Wave Scattering by a Single Particle

Authored by: Guifu Zhang

Print publication date:  August  2016
Online publication date:  August  2016

Print ISBN: 9781439869581
eBook ISBN: 9781315374666

10.1201/9781315374666-4

Abstract

Weather radars measure electromagnetic (EM) wave scattering by the hydrometeors that are in clouds and precipitation. To understand polarimetric radar signatures as they relate to the shape, orientation, and composition of these hydrometeors for specific weather conditions, it is important to understand EM wave scattering by various types of hydrometeors. This chapter deals with wave scattering by a single particle, focusing on the sphere and the spheroid shapes on which many hydrometeors can be modeled. This chapter describes the basic physics behind and mathematical representation of EM waves and wave scattering. The approaches for scattering calculations, including the Rayleigh approximation, Mie theory, and T-matrix method, are provided. The basic scattering characteristics of hydrometeors are also described.

Wave Scattering by a Single Particle

Weather radars measure electromagnetic (EM) wave scattering by the hydrometeors that are in clouds and precipitation. To understand polarimetric radar signatures as they relate to the shape, orientation, and composition of these hydrometeors for specific weather conditions, it is important to understand EM wave scattering by various types of hydrometeors. This chapter deals with wave scattering by a single particle, focusing on the sphere and the spheroid shapes on which many hydrometeors can be modeled. This chapter describes the basic physics behind and mathematical representation of EM waves and wave scattering. The approaches for scattering calculations, including the Rayleigh approximation, Mie theory, and T-matrix method, are provided. The basic scattering characteristics of hydrometeors are also described.

3.1  Wave and Electromagnetic Wave

A wave is the propagation of a vibration or oscillation and is an efficient way of transporting energy and information. Waves are omnipresent in our daily lives. We hear each other through sound, which are acoustic waves, and see our environment by light, which are EM waves at extremely high frequencies. An acoustic wave is a longitudinal wave, which vibrates air molecules along the direction of the wave propagation. As such, only a one-dimensional variable of the displacement from its reference is needed to describe wave characteristics. A longitudinal wave is also called a scalar wave. An EM wave, however, is a transverse wave, which vibrates electrons (if present) in a direction perpendicular to the direction of propagation. Since a two-dimensional vector is needed to define the vibration direction, a trans-verse wave is also called a vector wave.

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

3.1 $f = m d 2 z d t 2 = − κ z ,$

Figure 3.1   Conceptual illustration of a spring vibration. The restoring force is opposite to the displacement.

Figure 3.2   Sketch of a vibration with instantaneous displacement z as a function of time.

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

3.2

where ω2 = κ/m.

Solving Equation 3.2 yields

3.3

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 tx/v in Equation 3.3, yielding

3.4

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).

Figure 3.3   Sketch of wave propagation: asterisk (*) propagates in the x-direction.

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

3.5

where z(x) = Aej(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 [ejωt] 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: $∂ z ( x , t ) ∂ t ↔ j ω z ( x , t )$

.

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 $E → ( r → , t )$

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 $E → ( r → , t ) = Re[ E → ( r → ) e j ω t ]$ . In the case of the time-harmonic wave, the Maxwell equations (Equation 2.14) for the phasors of the electric and magnetic fields become

3.6 $∇ × E → = − j ω μ H →$
3.7 $∇ × H → = j ω ε E →$
3.8 $∇ • E → = 0$
3.9 $∇ • H → = 0.$

Taking the curl of Equation 3.6 and using the vector identity of $∇ × ∇ × E → = ∇ ( ∇ • E → ) − ∇ 2 E →$

, and using Equation 3.7, we have

3.10 $∇ 2 E → × k 2 E → = 0$

with the wave number

3.11 $k = ω μ ε = ω μ 0 ε 0 ε r = k 0 m = 2 π λ 0 m .$

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

3.12 $E → = e ^ E 0 e − j k → • r →$

and

3.13 $H → = k ^ × e ^ E 0 η e − j k → • r → ,$

where $η = μ ε$

is the intrinsic impedance of the medium, $e ^$ is the unit vector for polarization, $k → = k k ^$ is the wave vector, and $k ^$ is the unit vector of wave propagation (Figure 3.4).

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

3.14 $E → = e ^ E 0 e − j k r r$

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

Figure 3.4   A sketch of a uniform plane wave propagation, where phases of the wave are equal on a plane that is perpendicular to the wave vector.

Figure 3.5   A sketch of a spherical wave propagation, where equal phase surfaces are perpendicular to the wave vector.

3.1.4  Wave Polarization and Representation

An EM wave is a transverse wave, and the electric field $E →$

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 $E →$ 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 $E →$

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.

Figure 3.6   Polarization of an electric wave field. (a) Linear polarization, (b) circular polarization, and (c) elliptical polarization.

A mathematical representation of a polarized wave can be made by describing the wave field $E →$

vector in the polarization plane. Assuming that a plane wave propagates along the x-axis, the $E →$ vector will be in the yz plane:
3.15 $E → ( x , t ) = E y ( x , t ) y ^ + E z ( x , t ) z ^ = Re ( E → e j ω t )$

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

3.16 $E y ( x , t ) = A y cos ( ω t − k x + ϕ 01 )$

3.17 $E z ( x , t ) = A z cos ( ω t − k x + ϕ 02 ) ,$

where (ωtkx) is the time–space variable phase term; Ay and Az are the amplitudes for the y and z components, respectively and ϕ01 and ϕ02 are their corresponding initial phases. Because the variable phase (ωtkx) 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:

3.1.4.1  Linear Polarization

Let the phase difference be

3.18

Substituting Equations 3.16 and 3.17 into Equation 3.15 and using Equation 3.18, we obtain

3.19

which shows that the two components have the same phase term, and the locus is a straight line with a slope of Az/Ay, as shown in Figure 3.6a. Hence, the wave represented by Equation 3.19 is described as being linearly polarized.

