Global temperature reconstructions from those in Fig. 1 of the Summary for Policymakers of the sixth Assessment Report of the Intergovernmental Panel on Climate Change (IPCC) are used here for comparison purposes and to generate synthetic data to test the inversion methods . These reconstructions are based on multi-proxy systems, such as corals and tree rings, and were estimated by the 2k Network of the Past Global Changes organization (PAGES2k) . The global mean temperature anomaly is provided as a smoothed temporal series using a 10-year window; thus, data are available from year 5 to 1995 of the Common Era (CE).

PAGES2k reconstructions combine land and marine proxies to produce global mean temperatures, which we transform into global land temperatures. To this end, we apply the ratio between land and ocean temperature changes estimated in based on an ensemble of global climate simulations. This multimodel ensemble includes different forcing scenarios at several time scales, resulting in land surface temperature changes being ∼2.36 times larger than sea surface temperature changes. Therefore, the PAGES2k global temperature anomaly can be scaled to land temperature changes multiplying by ∼1.38. The same procedure is applied to the provided uncertainty range, as prescribed by the error propagation theory. The resulting temperature anomaly after the scaling is modified to have 1300–1700 CE as period of reference, the same as the inversions of subsurface temperature profiles (see below). The land temperature anomaly from the PAGES2k global temperature anomaly, labeled PAGES2k-Land temperature hereinafter, is used as the upper boundary condition to derive an artificial subsurface temperature profile designed to test the performance of the different methods considered here. Additionally, inversions of the Xibalbá database are also compared with these PAGES2k-Land temperatures.

Subsurface temperature profiles from the Xibalbá dataset are inverted to estimate global mean ground surface temperature histories. Xibalbá profiles were assembled using measurements from several sources, including the National Oceanic and Atmospheric Administration (NOAA) server and several publications . Each log was screened to ensure that the profiles did not contain obvious non-climatic signals due to water flow or other factors. All logs were truncated to contain depths between 15 and 300m to ensure that all profiles produced inversions relative to the same temporal period .

The dataset includes 1079 STPs distributed around the world, although with a lack of measurements in South America, most of Africa, the Middle East, and southeastern Asia see Fig. 1 in. Furthermore, all STPs were measured at different dates, with the majority of profiles obtained before the year 2000 CE. Hence, we aggregate the inversions of these logs, respecting their different logging years, thus obtaining results with a different number of inversions per year and focused on the period before the year 2000 CE.

An important caveat of the Xibalbá dataset is the lack of measurements of thermal properties at the location of the profiles. Most measurements of temperature profiles provide no thermal property data, thus requiring assumptions about thermal diffusivity and conductivity values in order to perform the analysis. The continental subsurface is typically considered as an homogeneous medium because of this limitation, which is supported by the few profiles including measurements of thermal properties. These measurements show that thermal properties vary by a relatively small amount around a mean value with depth (e.g., the Neil Well in the Arctic ). Additionally, the screening process for all individual logs (see above) helps to exclude profiles at locations with marked changes in thermal properties, since these changes would be reflected in the temperature records as non-climatic signals.

Ground temperature changes with depth as a response to the surface energy balance variations and the heat flow from the Earth's interior, constant at time scales of millions of years . These variations in the surface energy balance are propagated through the ground following the one-dimensional heat diffusion equation, altering the quasi-equilibrium temperature profile resulting from the long-term surface temperature (T0) and geothermal gradient (Γ0). Assuming constant thermal properties through the ground column, a subsurface temperature profile can be described as 1Tz,t=T0+Γ0⋅z+Ttz, where z is depth, the term T0+Γ0⋅z constitutes the quasi-equilibrium temperature profile, and the term Ttz represents the signature of recent changes in the surface energy balance.

