Conservative behavior of tracers used in the analysis was not tested. I had this concern before, but it was not addressed adequately or correctly in the current revision. I suggested to run DTMM first to determine the conservative behavior of all solutes (e.g., completely conservative, semi- or quasi-conservative, and non-conservative under a lower dimensionality) and then to include conservative ones in CHEMMA. Instead, authors ran DTMM after CHEMMA and just compared the outcomes. They found that the residuals of sulfate, magnesium, and calcium still maintain some structures in a two-dimensional mixing space and the residual structures persist until the dimension goes beyond five (Figure 4 b). If this result is true (I am not sure if Shapiro-Wilk test is appropriate for this analysis; see my comment on Figure 4 below), it strongly indicates some solutes are not completely conservative. With six solutes, you cannot go beyond five dimensions to determine conservative solutes and the number of end-members (see my comment in the second round of revision). Authors misunderstood how DTMM works.
With that said, I do not mean you cannot run CHEMMA with all six solutes together or with different groups of solutes. Instead, I do encourage authors to run different versions of CHEMMA with various combinations of solutes (following the DTMM results) and compare the outcomes, including the number of end-members.
As a research article, I strongly suggest to run DTMM first (following Hooper 2003), then EMMA (similar to Christophersen and Hooper 1992), and finally compare with the results of CHEMMA (e.g., groups of all six solutes and conservative solutes identified by DTMM as mentioned above). The comparison should not be limited to the end-member composition, but include the number of end-members, the fractional contributions of end-members, and the end-member distances. For example, how do the end-member distances from the end-member composition of CHEMMA compare to that of the measured ones? Is there an improvement in the end-member distances with CHEMMA? I did not keep track of whether or not Hooper (2003) used exactly the same data set as Hooper (1990). If so, you may not need to re-run both DTMM and EMMA but just summarize their results.
Addition of a test with varying sample sizes (Figure 6) is nice and very much appreciated. One result is very much promising (e.g., relatively stable compositions for end-members 2 and 3), but others are not (e.g., significant variation of composition of end-member 1; still many outliers for end-members 2 and 3). Together with significant variability in identifying end-members of the synthetic data (Figure 8) and high uncertainties of algorithm (Figure 7), it indicates that the data structure or the distribution of sample points determines the end-member composition. The role of CHEMMA in end-member mixing analysis is limited. This limitation should be explicitly discussed and stated in the abstract and conclusion. This does not downplay CHEMMA’s value, but simply tell the truth so that future users will not be misled. As a matter of fact, CHEMMA would be very helpful in identifying a missing end-member, guiding field sampling of end-members, and generating a hypothesis test.
The manuscript has been modified, with a new section (re: synthetic dataset) and a new analysis (re: varying number of samples) added, but the abstract was not updated to include the results from these analyses, nor was the limitation of the approach stated in the abstract.
Somewhere in the introduction, all of the mixing model assumptions needs to be explicitly listed (i.e., i. Tracers used in the mixing model must be conservative; ii. The number of end-members is known; iii. End-member compositions must be distinct for at least one tracer; iv. End-member compositions are spatiotemporally constant or their variations are known or treated as different end-members). These assumptions should be discussed, e.g., which ones have been addressed by which tools and which ones are still up for research. In my opinion, the first two assumptions have been resolved by the diagnostic tools of mixing models, which has to be acknowledged to respect the earlier study. This will pave a clear pathway for your own research. However, this does not mean DTMM cannot be challenged or improved.
Saying that including non-conservative solutes in the mixing models has not been resolved is inappropriate and misleading, as non-conservative solutes should not be included in the analysis based on the mixing model assumptions. This does not mean you cannot challenge the assumptions, but I do not think that is what your study aimed at.
How do the fractional contributions compare between your and Hooper’s results? Were the fractional contributions of the fourth and fifth end-members significant compared to the three end-members used by Hooper? How do the end-member distances change with the end-member composition of CHEMMA?
Is the Shapiro-Wilk test appropriate for this analysis, as it is usually used for normal distribution test?
Miscellaneous Comments and Suggested Edits
Title: Get rid of “observations”, which I think is redundant.
P1/L1: Delete “, and is”.
P1/L2-3: This statement is not completely true, not exactly part of the mixing model assumption (see one of my comments above). As long as the temporal/spatial variation is known, a hydrograph separation is still valid.
P1/L4: Change “additional measurements” to “samples”.
P1/L5 and also L12: I think this article is no longer a technical note.
P1/L16: Delete “profile”.
