal) directions with

where

Hence, the scattered wave field is related to the incident wave field by

and is generally elliptically polarized. The scattering matrix elements are discussed further in the following sections.

Whereas a wave field is represented by its complex scattering amplitude and scattering matrix, the wave power is characterized by the power flux density—the power passing through a unit area, which is the magnitude of the power density flux vector. The power flux density is also called the Poynting vector. For a time-harmonic wave, we have the mean Poynting vectors

for the incident wave and

for the scattered wave. The factor of 1/2 in Equations 3.43 and 3.44 accounts for the difference between the averaged power and the peak power for time-harmonic waves (Shen and Kong 1983).

Hence, the differential power dP_s for the scattered wave through a differential area da = r²dΩ is

where dΩ = sinθdθdϕ is the differential solid angle. Using $S_{\text{i}}=\frac{|E_0{|}^2}{2{\eta }_0}$

in Equation 3.45, we have the scattered differential power normalized by the incident power flux density

where ${\sigma }_{\text{d}}\left({\widehat{k}}_{\text{s}},{\widehat{k}}_{\text{i}}\right)=|S\left({\widehat{k}}_{\text{s}},{\widehat{k}}_{\text{i}}\right){|}^2$

is called the differential cross section because it represents scattered wave power in a unit solid angle for an incident wave having a unit of power density and has a unit of area (section). In radar applications, a factor of 4π is multiplied to obtain a bistatic and backscattering radar cross section

so that it can be compared with the total scattered power in the 4π steradians of a solid angle.

The scattering cross section that represents power loss for the scattered power going to all the angles is the integral of Equation 3.46, specifically

Besides the scattered power loss, another part of the wave power is absorbed by the particle and dissipated. This part can be calculated by the volume integral of the power loss inside the particle. Consider that the differential power is a product of different current and voltage:

where the conductivity is σ = ωε″ε₀. Taking the factor of $\frac{1}{2}$

into account for the difference between the mean and peak powers, the absorption power is normalized by the incident power flux density to obtain the absorption cross section as

The sum of the scattering and absorption represents the total power loss due to the scattering process; then the total cross section is the sum of the scattering cross section and absorption cross section:

where $\left({\widehat{k}}_{\text{i}},{\widehat{k}}_{\text{i}}\right)$

is the forward scattering amplitude; σ_t ≡ σ_e is also called the extinction cross section. The second equity of Equation 3.51 shows that the extinction cross section is proportional to the imaginary part of the forward scattering amplitude; it is known as the optical theorem (see Appendix 3A). The ratio between the scattering cross section (Equation 3.48) and the extinction cross section, w = σ_s/σ_t, is called albedo.

For convenience of comparison, the efficiency factors for extinction, scattering, and absorption are defined (Bohren and Huffman 1983) as the cross sections normalized as follows:

where σ_g is the geometric cross section, and σ_g = πa² for a sphere of radius a.

Figure 3.10 shows a plot of the efficiency factors for water spheres, specifically, the normalized cross sections as functions of the electric size (ka). When a particle is very small (ka ≪ 1), the efficiency factors monotonically increase as the size increases. This is called the Rayleigh scattering regime. When a particle is about the size of the wavelength (ka ∼ 1), the normalized cross sections oscillate, which is called resonance or the Mie scattering regime. When the particle size is very large compared with the wavelength (ka ≫ 1), each of the efficiency factors approaches a constant, where the theory of geometric optics applies. It is interesting to note that, in the geometric optical regime, the total (or extinction) cross section is twice that of the geometric cross section for very large particles. This is contrary to our intuition in which an object blocks and makes extinct an area that exactly corresponds to its geometric cross section. This difference between what our intuition tells us and the wave theory is called the extinction paradox (Van De Hulst 1957). Next, we discuss the scattering calculations in the aforementioned regimes.

3.3.1  Original Statement

The law of Rayleigh scattering was initially developed by Rayleigh (1871) through dimension matching, which was cited by Bohren and Huffman (1983) in the following statement: “When light is scattered by particles which are very small compared with any of the wavelengths, the ratio of the amplitudes of the vibrations of the scattered and incident light varies inversely as the square of the wavelength and the intensity of the lights themselves as the inverse fourth power.”

