A number of 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 as their input parameters in addition to air or ground temperatures. However, ground physical properties are frequently unavailable or unrepresentative and therefore need to be estimated, which introduces uncertainties into model outputs. Hence, we devised two simple analytical–statistical models (ASMs) for MAPT and ALT, which are driven solely by thawing and freezing indices from two depth levels within the active layer, while no ground physical properties are required. ASMs reproduced MAPT and ALT in the Earth's major permafrost regions with the total mean errors of less than 0.05 °C and 9 %, respectively. This is similar or better than other analytical or statistical models, which suggests that ASMs can be useful tools for estimating MAPT and ALT under a wide range of environmental conditions.
Grantová Agentura České RepublikyGM22-28659MGA25-18272SMinisterstvo Školství, Mládeže a TělovýchovyLL2505Introduction
Of ∼ 11 % of the Earth's exposed land surface underlain by permafrost , most seasonally thaws from the ground surface to a depth of up to several meters and then completely refreezes, which is mainly controlled by climate conditions and ground physical properties . This superficial active layer greatly influences the energy and mass transfer between the underlying permafrost, ground surface and the atmosphere, and is therefore critical for the dynamics of hydrological, geomorphic, pedogenic, biological and/or biogeochemical processes including greenhouse gas fluxes, as well as for human infrastructure in permafrost regions e.g.,. As climate is a first-order control on ground temperatures and thaw depth , the thermal state of permafrost and the thickness of the active layer have attracted a huge interest over recent decades because they are important indicators of how the climate system is evolving . Climate change has provoked permafrost warming and active-layer thickening at a global scale , which can have severe consequences on landscape and ecosystem stability as well as infrastructure integrity. Carbon release due to permafrost degradation is likely to trigger feedback mechanisms with impacts on the Earth's climate system . The permafrost and active-layer monitoring is therefore of utmost scientific and societal importance .
The thermal state of permafrost and the thickness of the active layer have been investigated by semi-continuous temperature measurements using data loggers with temperature sensors distributed in vertical arrays across the active layer and near-surface permafrost e.g.,, by periodic or semi-continuous geophysical measurements using electric, electromagnetic or seismic methods e.g.,, or by periodic thaw-depth measurements using physical probing with rigid rods or thaw-tube readings e.g.,. Of these methods, temperature measurements using data loggers are the most convenient in terms of accuracy, temporal resolution and/or logistics, which is well suitable for remote and poorly accessible permafrost regions that have limited or no technical infrastructure . However, ground temperatures are frequently measured only in the active layer, and therefore the permafrost temperatures and the active-layer thickness need to be estimated in these situations. This has been done using either statistical methods or numerical and analytical models of various complexity e.g.,.
Of these solutions, analytical models in particular have become popular for estimating the mean annual temperature at the top of permafrost (hereafter referred to as the mean annual permafrost table temperature, MAPT) and the active-layer thickness (ALT) because of their simplicity, small number of input parameters, computational efficiency and yet sufficient accuracy, which is advantageous for diverse permafrost regions and environmental settings e.g.,. These tools typically require at least a few ground physical properties, such as thermal conductivity, heat capacity, water content or bulk density, as their input parameters in addition to air or ground temperatures. However, ground physical properties are frequently unavailable or unrepresentative and therefore need to be estimated, which introduces uncertainties into model outputs. But even in situ observations of ground physical properties may not guarantee accurate model outputs either, as these properties are usually measured annually or less frequently and are then treated as constants in models, regardless of their temporal variability, which can be considerable e.g.,.
Here, we devise two novel analytical–statistical models (ASMs) for MAPT and ALT, which are driven solely by thawing and freezing indices from two depth levels within the active layer. ASMs are primarily intended to be used for MAPT or ALT estimates where ground temperature measurements are too shallow and MAPT or ALT therefore cannot be determined directly, while no information on ground physical properties exists. We evaluate ASMs against in situ ground temperature measurements from the Earth's major permafrost regions, and we discuss their performance, advantages and limitations.
