Simple analytical&ndash;statistical models (ASMs) for mean annual permafrost table temperature and active-layer thickness estimates

Uxa, Tomáš; Hrbáček, Filip; Kňažková, Michaela

doi:10.5194/egusphere-2024-2989

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https://doi.org/10.5194/egusphere-2024-2989

Preprints

18 Oct 2024

| 18 Oct 2024

Simple analytical–statistical models (ASMs) for mean annual permafrost table temperature and active-layer thickness estimates

Tomáš Uxa, Filip Hrbáček, and Michaela Kňažková

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.

Received: 24 Sep 2024 – Discussion started: 18 Oct 2024

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07 Jan 2026

Simple analytical–statistical models (ASMs) for mean annual permafrost table temperature and active-layer thickness estimates

Tomáš Uxa, Filip Hrbáček, and Michaela Kňažková

The Cryosphere, 20, 97–112, https://doi.org/10.5194/tc-20-97-2026,https://doi.org/10.5194/tc-20-97-2026, 2026

Short summary

Tomáš Uxa, Filip Hrbáček, and Michaela Kňažková

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2024-2989', Anonymous Referee #1, 08 Nov 2024

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 k_t is constant over time, both at seasonal and multi-annual time scale. The frozen soil thermal conductivity k_f is also considered as constant at multi-annual time scale I guess. Since k_t 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 k_t (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 k_t 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
- AC1: 'Reply on RC1', Tomáš Uxa, 20 Dec 2024
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-2989/egusphere-2024-2989-AC1-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2024-2989-AC1
RC2:
'Comment on egusphere-2024-2989', Anonymous Referee #2, 12 Nov 2024

