I was happy to see this submission out focusing on improving interpretation of data types such as obtained with a single particle soot photometer (SP2). As a specialist with this instrument, I can clearly understand (and appreciate) the value of this to the community of SP2 users. The SP2 measures a few quantities on a per-particle basis relevant to determining mixing state. First, it provides the refractory black carbon (rBC) mass content of a particle (within some range of mass). This is based on an optical measurement of thermal emission, and is quite robust. Secondly, with appropriate analysis and setup, the instrument *measures* the total particle optical size – it detects a scattered light signal, and quantifies it. This is also valid only over some range of particle optical size, and is specific to the geometry of detection of the SP2. Third, some groups use the optical size of *only* the rBC portion of the particle (which can often be measured after the detection technique evaporates non-rBC material). Finally, inspection of the evolution of scattered light as a particle interacts with the SP2 laser provides another indication of internal mixing of materials with rBC. These measurements have been dealt with at length in the literature.   After a measurement one can say: this particle had XX femtograms of rBC content, scattered as much light as a YY nm-diameter polystyrene latex sphere (PSL) into the SP2 scattering detector, and showed (or did not) evidence of shrinking during heating.  These measurements have statistical and systematic errors associated with them, but are independent of Mie or any other theory of light scattering from particles.  Now, the point at which this submitted manuscript becomes relevant is in the interpretation of those measured quantities. With knowledge about the amount of rBC mass and the total particle scattering signal, how can we interpret these quantities to infer conclusions about the amount of non-rBC material and its impacts on light absorption? 

Presently, the paper is presented as though dealing with “SP2 measurement errors”. This is not the case. Rather, it deals with assessing Mie-theory inadequacies for complex aerosols (which are highly relevant to SP2 analyses).  This is a more general topic of interest to a wider slice of the community than SP2 users/interpreters, but has been addressed often in the literature in the broad sense. Hence, I recommend maintaining the focus on the SP2 community.

Broad comments:

• The focus on “measurement errors” should be adjusted to more correctly address “interpretation effects”. This is important because there is nothing wrong with the measurements that are published, and which remain valid independent of the interpretation method. Note that this manuscript does not actually determine “errors” (which would require measurements for comparison) -  rather it determines differences between different optical calculations applied to interpretations.  I think this was reasonably summed up in our 2015 paper on measurements and interpretation with a humidified SP2, which I quote here in part to suggest some additional references that should be added to the paper and considered. The final sentence of this quoted section speaks directly to the value of the manuscript under consideration:

“The SP2 user community often relies on Mie theory to interpret SP2-measured particle scattering cross-section and rBC mass content for the amount of non-rBC material internally mixed with the particles [Schwarz et al., 2008]. Although there is mild experimental support for the SP2 determination of coating thickness via Mie-theory assuming a shell‐and-core morphology [Laborde et al, 2012], recent results hint at the uncertainties associated with this approach. Scarnato et al. [2013] used discrete dipole approximations as well as Mie theory to explore morphology effects on scattering and absorption of bare and internally mixed BC. They show (their Figure 3) that there can be considerable differences (~2X) between the exact numerical methods and the Mie-­‐theory approximation for light-­‐scattering at 1000 nm wavelength (near the SP2 wavelength of 1064 nm). Moteki et al. [2014] included comparison of SP2 light-­‐scattering observations from near core-­‐shell morphologies of BC coated with oleic acid via vapor deposition. They observed a bias up to 40% compared to Mie shell-­‐and-­‐core theory estimates constrained by total particle mass, rBC mass, and the (known) index of refraction of the oleic acid. Exploration of the validity of Mie theory approximations is beyond the scope of this manuscript, but is clearly relevant.”

• The paper does not specify if the results were generated using the specific detection geometry of the SP2, and hence I suspect that the results are based on using particle scattering cross-sections, rather than partial cross-sections only integrated over the SP2 detection angles. In my opinion, use of the partial cross-sections is a requirement for relevance to the SP2 community and for publication in AMT.

• A lot of value would be added to the paper for the SP2 community if, in addition to addressing this error, it was made easier for SP2 users to use the results of the numerical simulations. I’m suggesting that the authors consider including lookup tables that could be used by SP2 users (rather than the mie-theory look ups that are currently more commonly used). The format of these tables would be up to the authors, but I’d suggest something similar to what we use: a dimension for the rBC mass content (or volume-equivalent diameter for an assumed density) and a dimension for the amount of internally mixed material (a mass or volume ratio, again with an assumed density for the internally mixed material). In our lookups we also vary the real index of refraction of the internally mixed material as a third dimension, but this would likely be overkill here. Each entry of the table would then provide the partial scattering cross-section, as would be measured with the the LEO approach with the SP2. Different tables for the different fractal dimensions of the rBC could be used, or that could be added as an additional dimension of the table. Additional tables with absorption information would then complete the set that would be commonly used by the community. I don’t think this is necessary for publication, but would represent a great contribution and example for how future numerical studies could be more impactful, if the authors are willing to publish it. Note, too, that this would strengthen the relevance of the paper for AMT.