The law of Rayleigh scattering was derived by dimension matching for the amplitude ratio because it was suspected to be related to the physical parameters as follows:

where V is the volume of the particle, r is the range from the particle, λ is the wavelength, c is the velocity of the EM wave, and ρ_e is the density of the ether medium.^* In Equation 3.53, the velocity c has a dimension of [L T ⁻¹], but no other term contains the dimension of time [T]. Hence, because the amplitude ratio is dimensionless and there is no other term to cancel out the time dimension in c, c is not a component of $\frac{A_{\text{s}}}{A_{\text{i}}}$

. Similarly, the ether density ρ_e contains the dimension of mass [M]. However, there is no other term that contains the dimension of mass; therefore, the dependence on ρ_e should be removed. This leaves V, r, and λ on the right-hand side of Equation 3.53. If the ratio between the scattered wave amplitude and the incident wave amplitude is proportional to the volume of particle V and is inversely proportional to the range r, the wavelength squared is needed in the denominator, yielding

Taking the square of both sides of Equation 3.54 gives an intensity ratio of

Hence, Equations 3.54 and 3.55 constitute the mathematical expression for the law of Rayleigh scattering: the intensity ratio is inversely proportional to the fourth power of the wavelength. This law provides, among other things, the explanation for why the sky is blue and the sun is orange: When looking to the sky, because the wave scattering is stronger for short wavelength (blue) light, the sky is perceived as blue; when looking directly at the sun, because the short wavelength light is removed from the direct line of sight, what remains the long wavelength light and the sun therefore appears orange to our eyes.

Although the law of Rayleigh scattering derived from dimension-matching was successful in explaining natural phenomena, there were unresolved issues such as what the angular and polarization dependences were, and what the absorption was. These issues were addressed through a more rigorous formulation of wave scattering based on the Maxwell equations.

3.3.2  Scattering as Dipole Radiation

When the particle is small compared with the wavelength, the Rayleigh scattering can be understood as a dipole radiation. As illustrated in Figure 3.11, when a wave is incident on a small spherical particle (represented by the dashed circle), an electric field is applied to the particle and causes positive charges to move to one end and negative charges to move to the other end. The wave then changes phases (i.e., the direction of the electric field changes), and the charges move in the opposite directions. This process continues, causing the particle to act as a dipole antenna and radiate an EM wave after it is excited by the incident wave (Ishimaru 1991).

Therefore, the scattered wave field can be represented by the vector potential $\overrightarrow{A}$

(Ishimaru 1978, 1997) as

and

where r is the observation point and r′ is the source location inside the particle. The free space Green’s function is

and the equivalent current source is

which exists inside the particle only due to the excitation of the charges inside the particle.

The internal field can be represented by an incident wave field as follows (Stratton 1941, 205):

Substituting Equations 3.58 and 3.59 into Equations 3.56 and 3.57, assuming a unit wave of incidence, and using far-field approximation (r ≫ r′), we have

with the scattering amplitude as

which represents the amplitude of the scattered wave field with the unit wave of incidence on a small sphere. As expected, the scattering amplitude depends on the particle’s physical property, volume V, electric property ε_r, both the scattering direction and incident direction $\begin{array}{c}\left({\widehat{k}}_{\text{s}},{\widehat{k}}_{\text{i}}\right)\end{array}$

, and the polarization direction of the incident wave field ${\widehat{e}}_{\text{i}}$ . It is noted that χ is the angle between the scattering direction ${\widehat{k}}_{\text{s}}$ and the incident wave polarization ${\widehat{e}}_{\text{i}}$ , as shown in Figure 3.11. The sinχ term can be understood as the unit vector of incident wave polarization ${\widehat{e}}_{\text{i}}$ projected onto the scattered wave polarization of ${\widehat{e}}_{\text{s}}$ , which is perpendicular to the scattered wave propagation direction of ${\widehat{k}}_{\text{s}}$ , demonstrating the transverse nature of EM waves.

