Recent decades have witnessed substantial steps forward in knowledge of the climate system and ongoing anthropogenic climate change . An increase in observational datasets and progressively more realistic climate model ensembles allowed the attribution of a global mean near-surface air temperature rise of more than 1 ∘C since the end of the 19th century to human activities . This warming has been larger over land than over the ocean, with temperature values in 2011–2020 being 1.59 and 0.88 ∘C, respectively, higher than 1850–1900 . It is important to understand how this temperature rise has propagated into the soil and subsoil, which was shown to be important for land surface processes and meteorological extreme events .

Ground surface temperature (GST) and surface air temperature (SAT) are coupled at multi-decadal and centennial scales , and GST perturbations are subsequently propagated into depth by heat transfer . A warming of near-surface soil has an impact on its biogeochemical activity , enhancing the metabolic activity of microorganisms, mineral weathering, and decomposition of soil organic matter . It also affects soil hydrology and increases soil respiration , contributing to an acceleration of the terrestrial carbon cycle . Moreover, soil warming has relevant impacts on high-latitude regions, where permafrost thawing can produce changes in soil hydrology and eventually subsurface collapse , posing risks to infrastructure and reshaping landscapes. Moreover, permafrost thawing triggers the release of pollutants e.g., and massive amounts of carbon into the atmosphere, potentially affecting the global climate e.g.,. The subsurface is sensitive to not only present global warming but also to past climate changes. Deep subsurface temperatures allow recovery of past GST histories by borehole temperature profile (BTP) inversion techniques , which makes them a valuable source for climate reconstruction as far as the two fundamental hypotheses are sustained: SAT and GST changes are coupled at long-term scales and GST changes are propagated downwards by heat conduction, the main heat transfer mechanism in solids.

Support for SAT–GST coupling has been found so far from observational temperature data at some sites in North America e.g., and Europe e.g., as well as from climate simulations . These studies showed a strong coupling at long-term scales globally, with some degree of seasonal decoupling due to snow cover or changes in the surface energy balance at regional scales. Examining whether conduction is the main heat transport mechanism in the subsurface was usually tackled by assessing the propagation of temperature perturbations and verifying that they experience an amplitude damping and phase shift with depth to an extent controlled by thermal diffusivity, a parameter that governs heat conduction processes . Previous research assessed the propagation of temperature trends and thermal orbits using a priori defined constant values of thermal diffusivity. Other studies tackled the propagation of single-frequency harmonic signals with depth, i.e., the annual cycle , in order to derive estimates of the apparent thermal diffusivity. Nevertheless, this approach requires continuous monitoring of subsurface temperature for at least a relatively long lapse of time (at least a few years). This was only achieved in a low number of places in North America and Europe . Moreover, the analysis might be hampered by potential deviations from the conductive regime in the soil near the surface due to biological, chemical, and hydrological processes . A deeper insight into the soil thermal structure is required to understand changes in its thermal properties with depth, relevant for land surface models (LSMs) and therefore with implications for land–atmosphere interactions affecting Earth system and forecast models. A better characterization of subsurface thermal properties can also contribute to more accurate land heat uptake estimates, i.e., energy storage in the subsurface due to industrial warming, which is the second component contributing the most to terrestrial energy partitioning .

This paper assesses the validity of the SAT–GST coupling and conductive thermal propagation hypotheses in the penetration of SAT perturbations into the subsurface in a high-altitude area in central Spain. This is achieved by determining the apparent soil thermal diffusivity from four short (4–6-year) and two longer (ca. 20-year) subsurface temperature records obtained at six sites in the area of the Sierra de Guadarrama and studying temperature variability in time and depth through the soil and down to the first meters of bedrock. The dataset presented in this work is unique since it gathers shallow and deep subsurface information (down to 20 m) from different sites and sensors in a high-mountain area. The complex orography of the Sierra de Guadarrama was shown to dominate the spatial mean and variability of SAT, while temporal variability was found to be very homogeneous over the region . In a first stage, we explored the SAT–GST coupling at different sites, considering the influence of processes at seasonal timescales. The propagation of the annual wave with depth was subsequently studied. This allowed us to test the conductive hypothesis and to establish a comparison between different sites by using the standard approach of focusing on the propagation of the annual wave with depth. The analysis was further extended by assessing changes in thermal diffusivity in the subsurface temperature profile, which were linked to the soil–bedrock mineral composition transition observed from samples extracted at each of the sites.