The manuscript by Uxa et al. presents an approach for determining MAPT and ALT utilizing shallow ground temperatures at two depths. The MS is interesting and generally well written with a good description of the approach being proposed.
However, there are some concerns regarding the validation of the approach and evidence that it is novel given that the two variables being determined are commonly calculated by interpolation/extrapolation when shallow ground temperatures are available. It is unclear what advantage the method proposed has over this commonly used approach especially given that the authors acknowledge that their equation is in principle a linear extrapolation of the temperature indices. The MS would benefit from a comparison of their model results to those from interpolation/extrapolation of observed shallow ground temperatures. The authors may have done this when they compare their model results to observed MAPT and ALT, but it is not clear how the observed values were determined, and additional explanation is required (see additional comments below). The authors may also want to consult Riseborough (2008, ICOP) regarding use of interpolation and extrapolation to determine thaw depth and importance of spacing of sensor depths.
The analysis and conclusions would benefit from better descriptions of the field sites including material properties, vegetation, climate etc. The limited range in their characteristics is a concern as all the sites are in cold permafrost of the continuous zone and likely tundra sites. This limits the conclusions that can be made regarding model performance and the broader applicability of the approach. Consideration of sites in warmer permafrost in the discontinuous zone including those in forested and peatland terrain would be useful as this would further back up statements regarding model performance including statements made regarding warmer permafrost.
Additional comments related to the concerns raised above and other comments are provided below for the authors’ consideration.
Additional Comments
L21 – “indicators” might be better than “measures”
L22-23 – suggested revision: “Climate change has resulted in permafrost warming and active-layer thickening throughout the permafrost regions”. Biskaborn et al. is now out of date with respect to the trends. I suggest you include Smith et al. (2024, State of Climate- Arctic), along with Noetzli et al. (2024), as it provides the details for Arctic and is up to date.
L29-30 – It is important to note that active layer thickness is not determined directly from geophysical surveys but is interpreted and it is difficult to determine ALT in warm permafrost with high unfrozen water contents.
L27-34 –Smith and Brown (2009) outline the various methods used as does Streletskiy et al. (2022).
L34-37 – Even if ground temperatures are measured within shallow permafrost and the active layer the permafrost table temperature or active layer thickness still needs to be determined/calculated. Interpolation and extrapolation is the method usually used. I suggest you consult Riseborough (2008 ICOP) which describes the appropriateness of interpolation/extrapolation approaches.
L38-42 – The purpose of the model is important. The ones mentioned are generally used for predictive applications such as determining conditions with little information on the site conditions. What you seem to be proposing is away to determining ALT or MAPT based on having ground temperature measurements which is what we do when we use interpolation/extrapolation of shallow ground temperatures to determine ALT or MAPT (see comment above).
L45 – Is reference being made to air or ground temperatures here?
L46-49 – Some of the approaches mentioned do consider variable properties including thermal conductivity for thawed and frozen conditions (e.g. ratio between them is included in TTOP equation) or the variation of conductivity with temperature (and unfrozen water content).
L57 – Editorial suggestion – delete “Besides other solution (Garagulya, 1990)” – it is not adding anything.
L66 – Editorial suggestion – Delete first part of sentence: “Eq. (1) is also valid for temperatures measured…..layer, which is convenient because….”
L68 – Note that usually the reference to surface temperature measurements used in these equations is from a sensor in upper 3-5 cm of the ground so not using exact surface temperature. The surface temperature can be estimated using n-factor to provide input into the equation.
L84 – If this is essentially extrapolation how is this different from the approach others use to determine the depth of the permafrost table and MAPT when they have temperatures at two or more depths?
L97 – Editorial suggestion - Delete first part of sentence: “ALT (m) can be calculated using the….
L99 – missing word: “…simplest form is as….”
L103 – See earlier comment regarding estimates of surface temperature used by others.
L136 – Smith et al. (2009) is also relevant - used observed ALT from sites in various regions and environments (tundra, forest, peatland and mineral and organic soil) to show the range in the Edaphic factor.
L148 – See earlier comment regarding extrapolation
L159-197 – It seems that there are several assumptions being made as well as simplifications. For example, thermal conductivity changes with temperature due to change in unfrozen water but it appears that constant frozen and unfrozen conductivity are assumed. Are the results of the two models really comparable?
Table 2 – For years and seasons is the 2^nd number the total record length and the first number the number of years utilized in analysis? It might be clearer to refer to number of years used and refer to it as e.g. 6 of 6. Are the ALT values determined from temperature or through probing? How is MAPT determined? It is unclear what you are comparing your modelled values with? I might have missed something here but maybe there needs to be a clearer explanation
L200-201 – No information on the sites is provided so the reader doesn’t know how diverse they are. There is no information provided on material characteristics or vegetation. The Alaskan sites are all on the North Slope in the continuous permafrost zone and likely in tundra environments, so conditions are not that diverse with respect to climate and vegetation. Using field data from sites in warmer permafrost in discontinuous zone and for forested sites would provide more diverse conditions. This would help show if your approach is valid for a wide range in conditions.
L204 – Since you are referring to a depth it would be better to refer to permafrost table or base of the active layer. Do you mean the base of the active layer was above the shallowest sensor?
L206-2011 – See Riseborough et al. (2008 ICOP) regarding errors associated with different approaches (interpolation, extrapolation) to determine thaw depth/top of permafrost etc. and guidance on the best approach to use.
L222 – Wasn’t this exponential decrease already fairly well known? Doesn’t the magnitude of the decrease depend on the material properties?
L235 – See earlier comment – How were observed values determined?
L260 – See earlier comment – How were observed values determined?
L265-268 – Does the difference in error between Antarctica and Alaska sites have anything to do with the material properties. Was latent heat more of a factor for the AK sites?
L264-292 – The thermal offset depends on the ratio of thawed and frozen thermal conductivity which depends on the amount of moisture/ice in the ground. If the moisture content is low or arid conditions exist, then the offset will be low or positive. Is the site in McMurdo Sound a dry site? It would be useful to know this. It would have been good to use sites with warmer permafrost in your analysis to back up the comment that deviation in MAPT estimates would be larger.
L306-310 – It would be useful to have information on the material properties at the field sites to back up these statements.
L315-330 – Although these other approaches make assumptions regarding thermal properties etc. based on general site characteristics, information on ground temperature is not required and the models determine the ground temperatures. This makes them useful for determining current and future conditions. This might make them more broadly applicable.
L327-330 – If temperature below the permafrost table was available would it be used if there were only one sensor at a shallower depth? You state that inputs can be any depth combination within the active layer based on temperature data availability and site characteristics. What are the site characteristics being referred to?
L331-334 – Aren’t these products based on modelling with various assumptions made regarding ground properties etc.
L338-340 – This is likely one of the primary sources of error especially with respect to moisture/ice contents and latent heat effects as discussed in Riseborough (2003).
L340-348 – Riseborough (2008) is probably relevant here especially with respect to spacing of temperature measurements etc. in determining thaw depth.
References
Riseborough, D.W. 2008. Estimating active layer and talik thickness from temperature data: implications from modeling results. In Ninth International Conference on Permafrost. Edited by D.L. Kane and K.M. Hinkel. Fairbanks, Alaska. Institute of Northern Engineering, University of Alaska Fairbanks, Vol.2, pp. 1487-1492.
Smith, S.L., Romanovsky, V.E., Isaksen, K., Nyland, K., Shiklomanov, N.I., Streletskiy, D.A., and Christiansen, H.H. 2024. Permafrost (Arctic) [in "State of the Climate in 2023"]. Bulletin of the American Meteorological Society (supplement), 105(8): S314-S317. doi:10.1175/BAMS-D-24-0101.1
Smith, S.L., Wolfe, S.A., Riseborough, D.W., and Nixon, F.M. 2009. Active-layer characteristics and summer climatic indices, Mackenzie Valley, Northwest Territories, Canada. Permafrost and Periglacial Processes, 20(2): 201-220. doi:10.1002/ppp.651
Smith, S. and Brown, J. 2009: Assessment of the status of the development of the standards for the Terrestrial Essential Climate Variables - T7 - Permafrost and seasonally frozen ground. GTOS 62 Essential Climate Variables
Streletskiy et al. 2022 Measurement Recommendations and Guidelines for the Global Terrestrial Network for Permafrost (GTN-P). DOI: 10.5281/zenodo.5973079

