## Abstract

This paper describes a simple relationship between the slope of particulate optical depth as a function of wavelength and the size distribution of spherical particles. It is based on approximating extinction using a truncated geometric optics relationship and is applicable when optical depth decreases with wavelength. The new relationship suggests that extinction versus wavelength measurements are most sensitive to particles that are comparable in size to the wavelength. When optical depth is expressed as a power-law function of wavelength, the resulting particle size distribution is also a power-law function of size, with the two exponents reproducing the well-known relationship between the Ångström and Junge exponents. Examples of applying the new relationship are shown using both numerical calculations based on Mie theory and measurements from the Aerosol Robotic Network (AERONET) sun photometer at NASA Goddard Space Flight Center (GSFC). Since the truncated geometric approximation makes no assumptions per se concerning the form of the particle size distribution, it may find application in supplementing solar aureole profile measurements in retrieving the size distributions of particles in thin clouds—for example, cirrus—or when they are present.

## 1. Introduction

The shape of solar aureole profiles—for example, as seen in cirrus—depends sensitively on the size distribution of particles comparable to and larger than the wavelength of sunlight. Recognizing the key role that diffraction plays in forming a solar aureole, DeVore et al. (2009) approximated diffraction using a rectangular function, which led to a simple, robust algorithm relating the size distribution of large particles to the shape of the aureole radiance profile. An analogous situation exists with the shape of solar extinction spectra—for example, as seen through aerosol layers. In this case the wavelength dependence of the aerosol optical depth depends sensitively on the size distribution of particles comparable to and smaller than the wavelength of sunlight. Many different algorithms have been developed (see for example King et al. 1978) exploiting this dependence since the pioneering work of Ångström (1929).

Solar extinction spectra are measured routinely by hundreds of automated sun photometers around the world (see for example Holben et al. 1998). These instruments generally also make scans of diffusely scattered radiance in the principal plane (the vertical plane through the sun) and in azimuth at the solar elevation angle (almucantar scans), which are more useful for retrieving aerosol particle size distributions. However, the requirement that the scans be made under cloud-free conditions means that they frequently cannot be used for retrieving particle size. Therefore it is useful to consider solar extinction spectra further.

This paper starts with a review of the wavelength dependence of small particle cross sections based on Mie theory. It then presents a simple approximation to this dependence and uses it to develop an algorithm relating the size distribution of small particles to the wavelength dependence of solar extinction. The resulting algorithm is able to reproduce the well-known relationship between the Ångström and Junge exponents. Two numerical implementations of the algorithm are provided. Examples of retrieved particle size distributions are comparable to ones based on almucantar scan measurements where they overlap and when extinction decreases with wavelength.

## 2. The truncated geometric approximation

Consider the dependence of the extinction cross section *σ*_{ext}(*r*, *λ*, *m*) of a spherical particle on its radius *r*, the wavelength *λ* of light, and the complex index of refraction *m* of the material. Using Mie theory (Mie 1908) to calculate this dependence, it is convenient to work with the extinction efficiency *Q*_{ext}(*r*, *λ*, *m*) given by the ratio of *σ*_{ext} to the geometric cross section *πr*^{2}:

In Mie theory, the radius *r* and wavelength *λ* appear together in the combination *X* = 2*πr*/*λ*. Figure 1 shows calculations of *Q*_{ext}(*X*) as a function of *X* using the code presented by Bohren and Huffman (1983). Calculations for both water droplets and spherical ice particles are shown. The separation of the curves comes about from the dependence of *Q*_{ext} on the complex index of refraction *m*. The wavelength range from 0.40 to 1.60 *μ*m represents that typically covered by sun photometers and therefore also the range of values of *m* of interest.

Note that the dependence of *Q*_{ext} on *m* is small relative to that for *X*. For small *X*, or equivalently small *r* for a given *λ*, *Q*_{ext} asymptotically follows Rayleigh’s law (Rayleigh 1871) *Q*_{ext} (*X*) ∝ *X*^{4} (for fixed *m*). For large *X*, *Q*_{ext} → 2. Hence particles small compared with the wavelength of light interact very weakly, while large particles interact roughly in accord with twice their geometric size.