The quasi-equilibrium temperature profile represents the long-term mean thermal state of the subsurface without changes in the surface energy balance and in the geothermal flux from the Earth's interior – that is, the profile resulting from a constant surface temperature and a constant geothermal gradient. Changes in the surface energy balance propagate through the ground by conduction and are recorded as alterations of this quasi-equilibrium profile. The propagation of a perturbation modeled as a step change in surface temperature (ΔT) at a certain time t in the past through a homogeneous medium can be described as 2Tz=ΔT⋅erfcz2κ⋅t, where erfc is the complementary error function, κ is the thermal diffusivity of the medium, and z is depth. Therefore, the final anomaly profile caused by the propagation of a series of step changes in surface temperatures (ΔTi) through the ground can be expressed as 3Ttz=∑i=1NΔTi⋅erfcz2κ⋅ti-erfcz2κ⋅ti-1, which represents the depth-varying temperature term in Eq. (). Ttz is the solution of the forward problem – that is, for a given surface temperature time series. Equation () provides the final perturbation of the subsurface profile in response to changes in the upper boundary condition. Therefore, Eq. () constitutes a one-dimensional forward model of heat diffusion through the ground.

An artificial STP is generated using the PAGES2k-Land temperature as the upper boundary condition for a purely conductive, homogeneous, half-space forward model (see Eq. ). From the PAGES2k-Land temperatures, this forward model generates a temperature profile that contains the changes in subsurface temperatures as a response to the prescribed changes in surface temperatures with time. A quasi-equilibrium temperature profile consisting of a long-term surface temperature of 8∘C and a long-term geothermal gradient of 20∘Ckm-1 is added to the profile in addition to Gaussian noise with zero mean and a standard deviation of 0.02 ∘C to account for measurement uncertainties in real profiles . Ground surface temperature histories are estimated from the inversion of the resulting artificial profile (Fig. ) using the three techniques detailed below. These temperature histories are then evaluated against the original PAGES2k-Land temperatures that were used as boundary conditions to generate the synthetic data to evaluate the performance of each technique.

The inversion problem for STPs consists of estimating the magnitude of the past changes in surface temperature that gave rise to the observed variation of temperature with depth. Such changes in surface temperature are typically modeled as a series of step changes whose magnitude is determined by the inversion technique, with the length of the time step set as a parameter. One of the standard methodologies to perform these inversions is based on solving a system of equations given by the observed profile and the combination of Eqs. () and () . This approach uses a singular value decomposition (SVD) algorithm to solve this system, and it is known as the SVD inversion method (Fig. ). The system considered in this inversion method can be expressed as 4Tobs=MTmodel, where Tobs is the anomaly temperature profile, Tmodel is a vector containing the step change model of the surface temperature to be determined, and M is the matrix containing the coefficients given by 5Mi,j=erfczi2κtj-erfczi2κtj-1. This system of equations, nevertheless, is overdetermined. This means that there are more equations in the system than parameters to be determined; thus, the solution is non-unique. A singular value decomposition (SVD) algorithm has been extensively used to solve these overdetermined systems e.g.,. The SVD algorithm is based on decomposing the matrix of coefficients M into two orthogonal matrices (U and V) and a rectangular matrix (S) containing the eigenvalues in the diagonal: 6M=USVT. Thereby, the solution of the system can be found as 7Tmodel=VS-1UTTobs. Finally, small eigenvalues are removed from S-1 in order to stabilize the solution, though this is at the cost of losing temporal resolution in the model. The final number of eigenvalues used to retrieve the solution is an important parameter, typically retaining two or three eigenvalues. More details about SVD inversions can be found in the literature .

The SVD inversion (Fig. ) is applied to the anomaly profiles – that is, the resulting perturbation profiles after removing the quasi-equilibrium profile from the measured log (Fig. b, c). The quasi-equilibrium profile – in other words the term T0+Γ0⋅z in Eq. () – is estimated using a linear regression analysis of the deepest 100m of the corresponding profile, which is the part of the measured profile not affected by recent changes in surface conditions. The errors in the estimates of the long-term surface temperature (T0) and the geothermal gradient (Γ0) have been typically included in the analysis by inverting three anomaly profiles: the anomaly profile resulting from subtracting the best estimate of T0 and Γ0 from the measured profile (black dots in Fig. b), and two extremal profiles resulting from subtracting the quasi-equilibrium profiles determined by the errors in T0 and Γ0 (red and blue dots in Fig. b). These extremal profiles are derived by subtracting and adding the corresponding two sigma values to the best estimate of T0 and Γ0. The inversions of the extremal anomaly profiles constitute the upper and lower boundaries of the uncertainty range in the retrieved ground surface temperature history of the corresponding log (red and blue lines in Fig. a).

