Debris-covered glaciers can be found in tectonically active mountain regions, such as Alaska, the European Alps, High-mountain Asia, or New Zealand , where large amounts of debris migrate into the ice via glacial and periglacial processes . Debris falling onto the ablation zone contributes directly to any surface debris load, while debris added to the glacier surface in the accumulation zone or sourced subglacially is transported englacially to the ablation area of the glacier, where it melts out and contributes additional debris load , as shown in Fig. a. In comparison to clean ice, thin or patchy debris amplifies ice melt due to its higher absorptivity of short-wave radiation, while thicker debris layers reduce ice melt due to the insulation and attenuation of the diurnal heating signal . The relationship between debris thickness and ablation rate varies for different debris layer compositions and prevailing climatological conditions but retains the same character (Fig. b). The critical debris thickness beyond which sub-debris ice ablation is inhibited compared to clean-ice ablation ranges from 15 to 115 mm depending on the optical and thermal properties of the debris and the ambient climate . Therefore, in contrast to clean-ice glaciers, where the melt increases towards the glacier tongue in response to typical environmental temperature lapse rates, the spatial pattern of melt of debris-covered glaciers depends more on the debris thickness than on the elevation e.g.. found that 7.3 % ± 3.3 % of all mountain glacier area is covered by a rock debris cover, which, at a global scale, delays the loss of debris-covered glaciers for the coming decades . With continued glacier decline, debris-covered glacier surfaces are expected to increase in absolute and percentage terms in the future , highlighting the need for accurate modelling of sub-debris ice melt to be included in future glacier projections .

Although, under certain circumstances, heat can be transferred through the debris by convection, advection, and radiation, observations e.g. show that the system often, and especially under dry stable meteorological conditions, approximates Fourier's law of conduction, q=-k∇T, where q represents the local heat flux density, k represents the thermal conductivity, and T represents the temperature . Consequently, in models of glacier ice melt, the energy supply for ice melt beneath the debris cover is typically treated as if it were heat conduction only e.g., driven by the surface temperature, the debris thickness, and a value of debris thermal conductivity to be supplied as a model parameter. As a second consequence, Fourier's law of conduction has also been used to derive representative parameter values of effective debris thermal conductivity for horizontally homogeneous debris layers from field observations of spatiotemporal variations in debris temperature. To do this, the one-dimensional heat conduction equation for a homogeneous, isotropic medium (Eq. ) is used to derive the apparent thermal diffusivity, κ, from the spatiotemporal variation in a vertical profile of temperature measurements by finding the gradient of the regression line between the first derivative of temperature with time and the second derivative of temperature with depth . Effective thermal conductivity k can then be calculated from κ and the volumetric heat capacity of the debris, given by the specific heat capacity cs and the material density ρ (Eq. ), including the porosity for a granular material. 1∂T∂t=κ∂2T∂x2+const.,2κ=kρ⋅cs←thermal conductivity←heat capacity

Application of this method therefore requires (1) a vertical array of temperature measurements through the supraglacial debris cover (Fig. a) for conditions in which the debris heat transfer closely approximates that of a conductive system from which the apparent κ is derived and (2) an estimate of the volumetric heat capacity of the debris used to convert the apparent κ into effective conductivity.

To meet the first requirement, a sample site must be chosen for which lateral heat transfer can reasonably be expected to be negligible, so a site that is horizontally homogeneous in factors such as slope, debris type, and thickness and without evidence of any hydrological heat transfer. Then, the observed temperatures must be evaluated to find specific time periods and vertical subsets of debris temperature profile data that are identified as being “well-behaved” approximations of a conductive system; data that show evidence of non-conductive processes can be excluded from subsequent analysis . To meet the second requirement, estimates of the debris porosity and the rock density thermal properties must be made. Commonly used values for these terms are porosity of 0.3, rock density of 2700 kgm-3, and rock specific heat capacity of 750 Jkg-1K-1, with a 10% error applied to the combined terms . Most studies assume that the pore spaces are air-filled when calculating the volumetric heat capacity, but, in principle, if the debris cover is known to be fully saturated, a water-filled case can be used to obtain the volumetric heat capacity of the sampled debris layer . In an ideal case, this workflow can yield a reliable estimate of effective thermal conductivity from a homogenous dry portion of the debris with stable meteorological forcing conditions and minimal non-conductive processes. Further use of these effective dry-debris thermal conductivity data in surface energy balance models can allow non-conductive processes and non-uniform debris layers to be included in the model structure by, for example, accounting for stratification in the debris porosity and air flow through the debris , stratification of moisture content, and associated phase changes within the debris layer .

