Measuring trends in MAGT is one of the most common ways to quantify permafrost change. The measurement of ground temperatures at or near dza is established practice and effective at smoothing out inter-annual and fine-scale spatial variability. Typically, permafrost warming trends are calculated from sensors 10–25 m deep using linear regression of temperature time series or Bayesian methods .

We distinguish between MAGT measured at a specific fixed depth (denoted Td at depth d) and mean annual ground temperature measured at the true dza, the position of which is dynamic over time. We denote this latter metric Tza and discuss it below.

Observed rates of MAGT change are typically lower than 0.3 °C decade-1 for warm permafrost (>-2 °C) and lower than 1 °C decade-1 for cold permafrost (<-2 °C) . estimated average warming rates as 0.39 °C decade-1 in continuous permafrost, 0.20 °C decade-1 in discontinuous permafrost, and 0.29 °C decade-1 globally.

MAGSTs provide information on changes to the upper boundary of a soil column, which propagate downwards to affect permafrost. Because they do not require costly drilling, MAGST measurements can be collected more economically than thermistor strings in boreholes, and enable denser data collection e.g.. We calculate MAGST using model output at 0.1 m.

Systematic reporting of MAGST is less common than for MAGT or active layer thickness at permafrost research sites . In Switzerland, MAGST warming rates over permafrost have been estimated as 0.4–0.6 °C decade-1 1998–2022, . On the Tibetan Plateau, MAGST warming rates were estimated as 0.16–0.60 °C decade-1 1980–2015, and 0.60 °C decade-1 on average 1980–2007,. Average trends across China (including non-permafrost regions) were reported as 0.20 ± 0.02 °C decade-1 1956–2022,.

ALT is one of three products used to monitor the permafrost Essential Climate Variable . Strictly speaking, its vertical extent is defined by the the greatest thaw penetration depth in a year, the maximum extent of the zero-degree isotherm is sometimes used as a thermal approximation of the active layer . The thermally defined ALT has the advantage that it can be estimated using ground temperature records. This is often done by interpolating the two observations above and below the active layer . However, found that the lowest error (equal to about 20 % of node spacing) was obtained by extrapolating from above using the two lowest sensors in the active layer, and by using instantaneous profiles rather than annual envelopes. The worst results were obtained when extrapolating from measurements below the active layer. Alternatively, some authors suggest fitting exponential curves to the ground temperature envelope .

To estimate active layer thickness using the “extrapolation from above” method described by , we first calculate annual maximum temperatures at each sensor depth and identify the index (n) of the deepest sensor above permafrost where temperatures exceed 0 °C during the year.

Next, for each time step we determine the extrapolated thermal gradient using the deepest (zn) and second-deepest (zn-1) sensors above the isotherm 1dTdz(t)=T(zn,t)-T(zn-1,t)zn-zn-1. The depth intercept of the zero-degree isotherm is then calculated as follows: 2zT=0(t)=zn+0-T(zn,t)dTdz(t). Next, the process is repeated, but using the deepest sensor above permafrost with minimum temperature below 0 °C. This is repeated for all time steps in the year, and the active layer thickness is chosen as the greatest depth intercept. To estimate ALT by interpolation (ALT^↕), we use a similar approach, but calculate the thermal gradient using the nth and (n+1)st sensors. Additionally, the position of the permafrost table can also be estimated by using only the maximum temperature in Eqs. () and (). Finally, we average the two estimates of ALT. Long-term trends in ALT provide a minimum estimate for how much permafrost is lost due to thaw, but are insensitive to any additional ground lost due to thaw subsidence .

Observed ALT trends are typically low. Data from 109 active layer monitoring sites shows rates of increase generally between -0.2 to +0.4 m decade-1 depending on the region, but up to +3.9 m decade-1 in the Swiss Alps .

In cold permafrost, our estimate of the ALT coincides with the depth to the permafrost table, or top of permafrost (TOP). However, if a supra-permafrost talik develops, this is no longer true. In that case, TOP continues to deepen independently of ALT causing the two metrics to differ by an amount equal to the talik thickness. Here, we neglect any differences between the thermal (cryotic) and physical (freeze-thaw) definitions of talik or active layer boundaries .

It is possible for TOP to change independently of ALT, so we consider TOP as an additional metric. To calculate TOP we use Eqs. () and () to estimate the position of the zero-degree isotherm.

While change to ALT is commonly used as an indicator of thaw , it does not include any information about the duration of thaw. Both factors have implications for biological activity, terrain hazards, and carbon cycling. define (D‾), a time- and depth-integrated value as 3D‾=1365∬H(T(z,t))dzdt, where T(z,t) describes the mean soil temperature at an arbitrary depth (m) and time (d) and H is the Heaviside step function. For each year, we select integration bounds from the ground surface to the top of permafrost, and from 0 to 365 d (simulations do not consider leap years). Practically, for daily observational data, we use a discretized form of this equation 4D‾=1365∑j=1365∑i=1NH(τ(zi,tj))Δz. Here τ is a linearly interpolated function of temperature with depth, discretized into N elements, and Δz is an arbitrary thickness increment, for which we use 0.01 m. Note that the use of interpolation rather than extrapolation across the zero-degree isotherm at the thaw depth may introduce a slight bias here .

