the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Simple analytical–statistical models (ASMs) for mean annual permafrost table temperature and active-layer thickness estimates
Abstract. A variety of numerical, analytical and statistical models have been developed for estimating the mean annual permafrost table temperature (MAPT) and active-layer thickness (ALT). These tools typically require at least a few ground physical properties, such as thermal conductivity, heat capacity, water content or bulk density, as input parameters in addition to temperature variables, which are, however, unavailable or unrepresentative at most sites. Ground physical properties are therefore commonly estimated, which may yield model outputs of unknown validity. Hence, we devised two simple analytical–statistical models (ASMs) for estimating MAPT and ALT, which are driven solely by pairwise combinations of thawing and freezing indices in the active layer; no ground physical properties are required. ASMs reproduced MAPT and ALT well in most numerical validations, which corroborated their theoretical assumptions under idealized scenarios. Under field conditions of Antarctica and Alaska, the mean ASMs deviations in MAPT and ALT were less than 0.03 °C and 5 %, respectively, which is similar or better than other analytical or statistical models. This suggests that ASMs can be useful tools for estimating MAPT and ALT under a wide range of climates and ground physical conditions.
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RC1: 'Comment on egusphere-2024-2989', Anonymous Referee #1, 08 Nov 2024
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GENERAL COMMENTS
The authors propose a simple method for estimating mean annual permafrost table temperature and active layer thickness solely based on temperature monitoring at two depths within the active layer. The approach, based on the TTOP formula, is elegant and could be helpful for the interpretation of field data. The main advantage is that, considering yearly integrated observations of soil temperature at two depths, the proposed method avoids the need of using ground properties measurements, at the price nevertheless of strong simplifying assumptions.
However the assessment of the performance of the proposed method should be thoroughly improved. The validation based on numerical simulations in idealized cases is not relevant, using an outdated modelling approach as the reference. The validation against field data is good, but too few sites are considered. Once these problems solved, the discussion of the limitations related to the strong underlying assumptions (e.g.: constant ground properties) should be carefully made.
Thus I recommend major revisions of this manuscript prior to consider its publication in The Cryosphere.
SPECIFIC COMMENTS
- l 8: “which corroborated their theoretical assumptions under idealized scenarios” ; unclear, please rephrase.
- l 66: “Besides surface temperatures, Eq. (1) is valid for temperatures measured at any depth in the active layer”. Please clarify here what is exactly meant by ‘is valid’ ; has it been validated against field data? With which general procedure?
- l 76: Eq. (5) (and generally the TTOP formula used here) imply the assumption that the thawed soil thermal conductivity kt is constant over time, both at seasonal and multi-annual time scale. The frozen soil thermal conductivity kf is also considered as constant at multi-annual time scale I guess. Since kt does depends on the soil water content, it varies within an active season and along the years according to the variability of precipitations (and thus infiltration, and thus soil water content). This is a strong assumption that must be pointed out here and extensively discussed in the paper.
- l 94-95: “This documents that Eq. (8) for MAPT is analytical and statistical at the same time because it integrates both approaches.” ; I don’t understand.
- l 100: Eq. (13) implies that the volumetric water content φ is constant over time, although it varies within an active season and at multi-annual time scale depending on precipitations. Same remark for kt (see also my specific comment at line 76). This is a strong assumption that must be pointed out here and extensively discussed in the paper.
- l 128-129: “Usually, Eq. (21) has been referred to as the modified Stefan model and proved to be useful in situations where the ground physical properties were unavailable and/or for spatial modelling of ALT”. Eq. (21) and eq. (13) are strictly equivalent. May be that the difference that the authors want to point out is that the edaphic term in (21) maybe calibrated in itself, without estimating the thawed heat conductivity and the volumetric water content separately. But would it be really different to make a two parameters calibration for kt and φ? Anyway these ones would be estimated averages, probably calibrated as well, since these quantities do vary in time (see specific points l 76 and l 100).
- l 157-158: “As with Eq. (8), this documents that Eq. (27) for ALT is analytical and statistical at the same time because it integrates both approaches.” ; I don’t understand.
- l 168: The numerical model used for solving heat transfer in the active layer is a very old fashioned one (Carslaw and Jaeger, 1959). Since then numerous modelling works as been done for the simulation of heat and water transfers in soils with freeze-thaw (see for instance the benchmark of Grenier et al., 2018, or the reviews of Bui et al., 2020 and Hu et al., 2023). A more up to date model should be used.