3.1.4.2  Circular Polarization

Let the phase difference be

3.20 $δ = ϕ 02 − ϕ 01 = ± π 2$

and the amplitude ratio is unity, specifically, Az/Ay = 1. Substituting Equations 3.16 and 3.17 into Equation 3.15 and using these conditions, we have

3.21

Equation 3.21 is clearly an equation for a circle in a 2D plane, hence representing the circular polarization illustrated in Figure 3.6b. Because the wave is coming out of the page (the x direction), δ = π/2 represents left-hand circular polarization and δ = −π/2 represents right-hand circular polarization.

3.1.4.3  Elliptical Polarization

The wave represented by the general equations of (3.11) and (3.12) without further conditions is elliptically polarized. Combining Equations 3.15 through 3.17 yields

3.22

The tip of the vector forms an ellipse, which can also be represented by cancelling the varying phase term (ωtkx) as

3.23 $E y 2 A y 2 + E z 2 A z 2 − 2 E y E z A y A z cosδ = sin 2 δ ,$

Figure 3.7   Representation of an elliptically polarized wave by the semimajor (a) and semiminor (b) axes of the ellipse and the orientation of ψ.

where δ = ϕ02 − ϕ01 is the phase difference. Three parameters (Ay, Az, δ) describe an elliptically polarized wave field. The ellipse represented by Equation 3.23 is in general a tilted ellipse, which can also be described by the semimajor (a) and semiminor (b) axes of the ellipse and the orientation of ψ, as shown in Figure 3.7, where the ellipticity angle is defined as tan $χ ± b a$

with the positive sign for left-hand polarization and the negative sign for right-hand polarization (Shen and Kong 1983).

Another way to represent an elliptically polarized wave is to use four parameters with the same dimension, which has been widely used in the remote sensing community but has not received much attention in the radar meteorology community. This parameterization was introduced by G. G. Stokes (1852a, 1852b) and is called the Stokes parameters:

3.24 $I = | E y | 2 + | E z | 2 = A y 2 + A z 2$

3.25
3.26
3.27

Note that there is a relationship among the four parameters, which is

3.28 $I 2 = Q 2 + U 2 + V 2$

for a fully polarized wave. Hence, the four-parameter Stokes notation of [I,Q,U,V] is equivalent to the representation by the three parameters (Ay,Az,δ) in this case.

In the case of a partially polarized or unpolarized wave, the amplitudes and the phase difference are random variables. Then, the definitions of the Stokes parameters (Equations 3.24 through 3.27) are replaced by their respective averages. In this case, it can be shown that Equation 3.28 is replaced by

3.29 $⟨ I 2 ⟩ > ⟨ Q 2 ⟩ + ⟨ U 2 ⟩ + ⟨ V 2 ⟩ ,$

The ratio between the right-hand and left-hand sides, p = (⟨Q2⟩+⟨U2⟩+⟨V2⟩)/⟨I2⟩, is the degree of polarization. For elliptically polarized waves, p = 1; 0 < p < 1 for partially polarized waves, and p = 0 for unpolarized waves. In weather radar polarimetry, we usually deal with a high degree of polarization (except for those from clutter or biological objects)—as shown in Chapter 4, the degree of polarization is closely related to the co-polar cross correlation coefficient (ρhv).

3.2  Scattering Fundamentals

Scattering is a physical process in which an object or objects, called scatterers (e.g., hydrometeors), redirect the wave incidence on them into all directions. Figure 3.8 shows a sketch of wave scattering by a single particle. The question is how to conceptually understand and quantitatively represent the scattering process by a single particle, which is discussed in this section.

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

3.30 $ɛ ( r → ) = ɛ r ( r → ) ɛ 0 = ( ɛ ′ ( r → ) − j ɛ " ( r → ) ) ɛ 0$
3.31 $μ ( r → ) = μ 0 ,$

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

Figure 3.8   Conceptual sketch of wave scattering by a hydrometeor particle.

Assume that the incident wave is a linearly polarized plane wave propagating in free space with a permittivity and permeability of (ε00). Hence, the phasor representation of the plane wave is Equation 3.12, which is

3.32 $E → i = e ^ i E 0 e − j k ^ i • r ^ ,$

where $e ^ i$

is the unit vector for the incident wave polarization; E0 is the amplitude; and $k → i = k k ^ i$ is the incident wave vector, with k as the wave number for the background medium and the unit vector $k ^ i$ 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:

3.33 $E → s = S → ( k ^ s , k ^ i ) E 0 e − j k r r ,$

where $S → ( k ^ s , k ^ i ) = S ( k ^ s , k ^ i ) e ^ s$

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 $( e ^ i 1 , e ^ i2 )$

are the reference unit vectors for the incident wave field and $( e ^ s 1 , e ^ s2 )$ are the reference unit vectors for the scattered wave field, as shown in Figure 3.9, we have

3.34 $E → i = e ̂ i E 0 e − j k → i • r → = ( E i 1 e ̂ i 1 + E i 2 e ̂ i 2 ) e − j k ̂ i • r →$
3.35 $E → s = e ^ s E 0 s e − j k r r = ( E s 1 e ^ s 1 + E s2 e ^ s2 ) .$

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

3.36 $[ E s1 E s 2 ] = e − j k r r [ S 11 ( k ^ s , k ^ i ) S 12 ( k ^ s , k ^ i ) S 21 ( k ^ s , k ^ i ) S 22 ( k ^ s , k ^ i ) ] [ E i1 E i 2 ] .$

Figure 3.9   Coordinate systems for the scattering matrix of a hydrometeor particle.