The depth at which the annual temperature amplitude is completely attenuated is denoted (dza). A cutoff value of 0.1 °C in amplitude is typically used as a practical threshold.

Intuitively, we expect that dza will be shallower for boreholes in which greater amounts of thaw take place, and in ice-rich boreholes where seasonal freezing and thawing decrease the apparent thermal diffusivity. This latter case indicates a greater potential for thaw. Over longer periods of time, increasing dza should be expected to indicate a change to an increasingly latent heat-dominated system.

To estimate dza we use the method described by . Additional considerations for this calculation are described in Appendix . First, the annual amplitude Az at each observation depth, z, is calculated as half the difference between the annual minimum and maximum. Next, we fit coefficients x0 and x1 using least squares, 5ln⁡AA0=x0-x1ZZ0. where, A0 and Z0 are units of 1 °C and 1 m. For this step, we restrict the data by only using depth and amplitude pairs where A>0.5°C so that temperature trends at depth do not add noise by inflating the amplitude. Finally, we calculate dza using the fitted coefficients for each year of data 6dzaZ0=x0-ln⁡(0.1)x1. We are not aware of any long-term observations tracking dza change over time. For the most part, dza is treated as a static property to characterize or compare sites.

Although MAGT is defined as the temperature at the depth of zero annual amplitude, it is typically recorded at a fixed depth below dza in the range of 10 to 25 m.

We calculate a dynamic mean annual temperature at the depth of zero annual amplitude (Tza) whose position corresponds to the best estimate of dza calculated above. After determining dza, we interpolate the MAGT linearly to that depth to obtain Tza. Although this measurement technically describes a single depth, we consider it a borehole-aggregated metric because it is uniquely defined for any location, there are no choices to be made about its position, and it derives from more than one sensor.

As will be shown, Tza is the first ground temperature metric to become isothermal, and therefore can be used as a way to further classify borehole behaviour.

For clarity, we will use the acronym MAGT to refer to mean annual ground temperatures at a fixed-depth, and refer explicitly to Tza as necessary.

Monitoring ground temperature at a single depth provides a convenient summary statistic. However, it neglects data at other depths making it susceptible to sensor damage. Furthermore, the effect of unequal sensor depths between sites complicates meaningful comparison of warming rates between sites.

To investigate an alternative, we define the thermal integral (τ‾) as the depth-integrated temperature evaluated between a near-surface sensor (z1) and an arbitrary depth (zn) of the mean annual temperature profile. This value is normalized by the integration depth, effectively yielding a mean column temperature. 7τ‾znz1=1(zn-z1)∫z1znT(z)dz. Trends in the thermal integral correspond to mean column warming rates. Our hypothesis is that the trends will be more comparable between boreholes with unequal sensor spacing provided that the integration depths are similar.

Practically, to estimate τ‾ from observations at discrete depths rather than from a smooth function T(z), we use the trapezoidal rule: 8τ‾znz1≈12zn-z1∑i=1n-1T(zi)+T(zi+1)zi+1-zi. where zi is the depth of the ith sensor.

Our processing uses daily T(z,t) data; sub-daily values are aggregated to daily averages first. For each annual period evaluated with TSP metrics, the number of daily values is reported and we exclude years with less than 95 % data completeness. We consider T(z,t) data to be reported for constant depths relative to the ground surface. In practice, ground subsidence may result in sensor depths changing over time. Multi-year trends in TSP metrics are calculated using ordinary least squares regression. For the model experiments, we calculate rolling trend windows of 5-, 10- and 20-year duration, representing data durations commonly available from permafrost monitoring today.

We evaluate the effect of observation depth on MAGT trends and compare it to the effect of total integration depth on trends in τ‾. For each trend window, the maximum difference in change rates of all depths between 10 and 20 m is calculated. For MAGT, this is done in two ways: first, by including all data, and again excluding any depths at which permafrost has degraded completely. Finally, empirical cumulative distribution functions are generated to quantify the effect of observation depth.

To simulate transient ground temperatures in a one-dimensional configuration, we use the model FreeThawXice1D – a numerical model of heat conduction with freezing and thawing in soils without water flow . Energy conservation and model convergence is guaranteed at large time steps, making it suitable for ice-rich simulations requiring long spin up. Liquid water content, and thus freeze-thaw energy, is represented in the model using temperature-dependent soil freezing characteristic curves (SFCC). For our application we used the van Genuchten SFCC parametrization presented in . FreeThawXice1D represents the effects of subsidence caused by ground-ice loss and accurately tracks 0 °C isotherms via local mesh refinement. Although advective water transport is not included in the governing equations, liquid water volume caused by excess ice melt and the associated latent heat is removed directly from the simulation.

