Wellman et al. (2013) describe numerical simulations of lake-talik formation in watersheds modeled after those found in the lake-rich Yukon Flats of interior Alaska. Modeling experiments used the SUTRA groundwater modeling code (Voss and Provost, 2002) enhanced with capabilities to simulate freeze–thaw processes (McKenzie and Voss, 2013). The phase change between ice and liquid water occurs over a specified temperature range and accounts for latent heat of fusion as well as changes in thermal conductivity and heat capacity for ice–water mixtures. Ice content also changes the effective permeability, thereby altering subsurface flowpaths and enforcing a strong coupling between hydraulic and thermal processes. The modeling domain, which is adapted for this study, is axis-symmetric with a central lake and upwards-sloping ground surface that rises from an elevation of 500 m at r=0 to 520 m at the outer extent of the model, r=1800 m (Fig. 1). The model uses a layered geology consistent with the Yukon Flats (Minsley et al., 2012; Williams, 1962), with defined hydrologic and geophysical parameters for each layer summarized in Table 1. Initial permafrost conditions prior to lake formation were established by running the model to steady state under hydrostatic conditions with a constant temperature of -2.25 ∘C applied to the land surface, which produces a laterally continuous permafrost layer extending to a depth of about 90 m.

Subsequent hydrologic simulations assume fully saturated conditions and are performed over a 1000-year period under 36 different scenarios of climate (warmer than, colder than, and similar-to-present conditions), hydrologic gradient (hydrostatic, gaining, and losing lake conditions), and lake depth/extent (3, 6, 9, and 12 m deep lakes that intersect the ground surface at increasing distance, as shown in Fig. 1). Complete details and results of the hydrologic simulations can be found in Wellman et al. (2013). At each simulation time step, the SUTRA model outputs temperature, pressure, and ice saturation. Conversion of these hydrologic variables to electrical resistivity – the geophysical property needed to simulate electromagnetic data considered here – is described below.

Electrical resistivity is the primary geophysical property of interest for the electromagnetic geophysical methods used in this study. It is well established that resistivity is sensitive to basic physical properties such as unfrozen water content, soil or rock texture, and salinity (Palacky, 1987). Here, we build on earlier efforts to simulate the electrical properties of ice-saturated media (Hauck et al., 2011) by using a modified form of Archie's Law (Archie, 1942) that also incorporates surface conduction effects (Revil, 2012) to predict the dynamic electrical resistivity structure for the evolving state of temperature and ice saturation (Si) in the talik simulations. Bulk electrical conductivity is described by Revil (2012) as σ=SwnFσf+mSw-nF-1σs, where σ is the bulk electrical conductivity [Sm-1]; Sw is the fractional water saturation [–] in the pore space, where Sw=1-Si; σf is the conductivity of the saturating pore fluid [Sm-1]; m is the Archie cementation exponent [–]; n is the Archie saturation exponent [–]; F is the formation factor [–], where F=ϕ-m and ϕ is the matrix porosity [–]; and σs is the conductivity [Sm-1] associated with grain surfaces. The Archie exponents m and n are known to vary as a function of pore geometry; here, we use m=n=1.5, which is appropriate for unconsolidated sediments (Sen et al., 1981). Simulation results are presented as electrical resistivity [Ωm], which is the inverse of the conductivity, i.e., ρ=1/σ.

The first term in Eq. (1) describes electrical conduction within the pore fluid, where fluid conductivity is defined as σf=Fc∑iβiziCi. The summation in Eq. (2) is over all dissolved ionic species (Na+ and Cl- are assumed to be the primary constituents in this study), where Fc is Faraday's constant [Cmol-1] and Ci, βi, and zi are the concentration [molL-1], ionic mobility [m2V-1s-1], and valence of the ith species, respectively.

Surface conduction effects, described by the second term in Eq. (1), are related to the chemistry at the pore–water interface, and can be important in fresh water (low conductivity) systems at low porosity (high ice saturation). Additionally, the surface conduction term provides a means for describing the conductivity behavior for different lithologies, as will be described below. The surface conductivity is given by σs=23ϕ1-ϕβsQv, where βs is the cation mobility [m2V-1s-1] for counterions in the electrical double layer at the grain-water interface (Revil et al., 1998) and Qv is the excess electrical charge density [Cm-3] in the pore volume, and Qv=Sw-1ρg1-ϕϕχ, where ρg is the mass density of the grains [kgm-3] and χ is the cation exchange capacity [Ckg-1]. Changes in χ, representative of bulk differences in clay mineral content, are used to differentiate the electrical signatures of the lithologic units in this study (Table 1).

