We thank the reviewer for their comments, we will modify the manuscript in order to take them into account.

In the meantime, here are a few thoughts on the main points raised.

Regarding point (1) and (3) - i.e. the fact that the refractive index and the number of modes must be known a priori - we are working at a generalization of the method where these variables are estimated from the data as well. However, we thought the method in its current version is already interesting. The title can be modified to account for these limitations.



The log-normal assumption (point 2) is used in many studies; we think it is a reasonable compromise that allows to make the inverse problem less ill-posed.

Regarding point (4), r_min and r_max need to be known but we use quite large intervals; the main idea is to avoid superposition of the means, i.e. to look for modes in different intervals when multiple modes are sought.

We totally agree with point (5), a 5% of standard deviation is too small in general; however, as the method applies to a single set of optical parameters, this can be obtained by averaging (either in time or space, or both).

We thank the reviewer for their comments. We will modify the manuscript in order to take them into account.

In the meantime, here are few thoughts on the main points raised by the reviewer.

We will clarify the mathematical aspects of the problem better. Indeed, as suggested by the reviewer, the problem is generally under determined, with the only exception being when one looks for a monomodal distribution: in this case, one has three unknowns and 5 data points, i.e. the problem is over determined.

We agree that 5% of noise is too small for typical lidat signals, however, averaging in time and space can help to increase the signal-to-noise ratio [see also comment to Reviewer 1]. The level of noise on input data does deteriorate the retrieval; however, the use of a parametric model for the number size distribution reduces the impact of noise compared to. methods that work with "generic" shapes, which tend to be affected more by noise than methods working with parametric models. Of course, more noise implies more uncertainty on the estimated parameters. We reckon a thorough study of the impact of noise is however out of the scope of the present manuscript, as it takes a substantial amount of space, and will be the topic of future studies.

Finally, it is true that estimating the RI is a tricky point, and requires careful modeling and powerful algorithms to provide, possibly, multiple solutions compatible with the data This is also part of future work.