Proxy-based climate reconstructions are major tools in understanding the past climate system and predicting its future variability. Target regions, spatial density and temporal coverage of the proxy network vary between the studies, with a general trend towards more comprehensive networks and sophisticated reconstruction techniques used. For example, , , , , , , and present reconstructions of surface air temperatures (SAT) for different spatial and temporal domains. On the most detailed level, the available reconstructions tend to disagree on aspects such as specific timing, duration and amplitude of warm/cold periods, due to different methods, types and number of proxies, and to different regional delimitation used in the different studies . There are also alternative viewpoints on a more fundamental basis considering how the level of high-frequency versus low-frequency variability is best represented; see, e.g., . In this context, differences between reconstructions can occur due to shortcomings of the reconstruction techniques, such as regression causing variance losses back in time and bias of the target variable mean. These artifacts can appear as a consequence of noisy measurements used as predictors in regression techniques based on ordinary least squares , though ordinary least squares still provides optimal parameter estimation when the predictor variable has error . The level of high- and low-frequency variability in reconstructions also depends on the type and quality of the proxy data used as input .

The concept of pseudo-proxy experiments was introduced after millennium-long paleoclimate simulations from general circulation models (GCMs) first became available, and has been developed and applied over the last 2 decades . Pseudo-proxy experiments are used to test the skill of reconstruction methods and the sensitivity to the proxy network used; see for a review. The idea behind idealized pseudo-proxy experiments is to extract target data of an environmental variable of interest from long paleoclimate model simulations. The target data are then sampled in a spatiotemporal pattern that simulates real proxy networks and instrumental data. The target data representing the proxy period are further perturbed with noise to simulate real proxy data in a systematic manner, while the pseudo-instrumental data are left unchanged or only weakly perturbed with noise of magnitude typical for the real-world instrumental data. The surrogate pseudo-proxy and pseudo-instrumental data are used as input to one or more reconstruction techniques, and the resulting reconstruction is then compared with the true target from the simulation. The reconstruction skill is quantified through statistical metrics, both for a calibration interval and a much longer validation interval.

Available pseudo-proxy studies have to a large extent used target data from the same GCM model simulations, subsets of the same spatially distributed proxy network and a temporally invariant pseudo-proxy network . In the present paper we extend the domain of pseudo-proxy experiments to allow more flexible target data, with a range of explicitly controlled spatiotemporal characteristics. Instead of employing surrogate data from paleoclimate GCM simulations, ensembles of target fields are drawn from a field of stochastic processes with prescribed dependencies in space and time. In the framework of such an experiment design, the idealized temperature field can be thought of as a (unforced) control simulation of the Earth's surface temperature field with a simplified spatiotemporal covariance structure. The primary goal of using these target fields is to test the ability of the reconstruction method to preserve the spatiotemporal covariance structure of the surrogates in the climate field reconstruction. Earlier, generated stochastic target fields based on the AR(1) process model equations of BARCAST (Bayesian Algorithm for Reconstructing Climate Anomalies in Space and Time) introduced in Sect. . We present for the first time a data generation technique for fields of long-range memory (LRM) target data.

Additionally, we test the reconstruction skill on an ensemble member basis using standard metrics including the correlation coefficient and the root-mean-squared error (RMSE). The continuous ranked probability score (CRPS) is also employed; this is a skill metric composed of two subcomponents recently introduced for ensemble-based reconstructions .

Temporal dependence in a stochastic process over time t is described as persistence or memory. An LRM stochastic process exhibits an autocorrelation function (ACF) and a power spectral density (PSD) of a power law form: C(t)∼tβ-1 and S(f)∼f-β, respectively. The power law behavior of the ACF and the PSD indicates the absence of a characteristic timescale in the time series; the record is scale invariant (or just scaling). The spectral exponent β determines the strength of the persistence. The special case β=0 is the white noise process, which has a uniform PSD over the range of frequencies. For comparison, another model often used to describe the background variability of the Earth's SAT is the autoregressive process of order 1 (AR1; ). This process has a Lorentzian power spectrum (steep slope at high frequencies, constant at low frequencies) and thereby does not exhibit long-range correlations.

For the instrumental time period, studies have shown that detrended local and spatially averaged surface temperature data exhibit LRM properties on timescales from months up to decades . For proxy/multi-proxy SAT reconstructions, studies indicate persistence up to a few centuries or millennia . The exact strength of persistence varies between data sets and depends on the degree of spatial averaging, but in general 0<β<1.3 is adequate. The value of β>1 is usually associated with sea surface temperature, which features stronger persistence due to effects of oceanic heat capacity .

