Anomalous diffraction approximation to the light scattering coefficient spectra of marine particles with power-law size distribution

Maciej Matciak

doi:10.1364/OE.20.027603

1. Introduction

Particulate matter, commonly occurring in the sea water, participates in the attenuation of the light energy and therefore influences the radiance and irradiance fields in the sea. Particles are very efficient scatterers and they are mainly responsible for the diffusive component of these fields, especially the upwelling radiant flux, which eventually leaves the sea surface [1]. Apart from the optical properties of particles, the light attenuation by marine suspended matterdepends on their sizes and concentrations, which are taken into account through the particle size distribution (PSD). PSD represents the number of particles per unit particle size and the suspension volume [2]. At large-size scale most of marine PSDs display a rapid decrease in the number of large particles. To a first approximation such a trend can be described by a decreasing power-law function [2–5]:

P S D = m {(\frac{D}{D_{o}})}^{- k},

where D is a characteristic particle size, m and k (the slope coefficient) are positive constants. Measurements of size distributions of irregular in shape particles are usually related to the particle’s volume (e.g. when using Coulter counter) and then the sphere volume-equivalent diameter is considered to be the characteristic size [2]. The unit size D_o renders the ratio

D / D_{o}

and slope coefficient non-dimensional. Since this approximation represents the basic common feature of marine PSDs it is frequently taken into account in studies on particulate light attenuation in the sea [6–9].

The basic parameter characterizing the ability of particles to scatter the light is the spectral scattering coefficient. It is defined as a fraction of monochromatic light power scattered out of narrow collimated beam by particles per unit distance travelled by the beam in the suspension [10]. When spherical particles with power-law PSD in the infinite diameter size intervals (0, ∞ µm) are non-absorbing and their index of refraction is constant, the scattering coefficient spectrum is described by a power function of the light wavelength: $λ^{- (k - 3)}$ [11]. This well-known dependency was derived based on the exact Mie solution to the light scattering assuming that $3 < k < 7$ when the values of the scattering coefficient are finite. Actually, the particle absorption can cause distinct deviations from the aforementioned power dependency [8, 12]. For absorbing particles the exact solutions are very complex [13]. This definitely makes the interpretation of the spectral variability of the scattering coefficient much more difficult. Some progress in this matter can be achieved by means of simplified approaches, such as the anomalous diffraction approximation (ADA). ADA was developed in order to determine the light attenuation efficiencies of optically “soft” particles whose refractive index relative to the surrounding medium is close to unity [14]. Most marine particles satisfy this condition [2]. Particularly in the case of polydispersive particles ADA offers a significant simplification and reasonable accuracy in calculating their optical attenuation. In the case of non-absorbing particles described above ADA predicts the same spectral dependency. However, it can be shown that scattering coefficients obtained with the use of this approximation are finite for the values of k varying within a narrower range, i.e. $3 < k < 5$ . When the power-law PSD slope coefficient is higher ( $5 \leq k < 7$ ) the scattering coefficients become infinitely large due to the fact that ADA overestimates the scattering intensity of the smallest particles in the so-called Rayleigh size domain [14]. Fortunately, natural marine particle assemblages are usually characterized by the values of k ranging between 3 and 5 [4, 5, 12].

This study presents how to obtain a general analytical expressions for scattering coefficient of the particles with power law PSD in the infinite interval of their sizes based on ADA. This approach was examined against the exact solution to the light scattering for spherical particles in the (0.02, 500 μm) diameter range. It is considered that such a range of sphere volume-equivalent diameters covers most of the optically significant marine particles [15].

