# Probabilistic reconstructions of local temperature and soil moisture from tree-ring data with potentially time-varying climatic response

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s00382-014-2139-z

- Cite this article as:
- Tolwinski-Ward, S.E., Tingley, M.P., Evans, M.N. et al. Clim Dyn (2015) 44: 791. doi:10.1007/s00382-014-2139-z

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## Abstract

We explore a probabilistic, hierarchical Bayesian approach to the simultaneous reconstruction of local temperature and soil moisture from tree-ring width observations. The model explicitly allows for differing calibration and reconstruction interval responses of the ring-width series to climate due to slow changes in climatology coupled with the biological climate thresholds underlying tree-ring growth. A numerical experiment performed using synthetically generated data demonstrates that bimodality can occur in posterior estimates of past climate when the data do not contain enough information to determine whether temperature or moisture limitation controlled reconstruction-interval tree-ring variability. This manifestation of nonidentifiability is a result of the many-to-one mapping from bivariate climate to time series of tree-ring widths. The methodology is applied to reconstruct temperature and soil moisture conditions over the 1080–1129 C.E. interval at Methusalah Walk in the White Mountains of California, where co-located isotopic dendrochronologies suggest that observed moisture limitations on tree growth may have been alleviated. Our model allows for assimilation of both data sources, and computation of the probability of a change in the climatic controls on ring-width relative to those observed in the calibration period. While the probability of a change in control is sensitive to the choice of prior distribution, the inference that conditions were moist and cool at Methuselah Walk during the 1080–1129 C.E. interval is robust. Results also illustrate the power of combining multiple proxy data sets to reduce uncertainty in reconstructions of paleoclimate.

### Keywords

Bayesian hierarchical modeling Biological–statistical modeling Multiproxy paleoclimate reconstruction Tree-ring width Time-varying climate-paleodata relationship## 1 Introduction

Time series of tree-ring widths (TRWs) provide one of the best-dated and most spatially extensive records of paleoclimatic variability (Jansen et al. 2007), and most attempts to reconstruct climate at global or hemispheric scales rely heavily upon them (Jones et al. 1998; Crowley and Lowery 2000; Moberg et al. 2005; D’Arrigo et al. 2006; Mann et al. 2008; Christiansen and Ljungqvist 2011). Such reconstructions are often based on establishing linear, empirical–statistical relationships between the paleo-observations and a single climatic variable during a calibration period, and assuming the climate-paleodata relationship applies in earlier times without modification. Researchers have long recognized that the latter assumption, known as ‘the principle of uniformitarianism’ (Bradley 1999, p. 4), should be tested (e.g. Hughes and Ammann 2009), especially when past values of the climatic drivers of the paleodata may have been outside the range observed in the calibration period. Such tests require methods that deal explicitly with changes in the response by modeling the causal effect of a changing environment on the proxy archive, rather than treating the data as if they were produced by using thermometers or rain gauges.

Inferring climate from ring-width data is challenging because trees are fundamentally lossy recorders of climate. Information about the climatic conditions at the time of growth is lost in part because growing trees integrate subannually-resolved climatic conditions (Bradley 2011), and because ring-width chronologies are standardized to remove non-climatic effects on growth (Cook and Kairiukstis 1990). A larger source of information loss arises from the joint influence of both temperature and moisture on tree growth. Without assuming that a site’s climatology was the same in the past as during instrumentally observed times, it is impossible to attribute tree ring width variability to variations in one climatic variable or the other. In mathematical terms, the transformation from the space of all possible climate histories to the space of tree-ring with series is not one-to-one. This hallmark feature of inverse problems means that regularization, or the incorporation of information from an informative prior or an additional type of observation, is required to enable mathematical solution.

In traditional regression-based climate reconstructions, the problem is implicitly regularized by restricting analysis to sets of data that roughly conform to the assumption of a linear and stationary relationship between ring width and univariate climate. Indeed, site-selection techniques have been developed in dendrochronology to maximize the chances that the signal contained in ring-width samples is univariate (Stokes and Smiley 1968; Cook and Kairiukstis 1990. While this traditional approach yields a statistically straightforward solution, it sacrifices more nuanced, mechanisms-based information about paleodata formation. In particular, paleo-observations that originate from an evolving and/or nonlinear proxy system response to environmental forcing may bias reconstruction and uncertainty estimates derived by assuming a static and/or linear response.