Whereas the scattering amplitude represents the property of the scattered wave field, the scattered wave power distribution is represented by the differential scattering cross section as

The scattering field patterns and scattering power patterns are plotted in Figure 3.12. Whereas the scattering pattern appears in the sinχ distribution in the electric field plane (E-plane), the scattering is isotropic in the magnetic field plane (H-plane). This is why a single polarization monostatic radar tends to use horizontal polarization, but a bistatic radar like BINET (bistatic receiver network) uses vertical polarization (Wurman et al. 1993).

Once the scattering amplitude (Equation 3.62) and the differential scattering cross section (Equation 3.63) are known, the total scattering cross section is the integral over the 4π steradians of a solid angle, yielding

The backscattering radar cross section is then

Note that the backscattering radar cross section (Equation 3.65) is larger than the total scattering cross section (Equation 3.64). This is because the radar cross section is defined as 4π times the differential scattering cross section, for which both the maximum and the backscattering direction occur at the equator.

Following Equations 3.49 and 3.50, the absorption cross section is

Whereas the scattering cross sections (Equations 3.64 and 3.65) are proportional to the volume squared (∼a⁶ = (D/2)⁶), the absorption cross section (Equation 3.66) is proportional linearly to the volume (a³ = (D/2)³) of the particle, in the case of Rayleigh scattering. Because the internal wave field has been assumed to be constant in the derivations of the scattering amplitude (Equation 3.62) and cross sections (Equations 3.63 through 3.66), the derived formulas apply only to the electrically small scatterers/particles (ka ≪ 1, i.e., a ≪ λ). The valid regime for Rayleigh scattering can be seen more clearly in comparison with the Mie theory discussed in Section 3.4.

3.4.1  Conceptual Description

The exact solution for wave scattering by a sphere was developed in 1908 by Gustav Mie and is called the Mie theory. The Mie theory allows for the accurate calculation of wave scattering by a uniform sphere of any size and dielectric constant; these calculations have been well documented in multiple textbooks (Bohren and Huffman 1983; Kerker 1969). The results of the Mie scattering calculation also allow for a determination of the valid regime of the Rayleigh scattering approximation.

In the Rayleigh scattering approximation, the internal wave field is assumed to be constant and the wave scattering is treated as if it were radiation from a dipole antenna. By contrast, the Mie theory allows for the wave field to vary inside the sphere and for not only the dipole mode, but the quadrupole, hexapole, and higher-order pole modes as well (a dipole, a quadrupole, and a hexapole are illustrated in Figure 3.13).

In general, the wave scattering by a sphere is the superposition of all order-pole modes that are resonating in the scatterer, and hence its radiation field has a more complex pattern than that of Rayleigh scattering.

3.4.2  Mathematical Expression and Sample Results

The purpose of the mathematical representation for Mie scattering is to solve the EM boundary problem. That is, under the wave incidence of ${\overrightarrow{E}}_{\text{i}}$

, the internal wave field inside the sphere is ${\overrightarrow{E}}_{\text{int}}$ , and the wave field outside is the summation of the incident wave field ${\overrightarrow{E}}_{\text{i}}$ and the scattered wave field ${\overrightarrow{E}}_{\text{s}}$ , specifically ${\overrightarrow{E}}_{\text{i}}+{\overrightarrow{E}}_{\text{s}}$ . The first and second Maxwell equations require that the tangential components of the electric and magnetic fields must be continuous. Let us use a spherical coordinate system (r, θ, ϕ) with the sphere center as the origin. We have

Each part of Equations 3.67 and 3.68 represents two equations if it is expressed in component form in the θ and ϕ directions, totaling a set of four equations.

Because the wave fields can be expanded in vector spherical harmonics, the incident, scattered, and internal wave fields are expressed as

where C_n = (−j)ⁿE₀(2n + 1)/[n(n + 1)] are the expansion coefficients for the incident field, and (a_n, b_n) and (c_n, d_n) are those for the scattered and internal wave fields, respectively. ${\overrightarrow{M}}_{mn}$

and ${\overrightarrow{N}}_{mn}$ are the vector spherical harmonics (see their expressions in Appendix 3B), and the subscripts e and o indicate the even and odd modes, respectively.