2. ADA scattering coefficients

The anomalous diffraction approximation for the light attenuation of arbitrarily shaped homogenous particle in the water allows to determine the extinction (C_e) and absorption (C_a) cross sections by means of the surface integrals [16]:

C_{e} = Re \iint_{P} 2 {1 - \exp [- i \frac{2 π n_{w} r}{λ} (n - i n' - 1)]} d P,

C_{a} = \iint_{P} [1 - \exp (- \frac{4 π n_{w} r n'}{λ})] d P,

where λ is the light wavelength in the vacuum,

n_{w}

(\approx 1.33)

is the real part of the absolute refractive index of water, r is the light ray path through the particle, n - in' is the wavelength dependent particle’s complex refractive index relative to water, and dP is an element of the particle projected area (P) on the plane perpendicular to the direction of the incident light. The ray paths are line segments within a particle and their direction is the same as the direction of the incident light. Neglecting the light refraction and reflection on a particle's surface results from the ADA assumption that the particle complex index of refraction is close to that of water that is, its real part is close to 1 and the imaginary part is close to 0. In this approach the light extinction is caused by the absorption of the light rays passing through a particle and by interference of those ones which were transmitted through a particle and those passing around the particle [14]. The difference between the cross sections expressed by Eqs. (2) and (3) gives the scattering cross-section.

Consider polydispersive particles of the same refractive index and shape, whose size can be described by one chosen linear dimension varying in a range $(D_{min}, D_{max})$ and which are characterized by a power-law PSD given by Eq. (1). Then the scattering coefficient (b) can be determined as follows [2]:

b = 〈 m \int_{D_{min}}^{D_{max}} (C_{e} - C_{a}) D^{- k} d D 〉,

where the unit size D_o was omitted for simplicity and

〈 . 〉

denotes the average over different particles' orientations with respect to the direction of the incident light. After substituting Eqs. (2) and (3) into Eq. (4) the sequence of integration is changed and the variables are arranged in such a way to make the surface integration independent on particles' size:

\begin{array}{l} b = m 〈 \iint_{\frac{P}{D^{2}}} ({Re \int_{D_{min}}^{D_{max}} 2 [1 - \exp (- 0.5 A D - i B D)] D^{2 - k} d D} \\ - \int_{D_{min}}^{D_{max}} [1 - \exp (- A D)] D^{2 - k} d D) d (\frac{P}{D^{2}}) 〉, \end{array}

where

A = 4 π n_{w} (\frac{n'}{λ}) (\frac{r}{D}),

B = 2 π n_{w} (\frac{n - 1}{λ}) (\frac{r}{D}),

and the ratios

P / D^{2}

and

r / D

do not depend on D. Accounting for the fact that the infinitely large size (

D_{max} \to \infty

) will be considered, the following integral is essential in the analysis:

I_{k} (Z, D_{min}) = \int_{D_{min}}^{\infty} [1 - \exp (- Z D)] D^{2 - k} d D,

where Z can equal

0.5 A + i B

or A. The integral

I_{k} (Z, D_{min})

exists (is convergent) for

3 < k

. Whereas when

D_{min} \to 0

at the same time then additionally k must be lower than 4. Therefore, this range of the PSD slope values i.e.

3 < k < 4

will be considered first. After integration by parts

I_{k} (Z, D_{min})

can be written as:

I_{(3 - 4)} (Z, D_{min}) = \frac{D_{min}^{3 - k}}{k - 3} [1 - \exp (- Z D {}_{min})] + \frac{Z}{k - 3} \int_{D_{min}}^{\infty} \exp (- Z D) D^{3 - k} d D .

Using the gamma function [17]:

Γ (x) = Z^{x} \int_{0}^{\infty} D^{x - 1} \exp (- Z D) d D

with positive x, which in this case is equal to 4 - k, it is easy to find the limit:

\lim_{D_{min} \to 0} I_{(3 - 4)} (Z, D_{min}) = \frac{Γ (4 - k)}{k - 3} Z^{k - 3} .

If we introduce this result into Eq. (5), the scattering coefficient accounting for all possible particle sizes equals:

b = m \frac{Γ (4 - k)}{k - 3} 〈 \iint_{\frac{P}{D^{2}}} {Re [2 {(0.5 A + i B)}^{k - 3}] - A^{k - 3}} d (\frac{P}{D^{2}}) 〉 .