The study of the conductive regime also explored changes in apparent thermal diffusivity with time. This was feasible for the longer records for which the analysis can be performed, focusing on different multi-annual temporal intervals. This approach was however hampered by the short time span of observations at most sites considered herein. This problem was overcome by introducing a new method that makes use of the information in the spectral domain. Instead of considering only the annual cycle, the propagation of the complete range of harmonics of different frequencies that contribute to temperature variability was used. This allowed for focusing on intra-annual timescales and using shorter time series to estimate soil thermal diffusivity near the surface. The fact that apparent thermal diffusivity values can be retrieved from shorter time spans also allowed diffusivity changes to be evaluated through time and connected to changes in soil moisture content .

The subsurface data analyzed hereinafter consist of temperature time series from six locations distributed over the Sierra de Guadarrama. Table 1 includes a more detailed description of the name, code, geographical position, and date range of available data of each location. The Sierra de Guadarrama is part of the Sistema Central, a mountain system that splits the Spanish Central Plateau into a southeastern side, with altitudes of around 600 m a.s.l. (above sea level), and a northwestern side, of higher elevation (ca. 750 m a.s.l.). Elevations in the sierra span from 900 at the foothills to ca. 2200–2400 m a.s.l. at the summits. Two sites, Puerto de Navacerrada and Segovia, belong to the Spanish Meteorological Service (Agencia Estatal de Meteorología, AEMET), and the rest are part of the Guadarrama Monitoring Network (GuMNet). The latter was created in 2014 with the aim of gaining further insights into climate variability in mountain environments in central Spain . GuMNet consists of 10 atmosphere and subsurface monitoring stations, which cover a vertical gradient ranging from 900 to 2200 m a.s.l., spreading over the mountainous terrain of the Sierra de Guadarrama. In this work, four stations were selected that provide information on atmospheric variables and subsurface temperatures (Fig. a). In all the cases, vegetation cover at the sites consists of short grass that changes minimally during the year. No site has trees or shrubs within the fenced perimeter of the monitoring station.

The subsurface thermal regime is monitored by two different arrangements. Near-surface soil temperatures are measured at various levels in trenches (TRCH; Fig. b) at all the sites. Trenches are excavations forming a slope in which the frontal wall was used to analyze the vertical structure of soil horizons and to insert soil temperature reflectometer sensors down to a depth of about 1 m. Trenches were refilled with the previously extracted material. Deeper temperatures in the soil and bedrock are monitored within cased boreholes. They consist of a small diameter (ca. 76 mm) cylindrical drill in depth where a casing was inserted and afterward filled up with a silicone gel. Temperatures therein are measured with calibrated platinum resistance thermometers embedded inside the casing (BRH; Fig. c, d). For the sake of having a finer vertical resolution in the soil near the surface and some level of redundancy, two boreholes were installed at these sites: a deep one going down to 20 m (BRH 20 m in Fig.  and Table ) and a shallow one of 2 m (BRH 2 m in Fig. ). Table  provides a detailed list of the depths at which subsurface temperature sensors were installed. Levels with malfunctioning sensors are marked with a dash.

During the borehole drilling process at the GuMNet sites, the extracted cores were preserved. They were subsequently sampled and characterized. The result is shown in Fig. , where the limits between the soil, sediment, and rock layers were determined. The superficial soil layer is made up of different soil horizons containing an O and/or A horizon (i.e., topsoil) followed by E and/or B (i.e., subsoil) and several C (i.e., parent material) horizons that reach a depth of approximately 1 m. The underlying sediment layer contains eroded material of loose debris rocks that have accumulated at the corresponding locations. The Cotos (CTS), Hoyas (HYS), and Raso del Pino I (RPI) profiles are located over gneiss rock and Herrería (HRR) on granite rock. In all the profiles, the upper layers have weathered bedrock, and with depth this bedrock is unweathered and compact. This information was used in Sect.  to compare changes in apparent thermal diffusivity to changes in mineral composition and textures with depth.