The matrix $[ S ] = [ S 11 S 12 S 21 S 22 ]$

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 $( e ^ 1 , e ^ 2 )$ 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 $e ^ 1 = e ^ ⊥ = k ^ s × k ^ i | k ^ s × k ^ i |$

. The vectors on the scattering plane are defined as $e ^ i | | = k ^ i × e ^ i ⊥$ and $e ^ s | | = k ^ s × e ^ s ⊥$ for the incident and scattered waves, respectively. In this case, Equation 3.36 becomes

3.37 $[ E s ⊥ E s | | ] = e − j k r r [ S ⊥ ⊥ ( k ^ s , k ^ i ) S ⊥ | | | ( k ^ s , k ^ i ) S | | ⊥ ( k ^ s , k ^ i ) S | | | | ( k ^ s , k ^ i ) ] [ E i ⊥ E i | | ] .$

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

3.38

where

3.39
3.40
3.41

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

3.42 $[ E sh E sv ] = e − j k r r [ S hh ( k ^ s , k ^ i ) S hv ( k ^ s , k ^ i ) S vh ( k ^ s , k ^ i ) S vv ( k ^ s , k ^ i ) ] [ E ih E iv ]$

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

3.43 $S → i = 1 2 Re ( E → i × H → i * ) = | E 0 | 2 2 η 0 k ^ i$

for the incident wave and

3.44 $S → s = 1 2 Re ( E → s × H → s * ) = | E s | 2 2 η 0 k ^ s = | S ( k ^ s , k ^ i ) | 2 r 2 | E 0 | 2 2 η 0 k ^ s$

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 dPs for the scattered wave through a differential area da = r2dΩ is

3.45 $d P s = S s d a = | s ( k ^ s , k ^ i ) | 2 r 2 | E 0 | 2 2 η 0 r 2 d Ω ,$

where dΩ = sinθdθdϕ is the differential solid angle. Using $S i = | E 0 | 2 2 η 0$

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

3.46 $d P s S i − | S ( k ^ s , k ^ i ) | 2 d Ω = σ d ( k ^ s , k ^ i ) d Ω ,$

where $σ d ( k ^ s , k ^ i ) = | S ( k ^ s , k ^ i ) | 2$

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

3.47 $σ bi ( k ^ s , k ^ i ) = 4 π σ d ( k ^ s , k ^ i ) , σ b = 4 π σ b ( − k ^ i , k ^ i ) ,$

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

3.48 $σ s = ∫ 4 π σ d ( k ^ s , k ^ i ) d Ω .$

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:

3.49 $d P = d I × d V = ( J d a ) × ( E d ℓ ) = σ E 2 d a d ℓ ,$

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

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

3.50

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:

3.51

where $( k ^ i , k ^ i )$

is the forward scattering amplitude; σt ≡ σe 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 = σst, 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:

3.52 $Q t = σ t / σ g , Q s = σ s / σ g , Q a = σ a / σ g ,$

where σg is the geometric cross section, and σg = 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.

Figure 3.10   Normalized extinction cross section Qt, absorption cross section Qa, and total scattering cross section Qs of water spheres.

3.3  Rayleigh Scattering

As mentioned earlier, wave scattering by a small particle whose size is much smaller than a wavelength is known as Rayleigh scattering. Rayleigh scattering is discussed based on the conceptual statement and mathematical representation that follow.

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:

3.53 $A s A i : V , r , λ , c , ρ e ,$

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 ρe 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 $A s A i$

. Similarly, the ether density ρe contains the dimension of mass [M]. However, there is no other term that contains the dimension of mass; therefore, the dependence on ρe 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

3.54 $A s A i ∝ V λ 2 r .$

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

3.55 $I s I i ∝ V 2 λ 4 r 2 .$

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.

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).

Figure 3.11   Illustration of wave scattering from a small sphere as a dipole radiation; (a) scattering configuration and (b) sketched process for dipole vibration and radiation.

Therefore, the scattered wave field can be represented by the vector potential $A →$

(Ishimaru 1978, 1997) as

3.56 $E → s ( r → ) = 1 j ωμ 0 ε 0 ∇ × ∇ × A → ( r → )$

and

3.57

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

3.58 $G ( r , r ′ ) = exp ( − j k | r → − r → ′ | ) / ( 4 π | r → − r → ′ | ) ,$

and the equivalent current source is

3.59 $J eq ( r → ′ ) = − j ωε 0 [ ε r ( r → ′ ) − 1 ] E → int ,$

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):

3.60 $E → int = 3 ε r + 2 E → i .$

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 (rr′), we have

3.61 $E → s ( r → ) = e − j k r r E 0 S → ( k ^ s , k ^ i )$

with the scattering amplitude as

3.62

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 εr, both the scattering direction and incident direction $( k ^ s , k ^ i )$

, and the polarization direction of the incident wave field $e ^ i$ . It is noted that χ is the angle between the scattering direction $k ^ s$ and the incident wave polarization $e ^ i$ , as shown in Figure 3.11. The sinχ term can be understood as the unit vector of incident wave polarization $e ^ i$ projected onto the scattered wave polarization of $e ^ s$ , which is perpendicular to the scattered wave propagation direction of $k ^ s$ , 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

3.63

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

3.64

Figure 3.12   Normalized scattering field patterns (top row) and power pattern (bottom row). The left column is in the plane containing the incident wave polarization (E-plane), and the right column is the plane perpendicular to the incident wave polarization (H-plane).