Specific comments:

• The authors make the point that aging leads to more compact particles. I think it would be good to also cite China et al, “Morphology and Mixing State of Aged Soot Particles at a Remote Marine Free-tropospheric Site: Implications to Optical Properties”, 2015 for context here (with their conclusion that Mie theory is within 12% of DDA for the older rBC-containing aerosol).
• Line 44 – there has also been a fair number of publications using the SP2 fraction of rBC-containing particles that show evidence of being internally mixed (often referred to as “thinly vs thickly coated rBC”.
• Line 67: this connects to my first broad comment. Sp2 does not measure Dp/Dc, and does not have unavoidable errors in the LEO scattering measurement. The Mie theory interpretations do not destroy the information in the quantities measured by the SP2, they only transform them into different spaces (coating thickness or Dp/Dc), which can still be used to infer the original observed quantities, and allow reinterpretation with another optical model. Similarly, the table headings titled “SP2 retrieved core-shell models” – these are Mie-theory core-shell models. (Note that we have also used RDG to interpret SP2 data… Mie theory is not tied to the SP2 or vice versa.)

To summarize – this is a very promising entry that could provide a lot of value to the SP2 community. Making sure that the calculations are as relevant as possible to the geometry of the SP2 is one requirement. Another is correcting the association of interpretation differences to instrumental error. The authors also have the opportunity to provide a data set that I suspect would be broadly used in SP2-science.

Thanks for submitting this manuscript –

Joshua (Shuka) Schwarz

A review of “Numerical investigation on measurement errors of mixing states of fractal black carbon aerosols using single-particle soot photometer and the effects on radiative forcing estimation” by Jia Liu et al.

General comments:

This study aims to evaluate the uncertainty of the amount of coating material derived from the measured scattering-cross section by SP2 depending on the particle shape assumption. In particular, the authors focused on evaluations of the systematic error due to the assumption of the shell-core particle shape model, which has been widely used but unrealistic for real-world BC-containing particles. The research focus is of significance to every SP2 user. However, I found some issues which must be addressed (corrected) before considering publication.

1.    The “closed-cell model” adopted for thinly-coated BC looks very different from the real-world BC-containing particles. I can’t agree with using such a fictitious model as a “reference” for quantifying the error of the conventional shell-core model. In my opinion, the authors should use another more realistic shape model (e.g., according to TEM images) for thinly-coated BC. The distance between neighboring BC monomers, which artificially depends on the BC volume fraction in the closed-cell model, can be a critical factor in determining the optical properties of BC aggregates because the multipole interaction between monomers strongly depends on the distance, as implied by, for example, Mackowski 1995 https://doi.org/10.1364/AO.34.003535. If the authors are using “the closed-cell model”, the authors should show evidence of its accuracy. 

2.    The authors seem to (implicitly) assume the scattering cross-section measured by the SP2 as if it is the total scattering cross-section. In fact, the scattering cross-section measured by the SP2 is only a small fraction of the “4pi str integral” of the differential scattering cross-section. The authors should explain this fact and evaluate the maximum error caused by the assumption.

Specific comments:

L11: “measured mixing state” ->

The meaning of the “mixing state” could depend on the context and it is not obvious if it is a well-defined physical quantity (such as “volume”). I recommend using “volume of coating” or “volume fraction of coating” for clarity throughout the manuscript.

L13: ”thinly and heavily” -> 

“thinly and thickly” or “lightly and heavily” sounds more accurate.

L15: “the diameter of BC core (Dc)” ->

Is it volume-equivalent diameter? Please clarify.

L25: “is considered to be the second most important factor affecting global warming after carbon dioxide (Zhang et al., 2021)” -> I'm not sure how this statement is still supported by recent climate research. In the IPCC AR6 report, methane was considered to have a larger positive effective radiative forcing than BC.

L34: “affects the vertical diffusion” -> suppress the vertical diffusion?

L57: “single-particle soot photometer” -> You should use “SP2” which has already been defined.

L59: “The scattering cross-section of the BC particle can be rapidly retrieved based on the measurement results of the scattering signal detectors (Schwarz et al., 2006).” -> The methods for retrieval of the scattering cross-section (integrated over the solid angle of light collection) using SP2 were introduced by Gao et al. 2007 AST (with position-sensitive detector) and Moteki and Kondo 2008 JAS (without position-sensitive detector). Please refer to at least one of these papers here.

L62: “The intensity of the incandescent light signal is proportional to the mass of rBC” -> This statement sounds like oversimplifying the truth. The linear proportionality between the LII signal and BC mass is only valid under a limited condition of BC size and LII detection wavelengths. 

Pease see Moteki and Kondo 2010 AS&T https://doi.org/10.1080/02786826.2010.484450. 

L217: “The distribution of retrieved results of mixing states for single-particle with different fractal dimensions over the entire particle size range is shown in Figure 5, and the filling width represents the probability distribution of retrieved Dp/Dc.” -> I’m not sure if this can be regarded as a “probability” distribution. I guess from the context that each violin plot in Figure 5 shows a histogram of the retrieved Dp/Dc values for uniformly-sampled Dc,v ordinate. Is my guess correct? If so, Figure 5 is just a different plot of the data shown in Figures 3 and 4?

Please find my comments attached.