C. Grenier, H. Anbergen, V. Bense, et al., Adv. Water Resour. 114 (2018) 196–218, https://doi.org/10.1016/j.advwatres.2018.02.001
M.T. Bui, J. Lu, L. Nie, A review of hydrological models applied in the permafrost-dominated Arctic region, Geosciences 10 (2020) 401, https://doi.org/10.3390/geosciences10100401
Hu G., Zhao L, Li R., Park H., Wu X., Su Y., Guggenberger G., Wu T., Zou D., Zhu X., Zhang W., Wu Y., Hao J.: Water and heat coupling processes and its simulation in frozen soils: Current status and future research directions, CATENA, Volume 222, 106844, ISSN 0341-8162, doi:10.1016/j.catena.2022.106844, 2023.
- l 179: Using eq. (32) for the reference numerical simulations prevent to consider the effect of the coupling of water flow and heat transfer, since volumetric water content is considered as constant (see table 1). Meanwhile, spatial and temporal variations of water content may be of primary importance for soil thermal regime (see for instance Kurylyk and Watanabe 2013, Sjöberg et al. 2016, Orgogozo et al., 2019). A more complete model should be used.
Kurylyk B.L., Watanabe K., 2013. The mathematical representation of freezing and thawing processes in variably-saturated, non-deformable soils, Advances in Water Resources, Volume 60, Pages 160-177, ISSN 0309-1708, doi:10.1016/j.advwatres.2013.07.016., 2013.
Sjöberg Y., Coon E., Sannel A. B. K., Pannetier R., Harp D., Frampton A., Painter S. L. and Lyon S. W., 2016. Thermal effects of groundwater flow through subarctic fens: A case study based on field observations. doi:10.1002/2015WR017571, 2016.
L. Orgogozo, A.S. Prokushkin, O.S. Pokrovsky, C. Grenier, M. Quintard, J. Viers, S. Audry, Permafr. Periglac. Process. 30 (2019) 75–89, https://doi.org/10.1002/ppp.1995
- l 256: “Overall, however, these findings corroborate the theoretical assumptions outlined in Sect. 2.2” Please be more specific. Which precise assumptions?
- l 263: “ALT estimates by Eq. (27) were more accurate in Antarctica” I think that this is due to the fact that soil water content varies much more in the Alaskan sites that in the Antarctica sites. The bimodal distribution in the bottom left graph of Figure 5 is maybe also due to this.
- l 271: “a reasonable accuracy” ; reasonable according to which criterium ?
- l 272: “which corroborated their theoretical assumptions (see Sect. 2.1 and 2.2)” ; unclear, please rephrase.
- l 273: “they can work reasonably well under a wide range of climates and ground physical conditions” ; this has to be demonstrated by the discussion.
- l 277-278: “Under field conditions, the ASM deviations were close to zero on average” ; this statement seems not in line with the results shown in Figure 5.
- l 315-317: “Since ASMs build solely on thawing and freezing indices at two distinct depths in the active layer, the values of which reflect the rate of heat transfer across their intermediate layer, the solutions also intrinsically account for the temporal variability of ground physical properties.” ; I do not agree. According to eq. (1), the TTOP formula on which is based the ASMs does not take into account these temporal variabilities.
- l 317-319: “Likewise, they consider latent and sensible heat and any other factors that might affect the heat transfer in the active layer, some of which other models do not explicitly account for.” ; same thing that the previous comment.
- l 325-326: “Ground physical properties also commonly show more or less variability on seasonal and annual time scales […] which most other models cannot handle because they typically treat ground physical properties as constants.” ; I think that it is also the case here, according to eq. (1).
- l 359-360: “ASMs for estimating MAPT and ALT can find applications under a wide range of climates and ground physical conditions” ; it sounds to me like an overstatement. Two sites where investigated, largely not enough to sample the variety of permafrost environments: continuous/discontinuous/sporadic, in various environments such as for instance tundra or boreal forest, with diverse lithology and pedology, under various climatic (e.g. precipitation) conditions, etc.
TECHNICAL CORRECTIONS
- l 288-289: “in the order of tenths to first degrees Celsius” ; english language problem.
- l 346-350: Not necessary.
Citation: https://doi.org/10.5194/egusphere-2024-2989-RC1
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