Ground heat flux histories for each STP are estimated from ground surface temperature histories (see Sect. ). Concretely, a flux history is estimated from each temperature history retrieved from each STP – that is, from the best estimate and the inversion of the two extremal anomaly profiles. Flux histories can also be retrieved from flux profiles, which can be estimated from the measured STPs by applying the Fourier equation . Inversions of flux profiles are, nevertheless, noisier than inversions of temperature profiles; thus, we decided to derive heat flux histories from temperature histories in this analysis.

Please note that the method described in this section is applied to individual STPs – see Sect.  for a description of the approach used to obtain results from a set of profiles.

Inverting the extremal anomaly profiles obtained from the errors in estimating the long-term mean surface temperature and mean geothermal gradient allows one to account for the uncertainty in the determination of the quasi-equilibrium temperature profile in the SVD approach described above. Nevertheless, the SVD approach is not able to include the uncertainty due to the unknown thermal properties at each site. Hence, a more comprehensive way to estimate uncertainties in STP inversions was developed in . This approach, called perturbed parameter inversion (PPI), consists of generating a large ensemble of SVD inversions from each subsurface temperature profile by varying the values of the inversion parameters: time step of the surface signal, thermal diffusivity and conductivity, and the number of eigenvalues retained in Eq. () (Fig. ). This process is repeated for the three anomaly profiles retrieved to characterize the uncertainty in the determination of the quasi-equilibrium temperature profile (see SVD technique above). We perturb only the parameters related to thermal properties and the quasi-equilibrium temperature profile in this study, as these are the most important sources of uncertainty . That is, we generate an ensemble of inversions using the three anomaly profiles obtained from the best estimate and two sigma values of the intercept (T0) and slope (Γ0) determined from the linear regression of the deepest part of the profile, and considering a different thermal diffusivity for each inversion (NDiffusivity in Fig. ). Thereby, the ensemble contains 3×NDiffusivity elements. The 2.5th, 50th, and 97.5th percentiles of all retrieved temperature histories are then estimated in order to provide a best estimate (the median) and a 95 % confidence interval for the ground surface temperature history of each profile.

The PPI method described in removed highly unlikely individual ground surface temperature histories from the final ensemble. However, here, we consider all possible histories in order to explore the full range of variability in the inversions, including unlikely cases. For the same reason, forward models of individual histories are not compared with the anomaly profiles, and thus, each history is weighted equal in the estimated percentiles, in contrast to the approach in . Thereby, the full effect of the unknown thermal properties in the estimated uncertainty can be assessed by comparing PPI solutions with SVD inversions, since the only difference between these two techniques is the use of a range of diffusivities in the PPI method.

Ground heat flux histories for each STP are estimated from ground surface temperature histories (see Sect. ), similarly to the SVD case but with some differences. The PPI method retrieves an ensemble of temperature histories for each STP. We estimate another ensemble containing flux histories using each of the temperature histories in the PPI ensemble, obtaining the 2.5th, 50th, and 97.5th percentiles that constitute the best estimate and uncertainty range for the ground heat flux history of the corresponding profile.

Note that this method is applied to individual STPs – see Sect.  for a description of the approach used to obtain results from a set of profiles.

As explained in the Introduction, the SVD and PPI techniques do not provide a correct statistical framework to estimate the uncertainty resulting from aggregating inversions from several STPs. In order to overcome this problem, we have developed a new method to retrieve ground surface temperature histories combining the SVD algorithm described in Sect.  and a bootstrapping sampling strategy that provides a meaningful statistical method to estimate uncertainty ranges for aggregations of any number of individual profiles .