The temperature, T [C], dependence of ionic mobility affects both the fluid conductivity (Eq. 2) and surface conductivity (Eq. 3), where mobility is approximated as a linear function of temperature (Keller and Frischknecht, 1966; Sen and Goode, 1992) as βT=βT=25∘C1+0.019T-25. Finally, we consider the effect of increasing ice saturation on salinity. Because salts are generally excluded as freezing occurs, salinity of the remaining unfrozen pore water is expected to increase with increasing ice content (Marion, 1995), leading to a corresponding increase in fluid conductivity according to Eq. (2). To describe this dependence of salinity on ice saturation, C(Si), we use the expression CSi=CSi=0Sw-α, where α∼ 0.8 accounts for loss of solute from the pore space due to diffusion or other transport processes and Si=1-Sw.

Information about the different lithologic units described by Wellman et al. (2013) that are also summarized in Table 1 are used to define static model properties such as porosity, grain mass density, cation exchange capacity, and Archie's exponents. Dynamic outputs from the SUTRA simulations, including temperature and ice saturation, are combined with the static variables in Eqs. (1)–(6) to predict the evolving electrical resistivity structure.

Synthetic AEM data are simulated for each snapshot of predicted bulk resistivity values using nominal system parameters based on the Fugro RESOLVE

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The vertical profile of resistivity as a function of depth is extracted at each survey location and is used to simulate forward geophysical responses. There are 181 sounding locations for each axis-symmetric model, starting at the center of the lake (r=0 m) and moving to the edge of the model domain (r=1800 m) in 10 m increments. Each vertical resistivity profile extends to 200 m depth, which is well beyond the depth to which we expect to recover parameters in the geophysical inversion step. A center-weighted five-point filter with weights equal to [0.0625, 0.25, 0.375, 0.25, 0.0625] is used to average neighboring bulk resistivity values at each depth before modeling in order to partly account for the lateral sensitivity of AEM systems (Beamish, 2003). Forward simulations are repeated for each of the 50 simulation times between output of 0 and 1000 years from SUTRA, resulting in 9050 data locations per modeling scenario.

Synthetic ground-based electromagnetic data presented in Sect. 3.3 are simulated using nominal system parameters based on the GEM-2 instrument (Huang and Won, 2003). The GEM-2 has a single HCP transmitter–receiver pair separated by 1.66 m, and data are simulated at six frequencies: 1.5, 3.5, 8.1, 19, 43, and 100 kHz. A system elevation of 1 m above ground is assumed, which is typical for this hand-carried instrument.

The inverse problem involves estimating subsurface resistivity values given the simulated forward responses and realistic assumptions about data errors. Geophysical inversion is inherently uncertain; there are many plausible resistivity models that are consistent with the measured data. In addition, the ability to resolve true resistivity values is limited both by the physics of the AEM method and the level of noise in the data. Here, we use a Bayesian Markov chain Monte Carlo (McMC) algorithm developed for frequency-domain electromagnetic data (Minsley, 2011) to explore the ability of simulated AEM data to recover the true distribution of subsurface resistivity values at 20-year intervals within the 1000-year lake-talik simulations. This McMC approach is an alternative to traditional inversion methods that find a single “optimal” model that minimizes a combined measure of data fit and model regularization (Aster et al., 2005). Although computationally more demanding, McMC methods allow for comprehensive model appraisal and uncertainty quantification. AEM-derived resistivity estimates for the simulations considered here will help guide interpretations of future field data sets, identifying the characteristics of relatively young versus established thaw under different hydrologic conditions.

The McMC algorithm provides comprehensive model assessment and uncertainty analysis and is useful in diagnosing the ability to resolve various features of interest. At every data location along the survey profile, an ensemble of 100 000 resistivity models is generated according to the Metropolis–Hastings algorithm (Hastings, 1970; Metropolis et al., 1953). According to Bayes' theorem, each model is assigned a posterior probability that is a measure of (1) its prior probability which, in this case, is used to penalize models with unrealistically large contrasts in resistivity over thin layers; and (2) its data likelihood, which is a measure of how well the predicted data for a given resistivity model match the observed data within data errors. A unique aspect of this algorithm is that it does not presuppose the number of layers needed to fit the observed data, which helps avoid biases due to assumptions about model parameterization. Instead, trans-dimensional sampling rules (Green, 1995; Sambridge et al., 2013) are used to allow the number of unknown layers to be one of the unknowns. That is, the unknown parameters for each model include the number of layers, layer interface depths, and resistivity values for each layer.