Here we develop, test, and apply a Bayesian hierarchical model for inferring past climate, based on a mechanistic understanding of the tree ring response to soil moisture and temperature variability. Bayesian hierarchical models (BHMs) permit a fundamentally different solution to the inverse problem, and are powerful and well-established tools in the statistical sciences (Gelman et al. 2003). Similar tools have recently been proposed (Hughes and Ammann 2009; Tingley et al. 2012) and used (Li et al. 2010; Tingley and Huybers 2010a, b, 2013; Haslett et al. 2006; Wahl et al. 2011; Guiot et al. 2009; Parnell et al. 2008; Yu et al. 2012; Garetta et al. 2012) for aspects of paleoclimate reconstructions. This inference framework enables coupling of forward models of the lossy ring-width formation process to stochastic models of climate. “Inverting” the mechanistically based forward models using Bayes’ law can then provide a different avenue for regularizing the reconstruction problem and lending increased scientific interpretability to results (Guiot et al. 2000; Hughes and Ammann 2009). In particular, use of a validated forward model of tree-ring width growth that models the potential of the biological system to change its response to climate may provide a more defensible interpretation of ring-width data. Within the Bayesian framework, the relative probabilities of distinct climate histories that map to the same paleodata series under the forward model are weighted according to a formal model for the prior information about the site climatology.

We develop a Bayesian hierarchical model that employs a mechanisms-based forward model of tree-ring growth to tie the data to past variations in temperature and soil moisture (Sect. 2). For a limited range of climatic inputs, the model can produce synthetic tree-ring responses that appear linear and univariate, but for more widely varying climatic time series, it can produce switching in the response of paleodata from temperature to soil moisture or vice versa. The forward model can thus be viewed as a more general representation of tree-ring growth that subsumes models expressing ring width as a linear function of univariate climate. In the inverse reconstruction problem, this forward response-switching capability manifests itself in non-Gaussian, bimodal posterior probability densities, which we explore in a controlled “pseudoproxy” reconstruction experiment (Sect. 3). We use synthetic target climate series and simulated tree-ring data engineered such that the climatic controls on the data are known to differ in the calibration- and reconstruction-intervals.

We use the same approach to infer pre-instrumental climate from Methuselah Walk, a site close to the lower elevational limit of Great Basin bristlecone pine (*Pinus longaeva* DK Bailey) in the White Mountains of California (Sect. 4). The ring-width data from this specific site are known to be primarily indicative of local moisture variability (Hughes and Funkhouser 1998), but isotopic data from trees at the same site indicate that a 50-year medieval interval was anomalously wet (Leavitt 1994). In this real-data context, the goal of our research is threefold. First, we seek an estimate of the paleoclimatology of the interval 1080–1129 AD that accounts for the possibility that the past ring-width signal may contain information about temperature or about soil moisture. Secondly, we wish to elucidate the importance of prior information on the inference when a more general model of proxy response to climate is used to interpret the tree-ring data. Finally, we explore the utility of an additional source of proxy information, in this case isotope dendrochronologies, to reduce reconstruction uncertainty when the ring-width indices are allowed to be interpreted more broadly than as a linear function of univariate climate. We end with discussion and conclusions (Sect. 5).

## 2 A hierarchical model for climate and tree ring widths

*process level*and denoted \([X|\theta _X]\), where \(X\) is the process of interest and \(\theta _X\) is a parameter or set of parameters on which the prior model may depend. A

*data-level model*\([Y|X,\theta _Y]\) describes the dependence of the data \(Y\) on \(X\), and includes a set of data-level model parameters \(\theta _Y\). In the Bayesian framework, the parameters of the data- and process-level models are also given random interpretation and

*parameter-level*prior distributions \([\theta _X]\) and \([\theta _Y]\). The object of our analysis is the

*posterior distribution*\([X,\theta _X,\theta _Y|Y]\), the joint distribution of the process of interest and model parameters conditional on the data, and is defined through Bayes’ law:

Model variables and parameters

| |

\(W(t)\) | Ring-width index (unitless, annual resolution) |

| |

\(T_1\), \(T_2\) | Threshold temp. for nonzero and optimal growth, resp. (\(^\circ \)C) |

\(M_1\), \(M_2\) | Threshold soil moist. for nonzero and optimal growth, resp. (v/v) |