Model derivationMean annual permafrost table temperature
MAPT [°C] can be calculated using the TTOP model , which assumes that the ratio of thawed and frozen thermal conductivity and the effects of latent heat produce the difference between MAPT and the mean annual ground surface temperature (thermal offset). The TTOP formula for permafrost conditions (MAPT ≤ 0 °C) is as follows MAPT=ktkfIts-IfsP,
where kt [W m−1 K−1] and kf [W m−1 K−1] is the thawed and frozen thermal conductivity, respectively, that defines the thermal conductivity ratio, Its [°C d] and Ifs [°C d] is the ground surface thawing and freezing index, respectively (both assumed in absolute values), and P [365 d] is the length of one year.
However, Eq. () can work with thawing and freezing index observed at any depth within the active layer . This is highly convenient because ground surface temperatures are difficult to measure due to radiative and convective energy fluxes and problematic fixing of temperature sensors exactly at the ground surface . Using ground temperatures observed at two depth levels within the active layer z1 and z2 (z1<z2< ALT), MAPT can therefore be expressed as
2MAPT=ktkfItz1-Ifz1P,3MAPT=ktkfItz2-Ifz2P,
where Itz1 [°C d] and Ifz1 [°C d] is the thawing and freezing index at the depth z1, and Itz2 [°C d] and Ifz2 [°C d] is the thawing and freezing index at the depth z2. This implies that Eqs. () and () are equivalent:
ktkfItz1-Ifz1P=ktkfItz2-Ifz2P.
Solving Eq. () for the thermal conductivity ratio yields
ktkf=Ifz1-Ifz2Itz1-Itz2.
Equation () can be substituted for the thermal conductivity ratio in Eqs. () and () as follows
6MAPT=Ifz1-Ifz2Itz1-Itz2Itz1-Ifz1P,7MAPT=Ifz1-Ifz2Itz1-Itz2Itz2-Ifz2P.
Simplifying Eqs. () and () then produces the same formula for MAPT:
MAPT=Ifz1Itz2-Ifz2Itz1Itz1-Itz2P.
Substantially, Eq. () implies that MAPT can be simply estimated using thawing and freezing indices from two depth levels within the active layer alone, that is, without knowing the thermal conductivity ratio.
Since Eq. () was derived from Eq. (), it has a physical basis cf.. However, it can be shown that it is in principle a linear extrapolation of the freezing index to the depth, where the thawing index becomes zero, and dividing it by the length of one year. Using the same notation as before, this can be expressed as
9Ifz1-IfALTItz1-ItALT=Ifz1-Ifz2Itz1-Itz2,10Ifz2-IfALTItz2-ItALT=Ifz1-Ifz2Itz1-Itz2,
where ItALT [°C d] and IfALT [°C d] represents the thawing and freezing index at the base of the active layer. Note that the slope of the relationship is determined by the thermal conductivity ratio. Solving Eqs. () and () for IfALT gives
11-IfALT=Ifz1-Ifz2Itz1-Itz2Itz1-ItALT-Ifz1,12-IfALT=Ifz1-Ifz2Itz1-Itz2Itz2-ItALT-Ifz2.
Since the thawing index at the base of the active layer is zero, Eqs. () and () become equivalent to Eqs. () and (), respectively, when divided by the length of one year, and both simplify to Eq. (). This documents that Eq. () can be derived in two alternative manners consisting of analytical and statistical procedures.
Active-layer thickness
ALT [m] can be calculated using the model, which builds on the premise that the conductive heat flux above the thaw front equals to the rate at which latent heat is absorbed as the thaw front propagates downwards. Its simplest form is as follows ALT=2ktItsLϕ,
where L [3.34 × 108 J m−3] is the volumetric latent heat of fusion of water and ϕ [–] is the volumetric water content. Note that the thawing index must be multiplied by the scaling factor of 86 400 s d−1. As stated previously (Sect. ), ground surface temperatures are difficult to measure , and therefore the Stefan model has commonly been forced by ground temperatures collected at some depth within the active layer. However, this has rarely been accounted for, although it has been shown to substantially affect the model outputs . Yet, it can be easily implemented as follows ALT=z+2ktItzLϕ,
where z [m] is the depth at which the thawing index Itz [°C d] is observed. Using ground temperatures observed at two depth levels within the active layer z1 and z2 (z1<z2< ALT), ALT can therefore be expressed as
15ALT=z1+2ktItz1Lϕ,16ALT=z2+2ktItz2Lϕ.