Numerous measures and statistics are generated from the ensemble of plausible resistivity models, such as the single most-probable model, the probability distribution of resistivity values at any depth, the probability distribution of where layer interfaces occur as a function of depth, and the probability distribution of the number of layers (model complexity) needed to fit the measured data. A detailed description of the McMC algorithm can be found in Minsley (2011). Finally, probability distributions of resistivity are combined with assumptions about the distribution of resistivity values for any lithology and/or ice content in order to make a probabilistic assessment of lithology or ice content, as illustrated below.

Information about the different lithologic units described by Wellman et al. (2013) that are also summarized in Table 1 are used to define static model properties such as porosity, grain mass density, cation exchange capacity, and Archie's exponents. Dynamic outputs from the SUTRA simulations, including temperature and ice saturation, are combined with the static variables in Eqs. (1)–(6) to predict the evolving electrical resistivity structure. The behavior of bulk resistivity as a function of ice saturation is shown in Fig. 2. Separate curves are shown for a range of χ (cation exchange capacity) values, which are the primary control in defining offset resistivity curves for different lithologies, where increasing χ is generally associated with more fine-grained material such as silt or clay.

For each of the 1000-year simulations, the static variables summarized in Table 1 are combined with the spatially and temporally variable state variables T and Si output by SUTRA to predict the distribution of bulk resistivity at each time step using Eqs. (1)–(6). An example of SUTRA output variables for the 6 m deep gaining lake scenario at 240 years (the approximate sub-lake talik breakthrough time for that scenario) is shown in Fig. 3a and b, and the predicted resistivity for this simulation step is shown in Fig. 3c. The influence of different lithologic units is clearly manifested in the predicted resistivity values, whereas lithology is not overly evident in the SUTRA state variables. For a single unit, there is a clear difference in resistivity for frozen versus unfrozen conditions. Across different units, there is a contrast in resistivity when both units are frozen or unfrozen. Resistivity can therefore be a valuable indicator of both geologic and ice content variability. However, there is also ambiguity in resistivity values as both unfrozen unit 2 and frozen unit 3 appear to have intermediate resistivity values of approximately 100–300 Ωm (Fig. 3c) and cannot be characterized by their resistivity values alone. This ambiguity in resistivity can only be overcome by additional information such as borehole data or prior knowledge of geologic structure. Synthetic bulk resistivity values according to Eq. (1) are shown in Fig. 4 for the four different lake depths (3, 6, 9, and 12 m) at three different simulation times (100, 240, and 1000 years) output from the hydrostatic/current climate condition simulations.

Lithology and ice saturation are the primary factors that control simulated resistivity values (Fig. 2), though ice saturation is a function of temperature. The empirical relation between temperature and bulk resistivity is shown in Fig. 3d by cross-plotting values from Fig. 3b and c. Within each lithology resistivity is relatively constant above 0 ∘C, with a rapid increase in resistivity for temperatures below 0 ∘C. This result is very similar to the temperature–resistivity relationships illustrated by Hoekstra et al. (1975, Fig. 1), lending confidence to our physical property definitions described earlier. Above 0 ∘C, the slight decrease in resistivity is due to the temperature dependency of fluid resistivity. The rapid increase in resistivity below 0 ∘C is primarily caused by reductions in effective porosity due to increasing ice saturation, though changes in surface conductivity and salinity at increasing ice saturation are also contributing factors. Below -1 ∘C, the change in resistivity values as a function of temperature rapidly decreases. This is an artifact caused by the imposed temperature–ice saturation relationship defined in SUTRA that, for these examples, enforces 99 % ice saturation at -1 ∘C. It is more likely that ice saturation continues to increase asymptotically over a larger range of temperatures below 0 ∘C, with corresponding increases in electrical resistivity. However, because AEM methods are limited in their ability to discern differences among very high resistivity values, as discussed later, this artifact does not significantly impact the results presented here.

