The Rupelian clay in the Netherlands is currently the subject of a
feasibility study with respect to the storage of radioactive waste in the
Netherlands (OPERA-project). Many features need to be considered in the
assessment of the long-term evolution of the natural environment surrounding
a geological waste disposal facility. One of these is permafrost development
as it may have an impact on various components of the disposal system,
including the natural environment (hydrogeology), the natural barrier (clay)
and the engineered barrier. Determining how deep permafrost might develop in
the future is desirable in order to properly address the possible impact on
the various components. It is expected that periglacial conditions will
reappear at some point during the next several hundred thousands of years, a
typical time frame considered in geological waste disposal feasibility
studies. In this study, the Weichselian glaciation is used as an analogue
for future permafrost development. Permafrost depth modelling using a best
estimate temperature curve of the Weichselian indicates that permafrost
would reach depths between 155 and 195 m. Without imposing a climatic
gradient over the country, deepest permafrost is expected in the south due
to the lower geothermal heat flux and higher average sand content of the
post-Rupelian overburden. Accounting for various sources of uncertainty,
such as type and impact of vegetation, snow cover, surface temperature
gradients across the country, possible errors in palaeoclimate
reconstructions, porosity, lithology and geothermal heat flux, stochastic
calculations point out that permafrost depth during the coldest stages of a
glacial cycle such as the Weichselian, for any location in the Netherlands,
would be 130–210 m at the 2

Northern hemispheric permafrost is presently restricted to areas in close
proximity to the Arctic, generally north of 60

Many of the countries that experienced permafrost in the past, such as the Netherlands, Belgium, United Kingdom, France and Germany, are currently running feasibility studies for the disposal of radioactive waste in deep geological repositories (Grupa, 2014). The concept of geological disposal is based on a multi-barrier system. Engineered barriers, consisting of steel components, concrete and/or bentonite are designed to contain the nuclear waste. Subsequently, the engineered barrier is placed in a natural barrier consisting of a low-permeability host rock that is situated at sufficient depth and located in a stable geological environment. Permafrost development may have an impact on various components of the disposal system, including the natural environment, the natural barrier and the engineered barrier. The hydrological cycle will completely change under permafrost conditions, up to the point where surface and subsurface hydrology become completely independent, and groundwater flow is drastically changed (Weert et al., 1997). Furthermore, landscape, vegetation and thus the biosphere will be completely different under permafrost conditions (Beerten et al., 2014). Hydraulic properties of aquifer sands and aquitard clays might be affected during permafrost conditions, especially if accompanied by repeated freeze–thaw cycles (Othman and Benson, 1993; McCauley et al., 2002). Migration of radionuclides through the host rock may be altered and the mechanical properties of the engineered barriers may be affected due to freeze–thaw cycles (Busby et al., 2015). Pore-water chemistry might change due to outfreezing of salts, and gas hydrates may develop (Stotler et al., 2009). Finally, microbial activity in the host rock will likely be affected. It is thus important to determine how deep permafrost will develop in order to properly address the possible impact on the various components. Thick and deep low-permeability clay formations are often good candidates for host rocks. The Rupelian clay, which is subcropping in most of the northwestern European countries mentioned above (Vandenberghe and Mertens, 2013), is considered in the Dutch geological disposal project (Onderzoeksprogramma eindberging radioactief afval, “OPERA”) as a candidate host-rock (Grupa, 2014). In the present northwestern European context it provides an interesting case study because it is distributed over a wide range of settings with respect to overburden thickness, lithology, porosity and geothermal heat flux.

Usually, uncertainties in permafrost modelling studies in the context of radioactive waste disposal are not addressed in a systematic way (e.g. Busby et al., 2015; Holmen et al., 2011; Hartikainen et al., 2010). The sensitivity of permafrost models is often assessed in terms of variations to single parameters like porosity or surface temperature one at a time (Kitover et al., 2013). These kinds of local sensitivity analyses (SAs) assess the response of the model output to a small perturbation of single parameters, one at a time, around a nominal value. The main disadvantage of this method is that information about the sensitivity is only valid for this very specific location in the parameter space, which is usually not representative of the physically possible parameter space, and becomes problematic especially in the case of non-linear models. To overcome this problem, a global stochastic sensitivity analysis is used in this work, where multiple locations in the physically possible parameter space are evaluated at the same time.