Using Equations 3.69 through 3.71 and their derived expressions for magnetic fields from Equations 3.67 and 3.68 and solving the equations yields the scattering coefficients:

where ${\text{Ψ}}_n\left(x\right)=x{j}_n\left(x\right)=\sqrt{\text{π}x/2}{J}_{n+1/2}\left(x\right)$

is the Riccati–Bessel function, and j_n(x) is the spherical Bessel function. Then, the scattered wave fields are

where

and

with $P_n^1\left(\text{cos θ}\right)=\frac{d}{d\text{θ}}{P}_n\left(\text{cos θ}\right)$

and P_n (cos θ) as the nth order Legendre polynomial.

The sample calculation results for the amplitude patterns at the X-band of two water spheres with diameters of 2 mm (top row) and 2 cm (bottom row) are shown in Figure 3.14. It is apparent that the pattern for the sphere with a diameter of 2 mm is stronger in the forward direction than the pattern in the backward direction (the pattern magnitude is ~10% different between forward and backward scattering), but is somewhat similar to those patterns of Rayleigh scattering shown in Figure 3.12. (For hydrometeors of diameters larger than 2 mm, the Rayleigh scattering approximation starts to become invalid.) As is evident, for the sphere with a diameter of 2 cm, the forward scattering pattern is dominant and is of an order larger than that of backscattering. Note also that the backscattering by the 2-cm sphere is still two orders larger than the backscattering by the 2-mm sphere, although the main scattered energy is in the forward direction. This is because the total scattering cross section increases by many orders when the sphere increases in size from 2 mm to 2 cm.

Using the scattering amplitude (Equations 3.75 and 3.76) in the expressions in Equations 3.47, 3.48, and 3.51, we have the cross sections for Mie scattering as follows:

The radar backscattering cross section is

The total scattering cross section is

and the extinction cross section is

Now, we have cross sections from both the Rayleigh scattering approximation given in Equations 3.64 through 3.66 and the rigorous calculation given by the Mie theory (Equations 3.78 through 3.80). The efficiency factors, cross sections normalized by the geometric cross section, are calculated using the two aforementioned approaches and compared in Figure 3.15.

The calculations are performed for water spheres at X-band (wavelength: λ = 3 cm) with a dielectric constant of (44 − j43). It is evident that the scattering efficiency factors of Rayleigh scattering and those of Mie theory agree well up to a diameter of 2 mm. That means Rayleigh scattering approximation is valid for a sphere whose diameter D < λ/16, which is equivalent to 2kD < π/4, meaning that the wave field inside the particle can be treated as a constant. However, the results of absorption and extinction start to differ at smaller sizes, D < 1 mm. Typically, there is a more stringent requirement for Rayleigh scattering to be valid, specifically D < λ/50. This value is close to k(m′ − 1)D < π/4 (with m′ = 7), which requires that the phase difference between a wave passing through the particle and a wave in free space is small.

At the other end of the extreme, when a sphere is much larger than the wavelength (D > 100λ), geometric optics is a good approximation. However, the extinction cross section is twice as large as the geometric cross section, which is counterintuitive. This unexpected result is called the extinction paradox, as noted in Section 3.2. The extinction paradox can be resolved if we note that (i) anything removed from the forward direction is considered part of scattering and (ii) the extinction cross section is defined in the far field, and any scatterer, no matter how big it is, has an edge that can alter wave propagation and can cause diffraction that geometric optics does not take into account. Hence, there is a difference between our intuition and the full wave theory.

As for the resonance effect of the up and down changes in the normalized backscattering cross section, the intuitive explanation is constructive and destructive summation of scattered waves going through different paths. Figure 3.16 shows two conceptual models: (a) the reflection model and (b) the reflection and creeping wave model. The reflection model is shown in Figure 3.16a: one wave is reflected back from the front edge, the other penetrates into the sphere and is then reflected from the back edge. In Figure 3.16b, Path 1 is the same as that in Figure 3.16a, but the second wave propagates around the half-sphere and then comes out in the opposite direction.