AEM data (not shown) are simulated for each of the electrical resistivity models (e.g., Fig. 4) using the methods described in Sect. 2.3. The simulated data are then used to recover estimates of the original resistivity values according to the approach outlined in Sect. 2.4, assuming 4 % data error with an absolute error floor of 5 ppm. Resistivity parameter estimation results for the 6 m deep hydrostatic lake scenario (Fig. 4d–f) are shown in Fig. 5. At each location along the profile, the average resistivity model as a function of depth is calculated from the McMC ensemble of 100 000 plausible models. The overall pattern of different lithologic units and frozen/unfrozen regions is accurately depicted in Fig. 5, with two exceptions that will be discussed in greater detail: (1) the specific distribution of partial ice saturation beneath the lake before thaw has equilibrated (Fig. 5a and b) and (2) the shallow sand layer (unit 1) that is generally too thin to be resolved using AEM data.

A point-by-point comparison of true (Fig. 4f) versus predicted (Fig. 5c) resistivity values for the hydrostatic 6 m deep lake scenario at the simulation time 1000 years is shown in Fig. 6a. The cross-plot of true versus estimated resistivity values generally fall along the 1:1 line, providing a more quantitative indication of the ability to estimate the subsurface resistivity structure. Estimates of the true resistivity values for each lithology and freeze–thaw state (Fig. 6b) tend to be indistinct, appearing as a vertical range of possible values in Fig. 6a due to the inherent resolution limitations of inverse methods and parameter tradeoffs (Day-Lewis et al., 2005; Oldenborger and Routh, 2009). Although the greatest point density for both frozen and unfrozen silts (unit 3) falls along the 1:1 line, resistivity values for these components of the model are also often overestimated; this is likely due to uncertainties in the location of the interface between the silt and gravel units. This is in contrast with the systematic underestimation of frozen gravel resistivity values due to the inability to discriminate very high resistivity values using electromagnetic methods (Ward and Hohmann, 1988). Frozen sands (true log resistivity ∼ 2.8 in Fig. 6b) are also systematically overestimated in Fig. 6a; in this case, due to the inability to resolve this relatively thin resistive layer.

While useful, single “best” estimates of resistivity values at any location (Fig. 6) are not fully representative of the information contained in the AEM data and associated model uncertainty. From the McMC analysis of 100 000 models at each data location, estimates of the posterior probability density function (pdf) of resistivity are generated for each point in the model. Probability distributions are extracted from a depth of 15 m, within the gravel layer (unit 2), at one location where unfrozen conditions exist (r=0 m) and a second location outside the lake extent (r=750 m) where the ground remains frozen (Fig. 7a). Results from a depth of 50 m, within the silt layer (unit 3), are shown in Fig. 7b. With the exception of the frozen gravels, the resistivity of which tends to be underestimated, the peak of each pdf is a good estimate of the true resistivity value at that location.

Resistivity values are translated to estimates of ice saturation, which is displayed on the upper axis of each panel in Fig. 7, using the appropriate lithology curve from Fig. 2. Using the ice-saturation-transformed pdfs, quantitative inferences can be made about the probability of the presence or absence of permafrost. For example, the probability of ice content being less than 50 % is estimated by calculating the fractional area under each distribution for ice-content values less than 0.5. Probability estimates of ice content less than 50 % and greater than 95 % for the four distributions shown in Fig. 7 are summarized in Table 2. High probabilities of ice content exceeding 95 % are associated with the r=750 m location outside the lake extent, whereas high probability of ice content below the 50 % threshold are observed at r=0 beneath the center of the lake. The pdfs for each lithology shown in Fig. 7 are end-member examples of frozen and unfrozen conditions. Within a given lithology, a smooth transition from the frozen-state pdf to the unfrozen-state pdf is observed as thaw occurs, with corresponding transitions in the calculated ice threshold probabilities.