Citation: https://doi.org/10.5194/egusphere-2024-2989-RC2
- AC2: 'Reply on RC2', Tomáš Uxa, 20 Dec 2024
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-2989/egusphere-2024-2989-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2024-2989-AC2

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2024-2989', Anonymous Referee #1, 08 Nov 2024

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 k_t is constant over time, both at seasonal and multi-annual time scale. The frozen soil thermal conductivity k_f is also considered as constant at multi-annual time scale I guess. Since k_t 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 k_t (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 k_t 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
- AC1: 'Reply on RC1', Tomáš Uxa, 20 Dec 2024
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-2989/egusphere-2024-2989-AC1-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2024-2989-AC1
RC2:
'Comment on egusphere-2024-2989', Anonymous Referee #2, 12 Nov 2024

The manuscript by Uxa et al. presents an approach for determining MAPT and ALT utilizing shallow ground temperatures at two depths. The MS is interesting and generally well written with a good description of the approach being proposed.
However, there are some concerns regarding the validation of the approach and evidence that it is novel given that the two variables being determined are commonly calculated by interpolation/extrapolation when shallow ground temperatures are available. It is unclear what advantage the method proposed has over this commonly used approach especially given that the authors acknowledge that their equation is in principle a linear extrapolation of the temperature indices. The MS would benefit from a comparison of their model results to those from interpolation/extrapolation of observed shallow ground temperatures. The authors may have done this when they compare their model results to observed MAPT and ALT, but it is not clear how the observed values were determined, and additional explanation is required (see additional comments below). The authors may also want to consult Riseborough (2008, ICOP) regarding use of interpolation and extrapolation to determine thaw depth and importance of spacing of sensor depths.
The analysis and conclusions would benefit from better descriptions of the field sites including material properties, vegetation, climate etc. The limited range in their characteristics is a concern as all the sites are in cold permafrost of the continuous zone and likely tundra sites. This limits the conclusions that can be made regarding model performance and the broader applicability of the approach. Consideration of sites in warmer permafrost in the discontinuous zone including those in forested and peatland terrain would be useful as this would further back up statements regarding model performance including statements made regarding warmer permafrost.
Additional comments related to the concerns raised above and other comments are provided below for the authors’ consideration.
Additional Comments
L21 – “indicators” might be better than “measures”
L22-23 – suggested revision: “Climate change has resulted in permafrost warming and active-layer thickening throughout the permafrost regions”. Biskaborn et al. is now out of date with respect to the trends. I suggest you include Smith et al. (2024, State of Climate- Arctic), along with Noetzli et al. (2024), as it provides the details for Arctic and is up to date.
L29-30 – It is important to note that active layer thickness is not determined directly from geophysical surveys but is interpreted and it is difficult to determine ALT in warm permafrost with high unfrozen water contents.
L27-34 –Smith and Brown (2009) outline the various methods used as does Streletskiy et al. (2022).
L34-37 – Even if ground temperatures are measured within shallow permafrost and the active layer the permafrost table temperature or active layer thickness still needs to be determined/calculated. Interpolation and extrapolation is the method usually used. I suggest you consult Riseborough (2008 ICOP) which describes the appropriateness of interpolation/extrapolation approaches.
L38-42 – The purpose of the model is important. The ones mentioned are generally used for predictive applications such as determining conditions with little information on the site conditions. What you seem to be proposing is away to determining ALT or MAPT based on having ground temperature measurements which is what we do when we use interpolation/extrapolation of shallow ground temperatures to determine ALT or MAPT (see comment above).
L45 – Is reference being made to air or ground temperatures here?
L46-49 – Some of the approaches mentioned do consider variable properties including thermal conductivity for thawed and frozen conditions (e.g. ratio between them is included in TTOP equation) or the variation of conductivity with temperature (and unfrozen water content).
L57 – Editorial suggestion – delete “Besides other solution (Garagulya, 1990)” – it is not adding anything.
L66 – Editorial suggestion – Delete first part of sentence: “Eq. (1) is also valid for temperatures measured…..layer, which is convenient because….”
L68 – Note that usually the reference to surface temperature measurements used in these equations is from a sensor in upper 3-5 cm of the ground so not using exact surface temperature. The surface temperature can be estimated using n-factor to provide input into the equation.
L84 – If this is essentially extrapolation how is this different from the approach others use to determine the depth of the permafrost table and MAPT when they have temperatures at two or more depths?