Consider an extremely simplified model of this behavior shown by the dashed line in Fig. 1. The simplification consists in modeling large particles using twice their geometric areas and ignoring small particles. Termed the “truncated geometric approximation” herein, *Q*_{tga} is represented analytically as follows:

Note that the condition *X* ≥ 2 is equivalent to *r* ≥ *λ*/*π*.

Although this model admits factor of 2 errors near the peak of the extinction efficiency curve, it does capture the gross dependence of *Q*_{ext} on particle size. Furthermore, when *σ*_{ext} is plotted rather than *Q*_{ext}, the additional factor of *r*^{2} makes the approximation appear better. However, it should be clear that applicability of this model is limited. For example, it would be inappropriate to apply it to particle size distributions that peak near *X* ~ 2 (or *r* ~ *λ*/*π*). Nevertheless, the simplicity of this model leads to some interesting relationships that can be derived analytically. Let us begin by considering the dependence of optical depth on wavelength.

The optical depth *τ*(*λ*) of a path through a particulate layer is given by the integral over particle radius *r* of the product of the cross section *σ*_{tga}(*r*, *λ*) and the size distribution of particles *dN*(*r*)/*dr* (*μ*m^{−2 }*μ*m^{−1}):

where all of the length units are expressed in microns for convenience and the cross section is modeled using the truncated geometric approximation. Note that *τ*(*λ*) as used in this paper represents the column optical depth resulting from extinction by atmospheric particles; that is, measurements of *τ*(*λ*) are assumed to have been corrected for molecular scattering and gaseous absorption, for example, the broad Chappuis band of ozone. Substitution of Eq. (2) into Eq. (3) using Eq. (1) gives

Next differentiate Eq. (4) with respect to *λ*:

Solving for *dN*(*r* = *λ*/*π*)/*dr* gives

Equation (6) is the primary result of the truncated geometric approximation and the basis for retrieval algorithms. It relates the spectral density of particles of radius *r* = *λ*/*π* to the derivative of the optical depth *τ*(*λ*) with respect to wavelength *λ*. Since it gives an erroneous result if *τ*(*λ*) increases with *λ*, we can conclude that the truncated geometric approximation is inappropriate in such cases.

## 3. Relationship between the Ångström and Junge exponents

Consider what the truncated geometric approximation says about an optical depth variation characterized by a power-law dependence on wavelength. Ångström (1929) first documented the (approximate) power-law behavior of aerosol optical depth with wavelength:

where *τ*_{1} is the aerosol optical depth at wavelength 1 *μ*m and the parameter *α* is known as the Ångström exponent. Junge (1963) assumed a power-law distribution of aerosols *dN*(*r*)/*d*ln*r*:

where *A* is a constant and *ν* is known as the Junge exponent. For nonabsorbing aerosols with *α* > 1 Junge showed that *ν* ≈ *α* + 2.

Using the relationship *r* = *λ*/*π* this equation becomes

If the particle size distribution is expressed using the derivative relative to the logarithm of radius *dN*(*r*)/*d*ln*r*, Eq. (10) becomes

Comparison of Eq. (11) with Eq. (8) shows that the truncated geometric approximation reproduces Junge’s relationship between the Ångström and Junge exponents. This relationship is well known and has been studied extensively (e.g., Tomasi et al. 1983). However, as Schuster et al. (2006) note, the relationship is mainly of historical interest since it does not hold when *α* ≲ 1 or for absorbing aerosols. Moreover aerosols are frequently better represented by more complicated (e.g., bimodal) size distributions. Nevertheless, since the Ångström exponent remains an operationally useful parameter for aerosol size information (e.g., O’Neill et al. 2001; Schuster et al. 2006), there is the possibility that the truncated geometric approximation and Eq. (6) may be of some use.

## 4. Two implementations

Application of Eq. (6) to retrieve *dN*(*r*)/*dr* from measurements of *τ*(*λ*) must deal with the fact that the latter do not form a continuous function of *λ* because of the presence of strong gaseous absorption features in the measured spectra. It is useful to correct for the broad Chappuis band of ozone, which is situated in the middle of the visible spectral region (King and Byrne 1976). Otherwise one usually works with measurements in the window regions. One can fit ln[*τ*(*λ*)] with a polynomial in ln(*λ*) as King and Byrne did and then differentiate it analytically. Alternatively, one can simply form differences using the measured values. In either case, one starts with a set of *n* measurements: *τ*(*λ*_{1}), *τ*(*λ*_{2}), … , *τ*(*λ _{n}*).