The power function

{(0.5 A + i B)}^{k - 3}

is multivalued and its principal value is taken for all further calculations [18]:

{(0.5 A + i B)}^{k - 3} = {| 0.5 A + i B |}^{k - 3} \exp [(k - 3) φ i],

where

φ = arc \tan (\frac{B}{0.5 A}) .

After expressing the A and B parameters by means of Eqs. (6) and (7) the final formula for the scattering coefficient is given by:

\begin{array}{l} b = m S \frac{Γ (4 - k)}{k - 3} {(2 π n_{w} \frac{n - 1}{λ})}^{k - 3} \\ \times {2 {[{(\frac{n'}{n - 1})}^{2} + 1]}^{(k - 3) / 2} \cos [(k - 3) arc \tan (\frac{n - 1}{n'})] - {(2 \frac{n'}{n - 1})}^{k - 3}}, \end{array}

where

S = 〈 \iint_{\frac{P}{D^{2}}} {(\frac{r}{D})}^{k - 3} d (\frac{P}{D^{2}}) 〉 .

This non-dimensional integral is the only factor in Eq. (15) which depends on the particles' shape. The scattering coefficient must be a continuous function of k. Therefore, to obtain an expression for the scattering coefficient when k = 4, it is sufficient to determine the limit in Eq. (15) when k approaches 4 through values less than 4. Then the factor in the brace approaches 0 whereas

Γ (4 - k) \to \infty

and their product assumes the indeterminate form. Its limit was determined with the use of Mathematica software:

\begin{array}{l} \lim_{k \to 4^{-}} Γ (4 - k)) {2 {[{(\frac{n'}{n - 1})}^{2} + 1]}^{(k - 3) / 2} \cos [(k - 3) arc \tan (\frac{n - 1}{n'})] - {(2 \frac{n'}{n - 1})}^{k - 3}} \\ = 2 arc \tan (\frac{n - 1}{n'}) + \frac{n'}{n - 1} \ln [\frac{4 n'^{2}}{n'^{2} + {(n - 1)}^{2}}] . \end{array}

Then, for

k = 4

Eq. (15) takes the following form:

b = m S (2 π n_{w} \frac{n - 1}{λ}) {2 arc \tan (\frac{n - 1}{n'}) + \frac{n'}{n - 1} \ln [\frac{4 n'^{2}}{n'^{2} + {(n - 1)}^{2}}]} .

In this case the interpretation of S is very simple. It is the volume of a particle divided by its characteristic size raised to the third power. When 4 < k < 5 the integral in Eq. (9) is integrated by parts once again to obtain:

\begin{array}{l} I_{(4 - 5)} (Z, D_{min}) = \frac{D_{min}^{3 - k}}{k - 3} [1 - \exp (- Z D {}_{min})] + \frac{1}{k - 4} \frac{D_{min}^{4 - k}}{k - 3} \exp (- Z D {}_{min}) \\ - \frac{1}{k - 4} \frac{Z}{k - 3} \int_{D_{min}}^{\infty} \exp (- Z D) D^{4 - k} d D . \end{array}

Using this expression and the Eq. (10), the limit of the difference can be calculated:

\begin{array}{l} \lim_{D_{min} \to 0} {Re [2 I_{(4 - 5)} (0.5 A + i B, D_{min})] - I_{(4 - 5)} (A, D_{min})} = \\ = Re [- 2 \frac{Γ (5 - k)}{k - 4} \frac{{(0.5 A + i B)}^{k - 3}}{k - 3}] + \frac{Γ (5 - k)}{k - 4} \frac{A^{k - 3}}{k - 3} . \end{array}

Replacing Γ(5 - k) by (4 - k) Γ(4 - k) and accounting for the above limit Eq. (5) finally gives the same result as Eq. (15). Note that in this case Γ(4 - k) and the factor in the brace in Eq. (15) are both negative. Letting

n' = 0

in Eqs. (15) and (18) one derives expressions for the scattering coefficient of non-absorbing particles:

b = 2 m S \cos [0.5 π (k - 3)] \frac{Γ (4 - k)}{k - 3} {(2 π n_{w} \frac{n - 1}{λ})}^{k - 3}

when 3 < k < 4 or 4 < k < 5, and

b = π m S (2 π n_{w} \frac{n - 1}{λ})

when k = 4.