\(\sigma ^2_w\) | Variance of data model noise (unitless) |

| |

\(T(s,t)\) | Temperature (\(^\circ \)C, monthly resolution) |

\(M(s,t)\) | Soil Moisture (v/v, monthly resolution) |

| |

\(\varDelta T, \varDelta M\) | Mean shifts in \(T\), \(M\) climatology (calibr. interval std. deviations) |

| |

\(\mu _T(s)\) | Mean temperature in month \(s\) (\(^\circ \)C) |

\(\sigma ^2_T(s)\) | Temperature variance in month \(s\) (\(^\circ \)C\(^2\)) |

\((\alpha _M(s),\beta _M(s))\) | Soil moisture shape parameters in month \(s\) (unitless) |

\((M_n,M_x)\) | Minimum and maximum allowable soil moisture values (v/v) |

\(S_s\) | Covariance matrix for temperature and soil moisture in month \(s\) |

\((\phi _1, \phi _2)\) | AR(1) parameters underlying temporal structure of climate (unitless) |

### 2.1 Modeling the data level variable

The time series of ring-width indices that we take as our starting point is two steps removed from raw measurements. Generally ‘site chronologies’ are derived from the combination of many replicated measurements from multiple co-located trees, with repeated samples taken from each tree. In this case we used such a mean series for the Methuselah Walk location. It was developed by D.A. Graybill and is available as record CA535 obtained from the International Tree-Ring Data Bank, as are the raw measurements from which it was calculated.^{1} In order to remove non-climatic variation in ring width associated with tree size or age, each sample’s annual ring-width series was detrended. This was done by fitting a negative exponential curve or a straight line of zero or negative trend and then deriving dimensionless tree-ring width indices by dividing the actual ring-width by that year’s value for the fitted line. These individual sample tree ring-width indices were then averaged to form the site chronology. There were at least 31 samples represented for every year of the period of interest in this case (1080–1129 C.E.). Stokes and Smiley (1968) and Cook and Kairiukstis (1990) provide further background on the development of dendrochronologies. While others have worked toward explicit statistical models of the chronology from raw measurements,^{2} here we use the derived series and neglect any sources of error deriving from the data processing.

The minimum in Eq. 5 represents a second nonlinearity not typically represented in traditional reconstructions of climate. This model feature is meant to mimic the Principle of Limiting Factors, which states that tree ring growth is constrained by the environmental variable that is most limiting (Fritts 2001). The minimum endows the model with a “switching” ability, in which the response to climate in a given month may be controlled by either temperature or soil moisture depending on their relative values. The sum over months integrates subannual climatic influences on growth over the growing season on the modeled signal. The resulting series \(\left\{ \varGamma (t), t = 1,2,\ldots ,T\right\}\) is standardized over the \(T\) simulation years to produce a simulated ring width index series \(\hat{\varvec{W}}_{VSL}\) comparable with observed standardized chronologies over the same time interval. The VS-Lite model has been validated for a range of sites across North America (Tolwinski-Ward et al. 2011) as well as globally (Breitenmoser et al. 2013).

### 2.2 Joint modeling of temperature and moisture

The parameters \(\varDelta T\) and \(\varDelta M\) describe changes in the climatology compared to the calibration interval parameterization, and are treated as random variables. When \(\varDelta T = \varDelta M = 0\), as in the calibration interval itself, the transformation of temperature depends only on seasonally dependent parameters \(\mu _T(s)\) and \(\sigma _T(s)\), which are given by the sample mean and standard deviation of the monthly calibration-interval temperatures in month \(s\) across all calibration-interval years. Likewise, when \(\varDelta M = 0\), Eq. 8 transforms the four-parameter beta-distributed soil moisture into the standard-normal \(M'\) process via anamorphosis (Chiles and Delfiner 1999, section 6.2). In contrast, nonzero values of \(\varDelta T\) and \(\varDelta M\) shift the mean of the perturbed variables \(T'\) and \(M'\) away from zero, allowing for climatic conditions that are substantially different from the calibration interval. The presence of these parameters relaxes the often implicitly-made assumption in more traditional reconstructions that the reconstruction interval has the same climatology as that observed in the instrumental period.