We use GlobSim to generate meteorological forcing data from the ERA5 reanalysis to drive the model at the upper boundary. This tool streamlines the download of reanalysis data, interpolates grid cells to point-scale to make data suitable for 1D simulation, standardizes units and time steps, and performs heuristic downscaling to account for terrain and other local effects. Future conditions are simulated by repeating several years of data with an added linear warming trend. More details on the simulation and the evaluation of resulting temperatures are presented in Appendix .

Simulation results are near-perfect data, with limitations related to model assumptions and numerical imprecision. To assess the sensitivity of TSP metrics to the quality limitations of real measurement systems, we degrade simulation results accounting for the accuracy (bias), drift, and precision (noise) of typical sensing systems (Appendix ). Based on typical performance characteristics, we create two additional data sets (Table ) that emulate an excellent (Q1) and a good (Q2) commercial measurement system. The original model output is denoted (Q0).

For each time window, TSP metrics are regressed against the change in heat content (sensible, latent, total). The strength of the relation – represented by the regression slope – is normalized by the standard error. The resulting values for the t-statistic, here analogous to a signal-to-noise ratio, are summarized in a histogram. Histograms are displayed to distinguish regression results by the statistical significance (p<0.05) and sign of the resulting trend.

The distribution of t-statistics across all trend windows is used as a measure of the reliability of a TSP metric as a way to detect heat-content changes in the ground. Positive slopes indicate true positives (sensitivity when expressed as a rate) and negative slopes indicate false positives (specificity when expressed as a rate). Spuriously significant regression can occur given the non-stationarity of our time series. Because all simulations are subject to similar levels of non-stationarity, the effects on relative efficacy of metrics is likely small.

As will be shown below, Tza is the first mean temperature metric to reach isothermal behaviour near 0 °C. We investigate whether this metric can be used to partition the simulations into two distinct phases of thaw. For this particular experiment, we exclude bedrock simulations which have no appreciable ice content. For each simulation, we identify the breakpoint in slope (e.g., Fig. i) visually. Once the date is established, the mean trend in Hs and Hl are computed for the period before and after. Finally, the relative change in slope is computed as: 9Mean Trend Ratio=maftermbefore, where m is the corresponding mean trend. We also aim to develop a way to automatically determine the date of the breakpoint. For this, we use a quantitative threshold of four consecutive years of less than 0.1 °C of change in Tza. Finally, for comparison, we perform the same analysis using the commonly-used criterion to distinguish warm permafrost based on MAGT. That is, when MAGT of permafrost is between -2 and 0. We define the boundary as when T20 reaches -2 °C for the first time; in our simulations this transition happens before the Tza breakpoint.

We present the results of two simulations as exemplars of the simulation ensemble and the behaviour of TSP metrics. The first simulation (Fig. a–f) is for a cold soil column containing no excess ice with an initial MAGT near -6 °C. Heat gain throughout the simulation is predominantly through sensible heat and approximately linear in time. All metrics exhibit a similarly linear response.

The second illustrative simulation (Fig. g–l) is for a column containing excess ice with an initial MAGT of around -3 °C. The loss of excess ice causes a decreasing ground surface height accompanied by an approximately linear increase in heat content. Unsurprisingly, the warming rate of MAGT is strongly dampened late in the simulation, demonstrating the challenge associated with interpreting this metric in the presence of latent heat transfer.

The ALT trend can be positive or negative in warm permafrost. When a talik develops, this causes a discontinuity in ALT followed by a reverse in the direction of the trend (Fig. i). This generally occurs in three stages: (1) An initial decoupling of ALT from the permafrost table when the ground no longer refreezes completely and a residual thaw layer persists. (2) A period when ground temperatures near 0 °C and multi-year temperature variability cause ALT to vary strongly, with occasional large jumps. (3) A more stable period (not shown in Fig. g–l) during which ALT decreases once the talik is well developed. ALT thinning has also been observed in the field .

In contrast to ALT, TOP declines monotonically during the warming period, indicating the loss of permafrost volume with less ambiguity as it is not reversing with the formation of a talik. Similarly, D‾ is monotonic in its reaction to sustained warming. For the remaining analysis, we use TOP as the metric of choice, intuitively summarizing changes to the upper boundary of permafrost. Before the formation of a talik, changes in TOP, ALT, and D‾ are very similar. This is not surprising given the considerable overlap between the definitions of these three metrics.

In warm simulations, we see that ALT, D‾, and TOP change gradually for some time and then the rate of change begins to increase dramatically. This regime change between cold and warm permafrost has been previously described .

Visually, MAGT behaves most similarly to borehole sensible heat content until it nears 0 °C, when warming rates decline in simulations with ground ice. Trends in Tza are similar to those of MAGT. However, in warming simulations with ice, Tza is the first depth to become near-isothermal because dza becomes shallower as phase change takes place and temperature fluctuations are damped by latent heat transfer. The magnitudes of simulated trends in Tza and MAGT are similar.