The backscattering radar cross section is then

3.65 $σ b = 4 π σ d ( − k ^ i , k ^ i ) = 4 π k 4 a 6 | ε r − 1 ε r + 2 | 2 .$

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

3.66

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 $E → i$

, the internal wave field inside the sphere is $E → int$ , and the wave field outside is the summation of the incident wave field $E → i$ and the scattered wave field $E → s$ , specifically $E → i + E → s$ . 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

3.67 $r ^ × E → int | r=a = r ^ × [ E → i + E → s ] | r=a$
3.68 $r ^ × H → int | r=a = r ^ × [ H → i + H → s ] | r=a ⋅$

Figure 3.13   Conceptual sketch of the electric field inside a scatterer excited by the incident wave field as dipole (a), quadrupole (b), and hexapole (c) radiation.

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

3.69 $E → i ( r → ) = Σ n = 1 ∞ C n [ M → o 1 n ( 1 ) ( k r , θ , ϕ ) + j N → e 1 n ( 1 ) ( k r , θ , ϕ ) ]$
3.70 $E → s ( r → ) = Σ n = 1 ∞ C n [ − b n M → o 1 n ( 4 ) ( k r , θ , ϕ ) − j a n N → e 1 n ( 4 ) ( k r , θ , ϕ ) ]$
3.71 $E → int ( r → ) = Σ n = 1 ∞ C n [ c n M → o 1 n ( 1 ) ( k ′ r , θ , ϕ ) + d n N → e 1 n ( 1 ) ( k ′ r , θ , ϕ ) ] ,$

where Cn = (−j)nE0(2n + 1)/[n(n + 1)] are the expansion coefficients for the incident field, and (an, bn) and (cn, dn) are those for the scattered and internal wave fields, respectively. $M → m n$

and $N → m n$ 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:

3.72 $a n = Ψ n ( k a ) Ψ n ′ ( k m a ) − m Ψ n ( k m a ) Ψ n ′ ( k a ) ζ n ( k a ) Ψ n ′ ( k m a ) − m Ψ n ( k m a ) ζ n ′ ( k a )$

3.73 $b n = m Ψ n ( k a ) Ψ n ′ ( k m a ) − Ψ n ( k m a ) Ψ n ′ ( k a ) m ζ n ( k a ) Ψ n ′ ( k m a ) − Ψ n ( k m a ) ζ n ′ ( k a ) ,$

where $Ψ n ( x ) = x j n ( x ) = π x / 2 J n + 1 / 2 ( x )$

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

3.74 $[ E s ⊥ E s | | ] = e − j k r r [ S ⊥ ⊥ ( θ ) 0 0 S | | | | ( θ ) ] [ E i ⊥ E i | | ] ,$

where

3.75
3.76

and

3.77

with

and Pn (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

3.78 $σ b = π k 2 | Σ n = 1 ∞ ( 2 n + 1 ) ( − 1 ) n ( a n − b n ) | 2 ,$

Figure 3.14   Sample amplitude patterns for water sphere scattering at X-band: (a) Top row for a sphere with a 2-mm diameter and (b) bottom row for a sphere with a 2-cm diameter.

Figure 3.15   Dependence of efficiency factors on water sphere diameter at X-band.

The total scattering cross section is

3.79 $σ s = 2 π k 2 | Σ n = 1 ∞ ( 2 n + 1 ) ( | a n | 2 + | b n | 2 )$

and the extinction cross section is

3.80 $σ t = 4 π k Im ( s ( 0 ) jk ) = 2 π k 2 Σ n = 1 ∞ ( 2 n + 1 ) Re [ a n + b n ]$

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.

Figure 3.16   Conceptual sketches for resonance effect: (a) reflection model and (b) reflection and creeping wave model.

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  Scattering Calculations For A Nonspherical Particle

As discussed earlier, backscattering by a homogenous sphere exhibits no difference in polarization, meaning that the backscattering amplitudes and the radar cross sections are the same for horizontally and vertically polarized waves. However, most hydro-meteors are not spherical, meaning that they give different radar returns and allow for additional information from polarimetric radar measurements. Wave scattering by a single nonspherical particle is fundamental to understanding and interpreting the polarimetric measurements. Shape and orientation are the two key factors that determine wave-scattering characteristics. Once the shape and orientation of a scatterer are determined, the scattering amplitudes can be calculated using the Rayleigh scattering approximation, T-matrix method, or other numerical approach, depending on its electric size and the complexity of its shape.

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.

Figure 3.17   Basic shape models for hydrometeor scattering: (a) oblate spheroid; (b) prolate spheroid.

