Appendix K — Aerosol: Model Representation

Implementation status

Model	DITL	PACE V3	PACE V4	Evaluation	Planned
Aer-1	x	x	x	—	—
Aer-2/3	—	—	—	x	—

This chapter describes the aerosol representations used in FastMAPOL and the aerosol properties derived from the retrieved state vector. The mathematical formulation of log-normal size distributions, including conversions between number- and volume-density representations and the calculation of distribution moments, is provided separately in Appendix O.

K.1 Aerosol Model Overview

FastMAPOL supports several aerosol representations with increasing complexity:

• Aer-1: five predefined log-normal modes
• Aer-2: six predefined log-normal modes
• Aer-3: aerosol components represented by chemical composition

The currently released HARP2 and SPEXone FastMAPOL products (V1–V4) use the five-mode Aer-1 representation (Gao et al. 2021, 2023; Gao et al. 2026). The six-mode Aer-2 representation was developed for retrievals using RSP observations that include shortwave-infrared (SWIR) measurements (Gao et al. 2018). Aer-3 represents aerosols through aerosol components and has been applied to HARP2 and SPEXone for advanced retrieval studies (Aryal et al. 2026). The component-based aerosol model is described with more details in Appendix L.

In general, the effective radius and effective variance can be derived from any aerosol size distribution, as described in Appendix O. Conversely, when the effective radius and other size-distribution parameters are known, the aerosol mode volume densities can be derived by solving a linear system, as discussed in Appendix P. This can be useful for cross-comparisons of data products based on different aerosol size representations.

This chapter focuses primarily on the Aer-1 and Aer-2 representations under the assumption of spherical particles. For PACE HARP2 and SPEXone data product, aerosol particle shape is represented as a mixture of spherical and nonspherical particles, as implemented in the current FastMAPOL V3 and V4 products (Gao et al. 2026). The treatment of aerosol particle shape and nonsphericity is described in detail in Appendix N.

K.2 Log-Normal Aerosol Modes

In Aer-1 and Aer-2, the aerosol size distribution is represented as a linear combination of predefined log-normal modes. Each mode has a prescribed characteristic radius and distribution width, while its column volume density is varied in the forward model and retrieval.

The total volume size distribution is

\[ \frac{dV(r)}{d\ln r} = \sum_i \frac{V_{0,i}}{\sqrt{2\pi}\sigma_i} \exp \left[ -\frac{(\ln r-\ln r_{v,i})^2}{2\sigma_i^2} \right], \]

where \(V_{0,i}\) is the column volume density of mode \(i\), \(r_{v,i}\) is its volume-mean radius, and \(\sigma_i\) is the logarithmic standard deviation.

Because the mode radii and widths are prescribed, the retrieved mode volume densities determine the shape and magnitude of the overall aerosol size distribution.

K.3 Aer-1: Five-Mode Aerosol Model

The default FastMAPOL aerosol representation, Aer-1, follows the multimode parameterization adopted from Dubovik et al. (2011) and Xu et al. (2016).

The five volume-mode radii are

\[ 0.1,\ 0.1732,\ 0.3,\ 1.0,\ 2.9\ \mu\mathrm{m}, \]

with corresponding logarithmic standard deviations

\[ 0.35,\ 0.35,\ 0.35,\ 0.5,\ 0.5. \]

The first three modes represent fine-mode aerosols, while the last two represent coarse-mode aerosols:

• Fine mode: \(r_v=0.1\), \(0.1732\), and \(0.3\ \mu\mathrm{m}\)
• Coarse mode: \(r_v=1.0\) and \(2.9\ \mu\mathrm{m}\)

The relative contributions of these predefined modes allow FastMAPOL to represent a range of aerosol size distributions without directly retrieving the radius and width of individual log-normal modes.

The impact of the number of aerosol modes on retrieval performance is discussed in Fu et al. (2020).

K.4 Aer-2: Six-Mode Aerosol Model

Aer-2 extends Aer-1 by adding an additional large coarse mode. This representation was introduced in Gao et al. (2018) for RSP retrievals, where SWIR observations provide additional sensitivity to large aerosol particles.