3. Comparison between ADA and Mie visible scattering coefficients

3.1 Calculation assumptions

The comparison between ADA and Mie visible scattering coefficients was performed for hypothetical populations of particles which were not intended to simulate any natural situations. Three main types of marine suspended matter, phytoplankton, detritus, and mineral particles, were considered. As in Babin et al. [12], the optical properties of phytoplankton were represented by Prorocentrum micans (Dinophyceae). The data on the refractive index in the visible band (0.4 - 0.75 µm) were obtained by means of digitalization of proper graphs (Figs. 3(a) and 3(b)) in Ahn et al. [19]. The imaginary part of the refractive index shows typical for phytoplankton maxima in the blue and red part of the spectrum (Fig. 1 ). Within these wavebands the light absorption is enhanced. The real part of the Prorocentrum micans refractive index shows only small fluctuations around the value of 1.05.

Fig. 1 Refractive index of marine particulates (relative to water).

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The imaginary part of the refractive index of the detrital and mineral particles was described by the same decreasing function:

n' = 0.007954 \exp (- 7.186 λ)

with λ in [μm], and the real parts are wavelength-independent and equal 1.04 and 1.18 respectively (Fig. 1). It is believed that such optical properties are characteristic for nonliving particles [20, 21].

The scattering spectra were determined for each type of particulates based on Eqs. (15) and (18). It was assumed that the particles are spherical and the characteristic size is their diameter. In this case the integral S equals 0.5π/(k - 1). The value of m in the power-law PSD (Eq. (1)) was taken to be 10⁵ cm⁻³ μm⁻¹. It is a typical order of magnitude of this parameter [2]. The values of the PSD slope coefficient ranged between 3 and 5. This analytical approach was examined against the exact Mie solution to the light attenuation cross-sections. In this case the integral in Eq. (4) was evaluated numerically by means of a Matlab script [22] with lower and upper limits of particle diameters of 0.02 and 500 μm, respectively. The calculations were performed in the visible band (0.35 - 0.75 µm) with a 0.01-µm step.

3.2 Results and discussion

The computation results shown in Fig. 2 were first analyzed with the use of the following relationship:

b_{A D A} (λ) = α b_{M i e} (λ)

The magnitude of the proportionality factor

α

, obtained by means of the least square method, estimates the average ratio between the calculated scattering coefficients whereas the coefficient of determination (

R^{2}

- squared correlation coefficient) describes the similarity of the shapes of obtained spectra.

Fig. 2 Visible spectra of the scattering coefficient of spherical particles with power-law PSD determined by means of ADA in the infinite diameter interval (0, ∞ µm) and Mie solution in the (0.02, 500 µm) diameters interval.

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Among all of selected values of the power-law PSD slope coefficients, the best agreement between the magnitude of the scattering coefficients, regardless of the type of the particulate matter, was observed when $k = 3.5$ or 4.0. Then $α$ and $R^{2}$ were close to 1 (Table 1 ).

Table 1. Results of the analysis of the relationship between ADA¹ and Mie² visible particles' scattering coefficients (Eq. (24).

