The model requires inputs on the palaeo-active-layer thickness ξ [m], the volumetric ground moisture content ϕ [-], the dry ground bulk density ρ [kgm-3], the ground quartz content q [-], the ground grain-size class f/c [-], the ground-surface thawing n factor nt [-], and the annual air temperature range Aa [∘C] (Table ), which corresponds to the difference between the mean air temperature of the warmest and coldest month. The ground physical properties are assumed to characterize the entire modelling domain (∼active layer) and are treated as constant over time.

The Stefan equation for calculating the active-layer thickness in a homogeneous substratum with constant physical properties has the following form : 1ξ=2ktItsLϕρw, where kt [Wm-1K-1] is the thermal conductivity of the thawed ground calculated here as a function of its dry bulk density and volumetric moisture content using the thermal conductivity model (Appendix ), Its [∘Cd] is the ground-surface thawing index defined as a sum of positive daily ground-surface temperatures in the thawing season (Fig. ), L [334 000 Jkg-1] is the specific latent heat of fusion of water, and ρw [1000 kgm-3] is the density of water. Note that Its must be multiplied by the scaling factor of 86 400 sd-1 in Eq. () to obtain the active-layer thickness in metres. Also, the product of ϕ and ρw can be alternatively substituted by that of the gravimetric ground moisture content and dry ground bulk density as their results are identical.

The ground-surface thawing index required to reach a given active-layer thickness can be obtained if Eq. () is rearranged as 2Its=ξ2Lϕρw2kt. The ground-surface thawing index can then be converted into the air thawing index Ita [∘Cd] through the ground-surface thawing n factor , which is a simple empirical transfer function that has been widely used to parametrize the thawing season air–ground temperature relations across permafrost landscapes e.g.: 3Ita=Itsnt. Note that the effect of snow cover does not have to be accounted for because the Stefan equation (Eq. ) considers solely the thawing season temperatures responsible for the active-layer thawing. Likewise, the resulting air thawing index is later used only to calculate other air temperature characteristics, and these are not affected by snow in any way. Such a scheme is particularly advantageous because it keeps the number of inputs small. Yet, it should be borne in mind that it is especially suitable for locations without long-lasting snow cover, which might disrupt the coupling between the air and ground temperatures during prolonged snow melting periods e.g..

The temporal evolution of air temperature over the year Ta(t) [∘C] can be described by a sine wave (Fig. ) such as 4Ta(t)=MAAT+Aa2sin⁡2πtP, where t [d] is the time and P [365 d] is the period of air temperature oscillations. Note that the annual air temperature range must be halved in Eq. () and the subsequent equations in order to characterize the annual temperature variations around MAAT (Fig. ).

The air thawing index represents the positive area under the annual air temperature curve (Fig. ) and can be calculated by integrating Eq. () over the thawing season as follows: 5Ita=∫t1t2Ta(t)dt, with 6t1=arcsin⁡-MAATAa2P2π,7t2=π-arcsin⁡-MAATAa2P2π, where t1 is the time when the air temperature curve crosses the 0 ∘C level from below (∼thawing season begins), while t2 is the time when it crosses this level from above (∼ thawing season ends) e.g.. Unfortunately, Eq. () has no analytical solution for MAAT. However, it can be derived from a nomogram (Fig. ) or, as here, it can be calculated numerically using the bisection root-finding method searching for MAAT such that -Aa2<MAAT≤0. This condition ensures that both positive and negative air temperatures have occurred during the year, which is an essential prerequisite for the active layer to form. Admittedly, it is simplistic because air–ground temperatures are modulated by surface and subsurface offsets so that the permafrost–seasonal frost boundary usually occurs at slightly negative MAAT . Consequently, there might be a risk that the model is incorrectly applied to seasonal frost conditions. However, this can be easily prevented if periglacial features that have indisputably developed in the presence of permafrost are examined.