In cold or moisture-poor permafrost, MAGT and Tza both generally follow a similar pattern as Hs. It is the effect of latent heat near 0 °C that dampens the warming trend in other cases. Because Tza is the first metric to reach isothermal behaviour, it is a good candidate for an additional metric. Tza also does not involve an arbitrary choice of depth but is derived from the behaviour of all observations. Otherwise, observations at a fixed depth, as commonly used for MAGT, are preferable to Tza because they are simpler to produce and understand.

When simulations are partitioned into warm (MAGT ≥-2°C) and cold (MAGT <-2°C) permafrost, the mean Hl trend is significantly greater in warm permafrost than in cold permafrost (Fig. ). This is not true of the Hs trend, for which the 95 % confidence interval of the mean trend ratio includes 1. On the other hand, when using a visually picked breakpoint in Tza (e.g., Fig. i) to distinguish two stages of thaw, we see that Hl trends are significantly greater in the later stages of thaw and Hs are significantly reduced. This is also true when the Tza breakpoint is estimated quantitatively rather than visually.

Another advantage of the Tza classification is that, because no specific depth must be chosen, the classification is valid for the entire borehole. Since dza moves towards the surface, it also more likely that when the transition from one stage to another takes place, the sensors will be placed deep enough to measure this.

While we generally expect Hs trends to decrease and Hl trends to increase as icy permafrost warms, the choice of where to split up different stages of thaw can result in differences when comparing trends in the two stages. Soil freezing characteristic curves generally start increasing around -2, so it is intuitive that the warm-cold permafrost distinction captures the increase in Hl trends. However, while the distinction between warm and cold permafrost may be effective at identifying when ground ice begins to melt, the breakpoint in Tza signals more of a regime shift at which point the warming trends and concomitant Hs trends become significantly diminished, while the changes in Hl also continue to increase.

In all simulations, trends in MAGST resemble those of total sensible heat. MAGST shows strong interannual variability. In our simulations, MAGST trends are not visibly affected by phase change near 0 °C even when warming at depth decreases significantly. MAGST trends were roughly 0.40 °C decade-1.

At low temperatures, the difference between MAGST and MAGT or Tza remains stationary slightly above 0 °C. When the ground warms to near 0 °C, MAGT trends within permafrost with some ice reduce in magnitude, while MAGST trends are unaffected. This period is also accompanied by inflection points in the borehole latent heat trends. Therefore, MAGST-Tza can become an additional TSP metric where an increasing trend indicating increased latent heat uptake, while deeper observations remain isothermal at 0 °C.

Conceptually, treating MAGST change as a proxy for permafrost change is not entirely appropriate because MAGST is observed in the active layer and changes in permafrost respond with some delay to surface change. However, MAGST may still be useful as an indicator of subsurface heat gain for several reasons: (1) Averaged over longer periods, the impact of lag time is reduced. (2) MAGST is minimally affected by latent heat; in our simulations, even during periods of significant phase change, we do not observe a reduction in MAGST trends. (3) Many thaw phenomena occur closer to the ground surface than to a depth of 10 m and MAGST can give more direct insight into the lateral variation of temperature and temperature trends that may drive those changes than deeper, and much more costly, MAGT. The strong lateral variation of MAGST can be addressed with dedicated sampling e.g.,.

We expected that dza would provide additional information about changes in Hl because the attenuation of the annual temperature wave is governed in part by phase change affecting the apparent thermal diffusivity.

Our simulations show that this this relationship is not linear: large increases in dza can occur even when there is relatively little change in Hl and large increases in Hl occur even with small changes in dza (Fig. ). Therefore, we interpret changes to dza as more accurately reflecting changes in the annual range of liquid water content, vertically integrated from the surface downward. However, this can still provide information on liquid water content in some cases. If we assume that for warming permafrost, an increase in temperature range is the result of a greater maximum annual water content (and not a decrease in the annual minimum), then the annual mean water content must also increase.

Conversely, without change in Hl, we do not observe changes to dza: in all simulations, bedrock profiles – which contain no moisture – exhibit no long term trend in dza regardless of temperature.

The relationship between patterns of dza change and column heat gain is distinctly temperature-dependent. In cold permafrost, dza becomes more shallow as permafrost warms. At a certain point, a borehole becomes completely isothermal and dza stays relatively constant. Following this, the ALT becomes more shallow while dza and TOP deepen. The exact point at which the sign of dza changes during warming depends on the configuration of climate and ground simulated, but it should be expected when Tza is close to 0 °C and when dza is close to ALT. In icy soil, we found dza to reach as shallow as 3.1 m before deepening in the late-stage warming phase. The shallow upper limit reached by dza in Fig. i reflects the choice of cutoff used to define zero amplitude (0.1 °C in this case) and a smaller cutoff value would result in greater sensitivity to change for shallow dza values because it would take longer to reach the limit.

In contrast, for bedrock simulations the dza is stationary about some mean value, with fluctuations due to interannual changes in surface amplitude.

On final consideration for interpretation is that the metric only responds to changes in liquid water above dza whereas large increases in latent heat (e.g. as seen in Fig. h) may be due to melting taking place deeper in the ground.