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

3.81

where the internal field is

3.82 $E → int = E → i + E → p .$

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):

3.83 $E p = ∫ S d q 4 πε 0 r 2 ( − r ^ • p ^ ) = − L ε 0 P = − L ( ε r − 1 ) E int .$

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):

3.84

where

3.85 $L x = ∫ 0 ∞ a 1 a 2 b 2 ( s + a 1 2 ) [ ( s + a 1 2 ) ( s + a 2 2 ) ( s + b 2 ) ] 1 / 2 d s ,$
3.86 $L y = ∫ 0 ∞ a 1 a 2 b 2 ( s + a 2 2 ) [ ( s + a 1 2 ) ( s + a 2 2 ) ( s + b 2 ) ] 1 / 2 d s ,$
3.87 $L z = ∫ 0 ∞ a 1 a 2 b 2 ( s + b 2 ) [ ( s + a 1 2 ) ( s + a 2 2 ) ( s + b 2 ) ] 1 / 2 d s$

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

3.88 $L x + L y + L z = 1.$

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

3.89 $L x : L y : L z ≈ 1 / a 1 : 1 / a 2 : 1 / b .$

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

3.90 $L x = L y = 1 2 ( 1 − L z )$

and

3.91

for an oblate spheroid (a > b) and

3.92

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 Ly = La and Lz = Lb as follows:

3.93

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 $ε r − 1 3 [ 1 + L a,b ( ε r − 1 ) ]$

, instead of $ε r − 1 ε r + 2$ , 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 $( k ^ s = k ^ i ; Θ = 0 )$

or backward $( k ^ s = − k ^ i ; Θ = π )$ 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 (sa: solid line) is larger than for polarization on the minor axis (sb: 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.

Figure 3.18   Scattering amplitudes of raindrops as a function of equivolume diameter for polarization at major (sa) and minor (sb) axes, respectively.

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 $E → i ( r → )$

, there is the scattered wave field $E → s ( r → )$ outside the particle and the internal wave field $E → int ( r → )$ inside the particle. These wave fields are expanded in vector spherical harmonics

3.94 $E → s ( r → ) = Σ m , n [ a m n M → m n ( 4 ) ( k r , θ , ϕ ) + b m n N → m n ( 4 ) ( k r , θ , ϕ ) ]$
3.95 $E → int ( r → ) = Σ m , n [ c m n M → m n ( 1 ) ( k r , θ , ϕ ) + d m n N → m n ( 1 ) ( k r , θ , ϕ ) ]$
3.96 $E → i ( r → ) = Σ m , n [ e m n M → m n ( 1 ) ( k ′ r , θ , ϕ ) + f m n N → m n ( 1 ) ( k ′ r , θ , ϕ ) ] .$

Because the incident wave is known, the expansion coefficients (emn, fmn) are also known. That leaves four sets of expansion coefficients, including the scattering coefficients (amn, bmn) and the internal coefficients (cmn, dmn), 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.

Figure 3.19   Illustration of wave scattering by a nonspherical scatterer and the concept of extended boundary conditions.

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

3.97

where $G ¯ ¯ ( r → , r → ′ )$

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

3.98 $[ e m n f m n ] = [ A ] [ c m n d m n ] ,$

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

3.99

Using Equations 3.94 and 3.95 in Equation 3.99, we obtain

3.100 $[ a m n b m n ] = [ B ] [ c m n d m n ] .$

Solving (cmn, dmn) from Equation 3.98 and substituting them into Equation 3.100, we have

3.101 $[ a m n b m n ] = [ B ] [ A ] − 1 [ e m n f m n ] = [ T ] [ e m n f m n ] ,$

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 (amn, bmn) are known, the scattered wave field and the scattering amplitude/matrix can be found.

Figure 3.20   Magnitudes and phases of scattering amplitudes as a function of raindrop size at S-band (a), C-band (b), and X-band (c) frequencies.

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   Normalized backscattering cross section (a), backscattering magnitude ratios (b), backscattering phase differences (c), and the real part of the forward scattering amplitudes (d) as a function of raindrop size at S-band, C-band, and X-band frequencies.

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 (ρhv) between the dual-polarizations. The real part of the forward scattering amplitudes is associated with the specific differential phase (KDP). These are described in Chapter 4.

Figure 3.22   Magnitudes (left column) and ratios (right column) of scattering amplitudes as a function of particle size at S-band for snow (top row) and hail (bottom row).

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.

3.5.4  Other Numerical Methods for Scattering Calculations

Besides T-matrix methods, other numerical methods have also been developed for scattering calculations. These include physical optics (Born and Wolf 1999), method of moment (Harrington 1968), and discrete dipole approximation (Goodman et al. 1991; Purcell and Pennypacker 1973). The numerical methods can be used to calculate wave scattering by irregularly shaped objects such as terrain, vegetation, and biological objects (Zhang 1998; Zhang et al. 1996).

3.6  Scattering for Arbitrary Orientations

The scattering amplitudes sa and sb represent wave scattering for polarizations in the major and minor axes of a spheroid (Figure 3.23). Natural hydrometeors, however, occur in random orientations, and their major and minor axes don’t necessarily align with the radar polarization base directions (typically horizontal and vertical). Whereas raindrops fall with their major axis aligned mostly horizontally, hailstones tumble as they fall, yielding random orientations. Hence, the scattering amplitudes for arbitrary orientation are needed. In general, the problem needs to be solved by the EM boundary condition for such scattering configurations. In special cases, wave scattering by a canted particle can be derived with the simple approaches discussed below.

3.6.1  Scattering Formulation Through Coordinate Transformation

When a scatterer cants in the polarization plane with an angle of ϕ, wave scattering can be represented by either local (body) polarization reference $( h ^ ′ , v ^ ′ )$

(symmetry axis of the scatterer) or global (radar) polarization reference $( h ^ , v ^ )$ . As shown earlier, with the local polarization reference, the wave scattering can be expressed by

Figure 3.23   Coordinate systems for wave scattering by a scatterer canted in polarization plane with an orientation angle ϕ; (h^′,v^′) is the local polarization reference on its major and minor axis, and (h^,v^) is the radar polarization reference.