To develop a realistic stochastic description of monthly resolved temperature and soil moisture, we fit the process-level model to the gridded Parameter–Elevation Regressions on Independent Slopes Model (PRISM) data product (Daly et al. 2008). The gridded data product consists of monthly means over the daily recorded minimum temperatures, monthly means over the daily recorded maximum termperatures, and monthly accumulated precipitation, interpolated to \(4\text{ km}\times 4\text{ km}\) resolution across the continental United States. The model for temperature is fitted on the mean of the two temperature fields. We derive estimates of soil moisture by using PRISM as inputs to the Climate Prediction Center’s “Leaky Bucket” monthly time-step model of hydrology (Huang et al. 1996), a physically-based water balance model with Markov structure. Measurement and model error inherent in the PRISM data product and the Leaky Bucket model are both neglected.

We take an empirical Bayes’ approach to the process-level parameters, estimating parameters of the transforms, the seasonality, temporal persistence, and covariance of temperature and soil moisture “offline” and holding them fixed thereafter (Robert 2007, section 10.4). We estimate soil moisture parameters using maximum likelihood based on linear transformations to the unit interval of calibration-interval soil moisture computed in month \(s\). We use the sample covariance matrix of deseasonalized time series of temperature and soil moisture anomalies in month \(s\) over the calibration interval for \(S_s\), and \(\phi _1\) and \(\phi _2\) are least-squares estimates of the AR(1) coefficients.

### 2.3 Modeling the parameter level variables

The parameters \(T_1, T_2, M_1, M_2\) of the data-level model are loosely interpretable as temperature and soil moisture thresholds above which growth begins or is no longer sensitive to climatic fluctuations. They are given priors based on current scientific understanding of thresholds for tree growth following Tolwinski-Ward et al. (2013). We use a noninformative uniform prior distribution over the unit interval for the error model variance \(\sigma ^2_w\). Given the scaling of the VS-Lite estimate in the data-level model (Eq. 6), values of \(\sigma ^2_w\) near zero (one) would indicate a tree-ring width series with high (low) signal-to-noise ratio.

## 3 Reconstruction from synthetic data with time-varying climate response

We first present a numerical experiment to explore the switching behavior of the hierarchical model. Simulated, or “pseudoproxy”, tree-ring-width data are constructed such that cool, moist conditions drive temperature-limited growth in the calibration interval, but warmer and drier conditions in the reconstruction interval cause a predominantly moisture-driven signal. The reconstruction experiment explores the climatic inferences that are possible from the model using data near the response space boundaries described by VS-Lite (Fig. 2). This “PseudoProxy Experiment” is hereafter referred to as the PPE.

### 3.1 Methods and design

We choose process-level parameters (Sect. 2.2) to simulate a location with cool moist climatology, and use the resulting model to simulate a 30-year calibration interval. A 30-year reconstruction interval was also simulated by using the same model but with climate change parameters set to \(\varDelta T = 1.75, \varDelta M = -1.5\), corresponding to warming and drying. Parameters for VS-Lite were chosen such that simulated growth under the synthetic calibration interval is temperature-limited, while moisture-limited growth results for the shifted reconstruction-interval climatology (Fig. 3, top panel). The simulated calibration and reconstruction interval data series are comparable in mean and variability (Fig. 3, lower panels), and so there is no way to tell without further information whether each series is the result of temperature- or moisture-limited growth.

The value of the data-level model noise \(\sigma ^2_W\) was set to 0.5 to give a signal-to-noise ratio of 1, which is considered optimistic for real paleoclimate proxies (Smerdon 2012). The reconstruction was performed with data-level parameters fixed at the values used to generate the pseudoproxy data. The only model parameters inferred are the process-level parameters \(\varvec{\theta }_{T,M} = (\varDelta T, \varDelta M)^T\). We choose the spread of the prior to demonstrate whether the simulated change can be recovered by the model when it is unlikely, but not outside the realm of possibilities allowed by the prior (Eq. 10). To do so we set the prior variance \(\sigma ^2_\varDelta\) such that the simulated climate change falls outside the contour containing the 75 % most probable climate changes, but is still contained within the contour containing 95 % of the prior probability mass.

### 3.2 Results

The PPE reconstruction consists of a probabilistic ensemble of 2,000 realizations from the posterior distribution of the climate history given the paleodata and data- and process- level model structures (further details of the sampling are given in the “Appendix”). Of principal interest is the posterior reconstruction of the climatic shifts \(\varDelta T\) and \(\varDelta M\) relative to the calibration interval, but we also look at the reconstructed interannual variations in temperature and moisture in order to better interpret our results. Although the reconstruction realizations are resolved at a monthly time-step, the posterior climatic variability could only be meaningfully distinguished at seasonal timescales, reflecting the fact that individual monthly variations are not identifiable from annually-resolved data. We explore the reconstructed interannual variability by looking at the ensemble of means over June, July and August, when the longer relative day lengths result in the bulk of modeled growth.

