This works provides an analysis of tritium observations to estimate groundwater mean transit times in the Luxembourg Sandstone aquifer. The analysis benefits from the interpretation of a large tritium observation dataset that is distributed in both space and time. The authors apply an uncertainty analysis framework to quantify some of the salient sources of uncertainty in estimating mean groundwater transit times from the tritium observations. A powerful component of this work is the assimilation of spring discharge response times and major ion solute chemistry to regularize the tritium analysis. The primary conclusion from the analysis is that groundwater mean transit times can successfully be interpreted at this field study site, despite numerous sources of uncertainty and short observation timeseries. While the manuscript is easy to follow, it often lacks clarity and work is needed to update the language.

My main concern is that this work does not provide novel nor significant insights into the interpretation of tritium observations nor hydrologic processes at the investigated site. Rather, it is generally an application of established methods to estimate mean groundwater transit times. There is little interpretation into what the inferred mean transit times add to our understanding of this aquifer. Furthermore, it is not clear whether the uncertainty analysis applied in this work results in a substantial advantage over the already established tritium interpretation techniques, many of which also propagate uncertainties.

Major Points:

While this work propagates the analytic tritium errors in the uncertainty analysis, I feel that it misses the key point that mean groundwater transit times can be non-unique because tritium is insensitive to the long residence time groundwater fractions (the tails of the transit time distribution). This non-uniqueness poses a major challenge in estimating mean transit times with tritium alone and is potentially the larger source of uncertainty compared to analytical errors. I think that the introduction can be revised to better account for the multiple sources of uncertainty in tritium mean transit times and clearly state in what circumstances having a tritium timeseries is most likely to reduce this uncertainty and in what situations it cannot.

Estimating groundwater mean transit times with tritium and the convolution integral is not new. Thus, I do not entirely agree with the claim that this work is pioneering interpretation of sparse tritium measurements in the northern hemisphere. I suggest revising the manuscript to focus more on the new hydrologic processes we learn from the tritium analysis, rather than inferring mean transit times from tritium. Alternatively, if the more accurate tritium measurements are proving to be essential to estimate mean travel times compared to previous studies, I think this needs to be explicitly shown in this work. While the mean travel time estimates in the Luxembourg aquifer can have value, I feel any innovation in this work is not properly highlighted.

I have concerns on the authors approach to constrain the parameters during the Monte Carlo uncertainty analysis. For instance, the mean transit time parameter is forced to be less than 35 years during the Monte Carlo sampling. Yet, the evidence and rationale to make this assumption is not strong and needs further discussion. Falsifying old transit times due to the high storage volumes is one line of evidence; however, I feel the highly simplified interpretation of storage volumes in this work is not a particularly strong argument and further analysis or comparison to alternative environmental tracers is needed. Furthermore, if it is evident that mean transit times cannot be older than 35 years, it is unclear why the first Monte Carlo was performed. Generally, our prior bounds should encompass our best estimate of the uncertainty ranges for the given parameter. If the reader is to trust the uncertainty analysis, I suggest that the implications of these strong assumptions on the plausible mean transit times need to be further explored and presented.

It is unclear whether the mass balance is closed using the spring discharge measurements alone, or if river discharge or regional groundwater flow are additional fluxes that need to be considered. In general, there needs to be more explanation of the soil model that calculates the ET fluxes and how the uncertainties in the effective recharge rates impact the mean transit times. This uncertainty can potentially be more important than the tritium analytical error. While it can be argued this point is beyond the scope of this manuscript, I think this work needs to better acknowledge the many sources of uncertainty when quantifying mean transit times with tritium observations.

My opinion is that major revisions are required to address the comments above and to highlight the novelty in this work.

Specific Comments:

L19: I suggest that the abstract needs to explicitly state what the identified mean transit times are.

29: Duplicated references.

L34: The use of ‘by far’ and ‘obviously’ are too strong of descriptions and are not needed. These are examples of the many places where the clarity of the manuscript can be improved.

L67: I think at this point the Introduction needs to mention the power of 3H/3He dating, which has largely super-seeded using 3H alone. Using 3H alone represents a major limitation of this study in my opinion.

L208: The information “initiated in 1958 by the IAEA (International Atomic Energy Agency) and the WMO (World Meteorological Organization)” is not needed.

L218: Why was a water-holding capacity of 100 mm assumed? In general, there needs to much more information on the soil model.

L225: Equation 3 for the tritium input weighting does not seem correct. Will P_eff just not cancel out because it is in the numerator and denominator unchanged?

L241: It states that the input signal uncertainties are propagated by randomly sampling from a normal distribution; yet there is no information on what the distribution parameters are.

L255: It is not explained why the comparison to hydraulic response times and major ions is done in first place. Adding some rationale for why this analysis was performed into the introduction seems necessary.

L345: It is not clear how the measured tritium suggests the presence of bomb-peak water. For instance, it seems that the measured values could be obtained through recent recharge and radioactive decay.

L357: I think there needs to be much more discussion on how these samples with old mean transit times can be confidently rejected. I am not convinced of this assumption given the terse description provided.