Subsurface temperature measurements are taken at a 10 min time resolution at every GuMNet station. As a first step prior to the analysis carried out in this work, subsurface data were subjected to quality-control and resampling procedures. AEMET stations Puerto de Navacerrada and Segovia (NVC and SGV) were independently processed and resampled to daily resolution as described in . The quality control of GuMNet sites focused first on removing data outliers. Since subsurface temperature sensors are connected in series in both boreholes and trenches (see Fig. b–d), most often erroneous extreme values are recorded at all levels simultaneously. Thus, to avoid discarding values that might be correct meteorological extremes, this correction was applied for both boreholes and trenches using the deepest time series to detect the erroneous values. The lowermost level was used because it shows the lowest temperature variability due to conductive damping of the surface signal. This reduces the range of high-frequency variability in the series except for the erroneous data, which makes non-reliable outliers easier to detect (Fig. ). First, outliers flagged as physically implausible were removed, i.e., values falling below 230 K or exceeding 350 K. The detection of outliers below 230 K can be considered redundant since the next step would have detected all the values lying beneath this threshold. Second, the remaining spikes were flagged and discarded. Spikes were detected by calculating the sequential differences of each value minus its preceding one and screening the values in the interval (Q1-3IQR,Q3+3IQR), where Q1, Q3, and IQR are the first and third quartiles and the interquartile range of the distribution of temperature differences . Time steps whose difference values were missing or outside of the screening interval were removed. Figure  shows an example of such spike screening for the BRH 2 m at CTS. The outliers depict coinciding failures at all the levels. The procedure is very conservative as the bottom levels have very low temperature variability and all spikes are identified and removed at each level. Note the decrease in high-frequency variability with subsurface depth, which is consistent with a heat conduction process . Once outliers were removed, time series were subjected to a one-by-one inspection, and periods with clear drifts or strong changes in variability were manually removed (check Table  for a list of removed periods at CTS and HYS). Figure  also shows an example of one such period for the 1.5 m depth temperatures at CTS (green arrow).

After applying the aforementioned quality-control procedures, the GuMNet series were resampled at hourly resolution, which filtered out noisy intra-hourly variability unnecessary for the subsequent analysis of the data. This was attained by calculating pseudo-hourly time series, which was carried out with a nearest-neighbor resampling technique by selecting the 10 min data closest to the hour. Finally, the resulting series were averaged to obtain daily mean subsurface temperatures that were subsequently used to analyze the propagation of the annual cycle.

In addition to subsurface temperature data, all the stations include SAT measurements as well as soil moisture content within the trench-monitoring levels. SAT is recorded at 2 m above the ground surface at all the sites using a temperature probe encapsulated within a shield. Snow cover data are also available at CTS, HYS, and RPI (see the codes in Table ). Snow depth is measured with an ultrasonic snow depth sensor that captures the distance to the surface. Depth values are obtained from the subtraction of the surface level measured from the reference ground surface level value. These additional data were used to explore potential drivers of the SAT–GST seasonal decoupling and temporal changes in soil apparent thermal diffusivity at some sites.

The SAT–GST seasonal coupling was explored by quantifying the offset and correlation of seasonal (December–January–February, DJF; June–July–August, JJA) and annual temperatures at 2 m above the ground surface with GSTs taken from the top-level temperatures in the BRH 2 m setup. The estimates were obtained from the complete time span of available data at each site (Tables 1 and 2).

Furthermore, the subsurface temperature distribution was analyzed by assuming that GST perturbations are propagated downwards by heat conduction. Considering that every level is at a thermal equilibrium state, horizontal heat transport may be neglected. Thus, the problem is reduced to resolve the one-dimensional time-dependent heat conduction equation, ∂T/∂t=α∂2T/∂z2. If GST is assumed to follow a sinusoidal (e.g., annual) cycle, subsurface temperatures at any depth z can be derived as follows : 1T(z,t)=T0+A0e-zπfa/αcos⁡(2πfat-zπfa/α), where T0 and A0 are the mean temperature and wave amplitude of the annual cycle at the ground surface, respectively, fa is the frequency of the annual cycle, and α is the thermal diffusivity, which is a direct function of thermal conductivity (k) and the inverse of density (ρ) and heat capacity (c), i.e., α=k/ρc. Equation () shows that the annual cycle amplitude is attenuated and phase-shifted with depth. The amplitude attenuation is exponential, and the phase shift is linear with depth and dependent on the thermal diffusivity; i.e., increasing this parameter produces both a greater amplitude attenuation and a phase shift.