### a. Numerical difference implementation

Consider the pair of measurements *τ*(*λ _{i}*) and

*τ*(

*λ*

_{i}_{+1}). The derivative

*dτ*(

*λ*)/

*dλ*can be approximated by the numerical difference

where the evaluation wavelength *λ*_{i+1/2} is taken as the arithmetic mean of the two wavelengths:

Given a set of measurements of *τ*(*λ _{i}*) at a discrete set of wavelengths

*λ*, the numerical difference implementation in Eq. (14) provides the spectral column density of scatterers at discrete radii

_{i}*r*

_{i+1/2}=

*λ*

_{i+1/2}/

*π*.

### b. Polynomial fit implementation

Measurements of *τ*(*λ*) in window spectral regions can be fit with a polynomial in log–log space:

where the coefficients *a*_{0}, *a*_{1}, *a*_{2}, … are found using the method of least squares. Then *τ*(*λ*) is given by

Differentiate Eq. (16) with respect to *λ*:

The polynomial implementation of the truncated diffraction approximation entails fitting the optical depth measurements using Eq. (15) and then using Eq. (18) to retrieve the spectral column density of scatterers at values of *r* within the range *r*_{min} = *λ*_{min}/*π* ≤ *r* = *λ*/*π* ≤ *r*_{max} = *λ*_{max}/*π*, where *λ*_{min} and *λ*_{max} represent the limits of the range of measurements of *τ*(*λ*). One must be careful not to use a polynomial with too high order lest one end up fitting measurement noise (see for example Shifrin 1995). In the examples that follow a second-order polynomial was used, and the values of *dN*(*r _{i}*)/

*dr*obtained by the method were evaluated at radii

*r*corresponding to the measurement wavelengths

_{i}*λ*through

_{i}*r*=

_{i}*λ*/

_{i}*π*.

## 5. Example retrievals

The algorithms presented above were applied both to model calculations using Junge and lognormal size distributions and to Aerosol Robotic Network (AERONET) measurements.

### a. Junge model comparisons

Mie theory was used to calculate examples of *τ*(*λ*) for Junge size distributions of water droplets with 2 ≤ *ν* ≤ 5 and for 0.1 ≤ *r* ≤ 5.0 *μ*m. The values of *λ* employed were those frequently used by AERONET (Holben et al. 1998): 0.340, 0.380, 0.440, 0.500, 0.675, 0.870, 1.020, and 1.640 *μ*m. The total number of particles was adjusted to make *τ*(*λ* = 0.34 *μ*m) = 0.5.

Figure 2 compares the results of particle size retrievals using Eqs. (14) and (18) with the model distributions. The two implementations give nearly similar results. As the particle size distribution steepens, both implementations tend to overestimate particle density. This is not surprising because as the distribution steepens the small particle contribution to the extinction cross section that the approximation ignores becomes relatively more important. Figure 3 compares Junge exponents calculated for the retrieved particle size distributions with the model values. Both implementations show high biases that increase as the size distribution steepens.

### b. Lognormal model comparisons

In many cases lognormal size distributions are favored for representing aerosols (Whitby 1978). Consider the lognormal particle size distribution *dN*_{ln}(*r*)/*dr* representing a single mode:

where *N _{c}* (

*μ*m

^{−2}) is the column density of particles,

*r*(

_{m}*μ*m) is the median or geometric mean radius, and

*σ*(

_{r}*μ*m) is the variance or width of the distribution. Mie theory was used to calculate examples of

*τ*(

*λ*) for two lognormal size distributions of water droplets with

*r*= 0.03 and 0.10

_{m}*μ*m and

*σ*= 0.30

_{r}*μ*m. Here

*N*was adjusted so that

_{c}*τ*(

*λ*= 0.34

*μ*m) = 0.5. The values of

*λ*used were the same as those used in section 5a. The light gray curves in Fig. 4 showing the model size distributions are compared with retrievals using Eq. (18). Note that the retrieval for the case with

*r*= 0.10

_{m}*μ*m is cut off for

*r*≲ 0.20

*μ*m. Figure 5 shows that

*dτ*(

*λ*)/

*dλ*≲ 0 for

*λ*≲ 0.67

*μ*m in this case.