L97 – Editorial suggestion - Delete first part of sentence: “ALT (m) can be calculated using the….
L99 – missing word: “…simplest form is as….”
L103 – See earlier comment regarding estimates of surface temperature used by others.
L136 – Smith et al. (2009) is also relevant - used observed ALT from sites in various regions and environments (tundra, forest, peatland and mineral and organic soil) to show the range in the Edaphic factor.
L148 – See earlier comment regarding extrapolation
L159-197 – It seems that there are several assumptions being made as well as simplifications. For example, thermal conductivity changes with temperature due to change in unfrozen water but it appears that constant frozen and unfrozen conductivity are assumed. Are the results of the two models really comparable?
Table 2 – For years and seasons is the 2^nd number the total record length and the first number the number of years utilized in analysis? It might be clearer to refer to number of years used and refer to it as e.g. 6 of 6. Are the ALT values determined from temperature or through probing? How is MAPT determined? It is unclear what you are comparing your modelled values with? I might have missed something here but maybe there needs to be a clearer explanation
L200-201 – No information on the sites is provided so the reader doesn’t know how diverse they are. There is no information provided on material characteristics or vegetation. The Alaskan sites are all on the North Slope in the continuous permafrost zone and likely in tundra environments, so conditions are not that diverse with respect to climate and vegetation. Using field data from sites in warmer permafrost in discontinuous zone and for forested sites would provide more diverse conditions. This would help show if your approach is valid for a wide range in conditions.
L204 – Since you are referring to a depth it would be better to refer to permafrost table or base of the active layer. Do you mean the base of the active layer was above the shallowest sensor?
L206-2011 – See Riseborough et al. (2008 ICOP) regarding errors associated with different approaches (interpolation, extrapolation) to determine thaw depth/top of permafrost etc. and guidance on the best approach to use.
L222 – Wasn’t this exponential decrease already fairly well known? Doesn’t the magnitude of the decrease depend on the material properties?
L235 – See earlier comment – How were observed values determined?
L260 – See earlier comment – How were observed values determined?
L265-268 – Does the difference in error between Antarctica and Alaska sites have anything to do with the material properties. Was latent heat more of a factor for the AK sites?
L264-292 – The thermal offset depends on the ratio of thawed and frozen thermal conductivity which depends on the amount of moisture/ice in the ground. If the moisture content is low or arid conditions exist, then the offset will be low or positive. Is the site in McMurdo Sound a dry site? It would be useful to know this. It would have been good to use sites with warmer permafrost in your analysis to back up the comment that deviation in MAPT estimates would be larger.
L306-310 – It would be useful to have information on the material properties at the field sites to back up these statements.
L315-330 – Although these other approaches make assumptions regarding thermal properties etc. based on general site characteristics, information on ground temperature is not required and the models determine the ground temperatures. This makes them useful for determining current and future conditions. This might make them more broadly applicable.
L327-330 – If temperature below the permafrost table was available would it be used if there were only one sensor at a shallower depth? You state that inputs can be any depth combination within the active layer based on temperature data availability and site characteristics. What are the site characteristics being referred to?
L331-334 – Aren’t these products based on modelling with various assumptions made regarding ground properties etc.
L338-340 – This is likely one of the primary sources of error especially with respect to moisture/ice contents and latent heat effects as discussed in Riseborough (2003).
L340-348 – Riseborough (2008) is probably relevant here especially with respect to spacing of temperature measurements etc. in determining thaw depth.
References
Riseborough, D.W. 2008. Estimating active layer and talik thickness from temperature data: implications from modeling results. In Ninth International Conference on Permafrost. Edited by D.L. Kane and K.M. Hinkel. Fairbanks, Alaska. Institute of Northern Engineering, University of Alaska Fairbanks, Vol.2, pp. 1487-1492.
Smith, S.L., Romanovsky, V.E., Isaksen, K., Nyland, K., Shiklomanov, N.I., Streletskiy, D.A., and Christiansen, H.H. 2024. Permafrost (Arctic) [in "State of the Climate in 2023"]. Bulletin of the American Meteorological Society (supplement), 105(8): S314-S317. doi:10.1175/BAMS-D-24-0101.1
Smith, S.L., Wolfe, S.A., Riseborough, D.W., and Nixon, F.M. 2009. Active-layer characteristics and summer climatic indices, Mackenzie Valley, Northwest Territories, Canada. Permafrost and Periglacial Processes, 20(2): 201-220. doi:10.1002/ppp.651
Smith, S. and Brown, J. 2009: Assessment of the status of the development of the standards for the Terrestrial Essential Climate Variables - T7 - Permafrost and seasonally frozen ground. GTOS 62 Essential Climate Variables
Streletskiy et al. 2022 Measurement Recommendations and Guidelines for the Global Terrestrial Network for Permafrost (GTN-P). DOI: 10.5281/zenodo.5973079