The additional mode has

\[ r_v=8.4\ \mu\mathrm{m}, \qquad \sigma=0.5. \]

Aer-2 therefore contains the same three fine modes as Aer-1 but extends the coarse-mode representation to

\[ 1.0,\ 2.9,\ 8.4\ \mu\mathrm{m}. \]

The additional large-particle mode provides greater flexibility for representing coarse aerosols when sufficient SWIR information is available.

K.5 Fine and Coarse Aerosol Partitioning

For Aer-1 and Aer-2, fine- and coarse-mode aerosol properties are calculated by grouping the predefined modes:

• Fine aerosol: modes 1–3
• Coarse aerosol: modes 4 and above

The total column volume density is

\[ V_0=\sum_i V_{0,i}. \]

The fine-volume fraction (FVF) is therefore

\[ \mathrm{FVF} = \frac{ \sum_{i\in\mathrm{fine}}V_{0,i} }{ \sum_i V_{0,i} }. \]

In contrast, the fine-mode fraction (FMF) is defined using aerosol optical depth:

\[ \mathrm{FMF}(\lambda) = \frac{ \tau_{\mathrm{fine}}(\lambda) }{ \tau_{\mathrm{total}}(\lambda) }. \]

FVF and FMF describe different aspects of the aerosol distribution. FVF is a particle-volume fraction, whereas FMF is an optically weighted quantity and therefore depends on wavelength, particle size, and aerosol optical properties.

K.6 Aerosol Optical Depth

The aerosol optical depth is obtained by integrating the extinction contribution from all aerosol modes:

\[ \tau_a(\lambda) = \sum_i C_{\mathrm{ext},i}(\lambda)N_{0,i}, \]

where \(C_{\mathrm{ext},i}\) is the size-averaged extinction cross section and \(N_{0,i}\) is the column number density of mode \(i\).

Because FastMAPOL retrieves column volume density, the corresponding column number density is calculated from

\[ N_{0,i} = V_{0,i} \frac{3}{4\pi r_{v,i}^3} \exp(4.5\sigma_i^2). \]

Thus,

\[ \tau_a(\lambda) = \sum_i \left[ C_{\mathrm{ext},i}(\lambda) \frac{3}{4\pi r_{v,i}^3} \exp(4.5\sigma_i^2) \right] V_{0,i}. \]

The mathematical derivation of the conversion between volume and number distributions is provided in Appendix O.

K.7 Effective Radius and Effective Variance

Bulk effective radius and effective variance are calculated from the combined aerosol size distribution. For a multimode aerosol,

\[ r_{\mathrm{eff}} = \frac{ \int r^3(dN/dr)\,dr }{ \int r^2(dN/dr)\,dr } \]

and

\[ \nu_{\mathrm{eff}} = \frac{ \int (r-r_{\mathrm{eff}})^2 r^2(dN/dr)\,dr }{ r_{\mathrm{eff}}^2 \int r^2(dN/dr)\,dr }. \]

These quantities can be calculated for the total aerosol distribution or separately for the fine and coarse modes by restricting the mode summation Appendix O.

Although the three fine modes in Aer-1 and Aer-2 have the same logarithmic standard deviation, the effective radius and effective variance of the combined fine-mode distribution vary according to the relative retrieved volume densities of the three modes.

Detailed derivations of these expressions are provided in Appendix O.

K.8 Aerosol Products

The retrieved mode volume densities, together with the retrieved aerosol optical properties, are used to derive a range of aerosol products, including:

• aerosol optical depth (AOD)
• fine-mode and coarse-mode AOD
• fine-mode fraction (FMF)
• fine-volume fraction (FVF)
• effective radius
• effective variance
• single-scattering albedo (SSA)
• spectrally dependent aerosol optical properties

These quantities may be calculated for the total aerosol distribution or separately for the fine and coarse modes.

The predefined multimode representation therefore provides a flexible connection between the FastMAPOL retrieval state vector and physically interpretable aerosol size and optical properties.

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