Our basic assumption is that the background temporal evolution of Earth's surface air temperature can be modeled by the persistent Gaussian stochastic model known as the fractional Gaussian noise (fGn; Chapter 1 and 2 in ; ). This process is stationary, and the persistence is defined by the spectral exponent 0<β<1. The synthetic target data are designed as ensembles of fGn processes in time, with an exponentially decaying spatial covariance structure. In contrast to using target data from GCM simulations, this gives us the opportunity to vary the strength of persistence in the target data, retaining a simplistic and temporally persistent model for the signal covariance structure. The persistence is varied systematically to mimic the range observed in actual observations over land, typically 0<β<1 . The pseudo-proxy data quality is also varied by adding levels of white noise corresponding to signal-to-noise ratios by standard deviation SNR=∞,3,1,0.3. For comparison, the signal-to-noise ratio of observed proxy data is normally between 0.5 and 0.25 . However, in , most tree-ring series were found to have SNR > 1.

The fGn model is appropriate for many observations of SAT data, but there are also some deviations. The theoretical fGn follow a Gaussian distribution, but for instrumental SAT data the deviation from Gaussianity varies with latitude . Some temperature-sensitive proxy types are also characterized by nonlinearities and non-Gaussianity .

Since the target data are represented as an ensemble of independent members generated from the same stochastic process, there is little value in estimating and analyzing ensemble means from the target and reconstructed time series themselves. Anomalies across the ensemble members will average out, and the ensemble mean will simply be a time series with non-representative variability across scales. Instead we will focus on averages in the spectral sense. The median of the ensemble member-based metrics are used to quantify the reconstruction skill.

The reconstruction method to be tested, Bayesian Algorithm for Reconstructing Climate Anomalies in Space and Time (BARCAST), is based on a Bayesian hierarchical model . This is a state-of-the-art paleoclimate reconstruction technique, described in further detail in Sect. . The motivation for using this particular reconstruction technique in the present study is the contrasting background assumptions for the temporal covariance structure. BARCAST assumes that the temperature evolution follows an AR(1) process in time, while the target data are generated according to the fGn model. The consequences of using an incorrect null hypothesis for the temporal data structure are illustrated in Fig. . Here, the original time series in Fig. a follows an fGn structure. The corresponding 95 % confidence range of power spectra is plotted in blue in Fig. c. Using the incorrect null hypothesis that the data are generated from an AR(1) model, we estimate the AR(1) parameters from the time series in Fig. a using maximum likelihood estimation. A realization of an AR(1) process with these parameters is plotted in Fig. b, with the 95 % confidence range of power spectra shown in red in Fig. c. The characteristic timescale indicating the memory limit of the system is evident as a break in the red AR(1) spectrum. This is an artifact that does not stem from the original data, but simply occurs because an incorrect assumption was used for the temporal covariance structure.

A particular advantage of BARCAST as a probabilistic reconstruction technique lies in its capability to provide an objective error estimate as the result of generating a distribution of solutions for each set of initial conditions. The reconstruction skill of the method has been tested earlier and compared against a few other climate field reconstruction (CFR) techniques using pseudo-proxy experiments. use instrumental temperature data for North America, and construct pseudo-proxy data from some of the longest time series. BARCAST is then compared to the RegEM method used by . The findings are that BARCAST is more skillful than RegEM if the assumptions for the method are not strongly violated. The uncertainty bands are also narrower. Another pseudo-proxy study is described in , where BARCAST is compared against the canonical correlation analysis (CCA) CFR method. The pseudo-proxies in that paper were constructed from a millennium-long forced run of the NCAR CCSM4 model. The results showed that BARCAST outperformed the CCA method over the entire reconstruction domain, being similar in areas with good data coverage. There is an additional pseudo-proxy study by , targeting precipitation which has a more complex spatial covariance structure than SAT anomalies. In that study, BARCAST was not found to outperform the other methods.

In the following, we describe the methodology of BARCAST and the target data generation in Sect. 2. The spectral estimator used for persistence analyses is also introduced here. Sect. 3 is comprised of an overview of the experiment setup and explains the hypothesis testing procedure. Results are presented in Sect. 4 after performing hypothesis testing in the spectral domain of persistence properties in the local and spatial mean reconstructions. The skill metric results are also summarized. Finally, Sect. 5 discusses the implications of our results and provides concluding remarks.

The experiment domain configuration is selected to resemble that of the continental landmass of Europe, with N=56 grid cells of size 5∘×5∘. The reconstruction period is 1000 years. reflecting the last millennium. The reconstruction region and period are inspired by the BARCAST reconstructions in and approximate the density of instrumental and proxy data in reconstructions of the European climate of the last millennium. The temporal resolution for all types of data is annual. By design, the target fGn data are meant to be an analogue of the unforced SAT field. We will study both the field and spatial mean reconstruction.