Once MAAT is known, the air freezing index Ifa [∘Cd] can be simply computed as 8Ifa=MAATP-Ita. Furthermore, the mean air temperature of the warmest MATWM [∘C] and coldest MATCM [∘C] month is calculated as 9MATWM=MAAT+Aa2,10MATCM=MAAT-Aa2. The mean air temperature of the thawing MATTS [∘C] and freezing MATFS [∘C] season is defined as 11MATTS=ItaLt,12MATFS=IfaLf, where Lt [d] and Lf [d] are the duration of the thawing and freezing season, respectively, which is expressed from Eqs. () and () as 13Lt=t2-t1=π-2arcsin⁡-MAATAa2P2π,14Lf=P-Lt. Unconventionally, the model can also be solved with the range of annual air temperature oscillations defined by MATWM see if its difference from MAAT (i.e. MATWM-MAAT) is substituted for Aa2 in Eq. () and elsewhere (Appendix ). Unsurprisingly but importantly, solutions based on alternate model input variables, as suggested above, produce identical outputs, allowing model adaptations to specific situations and available data.

We consider two study sites in the Czech Republic where vertical cross sections through cryoturbated horizons, portraying the palaeo-active layers (Fig. ), were exposed and sampled for attributes that allowed us to define the most plausible ranges of the model input variables.

The Brno–Černovice site (49∘10′43′′ N, 16∘38′56′′ E; 240 m a.s.l.) is an active sand–gravel pit situated about 3 km southeast of the centre of Brno, southern Moravia. The mine is embedded within sands and gravels of the Tuřany terrace of the Svitava River, in reality a highly flattened alluvial fan tentatively attributed to the Günz glaciation (∼Marine Isotope Stage 22–11), which is located about 1.5 km east of the present channelized riverbed and ∼40 m higher. The material tends to coarsen down to a depth of 6–13.5 m, under which the Neogene clayey sands and gravels emerge . The cryoturbations mostly consist of sands to gravels that constitute ∼80 % and ∼14 % of the substrate, respectively, and are deformed by numerous injection tongues (Fig. ). Currently, the cryoturbations rest ∼1.6–3.6 m under the ground surface, but we suppose they were just below it when they developed.

The Nebanice site (50∘07′13′′ N, 12∘27′09′′ E; 430 m a.s.l.) is a currently inactive sand–gravel pit about 1.5 km west of the centre of Nebanice, western Bohemia. It is set in sands and gravels of the Nebanice terrace of the Ohře River, which is thought to have originated during the Mindel–Riss glaciations (∼ Marine Isotope Stage 10) and is now situated about 200 m north of the riverbed at a relative height of ∼10 m . The terrace sediments have a thickness of 1.7–3.8 m and are underlain by the Pliocene–lower Pleistocene clays, sands, and gravels . The cryoturbations are mostly composed of sands and gravels that comprise ∼67 % and ∼29 % of the material, respectively, and take the form of festoons and ball-and-pillow structures (Fig. ) emerging in the uppermost part of the terrace ∼1.1–2.8 m below the modern ground surface as they were later overlaid by younger sediments and soils as well.

Cryoturbations, like other permafrost-related features, in the area of the Czech Republic have been tentatively attributed to the last glacial period . We believe that the studied relict cryoturbations indicate past environmental conditions similar to those at the end of the Last Glacial Maximum (LGM) when analogous features formed in nearby regions .

Most modelled palaeo-air temperature characteristics changed to various extents depending on the scenarios of the annual air temperature ranges, the higher values of which generally caused colder and/or more continental conditions as well as larger scatter in the model outputs. This was especially true for MAAT and the freezing season characteristics, whereas the changes were comparatively milder for the thawing season attributes (Fig. ).