Citation: https://doi.org/10.5194/egusphere-2024-2989-RC2
- AC2: 'Reply on RC2', Tomáš Uxa, 20 Dec 2024
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-2989/egusphere-2024-2989-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2024-2989-AC2

Peer review completion

AR – Author's response | RR – Referee report | ED – Editor decision | EF – Editorial file upload

ED: Reconsider after major revisions (further review by editor and referees) (08 Jan 2025) by Johannes J. Fürst

AR by Tomáš Uxa on behalf of the Authors (26 May 2025) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (28 May 2025) by Johannes J. Fürst

RR by Anonymous Referee #1 (10 Jun 2025)

Suggestions for revision or reasons for rejection

The manuscript has been significantly improved by the revision work. The presentation and discussion of the proposed methodology is clearer, and the validation is more convincing. Meanwhile I have a few minor concerns regarding the discussion of the underlying assumptions and the chosen set of validation sites, see my comments below. Thus I recommend a minor revision of the manuscript prior to its publication in TC.

l 255-256: “Although ASMs utilize only thawing and freezing indices from two depth levels within the active layer as inputs, they inherently account for the natural variability of ground physical properties in the intermediate layer between these two depths.” I understand better now what the authors mean here, nevertheless I think that it should be highlighted that with the developed methodology the variability of ground physical properties is handled by means of an averaging operation, resulting in a temporally (seasonal) and spatially (within all the layer between z1 and z2) average of kt/kf.

l 269-270: “Of course, ASMs in principle also treat them as constants, but their values are representative for individual years (Eq. 8) or thawing seasons (Eq. 27)” Here the sources of temporal variability (e.g.: temporal variation of water content) should be explicated.

Table C1 – the increase in number and diversity of the considered validation sites is a strong improvement of the manuscript. Nevertheless the proportion of forested sites is small (5/43, ~12%) compared to the proportion of the permafrost area covered by boreal forest (55%, Stuenzi et al., 2021). Additional sites in boreal forest areas should be added, or at least the possible biases linked to the used sampling of permafrost areas conditions should be discussed.
Stuenzi, S. M., Boike, J., Gädeke, A., Herzschuh, U., Kruse, S., Pestryakova, L. A., Westermann, S., and Langer, M.: Sensitivity of ecosystem-protected permafrost under changing boreal forest structures, Environ. Res. Lett., 16, 084045, https://doi.org/10.1088/1748-9326/ac153d, 2021.