Separating the posterior realizations of bivariate time series into those that imply differing controls on tree-ring growth can be used to explain the bimodality in the posterior. We classify each climate history by computing the fraction of reconstruction years that fall into each of four categories: (1) the mean summer growth response to temperature is nonzero but less than the growth response to moisture; (2) the growth response to moisture is nonzero but less than the temperature response; (3) both growth responses are one; and (4) both responses are zero. We classify realizations for which these fractions are statistically significantly greater than 0.5 as predominantly temperature-limited, moisture-limited, complacent, or inhospitable for growth, respectively. By this scheme, 41.8 % of the posterior contains temperature-limited climate histories, 48.7 % contains moisture-limited histories, 0 % produce complacent histories, and 0 % produce histories which tended not to grow at all. The remaining 9.5 % of histories had growth which was not dominated by any single control over the reconstructed years. The methodology thus estimates the probability of reconstruction interval climate controls on tree growth that differ from those “observed” during calibration at 58.2 %.

## 4 Multiproxy reconstruction at Methuselah Walk

As a real-world application, we reconstruct the shifts in temperature and soil moisture climatology during the interval 1080–1129 C.E. using a bristlecone pine ring-width chronology at the Methuselah Walk site in the White Mountains of California (37.26N, \(-\)118.10E). This location is characterized by a dry climate, and the Methuselah Walk chronology, which sits more than 700 m below treeline, has positive statistical association with local precipitation and negative association with temperature (Salzer et al. 2009). LaMarche (1974) interpreted tree-ring width records from Methuselah Walk as indicating wet conditions in late eleventh and early twelfth centuries, and also characterized this period as ‘warm’ based on other material at and near tree-line in the same mountain range. More recently, Salzer et al. (2009) found that the response of bristlecone pine to environmental forcing switches from a negative to a positive relationship with temperature in a zone very close to upper treeline. Based on this result, Salzer et al. (2013) restricted their analysis of past temperatures to tree-ring width data from within 100 m of the contemporaneous upper treeline, and reconstructed colder conditions in the first decades of the twelfth century. We note that the skill of the VS-Lite model at simulating the Methuselah Walk chronology was previously demonstrated by Tolwinski-Ward et al. (2011).

Isotope ratios of \(^{13}C/^{12}C\) from the area may also be interpreted as indicators of moisture conditions, and we explore the effects of additionally assimilating this source of information into the reconstruction. In general, depleted (enriched) values of the isotope ratio \(\delta ^{13}C\) at arid sites are indicative of drier (wetter) conditions (McCarroll and Loader 2004). Leavitt (1994) detrended an isotope chronology from the site to remove low-frequency changes associated with non-climatic variations such as atmospheric isotopic concentration, and found the resulting series was closely tied to moisture conditions. Using this time series, Leavitt (1994) inferred anomalously wet conditions in the period 1080–1129 C.E. The \(\delta ^{13}\)C results of Leavitt (1994) have since been independently replicated and extended to annual resolution at a nearby location, with two clusters of inferred wet years centered on the late 1080s and the early 1120s C.E. (Bale et al. 2011, Fig. 4). We use our hierarchical model to explore the possibility that the moisture controls on ring width growth observed during the calibration interval at Methuselah Walk may have been alleviated during this 1080–1129 C.E. interval due to increased moisture availability.

### 4.1 Methods and design

In this real-data application, the unknown \(\varvec{\theta }_W\) increases both the dimensionality and uncertainty of the problem. Although it is still computationally feasible to reconstruct the annual and subannual climatic variations, we focus on the question of what can be learned about the lower-dimensional shifts in climatology \((\varDelta T, \varDelta M)\). To explore the dependence of the answer on paleoproxy information, we compare the inferences that can be made from the tree-ring width data and isotopic data separately, and by using the two in combination.