3.102 $[ E sh ′ E sv ′ ] = e − j k r r [ S a 0 0 S b ] [ E ih ′ E iv ′ ] = e − j k r r S ¯ ¯ ( b ) E ¯ i ′ .$

There is no cross-polarization in this case because the polarization reference is on the scatterer’s symmetry axis.

Because the local reference is a rotation of the global reference system by an angle of ϕ, the relation of the wave field represented in the two polarization references is

3.103

Using Equations 3.103 and 3.102, we have the scattered wave field in the global reference system as

3.104 $E ¯ s = R ¯ ¯ ( ϕ ) E ¯ s ′ = e − j k r r R ¯ ¯ ( ϕ ) S ¯ ¯ ( b ) E ¯ i ′ = e − j k r r R ¯ ¯ ( ϕ ) S ¯ ¯ ( b ) R ¯ ¯ − 1 ( ϕ ) E ¯ i ≡ e − j k r r S ¯ ¯ E ¯ i ,$

where

3.105

is the scattering matrix with the canting angle taken into account.

3.6.2  General Expression for Rayleigh Scattering

In the case of the Rayleigh approximation, wave scattering for an arbitrary orientation can be described through the projection of a dipole radiation as follows.

As shown in Figure 3.24, a scatterer has its principal axis in the zb direction of the scattering body coordinate system (xb, yb, zb), which has an orientation (θb, ϕb) in the global coordinate system (x, y, z). The two coordinate systems are related by

3.106

According to the Rayleigh scattering approximation, as expressed in Equations 3.81 through 3.84, the scattering amplitude can be written as

3.107 $s → ( k ^ s , k ^ i ) = k 2 4 πε 0 [ p → − k ^ s ( k ^ s • p → ) ] ,$

and the scatterer polarization $p →$

can be expressed in the scatterer body coordinate system as

3.108 $[ p x p y p z ] = [ α x 0 0 0 α y 0 0 0 α z ] [ E ix E iy E iz ] ,$

Figure 3.24   Coordinate systems for wave scattering by a spheroidal scatterer with orientation angle (θb, ϕb); (x, y, z) is the global coordinate system, and (xb, yb, zb) is the scatterer body coordinate system.

where

3.109 $α j = V ε 0 ε r − 1 1 + L j ( ε r − 1 ) .$

Projecting the scattering amplitude to the reference (horizontal and vertical) polarization directions $( h ^ , v ^ )$

defined in Equation 3.38 and then writing them in matrix form, we obtain the scattering matrix

3.110 $S ¯ ¯ = k 2 4 πε 0 P ¯ ¯ s α ¯ ¯ ( b ) P ¯ ¯ i ,$

where the projection matrix

3.111 $P ¯ ¯ s = [ h ^ s • x ^ b h ^ s • y ^ b h ^ s • z ^ b v ^ s • x ^ b v ^ s • y ^ b v ^ s • z ^ b ]$

for the scattered wave field, and the projection matrix

3.112 $C ¯ ¯ i = [ h ^ s • x ^ b v ^ s • x ^ b h ^ s • y ^ b v ^ s • y ^ b h ^ s • z ^ b v ^ s • z ^ b ]$

for the incident wave field. It can be shown that each element is

3.113 $S hh = k 2 4 πε 0 ( α x ( h ^ s • x ^ b ) ( h ^ i • x ^ b ) + α y ( h ^ s • y ^ b ) ( h ^ i • y ^ b ) + α z ( h ^ s • z ^ b ) ( h ^ i • z ^ b ) )$
3.114 $S hv = k 2 4 πε 0 ( α x ( h ^ s • x ^ b ) ( v ^ i • x ^ b ) + α y ( h ^ s • y ^ b ) ( v ^ i • y ^ b ) + α z ( h ^ s • z ^ b ) ( v ^ i • z ^ b ) )$
3.115 $S vh = k 2 4 πε 0 ( α x ( v ^ s • x ^ b ) ( h ^ i • x ^ b ) + α y ( v ^ s • y ^ b ) ( h ^ i • y ^ b ) + α z ( v ^ s • z ^ b ) ( h ^ i • z ^ b ) )$
3.116 $S vv = k 2 4 πε 0 ( α x ( v ^ s • x ^ b ) ( v ^ i • x ^ b ) + α y ( v ^ s • y ^ b ) ( v ^ i • y ^ b ) + α z ( v ^ s • z ^ b ) ( v ^ i • z ^ b ) ) .$

3.6.3  Backscattering Matrix for a Spheroid

In the case of a spheroid, we have αx = αy = αa and αz = αb. To simplify the expressions in Equations 3.113 through 3.116 for backscattering with horizontal incidence, let θi = π/2, ϕi = π, θs = π/2, and ϕs = 0; we have

3.117 $h ^ i = − y ^ ; v ^ i = z ^$
3.118 $h ^ s = y ^ ; v ^ s = z ^ .$

From Equation 3.106, we have the body orientation vectors

3.119
3.120
3.121

Applying Equations 3.117 through 3.121 to Equations 3.113 through 3.116, we obtain the scattering amplitude elements as follows:

3.122

3.123
3.124

3.6.4  Forward Scattering Alignment Versus Backscattering Alignment

Thus far, we have used the scatterer location as the origin of the coordinate system and with reference vectors of $( h ^ i , v ^ i , k ^ i )$

and $( h ^ s , v ^ s , k ^ s )$ , which is convenient in theoretical scattering calculations. In monostatic radar applications, however, this is not convenient because it requires a change in the reference polarization and wave vector (Bringi and Chandrasekar 2001; Ulaby and Elach 1990), as shown in Figure 3.25. This is called the forward scattering alignment (FSA: $( h ^ s , v ^ s , k ^ s )$ ) convention.