In either case, we can expect a maximum if the two waves are in phase and constructively add, meaning the path difference is an integer of the wavelength, but a minimum if the difference is an odd number of a half of a wavelength. Using the reflection and creeping wave model in Figure 3.16b, the first maximum that satisfies the condition of k(ℓ₂ − ℓ₁) = k(D + πD/2) = 2π, results in D = 0.39λ. In the case of the result shown in Figure 3.15 for an X-band with λ = 3 cm, we have the maximum at D = 1.17 cm, which is close to that of ~1.0 cm, shown in the figure.

3.5.1  Basic Nonspherical Shape: Spheroid

As described in Chapter 2, there are a variety of shapes for hydrometeors: from a spherical shape for cloud droplets and small rain drops, to an oblate spheroidal shape with a flattened base for large raindrops, to an irregular shape for snowflakes and hailstones. Furthermore, an ice crystal in a snowflake can be in the shape of a needle, disk, plate, cylinder, dendrite, or a combination of these shapes. It is difficult to rigorously model the irregular shape of snowflakes and hailstones with a few parameters. Such rigorous modeling is unnecessary in radar polarimetry because there are, after all, only a few types of radar measurements. These measurements can capture the main statistical properties of the hydrometeors without requiring detailed structural information about the hydrometeors. Hence, a simple spheroidal shape model, as long as it is not based on a perfect sphere, is usually sufficient for interpreting polarimetric radar data. This is one reason the spheroid model is often used. Another reason that the spheroid is used to model hydrometeors is because it is one of the basic shapes for which the EM scattering problem can be analytically solved for small scatterers and numerically solved (with relatively low computation costs) for large (D ∼ λ) scatterers.

A spheroid is a special case of an ellipsoid; its two semidiameters are equal, which can be obtained by rotating an ellipse around its principal z-axis. When the ellipse rotates around its minor axis, we obtain an oblate spheroid (a₁ = a₂ > b); when the ellipse rotates around its major axis, we have a prolate spheroid (a₁ = a₂ < b). Both are illustrated in Figure 3.17.

Oblate spheroids are used to model raindrops, snowflakes, hailstones, and ice crystal plates and dendrites, whereas prolate spheroids can be used to represent needles and columns of ice crystals.

3.5.2  Rayleigh Scattering Approximation for Spheroids

The mathematical representation of Rayleigh scattering by a sphere was provided in Section 3.3.2. The expression of scattering amplitude provided there (Equation 3.62) can be rewritten as

where the internal field is

Although the internal field for a sphere can be easily represented by the incident wave field (Equation 3.60), the internal field for an ellipsoid (spheroid) is more difficult to find because it depends on the incident wave polarization and the particle’s orientation. However, the polarized wave field for each component can be expressed by the following (Stratton 1941):

Combining Equations 3.82 and 3.83 and solving for the internal field for each polarization component on its symmetry axis, we obtain the following (Van De Hulst 1957):

where

are factors that depend on the shape of the scatterer. It can be shown (Stratton 1941) that the shape factors follow an equality of

In general, the shape factors are inversely proportional to the corresponding dimension. If the scatterer is sufficiently spheroidal, that is, the axis ratio does not differ from the unity too much (0.5 < b/a < 2), an approximate relation exists

In the case of spheroids, the shape factors can be calculated by using Equation 3.88 and performing the integral of Equation 3.87 with (a₁ = a₂ = a), giving

and

for an oblate spheroid (a > b) and

for a prolate spheroid. Once the shape factors (L) and the dielectric constant are known, the scattering amplitude can be calculated. Consider a wave incident on a spheroid in the x-direction: the scattering amplitudes in the scattering plane for wave polarization aligned at the major and minor axes are obtained by substituting Equation 3.84 into Equations 3.81 through 3.83 and letting L_y = L_a and L_z = L_b as follows:

It is evident that the scattering amplitude of Equation 3.93 for a spheroid is similar to the scattering amplitude of Equation 3.62 for a sphere. The normalized amplitude patterns are the same with the term sin χ as shown in Figure 3.11. The magnitude is different with $\frac{{\text{ε}}_{\text{r}}-1}{3\left[1+L_{\text{a,b}}\left({\text{ε}}_{\text{r}}-1\right)\right]}$

, instead of $\frac{{\text{ε}}_{\text{r}}-1}{{\text{ε}}_{\text{r}}+2}$ , which depends on the polarization. The backward and forward scattering amplitudes are equal.

Using the axis ratio of Equation 2.16 for raindrops (Equations 3.90 and 3.91), the forward $\left({\widehat{k}}_{\text{s}}={\widehat{k}}_{\text{i}};\text{Θ}=0\right)$

or backward $\left({\widehat{k}}_{\text{s}}=-{\widehat{k}}_{\text{i}};\text{Θ}=\text{π}\right)$ scattering amplitudes for polarizations on the major and minor axes are plotted in Figure 3.18. The effective shapes of a few raindrops are also shown in the figure. As expected, the scattering amplitude with polarization on the major axis (s_a: solid line) is larger than for polarization on the minor axis (s_b: dashed line); the larger dimension of the scatterer causes stronger scattering because a larger dipole moment is formed. The difference in the scattering amplitudes between the two polarizations increases as raindrops become more oblate with size increases. The Rayleigh scattering approximation for spheroids is valid for clouds, rain, and dry snow at the S-band, but is not applicable for hail and melting snow, which require more accurate calculations for scattering amplitudes. At C-band and X-band or higher frequencies, wave scattering from all precipitation particles require more accurate calculation than is provided by Rayleigh scattering approximations.

3.5.3  T-Matrix Method

When a hydrometeor’s size is comparable to its wavelength, the Rayleigh scattering approximation introduced above becomes invalid, and an analytical solution does not exist. There is thus a need for a rigorous method to numerically calculate wave scattering. The T-matrix method is a numerical method that has been successfully developed and widely used in the radar meteorology community (Barber and Yeh 1975; Seliga and Bringi 1976; Vivekanandan et al. 1991; Waterman 1965). The idea of the T-matrix method is (i) to expand the incident, scattered, and internal wave fields in terms of their vector spherical harmonics and (ii) to use extended boundary conditions to determine the expansion coefficients through a transition matrix, which is briefly described in the following.

Wave scattering by an irregularly shaped particle is illustrated in Figure 3.19. Under the wave incidence ${\overrightarrow{E}}_{\text{i}}\left(\overrightarrow{r}\right)$

, there is the scattered wave field ${\overrightarrow{E}}_{\text{s}}\left(\overrightarrow{r}\right)$ outside the particle and the internal wave field ${\overrightarrow{E}}_{\text{int}}\left(\overrightarrow{r}\right)$ inside the particle. These wave fields are expanded in vector spherical harmonics

Because the incident wave is known, the expansion coefficients (e_mn, f_mn) are also known. That leaves four sets of expansion coefficients, including the scattering coefficients (a_mn, b_mn) and the internal coefficients (c_mn, d_mn), to be determined using the boundary conditions. This problem is similar to the problem set up by the Mie theory, but is more difficult to solve because we cannot apply the continuity conditions at r = constant as we could in the case of a sphere, which complicates the angular dependence of the wave fields.

To address the difficulty in solving the boundary problem, the concept of extended boundary conditions was introduced by Waterman (1965, 1969), and the extinction theorem and Huygens principle are used. As illustrated in Figure 3.19, two spheres are drawn in dashed lines: the inner sphere and the outer sphere circumscribing the scatterer. Inside the inner sphere, the scattered wave field cancels out the incident wave field, which is written as

where $\overline{\overline{G}}\left(\overrightarrow{r},\overrightarrow{r}\prime \right)$

is the dyadic Green’s function that can be represented by the vector spherical harmonics. Substitution of Equations 3.95 and 3.96 into Equation 3.97 yields

which links the expansion coefficients between the incident wave field and the internal field.