This implies that Eqs. () and () are equivalent:
z1+2ktItz1Lϕ=z2+2ktItz2Lϕ.
The vertical distance between z2 and z1 can be expressed as
z2-z1=2ktItz1Lϕ-2ktItz2Lϕ,
which simplifies to
z2-z1=2ktLϕItz1-Itz2.
Subsequently rearranging Eq. () gives
z2-z1Itz1-Itz2=2ktLϕ,
where the right-hand side corresponds to the so-called edaphic term , which has been used to combine the thawed thermal conductivity and volumetric water content into a single variable in the modified Stefan model:
ALT=EIts,
where E [m °C−0.5 d−0.5] denotes the edaphic term given by
E=2ktLϕ.
Although Eq. () is equivalent to Eq. (), it has frequently been preferred for estimating ALT because the edaphic term can be calibrated based on the relationship between ALT and thawing index, that is, without knowing the thawed thermal conductivity and volumetric water content . The edaphic term can be implemented in Eqs. () and () as follows
23ALT=z1+EItz1,24ALT=z2+EItz2.
Substituting the left-hand side of Eq. () for the edaphic term in Eqs. () and () yields
25ALT=z1+z2-z1Itz1-Itz2Itz1,26ALT=z2+z2-z1Itz1-Itz2Itz2.
Simplifying Eqs. () and () then produces the same formula for ALT:
ALT=z2Itz1-z1Itz2Itz1-Itz2.
Substantially, Eq. () implies that ALT can be simply estimated using thawing indices from two depth levels within the active layer alone, that is, without knowing the thawed thermal conductivity and volumetric water content or the edaphic term.
Since Eq. () was derived from Eq. (), it has a physical basis cf.. However, it can also be shown that it is in principle a linear extrapolation of the depth where the square root of the thawing index becomes zero cf.. This can be expressed as
28ALT-z1Itz1-ItALT=z2-z1Itz1-Itz2,29ALT-z2Itz2-ItALT=z2-z1Itz1-Itz2.
Note that the slope of the relationship is determined by the edaphic term. Solving Eqs. () and () for ALT gives
30ALT=z1+z2-z1Itz1-Itz2Itz1-ItALT,31ALT=z2+z2-z1Itz1-Itz2Itz2-ItALT.
Since the thawing index at the base of the active layer is zero, Eqs. () and () are equivalent to Eqs. () and (), respectively, and both simplify to Eq. (). As with Eq. (), this documents that Eq. () can also be derived in two alternative manners consisting of analytical and statistical procedures.
Model evaluation
ASMs for estimating MAPT and ALT were evaluated using in situ ground temperature measurements from the Earth's major permafrost regions that differ in climate, permafrost zone, ground surface cover and/or ground physical properties and their distribution within the active layer to enhance the robustness of the model evaluation. Unlike manual thaw-depth measurements, such as those from the Circumpolar Active Layer Monitoring (CALM) network , ground temperature measurements with sensors distributed in vertical arrays across the active layer and near-surface permafrost provide high temporal and depth resolutions, which enable consistent determination of MAPT and ALT using a uniform procedure at all sites and ensure the homogeneity of the validation dataset. Since the accuracy of these MAPT and ALT values depends on the spacing of the ground temperature sensors , we attempted to keep their maximum distances at 25 and 50 cm for ALT of <1 and >1 m, respectively. While this requirement excluded numerous sites, it ensured that the benchmark values for MAPT and ALT could be established as accurately as possible.
We collected ground temperature data for a total of 55 sites from monitoring networks and public databases of the Polar-Geo-Lab of the Masaryk University (MU) e.g.,, Global Terrestrial Network for Permafrost (GTN-P; http://gtnpdatabase.org, last access: 20 November 2024), Natural Resources Conservation Service of the United States Department of Agriculture (USDA; https://www.nrcs.usda.gov/resources/data-and-reports/soil-climate-research-stations, last access: 19 September 2024), Geophysical Institute Permafrost Laboratory of the University of Alaska Fairbanks (GI-UAF, https://permafrost.gi.alaska.edu, last access: 25 July 2025), Yukon Permafrost Database (YPD, https://service.yukon.ca/permafrost/, last access: 25 July 2025), Nordicana D of the Centre for Northern Studies (ND, https://nordicana.cen.ulaval.ca/en/, last access: 15 July 2025), and National Tibetan Plateau/Third Pole Environment Data Center (NTP/TPEDC; https://data.tpdc.ac.cn/en/disallow/789e838e-16ac-4539-bb7e-906217305a1d, last access: 21 November 2024) . The dataset comprised five different ground surface covers and four permafrost zones, spanned variable time periods during 1997–2023, and exhibited a wide range of MAPT and ALT from ∼-19 to ∼ 0 °C and ∼ 40 to ∼ 310 cm, respectively (Table ).