Pseudo-instrumental data cover the entire reconstruction region for the time period 850–1000 and are identical to the noise-free values of the true target variables. The spatial distribution of the pseudo-proxy network is highly idealized as illustrated in Fig. , the data covers every fourth grid cell for the time period 1–1000. The pseudo-proxies are constructed by perturbing the target data with white noise according to Eq. (). The variance of the proxy observations is τP2, and the SNR is calculated as 12SNR=β12Var(Tt)τP2.

Our set of experiments is summarized in Table  and comprises target data with three different strengths of persistence, β=0.55,0.75,0.95 and pseudo-proxies with SNR =∞,3,1, and 0.3. In total, 20 realizations of target pseudo-proxy and pseudo-instrumental data are generated for each combination of β and SNR and used as input to BARCAST. The reconstruction method is probabilistic and generates ensembles of reconstructions for each input data realization. In total, 30 000 ensemble members are constructed for every parameter setup.

BARCAST successfully estimates posterior distributions for all reconstructed temperature fields and scalar parameters. Convergence is reached for the scalar parameters despite the inconsistency of the input data temporal covariance structure with the default assumption of BARCAST. Table  lists the true parameter values used for the target data generation, and Table  summarizes the mean of the posterior distributions estimated from BARCAST. Studying the parameter dependencies, it is clear that the posterior distributions of α and σ2 depend on the prescribed β and, to a lesser extent, SNR for the target data. Instead of listing the posterior distributions of τP2 and β1 we have estimated the local reconstructed SNR at each proxy location using Eq. ().

Further results concern the spectral analyses and skill metrics. All references to spectra in the following correspond to mean spectra. Analyses of the reconstruction skill presented below are performed on a grid point basis, and for the correlation and RMSE also for the spatial mean reconstruction. While the latter provides an aggregate summary of the method's ability to reproduce specified properties of the climate process on a global scale, the former evaluates BARCAST's spatial performance.

In this study we have tested the capability of BARCAST to preserve temporal LRM properties of reconstructed data. Pseudo-proxy and pseudo-instrumental data were generated with a prescribed spatial covariance structure and LRM temporal persistence using a new method. The data were then used as input to the BARCAST reconstruction algorithm, which by design use an AR(1) model for temporal dependencies in the input/output data. The spatiotemporal availability of observational data was kept the same for all experiments in order to isolate the effect of the added noise level and the strength of persistence in the target data. The mean spectra of the reconstructions were tested against the null hypotheses that the reconstructed data can be represented as LRM processes using the parameters specified for the target data, or as AR(1) processes using the parameter estimated from BARCAST.

We found that despite the default assumptions in BARCAST, not all local and spatial mean reconstructions were consistent with the AR(1) model. Figures – and Tables – summarize the hypothesis testing results; the local reconstructions at grid cells between proxy locations follow to large extent the AR(1) model, while the local reconstructions directly at proxy locations are more similar to the original fGn data. However, the simulated proxy quality is crucial for the spectral shape of the local reconstructions, with higher noise levels indicating better agreement with the AR(1) model than fGn.

All spatial mean reconstructions are consistent with the fGn process null hypothesis according to Table . For the two cases β=0.55,0.75, SNR = 0.3, the spatial mean reconstructions are also consistent with the AR(1) process null hypothesis. This is clear from the spatial mean reconstruction spectra (gray curves in Fig. ) and from comparing the hypothesis testing results in Tables  and . The improvement in scaling behavior with spatial averaging is expected, as the small-scale variability denoted by ϵt in Eq. () is averaged out. Eliminating local disturbances naturally results in a more coherent signal. However, the spatial mean of the target data set does not have a significantly higher β than local target values. This is due to the relatively short spatial correlation length chosen: 1/ϕ=1000 km. In observed temperature data, spatial averaging tends to increase the scaling parameter β .

The power spectra in Figs. –b–d show that the temporal covariance structure of the reconstructions is altered compared with the target data for all experiments where noisy input data were used. Furthermore, the spectra of the pseudo-proxies in Fig. b–d all deviate from the target in the high-frequency range, but for a different reason. The pseudo-proxy data deviate from the target due to the white proxy noise component, while the reconstruction deviate because BARCAST quantifies the proxy noise from an AR(1) assumption. Real-world proxy data are generally noisy, and the noise level is normally at the high end of the range studied here. We demonstrate that the variability level of the reconstructions does not exclusively reflect the characteristics of the target data, but is also influenced by the fitting of noisy data to a model that is not necessarily correct. At present, there exists no reconstruction technique assuming explicitly that the climate variable follows an LRM process.

