A Discussion on the Applicable Condition of Rayleigh Scattering

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International Journal of Remote Sensing Applications (IJRSA) Volume 5, 2015 doi: 10.14355/ijrsa.2015.05.007

A Discussion on the Applicable Condition of Rayleigh Scattering Nan Li*1, Yiqing Zhu2, Zhenhui Wang3 Key Laboratory for Aerosol-Cloud-Precipitation of China Meteorological Administration, School of Atmospheric Physics, Nanjing University of Information Science & Technology, No.219, Ningliu Road, Nanjing, China shangjineh@163.com; 21316jessica@sina.com; 3eiap@nuist.edu.cn

*1

Abstract Based on the mechanism of particle scattering, rain detection radars are able to receive the backscattering and thus can detect the precipitation particles. For radars with different wavelength, scattering from precipitation particles may be approximated with different kinds of scattering theory, i.e. Mie scattering and Rayleigh scattering. When Mie scattering is used, the computation of the physical quantity that characterizes backscattering of the particle is completed, and the computation is much simpler when Rayleigh scattering is used. In traditional methods, a fixed threshold of the particle scale parameter is used as the criterion to discriminate Rayleigh scattering, that is, the particle size should be smaller than the wavelength of electromagnetic wave. In this work, the analysis on raindrop scattering against radars with different wavelength are discussed. It is concluded that the backscattering cross-section ratio of Mie scattering to Rayleigh scattering is more reasonable than the particle scale parameter for the measure of the criterion to discriminate Rayleigh scattering. Moreover, a small particle compared with the wavelength is a sufficient but not necessary condition for Rayleigh scattering.

Keywords Mie Scattering; Rayleigh Scattering; Applicable Condition

Introduction The earth's atmosphere contains large amounts of gas molecules, aerosols, and cloud and precipitation particles.

These particles will produce scattering when the electromagnetic wave (such as sunlight and the microwave emitted by radars) propagates in the atmosphere and encounters the particles. The incident electromagnetic wave makes the formation of the charge and current distribution in a particle which contributes to the electric multiple moment and the magnetic multiple moment. Since the electromagnetic field of the incident electromagnetic wave is alternating, the multiple moment and the magnetic multiple moment are also alternated in the particle, and thus electromagnetic wave radiate outward, i.e. the scattering electromagnetic wave. The scattering of the particle only changes the propagation direction of the incident electromagnetic wave and does not convert the electromagnetic wave into other forms of energy. The result of the particle scattering is that radars can receive echoes. The main particles that can produce scattering for the electromagnetic wave emitted by rain detection radars are precipitation particles [1].

The scattering characteristics are related to the size, shape, and other physical properties of the particle, in addition to the incident electromagnetic wave. Therefore, characteristics of the scattering wave can be learned if the physical properties of the particle are learned; conversely, the physical properties of the particle can be inferred if characteristics of the scattering wave are learned. Radar generally only receives the part of the scattering wave that transmits in return along the direction of the incident wave emitted by the radar antenna, i.e., the backscattering, so properties of the particle can be studied through the analysis of the backscattering. The physical quantity to characterize the backscattering ability of the particle is the backscattering cross-section (also called radar crosssection).

In addition, the dimensionless scale parameter α is a frequently used factor to study physical properties including the backscattering of the particle [2], = α

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2π r π d =

λ

λ

(1)

International Journal of Remote Sensing Applications (IJRSA) Volume 5, 2015

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where r is the radius of the particle, d is the diameter of the particle, λ is the wavelength of the incident electromagnetic wave. Through α, the scale of the particle is measured relatively to the wavelength of electromagnetic wave in scattering study.

Mie Scattering Mie scattering theory was established by Gustav Mie when he studied the scattering of colloidal metal particles [3], and has been studied by many others [4,5]. From the Maxwell equations, he derived the exact solution of the scattering wave of a homogeneous spherical particle against plane electromagnetic wave. The solution is a mathematical series that gives the amplitude of the scattering electromagnetic field of the particle as well as the amplitude of the electromagnetic field in the particle in far field, called Mie scattering coefficients. (2a) and (2b) are the Mie scattering coefficients of single spherical particle [6] an =

µ m 2 jn ( mα )[α jn (α )]'− µ1 jn (α )[mα jn ( mα )]' µ m 2 jn ( mα )[α hn(1) (α )]'− µ hn(1) (α )[mα jn ( mα )]'

(2a)

bn =

µ1 jn ( mα )[α jn (α )]'− µ jn (α )[mα jn ( mα )]' µ1 jn ( mα )[α hn(1) (α )]'− µ hn(1) (α )[mα jn (mα )]'

(2b)

where α is the scale parameter, m is the complex refractive index, ε1 and μ1 are the permittivity and permeability of the particle and ε and μ are the permittivity and permeability of the ambient medium, jn and yn, and hn(1)=jn+iyn, are spherical Bessel functions of order n of the arguments, n is ranging from 1 to ∞. A large number can be adopted for n when Mie scattering coefficients are actually calculated [7].