As natural debris covers often show vertical variation in porosity, grain size, and moisture content, recent studies have explored multi-layered applications of the thermal diffusion representation of the debris layer. perform multiple rather than single regression analysis to account for (i) unknown depth variation in κ in a two-layer model and (ii) non-conductive heat sources/sinks. They apply various methods to synthetic datasets to highlight that applying the original method of produces large errors when trying to recover a target κ that varies with depth and that unequally spaced temperature measurements introduce substantial truncation errors. If unequal spacing of measurements cannot be avoided, their new Bayesian method of determining κ outperforms that of . also include a term for depth-varying κ into the heat conduction equation and perform multiple linear regression to solve for its variation with depth in natural debris cover, identifying non-conductive processes as the residual from a comparison of the observed and modelled time-dependent temperature evolution. They find non-negligible heat transfer related to air motion and latent heat fluxes within the debris on Kennicott Glacier. These approaches offer solutions for the potential of vertically varying debris properties and allow quantified assessment of non-conductive processes in measured field sites.

Despite these new developments, the method of has been historically widely used e.g. and has provided the majority of published debris thermal conductivity values used in generalized surface energy balance models and for regional intercomparisons of supraglacial debris properties . Thus, many studies of debris-covered glaciers rely upon the robustness of debris thermal properties produced following . The limited number of datasets used to provide generalized values of effective thermal conductivity have deployed very different field and analytical strategies, with temporal and spatial sampling intervals, thermistor placement within the debris, debris depth of the sampled site, and sensor precision all selected ad hoc in different studies and differing from measurement site to measurement site e.g.. For example, spatiotemporal sampling intervals range from 2 cm to tens of centimetres and from 5 min to 6 h, sometimes including time-averaged rather than sampled temperatures (Appendix ). The impact of these choices on the derived κ values is not well addressed in the published literature, but, for example, the same data from Imja Glacier in Nepal analysed at 30 min and 60 min intervals yielded thermal conductivity values that differed by almost 40 % despite the same properties being used to derive thermal conductivity from κ. This highlights that baseline literature values that are used in surface energy balance modelling may be differently influenced by sensor, installation, and numerical truncation errors and indicates that care should be taken when comparing across sites for which different instrumental and analytical choices have been made e.g.. Therefore, a deeper exploration of the error sources of this method is warranted, and it would be advantageous to develop standardized field and analytical implementation strategies.

This study explores the effect of measurement setup on κ values derived using the method of in order to highlight the potential dependency of published values of thermal conductivity on the spatiotemporal intervals chosen for the analysis and on the sensor precision and locational accuracy. To achieve this, we apply the method of to data generated using a forward diffusivity model for a purely conductive system with a specified value of κ and assess how closely the known κ is recovered when varying choices of instrumental and analytical setups. Since the approach recommended by is only valid for conductive systems, we focus our study on a purely conductive system to provide a baseline reference for individual method-related error sources, expanding the analysis of the impact of irregular spacings performed in to include an assessment of a wider range of field measurement choices. By isolating the individual roles of these different error sources, they can be quantified and their tendencies can be understood, thereby making possible a more critical reassessment of the extent to which differences in published effective thermal conductivity values reflect real-world differences in debris properties or instrumental and analytical choices. We provide an interactive tool (https://github.com/calvinbeck/TC-DTD, last access: 30 June 2025) to allow analysis of the combined errors for any given measurement procedure and a best-practice guideline on how to minimize the systematic errors of using this method (Appendix ).

While the interactive tool provided allows a full range of sampling strategies to be explored, here we present results for selected cases within the range of realistic instrumental setups. Our focus is to provide illustrative examples that characterize the error properties of each individual source.