P1/L17-18: This is not exactly true. The end-member composition does not have to be constant. Delete the phrase.
P2/L26: Again, it does not have to be temporally invariant.
P2/L26: Change “observations’ to “samples”.
P2/L28-30: This is probably where you state all the assumptions of mixing models as I suggested earlier.
P2/L33-35: Any citation(s) for this statement? This is the most important statement to justify your study.
P2/L40: Add “estimate uncertainties of ” after “to” if those are true.
P3/L61-63: This is only one of a few criteria used to screen end-members (see Hooper, 2003; Liu et al., 2008 and 2020).
P3/L63-65: This is not what Hooper (2003) means. Instead, he demonstrated the filature of using the rule of one. He suggested to use residual distribution pattern. Indeed, their 1992 paper followed the rule of one (as you stated in the sentence following this one). You have to follow the temporal evolution of EMMA and cannot use an early one to reject a later one.
P3/L69-74: With DTMM, the #2 is not true. The #3 should not be stated as mixing is subject to conservative tracers and so is CHEMMA as I talked above. I do not think CHEMMA can include non-conservative solutes.
P3/L81-82: Change “allows for identification of” to “aims at identifying”, as by far you have not yet demonstrated if you can.
P3/L84-85: I think what you want to say here is that end-member composition does not have to be distinct for all tracers (assumption iii above).
P4/L95: Change “find” to “determine”.
P4/L113: Why not running DTMM before CHEMMA?
P4/L114: Based on Figure 1, “…projected into the 2D subspace spanned by pair-wise PCs”?
P4/L115: Add “at each 2D subspace” at the end of the sentence. Then, get rid of the middle sentence if “pair-wise” is added as suggested above.
P5/L (not clear which line but the statement following “Result”): Does x-matrix contain the standardized values or original concentrations? Need to specify.
P5/L (#4 in the table): Change “needed” to “found” because the number is variable and it does not matter for how many to be found.
P5/L (#5 in the table): I am sure both I-vector and J-vector were explained in the text, but in the table their function still needs to be specified so that readers understand what SI-matrix means.
P5/L (#6 in the table): Need to say h and H represent for fractional contributions.
P5-6: After reading the text, it is still not clear how both I- and J-vectors were generated. Through an optimization procedure itself?
P6/L158-163: Kind of arguments are needed to set the stage for multiple runs. But the exact statements here fit better to discussion.
P7/L187: You cannot just cite Hooper et al. (1990). DTMM has to run to identify conservative solutes and the number of end-members.
P7-8/L193-215: Most if not all of them should be presented under 3.1.
P9/L253-259: Some of them are very much speculative.
P11/L325-330: These are not conclusion.
Figure 1: What are the red crosses in c? Why are they in different positions from those in b?
Also, need to say this uses 3D as an example.
Figure 2: Do not use the same colors for the PC axis’s of solutes as for the triangles; way too confusing. You used “diamonds” not “squares”. In the case of four end-members, this is not exactly the convex hull but the projected convex hull into 2D PC subspace. For four end-members, it needs three PC subspaces. This should be specified.
Figure 3: Do the colors shown in legend match those points on the plot? I do not see blue and red colors in the legend.
Also, the same legend used in all cases regardless of the number of endmembers. Is it necessary to show all four (maybe five) when you are seeking for only three endmembers?
Figure 4: Cannot be self-explained. Averaged from 100 runs? Residuals of what? What does five-fold cross validation mean?
Font size too small; resolution too low; symbol size too small too.
Very hard to distinguish Ca and Si curves.
Figure 5: Observed in both stream water and end-members? Do not use "observation" for "stream water" as observed end-members were also shown.
Figure 6: You mean "the same size ..."? Also, change "sample" to "samples". Font size too small.
Figure 7: How were algorithm and data uncertainties calculated? I do not think they were covered in the text.
Figure 8: (top) Different number of samples? Or, different sample distributions? (caption) Do they all have the same number of samples?
Figure 9: I cannot follow the definition of "percent end-member limited". If samples were generated from three synthetic end-members with constrains of each within the fraction of 0-1 and all summing to 1, how come are some outside of the triangle?
Standard deviation or normalized uncertainties? Also need to explain if the normalization is the same as the one shown in an earlier figure.
What do components X and Y mean here? PC components? If so, say so!
Table 1: Title should be placed above the table (different from figures) unless the journal requires to be placed under.
Indicate the number of end-members for each block.
What are their fractional contributions? Are the contributions from fourth and fifth end-members significant?
Table 2: Title should be above the table if the journal does not require to be under.