The backscattering cross-section of the particle can be calculated through (2),

σ M=

λ2 4π

∞

∑ (−1) (2n + 1)(a

2

n

n =1

n

− bn )

(3)

Rayleigh Scattering Mie scattering can be used for all size particles, but its computation is complex. When the scale parameter α is very small, i.e. the particle is much smaller than the wavelength of the incident electromagnetic wave, Rayleigh scattering can replace Mie scattering in computation. Neglecting terms of higher than sixth power of α in (2), only a1, b1 and b2 are significant, and they are equivalent to Rayleigh scattering proposed by Lord Rayleigh [8,9] i a1 = − ( m 2 − 1)α 5 45 2i m 2 − 1 3 3 m 2 − 2 2 2i m 2 − 1 b1 = − ) α (1 + α − 2 3 m +2 5 m2 + 2 3 m2 + 2

(4a) (4b)

i m2 − 1 5 (4c) α 15 2m 2 + 3 (4) is substituted into (3) and the backscattering cross-section given by Mie scattering would be approximated by Rayleigh scattering such as b2 =

2

= σR

2

λ 2 6 m2 − 1 π 5 m2 − 1 6 = α d 2 m +2 λ 4 m2 + 2 4π

(5)

The Applicable Condition Of Rayleigh Scattering For the same particle, the backscattering cross-sections calculated by Mie scattering and Rayleigh scattering are different, and Rayleigh scattering is the approximation of Mie scattering. Through the comparison of (5) and (3), it can be found that the calculation of Rayleigh scattering is much simpler than Mie scattering. However, only when the scale of the particle is in certain ranges, can Rayleigh scattering be adopted in computation, otherwise, Mie scattering must be used. In this study, 4 radars to detect raindrops with different wavelength are taken as examples to discuss the applicable condition of Rayleigh scattering. They are an S band (10 cm), a C band (5 cm), and an X

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International Journal of Remote Sensing Applications (IJRSA) Volume 5, 2015

band (3 cm) ground-based radar and a Ku band (2.2 cm) satellite-based radar.

The Use of the Scale Parameter Α Traditional methods use a threshold of the scale parameter α as the criterion to discriminate Rayleigh scattering.

They emphasized that when α was much smaller than 1, the Rayleigh approximation can be used without correction [10-12]. Some scholars discussed the range of Rayleigh scattering for different wavelengths and phase states [9,13-16], and set their sights on the range of small α. In general, 0.13 is generally employed. When α<0.13, Rayleigh scattering can be adopted for approximation calculation and when α>0.13, Mie scattering should be used.

Corresponding to this threshold, when the particle is small enough compared with the wavelength of electromagnetic wave, Rayleigh scattering can be adopted.

Table 1 gives the critical diameter of the raindrop when α<0.13 is adopted as the applicable condition of Rayleigh scattering [10,17]. It can be seen from Table 1 that for the same scale parameter, the critical sizes of raindrops are different for different wavelength. Meanwhile, it should be noted that the backscattering cross-section ratios of Mie scattering to Rayleigh scattering are different for different wavelength. This means that when a fixed threshold of α is used as the criterion, the approximation of Rayleigh scattering is not consistent for different wavelength. For example, there can be about an error of 5% error for the Ku band radar while an error of 8% for the S band radar when Rayleigh scattering is adopted.

In fact, the backscattering of a particle changes gradually with its scale, and a scale parameter threshold of 0.13 is a general empirical criterion. Since the aim to give applicable condition of Rayleigh scattering is the approximate calculation of the backscattering, a more reasonable and consistent criterion might be provided for Rayleigh scattering.

TABLE 1 THE CRITICAL DIAMETER OF THE RAINDROP WHEN Α<0.13 IS ADOPTED AS THE APPLICABLE CONDITION OF RAYLEIGH SCATTERING

Band

λ (cm)

S

10

C

5

X

3

Ku

2.2

α

0.13

d(mm)

σM/σR

4.138

0.9193

2.069

0.9274

1.2414

0.9408

0.9104

0.953

The Use of the Backscattering Cross-Section Ratio of Mie Scattering to Rayleigh Scattering Because Rayleigh scattering is an approximation of Mie scattering, the backscattering cross-section ratio of Mie scattering to Rayleigh scattering can be used as an indication of the approximation degree of Rayleigh scattering.