MAAT was modelled at -6.6±2.7 to -3.3±2.0 ∘C and -7.0±1.9 to -3.2±1.5 ∘C at the Brno–Černovice and Nebanice sites, respectively (Fig. ), which corresponds to its average decline of -16.0 to -12.7 ∘C and -15.1 to -11.3 ∘C, respectively, in comparison with the 1981–2010 period. The average contrasts between the warmest and coldest MAAT scenarios based on the current and assumed past annual air temperature ranges were 3.3 and 3.8 ∘C at the Brno–Černovice and Nebanice sites, respectively, which means that MAAT changed by 33 % and 38 % of the change in the annual air temperature range.

The thawing seasons had MATTS of 5.4±1.2 to 6.5±1.6 ∘C and 4.7±0.8 to 5.5±1.0 ∘C at the Brno–Černovice and Nebanice sites, respectively, and culminated with MATWM of 8.4±1.9 to 10.1±2.6 ∘C and 7.3±1.3 to 8.5±1.6 ∘C (Fig. ), which suggests that MATTS changed on average by 11 % and 8 % of the magnitudes of the annual air temperature range perturbations, while MATWM varied on average by 17 % and 12 %. Ita showed even more consistent outputs as it reached 823±271 to 915±353 ∘Cd and 704±181 to 721±203 ∘Cd at the Brno–Černovice and Nebanice sites, respectively (Fig. ). As such, it varied by as little as ∼1.1 % and ∼0.2 % per 1 ∘C change in the annual air temperature range. Likewise, Lt was modelled at 135±20 to 149±20 d and 128±14 to 147±15 d (Fig. ), which yields respective changes of ∼1.0 % and ∼1.5 %.

The freezing seasons exhibited the largest scatter and the highest variability among the scenarios, with MATFS of -14.2±1.8 to -9.3±1.5 ∘C and -13.7±1.6 to -8.5±1.5 ∘C at the Brno–Černovice and Nebanice sites, respectively, and MATCM as low as -23.2±3.2 to -15.0±2.6 ∘C and -22.5±2.8 to -13.7±2.5 ∘C (Fig. ). It thus follows that their variations reached on average 49 % and 52 % of the annual air temperature range perturbations for MATFS and as much as 82 % and 88 % for MATCM. Ifa spanned -3309±667 to -2041±492 ∘Cd and -3270±523 to -1873±429 ∘Cd at the Brno–Černovice and Nebanice sites, respectively (Fig. ), which corresponds to average variations of ∼6.2 % and ∼7.5 % per 1 ∘C change in the annual air temperature range. On the other hand, Lf was modelled at 216±20 to 230±20 d and 218±15 to 237±14 d, respectively, and thus it varied on average by as little as ∼0.6 % and ∼0.9 % of the magnitudes of the annual air temperature range perturbations.

Generally, the model validation using the modern data (Sect. ) showed relatively high accuracy for most air temperature characteristics if suitable input variables were available. It is remarkable, though, given that active-layer thickness commonly exhibits rather moderate to low correlations with annual or freezing season air and ground temperature attributes, but on the other hand, it mostly strongly couples with thawing season air and ground temperature indices e.g.. Since the model builds on active-layer thickness–thawing season temperature relations, which it further converts into annual and freezing season air temperature characteristics, this scheme gives rise to reasonable accuracy. It also needs to be stressed that the model was successful regardless of the fact that the ground physical properties used have certainly undergone at least slight changes since sampling and varied over the validation period. More accurate and less scattered outputs at the James Ross Island sites were largely due to a rather homogeneous distribution of ground physical properties within the active layer there . By contrast, the Alaskan profiles have a two-layer composition with peat over mineral soil . Additionally, the model parameterizes the air temperature behaviour with a sine wave, which simplifies its actual evolution over the year and completely ignores sub-annual variations.