Despite its complexity, dza can be used to infer changes in Hl and freeze-thaw behaviour and its behaviour is qualitatively distinct from ground temperature, providing unique insights.

The mean annual borehole temperature τ‾ over a chosen integration depth exhibits behaviour that is often intermediate between the rapidly fluctuating MAGST and the much steadier MAGT. Qualitatively, τ‾ time series correlate with changes in Hs. During near-isothermal periods, τ‾ often resembles Hs more than MAGT. Although no published τ‾ trends exist, the magnitude of modeled trends can be reasonably compared to MAGT or MAGST trends because τ‾ is normalized to represent an average borehole warming.

The balanced influence of temperature trends over a range of depths makes τ‾ less susceptible than MAGT to producing undetectable trends when boreholes become near-isothermal. This comes at the cost of greater interannual variability because of the incorporation of near-surface temperatures.

In simulations with appreciable ice content, we observe a transition from moderate to accelerated permafrost degradation as the permafrost becomes very warm (e.g., ca. 2060 in Fig. g–l). This occurs after the permafrost becomes sufficiently warm, coinciding with the reversal of the dza trend.

We interpret the accelerated degradation to be caused primarily by the development of isothermal conditions which limits the re-establishment of a temperature gradient in winter to draw heat out of the ground . This phenomenon has been discussed in the context of talik formation as a driver of tipping-point behaviour in peatlands and discontinuous permafrost . Because our simulations do not consider water transport out of – and the subsequent drying of – the active layer, any melted ice persists as water; we expect this would accentuate the inhibition of freezing due to latent heat.

The differences in 10-year MAGT warming rates between 10 and 20 m are positively skewed (Fig. a) with modal values of less than 0.1 °C decade-1. Nevertheless, across all soil types, 50 % of observation windows have more than 0.23 °C decade-1, 10 % have more than 0.60 °C decade-1, and 5 % have more than 0.72 °C decade-1. Very low trend differences are attributed to low warming rates associated with warm, near-isothermal conditions in icy materials.

When trend differences are normalized their distribution becomes less skewed (Fig. b) and the effect of ground materials is reduced. 50 % of windows have MAGT trend differences greater than 73 %, 10 % have differences greater than 291 % and 5 % have differences greater than 555 % .

In τ‾ trends, the impact of different integration depths and the effect of terrain type is weaker than in MAGT trends (Fig. c). However, icier locations still exhibit reduced differences. Compared to MAGT, the distribution of values is less positively skewed. The variability at 50 %, 90 % and 95 % probability is 0.17, 0.45, and 0.53 respectively.

When τ‾ trend differences are normalized, there is virtually no difference between different ground types (Fig.  d). The differences at 50 %, 90 % and 95 % probability are 44 %, 156 %, and 319 %, respectively.

In summary, a difference in observation depth can introduce meaningful differences in the revealed decadal warming trends (Fig. ). It is greater than 0.23 °C decade-1 in 50 % of our simulated trend windows across all ground types. To contextualize this, report global average warming rates of 0.39±0.15 °C decade-1 in continuous permafrost and 0.20±0.10 °C decade-1 in discontinuous permafrost, based on sensors that are between 5 and 24.5 m deep. The magnitude and uncertainty of warming rates presented there are commensurate with the uncertainties due to sensor position we show.

The differences of τ‾ trends caused by integration depth are lower than for MAGT observation depths (Fig. c, d). Additionally, sites can be more meaningfully compared even when the sensors may not be at the same depth because τ‾ can be interpolated to a common depth where sensors at the same depths are not available.

For each TSP metric and heat content variable (Hs, Hl, and Ht), we evaluate the signal to noise ratio (SNR) and reliability of the relation. We use the regression t-statistic (i.e. coefficient estimate normalized by the standard error) and significance to measure SNR; larger percentages of significant t-statistics provide evidence of high SNR. Note that this does not tell us about the strength of the relation, but rather how much we can trust the direction and existence of change in borehole heat based on an observed change in a metric. We use the distribution of positive vs negative values to measure directional reliability; more consistently positive or consistently negative coefficients provide evidence for greater directional reliability.

For each 10-year moving window in the simulation results, we calculate a linear regression between the metric and the heat content variable. The t-statistics and levels of significance are then aggregated and summarized by the percentage of significantly positive and significantly negative coefficients. This is repeated for different levels of sensor quality (Q0, Q1, Q2) and different observation lengths (5, 10, and 20 years) to describe the impact of these variables on reliability and SNR.

A detailed example for a single metric and heat variable is shown in Fig. . In this example, most correlations are positive. However, in as many as 2 % of 5-year windows and 4 % of 20-year windows, this correlation is negative (decreased reliability). Results for all simulations are summarized in Table  and the remaining figures are included in the Supplement (S1).