L366: Why were only a third of the samples retained? The seemingly subjective ‘throwing away’ of samples concerns me and seems to deteriorate the uncertainty quantification significantly. The assumptions and rationale need to be explained better.

L369: Same comment as above. Is keeping only the dominant population actually providing a robust measure of the uncertainty?

L472: This sentence is awkward and is not needed.

L522: Is this assumption of vadose zone that is the entire thickness of the aquifer reasonable? I think that much more information about the vadose is needed. For instance, are there any wells that can constrain the vadose zone thickness?

General remarks

The paper deals with a relevant topic. It is well written, straight to the point and interesting for both the specialists and the less experienced audience. For these reasons I recommend minor revisions, that mostly concern the description of the aquifer properties of the Luxembourg Sandstone Formation. As a matter of fact, sedimentary, diagenetic, structural and geomorphological (i.e. diffuse karstification) properties have been too poorly described, sometimes in a misleading way. I suggest to revise deeply section 2, removing the misleading sentences, introducing the most relevant hydrostratigraphic and hydrogeological features and completing/updating the references to the most relevant papers. This will make more convincing the discussion and conclusion sections, permitting to establish reliable comparisons between the modelling results and the heterogeneity and anisotropy of the real world aquifer.

Specific remarks

Section 2.1: the part of this section dedicated to the geological properties of the Luxembourg Sandstone is definitely too poorly informative and insufficient to characterize the heterogeneity and the anisotropy of the aquifer under investigation. It also contains some incorrect statements (see the following remarks). Please update the references about the formation, that are incomplete and in some cases outdated. The most relevant stratigraphic, sedimentological, compositional and diagenetic properties should be mentioned to describe shortly how they control the modes and paths of groundwater flow through the porous/fractured/karstic medium. The superimposed structural pattern of faults and fractures should be also introduced to mention how it contributes to the duality of groundwater circulation (fissured - porous rocks) that You assess. In its present from the description of fractures is almost useless. The presence of widespread karst features, reported by some literature, should be mentioned and commented, also considering the impact of these features on the “dual” circulation system (is it really dual?). In addition, the presence of aquitards within the aquifer group should be introduced before the discussion section. In its present form this section conveys the wrong idea of an almost homogeneous and isotropic sandstone body, with uniform facies/hydrostratigraphic properties through space, that is not the case. It should also be mentioned which members, i.e. which aquifer systems within the group, and at which sites have been sampled and studied. This latter part totally relies on the technical notes by public agencies, that do not permit to figure out the geometry of the aquifers composing the group, of the compartments within them and of their recharge areas. As a matter of fact, the identification of the groundwater bodies that You are studying is not sufficiently clear to the reader and reliable. As You state in Your discussion section, the integration of Your modeling results with the knowledge on aquifer geometry and physical behavior would lead to “more accurately represent the multi-scale complexity of the Luxembourg Sandstone bedrock aquifer”. So why don’t You start by incorporating a very short synthesis of the most relevant hydrostratigraphic knowledge, in terms of heterogeneity and anisotropy of the rocks and identification of the major groundwater bodies within them in Your paper?

Line 102: no need to indicate SiO2 and CaC03, just state “quartz” and “calcite”

Line 103: really the Luxembourg Sandstone is just a calcite-cemented pure quartzarenite (see for instance Berners, 1983)?

Line 103: is the average bulk chemical composition relevant to describe aquifer heterogeneity? The Luxembourg sandstone is an outstanding example of how diagenesis determined the poro-perm properties under control of the composition of the framework grains and the changes of texture and sedimentary structures (i.e. facies associations).

Line 103: please replace the reference to the unpublished PhD thesis with the reference to Van Den Brill and Swennen (2009), or at least quote also the published paper.

Line 104: what do You mean by “… crossed by beds of sandy marls”? Are these beds neptunian dykes? Please describe the stratigraphy of the formation properly: there are marly units separating sandstone bedsests, facies and compositional changes (framework grains, cements, matrixes) occur through the sandstone divisions, many bedsets are almost limestones, owing to primary composition and diagenetic replacement, so karst features are widespread in some bodies at some sites (see for instance Meus and Willems, 2021). Do You really would describe this formation as a “uniform” unit? On the contrary it is highly heterogeneous and anisotropic, that implies relevant bearings on Your experiments.

Line 115: Meus and Willems (2021) is missing in the reference list

Line 131: considering the regional geology, are You pretty sure that recharge occurs only through the outcrop area of the Formation?

Fig.1 is almost useless to describe aquifer architecture and heterogeneity. Stratigraphic logs introducing the general features of the formation should be added.

Fig.2 is very difficult to read. The geological attributes are hidden by the elevation map of the formation base (please specify m “above sea level” in the color scale). The legend of the geological features is obscure (alluvial materials? What do You mean? Quaternary? Which formations are involved?). Where do we read these features? Where are represented the fault/fracture systems? Which hydrogeological features are shown? The sampled springs are sparse at different settings. Which units of the Formation have been sampled? All belong to the same sandbar systems? Are some springs located in the limestone (karstified) units? Are there marlstone beds (aquitards) at some locations?