Overall, however, the model underestimated most air temperature characteristics (Fig. ). The underestimation is attributed to intrinsic shortcomings of the Stefan equation (Eq. ) that tends to deviate inversely proportional to the moisture content in the active layer and the active-layer temperature at the onset of thawing . Also, it is associated with the () thermal conductivity model (Appendix ), which tended to produce overly high thermal conductivity values at the James Ross Island sites. Surely, the Stefan equation (Eq. ) might be improved by a number of correction factors, but these require additional inputs, such as frozen thermal conductivity, thawed and frozen volumetric heat capacity, or active-layer temperature at the start of thawing . As such, the corrections are frequently difficult to implement even in many present-day situations and are definitely much less viable for palaeo-applications. Moreover, their inverse solution would not be straightforward and would probably demand iterative techniques. Similarly, the () thermal conductivity model is advantageous in that it requires fewer inputs compared with other solutions while having comparable accuracy e.g.. As such, it is also difficult to replace by other thermal conductivity schemes.

It should also be highlighted that for the model validation cases the active-layer thickness was reduced by the depth of the temperature sensor, which was used to determine the ground-surface thawing n factor (see Appendix ). However, if this treatment was not done, the model outputs improved, with MAAT, MATWM, MATCM, MATTS, and MATFS showing average errors ≤1 ∘C, Ita and Ifa deviating on average by -1 % and -8 %, respectively, and Lt and Lf by -16 % and 6 %, respectively, because this compensated for the intrinsic deviations of the Stefan equation (Eq. ) described in the previous paragraph. Since the published ground-surface thawing n factors used for the palaeo-applications built on various near-surface depths of ground temperature sensors, we did not adjust the active-layer thickness in these cases, and the modelled palaeo-air temperature characteristics could thus have slightly increased accuracy as well.

The palaeo-MAAT modelled for two sites in the Czech Republic (Sect. ) was between -7.0±1.9 and -3.2±1.5 ∘C and its corresponding reduction between -16.0 and -11.3 ∘C in comparison with the 1981–2010 period, which is relatively consistent with earlier reconstructions utilizing various relict periglacial features in the central European lowlands that suggested MAAT depressions mostly between -17 and -12 ∘C . By contrast, it disagrees with slightly milder MAAT reductions of at least -7 and -13 to -6 ∘C derived from groundwater data and borehole temperature logs , but these may not necessarily correspond to the lowest temperatures because groundwater cycling has been slowed or interrupted by permafrost, while ground temperature history may have been partly masked by latent heat effects. Similarly, the modelled MAAT differs rather highly from simulations of three global climate models (GCMs), namely the Community Climate System Model 4, the Model for Interdisciplinary Research on Climate Earth System Model, and the Max Planck Institute Earth System Model Paleo, capturing the LGM at ∼22 ka, which were originally released by the Coupled Model Intercomparison Project 5 in coarse resolutions but later downscaled to 2′30′′ and made available via the WorldClim 1.4 dataset at https://www.worldclim.org (last access: 8 August 2016) . The GCMs suggest that MAAT was between -4.5 and -0.4 ∘C at the study sites, which corresponds to its reduction of between -12.6 and -9 ∘C in comparison with the 1981–2010 period. Such relatively high temperatures, however, contrast with many proxy records and are thus suspected to be unrepresentative of the coldest LGM conditions when continuous permafrost presumably occurred there .

The modelled MATWM of 7.3±1.3 to 10.1±2.6 ∘C is well in the range of proxy-based MATWM of 5 to 13 ∘C that has been reconstructed for the central European lowlands . By contrast, it deviates greatly from the GCM outputs, being as high as 11.1 to 18 ∘C. The modelled MATCM ranged widely from -23.2±3.2 to -13.7±2.5 ∘C, which, however, also pertains to other proxy records that show somewhat lower values of -27 to -16.5 ∘C . The GCM-based MATCM exhibits less variability of -21.6 to -17.7 ∘C, but its average is generally consistent with that modelled by this study. Unfortunately, other modelled palaeo-air temperature characteristics cannot be validated directly because no reliable proxy records or GCM outputs are available for them. However, as they are closely related to MAAT, MATWM, and MATCM, we believe that those attributes have similar plausibility.

