The authors introduce a modification and improvement of numerical advection algorithms by merging existing techniques in a way the optimizes the numerical approximation of advection. They also introduce a new error metric that assesses not only local extrema, but also the distribution and range of concentration distributions that are advected. My main complaint about the study is the limited nature of the test presented (only one Courant number, and one fairly well-resolved gaussian hill shape. Aside from a few minor pounts that need to be mentioned, the paper is publishable 

1) typo line 250. How can fluid density everywhere be zero. is this a typo"

2) using these deformational flows, figures like 8c and others do not properly quantify the true "accuracy" of the scheme since the shape is advected/sheared into a shape that has scale features that are smaller than the 4 km resolution of this experiment. If this entire study were performed at 1 or 0.1 km resolution, the error measures would change. Basically, the simulation called "base" in these figures is not a true average of the 4x4km resolution "base". When the peak of the "base" Guassuan shape is sheared into successively smaller and smaller cells, then averaged BACK to the 4 km coarse T=0 4km-resolution discrete grid, then the simulation called "base" should change with time also. This applies to the figures which show the distribution at the T/2 times at the point where the. Here the authors could do a simulation at spatial resolution of 0.4 km (much smaller than 4 km), then MAP the 0.4 km simulation back onto the 4km "base" grid by averaging 10X10 grid cells into a single 4 km grid, then compare.

3) one of the problems with the Walcek peak scheme is the inability to preserve lumped but conserved species since there it is guaranteed that numerical treatment around local maxima are NOT treated identical in SUMS of conserved species. The "peaks" in NO, NO2 and HNO3 will occur at different places than the combined NO+NO2+HNO3. In the context of the problem presented here, total nitrogen should be conserved (NO+NO2+HNO3). The authors should be able to show that there is non-monotonic behavior of any advection scheme which treats local EXTREMES algebraically differently. Please show graphs of NO+NO2+HNO3

The authors integrated PPM with the flux adjustment of Walcek method to improve the representation of local extrema which results in overall improvement without an increase in computational intensity. To evaluate their method they performed a simulation with different advection schemes with a chemical mechanism. They introduced the signature function as an alternative error statistic which basically evaluates whether the concentration probability distribution remains unchanged after advection. I think the advection scheme they suggested here is certainly an advancement for chemical transport modeling but the way they evaluated the method is in question.

The chemical transport model deals with the various velocity fields and initial conditions. I don’t think the authors need to test their model with very realistic data, but just one simulation is not sufficient. For example, the key advancement of their method is on how to address local maxima, but there is only one local maxima in their initial condition. I’m curious how much the method will be effective if there are more stiff gradients–even testing with those scenarios can give them more benefits in computational cost.

They alleged that theoretically, the signature function will remain the same if there is no numerical error. I think they can change if the wind field has divergence. For example, if there is a negative divergence in a certain region, the concentration can be accumulated and the number of cells with high concentration can increase and the maximum range can increase. In case of the positive divergence, the wind will make scalars dispersed away so the number of cells with low concentration increases and the number of high concentration cells will decrease. The negative divergence scenario is relevant to high pollution episodes trapped by temperature inversion like LA smog while the positive divergence scenario is like the radial spread of volcanic ash. Even though those might be a bit extreme cases, it is the role of a chemical transport model to address those cases and in this regard the signature function cannot guarantee a proper evaluation.

If authors admit that the signature function works well on non-divergent wind scenarios including their simulation here, then it can make sense. However, the assertion that the function as a good method working universally is not true in my opinion.

Below are minor details authors can correct:

L75 typo: … the three reactions wthat …

L80-90: R3 and R8 are incorrect in stoichiometry

Table 1: mean with standard deviation will be better to represent execution time

L256: The variable X should be explained.

Equation 11: H is non-italic here but in L259 it’s italic.

L264: X is non-italic here.

L300: This sentence does not make sense to my perspective as I explained above.

L320: For similar reason, I don’t think the maximum or minimum will not change. It is more natural to evolve with time when there is divergence.

L334: There might need a period in the middle. Looks like two different sentences are not properly separated.

L361: berformance looks like a typo of performance

L364~365: The conclusions drawn from the analysis… but the evidence is only a single experiment?

L375: Is a signature function really invariant? (same for other invariants authors mentioned)

This is a very nice paper that describes evaluation of tracer transport using a signature function, as well as the improved PPM scheme mentioned in the title. The signature function gives a lot of information about the evolution of transport errors without the need for an analytic solution. I like it and I am surprised that it is not already widely used.

I have two main comments which the authors may argue is beyond the scope, but these comments, I think, would make the paper even more convincing.

1. A new advection scheme is introduced - PPM+W. The results of your test cases look good but you are comparing against some old advection schemes using a non-standard test case. Please include results using a standard test case so that you can refer to other papers that include results of exactly the same test cases and the reader can easily compare errors.

2. You propose a new test case. There isn't enough evidence presented to demonstrate that a new test case is needed.

Minor comments:

1. "Non-monotonic" rather than "non-monotonous".

2. The section heading 2.2 "Flux description" is not well named. This is really "Definition of Test Case". 

3. It is not clear why the test case described in sections 2.2.1-2.3 is not exactly the same as LeVeque (1996). Using dimensioned rather than non-dimensional variables does not constitute an effective change.

4. Line 147. Replace with "PPM (Colella and Woodward, 1984)" as that is how you refer to it elsewhere.

5. The errors of van Leer and PPM are not obvious in figures 3 and 4. These schemes look better than the +W schemes. Would it be possible or helpful to compare the scheme cell averages with the exact cell averages?

6. Is it usual for chemistry schemes to violate machine precision conservation? Provide references.

7. You often start a new paragraph where there shouldn't be a new paragraph. Eg line 224 and 259. Note that any blank line in LaTeX will create a new paragraph. So don't put blank lines around equations unless you want a new paragraph.

8. Equations 9 and 10 need u to be displayed as a vector and equation 10 needs a dot between u and grad.