Ground temperature data were first checked for quality and then daily means were calculated for all available depths before further processing. Thawing and freezing indices were calculated as annual sums of positive and negative mean daily ground temperatures, respectively, which were expressed in absolute values for convenience. Following standard procedures and monitoring guidelines , ALT was determined as the maximum annual depth of the 0 °C isotherm that was tracked by linear interpolation of mean daily ground temperatures within the measured profile. MAPT was calculated as the mean annual ground temperature, which was linearly interpolated to the depth that corresponds to ALT e.g.,. It is important to note that there is no universal method for interpolating between ground temperature sensors that works best, and therefore we used the linear interpolation, which is generally accepted e.g.,. Hereafter, these values are referred to as the observed MAPT and ALT. They were considered suitable for the model evaluation because ∼ 65 % of the observed MAPT differed by less than 0.1 °C from the temperature of the closest temperature sensor used for the interpolation and ∼ 80 % of the observed ALT were less than 10 cm from the closest temperature sensor, which sets their maximum possible deviations from the actual MAPT and ALT values (Fig. ).
Cumulative distributions of the temperature and depth differences between the observed MAPT and ALT and the closest temperature sensor used for the linear interpolation, which sets their maximum possible deviations from the actual MAPT and ALT values.
Subsequently, MAPT and ALT were also modelled using ASMs given by Eqs. () and () forced by the observed thawing and freezing indices from the depth intervals of 0–10, 25–35 and 45–55 cm, which were combined into three pairs of 5/30, 5/50 and 30/50 cm so that they were comparable across the validation sites. This provided us with three sets of MAPT and ALT estimates that allowed to determine which depth combinations worked best. The three depth pairs were situated within the active layer in all instances, and therefore differed from the temperature sensors used to determine the observed MAPT and ALT, so this did not invalidate the evaluation.
We compared the modelled MAPT and ALT directly with the observed MAPT and ALT, and evaluated the model accuracy for each site using common error metrics, such as mean error (ME), mean percentage error (MPE), mean absolute error (MAE), mean absolute percentage error (MAPE), and root-mean-square error (RMSE). The evaluation statistics were grouped by depth pairs and surface cover, as the latter also broadly captures the common characteristics of the validation sites in terms of climate and composition of the active layer.
ResultsMean annual permafrost table temperature
The MAPT modelled using ASM given by Eq. () based on the observed thawing and freezing indices for the depth pairs of 5/30, 5/50 and 30/50 cm showed the total site-weighted ME from 0.01 °C to 0.05 °C compared to the observed MAPT (Table ). Since the errors were scattered around zero (Fig. ), the total site-weighted MAE was somewhat larger and ranged from 0.11 to 0.16 °C, while the total site-weighted RMSE was 0.12 to 0.19 °C (Table ). The majority of errors were well within ±0.2 °C (Fig. ).
Evaluation statistics of MAPT modelled using ASM given by Eq. () based on the observed thawing and freezing indices for the depth pairs of 5/30, 5/50 and 30/50 cm and diverse surface covers.
Comparison of the observed MAPT and MAPT modelled using ASM given by Eq. () based on the observed thawing and freezing indices for the depth pairs of 5/30, 5/50 and 30/50 cm and diverse surface covers. The black solid and dashed lines in the upper plots represent the line of identity and the deviation of ±1 °C, respectively.
The accuracy of the modelled MAPT was similar for the three depth pairs, although 5/50 and 30/50 cm performed slightly better than 5/30 cm (Table ). Similarly, there were rather small differences between individual surface covers (Fig. ) that exhibited the site-weighted ME from -0.06 to 0.12 °C (Table ). However, the MAPT estimates were somewhat better at the vegetated sites, as the site-weighted MAE and RMSE there were mostly less than ∼ 0.15 °C, while the bedrock and bare-ground sites mostly showed the site-weighted MAE and RMSE greater than ∼ 0.15 °C (Table ). The site-weighted errors also tended to be somewhat larger at higher MAPT for all three depth pairs.