In addition to BARCAST, other reconstruction techniques that may experience similar deficiencies for LRM target data are the regularized expectation–maximization algorithm (RegEM; ; ) and all related models (CCA, PCA, GraphEM). These models assume observations at subsequent years are independent . The assumption of temporal independence corresponds to yet another incorrect statistical model for our target data, a white noise process in time. Note that for target variables/data sets consistent with a white noise process, these types of reconstruction methods are appropriate, as demonstrated using the truncated empirical orthogonal function (EOF) principal component spatial regression methodology on precipitation data in .

When an incorrect statistical model is used to reconstruct a climate signal, the temporal correlation structure is likely to be deteriorated in the process. For the range of different reconstructions available, such effects may contribute to discussions on a number of questions under study, including the possible existence of different scaling regimes in paleoclimate; see .

The criteria for the hypothesis testing used in this study are strict, and may be modified if reasonable arguments are provided. For example, if the first null hypothesis used here was modified so that only the low-frequency components of the spectra were required to fall within the confidence ranges, more of the reconstructions would be consistent with the fGn model. However, from studying the spectra in Figs. –, it is generally unclear where one should set a threshold, since the spectra show a gradual change with a lack of any abrupt breaks. Considering real-world proxy records, the noise color and level is generally unknown or not quantified. We know there are certain sources of noise influencing different frequency ranges that are unrelated to climate, but it is difficult to decide when the noise becomes negligible compared with the effects of climate driven processes. The decision to use all frequencies for the hypothesis testing in this idealized study is therefore a conservative and objective choice.

The skill metrics used to validate the reconstruction skill are the Pearson's correlation coefficient r, the RMSE and CRPS‾, the latter divided into the CRPS‾pot and the Reli‾. Skill metric results are illustrated in Figs. – and summarized in Table . The reconstruction skill is sensitive to the proxy quality, and highest at sites with co-localized proxy information. This is an expected result, due to the BARCAST model formulation and our choice of a relatively short decorrelation length 1/ϕ∼1000 km. Contrasting results of high skill away from proxy sites and poor skill close to proxy sites have been documented in , although care must be taken in the comparison as that paper used a different reconstruction methodology (truncated EOF principal component spatial regression) and the target variable was precipitation/hydroclimate instead of SAT. Without performing dedicated pseudo-proxy experiments it is difficult to resolve the main cause of these contrasting results for spatial skill.

Several extensions to the presented work appear relevant for future studies, including (a) implementing external forcing and responses to this forcing in the target data to make the numerical experiments more realistic, (b) generating target data using a more complex model than described in Sect.  and (c) reformulating BARCAST model Eqs. (1)–(2) to account for LRM properties in the target data. In addition, there is the possibility of repeating the experiments from this study using a different reconstruction technique, and experiments with more complicated spatiotemporal design of the multi-proxy network can also be considered .

The alternatives (a) and (b) can be implemented together. Relevant advancements for target data generation can be obtained using the class of stochastic–diffusive models, such as the models described in . The alternative method for generating spatial covariance stands in contrast to what is done in the present study. The data generation technique used in this paper and also in and generates a signal without spatial dynamics, where the spatial covariance is defined through the noise term. On the other hand, the stochastic–diffusive models generate the spatial covariance through the diffusion, without spatial structure in the noise term. The latter model type may be considered more physically correct and intuitive than the simplistic model used here. use an exponential model for the temporal covariance structure, while use an LRM model.

For the BARCAST CFR methodology, reformulation of model Eqs. (1)–(2) would drastically improve the performance in our experiments. However, at present we cannot guarantee that modifications favoring LRM are practically feasible in the context of a Bayesian hierarchical model, due to higher computational demands. Changing the AR(1) model assumption to instead account for LRM would in the best scenario slow the algorithm down substantially, and in the worst scenario it would not converge at all. Some cut-off timescale would have to be chosen to ensure convergence. Regarding the spatial covariance structure, accounting for teleconnections introduces similar computational challenges. The more general Matérn covariance family form has already been implemented for BARCAST, but was not used in this study. Another problem is the potential temporal instability of teleconnections; it is possible that major climate modes might have changed their configuration through time. Therefore, setting additional a priori constraints on the model may not be considered justified. The use of exponential covariance structure appears to be a conservative choice in such a situation.

The pseudo-proxy study presented here sets a powerful example for how to construct and utilize an experimental structure to isolate specific properties of paleoclimate reconstruction techniques. The generation of the input data requires far less computation power and time than for GCM paleoclimatic simulations, but also results in less realistic target temperature fields. We demonstrate that there are many areas of use for these types of data, including statistical modeling and hypothesis testing.