In this paper, we will present permafrost depth calculations for a number of areas in the Netherlands that are representative for a specific geological setting using a best estimate temperature curve for a glacial cycle that is analogous to the last glacial period (Weichselian). In addition, results from stochastic simulations will be presented that give an indication of the probability of permafrost depths under a future glacial climate, taking into account various combinations of temperature, overburden lithology, porosity and geothermal heat flux. Furthermore, a sensitivity analysis is performed to identify the key model parameters that cause the uncertainty of the calculated permafrost depths. As such, the work performed in Govaerts et al. (2011), which was done for one potential site in the framework of the Belgian research programme on geological disposal, is taken a few steps further.

The results of the simulations can be used to assess if the foreseen depth of a future geological disposal facility in the Netherlands is sufficient to exclude possible adverse effects from the presence of developing permafrost. Until now, these kinds of simulations were not performed nationwide, and a more thorough uncertainty and sensitivity analysis adds to the robustness of the findings.

To describe heat transport in the subsurface of the Netherlands, the following one-dimensional enthalpy conservation equation is used with heat transport only occurring by conduction.

When modelling the thermal effects of freezing and thawing, Eq. (1) has to include three phases: rock matrix, fluid and ice. To achieve this, the following volume fractions are defined:

When a material changes phase, for instance from solid to liquid, energy is
added to the solid. This energy is the latent heat of phase change. Instead
of creating a temperature rise, the energy alters the material's molecular
structure. This latent heat of freezing and melting water,

Properties of the different components of the subsoil.

The values of the specific heat of the Boom Clay matrix are obtained from
Cheng et al. (2010). The equivalent, mass-based heat capacity then adds up
to 1443 and 981 J (kg K)

In case of phase change at a single temperature, thermal conductivity is not continuous with respect to temperature. However, considering the freezing range in rocks, we use Eqs. (2) and (4) for taking into account the contributions of the fluid and ice phases. Since the materials are assumed to be randomly distributed, the weighting between them is realized by the square-root mean, which is believed to have a greater physical basis than the geometric mean (Roy et al., 1981).

The values of the thermal conductivity of the rock matrix of Boom Clay and
sand are chosen in the same range of the values used by Bense et al. (2009)
and Mottaghy and Rath (2006), who used respectively 4.0 and 2.9 W (m K)

For Boom Clay, the equivalent thermal conductivity then adds up to 1.31 and 2.03 W (m K)

In sandy soils, the equivalent thermal conductivity is 2.05 and 2.80 W (m K)

The heat transport equation is implemented in COMSOL Multiphysics' Earth
Science Module (2008), together with all the correlations for the thermal
properties. Because the thermal properties differ between the frozen and
unfrozen state, a variable

Moreover, the freezing process is modelled using a gradual and not a sudden
uptake and release of the latent energy, starting at 0

Structural elements in the Dutch subsurface. LBH

Permafrost development is dependent on atmospheric and surface boundary conditions, mostly air temperature and vegetation, and subsurface properties such as lithology, porosity and geothermal heat flux. As such, it is the result of interactions between global changes (temperature) and local conditions (geology). The strategy adopted for this specific study consists of the following elements. First, we try to simulate a future glacial climate using the Weichselian glaciation (115–11 ka) as an analogue. Various temperature estimates are available for this glacial period, many of them being derived from palaeoclimatological archives in Belgium and the Netherlands. The forcing temperature is allowed to change temporally at the upper boundary but held spatially uniform for a given time step. Subsequently it will be used to force the permafrost model, which is fragmented into different representative polygons. The initial condition of the model is the steady state temperature profile based on the present day temperature gradient.