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For higher k ADA overestimated the magnitude of the scattering coefficients in relation to the Mie results. The overestimation got worse with increasing value of k. For example, when $k = 4.9$ $b_{A D A}$ was approximately 3.4 times higher than $b_{M i e}$ for detritus and algae and nearly 5 times higher for mineral particles which were characterized by the highest n value. Such a result is caused by two reasons. When the PSD slope coefficients becomes larger, the contribution of smallest particles in the light scattering gets more significant and their scattering cross-section according to ADA decreases slower (~ $D^{4}$ ) than in the case of the exact solution (~ $D^{6}$ ) [14]. Additionally, the magnitude of $b_{A D A}$ is also influenced by the presence of particles smaller than 0.02 μm, which are not included in the Mie calculations. Despite the tendency to overestimate the values of the scattering coefficient ADA reproduces the spectral shape of scattering very well, which was confirmed by the values of the determination coefficient higher than 0.999 for $k = 4.5$ or 4.9. For k = 3.1 the scattering coefficients calculated with the use of ADA were also distinctly higher than these obtained basing on the Mie solution (Table 1). Moreover, low values of the determination coefficients indicated significant differences between the shape of the scattering spectra. In this case, particles that are very large in comparison to the light wavelength, can contribute significantly into the light scattering. Therefore, when the upper limit of the particles' diameters was assumed to be 500 μm some of them were not taken into account in the Mie calculations. This resulted in such an unsatisfactory fit between obtained spectra. For example, after changing this upper limit to 5000 μm the recalculated value of $b_{M i e}$ for mineral particles was higher and the value of the α parameter decreased to 1.288 (previously $α = 1.430$ ) whereas $R^{2}$ increased from 0.203 to 0.973.

Equations (15) and (18) reconstructed reasonably the particles' scattering coefficient spectra with accuracy within a constant factor. Therefore it was worth analyzing them in detail. The expressions for b are products of two spectral functions. The first one is ${[(n - 1) / λ]}^{k - 3}$ and it determines the scattering spectra of non-absorbing particles (Eqs. (21)–(22)). Its variability is practically governed by $λ^{- (k - 3)}$ because the real part of the refractive index of all types of marine suspended matter is nearly constant [2]. The second, more complex, function is dependent on n'/(n - 1) and k - 3. The ratio n'/(n - 1) is the indicator characterizing the absorption-to-scattering properties of particles [14]. The higher it becomes, the more significant is the role of the absorption in the light attenuation. In such a case the absolute values of the latter function decrease, but the decrease is lower when k gets higher. The values of n'/(n - 1) are usually close to zero, even in the case of phytoplankton which strongly absorbs the light. Its maximum for Prorocentrum micans is 0.055 in the blue part of the visible band. Therefore generally the function of n'/(n - 1) does not change much and when k is sufficiently large it can be expected that the $λ^{- (k - 3)}$ dependency significantly influences the shape of the scattering spectra. The approximation with the use of the power function of the light wavelength was tested on the basis of the ADA results. The goodness of fit and the value of the non-dimensional spectral slope coefficient (ε) were estimated by means of the least squares method after an appropriate logarithmic transformation of variables:

\ln b_{A D A} (λ) = - ε \ln λ + c o n s t,

where the constant on the right hand side of the equation is the logarithm of proportionality coefficient between

b_{A D A} (λ)

and

λ^{- ε}

. The results presented in Table 2 show that the power relationship describes well the shape of majority of analyzed scattering spectra. The exception were the spectra of the phytoplankton when k was close to 3 and the determination coefficients were the lowest. In such a case the function

λ^{- (k - 3)}

is relatively flat and the changes in the magnitude of n'/(n - 1) cause greater effects.

Table 2. Power approximation to the ADA particles’ visible scattering coefficients spectra (Eq. (25).