Outside the scatter, by contrast, the scattered wave field is caused by the surface wave field. From the Huygens principle, we have

Using Equations 3.94 and 3.95 in Equation 3.99, we obtain

Solving (c_mn, d_mn) from Equation 3.98 and substituting them into Equation 3.100, we have

where [T] = [B][A]⁻¹ is the transition matrix that relates the coefficients of the scattered wave field to the coefficients of the incident wave field. It is called the T-matrix, and this method is known as the T-matrix method. Once the coefficients (a_mn, b_mn) are known, the scattered wave field and the scattering amplitude/matrix can be found.

Figure 3.20 shows the T-matrix calculations for backscattering amplitudes in magnitude (left column) and phase (right column) for wave scattering by spheroid raindrops at different frequencies, as compared with the backscattering amplitudes obtained with the Rayleigh scattering approximation. A temperature of 10°C was used for the calculation. At S-band, the T-matrix-calculated magnitudes agree well with their corresponding Rayleigh approximation results until the drop diameter increases to 6 mm. The phases of the scattering amplitudes are very small. At C-band, however, the results between the T-matrix calculations and the Rayleigh approximation start to differ at diameters of approximately 3 mm, because the electric size (ka) at C-band is double that at S-band. The T-matrix calculations show the resonance effect and substantial phase for the scattering amplitudes. At X-band, as the wavelength becomes even shorter, the resonance effect and scattering phase appear at an even smaller sizes (D ~ 2 mm); Rayleigh approximation then becomes invalid even for median-sized raindrops. The forward scattering amplitudes have similar properties; interested readers can download the T-matrix results and compare their own plots.

Figure 3.21 shows a summary of the results given in Figure 3.20 by plotting the normalized backscattering cross section (upper left), backscattering magnitude ratios (upper right), backscattering phase differences (bottom left), and the real part of the forward scattering amplitudes (bottom right). Chapter 4 shows that the normalized backscattering cross section and magnitude ratio are the reflectivity factor and differential reflectivity, respectively, for monodispersion drop size distributions. There is a peak in the C-band magnitude ratio, which is caused by the resonance effect. In general, the scattering phase difference increases as drop size and/or frequency increases. The phase difference is a main factor that causes signal decorrelation (ρ_hv) between the dual-polarizations. The real part of the forward scattering amplitudes is associated with the specific differential phase (K_DP). These are described in Chapter 4.

The scattering amplitudes of snowflakes and hailstones were also calculated using the T-matrix method, and the results are shown in Figure 3.22. Both snowflakes and hailstones are assumed to be oblate spheroids with an axis ratio of γ = 0.75. The difference between the snow and hail modeling is the density: the density of snowflakes is assumed to be 0.1 g/cm³, whereas hailstones have a density of 0.92 g/cm³. The cases for both dry and wet snow/hail are calculated. For wet snow/hail, 20% melting is used. A melting hailstone can be treated as a water-coated ice (two-layer) particle. At low frequencies, it can be shown that the water-coated ice is equivalent to an effective particle of water–ice mixture from the Maxwell-Garnett formula with water as background (see Problem 3.4). As in Figure 3.20, the left column shows the magnitudes of the backscattering amplitudes with polarizations aligned with the major and minor axes. A substantial difference is evident in the scattering amplitude between the wet and dry cases: there is almost an order of difference in the magnitudes between wet and dry snow. The difference between the polarizations in the major and minor axes is very small for dry snow. To better show the polarization difference, the right column contains the ratio of the scattering amplitude magnitudes squared in decibels. It is clear that the ratio between the two polarizations is much larger for wet snow than that for dry snow.

The scattering characteristics of hail are more complicated: there is a greater difference between the two polarizations, but a smaller difference between the wet and dry cases. The resonance effect is very pronounced for hail with a diameter greater than 2 cm, which could yield backscattering for the polarization on the major axis that is smaller than that on the minor axis. At higher frequencies (C- and X-bands), these differences are even greater.