The thickness of subsurface units and their lithofacies distribution are
considered relevant for permafrost modelling as this affects porosity and
the effective thermal properties. The selection of the different areas for
permafrost modelling is based on the presence of 17 structural elements,
including six highs, five basins and six platforms (Fig. 1). The rationale is that
these structural elements delineate differences in thickness and depths of
both Mesozoic and Cenozoic subsurface units. Subsequently, a geological
(property) model was constructed based on the surfaces of the deep
geological model (DGM) shallow subsurface model. For each unit, vertical
grid cells with a surface of 250 m

The reference calculation consists of 17 simulations of permafrost progradation and degradation during the last interglacial–glacial–interglacial cycle. Each calculation is performed for one of the 17 polygons. These are discussed in more detail in the following section. It should be noted that this method does not allow for local variations to be included, but rather serves to highlight regional trends over the Netherlands.

An important parameter for permafrost modelling is porosity. Porosity is directly linked with water content as full saturation is assumed. Consequently, thermal conductivity and the equivalent heat capacity of the soil correspond with these conditions (see Table 1 and Eq. 3). A porosity value is assigned to each of the lithostratigraphic units defined in the Digital Geological Model (DGM; Gunnink et al., 2013) of the shallow Dutch subsurface. The hydrogeological model REGIS (Vernes and van Doorn, 2005) provides a further subdivision and includes both aquifers (sand) and non-aquifer layers (clay). Using the REGIS information a percentage of clay vs. sand for each of the DGM units can be calculated. Based on a relatively small number of porosity measurements of the sand and clay layers in the stratigraphic interval above the Rupel Formation, a best fit, generally applicable porosity–depth relationship was established for all sandy and clayey depositional facies of the various units. This allows for making porosity predictions in non-studied domains.

Several trends can be observed from Fig. 2. The highest averaged mid-depth porosities for the post-Rupelian overburden are observed in the east and the southwest, reaching 50 %. The lowest values are found in the southeast (Ruhr Valley Graben, polygon RVG) and the northwest. The porosity is influenced by two other parameters: lithology and burial depth. The thicker the post-Rupelian overburden, the deeper the mid-depth for which the porosity is determined, and thus the lower the porosity. Lithology also has an influence on porosity because on average, clay has a higher porosity than sand. As such, the porosity map is a mirror image of a combination of lithology and overburden thickness. Note that the depth to the top of the Rupel Formation is representative for the total overburden depth, which is strongly coupled to the tectonic setting, i.e. thick in basins and grabens and relatively thin on structural highs (Figs. 1 and 2). Lithofacies (given as sand percentage) seem to be linked to certain structural elements. Notably in the southeastern Netherlands, the sand content is very high, reaching more than 80 % in the RVG and adjacent areas. These areas acted as traps for a thick series of Neogene continental sands.

The temperature curves and data used in this study are shown in Fig. 3. Best
estimates for the mean annual air temperature (MAAT) during MIS5 (marine
isotope stage 5) are based on pollen data from van Gijssel (1995), but
replotted against a more recent chronostratigraphic framework for the
Weichselian glaciation (see e.g. Busschers et al., 2007). The main features
of the MIS5 climate are the relatively mild stadials 5b and 5d, with a MAAT
of

Top: best estimate temperature evolution for the Weichselian
glaciation, which is taken as an analogue for a future glacial climate in
permafrost calculations. The curve is based on data from van Gijssel (1995),
Huijzer and Vandenberghe (1998), Renssen and Vandenberghe (2003), Busschers
et al. (2007) and Buylaert et al. (2008). Marine isotope stages (numbering 1
to 5e) are taken from Busschers et al. (2007) and references therein. The
oxygen isotope curve is reproduced from the North Greenland Ice Core Project (NGRIP) (2004) data. Bottom: upper and
lower bound values for stochastic permafrost calculations on a logarithmic
timescale. MIS5e is equivalent to the Eemian interglacial, while MIS1
corresponds to the Holocene (present interglacial). Y. Dryas