Amplitude and phase values of the annual cycle at every level were obtained by least-squares fitting subsurface temperature series at a daily resolution to an annual-period sinusoidal wave. Then, linear regression analysis of the annual-cycle amplitude (phase-shift) values with depth yielded estimates of the apparent thermal diffusivity of the subsurface, since the resulting slope is equal to πfa/α in Eq. (). Amplitude and phase-shift values coming from every installation (TRCH 1 m, BRH 2 m, and BRH 20 m) were independently adjusted and normalized to prevent any disruption due to blending data from different sources when assessing heat propagation from the ground surface to 20 m depth. Normalization was achieved by dividing the amplitude values of the annual cycle at every level by the amplitude value at the ground surface and by subtracting the phase-shift value at the ground surface from the phase-shift values of the annual cycle at every level. The values at the ground surface were obtained through a linear regression adjustment. Once normalized, all amplitude and phase-shift values were brought together and linearly adjusted to obtain a single estimate of thermal diffusivity for the whole subsurface profile at every station. A similar methodological approach based on the analysis of the annual cycle has been widely used in the literature e.g.,, and it will be referred to as the classic analytical approach (CA) hereafter. This CA framework was also used to estimate changes in apparent thermal diffusivity with depth, either by considering changes between pairs of subsurface levels or by identifying depths where significant changes in apparent thermal diffusivity occur. This was achieved using a two-phase regression analysis , which determines whether there is a change point for which linear fits to the segments before and after the change point significantly improve the results compared to a linear fit to the complete curve.

Beyond the regular nature of the annual cycle, GST variability encloses perturbations at different frequencies. If GST is considered a sum of temperature harmonics, each harmonic is also propagated with depth following : 2T(z,t)=T0+∑i=1NA0e-zπfi/αcos⁡(2πfi-zπfi/α), where fi is the frequency of each wave component in a spectral decomposition, as yielded by a Fourier periodogram for instance . Equation () shows that amplitude attenuation and phase shift are frequency-dependent. More precisely, amplitude attenuation grows exponentially with frequency; i.e., high-frequency harmonics decay more intensively with depth, whilst low-frequency perturbations can propagate more deeply. For instance, the daily cycle hardly reaches a depth of 50 cm, while the annual cycle propagates down to 15 m . The temperature spectral attenuation with depth can be expressed as 3ζ=Pz(ω)/PGST(ω)=e-zπf/α, where PGST(ω) and Pz(ω) are the spectral power densities of GST and that at a given level z, respectively: 4Pz(ω)=1Ns∑t=1NT(z,t)e-iωt2,withω=2πf, where Ns is the temporal resolution of the temperature time signal at z, T(z,t); f is the frequency; and ω is the angular frequency of the different wave components of T(z,t). In order to minimize the undesirable effect of Gibbs' noise at high frequencies, P was calculated by making use of Welch's modified periodogram . Because the periodogram calculation requires evenly spaced data, data gaps were previously filled by linear interpolation. As the spectral attenuation must follow an exponential relationship with frequency, the spectral amplitude attenuation curves attained from the analysis of Eq. (3) were subsequently linearized (i.e., taking the natural logarithm) and linearly fitted by least squares to the values obtained at all available depths: 5(ln⁡ζ)2=πz2αf.

