Please see attached file.

The paper fundamentally assumes a mu-lambda relationship which is not necessarily correct. But it can be accepted for publication in AMT after revision. The authors need to consider the following comments and revise the manuscript accordingly.

Abstract

Line 4:  There is no term used in the statistical gamma family of distributions that has the term "constrained gamma". The mu-lambda relation is an empirically derived based on measured DSDs. Since the measured DSDs are statistical (i.e. the parameters such as Dm can be treated as statistical) the mu-lambda is not a deterministic relation.

Line 12: Sentence beginning ‘The most difficult ..’ This is true of all retrievals of the DSD and R. It is not surprising that NT which is 0th moment of the DSD cannot be estimated accurately using higher order moments like Z=f(M6) and Dm=M4/M3.

Abstract, Last sentence: this increase in correlation from 0.12 to 0.24 is not a meaningful increase...the scatter still looks "random"

Line 33: Surely by now the entire DSD community is aware that N0-mu relation is not physical.

Line 46: I do not agree that calibration offsets in Zh and Zdr are often overlooked. The US Nexrad system has done extensie work to reduce the uncertainty of Zdr to within -+0.1 dB. To this, one can add the German DWD, and MeteoFrance as well.

Line 71: The instrument does not possess the resolution to measure the drizzle and very small drops. This is also termed as truncation of the DSD and the shape factor will be biased to strongly positive values with convex down shape at the small drop end.

Line 85: "comparable" is not the correct description…. you are only sampling in time to get 30 s sampling.

Line 98: fig 1 does not appear to have a clear melting layer....what is mean by clear? the vertical streaks of Z above the BB indicates vertical air motion.

Line 112: the BB does not look steady, rather the vertical streaks in Z well above the BB depict some vertical air motion.

Eq. 1: the use of NT was introduced by Chandrasekar and Bringi to emphasie that NT = 0th moment =total number density which makes this form similar to what statisticians would use.

Line 154: "empirical" or "statistical"?

Eq. 7: is there any physical basis for this power law?

Line 163: Dmax is approximately 3*Dm...see Carey and Petersen

Line 195: The critical aspect is that Parsivel cannot measure the drizzle or smalll dops with sufficient resolution causing truncation. This causes Dm to increase as well a tdecrease in the ahe spectral width sigma_m ..casing mu to decease.

Also, the stability of mu-lambda relation itself is not in question since it can be stable for the wrong reason.

Line 235: The NT is the same as M0 ie the total number density. It is not possible to estimate it from the higher order moments such as Nw or Dm. In fact the variability in NT of the DSD is larger than that of  Dm or mu. This is termed as number controlled DSDs.

Last sentence in 5.1.3: this is known as the point-to-area or non-uniform beam filling problem. This is very well known and has been addressed by several publications

Last sentence, 5.2: This is not surprising since NT is the M0th moment whereas Nw, Dm are of much higher order.

Line 405: no surprise here...unless one can measure M1, M2, there is no way to improve the estimate NT.

Line 447: The method of improving the correlation coeff especially for NT does not improve at al ...the corr~ 0.

This manuscript estimates variability of Lamba-mu relations of the assumed gamma-function DSD in observed liquid precipitation. The results obtained in this study may be useful for better understanding of uncertainties in these relations. I recommend a major revision of the manuscript having in mind comments below.

Main comments.

1. The authors should clarify their retrieval method described in section 3.3. They describe how they estimate mu (steps 2 and 3). How the corresponding Lambda value is then obtained? They state that they impose a fixed Lambda – mu relation with fixed coefficients (i.e., relation (7)). If they use this fixed relation then how different prefactors and exponents (alpha and beta in Table 1) are obtained?
2. Please provide a better description of the geometry of measurements. What are relative locations of the disdrometer and the radar? At what heights radar measurements are made? Is the disdrometer directly below the radar resolution volume? In other words, what are horizontal and vertical distance separations between the radar and disdrometer.
3. Are coefficients in (7) simple mean values or are they some kind of weighted mean values? (for example, weighted by event durations, etc.).
4. Equations (1) through (5) assume untruncated distributions. Do you have any estimates how truncation to Dmax in (9) and (10) would affect the results? I assume that this effect is mu-dependent.
5. Line 164: Eq.(3) from Unal (2015) shows only horizontal polarization backscatter cross section. Do you account for the elevation angle for the vertical polarization cross section? What were assumed drop orientations?
6. What are your estimates of uncertainties in the Lambda-mu estimates? Given the retrieval/measurement uncertainties, are the results for different events shown in Fig.3 really statistically different?
7. The correlation coefficients of 0.12 - 0.24 for retrieved Nt (as mentioned in the abstract) actually indicate no reliable correlation.
8. I suggest calculating a power-law correlation coefficient between Lambda and mu for each event and also RMSD between individual Lambda – mu points and the best fit. Showing these statistical metrics in in Table 1 would be beneficial.
9. Why not to use lower elevation angle for radar measurements to increase ZDR?

Minor comments

1. Since you use binned DSD information, you should probably use summations in (9) and (10) rather than integrals.
2. Equations (7) and (11) are repetitive.
3. The first line after (9): here capital Lambda size parameter and small lambda -wavelength are mixed up.
4. Add Zdr frame to Fig. 2.
5. Line 296 says: see Section 3a, but there is no section 3a in the paper. Is it 3.1 ? Also you are referring to section 3c in line 340 (and in other parts of the paper), but it probably should be section 3.3. Check the entire manuscript for consistency in referencing different sections.
6. Are sigma’ and sigma in lines 302-304 the same parameter?