Upper and lower bounds for these temperature data are given in Fig. 3. They
serve as input for the permafrost depth uncertainty analysis. Instead of
using one best estimate temperature evolution for the Weichselian glaciation
used as an analogue, a minimum and maximum temperature distribution for each
time frame is determined, which will be randomly sampled to produce various
combinations of upper and lower bound MAAT values in the stochastic
uncertainty analyses. Different sources of uncertainty are thus taken into
account, such as the reliability of the palaeotemperature proxy, the
transferability towards a future glacial climatic cycle, temperature
gradients across the country and atmosphere–soil temperature coupling. Lower
bound estimates are set ca. 3–4

The upper bound is based on a warm reconstruction of the Weichselian
climate, which is based on a pollen sequence in sediments from the crater
lake at La Grande Pile, situated ca. 500 km south of Amsterdam at an
altitude of ca. 350 m (Guiot et al., 1989). The mean MAAT estimate obtained
in that study is used here as an absolute maximum scenario for the
Weichselian glaciation in northwestern Europe given its location. The upper
bound for the time period around 20 ka, for which the pollen record gives no
solution, is set to

The temperature data presented here are directly used as soil surface
temperature data. However, vegetation, snow or ice would act as a shield
against penetration of cold air and hamper the evolution of permafrost with
depth. The effect of vegetation and snow can be addressed using the concept
of freezing and thawing

It has to be mentioned that the model neglects subsurface hydrology such as the vadose zone and groundwater flow. During very cold stadials, infiltration would probably be so low that the groundwater table would be significantly lower. Unsaturated soil slows down permafrost development because of the difference in thermal conductivity between air and water. Full saturation of the pore space is adopted in the present calculations, aiming at keeping the model conservative with respect to neglected processes. Next, groundwater flow is neglected but it is evident that this would slow down the speed of permafrost development as well because of the redistribution of heat (Kurylyk et al., 2014). Finally, outfreezing of pore water salt would lower the speed of permafrost development because more heat needs to be extracted to freeze water with elevated salt concentrations (Stotler et al., 2009).

For each of the 17 polygons, the mid-depth temperature and the surface
temperature were used to calculate the temperature gradient. These data are
then used to calculate the geothermal heat flux using the averaged
thermal conductivity values of the sandy overburden (Fig. 2d). Generally
speaking, the country is split up between a southeastern part with lower
values of the geothermal heat flux (0.06–0.07 W m

Model domain and boundary conditions.

The upper part of the one-dimensional domain is modelled as sandy soil for each of the 17 polygons as deep as the overburden reaches. The total vertical length of the one-dimensional lithological domain is then extended to at least 500 m with clay material, in case the overburden does not reach this depth. This implies assuming that the Rupelian layers underlying the overburden are sufficiently thick to bridge the distance from the bottom of the overburden to a depth of 500 m. The lower boundary condition needs to be imposed at a distance sufficiently far from the top to avoid artificial, numerical interaction with the surface temperature boundary condition. This is illustrated, together with the boundary conditions, in Fig. 4. Material parameters are held constant along the domain length for each soil type.

Due to the 1-D nature of the model, mesh size poses no strong obstacle. The results are checked for grid independency by systematically refining the mesh. As a result, a mesh of 500 equidistant elements is chosen as an optimal setting for the final simulations. Absolute and relative tolerances for convergence within each timestep are set to 1e-4. The COMSOL software adapts the size of the timestep in order to fulfill this requirement.

As an integral part of a safety case file, supporting calculations for radioactive waste disposal often involves the analysis of complex systems. Various types of uncertainty affect the results of the evaluations. An overview of the treatment of uncertainties in the disposal programmes of several European countries has been compiled within the PAMINA project (Marivoet et al., 2008).

The nature of the uncertainty can be stochastic (or aleatory) or subjective (or epistemic). Epistemic uncertainty derives from a lack of knowledge about the adequate value for a parameter, input or quantity that is assumed to be constant throughout model analysis. In contrast, a stochastic model will not produce the same output when repeated with the same inputs because of inherent randomness in the behaviour of the system. This type of uncertainty is termed aleatory or stochastic.