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RR by Anonymous Referee #3 (02 Jul 2025)

Suggestions for revision or reasons for rejection

In this paper, the authors propose set of semi-analytical models for estimating active layer thickness (ALT) and mean annual permafrost temperatures (MAPT) in permafrost-affected soils using only ground temperatures measured in the active layer. The presented equations are relatively straightforward algebraic derivations from well known equilibrium models of permafrost temperature (TTOP) and thaw (Stefan model). The authors validate their approach using ground temperature data from 43 sites in the Arctic spanning a wide range of surface conditions.

It is unclear to me whether or not the equations presented by the authors in this work are truly novel. I have not seen them before in this particular form anywhere in the literature, so I will give them the benefit of the doubt and assume that they are. However, the derivations are relatively straightforward and follow from long established equilibrium models, so I think that the added value is relatively low, all other assumptions and limitations of such equilibrium models notwithstanding.

The equations themselves are indeed perhaps useful in some problem settings and would certainly fit well in a technical report. However, I do not think that the level of novelty or scientific rigor of the validation meets the editorial standard for a full journal article.

From the scientific perspective, this article has two major flaws:

1. The “observed” MAPT and ALT are not actually observations, but are rather obtained by linear interpolation of observed ground temperatures in both the active layer and permafrost via linear interpolation. As far as I could tell, the authors do not report the depths of the temperature sensors which were used to calculate these pseudo-observations, which is also problematic. Such approximations could potentially be acceptable if the interpolation is over a short distance (<10 cm or so) but otherwise cannot be expected to realistically determine the actual MAPT and ALT.

This also likely explains why the model appears to perform so well at many of the sites. According to the authors’ own description, their method effectively amounts to a form of linear extrapolation from the two measured active layer temperatures. By generating psuedo-observations via linear interpolation, the authors are effectively generating data which, by construction, will be well predicted by such a linear extrapolation, in most cases. It is unclear why the authors did not attempt to use actual active layer measurements such as those from the Circum-arctic Active Layer Monitoring (CALM) network.

2. Although the validation is ostensibly extensive in terms of the number of field sites, the authors provide no baselines for comparisons. This is especially surprising considering that the authors’ method is derived from the TTOP and Stefan models, both of which would serve as natural baselines. The authors might respond that the novelty of their method is in the fact that they do not require estimates of the frozen vs. thawed thermal conductivities, information which they do not have for many or all of the sites. However, this would seem to ignore the fact that methods such as that of Hinkel and Nicholas (1995) and Romanovsky (1996), both of which are cited within, get around this simply by treating lumping these unknown terms into a linear parameter which can be estimated directly from the data. It would also be natural for the authors to compare their method to the similar and widely used approach of extrapolating the MAGT at the two measured points down to the zero-degree isotherm, which they themselves state is closely related.

In summary, although the equations derived by the authors may be of some interest to polar researchers and practitioners in the form of a technical report, the scientific novelty is modest at best, and the rigor of the validation is seriously lacking. Given that the paper has already gone through a full round of revision, I would recommend rejection.

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ED: Reconsider after major revisions (further review by editor and referees) (14 Jul 2025) by Johannes J. Fürst

AR by Tomáš Uxa on behalf of the Authors (07 Aug 2025) Author's response Author's tracked changes Manuscript

ED: Publish subject to revisions (further review by editor and referees) (20 Aug 2025) by Johannes J. Fürst

AR by Tomáš Uxa on behalf of the Authors (23 Aug 2025) Author's response

EF by Mario Ebel (25 Aug 2025) Manuscript Author's tracked changes

ED: Referee Nomination & Report Request started (03 Sep 2025) by Johannes J. Fürst

RR by Anonymous Referee #1 (08 Sep 2025)

ED: Publish as is (23 Sep 2025) by Johannes J. Fürst

AR by Tomáš Uxa on behalf of the Authors (03 Oct 2025)

Journal article(s) based on this preprint

07 Jan 2026

Simple analytical–statistical models (ASMs) for mean annual permafrost table temperature and active-layer thickness estimates

Tomáš Uxa, Filip Hrbáček, and Michaela Kňažková

The Cryosphere, 20, 97–112, https://doi.org/10.5194/tc-20-97-2026,https://doi.org/10.5194/tc-20-97-2026, 2026

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Tomáš Uxa, Filip Hrbáček, and Michaela Kňažková

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We devised two simple models for estimating the mean annual permafrost table temperature and active-layer thickness, which are driven solely by temperatures measured in the active layer; no ground physical properties are required. The models showed deviations of less than 0.03 °C and 5 %, and can therefore be useful tools for permafrost modelling under a wide range of environmental conditions.


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