The impact of sensor quality on SNR is within a few percentage points for most TSP metrics with a few exceptions. The SNR of TOP is decreased with decreased sensor quality (most notably for Q2) for all heat variables and for all window sizes. The SNR of MAGT (T10, T15, and T20) for Hl is also appreciably decreased with decreasing quality for 5-year windows, but not for longer window sizes. Additionally, the impact of sensor quality is greatest for larger observation depths (T20).

Reliability and SNR for dza is low overall, but when observation windows are split into warm and cold scenarios according whether tza is above or below -0.5 °C, the SNR and reliability both increase. For cold permafrost, dza achieves high SNR with up to 75 % significance for Hl in 20-year windows. In warm periods, dza achieves values up to 75 % for Hl in 20-year windows. When averaged over all time windows and levels of sensor quality, metrics describing annual temperature averages (MAGT and τ‾) rank among the highest for both Hl and Hs.

Without exception, the SNR of the metrics decreases for shorter temporal windows. Most metrics have scores of less than 30 % for 5-year windows. However, TOP-Hl is the exception, with a lowest value of 57 %. For short windows, τ‾ has higher SNR as predictors of Hs and Hl than single-depth temperature measurements, these also score more than 50 % in 5-year windows.

The SNR of most metrics is larger for Hs than for Hl but TOP-Hl is the exception to this. Interestingly, values for Ht are not simply weighted averages of Hl and Hs: the SNR of Ht is consistently greater than either Hl or Hs for T10, T15, Tza; smaller for MAGST; and intermediate between the two for τ‾100, τ‾150, and τ‾200. For dza, Ht is intermediate for 10- and 20-year windows, and greater for 5-year windows.

We interpret a higher SNR (larger number of significant values) to indicate that an increase in a given metric more reliably indicates an increase in the heat variable of interest over a randomly selected observation window. However, the time- and location-dependent relation between the metric and heat content means that we do not interpret these results in terms of the strength of the relation. Also, because we only simulate a warming trend, we generally interpret reversed correlations as a consequence of one of three factors: interannual variability, temporal window mismatches due to time lags, and errors due to decreased sensor quality. For example, the negative correlations in Fig.  would not be caused by an increase in the metric when Hs decreases within the window, but rather by a decrease in the metric (for one of the three reasons stated above) as the Hs increases.

In general, we see that our sensor imperfection model results in only a small effect on the effectiveness of metrics as predictors, except for short observation periods. For MAGT, we expect the effect of normally distributed noise is likely erased in individual sensor records because they are averaged over the year. We expect the bias and trend in Q2 also contributed to the lower performance.

Averaging multiple sensors in the calculation of τ‾ would also reduce the impact on the metric of sensor drift and bias in any individual sensor. On the other hand, the calculation of annual maxima required to estimate TOP relies on single data points and involves no averaging – this would be directly affected by sensor bias and noise. This would therefore more strongly affect metric performance by amplifying, rather than dampening, the effect of deviations from the perfect model data. dza also requires additional calculations, but the performance of this metric is not similarly affected; the method by which it is calculated also averages amplitude data from multiple sensors.

The large impact of decreased sensor quality on T20 which shifts the results towards a negative correlation may illustrate a weakness of the methodology which is that we only perform a single realization of the randomized sensor quality model. In this case, it may be that the realization produced significantly more negative trends at that depth. Alternatively, it is possible that the sensor bias and drift have a larger effect because the signal magnitude is much smaller at that depth.

dza exhibits a different signal than other metrics (Fig. c, i). However, the reliability is low, especially for short observation windows (Table ). We attribute this to the noisy nature of the metric, the period of stagnation prior to the reversal of the trend and the temperature-dependence of the metric overall. The higher interannual variability means that longer temporal windows are needed to obtain clear trends. More importantly, the results highlight the importance of defining a precise cutoff at which the behaviour of dza reverses.

τ‾ is a better indicator of Hs and Ht change over shorter observation periods than MAGT (e.g T20,T15,…). This is in spite of the greater interannual variability of τ‾ caused by the inclusion of near-surface measurements. Intuitively, we expected this variability to make trends harder to predict. We believe that this effect is outweighed by the diminished impact of latent heat in warm conditions, which causes MAGT trends to become greatly diminished. It is also interesting to note that although qualitatively τ‾ behaves similarly to a superimposition of MAGST and MAGT, both of those metrics are more strongly affected by shorter observation windows. We interpret this to mean τ‾ benefits from being depth-integrated, which more closely resembles how Hs would be calculated if heat capacities were known.

MAGT is traditionally the most widely used metric to report permafrost change. It is a direct measurement of the permafrost and it reveals how quickly the permafrost thermal state is changing at a specific depth. Trends correlate well with column heat contents, but the relationship is time-,depth-, and location-dependent. As a consequence, comparing (between locations or times) and aggregating MAGT trends usually involves blending dissimilar physical processes, producing a number with no meaningful connection to either heat gain or ice loss. Therefore, MAGT trends are most useful for detecting the presence and sign of a trend. The metric is also prone to periods of imperceptible change, making quantification difficult.