In general a distinction is made between three sources of uncertainty: uncertainty in scenario descriptions, including the evolution of the main components of the repository system; uncertainty in conceptual models; uncertainty in parameter values.

Although both types and even sources of uncertainties cannot be entirely separated, the work in this report deals mostly with subjective (or epistemic) uncertainties, which are reflected in the uncertainties in parameter values.

Parameters, initial and boundary conditions for mathematical models are seldom known with a high degree of certainty. The study of parameter uncertainty is usually subdivided into two closely related activities referred to here as uncertainty analysis and sensitivity analysis, where (i) uncertainty analysis (UA) involves the determination of the uncertainty in analysis results that derive from uncertainty in input parameters and (ii) sensitivity analysis (SA) involves the determination of relationships between the uncertainty in analysis results and the uncertainty in individual analysis input parameters. SA identifies the parameters for which the greatest reduction in uncertainty or variation in model output can be obtained if the correct value of this parameter could be determined more precisely.

To this end, a Monte Carlo simulation is based on performing multiple model evaluations using random or pseudorandom numbers to sample from probability distributions of model inputs. The results of these evaluations can be used to both determine the uncertainty in model output and perform SA. The popular and robust Monte Carlo (MC) method in combination with the efficient Latin hypercube sampling (LHS) is used here and is described in several review papers and textbooks (e.g. Marino et al., 2008; Helton, 1993). LHS requires fewer samples than simple random sampling and achieves the same level of accuracy.

Parameters and associated ranges used in the global UA and SA.,
T1 to T26, are used to control the magnitude of the various temperature
plateaus during the Weichselian temperature cycle (Fig. 3), ranging from

The calculations are done in three steps, using MATLAB 2012a (MathWorks) linked to the finite element PDE solver Comsol 3.5a (2008). First, values of all selected stochastic input variables are sampled for all the runs using built-in MATLAB functions. All variables are assumed to be independent so no in-between correlations were implemented. A number of simulations are then performed using the sampled parameter combinations. MATLAB was used to automate the simulations performed by the FE code COMSOL for all Monte Carlo runs. Tables of collected results produced by MATLAB can then be directly analysed to calculate and plot the percentiles and mean value of the permafrost depth as a function of time. MATLAB was then used again to compute standardized or partial correlation coefficients to investigate the parameter sensitivity. For the regression-based analyses, 1000 realizations are performed to obtain the results for each scenario. In order to guarantee stability of the output, enough realizations should be provided. The minimum number of realizations required to assure stable output depends on the system itself and the number of uncertain variables associated with it. Helton (2005) showed that 100–300 model runs were sufficient for stable results using a far more complex two-phase flow model with 37 uncertain variables.

Top: best estimate permafrost progradation during a simulation of the Weichselian glaciation cycle for the FRP polygon. Bottom: idem, for the LBH polygon. These two polygons (FRP and LBH) are at the low and high end of the resulting permafrost depths respectively.

The stochastic simulations give an indication of the probability of nationwide permafrost depths under a future glacial climate, taking into account various combinations of temperature, overburden lithology, porosity and geothermal heat flux.

A proper quantification of uncertainties in the form of probability density functions (PDFs) is an essential part of the uncertainty management and a prerequisite for probabilistic uncertainty and sensitivity analysis. As the actual knowledge about the statistical distribution of the parameters in question is limited, it is only possible to estimate a minimum, a maximum and a most probable value. In this case the triangular distribution is the most appropriate expression of the state of the knowledge (Bolado et al., 2009).

The parameters that are investigated in the stochastic analysis are shown in Table 2. Their minimum, maximum and mode values are used to build a triangular probability density function, which is sampled in the stochastic analysis. T1 to T26 are variables that are used to control the magnitude of the various temperature plateaus during the Weichselian temperature cycle. This allows to account for the actual parameter uncertainty for temperature as well as nationwide spatial parameter variability.

