Technical Note: Simple analytical relationships between Ångström coefficients of aerosol extinction, scattering, absorption, and single scattering albedo

Abstract. Angstrom coefficients are commonly used to parameterize the slow wavelength dependence of aerosol scattering, absorption, and extinction coefficients and single scattering albedo. Here we introduce simple analytical relationships between these coefficients that establish a framework for intercomparison between theory and experimental results from different instruments and platforms and allow for closure studies and improved physical understanding.


Introduction
The Ångström coefficient AC was originally introduced as a wavelength-independent constant in a power law to describe wavelength-dependent extinction (or optical depth) of light by aerosols ( Ångström, 1929).Since then, it has found additional extensive use in characterizing the "slow" wavelength dependence of scattering, absorption, and single scattering albedo (SSA) (Russell et al., 2010;Fischer et al., 2010;Virkkula et al., 2005;Flowers et al., 2010).ACs are generally not considered appropriate for "fast" wavelength dependences as encountered, for example, in the fast oscillations of scattering coefficients for individual, non-absorbing, spherical particles (Eversole et al., 1993).Symbols used in this paper are summarized in Appendix A.
While simple analytical relationships between extinction, scattering, and absorption coefficients and SSA exist (e.g.Moosmüller et al., 2009), we are not aware of corresponding relationships for ACs.Such relationships are useful to Correspondence to: H. Moosmüller (hansm@dri.edu ) compare ACs obtained from extinction, scattering, and absorption, including the ground truthing of remote sensing and satellite measurements.For example, aerosol optical depth (path-integrated extinction) can be obtained from groundbased and satellite remote sensing at multiple wavelengths yielding extinction Ångström coefficients EACs.Simple analytical relationships between EACs, scattering Ångström coefficients SACs, and absorption Ångström coefficients AACs will help attributing the EACs to the underlying physical phenomena, namely scattering and absorption, and analyzing closure between the different Ångström coefficients.In addition, SSA is the key parameter for obtaining the sign and magnitude of aerosol radiative forcing.SSA can be obtained at multiple wavelengths from in-situ measurements (Lewis et al., 2008;Virkkula et al., 2005;Flowers et al., 2010), ground-based remote sensing measurements (Dubovik et al., 1998;Dubovik and King, 2000), and potentially from satellite measurements (Mishchenko et al., 2007;Zhu et al., 2011).Relating the SSA Ångström coefficient SSAAC to the underlying SAC, AAC, and EAC will help with data interpretation and closure and physical understanding.
Ångström coefficients can be used to express the dependence of any parameter p(λ) on wavelength λ provided p(λ) can be approximated by a power-law function of wavelength.Conventionally, the two-wavelength AC is used to give the ratio of p(λ) at two wavelengths λ 1 and λ 2 as function of the ratio of these wavelengths as (Moosmüller et al., 2009) H. Moosmüller and R. K. Chakrabarty: Simple analytical relationships The AC can be written explicitly as which is the negative slope of p(λ) between wavelengths λ 1 and λ 2 on a log-log plot (Moosmüller et al., 2011).Such a slope obtained at a single wavelength λ defines the singlewavelength AC(λ) as (Moosmüller et al., 2011) In addition, ACs can be obtained from simple linear regression of data plotted on a log-log scale or more complicated non-linear fits of data that may also yield higher order terms such as curvature (Schuster et al., 2006).Relationships between different ACs that include the single scattering albedo (SSA) ω have only been derived by us for single-and twowavelength ACs, while for ACs obtained from linear or nonlinear fits the mathematics gets much more complicated due to the difficulty of appropriately attributing the influence of the SSA at different wavelengths.However, in most cases, the single-wavelength equations still give a good approximation.
In aerosol optics, ACs are of interest for scattering, absorption, and extinction coefficients and for the SSA ω.The relationships between these ACs are investigated using Eq.(1c).

Extinction Ångström coefficient EAC
The extinction coefficient γ is defined as the sum of absorption coefficient α and scattering coefficient β as (e.g.Moosmüller et al., 2009), Using Eq. (1c), the extinction Ångström coefficient EAC can be written as Rewriting Eq. (1c) for the absorption Ångström coefficient AAC yields (Moosmüller et al., 2011) with an equivalent expression for the scattering Ångström coefficient SAC Substituting Eqs.(2c, d) into Eq.(2b) yields With the SSA ω(λ) defined as and the single scattering co-albedo (SSCA) (λ) defined as Eq. ( 2e) can be written as where the extinction Ångström coefficient EAC is the sum of the -weighted absorption Ångström coefficient AAC, and the ω-weighted scattering Ångström coefficient SAC.As expected, the EAC is dominated by the AAC for mostly absorbing (i.e.black, ≈ 1) particles and by the SAC for mostly scattering (i.e.white, ω ≈ 1) particles.
Simple examples for specific cases include (a) if AAC(λ) = SAC(λ), the EAC can be written as EAC(λ) = AAC(λ) = SAC(λ) and (b) if AAC and SAC are constants, independent of wavelength λ, the wavelength-dependent part of EAC is the product of ω(λ) and the difference between SAC and AAC as shown in the second line of Eq. (2h).
In the Rayleigh regime, particle dimensions are small compared to the wavelength λ resulting in a size parameter x (i.e. for spherical particles, x is the ratio of particle circumference and λ) of x 1.It is well known that in the Rayleigh regime SAC Ray = 4 (why is the sky blue?) and AAC Ray = 1, under the assumption of wavelength independent refractive indices (Bohren and Huffman, 1998;Moosmüller and Arnott, 2009).Equation (2h) immediately yields the missing information on EAC Ray as where both EAC Ray and ω are written as functions of size parameter x, which incorporates both wavelength and particle diameter.In addition, in the Rayleigh regime ω can be expressed simply as function of size parameter x and complex refractive index m as (Bohren and Huffman, 1998;Moosmüller and Arnott, 2009) yielding a simple analytical expression for EAC Ray as where IM stands for the imaginary part.This nicely complements the well-known constants for SAC Ray and AAC Ray .

Single Scattering Albedo (SSA) Ångström coefficient SSAAC
Inserting the definition of SSA ω from Eq. (2f), into the general definition of AC given by Eq. (1c) and using the quotient rule, the SSA Ångström coefficient SSAAC can be expressed as In Eq. (4a), dα/dλ and dβ/dλ can be replaced using Eqs.( 2c) and (2d), respectively, yielding Some further rearrangement and use of the definition of SSA in Eq. (2f) yields and finally reduces to the form The SSAAC equals the difference between SAC and AAC multiplied with the SSCA .Note that for two-wavelength Ångström coefficients, the two-wavelength definition of SSCA in Eqs.(3a, b) must be used.
Simple analytical relations have been developed connecting ACs for aerosol extinction, scattering, and absorption coefficients, and single scattering albedo.These relationships will be useful for performing comparisons and closure between different measurements of the wavelength-dependent aerosol optical properties parameterized in terms of ACs.
They will be of special interest for ground-truthing of the wavelength dependence obtained from satellite aerosol optical depth measurements with that from ground-based and airborne measurements of scattering and absorption coefficients.Future work will include applications of these relationships to existing data of wavelength-dependent aerosol optics measurements.