Each reconstruction is also run with two priors distinguished by their variance (\(\sigma ^2_\varDelta\) in Eq. 10) to examine the influence of prior information on the results. The first prior, hereafter called the “7 \(^\circ\)C prior”, uses a value of the variance parameter such that \(\varDelta T = \sigma _\varDelta\) would result in a local temperature change equal or greater than 7 \(^\circ\)C. The 7 \(^\circ\)C prior allows for extremely large amplitude climatic change in the context of estimated global Holocene variability, as the upper bound on estimates of global temperature change since the Last Glacial Maximum are roughly 7 \(^\circ\)C (Jansen et al. 2007). This prior makes local climatic changes of this magnitude a relatively frequent “1-sigma” event. While the scales of local and global temperature changes differ, the 7 \(^\circ\)C prior is designed to demonstrate the effects of prior variance that is large in paleoclimatic terms on the inference from our model. The spread of the second prior is set such that \(\varDelta T = 2\sigma _\varDelta\) would induce a local temperature change in the least variable month at our site of 4.4 \(^\circ\)C. To provide a second paleoclimatic benchmark, 4.4 \(^\circ\)C is the upper bound on estimates of the temperature change associated with the so-called 8.2 ka event, the most extreme climatic excursion during the Holocene (Kobashi et al. 2007). This second prior represents a more conservative estimate of the potential magnitude of climatological shifts, with local climate change of the magnitude of the 8.2 ka event being a less frequent “2-sigma” event. We refer to this more informative prior as the “2.2 \(^\circ\)C prior”.

The data-level model for tree-ring width index is unchanged from the PPE. As our focus is on the climatic shifts, we do not directly estimate the monthly temperature and soil moisture anomalies. However, to ensure accurate uncertainty propagation, we integrate over Monte Carlo samples of \(\varvec{T}', \varvec{M}'\) from the same process-level model used in the PPE (see “Appendix” for further details). We additionally use calibration interval ring-width data and estimates of temperature and soil moisture derived from calibration-interval instrumental data, together generically denoted as \(D_{cal}\), to put informative priors on the parameters of the data-level model for tree-ring width (c.f. Tolwinski-Ward et al. 2013).

- 1.Isotope data alone:$${}[\varDelta T,\varDelta M|\overline{I_\delta },D_{cal}] \propto [\overline{I_\delta }|\varDelta M,D_{cal}][\varDelta T][\varDelta M],$$(12)
- 2.Tree-ring width alone:$$[\varDelta T,\varDelta M,\varvec{\theta }_W|\varvec{W},D_{cal}] \propto \int [\varvec{W}|\varvec{T}',\varvec{M}',\varDelta T, \varDelta M,\varvec{\theta }_W] [\varvec{T}',\varvec{M}'] [\varDelta T][\varDelta M][\varvec{\theta }_W|D_{cal}] d\varvec{T}' d\varvec{M}'.$$(13)
- 3.Both data types:$$[\varDelta T,\varDelta M,\varvec{\theta }_W|\varvec{W},\overline{I_\delta },D_{cal}] \propto \int [\varvec{W}|\varvec{T}',\varvec{M}',\varDelta T, \varDelta M,\varvec{\theta }_W][\varvec{T}',\varvec{M}'] [\varDelta T] [\overline{I_\delta }|\varDelta M,D_{cal}] [\varDelta M][\varvec{\theta }_W|D_{cal}] d\varvec{T}'d\varvec{M}'$$(14)

### 4.2 Results

The more informative 2.2 \(^\circ\)C prior rules out a priori the placement of probability mass near the boundary between regions II and III, and thus posteriors from uniproxy and multiproxy sources show the reconstruction interval climatology still firmly in the linear moisture-sensitive response regime of the trees (Fig. 6d–f). As with the 7 \(^\circ\)C prior, conditioning on only the isotope data results in higher estimates of the soil moisture climatology than compared to the calibration interval (Fig. 6d). The posterior given the ring-width data only also shows a tendency toward wetter conditions, as well as some negative association between shifts in temperature and moisture climatology (Fig. 6e). Combining both data sources again results in the greatest reduction of uncertainty (Fig. 6f) with estimated probabilities of the various growth regimes are 0.002 % for temperature-limited, 91.9 % moisture-limited, 8.091 % complacent, and 0 % for climate regimes that are inhospitable for growth. The probability of a change in tree growth controls is estimated in this case to be 8.1 %.