τ‾ is a good indicator of changes in borehole sensible heat (Hs), already over shorter time periods, outperforming MAGT in this regard. It is less prone to stagnant behaviour, making trend detection more reliable. It also takes advantage of all available sensors. Practically, it allows for compensation of unequal monitoring depths through standardized integration, improving comparability between boreholes. While being subject to the same limitations as MAGT in principle, the confounding effects of latent heat on the ability to compare or aggregate trends are reduced for τ‾.

Despite the common definition of MAGT as the temperature at dza, we find little support for using Tza as a TSP metric to monitor permafrost change. In addition to the greater complexity of calculating this metric, it is the first one to become isothermal in the simulations as the ground warms. It is also less predictive of column heat content compared to MAGT or τ‾ (Fig. ). However, it may be useful for (1) indicating near-isothermal behaviour in a borehole because it will be the first metric to achieve this, and (2) as a classifier for late-stage permafrost thaw; in our simulations, the point at which Tza reached approximately 0 °C coincided with the transition in sign of the dza-warming relation.

TOP is a straightforward metric for temperature-based monitoring with a clear physical meaning, describing changes in permafrost thickness. It behaves similarly to ALT or D‾ but with some clear advantages. The definition of TOP is purely thermal and less ambiguous than that of ALT, which is alternatively defined based on temperature or phase state . Additionally, the direction of ALT trends reverses after the formation of a supra-permafrost talik while TOP remains monotonic. Although D‾ may have inherent meaning, it is qualitatively similar enough to TOP that there is little to gain by including both in a parsimonious set of metrics. Adopting TOP as a primary metric also does not diminish the utility of existing ALT records and monitoring programs; in the absence of a talik, TOP can be considered equivalent to the purely thermally defined ALT.

Overall, TOP is a good proxy for Hl even in short time windows, even though the exact correspondence is unknown without quantification of ice content. In our simulations, it increases slowly in the early stages of warming and then accelerates in later stages of warming. TOP has an easily understood physical meaning, describing a change directly affecting permafrost thickness. However, the accuracy with which it can be estimated is more strongly affected by sensor spacing e.g., and quality than other metrics. Some confounding may originate from subsidence caused by a rising permafrost base.

dza can be used to infer changes to Hl, but it provides an incomplete picture: phase change below dza is invisible, the upper limit of dza is affected by the choice of cutoff for “zero amplitude”, and the sign of the correlation between dza and Hl is temperature-dependent. However, the behaviour of the metric is qualitatively distinct from ground temperature, which can be helpful in interpreting permafrost changes.

Given the above results, we recommend TOP, dza, and τ‾ in addition to MAGT as a parsimonious set of metrics for quantifying permafrost change. MAGST can be considered as an additional metric that, while not observed in permafrost, provides the clearest measure of how climate or disturbance drive changes at depth.

Simulations represent an ideal scenario with which to evaluate the metrics but they can only provide some of the complexity of the real environmental system. In practice, heterogeneous subsurface characteristics will mean observations at an individual depth are less well representative of permafrost above and below. Spatially variable ice content will amplify or dampen the response of these metrics to thaw, and this will affect rates of change. Instrumentation density has a major impact on the accuracy and precision of many metrics e.g., and will affect the resolution with which some of the metrics can be calculated.

Representing change as a single per-borehole statistic is challenging because of differences in data completeness and measurement density in boreholes. Observational records in boreholes differ in their vertical extent and duration. Averages can therefore mask locations with strong change when boreholes are very deep or have many sensors or exaggerate measures of change in boreholes with few sensors.

Our use of FreeThaw1D means we do not consider spatial effects or the soil water balance. Taken together, these limitations could affect our results and the applicability of the metrics.

Advective water transport could lead to local redistribution of heat that does not correspond to an overall net change in the larger area. This could lead to an overestimation of change by the metrics because they only measure local changes.

In our simulations, there is also no water flow within or out of the simulation with the exception of melted excess ice. Temporal variability in near-surface moisture can influence dza by altering the soil’s apparent thermal diffusivity independent of long-term change. The model’s fixed moisture conditions may smooth out these variations, leading to more stable dza trends than would occur in reality. This may cause dza to appear less sensitive to changes in Hl than it truly is.

This lack of water flow also means that during late-stage thaw in warm, icy simulations with a supra-permafrost talik, the dza becomes shallower than the top of permafrost, and is kept shallow by the buffering effect caused by freeze-thaw cycles of supra-permafrost water. It is plausible that an overall loss of moisture above the permafrost table during this period could prevent the dza from ever reaching the top of permafrost, or alternatively accelerate the late-stage shallowing of dza.

Despite these potential biases, our focus on the relative performance across metrics helps control for systematic errors. For example, if all metrics are similarly affected by missing processes advection, their relative performances remain valid.

Our results show that increasing MAGT is consistently indicative of increases in column Hl and Hs even in the presence of distortion from sensor noise, bias, and trends. However, the exact quantitative relationship is location- and time-dependent. In this regard, it may be the direction of the trend from which meaning can be most unambiguously derived. However, given that the vast majority of permafrost boreholes are warming , this may be an almost trivial conclusion.