Active-layer thickness
The ALT modelled using ASM given by Eq. () based on the observed thawing indices for the depth pairs of 5/30, 5/50 and 30/50 cm exhibited the total site-weighted ME from -11.5 cm (-9.3 %) to -1.6 cm (-1.2 %) compared to the observed ALT (Table ). The total site-weighted MAE was larger (Fig. ) and reached 13.1 cm (10.2 %) to 17.1 cm (19.8 %), while the total site-weighted RMSE was 14.2 cm to 18.2 cm (Table ).
Evaluation statistics of ALT modelled using ASM given by Eq. () based on the observed thawing and freezing indices for the depth pairs of 5/30, 5/50 and 30/50 cm and diverse surface covers.
Comparison of the observed ALT and ALT modelled using ASM given by Eq. () based on the observed thawing and freezing indices for the depth pairs of 5/30, 5/50 and 30/50 cm and diverse surface covers. The black solid and dashed lines in the upper plots represent the line of identity and the deviation of ±10 %, respectively.
The accuracy of the modelled ALT was higher for the depth pairs of 5/50 and 30/50 cm compared to 5/30 cm, especially at the bedrock, shrub and forest sites (Table ). Additionally, there were rather large differences between individual surface covers (Fig. ), among which the site-weighted ME ranged from -33.4 cm (-31.3 %) to 38.0 cm (33.8 %) (Table ). The most accurate ALT estimates were at the bare-ground sites and those with grass and shrub cover, as their site-weighted MAE ranged from 3.9 cm (6.0 %) to 22.0 cm (32.6 %), and the site-weighted RMSE was from 4.0 cm to 22.2 cm (Table ). Somewhat worse was the model performance at the bedrock and forest sites, with the site-weighted MAE from 9.0 cm (7.9 %) to 38.0 cm (33.8 %) and the site-weighted RMSE from 10.4 cm to 43.4 cm (Table ). The site-weighted errors were also larger at thicker ALT for all three depth pairs.
DiscussionMean annual permafrost table temperature
The modelled MAPT showed a relatively high accuracy for all three depth pairs and surface covers (Fig. ), with the mean errors close to zero and the majority of them within ±0.2 °C (Table ), which is similar or better than in most previous studies that used other analytical or statistical models for MAPT e.g.,.
Somewhat larger errors in the modelled MAPT arose especially under warmer conditions and within a thicker active layer where MAPT needs to be extrapolated to greater depth. Warmer climates are also dominated by vegetated sites (Table ) with well-developed soils and therefore a more heterogeneous active layer where MAPT estimates are more difficult. In addition, it may also be associated with increased complexity of the system at permafrost temperatures approaching 0 °C when simple models tend to fail to a greater extent . The worst MAPT estimates at the bedrock sites were also likely because active layer is thick there (Table ). Moreover, the boreholes were drilled into vertical rockwalls, and therefore it is possible that lateral flows of heat and moisture occur in the fractured bedrock, which further complicates MAPT estimates.
So far, models for estimating MAPT have typically assumed that the ratio of thawed and frozen thermal conductivity is less than or equal to 1, and that the thermal offset is therefore negative e.g.,, which would result in invalid MAPT estimates if the actual conditions were reversed. However, although nearly half of the bedrock and bare-ground sites exhibited a positive thermal offset with a thermal conductivity ratio above 1, the MAPT was modelled with similar accuracy at these locations as elsewhere (Table , Fig. ). This is because ASM utilizes thawing and freezing indices within the active layer and can therefore easily capture this behaviour. This is also demonstrated by the thermal conductivity ratios modelled using Eq. () for the three depth pairs that are close to those determined for the whole active layer (Fig. ) based on the relationship between MAPT and thawing and freezing indices . This is likely because the relationship between the thawing and freezing indices within the active layer is linear (see Sect. ) and its slope varies rather slightly with vertical changes in ground physical properties.