### c. AERONET aerosol comparisons

The algorithms presented above were also applied to aerosol optical depth measurements from the AERONET sun photometer at the National Aeronautics and Space Administration (NASA) Goddard Space Flight Center (GSFC) for two slightly hazy days. Figure 6 shows the level 1.0 AERONET spectral optical depth measurement data for 15.198 UTC 2 April 2010. The Ångström exponent, based on the 0.440- and 0.870-*μ*m measurements, is 1.305. Figure 7 compares the particle size distributions calculated using Eqs. (14) and (18) with the AERONET product based on almucantar scan data for 15.215 UTC. Only part of the latter is displayed since the almucantar scan contains information for a much broader range of particle sizes than does the spectral extinction measurement. There is general agreement among the three retrieved distributions.

Figure 8 shows level 1.0 AERONET spectral optical depth measurement data for 13.752 UTC 15 April 2010. The particle size distribution (Fig. 9) is steeper than the previous case as evidenced by the increase in the Ångström exponent to 1.785. The particle size retrievals using Eqs. (14) and (18) are in general agreement with the AERONET product based on almucantar scan data for 13.770 UTC.

### d. AERONET aerosol and cirrus comparison

Although AERONET applies various tests to check for cloud contamination, its measurements of *τ*(*λ*) can be used with the truncated geometric approximation to retrieve *dN*(*r*)/*dr* in the presence of thin clouds. Figure 10 shows AERONET measurements of *τ*(*λ*) at NASA GSFC from the afternoon of 11 November 2008 as cirrus arrived overhead. Note the increase and flattening of *τ*(*λ*) as the cirrus layer thickened.

Figure 11 shows retrievals of *dN*(*r*)/*dr* using Eq. (14) with the data shown in Fig. 10. The AERONET product based on almucantar scan data at 16.896 UTC, the last scan before the arrival of the cirrus, agrees fairly well with the results using the truncated geometric approximation. Note that the shape of the small particle size distribution changes very little as the optical depth increases between a half and a full order of magnitude during this time period, suggesting that the density of small particles associated with the cirrus is much smaller than that associated with the aerosol layer. At 18.878 UTC part of the *τ*(*λ*) curve has developed a positive slope and the truncated geometric approximation ceases to be applicable (the corresponding curve in Fig. 11 is cut off at *r* = 0.19 *μ*m).

## 6. Discussion and conclusions

The truncated geometric approximation for the wavelength dependence of the extinction efficiency of spherical water or ice particles was introduced. It enabled the analytical derivation of the well-known relationship between the Ångström and Junge exponents. Equation (6), along with the two numerical implementations in Eqs. (14) and (18), provided the basis for simple algorithms for calculating the size distribution of particles in a radius range consistent with the wavelength range of optical depth measurements. The algorithms were shown to be applicable to situations in which optical depth measurements decrease with wavelength. Example particle size retrievals were compared with retrievals based on almucantar scans. The example retrievals overlapped, but with the truncated geometric approximation the size range retrievable was limited to be commensurate with the spectral range of the measurements. Retrieval accuracy was shown to be affected by the steepness of the distribution.

The truncated geometric approximation makes no assumptions concerning the precise form of the particle size distribution, but is limited to situations in which the differential particle density decreases with increasing size. The truncated geometric approximation may be applicable to retrievals involving thin clouds, perhaps complementing retrievals using the diffraction approximation applied to measurements using the sun and aureole measurement (SAM) instrument (DeVore et al. 2009).

## Acknowledgments

The author notes that the numerical approach employed in the development of the truncated geometric approximation was inspired by Prof. Saul Rappaport’s application of a similar approach in the diffraction approximation (DeVore et al. 2009). The author thanks an anonymous reviewer who recommended starting the derivation of the truncated geometric approximation by considering the extinction efficiency rather than directly with the cross section; this both simplified and clarified the derivation. The author is pleased to acknowledge the NASA Science Mission Directorate and Dr. Hal Maring for their support through Research Opportunities in Space and Earth Science (ROSES) Grant NNX08AJ89G. The author also thanks Dr. Brent Holben and his staff for their work in establishing and maintaining AERONET and especially the site at Goddard Space Flight Center.