In previously published data, most apparent thermal diffusivity derived using the method of are below 10×10-7 m2s-1, typically ranging from 1 to 30, with some outlier values exceeding 100×10-7 m2s-1 see Table 2 in. The implementation errors that our analysis reveals are often comparable to this range of published values, highlighting how relevant it is to correctly consider the numerical errors in choosing how to apply this method.

While the interactive tool accompanying our analysis allows a wider range of the parameter space to be explored, the cases we present were chosen to characterize the main numerical error sources inherent in the method within the parameter space of published values (Appendix ). The numerical and measurement implementation error sources investigated here all tend to systematically underestimate κ, while the relative error associated with uneven thermistor spacing tested for a three-thermistor case by was previously identified to systematically overestimate κ by up to 50% at thermistor spacing ratios of 1:5.

In general, the numerical errors associated with applying this method are all related to how well the temperature gradients in space and time within the debris cover can be captured by the instrumental setup. Temporal truncation errors in the absence of statistical noise are typically <25% in most expected deployment settings at sampling intervals of ≤60 min. Near-surface measurements suffer more error because the diurnal temperature cycle at the surface is the most non-sinusoidal and therefore produces larger temporal truncation errors. Consequently, conditions that more closely approximate sinusoidal conditions (i.e. clear-sky stable atmospheric conditions) reduce the errors in the near-surface layers, but this becomes less relevant at depth, as surface noise introduced by weather is progressively smoothed out at greater depth in the debris. Spatial truncation due to the choice of thermistor spacing is not very sensitive to the non-sinusoidal forcing but becomes ≥25% at Δx above ∼25 cm for the range of κ reported in the literature, and the error is larger for smaller κ. The Δx range at which errors are small and similar regardless of the forcing and κ is ≤14 cm, providing a conservative upper bound to limit spatial truncation errors. Even though a Δt or Δx→0 would produce a minimal truncation error, sampling intervals that are too small can also produce erroneous results because, for a Δt→0, the linear regression coefficient of determination decreases strongly. In practice, this is not a problem for the temporal sampling, since short temporal sampling intervals can always be resampled afterwards. A more significant problem occurs if low-precision thermistors are positioned too close to each other, especially if the profile comprises only a few thermistors, making it impossible to spatially resample the temperature data. While this effect diminishes to a stable value of relative error ≥25% for Δx above ∼6 cm, with increasing depth, the thermistors must be further apart, otherwise the thermistor measurement uncertainty dominates the measurement. Therefore, although the highest-precision thermistors should always be chosen if possible, using thermistors with maximum precision becomes even more important at greater depths in the debris layer. The only error source investigated here that has the potential to overestimate κ is that due to inaccurate temperature measurement location. This can happen due to poorly measured positions or due to debris settling after sensor installation if the thermistor profiles are installed on a slope, which is subject to gradual gravitational sliding or reworking. In contrast to , who showed that a constant error in the thermistor position was not important to the analysis, we find that, at least for very small Δx, the calculated κ does depend on the thermistor positions relative to each other being correctly known and sustained over the measurement period. However, thermistors are typically placed more a than a few centimetres apart; this error source might be expected to have little effect if the κ is calculated at several levels in the debris cover, as the mean value of the location-perturbed cases recovers the target diffusivity. Introducing a statistical noise term highlights the manner in which noise degrades the temperature gradients that the method relies on, particularly at greater depths in the debris, where temperature variations in space and time are small compared to the introduced noise term. Thus, care must also be taken to assess if the method is being applied to portions of the debris layer where the gradients are well captured.