Comparison of the thermal conductivity ratio for the whole active layer determined using the rearranged Eq. () based on the observed MAPT and the observed thawing and freezing indices for the uppermost available sensors and the thermal conductivity ratio estimated using Eq. () based on the observed thawing and freezing indices for the depth pairs of 5/30, 5/50 and 30/50 cm and diverse surface covers. The black solid and dashed lines represent the line of identity and the deviation of ±0.1.
Active-layer thickness
Unlike MAPT, the modelled ALT showed variable performance for individual depth pairs and surface covers (Fig. , Table ). However, the errors were mostly well within ±20 %, which is also similar or better than in most previous studies that used other analytical or statistical models for ALT .
Notably, the modelled ALT showed variable accuracy for the depth pair of 5/30 cm (Table ). This is because the active layer is typically more heterogeneous at the vegetated sites and may often comprise a surface organic layer there, the physical properties of which strongly differ from the ground underneath. This alters the temperature gradient within the active layer and results in worse ALT estimates, which can be observed especially at the shrub and forest sites (Fig. ). By contrast, the ALT estimates showed substantially lower errors for the depth pairs of 5/50 and 30/50 cm (Fig. ), which largely to completely eliminated the influence of the surface layer. This also explains the consistently high accuracy of the modelled ALT at the bare-ground sites for all three depth pairs (Table ), as the active layer there is relatively homogeneous in terms of its stratigraphy and physical properties. The ALT estimates were also relatively accurate at the bedrock sites (Table ), but the same concern exists for them as for MAPT (see Sect. ). Similarly to MAPT, the modelled ALT tended to be less accurate under warmer conditions dominated by vegetated sites with a more heterogeneous and thick active layer (Table ) where ALT needs to be extrapolated to greater depth.
Previous studies have estimated the edaphic term based on the relationship between ALT and thawing index , which is restrictive because it requires ALT. However, the edaphic term modelled using Eq. () for the three depth levels was close to the edaphic term determined for the whole active layer (Fig. ) based on the relationship between ALT and thawing index . As with MAPT, this is because the square root of the thawing index within the active layer is linear (see Sect. ) and its slope varies rather slightly with vertical changes in ground physical properties .
Comparison of the observed edaphic term for the whole active layer determined using the rearranged Eq. () based on the observed ALT and the observed thawing index for the uppermost available sensor and the edaphic term estimated using Eq. () based on the observed thawing indices for the depth pairs of 5/30, 5/50 and 30/50 cm and diverse surface covers. The black solid and dashed lines represent the line of identity and the deviation of ± 1 cm °C−0.5 d−0.5.
Model advantages
Unlike other analytical or statistical models for MAPT e.g., and ALT e.g.,, ASMs given by Eqs. () and () can work in any grounds where conductive heat transfer prevails without knowing their physical properties.
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 that is expressed in terms of annual and seasonal means of the thermal conductivity ratio and the edaphic term, respectively. Similarly, ASMs consider latent and sensible heat or other factors that influence the thermal regime between the two depth levels, although these effects are not explicitly accounted for. This is because the relative values of the thawing and freezing indices at the two depth levels reflect the rate of heat transfer in the intermediate layer between them (see Eqs. and ) that is influenced by seasonal changes in ground physical properties. So in principle it is analogous to, for instance, the calculations of apparent thermal diffusivity, which are based on damping of temperature amplitude or phase lag between two depth levels .
This is highly convenient because ground physical properties, such as thermal conductivity, heat capacity, water content or bulk density, are frequently unavailable or unrepresentative. Ground physical properties in other models for MAPT and ALT have therefore been estimated empirically or based on published values with unknown validity e.g.,. Ground physical properties also show more or less variability on seasonal and annual time scales e.g.,, which most other models cannot handle because they typically treat ground physical properties as constants for whole modelling periods. Of course, ASMs also treat them as constants, but their values are annual or seasonal means that reflect the variations in ground physical properties over time mainly due to changes in water content and as such they are representative for individual years or thawing seasons. This is a major improvement over other analytical or statistical models for MAPT e.g., and ALT e.g.,, which can increase the spatial and/or temporal validity of modelled MAPT and ALT.