A wide range of SA methods exists but can generally be classified into Local and Global techniques. Local SA will be assessing the response of the model output to a small perturbation of single parameters (the so called one at a time method) around a nominal value. The main disadvantage of this method is that information about the sensitivity is only valid for this very specific location in the parameter space only, which is usually not representative of the physically possible parameter space, which becomes problematic especially in the case of non-linear models.

To deal with this problem global SA methods have been developed, where multiple locations in the physically possible parameter space are evaluated at the same time. The most frequently used global techniques are implemented using Monte Carlo simulations and are therefore called sampling-based methods. Global SA with regression-based methods rests on the estimation of linear models between parameters and model output. For linear trends, linear relationship measures that work well are the Pearson correlation coefficient (CC), partial correlation coefficients (PCCs) and standardized regression coefficients (SRC; Helton, 2005). In this study, the SRC will be used.

Interpolated best estimate maximum permafrost depth map for the
0

The definition of permafrost applied here is ground that remains at or
below 0

The progradation fronts (i.e. the location where the temperature reaches
0,

Simulated depth for different freezing states along a N–S transect from polygon centre FRP in the north to LBH in the south. See Fig. 1 for location of the profile.

The spatial distribution of maximum permafrost depth at any time during a
Weichselian climatic analogue is given in Fig. 6. The map is interpolated
(inverse distance weighted) from individual polygon results, and is the
result of model forcing by the best estimate climate evolution given in Fig. 3. The maximum permafrost depth generally corresponds with the coldest peak
in MIS2 (around 20 ka BP). The depth of the location where 0

Somewhat surprisingly, the calculated permafrost depth would be about 40 m less in the north. Intuitively, one would expect permafrost to reach greater depths in the north, because of the inferred temperature gradient over the country. As stated above, the forcing temperature was assumed to be spatially homogeneous for a given time step across the study area, such that the results can be interpreted solely in terms of subsurface properties. The spatial pattern of maximum permafrost depth is in fair agreement with the pattern of geothermal heat flux, as shown in Fig. 2, and a relationship with the weight fraction of sand can be observed as well. This seems logical since a higher geothermal heat flux imposes a stronger resistance against the intrusion of subzero temperatures into the soil. A higher sand fraction facilitates permafrost growth, as a sand matrix has a higher thermal conductivity, which allows a more rapid extraction of thermal energy towards the surface during cold periods. Thus, assuming a constant temperature evolution over the Netherlands, geothermal heat fluxes, and to a lesser extent sand percentage, seem to be the determining factor in explaining the N–S variability of the maximum permafrost depth. Parameter sensitivity will be addressed in more detail in the section on the sensitivity analysis.

The uncertainty analysis translates the uncertainty on the input parameters
into an uncertainty on the permafrost depth (0

All percentiles of permafrost front penetration depth during a stochastic nationwide simulation of the Weichselian glaciation.

SRCs as a function of time for a global sensitivity study of permafrost progradation during a Weichselian glaciation (top: physical parameters, bottom: selected set of temperature parameters T1 to T26, which are variables that are used to control the magnitude of the various temperature plateaus during the Weichselian temperature cycle).

The goal of this sensitivity analysis (SA) is to determine the relationships between the uncertainty in output and the uncertainty in individual input parameters. SA identifies the parameters for which the greatest reduction in uncertainty or variation in model output can be obtained if the correct value of this parameter could be determined more precisely. The results are analysed by looking at the evolution of the SRC and PCC coefficients. PCC and SRC provide related, but not identical, measures of the variable importance. If input factors are independent, PCC and SRC give the same ranking of variable importance. This is the case in the present study, and we will focus on the SRC to discuss the sensitivity analysis. A positive correlation coefficient (SRC and/or PCC) means that a higher value of the parameter will cause a larger permafrost depth and vice versa.