Combined with either prior, the information contained in the isotope and ring-width data together indicate that local conditions were most likely wetter and cooler from 1080–1129 C.E. at the Methuselah Walk site than instrumentally observed modern climates. However, the magnitude of the inferred change, as well as the most likely control on tree-ring growth, depends critically on the choice of prior.

## 5 Discussion and conclusions

Mechanisms-based forward models of paleodata formation can account for scientifically-understood complexities of data formation, but may result in statistical nonidentifiability in reconstructions of past climate. Using VS-Lite to perform the reconstruction, which provides a more nuanced model of the relationship of tree-ring growth to climate than linear functions of univariate climate, produces bimodality in posterior estimates of temperature and soil moisture. This feature of the posterior accurately reflects the nature of the information contained in the data and the understanding of tree ring formation as modeled by VS-Lite. Without regularization (i.e., additional sources of information), the paleo-observations provide only weak constraints on the solution and cannot distinguish between a set of reconstructed climate histories where past temperature variability explains the observed time series of ring width index, versus a set where moisture variations controlled growth. In situations where no further information exists, and where the data-level model provides the best possible description of the link between the climate and the data, such ambiguity should in fact characterize the most defensible inference that can be drawn from the available information.

Our results also demonstrate two avenues for regularizing such nonidentifiability: complementary paleodata sources, and increasing the information in the prior. The value of a second data source is most clearly demonstrated by comparing posteriors given the various combinations of data under the ‘7 \(^\circ\)C prior’. The posterior uncertainty given either data source alone is large compared to the uncertainty when the two data sets are used for combined inference (Fig. 6a–c). The potential of the prior to regularize the solution is illustrated by comparing the posterior of climatological shifts given tree-ring data using the extremely broad ‘7 \(^\circ\)C prior’ versus the much sharper ‘2.2 \(^\circ\)C’ prior. If one believes that shifts in the underlying climatology of the magnitude spanned by the former prior were feasible within the reconstructed interval, then one would have to accept that the data could potentially reflect variations in temperature or moisture (Fig. 6b). This degeneracy in the solution is regularized by using the more informative ‘2.2 \(^\circ\)C prior’, which transfers the information in the data to posterior inference on moisture conditions, rather than temperature, with nearly absolute certainty (Fig. 6e).

This dependence of the results on the prior underscores the importance of careful prior elicitation. While some argue that priors should contain as little information as possible, if any, so that the data can “speak for themselves”, we note that natural paleoclimatic data generally provide only very weak constraints to estimates of past climates precisely because of the complexities of the processes by which they are formed. This feature of the data necessitates informative priors in a paleoclimatic estimation context. Empirical–statistical reconstruction methods may arrive at the same conclusions as Bayesian analyses by making traditional assumptions *implicitly* before beginning an inference procedure. For example, such methods usually assume the relationship of climate to paleo-observation has not changed in the past, in contrast to the more general and scientifically-grounded data-level model used here. Additionally it is typically assumed that the amplitude of climatic variability captured by the data is contained in the range of instrumentally observed observations, which is analogous to an explicit assumption of a narrower prior in the Bayesian context. We argue that the explicit manner in which assumptions are laid out under the Bayesian framework provides greater transparency under which reconstruction approaches can be critically evaluated and improved as new understanding of climate processes and natural paleoclimate data become available.

The inference that the interval 1080–1129 C.E. was wet and cool at Methuselah Walk relative to the modern record is robust to the choice of prior when the analysis is conditioned on both types of paleodata, and is also consistent with several other lines of evidence. Stine (1990, 1994) used a combination of radiocarbon-dated and precisely located tufa-encrusted plant remains and geomorphic features to demonstrate the existence of ‘extremely severe drought conditions for more than two centuries before AD 1112 and for more than 140 years before AD 1350’ in this part of California, including Mono Lake, a basin with no outlet. Stine (1990, 1994) also noted that, between the two droughts, there was a ‘period of increased wetness’ in the eastern Sierra Nevada that resulted in a high stand of Mono Lake that has only been surpassed twice since, in the late fifteenth and mid seventeenth centuries C.E. (Stine 1990, Fig. 6). Subsequently, Graham and Hughes (2007) used hydrologic modeling to quantitatively link moisture-sensitive tree-ring width records in the White Mountains to Mono Lake level. Reconstructions of North American megadroughts from tree-ring networks by Cook et al. (2009) also find our interval to be comparably moist, while temperature reconstructions of LaMarche (1974) and Salzer et al. (2009, 2013) further support the inference of cooling conditions throughout the late eleventh/early twelfth century.