Estimates for the apparent thermal diffusivity of the soil layer in between the ground surface and level z can be retrieved by dividing πz2 by the regression line slope, πz2/α, in Eq. (5). To prevent poor signal-to-noise ratios at high frequencies from biasing the analysis, amplitude attenuation values at frequencies higher than the corresponding e2-fold decay cut-off frequency (i.e., spectral harmonics attenuated more than e2 times) were filtered out at every level. To determine this frequency at every site and level, near-surface soil thermal diffusivity values coming from the CA are used. However, it is possible to use a physically plausible prior thermal diffusivity value (ca. 10-6 m2 s-1) to calculate this frequency, leading to very similar results and thus making the spectral method (SpM) independent of the CA. This alternative approach to the CA will from now on be called the SpM. While the CA needs at least a few years of data to provide reliable fits of the annual cycle to subsequently retrieve the apparent thermal diffusivity, the SpM can be applied to data at shorter time intervals. This makes it suitable for analyzing short time series, which is the case with some of the time series presented in this work. Consequently, the SpM was used in this work to retrieve apparent thermal diffusivity values in the first meter of the soil at every station. These values were then compared to the estimates coming from the CA to assess the robustness of the results from both methods. Furthermore, the SpM allows some insight into short-term changes in soil apparent thermal diffusivity and its potential relation to changes in snow cover and soil moisture content.

The subsurface thermal structure at six sites in the Sierra de Guadarrama is consistent with the hypothesis of conduction being the main heat propagation mechanism of SAT changes into depth. On the one hand, SAT changes are transferred to the near surface at the ground. Although SAT and GST are coupled at long timescales, snow insulation and zero-curtain effects produce a decoupling in winter at the uppermost stations (CTS, HYS, and RPI), which are affected by the presence of a seasonal snow cap (Figs.  and ). This result agrees with previous studies carried out at cold and snowy sites, such as Fargo North Dakota, US;, Emigrant Pass Utah, US;, Bistriţa Romania;, and Hveravellir Iceland;.

On the other hand, GST changes are propagated to depth by heat conduction to an extent controlled by the apparent thermal diffusivity of the subsurface. A CA method based on the amplitude attenuation and phase shift of the annual cycle yielded values that are between 1 and 1.3×10-6 m2 s-1 for the whole soil column (i.e., 20 m) at most of the sites (Fig. ), which are in the range of typical diffusivity values for granite or gneiss, the main materials found in the Sierra de Guadarrama's subsurface (Fig. ). These values are higher than those observed at the other sites e.g., Fargo, Prague, or Cape Henlopen;, which include shallower measures. If apparent thermal diffusivity is assessed near the surface, the values are smaller than and closer to the previous literature (0.4 to 0.8×10-6 m2 s-1; Table ). A subsequent two-phase adjustment (Fig. ) and a profile derivation based on CA (Fig. ) portray an increase in thermal diffusivity with depth at all the sites, being more remarkable near the soil–bedrock transition. Evaluating thermal diffusivity changes with depth using a CA can therefore be a useful approach to determine soil depth at observational sites when no borehole rock samples are available. The deep monitoring profiles at the Sierra de Guadarrama are unique and provide a wider insight into subsurface thermal structure, which may have implications for calibrating soil and bedrock properties in models and monitoring land heat uptake in the terrestrial energy budget .

Near-surface (i.e., 1 m) soil apparent thermal diffusivity vertical and temporal heterogeneity is further explored by introducing a new SpM, which extends the analysis of temperature amplitude attenuation to the whole frequency spectrum. The SpM reports consistent apparent thermal diffusivity values in the first soil meter at SGV, NVC, and HRR with those previously derived with the CA for each of the respective sites regardless of the time resolution of the input data. Then, it was used in an intra-monthly (10-daily) sliding window to explore apparent thermal diffusivity changes in time at the ground surface at HRR (Fig. e). Results suggest that this parameter presents a seasonal cycle that is positively correlated with soil moisture, mostly under soil drought conditions. However, the fact that these changes are poorly correlated under high soil moisture contents might indicate that heat conduction is not the main heat transfer mechanism in wet soils. Heat transport via water advection and phase changes might not be negligible near the surface, so the thermal structure in the shallow soil could be better characterized using a modification of the SpM that complies with a conduction–convection heat transfer equation . Even so, the results show the potential of the SpM for inferring soil moisture content changes from soil temperatures, becoming a powerful tool for evaluating soil drought and water resource availability from observational and simulation-based subsurface temperature data.