Further illustration of the spatial and temporal changes in resistivity pdfs are shown in Fig. 8. The resistivity pdf is displayed as a function of distance from the lake center at the same depths (15 and 50 m) shown in Fig. 7, corresponding to gravel (Fig. 8a, c, and e) and silt (Fig. 8b, d, and f) locations. High probabilities, i.e., the peaks in Fig. 7, correspond to dark-shaded areas in Fig. 8. Images are shown for three different time steps in the SUTRA simulation for the hydrostatic 6 m deep lake scenario: 100 years (Fig. 8a and b), 240 years (Fig. 8c and d), and 1000 years (Fig. 8e and f). Approximate ice-saturation values, translated from the ice versus resistivity relationships for each lithology shown in Fig. 2, are displayed on the right axis of each panel in Fig. 8, and true resistivity values are plotted as a dashed line. Observations from Fig. 8 include the following:

Outside the lake boundary, pdfs are significantly more sharply peaked (darker shading) for the gravel unit than the silt unit, suggesting better resolution of shallower resistivity values within the gravel layer. It should be noted, however, that this improved resolution does not imply improved model accuracy; in fact, the highest probability region slightly underestimates the true resistivity value.

A more detailed analysis of the changes in resistivity and ice saturation as a function of time, and for the differences between hydrostatic and gaining lake conditions, is presented in Fig. 9. Average values of resistivity/ice saturation within 100 m of the lake center are shown within the gravel layer at a depth of 15 m (Fig. 9a) and a depth of 50 m within the silt layer (Fig. 9b) at 20-year time intervals. Outputs are displayed for both 6 m deep hydrostatic and gaining lake scenarios. Thawing due to conduction occurs over the first ∼ 200 years within the gravel layer (Fig. 9a), with similar trends for both the hydrostatic and gaining lake conditions and no clear relationship to the talik formation times indicated as vertical lines. Conduction-dominated thaw is observed for the gravel layer in the gaining lake scenario because significant advection does not occur until after the thaw bulb has extended beneath the gravel layer. In the deeper silt layer (Fig. 9b), however, very different trends are observed for the hydrostatic and gaining lake conditions. Ice content decreases gradually as thawing occurs in the hydrostatic scenario, consistent with conduction-dominated thaw, reaching a minimum near the time of talik formation at 687 years (Wellman et al., 2013, Table 3). In contrast, there is a rapid loss in ice content in the gaining lake scenario resulting from the influence of advective heat transport as groundwater is able to move upwards through the evolving talik beneath the lake. This rapid loss in ice content begins after the gravel layer thaws and reaches a minimum near the 258-year time of talik formation for this scenario. These trends, captured by the AEM-derived resistivity models, are consistent with the plots of change in ice volume output from the SUTRA simulations reported by Wellman et al. (2013, Fig. 3).

Finally, we focus on the upper sand layer (unit 1), which is generally too thin (2 m) and resistive (> 600 Ωm) to be resolved using AEM data, though may be imaged using other ground-based electrical or electromagnetic geophysical methods. Seasonal thaw and surface runoff cause locally reduced resistivity values in the upper 1 m, which is still too shallow to resolve adequately using AEM data. In practice, shallow thaw and sporadic permafrost trends are observed to greater depths in many locations, including inactive or abandoned channels (Jepsen et al., 2013b). To simulate these types of features, the shallow resistivity structure of the 6 m deep hydrostatic lake scenario at 1000 years is manually modified to include three synthetic “channels”. These channels are not intended to represent realistic pathways relative to the lake and the hydrologic simulations; they are solely for the purpose of illustrating the ability to resolve shallow resistivity features.

Figure 10a shows the three channels in a zoomed-in view of the uppermost portion of the model outside the lake extent. Each channel is 100 m wide but with different depths: 1 m (half the unit 1 thickness), 2 m (full unit 1 thickness), and 3 m (extending into the top of unit 2). Analysis of AEM data simulated for this model, presented as the McMC average model, is shown in Fig. 10b. All three channels are clearly identified, but their thicknesses and resistivity values are overestimated and cannot be distinguished from one another. To explore the possibility of better resolving these shallow features, synthetic electromagnetic data are simulated using the characteristics of a ground-based multi-frequency electromagnetic tool (the GEM-2 instrument) that can be hand carried or towed behind a vehicle and is commonly used for shallow investigations. The McMC average model result for the simulated shallow electromagnetic data is shown in Fig. 10c. An error model with 4 % relative data errors and an absolute error floor of 75 ppm was used for the GEM-2 data. Channel thicknesses and resistivity values are better resolved compared with the AEM result, though the 1 m deep channel near r=800 m appears both too thick and too resistive. In addition, the shallow electromagnetic data show some sensitivity to the interface at 2 m depth between frozen silty sands and frozen gravels, though the depth of this interface is overestimated due to the limited sensitivity to these very resistive features.