There are several ways the present model could be expanded for future applications. In the presence of more data, the inference could be made fully Bayesian, with all modeling parameters inferred rather than estimated and held fixed as we have done here. The priors on the process-level parameters \(\varDelta T\) and \(\varDelta M\) could also be made more informative given process-based understanding of the covariability and relative magnitude of likely lower-frequency changes in temperature and moisture at a given location. The development of a spatial extension of the current hierarchical model would enable the reconstruction of spatially explicit fields of covarying temperature and soil moisture given continental-scale networks of tree-ring width. Even without the expensive estimation of high-dimensional subannual climatic variations, such a scheme would be valuable for allowing probabilistic attribution of changes in ring-width mean and variance across networks of dendrochronologies to shifts in one of temperature, moisture, or a combination of both, while accounting for potential differences in the climatic controls on tree growth through time. This could be especially interesting during fixed temporal intervals of climatological interest, such as the Little Ice Age or Medieval Climate Anomaly. Independent data sources that constrain past climate with relative certainty, such as the extensive documentary records in Europe (Brazdil et al. 2005; Luterbacher et al. 2004), could be integrated into a hierarchical model that contains a VS-Lite data level in order to assess the probability of pre-instrumental changes in the response of tree-ring data in the same spatial domain.^{3} While our model for the observed data at Methuselah Walk accounts for some of the complexities in ring-width formation, a forward model of isotope fractionation, such as that by Farquhar et al. (1982), could be used to incorporate mechanistic understanding of the isotope signal formation, and take account of the second-order effects of temperature and sunlight on the data series and the uncertainty they induce in the reconstructed climate.

Finding candidate ring-width series for this study whose signal might be affected by interactions between low frequency climate variability and biological growth thresholds was a challenge. While this meta-result could indicate that this kind of switching is unlikely to occur in nature, it could also be interpreted as testimony to the techniques developed by dendroclimatologists to sample trees that are likely to contain a univariate signal. The carefully-selected samples are therefore unlikely to be near the boundaries of the different regions of response space depicted in Fig. 2. Indeed, our reconstruction using the more conservative ‘2.2 \(^\circ\)C prior’ and both sources of paleoproxy data lend support to the use of the uniformitarian principle in the case of the Methuselah Walk site, as tree growth appears to be solidly moisture limited even for relatively large changes in the background climate. Samples that do sit close to the biological margins for signal switching may also be unlikely to make their way into the literature, because they will be necessarily more difficult to interpret within the established framework of univariate response functions. Our method provides a means to broaden the set of sampled trees from which useful climatic information might be derived. It may also be of use in diagnosing genuine departures from linear tree growth responses to climate.

While incorporating the nonlinear response to two interacting variables makes our hierarchical model more complex than traditional linear-empirical models, the VS-Lite data level neglects many other real-world complexities of tree growth. Indeed, potential switching in growth response between temperature controls and moisture controls is only one of many mechanisms by which a tree’s response to climate can change over time, including changes in the timing of snowmelt (Vaganov et al. 1999), the effects of changes in solar radiation on photosynthesis (Stanhill and Cohen 2001), and end effects in the statistical detrending of ring-width series (D’Arrigo et al. 2004). Still, the present study gives a flavor of the statistical challenges that can arise as increasingly complex models of data formation are used to interpret paleoobservations in the inverse reconstruction context. Our results suggest that mechanistic-statistical Bayesian hierarchical modeling approaches may be most useful for climate reconstructions where researchers have either a strong hypothesis for the type of nonlinearity that may complicate the data, or else more data sources that can help constrain the likely state space.

Werner, J.P. and Tolwinski-Ward, S.E., poster PP51A-1913 of the American Geophysical Union Fall Meeting (2013).

## Acknowledgments

This work was supported in part by Grants NSF ATM-0724802, NSF ATM-0902715, NSF DMS-1204892, NSF AGS 1304309, and NOAA NA060OAR4310115. We thank Steve Leavitt for lending his isotope data as well as insights on their interpretation, Chris Daly and the PRISM project for making their work freely available, and Benno Blumenthal for making the PRISM data easily accessible on the IRI Data Library. We are also grateful for insightful comments from Chris Paciorek and one other anonymous reviewer, which substantially improved the final version of this paper.