It can be seen in Fig. 9 that the

The SRC indicates that the geothermal heat flux is the most important parameter, aside from temperature forcing. It is interesting to note that during permafrost growth (e.g. around 90 ka) the geothermal heat flux is equally as important as the porosity. However, when the surface temperature again rises and the permafrost starts to degrade, the geothermal heat flux acts as the main driving force of the thawing process at the base of the permafrost, resulting in a decrease of the permafrost depth.

During the course of simulation time, correlation coefficients can change their sign. During permafrost growth, at the initial phase of a subzero temperature period, a higher porosity will hamper permafrost growth. Because of the larger effective heat capacity, a larger pore water content means that a larger amount of energy needs to be removed from the subsoil in order to cool it down and to induce a phase change of the total amount of pore water. A larger water content also decreases the total effective thermal conductivity, which slows down the extraction of thermal energy towards the surface. Thereafter, during the subsequent warmer period, a higher ice content will require a larger amount of heat to be supplied to thaw the permafrost.

The sand fraction shows a relatively strong, positive influence on permafrost depth, which confirms the findings of the nationwide simulation. Compared to clay, sand has a higher thermal conductivity, which will cause a more rapid cooling of the subsurface during cold periods.

The overburden thickness only seems to play a role during moderately cold periods (e.g. MIS 5b and 5d). If the overburden thickness is lower than 500 m, the remaining part of the domain is repleted with Boom Clay type material, which has a slightly lower thermal conductivity than the often sandy overburden, which slows down permafrost formation. This only seems to be of any importance in periods when the surface temperature is only slightly below the freezing point. Later, when the cold periods become more extreme (e.g. MIS 2), the difference in thermal conductivity at the clay–sand interface plays a less prominent role, and the other parameters become more significant.

Finally, it is no surprise that, being the driving force in the formation of a permafrost, the surface temperature is crucial at the time it is imposed on the computational domain (Fig. 9; in order not to overcrowd the figure, only 8 of the 26 temperature variables are shown). It is important to note that a specific correlation coefficient becomes larger when that temperature is maintained for a longer period (e.g. T2 and T4). Closer to the present, the dynamics of the temperature evolution during the Weichselian are better captured in the proxy data, which translates itself to a more detailed temperature evolution during the last 50 ky (T12–T26). Subsequently, this makes the individual temperature parameters seem less important compared to the earlier temperatures, which can be seen as an artefact induced by the dynamics of the temperature curve.

Another interesting point to note is the fact that a low temperature during an early time frame can still manifest its influence thousands of years later. For instance, the SRC curves of T6 and T7 show a long tailing that still impacts the formation of the permafrost around 60 ka. This can be explained by the thermal inertia of the frozen soil, which has not been fully reverted to the initial temperatures at the start of a subsequent cold period (see also ter Voorde et al., 2014).

The best estimate permafrost depth values of 155–195 m (0

The stochastic approach applied in this study combines a large range of
parameter values and boundary conditions. Interestingly, this results in a
permafrost depth distribution ranging between 100 and 270 m for the LGM.
This window seems to cover any of the calculated permafrost depths
mentioned by previous studies. Some extreme values for the British Isles and
Sweden that go beyond the thickest permafrost calculated in the present
study are basically caused by much lower MAAT values for the coldest glacial
peaks. Indeed, some of the remaining uncertainties are the duration and
minimum MAAT values adopted for these cold peaks. Temperature
reconstructions, based on periglacial deformation phenomena, show that the
MAAT during the LGM in the Netherlands would have been equal to or lower
than

Permafrost depth modelling using a best estimate temperature curve of the
Weichselian as an analogue for the future indicates that the permafrost
front (0

The calculations presented here are robust and conservative.

Input data can be accessed at

The present work has been funded by COVRA, the Dutch Central Organisation for Radioactive Waste. The findings and conclusions in this paper are those of the authors and do not necessarily represent the official position of COVRA. We thank the co-editor-in-chief Stephan Gruber for the critical review of the originally submitted paper. We also want to express our gratitude to Daniella Kitover and the anonymous reviewer for their helpful suggestions and comments on the discussion paper. Edited by: S. Gruber Reviewed by: D. Kitover and one anonymous referee