In the case of FSA, both $h ^ s$

and $k ^ s$ have directions opposite to that of the incident wave in the back direction. To avoid the changing the directions of the reference vectors, a set of radar-based reference vectors $( h ^ r , v ^ r , k ^ r ) = ( − h ^ s , v ^ s , − k ^ s )$ are used, and

Figure 3.25   Diagram of coordinate systems for forward scattering alignment (FSA) and back scattering alignment (BSA) conventions.

it follows $( h ^ r , v ^ r , k ^ r ) = ( h ^ i , v ^ i , k ^ i )$

in backscattering, called the back scattering alignment (BSA) convention. Hence, the scattering amplitude/matrix in BSA is related to that in FSA by

3.125 $[ s hh s hv s vh s vv ] B = [ − 1 0 0 1 ] B [ s hh s hv s vh s vv ] = [ − s hh − s hv s vh s vv ] .$

Using Equations 3.122 through 3.124 in Equation 3.125, we obtain

3.126

It is worth discussing two extreme cases when the scatterer is oriented in a specific direction. If the scatterer is canted in the polarization (yz) plane, meaning ϕb = π/2. Let the canting angle in the polarization plane be θb ≡ ϕ; Equation 3.126 reduces to

3.127

This shows that both the horizontal and vertical scattering amplitudes change from those amplitudes based on the polarizations on the major and minor axes, and there is cross-polarization due to canting. This is expected, because the projected dimensions in the horizontal and vertical directions change from those of the major and minor axes, and the electric wave field inside the scatterer is no longer along the symmetry axis, hence yielding cross-polarization components. The co-polar ratio and cross-to-co-polar ratio are plotted in Figure 3.26.

Figure 3.26   Dependences of polarization ratio on canting angle: (a) co-polar ratio and (b) cross-to-co-polar ratio.

By contrast, if the scatterer is canted in the scattering (zx) plane, ϕb = 0. Let the canting in the scattering plane be θb ≡ ϑ; Equation 3.126 becomes

3.128

This makes physical sense, because the scatterer’s horizontal dimension does not change when it is canted in the scattering plane; hence, there is no change in scattering for the horizontal polarization, and there is no cross-polarization component.

3.6.5  Scattering Matrix by a Spheroid with any Orientation

In the last subsection, we obtained the scattering matrix for an arbitrary orientation expressed by either Equation 3.126 or by combining Equations 3.127 and 3.128. To combine Equations 3.127 and 3.128, we replace “sb” in Equation 3.127 with “sa sin2 ϑ + sb cos2 ϑ” based on Equation 3.128. Because we are using the backscattering alignment in this book, we can omit the subscript b and have

3.129

where the canting angle dependent factors are

3.130 $A = cos 2 ϕ + sin 2 ϑ sin 2 φ$
3.131 $B = cos 2 ϑ sin 2 φ$
3.132 $C = cos 2 ϑ cos 2 φ$
3.133 $D = sin 2 φ + sin 2 ϑ cos 2 φ .$

Once the canting angles (ϑ, ϕ) are known, the scattering matrix can be calculated using Equations 3.129 through 3.133 for a particle with any orientation. In reality, however, hydrometeors are randomly orientated; the canting angles (ϑ, ϕ) should be treated as random variables, and their statistics should be characterized—these points are addressed in Chapter 4.

Appendix 3A  Derivation Of Optical Theorem (Forward Scattering Theorem)

Consider a plane wave Ei incident on a particle (Figure 3A.1); the scattered wave field is Es, and the total wave field is

3A.1 $E → = E → i + E → s and H → = H → i + H → s .$

Then, the total absorbed power is

3A.2 $P a = − ∮ A 1 2 Re ( E → × H → * ) • d a → .$

Substitution of Equation 3A.1 into Equation 3A.2 yields

3A.3 $P a = − ∮ A 1 2 Re [ ( E → i + E → s ) × ( H → i + H → s ) * ] • d a → = ∮ A 1 2 Re ( E → i × H → i * + E → s × H → s * + E → i × H → s * + E → s × H → i * ) • d a → .$

Noting $S → i = 1 2 Re ( E → i × H → i * )$

and $S → s = 1 2 Re ( E → s × H → s * )$ , we have

3A.4
3A.5

Figure 3A.1   A sketch of wave scattering by a nonspherical scatterer.

Using Equations 3A.4 and 3A.5 in Equation 3A.3, we have

3A.6

Using the vector formula

3A.7 $∇ ⋅ ( A → × B → ) = B → • ∇ × A → − A → • ∇ × B → ,$

we have

3A.8 $∇⋅ ( E → i * × H → ) = H → • ∇× E → i * − E → i * • ∇× H → = H → • ( j ω μ 0 H → i ) * − E → i * • ( − j ω ɛ E → ) = − j ω μ 0 H → • H → i * + j ω ɛ r ɛ 0 E → • E → i *$
3A.9 $∇ ⋅ ( E → × H → i * ) = j ωμ 0 H → • H → i * − j ωε 0 E → • E → i * .$

Combining Equations 3A.8 and 3A.9, taking the real part, we have

3A.10 $Re [ ∇⋅ ( E → i * × H → + E → × H → i * ) ] = Re [ 0 − j ω ɛ 0 ( ɛ r − 1 ) E → • E → i * ] = − Im [ ω ɛ 0 ( ɛ r − 1 ) E → • E → i * ] .$