Moreover, we believe that, in addition to MAPT and ALT estimates, ASMs can also be useful for investigating the spatial and temporal variations in the thermal conductivity ratio (Fig. ) and the edaphic term (Fig. ) regardless of MAPT and ALT cf.. This could be done using networks of miniature temperature loggers collecting data only in shallow parts of the active layer because another advantage of ASMs is that their inputs can be any depth combinations from within the active layer. For most accurate outputs, however, we suggest using thawing and freezing indices from depth levels as close as possible to the permafrost table. For instance, this could improve ALT estimates at the bedrock sites where active layer is thick.
In addition to in situ ground temperature measurements, we suppose that ASMs could also be forced by diverse climate reanalyses or Earth system models, if these at least partially account for the physics of ground thawing and freezing. While these products have been widely used for permafrost applications e.g.,, they typically provide only ground surface and shallow active-layer temperatures with ground physical properties largely unknown, which is frequently insufficient to determine MAPT and ALT directly or using conventional models. If the active layer is thick, MAPT and ALT have therefore usually been confined to the deepest ground temperature level available in these products, which can obviously be misleading e.g.,. However, ASMs are designed so that they should be able to provide MAPT and ALT estimates even under these conditions.
Lastly, ASMs can also be easily reformulated to be used for estimating the mean annual temperature at the base of seasonally frozen ground and frost depth (see Appendices and ).
Model limitations
Since ASMs assume that active layer is vertically homogeneous, they can be biased if there are strong vertical changes in ground physical properties and/or higher ground-ice content near the base of the active layer . For instance, if temperature measurements are used from the topmost layer, whose physical properties differ from the rest of the active layer, ASMs may be inaccurate. Similarly, the modelled MAPT and ALT may be unreliable if only shallow temperature measurements in a thick active layer are used. This is because the estimates would be based on physical properties of a small portion of the active layer, which may be different in its deeper parts. Nevertheless, the natural variability of ground physical properties without sharp changes in their vertical distribution is unlikely to have a major influence on the MAPT and ALT estimates (see Figs. and , Tables and ).
Other downside of ASMs is that they require temperature measurements from two depth levels within the active layer, which may not be available at many sites.
Conclusions
We devised two novel analytical–statistical models (ASMs) for estimating MAPT and ALT given by Eqs. () and (), respectively, which are driven solely by thawing and freezing indices from two depth levels within the active layer, while no ground physical properties are required. ASMs reproduced MAPT and ALT in the Earth's major permafrost regions with the total mean errors of less than 0.05 °C and 9 %, respectively, which is very promising because it is similar or better than other analytical or statistical models. ASMs worked best in a homogeneous active layer with small vertical changes in ground physical properties and when permafrost table was close below the temperature sensors considered for MAPT and ALT estimates. By contrast, they performed worst in a heterogeneous and thick active layer when the topmost organic layer influenced the estimates.
We believe that ASMs can find useful applications under a wide range of climates, ground surface covers and ground physical conditions wherever at least two temperature measurements within the active layer are available. They are primarily intended to be used for MAPT or ALT estimates where ground temperature measurements are too shallow and MAPT or ALT therefore cannot be determined directly, but they can also be used to establish typical values of the thermal conductivity ratio and the edaphic term for MAPT and ALT estimates in the past and in the future or for modelling their spatial variations. In addition to in situ measurements, they could utilize diverse climate reanalyses or Earth system models. Lastly, they can be easily reformulated for estimating the mean annual temperature at the base of seasonally frozen ground and frost depth.
Derivation of ASM for mean annual temperature at the base of seasonally frozen ground
Similarly to Eq. (), the mean annual temperature at the base of seasonally frozen ground (MASFT > 0 °C) is calculated as follows MASFT=Its-kfktIfsP.
MASFT based on temperatures observed at two distinct depths in the seasonally freezing layer z1 and z2 (z1<z2< FD) can therefore be expressed as follows
A2MASFT=Itz1-kfktIfz1P,A3MASFT=Itz2-kfktIfz2P.
This implies that Eqs. () and () are equivalent:
Itz1-kfktIfz1P=Itz2-kfktIfz2P.
Solving Eq. () for the inverse of the thermal conductivity ratio yields
kfkt=Itz1-Itz2Ifz1-Ifz2.
Equation () can be then substituted for the thermal conductivity ratio in Eqs. () and () as follows
A6MASFT=Itz1-Itz1-Itz2Ifz1-Ifz2Ifz1P,A7MASFT=Itz2-Itz1-Itz2Ifz1-Ifz2Ifz2P.