Bootstrap Inversions (BTIs) are based on generating two ensembles – named Sampling and Bootstrapping ensembles (S and B ensembles in Fig. ) – constituted by inversions performed using the SVD algorithm. The Sampling ensemble consists of one SVD inversion per subsurface temperature profile in the considered population, with inversion parameters randomly selected from a range of possibilities (e.g., the value for thermal diffusivity – see below). One thousand different Sampling ensembles are created, with the mean ground surface temperature history of each Sampling ensemble constituting an element of the Bootstrapping ensemble (Fig. ). That is, the Bootstrapping ensemble includes one mean history per Sampling ensemble created. The median of the resulting Bootstrapping ensemble constitutes the best estimate for the mean ground surface temperature history of the considered population of logs, with the 95 % confidence interval given by the 2.5th and 97.5th percentiles of the Bootstrapping ensemble . Here, we create 1000 different Sampling ensembles in each BTI inversion; thus, the Bootstrapping ensemble includes 1000 averaged histories. Larger sizes for the Bootstrapping ensemble can be considered, but the results are approximately the same (see analysis below).

There are four parameters used to obtain surface temperature histories in the Sampling ensemble described above: the prescribed quasi-equilibrium temperature profile, the assumed thermal properties of the ground, the length of the prescribed step changes to model the retrieved surface temperature, and the number of eigenvalues retained in the solution. However, the largest sources of uncertainty are the determination of the quasi-equilibrium profile (i.e., the errors in T0 and Γ0) and the unknown thermal diffusivity and conductivity of the subsurface (Table ). For bootstrap inversions, all possible values for each parameter are equally probable, except for the quasi-equilibrium temperature profile. In this case, the long-term surface temperature and geothermal gradient are given by a Gaussian distribution, with mean and standard deviation corresponding to the best estimate and error of T0 and Γ0, which are retrieved from the linear regression analysis of the deepest part of each profile.

Similarly to the SVD and PPI cases, ground heat flux histories are derived from ground surface temperature histories for each STP. In this case, each temperature history considered in the Sampling ensemble is used to estimate a flux history, which is then averaged to obtain a mean ground heat flux history. The process is repeated 1000 times, as in the case of temperature histories, to generate a Bootstrapping ensemble containing mean flux histories, from which the 2.5th, 50th, and 97.5th percentiles are computed.

Ground heat flux histories are estimated using the method developed in by using a half-order derivative approach to estimate ground heat flux from soil temperatures in a homogeneous medium. Since SVD inversions also assume a homogeneous subsurface, this method can be applied to the retrieved ground surface temperature histories to obtain ground heat flux histories among others. Given a temperature history with equidistant time steps (tN), the corresponding heat flux history is estimated as 8GtN=2λπκΔt∑i=1N-1Ti+1-TiN-i-N-i+1, where, G is ground heat flux, κ is thermal diffusivity, λ is thermal conductivity, ΔT is the size of the time step, and Ti is the value of the temperature history at time step i. This equation is applied to ground surface temperature histories retrieved from SVD, PPI, and BTI techniques to estimate ground heat flux histories, with a range of plausible values for thermal conductivity and thermal diffusivity in the BTI case (Table ).