Substitution of Equation 3A.10 into Equation 3A.6 yields

3A.11 $P a + P s = Im ∫ V 1 2 ω 0 ε 0 ( ε r − 1 ) E → • E → i * • d v .$

From Equation 2.20 (Ishimaru 1997),

3A.12

In the forward direction, Equation 3A.12 becomes

$e ^ i s ( k ^ i , k ^ i ) = k 2 4 π E 0 ∫ V ( ε r − 1 ) E → ( r → ) • d v$

Hence,

3A.13 $∫ v ( ε r − 1 ) E → • E → i * • d v = 4 π | E 0 | 2 2 k 2 s ( k ^ i , k ^ i ) e ^ i • e ^ i .$

Substitution of Equation 3A.13 into 3A.11 leads to

3A.14 $P a + P s = 4 π ω ɛ 0 | E 0 | 2 2 k 2 Im [ s ( k ̂ i , k ̂ i ) ] = 4 π k | E 0 | 2 2 η 0 Im [ s ( k ̂ i , k ̂ i ) ]$
3A.15 $P t = S i σ t = S i 4 π k Im [ s ( k ^ i , k ^ i ) ] .$

Hence,

3A.16 $σ t = 4 π k Im [ s ( k ^ i , k ^ i ) ] .$

Appendix 3B  Vector Spherical Wave Harmonics

Let the eigen solution of a scalar wave equation in spherical coordinates be

3B.1

where zn(kr) is one of the four spherical Bessel functions: jn, yn, $h n ( 1 )$

, and $h n ( 2 )$ , depending on whether the wave is incoming or outgoing. Because we use the phasor of ejωt, $z n ( 1 ) = j n$ represents the incident and internal wave, and $z n ( 4 ) = h n ( 2 )$ represents the scattered wave. The vector spherical wave harmonics are defined based on ψmn as

3B.2 $M → m n = ∇ × ( Ψ m n r → )$
3A.3 $N → m n = 1 k ∇ × M → m n .$

Problems

• 3.1 As discussed in this chapter, a plane wave with any polarization can be represented by
3.134
Let Ay = Az, plot the trace of the tip of the electric field in the yz plane and give the name for each polarization for the following phase differences:
• δ2 − δ1 = 0 and δ2 − δ1 = π
• $δ 2 − δ 1 = ± π 2$
• $δ 2 − δ 1 = ± π 4$
• $δ 2 − δ 1 = ± 3 π 4 .$
What polarization (of the wave described above) is the polarimetric WSR-88D radar likely to transmit?
• 3.2 Using the given cross sections (hw2p2.dat) calculated from the Mie theory, plot the Q factors as functions of the particle diameter. The four columns are the particle diameter in millimeters, extinction cross section, scattering cross section, and backscattering cross section, respectively. It is known that the relative dielectric constant is (41, 41) and the wavelength is 3 cm. Calculate the Q factors using the Rayleigh scattering approximation and compare them with the results from the Mie theory. In addition, calculate and show the scattering albedo for both the Mie theory and the Rayleigh approximation. Discuss the valid regime for Rayleigh scattering, and explain the extinction paradox.
• 3.3 Assuming raindrops to be oblate spheroids with the following axis ratio (represented in polynomial form): γ = b/a = 0.995 + 0.251D − 0.0364D2 + 0.005303D3 − 0.0002492D4 and assuming the relative dielectric constant to be (80, 17), do the following:
• Using the Rayleigh scattering approximation, calculate the scattering amplitudes of the raindrops at S-band (2.8 GHz) and plot them as a function of equivolume diameter ranging from 0.1 to 8 mm. Using the provided results (hm4.dat, rigorously calculated with the T-matrix method), verify your calculations. Show the scattering magnitudes of both your results and the provided results in terms of real and imaginary parts as well as magnitude and phase and explain the similarities and differences.
• Write down the formula used in your calculation and discuss how the ratio of the scattering amplitudes |sa/sb| changes when the dielectric constant is replaced by that of dry snow with a fractional volume of 10%. Explain why this change occurred.
• 3.4 Show that the polarizability of a coated sphere (with the expression below) is equal to that of a mixture sphere calculated using the Maxwell-Garnett mixing formula with medium 2 as its background (the Maxwell-Garnett formula is discussed in Chapter 2). The expression of the coated sphere polarizability is
$α = V ε 0 × 3 ( ε 2 − 1 ) ( ε 1 + 2 ε 2 ) + f v ( ε 1 − ε 2 ) ( 1 + 2 ε 2 ) ( ε 2 + 2 ) ( ε 1 + 2 ε 2 ) + f v ( 2 ε 2 − 2 ) ( ε 1 − ε 2 ) ,$
where ε1 and ε2 are the relative dielectric constant of the inner sphere and outer shell, respectively. The fractional volume is fv = a3/a′3, where a and a′ are the radius for the inner and outer spheres, respectively. Use the expression $α = V ε 0 3 ( ε e − 1 ) ( ε e + 2 )$ for the effective polarizability of the coated sphere, where εe is the effective dielectric constant for the mixture sphere.
• 3.5 Assume that the backscattering amplitudes for polarizations on the major and minor axes are sa and sb, respectively. Show that the backscattering matrix in the reference of horizontal and vertical polarizations for a scatterer with a canting angle of ϕ in the polarization plane follows

At that time, it was not yet understood that the EM wave can propagate in free space. It was rather thought to propagate within a medium called “ether,” like a sound wave propagates in air.