In the best-practice guidelines (Appendix ), we address all sources of methodological error discussed in this paper, suggesting optimal implementation strategies for future field studies that wish to deploy these methods of analysing representative thermal conductivity of natural debris layers following the method of . Our recommendations differ somewhat from those of , as the purpose is different. While sought to determine the optimal method to determine sub-debris ablation rates directly from temperature sensors using a minimal number of thermistors, we seek to understand the best way to determine a representative κ from which effective thermal conductivity suitable for onward use in generalized surface energy balance models can be derived. For their purpose, they propose to “set the sensor spacing to be one-fifth of the debris thickness at the location”; however, the non-linear nature of the single-error sources presented in this paper indicates that we cannot generalize such statements if the goal is parameter determination rather than direct ablation determination. Furthermore, they stated “the top sensor should be placed approximately at the middle of the debris layer”, as this captures the relevant flux being delivered to the underlying ice. Our analysis indicates that, while it is true that thermistors too close to the surface produce large truncation errors, the same is valid for thermistors that are too deep, as the temperature gradient is too small relative to the thermistor precision. By providing an open-source interactive tool that can be used to explore all the methodological sources of error in implementing the most widely used method of determining κ, we offer a ready-to-use means for determining the field setup that minimizes these numerical methodological errors. The intention is that, prior to a new field deployment, the error response of the expected conditions of debris thickness, surface-forcing amplitude, sensor number, and precision can be explored and the best possible field deployment of sensors can be made.

In addition to the errors related to measurement setup and analysis procedure investigated in this study, non-conductive processes within the debris layer (e.g. rain, phase changes) can also be present . Unfortunately, it is not always clear in the published literature that the thermal diffusivities and associated thermal conductivity values were derived from optimal conditions sampled within the dataset. The suitability of the sampled debris temperature profiles for determining debris thermal parameters must be carefully evaluated on a case-by-case basis, using meteorological data and closely evaluating the measurements and their gradient functions in order to establish that the data subset represents predominantly conductive conditions, before applying the method of . Once a suitable effective thermal conductivity is established based on “well-behaved” conditions, these base values can be modified for implementation within a surface energy balance model to account for changes in the pore fluid type to allow simulation of varying wet-/dry-debris conditions .

The recently published database of supraglacial debris properties, DebDab v1 , reveals that, from the 176 values of debris thermal conductivity, only 33 report an associated uncertainty, and, while 121 include the debris layer thickness, only 23 report on details such as the thermistor depth. To facilitate the intercomparison of these data, it would be valuable to include the temporal sampling used, along with the rock properties and porosity used to convert κ to thermal conductivity. Deeper consideration and potential common reanalyses of these data would require the original thermistor data to be publicly available, which is not always the case. Reanalysing previously published vertical temperature profiles with common resampling strategies, based on the findings of this study, would facilitate intercomparison of κ values, while reanalysis using the methods of and/or might yield more robust and representative global values by providing respectively a more rigorous assessment of non-conductive processes and inclusion of multi-layered thermal properties within the natural debris layers that have been sampled.

provide a practical method to estimate thermal diffusivity values from a vertical array of thermistors in the supraglacial debris layer, which is applicable for spatially homogenous debris and behaves as a close approximation to a purely conductive system. Although this method has become the standard method for determining effective thermal conductivity to be used in surface energy balance models of sub-debris ice ablation e.g., our analysis demonstrates several ways in which the derived κ is sensitive to numerical errors related to instrumental setup and analysis choices, even when solving for a pure-conduction case. The method has regularly been used without considering these error sources, making it difficult to robustly compare published values derived using this method.

To address this, we provide an open-source tool (https://github.com/calvinbeck/TC-DTD, last access: 30 June 2025) where researchers can investigate the combined opportunities and limitations of applying the method by to glaciology and beyond. We hope this facilitates more consistent and rigorous experimental design in future field measurements determining debris thermal properties by allowing users to simulate their own artificial data, which most closely approximate their planned field site, and repeat all our analyses presented here with their own artificial or field datasets.

In this paper, we used this tool to provide illustrative examples of the magnitude and tendencies of the systematic errors associated with individual instrumental and analytical choices. Based upon our findings, we provide a set of best-practice guidelines (Appendix ) to minimize systematic errors in applying the method of . While recent publications highlight limitations of the simplest deployment of the heat diffusion equation in natural debris layers due to the role of non-conductive processes and internal debris stratification , our analysis and best-practice guidelines show the sampling strategies that will yield the best results, provided that the temperatures underpinning the analyses demonstrably sample conditions that closely approximate a homogeneous conductive system. Our analysis also highlights that it is challenging to interpret derived debris thermal properties if the sensor and the analysis system are not reported and accounted for. In the light of this, we encourage more rigorous reporting of implementation strategies and uncertainty in order to facilitate cross-comparison of reported results.