Global estimates of ground surface temperature from individual Xibalbá STPs are retrieved using the SVD, PPI, and BTI techniques. Global estimates from individual SVD inversions are obtained by averaging temperature and flux histories from each individual log, while the global uncertainty corresponds to the mean of the inversions of the extremal anomaly profiles. That is, the two extremal anomaly profiles derived by subtracting and adding the two sigma values of T0 and Γ0 to the best estimate of these parameters are inverted for each profile, constituting the upper and lower limits of the uncertainty range for the ground surface temperature history retrieved using the best estimate of T0 and Γ0 (i.e., the red and blue lines in Fig. ). The mean of all the upper limits (red line) and the mean of all the lower limits (blue line) estimated from the 1079 Xibalbá profiles constitute the uncertainty range for the global average. A similar approach is applied in the case of individual PPI solutions, but the 2.5th, 50th, and 97.5th percentiles of each STP in the Xibalbá dataset are averaged. That is, the means of all 50th percentiles are considered to be the global averaged history, and the interval given by the means of all 2.5th and all 97.5th percentiles are considered to be the corresponding uncertainty range. These approaches have been used in previous studies to derive global estimates of temperature and heat flux changes in the land surface from SVD and PPI inversions , although those are not correct statistical methods. As previously discussed, the BTI framework can yield results from any number of logs by modifying the size of the Sampling ensemble. Therefore, global averages of temperature histories are estimated by including an inversion from each of the 1079 Xibalbá STPs in the Sampling ensemble. The mean of these 1079 inversions constitutes one member of the Bootstrapping ensemble, which therefore consists of 1000 global averages, each one estimated from 1079 different inversions, one from each profile in the dataset. The final global results are obtained by computing the 2.5th, 50th, and 97.5th percentiles of the Bootstrapping ensemble. Furthermore, we have tested the effect of considering the effective number of spatial degrees of freedom in observed surface temperatures on the calculation of the global mean ground surface temperature histories, finding it to be negligible (see Supplement).

The same approaches are applied to ground heat flux histories from Xibalbá profiles to derive global averaged heat flux histories.

We apply the SVD, PPI, and BTI techniques described above to an artificial subsurface anomaly profile generated from the PAGES2k-Land temperatures (Fig. a) and to the 1079 worldwide subsurface temperature profiles included in the Xibalbá database. We consider a range of possible values for each relevant parameter (see Table ) in order to determine the uncertainty rising from poorly known quantities in the inversions.

The continental subsurface is considered to be homogeneous for all inversions performed here; thus, we assume no variation of the diffusivity or conductivity with depth. For SVD inversions, we use a constant thermal diffusivity of 1×10-6m2s-1, which is a typical value for these types of inversions e.g.,. Thermal conductivity is also considered to be constant through the subsurface, with a value of 3Wm-1K-1 that also matches the conductivity used in other works based on SVD inversions . For the PPI and BTI approaches, we consider a range of thermal diffusivities from 0.5×10-6 to 1.5×10-6m2s-1. The BTI technique also takes into account a range of thermal conductivities from 2.5 to 3.5Wm-1K-1 to estimate ground heat fluxes from temperature histories (Eq. ). Nevertheless, the PPI approach considers a constant thermal conductivity of 3Wm-1K-1 for estimating flux histories in order to simplify the comparison with results from SVD inversions. The considered ranges for thermal diffusivity and thermal conductivity for BTI inversions contain all plausible values addressed by the literature .

All SVD, PPI, and BTI inversions use the same model for the boundary condition, i.e., the temporal signal at the surface, consisting of a series of step changes, all with the same temporal length. We perform inversions with time steps of 10, 30, and 50 years to test the effect of this parameter on the results produced by the three techniques. We also obtain SVD, PPI, and BTI inversions retaining the two and three largest eigenvalues in the decomposition matrix (Eq. ) in order to test the dependency of the retrieved histories on this parameter.