Lastly, the modelled MAAT between -7.0±1.9 and -3.2±1.5 ∘C is relatively consistent with the MAAT threshold of <-4 ∘C commonly suggested for the formation of cryoturbation structures within fine-grained substrates . At the study sites, however, the cryoturbations consist of coarse-grained materials, for which a MAAT threshold as low as <-8 ∘C has been proposed . This study thus raises questions about the validity of the previously suggested MAAT thresholds for cryoturbation structures see and calls for their thorough revision.

Global sensitivity analysis using multiple regression suggested that the palaeo-active-layer thickness and annual air temperature range had a major impact on the modelled palaeo-air temperature characteristics at the Brno–Černovice and Nebanice sites (Fig. ). The palaeo-active-layer thickness importantly showed the highest values of the standardized regression coefficients (SRCs), especially for the annual and thawing season air temperature attributes. Similarly, the annual air temperature range also highly influenced MAAT, and, in particular, it had the utmost control over the freezing season air temperatures as its SRCs even tended to outweigh those for the palaeo-active-layer thickness at that time of the year. It thus follows that the freezing season characteristics may have limited accuracy if the annual air temperature range is uncertain, which indeed translates into their higher variance (Fig. ). On the other hand, the annual air temperature range slightly affected the thawing season air temperature attributes (Fig. ), and these are thus assumed to be the most plausible. Ground-surface and subsurface input variables, such as volumetric ground moisture content, ground dry bulk density, and ground-surface thawing n factor, had considerably lower, albeit stable, influences on most modelled palaeo-air temperature characteristics. Ground quartz content was the weakest of the input variables as it was responsible for only minor variability in the model outputs (Fig. ).

Besides cryoturbations, we assume that the model could be utilized to derive palaeo-air temperature characteristics on the basis of any relict periglacial features that can indicate former active-layer thickness, such as some kinds of patterned ground, some solifluction structures, frost-wedge tops, autochthonous blockfields, mountaintop detritus, active-layer detachment slides, up-frozen clasts, indurated horizons, and frost weathering microstructures . Also, it could be easily adapted for seasonal frost features, although the estimation of snow conditions would be complicated.

Since most periglacial features develop on at least decadal to centennial timescales e.g. inherently involving natural variations in climate and active-layer thickness, we hypothesize that their vertical extent frequently also includes the bottom transient layer in which the boundary between the active layer and permafrost fluctuates when periglacial features form see. This implies that the palaeo-active-layer thickness may appear as a dispersed rather than a sharp boundary in many vertical cross sections. Special attention must thus be paid to avoid any ambiguity in both the identification of relict periglacial features and the determination of the palaeo-active layer that may be severely degraded. Indeed, some periglacial features may be produced by seasonal frost alone, while some lookalike features may even have a non-periglacial origin . Nonetheless, even if identified correctly, problems may arise in places where the ground-surface level has changed over time due to sedimentation or erosion. High uncertainty can probably be expected for periglacial features composed of pebbly to bouldery materials, such as blockfields or mountaintop detritus, in which the vertical variability of ground physical properties is extremely high and difficult to describe by single-value variables. On slopes, these materials may also provoke non-conductive heat transfer processes, which give rise to high-magnitude variations in ground temperatures over short distances that cannot be addressed by simple heat conduction models .

Given the above considerations, modelling should ideally utilize coexisting periglacial features for more robust palaeo-air temperature estimates. Admittedly, it can capture only a snapshot of the temperature history, but this is no different from numerous other palaeo-indicators, such as various glacial deposits. Moreover, if relict periglacial assemblages of different ages occur in a given region, they may eventually provide a more complete record of former temperature conditions. Undoubtedly, dating of periglacial features is still challenging because they can have a highly complex formation history, which partly devaluates their palaeo-environmental importance. Nonetheless, this shortcoming is increasingly being suppressed by improved dating methods that bring more reliable periglacial chronologies e.g..