This was a worthwhile project, suitable for publication in TC.  I have a general comment, a major comment, and several minor comments.

General comment:  Many numbers are given to higher precision than warranted.  The precision of the average should not exceed the value of the standard deviation.  For example, on line 246, change “69.5±8.8” to “70±9”.  This change also makes the result easier to read.  As another example, on line 295, change “-14.6±7.6” to “-15±8”.

Major comment:  I am skeptical of the snow surface temperatures (SST) reported in Figure 5. The reported SST is often nearly 10 K colder than the air temperature, and is never warmer than the air temperature, although the temperature at 10-cm depth (measured by a different method) exceeds the air temperature until 30^th March.  How was the SST measured (maybe by IR emission)?  Has the SST method been verified by in-situ thermistor measurements? [This verification should be done at night, so that solar heating of the thermistor does not cause a bias.]

Minor comments:

Title (and elsewhere).  “Constrain” is a synonym of “inhibit”.  So to “constrain” the influence of near-surface melting suggests that you are looking for ways to reduce its influence.  I suggest changing “constrain” to “quantify” everywhere it appears.  Or say “determine”, as you do on line 394.

Throughout:  I suggest writing dD instead of d²H.  dD is commonly used, and d² looks like delta-squared.

Line 27. “increased by 4-5 C”. Since what year?

Line 36.  Explain why melting generally leads to ¹⁸O enrichment.

Line 78.  “snowpack covering 60-100% of Svalbard”.  Is this a seasonal variation, i.e. 100% in winter and 60% in summer?

Figure 1.  Point out that all these sites are near sea-level, none on the mountain or on a glacier.

Lines 171-172.  “maximum percolation depth . . . and average percolation depth”. Table E2 gives average but not maximum.

Line 176. Change “percolation depth dependent” to “percolation depth to be dependent”

Table 1, Liquid water content.  Give a reference for NESA-SM1, and ideally briefly say how the method works.

Line 226. Change “temperature gradient” to “temperature difference”.

Figure 4c (also on Figure C1). What is the green line plotting? Its horizontal position moves back and forth slightly, but there is no scale for it on the horizontal axis.

Figure 4 caption line 1. Change “with” to “versus”.

Line 253.  The LWC is not given on Figure C1.  Figure C1 includes a label “LWC”, but only gives a symbol, “D”, always the same, not a number. The symbol D is also undefined.

Line 295 says “average -14.6±7.6‰”, but line 297 says “mean of -14.55±2.5‰”. What distinction is being made here? They seem like two slightly different averages for the same thing.

Line 353 cites the extreme rainfall event of 24^th March 2007.  I actually was in Ny-Alesund during that event; it was dramatic. Then during the next two days the wet snow froze into a crust that the reindeer could not break through.  But the reindeer knew what to do; they walked up to higher elevation where it had snowed rather than rained. According to my log for that time, the rainstorm occurred in early evening of the 23^rd, not the 24^th.

Line 421. Define “matrix flow”.

Line 421. Define Ø.