An artificial profile generated from a known surface temperature allows one to evaluate the ability of the SVD, PPI, and BTI techniques described above to retrieve the original surface signal under perfect conditions (Fig. ). Ground surface temperature histories obtained by applying the SVD and PPI methodologies recover the main features of the PAGES2k-Land temperatures used as surface boundary condition to generate the artificial profile (purple and red lines in Fig. ). Inversions using two and three eigenvalues present similar temperature histories as well as similar uncertainty ranges, except at the end of the period when inversions performed with 10-year step changes in the surface signal with two eigenvalues yield larger uncertainties than inversions with three eigenvalues. Bootstrap inversions show similar temperature histories to those from SVD and PPI inversions, with uncertainty ranges slightly larger than those from SVD inversions and slightly lower than those from PPI inversions (Fig. ). This is an expected result, as the BTI approach considers a range of possible values for thermal diffusivity, while SVD inversions consider only one value for diffusivity (Table ). Also, PPI uncertainty ranges are expected to be larger than BTI confidence intervals, because the PPI approach considers the two sigma values in long-term surface temperature and geothermal gradient for determining the two extremal anomaly profiles (red and blue dots in Fig. b). That is, the extremal anomalies in the PPI method are always going to represent a highly improbable case of the Gaussian distribution of possible values for long-term surface temperature and geothermal gradient, while the BTI approach samples randomly the possible anomalies given by a Gaussian distribution around the regression line obtained to estimate the quasi-equilibrium profile; thus, the 2.5th and 97.5th percentiles for the BTI inversions are always contained in the inversions of the extremal anomalies in the PPI case.

Small differences in ground surface temperature histories are present in 30-year (Fig. c) and 50-year (Fig. e) time step inversions retaining the two largest eigenvalues in the solution (Eq. ). More importantly, the averaged ground surface temperature histories from inversions with time steps of 50-years and three eigenvalues show negative temperature changes around 1925 CE in SVD, PPI, and BTI cases, while the original surface temperature increases during the same period (Fig. f). This suggests that the third eigenvalue may be excessively small, leading to an unrealistic level of variability in the solutions of Eq. ().

Results from the new bootstrap technique generally recover the original surface temperature used to derive the artificial profile (Fig. ). The ground surface temperature histories retrieved by the BTI method, nevertheless, present a new parameter that affects the behavior of the retrieved histories: the size of the Bootstrapping ensemble (see Fig. , Table ). The bootstrap approach is based on randomly sampling with repetition a number of different populations from the original, finite data . An estimate of the probability density function for the desired statistic is then retrieved from these different populations, but the number of populations (i.e., the size of the Bootstrapping ensemble) is going to determine the final results. In the case of the retrieved temperature history from the artificial profile, we used a Bootstrapping ensemble with 1000 members, and an increase in the number of populations considered does not change the results (Fig. a). The retrieved uncertainty range is also sensible to the size of the Bootstrapping ensemble, but considering more than 1000 populations does not change the estimated uncertainty much (Fig. c).

Inversions of the artificial profile have shown that the SVD, PPI, and BTI methods are able to yield very similar averaged ground surface temperature histories for one log, in agreement with the prescribed surface temperatures. However, this is a theoretical case, and real case applications would involve more complex situations, including a higher number of profiles with more noisy temperature records. This section describes the results obtained from the application of these three techniques to the subsurface temperature profiles from the Xibalbá dataset, including the comparison with global land temperatures estimated from PAGES2k multi-proxy reconstructions and previous estimates from . Global ground heat flux histories are also retrieved and compared with estimates from .

Global averaged ground surface temperature histories from the Xibalbá dataset retrieved from the SVD, PPI, and BTI techniques are very similar for inversions considering time steps of 10, 30, and 50 years for the surface history and for inversions retaining the largest two and three eigenvalues (Fig. ). The most important difference between inversion methods is the estimated uncertainty ranges, with PPI results yielding the broadest interval and BTI inversions the narrowest interval. In fact, the uncertainty range of the bootstrap approach is much smaller than that from SVD and PPI inversions for the global case, which is in contrast with the similar values achieved by the three techniques for the artificial profile (Fig. ). This is caused by the differences in the methods used to estimate global uncertainties in the three approaches (see Sect. ). The SVD and PPI methods aggregate the uncertainty of individual profiles to derive global uncertainty estimates. Concretely, these approaches average the upper and lower bounds of the individual uncertainty ranges, which provide a conservative global uncertainty range. That is, it is highly probable that the global averaged ground surface temperature history is contained in this interval, because it is markedly broad by construction. Nevertheless, this uncertainty range is not interpretable statistically, thus hindering its assessment against other past temperature estimates from STP inversions using Bayes methods, climate simulations, or proxy-based reconstructions. In contrast, the bootstrapping approach provides a meaningful statistical framework to derive global uncertainties. That is, the global uncertainty range retrieved by the BTI technique can be interpreted as the 95 % confidence interval of the global averaged ground surface temperature history, as it considers different possible values of the global average estimated using different, but possible, values for the unknown parameters in the inversion. Therefore, results from the SVD and PPI techniques overestimate the global uncertainty due to an inadequate aggregation of individual uncertainties.