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It is apparent that local maxima of the imaginary part of the refractive index (absorption) (Fig. 1) are accompanied by minima of the scattering coefficient (Fig. 2(a)). Such an irregular spectral dependency, which has been observed under natural conditions [12], cannot be approximated by a simple power function. However, when the value of k increased the phytoplankton scattering spectra got smoother and more similar to the power function. On the other hand in the case of non-living particles (Figs. 2(b), 2(c)) n'/(n - 1) decreases exponentially because of the assumed spectrum of the imaginary part of the refractive index (Eq. (23)). Then the absorption reduces the magnitude of scattering in the short-wave part of the spectrum, and its influence gradually decreases with increasing light wavelength. For the lowest k value (3.1), it can cause a weak power-law increase (ε is negative) in the scattering coefficient with increasing wavelength. Beside this case the decreasing power function fitted excellently to the scattering spectra of non-living particles (Table 2). Generally, in such an approximation, the values of the ε exponent were lower than k - 3. This difference became smaller for particles with lower values of n'/(n - 1) ratio and when k was higher.

To conclude, the advantages of solutions derived on the basis of the anomalous diffraction approximation have to be emphasized. Most of all, they enable one to obtain reliable relative light scattering spectra of optically significant marine particles with power-law PSD without employing complicated calculation methods. Additionally, their analytical form is useful for better understanding how particles’ optical properties and power-law PSD slope influence the spectral shape of the scattering coefficient.

Acknowledgments

I would like to thank Bożena Wojtasiewicz for performing the calculations of the scattering spectra with the use of Mie theory. I also appreciate helpful comments and suggestions given by the reviewer. The financial support for editing this paper came from the statutory funds of the Department of Physical Oceanography at the University of Gdańsk.

References and links

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	Phytoplankton		Detritus		Mineral
PSD slope (k)	$α$	$R^{2}$	$α$	$R^{2}$	$α$	$R^{2}$
3.1	1.727	0.705	1.679	0.679	1.430	0.203
3.5	1.008	0.997	1.008	0.996	0.961	0.999<
4.0	1.010	0.999<	1.005	0.999<	1.035	0.999<
4.5	1.213	0.999<	1.192	0.999<	1.422	0.999<
4.9	3.486	0.999<	3.346	0.999<	4.922	0.999<

	Phytoplankton		Detritus		Mineral
PSD slope (k)	$ε$	$R^{2}$	$ε$	$R^{2}$	$ε$	$R^{2}$
3.1	−0.159	0.463	−0.073	0.988	−0.040	0.972
3.5	0.207	0.603	0.374	0.999<	0.441	0.999<
4.0	0.796	0.978	0.946	0.999<	0.984	0.999<
4.5	1.351	0.993	1.476	0.999<	1.494	0.999<
4.9	1.799	0.995	1.895	0.999<	1.899	0.999<

	Phytoplankton		Detritus		Mineral
PSD slope (k)	$α$	$R^{2}$	$α$	$R^{2}$	$α$	$R^{2}$
3.1	1.727	0.705	1.679	0.679	1.430	0.203
3.5	1.008	0.997	1.008	0.996	0.961	0.999<
4.0	1.010	0.999<	1.005	0.999<	1.035	0.999<
4.5	1.213	0.999<	1.192	0.999<	1.422	0.999<
4.9	3.486	0.999<	3.346	0.999<	4.922	0.999<

	Phytoplankton		Detritus		Mineral
PSD slope (k)	$ε$	$R^{2}$	$ε$	$R^{2}$	$ε$	$R^{2}$
3.1	−0.159	0.463	−0.073	0.988	−0.040	0.972
3.5	0.207	0.603	0.374	0.999<	0.441	0.999<
4.0	0.796	0.978	0.946	0.999<	0.984	0.999<
4.5	1.351	0.993	1.476	0.999<	1.494	0.999<
4.9	1.799	0.995	1.895	0.999<	1.899	0.999<

Anomalous diffraction approximation to the light scattering coefficient spectra of marine particles with power-law size distribution

Abstract

1. Introduction

2. ADA scattering coefficients

3. Comparison between ADA and Mie visible scattering coefficients

3.1 Calculation assumptions

3.2 Results and discussion

Acknowledgments

References and links

Cited By

Figures (2)

Tables (2)

Equations (25)

Optics Express