Section 2.2

Fig.3. This is a very general and unrealistic conceptual picture of the Luxembourg Sandstone Aquifer. It is portrayed as a uniform, homogeneous and isotropic medium, without bedding planes associated to litho-textural variations. Fractures are not drawn (the reader must assume two orthogonal vertical sets everywhere). “Slow infiltration through matrix” is declared in the unsaturated zone, with black lines maybe indicating strange infiltration paths (if I understand the picture) that would never permit the percolating water to reach the saturated zone. Moreover, what is the matrix? In the karstic aquifers, matrix is sometimes intended as the impervious rock with no circulation, that instead occurs through conduits, caves and fractures. In the list of “some numbers” You declare up to 40% porosity, so this would not be a matrix, neither from the lithological/sedimentological point of view (it would be a mudrock) nor from the hydrological point of view. In addition, it looks a little bit strange the use of the conceptual image of a carbonate karstic aquifer, without considering the karstic features of Your specific setting. I strongly suggest to redraw a realistic conceptual model of Your aquifer, with the true stratigraphic, lithotextural, structural and geomorphological features and inserting the plausible location of the clusters of springs You sampled. Please note that I am not asking for a more detailed or accurate picture, I just would like to see a very simple and general model showing the most relevant features of Your aquifer group.

Section 3.1

Lines 176-177: this statement should have been supported by the description of the conceptual model of the Luxembourg Sandstone in Chapter 2, that is unfortunately largely insufficient. Moreover, this assumption should be site-specific in such a large aquifer group as the one You are dealing with.

Line 190: Fig.2 does not explain the hydrogeological setting of the 32 sampled springs. Do they share the same recharge area, geological and hydrological conditions? Do the same approach apply to all the sampled springs? You rightly mention the hydrochemical and hydrogeological and exploitation variability among them, that should be better described and accounted for in Your approach.

Lines 200 and following: which data on average thickness and presence/absence of surface soils did You use? Which data for evapotranspiration?

Lines 245-250: an effort to estimate the aquifers volumes in order to obtain some independent numbers to evaluate the estimates of groundwater volumes would make this study a little more linked to the real world.

Section 3.2

Lines 289 – 290: here You refer to the current use of effective infiltration in karst aquifers after literature, but since this statement You did not consider the Luxembourg sandstone a karst aquifer. This issue must be addressed properly in section 2 where You should definitely characterize the aquifer group under investigation as karstified or not.

Lines 296-297: this assumption, in my opinion, is unrealistic and makes poorly reliable the use of effective infiltration.

Line 311: I suggest to make explicit the abbreviations at least at their first appearance (MRT)

Lines 323-324: might You consider the opportunity of inserting the Piper plots to better visualize and characterize the eventual variability of hydrochemical facies? At line 329 You state that the “… the spatial variability of the hydrochemistry … is stable over time… “, implicitly admitting that this variability do exist and might be clarified by these very simple and traditional plots.

Section 4

Lines 348 – 350: the recharge areas of the spring groups are not sufficiently described and commented by the previous sections and by Fig.2 that are based on the 1939 geological maps and on local RGD technical notes, that make it very difficult to obtain a reliable idea of the groundwater bodies and hydrogeological basins involved by the study.

Section 5

Through all the discussion section, several different properties of the real aquifer are mentioned for comparisons with the modelling results and the assessment of uncertainty (real sandstone thickness, vadose/saturated zone ratio, modes and times of transit through the vadose and saturated zones of the dual porous/fractured aquifer without considering the karst features, presence of local or widespread confining layers and internal compartments in the aquifer group, areas and lengths of the recharge regions and paths, hydrochemical properties and so on). In most cases You claim that these comparisons support the results, but most of these properties have been introduced and considered in a very rough and generic way, sometimes not truly coincident with the real world. In addition, You formerly used many of the same properties to take or to validate decisions, so introducing some circularity in Your line of reasoning. As a matter of fact, this generic use of a poorly presented knowledge on the aquifer group does not really support the results, on the contrary highlighting to the reader the distance between knowledge on the real-world aquifer heterogeneity and anisotropy and the presented modeling results. An example is given by the last part of the section (from line 552 to the end) where the obvious anisotropy of this kind of aquifers, that is portrayed by hundreds of papers of the current hydrostratigraphic literature, is introduced and discussed very roughly. Over these lines the Authors look to discover this property (that is shared by all the bedded aquifers) at this point of the paper (see also lines 590-592 in the Conclusions), arguing that this physical property might be better understood, so it should be studied in order to set up an integrated model (hydrostratigraphic, hydrogeological, hydrochemical, let’s say in 4D), that incorporates the tritium based LPM approach nicely proposed in this paper. So, why the Authors did not use all the existing literature on the Luxembourg sandstone aquifer group to make tight comparisons between the real world and their modelling results? How nice would have been to show that the well-known anisotropy of the bedded aquifer is mirrored by the computed anisotropy of water velocity through the vadose and saturated zones? 

Lines 475 – 476: I agree that presuming stationarity for groundwater is less critical than for stream water, but this does not mean that steady-state might be assumed safely for a heterogeneous, mixed karstic/fractured/porous aquifer group like the Luxembourg sandstone.