Subsequently, Eqs. () and () both simplify to the same formula for MASFT:
MASFT=Ifz1Itz2-Ifz2Itz1Ifz1-Ifz2P,
which only slightly differs from Eq. ().
Derivation of ASM for frost depth
Similarly to Eq. (), the frost depth (FD) can be calculated using the model as follows
FD=2kfIfsLϕ.
As with Eq. (), note that the freezing index must be multiplied by the scaling factor of 86 400 s d−1. FD estimated using freezing indices observed at two distinct depths z1 and z2 (z1<z2< FD) can be expressed as follows
B2FD=z1+2kfIfz1Lϕ,B3FD=z2+2kfIfz2Lϕ.
This implies that Eqs. () and () are equivalent:
z1+2kfIfz1Lϕ=z2+2kfIfz2Lϕ.
The vertical distance between z2 and z1 can be expressed as
z2-z1=2kfIfz1Lϕ-2kfIfz2Lϕ,
which simplifies to
z2-z1=2kfLϕIfz1-Ifz2.
Subsequently rearranging Eq. () gives
z2-z1Ifz1-Ifz2=2kfLϕ,
where the right-hand side corresponds to the edaphic term, which combines the ground physical properties in the Stefan model into a single variable. The edaphic term can be implemented in Eqs. () and () as
B8FD=z1+EIfz1,B9FD=z2+EIfz2.
Substituting the left-hand side of Eq. () for the edaphic term in Eqs. () and () yields
B10FD=z1+z2-z1Ifz1-Ifz2Ifz1,B11FD=z2+z2-z1Ifz1-Ifz2Ifz2.
Simplifying Eqs. () and () then produces the same formula for FD:
FD=z2Ifz1-z1Ifz2Ifz1-Ifz2,
which is the same as Eq. (), but with the freezing indices instead of the thawing ones.
GTN-P = Global Terrestrial Network for Permafrost, MU = Polar-Geo-Lab of the Masaryk University, USDA = Natural Resources Conservation Service of the United States Department of Agriculture, NTP/TPEDC = National Tibetan Plateau/Third Pole Environment Data Center, GI-UAF = Geophysical Institute Permafrost Laboratory of the University of Alaska Fairbanks, ND = Nordicana D of the Centre for Northern Studies, YPD = Yukon Permafrost Database.
Data availability
The validation data from James Ross Island and Petuniabukta are available upon request from Filip Hrbáček (hrbacekfilip@gmail.com) and Kamil Láska (laska@sci.muni.cz), respectively, while the other data are available from Global Terrestrial Network for Permafrost (http://gtnpdatabase.org, last access: 20 November 2024), Natural Resources Conservation Service of the United States Department of Agriculture (https://www.nrcs.usda.gov/resources/data-and-reports/soil-climate-research-stations, last access: 19 September 2023), Geophysical Institute Permafrost Laboratory of the University of Alaska Fairbanks (https://permafrost.gi.alaska.edu, last access: 25 July 2025), Yukon Permafrost Database (https://service.yukon.ca/permafrost/, last access: 25 July 2025), Nordicana D of the Centre for Northern Studies (https://nordicana.cen.ulaval.ca/en/, last access: 15 July 2025), and National Tibetan Plateau/Third Pole Environment Data Center (https://data.tpdc.ac.cn/en/disallow/789e838e-16ac-4539-bb7e-906217305a1d, last access: 21 November 2024).
The contact author has declared that none of the authors has any competing interests.
Disclaimer
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.
Acknowledgements
We thank Kamil Láska and acknowledge the Global Terrestrial Network for Permafrost, Natural Resources Conservation Service of the United States Department of Agriculture, Geophysical Institute Permafrost Laboratory of the University of Alaska Fairbanks, Yukon Permafrost Database, Centre for Northern Studies, and National Tibetan Plateau/Third Pole Environment Data Center for collecting long-term ground temperature data and disseminating them personally and/or publicly.
Financial support
The research was funded by the Czech Science Foundation (project numbers GM22-28659M and GA25-18272S) and by the Ministry of Education, Youth and Sports (project number LL2505).
Review statement
This paper was edited by Johannes J. Fürst and reviewed by three anonymous referees.
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