The closer σM/σR approaches 1, the better the degree of the approximation by Rayleigh scattering will be. The difference between backscattering cross-sections calculated by Mie scattering and Rayleigh scattering should be analyzed. Fig. 1 gives the backscattering cross-section ratios of Mie scattering to Rayleigh scattering for different wavelength against the raindrop diameter d. The maximum diameter for analysis is taken as 6.5 mm which is a little larger than the limit size of the raindrop [17,18].

It can be seen from Fig. 1 that (1) For the S band radar with 10 cm wavelength, the backscattering cross-section calculated by Rayleigh scattering is larger than Mie scattering, and the difference increases with the particle diameter. Only when the raindrop is small enough, can Rayleigh scattering give a good approximation to Mie scattering.

(2) For radars with 5cm, 3 cm and 2.2 cm wavelength, the backscattering cross-section calculated by Rayleigh scattering is smaller than Mie scattering at first but larger than Mie scattering later with the increasing raindrop diameter. There is more than one place where σM/σR is equal to 1, and around these places Rayleigh scattering gives a good approximation to Mie scattering. Therefore, not only very small raindrops also other conditional size raindrops are suitable for Rayleigh scattering.

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FIG. 1 THE BACKSCATTERING CROSS-SECTION RATIOS OF MIE SCATTERING TO RAYLEIGH SCATTERING FOR DIFFERENT WAVELENGTH AGAINST THE DIAMETER OF THE RAINDROP

When σM/σR is close enough to 1, Rayleigh scattering can be adopted without reference to the wavelength.

Therefore, ｜σM/σR-1︱<0.05 can be used as a uniform applicable condition of Rayleigh scattering. This threshold can ensure that the approximation degree of Rayleigh scattering is above 95% (i.e. the error is below 5%) for different wavelength.

Table 2 gives the scale range of raindrops when ｜σM/σR-1︱<0.05 is adopted as the applicable condition of Rayleigh scattering. It can be seen from Table 2 that with respect to a uniform approximation of Rayleigh scattering, critical values of the scale parameter α are different for radars with different wavelength. Furthermore, besides the scale range that α is very small, there is another scale range of raindrops for C, X and Ku band radar. With these scale ranges, the calculation of Rayleigh scattering is more accurate and consistent than traditional methods that directly use a fixed value of α as the criterion.

TABLE 2 THE SCALE RANGE OF A RAINDROP WHEN ｜ΣM/ΣR -1︱<0.05 IS ADOPTED AS THE APPLICABLE CONDITION OF RAYLEIGH SCATTERING

Band

λ (cm)

d(mm)

α

S

10

0-3.3312

0-0.1046

C

5

0-1.743

0-0.1095

5.1965-5.2492

0.3265-0.3298

X

3

0-1.146

0-0.12

3.0661-3.1469

0.3211-0.3295

Ku

2.2

0-0.9383

0-0.134

2.1925-2.3054

0.3131-0.3292

σM/σR

0.95-1.05

Conclusion Scattering from particles may be computed with different scattering theory, i.e. Mie scattering and Rayleigh scattering. The computation of Mie scattering is accurate but complex, while the computation of Rayleigh scattering is much simpler but it is an approximation of Mie scattering and is conditional. In this work, the applicable condition of Rayleigh scattering is discussed through the analysis on backscattering cross-sections of raindrops for four rain detection radars with different wavelength. Traditional methods use a fixed threshold of the particle scale parameter as a criterion to discriminate Rayleigh scattering but the approximation of Rayleigh scattering is not uniform for different wavelengths. In addition, traditional methods only give the scale range for

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International Journal of Remote Sensing Applications (IJRSA) Volume 5, 2015

Rayleigh scattering that the particle is very small compared with the wavelength. In this study, it should be emphasized that the backscattering cross-section ratio of Mie scattering to Rayleigh scattering is more reasonable than the particle scale parameter as a measure of the criterion. When the ratio is closer to 1, Rayleigh scattering is closer to Mie scattering, which is the essence of the applicable condition of Rayleigh scattering. By setting an allowable error of Rayleigh scattering compared with Mie scattering (e.g. 5%), scale ranges of the particle can be determined for Rayleigh scattering given the wavelength. Since scale ranges of the particle may be not confined to small size, a very small particle compared with the wavelength is a sufficient but not necessary condition for Rayleigh scattering.

ACKNOWLEDGMENT

This study was supported by the Special Scientific Research Fund of Meteorological Public Welfare Profession of China (GYHY201306078), the Young Scientists Fund of the National Natural Science Foundation of China (41305031), the Young Scientists Fund of the Natural Science Foundation of Jiangsu Province of China (BK2012466), and the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institution.

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