As in the case of the artificial profile, BTI results are sensitive to the number of re-samplings considered, i.e., the size of the Bootstrapping ensemble (Fig. ). The dependency on the size of the Bootstrapping ensemble of the global averaged temperature history and the global uncertainty range from Xibalbá profiles converges to a stable value when the number of samplings increases, with small changes after considering a Bootstrapping ensemble of 1000 members (Fig. b and d). Considering more than 1000 ensemble members, therefore, does not change the results much; thus, 1000 is the recommended size for the Bootstrapping ensemble when applying the BTI technique to individual profiles and large sets of profiles, given the results of Fig. .

Global averaged ground heat flux histories estimated from Xibalbá profiles show similar results using the SVD, PPI, and BTI methods (Fig. ). There is an unexpected decrease in the heat flux at the end of the 20th century for solutions considering 10-year step changes at the surface (Fig. a, b), which is not present in solutions with 30-year and 50-year time steps nor in previous estimates of global heat flux histories . Furthermore, this flux decrease is present in all three methods retaining the two and three largest eigenvalues in the inversions, indicating that a 10-year step change in the model for surface boundary conditions may be excessively small, at least to retrieve past histories of ground heat flux. There are also differences in the retrieved uncertainty range for the global mean from each technique, with BTI results displaying a much smaller uncertainty than SVD and PPI results. This is an expected result, since the ground heat flux histories from the three inversion methods are estimated from ground surface temperature histories, which also yielded these differences in the retrieved uncertainty range (Fig. ).

The assessment of ground surface temperature histories retrieved from Xibalbá STPs using the SVD, PPI, and BTI methods shows agreement with previous estimates from the same logs retrieved with a slightly different PPI technique (Fig. a and Table ). The main methodological difference between the PPI approach used here and the approach used in consists of filtering individual inversions to remove unrealistic histories in comparison with long-term changes in meteorological data. This additional screening removed the histories showing the largest variability with time, reducing the uncertainty range in the estimates of ground heat flux histories and ground surface temperature histories. Therefore, the lack of filtering in the PPI approach applied here explains the smaller uncertainty range in . Another remarkable result is the agreement in the temporal evolution of the PAGES2k-Land temperatures and the ground surface temperature histories (Fig. a). However, this agreement is not visible in the magnitude of temperature changes, particularly after 1800 CE., because of a temperature decrease recorded in the PAGES2k-Land dataset in 1820 CE. that is not captured in the STP inversions, which leads to smaller mean temperature changes from PAGES2k-land data than from the Xibalbá profiles from around 1800 to 2000 CE (Table ). Nevertheless, similar temperature trends can be observed for both proxy and STP datasets during this period. This difference in warming from multi-proxy data and subsurface temperature profiles is a known problem that needs further investigation in order to reconcile the estimated temperature evolution during the last millennium from both datasets. Nevertheless, results in Fig. a show smaller differences than expected in comparison with previous analyses . PAGES2k-Land temperatures have been horizontally displaced to have the same period of reference as the STP inversions – that is, 1300–1700 CE. Therefore, the small offset around 1600 CE is just caused by the natural variability of the PAGES2k series, which includes higher-frequency signals than STP results.

The averaged ground heat flux histories derived here are also in agreement with those from despite several methodological differences (Fig. b). As indicated above, the PPI method in removed individual inversions with high variability from the PPI ensemble, sampled a range of possible conductivities, and weighted each inversion in comparison with the measured profile. These three factors, particularly the sampling of a range of conductivities, lead to higher uncertainty in than in the PPI ground heat flux histories retrieved here.