Our results suggest that MAGT can contribute to a descriptive picture of permafrost change, but only when used in combination with other temperature-derived metrics. In isolation, the magnitude of MAGT trends can be attributable to both the effect of latent heat and the intensity of surface warming. With a more complete suite of indicators, the effect of latent heat can be made more explicit and permafrost change can be quantified more reliably.

Our results highlight the challenges of comparing trends between boreholes or regions. In our simulations, the behaviour of each metric generally follows a similar trajectory. For MAGT, this is: rapid warming, reduced warming, no warming, and finally rapid warming if thaw progresses to the observation depth. Most importantly, we do not change the magnitude of the warming trend at the surface, yet the trends measuring permafrost response show a great deal of variability across simulations and at different points in time. Therefore, in our experiments, MAGT tells us more about the configuration of the system and the stage of thaw than about the intensity of the warming trend. In this regard, the resolution of MAGT as a climate indicator may indeed be limited to the trend direction discussed above. In reality, locations will be subject to different trends in air temperature, snowpack and vegetation; all of which will affect warming rates at depth. However, this signal will be superimposed on the purely permafrost-dependent variability which, as we have shown, is on the same order of magnitude as observed trends.

Our results also provide some guidance on how to make MAGT trends more interpretable. If we are interested in understanding where permafrost change is happening faster or more slowly, groupings should account for both the ground conditions and some other indicator of the stage of warming. We show that Tza is a good candidate for this. The breakpoint in the Tza trend is able to separate warming into two stages with statistically distinct trends in Hs and Hl. The performance of the Tza breakpoint is superior to the traditional classification of warm permafrost (MAGT >-2 °C) which does not discriminate Hs trends in the two stages. This is significant because Hs trends are strongly linked to temperature. This approach also has the advantage that Tza often requires only relatively shallow observations because dza becomes shallower as the borehole warms.

Current comparisons and aggregations of MAGT trends are most commonly grouped by region or permafrost zone. This partially accounts for temperature and warming stage, but this is hampered by uneven and biased spatial sampling, particularly in the discontinuous permafrost zone where permafrost temperature may still be strongly affected by surface conditions. In these cases, more meaningful comparisons could be made by further subdividing datasets according to the ground conditions on a spectrum of ice content. However, such fine-scale subdivision may be limited by general lack of ground temperature data and result in too few boreholes in each category to meaningfully compare.

Despite its application as a climate indicator, permafrost temperature change is complicated by the effects of latent heat and we should expect permafrost warming rates to diminish as boreholes approach 0 °C. One possible solution is to consider using only boreholes with very little ice content, as would be expected in certain kinds of bedrock. Such an approach has been suggested by , who recommend monitoring in exposed bedrock to obtain the most direct climate signal from ground temperatures. The PACE project provides another example where boreholes in bedrock on mountain summits or plateaus were used for permafrost and climatic monitoring . In general, however, bedrock sites remain underrepresented in much of the permafrost monitoring data because of difficult drilling conditions, and because permafrost impacts are often caused by thaw of ice-rich ground.

also show observations of reversing dza behaviour and explain this behaviour with an equilibrium model (i.e., analytic heat conduction equation with a sinusoidal surface temperature variation). However, a limitation of this approach is that the transition in correlation sign cannot be examined – their warm, deep-dza simulation only exists as a fully thawed profile rather than one with dynamic behaviour. Our transient simulations show that this reversal can take place over several years during which the dza changes very little and the sign of the correlation between dza and air temperature is undefined.

This distinction is important because other studies report unidirectional (and conflicting) relationships between warming trends and dza e.g.. An important next step in this approach to monitoring will be developing additional criteria to more clearly delineate when a positive- or negative- correlation should be assumed.

We observe accelerated degradation (lowering of TOP) in warm permafrost as it becomes isothermal and develops a supra-permafrost talik. Similar discrepancies have been described elsewhere. On the Tibetan Plateau, ALT deepening rates have been shown to be greater in warm permafrost than in cold permafrost . In discontinuous permafrost and peatlands, taliks have been described as giving rise to tipping-point behaviour and kicking off rapid degradation e.g.,. We observe this phenomena across different stratigraphic types (Table ) and climates, representing a wide range of permafrost conditions, suggesting a greater abundance of this tipping point behaviour than may have previously been considered.

Certain TSP metrics may offer insight as to when this accelerated degradation may occur. In our simulations, it is preceded by the breakpoint in Tza and the halting of the dza shallowing trend. These can act as quantifiable indicators that permafrost is in late-stage thaw and that subsequent talik formation and rapid degradation are imminent.

Such rapid change could affect permafrost monitoring efforts by causing TOP to descend rapidly out of the measurement range of existing thaw tube installations . As a solution, movable thermistor chains with dense spacing near the permafrost table may help capture changes in TOP, and records of subsidence may inform estimates of excess ice loss. Similarly, methods of observing and reporting ALT may benefit from adjustments to remain relevant even with talik formation.