Abstract
We explore a probabilistic, hierarchical Bayesian approach to the simultaneous reconstruction of local temperature and soil moisture from treering width observations. The model explicitly allows for differing calibration and reconstruction interval responses of the ringwidth series to climate due to slow changes in climatology coupled with the biological climate thresholds underlying treering 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 reconstructioninterval treering variability. This manifestation of nonidentifiability is a result of the manytoone mapping from bivariate climate to time series of treering 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 colocated 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 ringwidth 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.
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Notes
 1.
 2.
M. Schofield et al., abstract 306105 of the 2012 Joint Statistical Meetings.
 3.
Werner, J.P. and TolwinskiWard, S.E., poster PP51A1913 of the American Geophysical Union Fall Meeting (2013).
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Acknowledgments
This work was supported in part by Grants NSF ATM0724802, NSF ATM0902715, NSF DMS1204892, 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.
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S.E. TolwinskiWard performed the research and submitted the article while at the Institute for Mathematics Applied to Geosciences, National Center for Atmospheric Research, Boulder, CO, 80305.
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Appendix: Sampling algorithm
Appendix: Sampling algorithm
Posterior sampling is complicated by the manytoone mapping from climate to treering width inherent to the VSLite model. As a result, it is usually possible to construct many hot, dry moisturelimited climate histories as well as many cool, wet temperaturelimited ones with the same likelihood from a given series of interannaul ringwidth variations. The nonidentifiability between hot and dry solutions versus cool and wet solutions is related to the switching in the controls on treering growth that can occur with large shifts in the lowfrequency climatology, and results in bimodality in the posterior distribution of \(\varDelta T\) and \(\varDelta M\). This feature of the problem precludes the use of standard Gibbs sampling approaches, in which each variable in the problem is iteratively sampled from its full conditional given the current value of all others unknowns (Gilks et al. 1996). Fixed values of the (2variable \(\times\) 12month \(\times\) Nyear) histories of climatic anomalies \(\varvec{T}', \varvec{M}'\) will only be consistent with either a hot, dry climate and moisturelimited growth, or with a cool, wet climate and temperaturelimited growth, and so a straightforward Gibbs sampler does not permit transitions between the two modes of the posterior of \(\varDelta T\) and \(\varDelta M\). The posterior is of the "doubly intractable" form discussed by Murray et al. (2006).
Further nonidentifiability stems from the finer timescale of the monthly climatic variations compared to the annual resolution of the data, as there are many ways to configure monthly climate variations resulting in the same annual growth. Thus, for a given shift in climatology captured by nonzero values of \(\varDelta T\) and \(\varDelta M\), and conditional on the parameters of VSLite controlling the growth response, a relatively long “burnin” period is necessary to wander through the (2variable \(\times\) 12month \(\times\) Nyear) space of \(\varvec{T}', \varvec{M}'\) histories to find the region of state space that is statistically consistent with both the data and the prior.
Given these complications, we employ a hybrid of Gibbs and Metropolis–Hastings sampling techniques, as well as an approximation of the likelihood using summary statistics and simulation as in Approximate Bayesian Computation approaches (Csillery et al. 2010) to numerically estimate the posterior. The approximation to the likelihood is based on the following intuition: variations in the climatological shifts \((\varDelta T, \varDelta M)\) tend to influence the mean and variance of simulated treering width time series, while the climatic anomalies \(T', M'\) mainly control the relative interannual variations in ring width index. By treating the ringwidth series as a set of variations about its sample mean with scale given by the sample standard deviation, and by treating the sample mean and variance as sufficient statistics for \(\varDelta T\) and \(\varDelta M\), we can solve an approximation to the problem, with the likelihood in Eq. 11 approximated by
The sampling for the pseudoproxy problem then proceeds in two separate steps:

1.
Draw a proposal from the prior \([\varDelta T, \varDelta M]\), and a Monte Carlo sample of size 100 from \([\varvec{T}',\varvec{M}']\) (sampled from Eq. 9 using the fixed values of covariance matrices \(S_s\) and temporal autocorrelation parameters \(\phi _1\) and \(\phi _2\) estimated in the calibration interval). Use these as inputs to the datalevel model Eq. 6 to compute a Monte Carlo sample \(\{\bar{W}_{MC},S^2_{MC}\}_{n = 1}^{100}\) of the sample mean and standard deviation of the simulated ringwidth index time series. Accept or reject the proposed values of \(\varDelta T, \varDelta M\) (as in a Metropolis–Hastings sampler) depending on the distance of the Monte Carlo sample means of the two sample statistics from the observed values. The likelihood is a normal distribution of the means of the Monte Carlo sample statistics about the observed values, with standard deviations given by the Monte Carlo standard errors. We run this step in series on a laptop. The resulting samples \(\{\varDelta T^{(n)},\varDelta M^{(n)}\}_{n=1}^N\) approximate the distribution \([\varDelta T, \varDelta M  \bar{W},s^2_W] \approx [\varDelta T, \varDelta M  \varvec{W}]\).

2.
Using the Metropolis–Hastings within Gibbs algorithm (Gilks et al. 1996), draw \(n = 1, \ldots , N_{MC}\) samples of the monthlyresolved bivariate time series from
$${}[\varvec{T}',\varvec{M}'\varDelta T^{(n)}, \varDelta M^{(n)},\varvec{W}] \propto [\varvec{W}\varvec{T}',\varvec{M}',\varDelta T^{(n)},\varDelta M^{(n)}][\varvec{T}',\varvec{M}']$$for each of the samples of the climatic shifts from Step 1, where the first factor on the righthand side is given by our VSLite datalevel model, and the second is given by the prior model (Eq. 9) for the climatic anomalies. We use a burnin period of 75 iterations, which was deemed appropriate after diagnosis of multiple MCMC chains examining trace plots, autocorrelation plots, and the \(\hat{R}\) statistic of Gelman and Rubin (1992) for our sample chains. We parallelize this step and draw the samples in a highperformance computing (HPC) environment.
The sampler for the reconstructions based on real treering data at Methuselah walk proceeds similarly, except that \(\varvec{\theta }_W\) must also be inferred. Because the elements of \(\varvec{\theta }_W\) depend only on calibrationinterval temperature, moisture, and ringwidth data (\(\varvec{W}_c,\varvec{T}_c,\varvec{M}_c\)), this sampling can be performed separately, and we draw samples from \([\varvec{\theta }_W  \varvec{W}_c,\varvec{T}_c,\varvec{M}_c]\) using the algorithm and freely available code detailed in TolwinskiWard et al. (2013). The likelihoods in Steps 1 and 2 also depend, in the realdata reconstructions, on the sampled values of (\(\varvec{\theta }_w, \varvec{W}_c,\varvec{T}_c,\varvec{M}_c\)), which serve as inputs to VSLite. For inference conditioned on both treering width and isotope data, the dependence of the sampler on the isotopes enters during the drawing of proposals for the Metropolis–Hastings sampler in Step 1, by drawing proposals of \(\varDelta M\) conditional on the mean delindex \(\bar{I}_\delta\).
All sampling was coded and run in MATLAB®. The CPU time for each step and each experiment is given in the Table 2.
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TolwinskiWard, S.E., Tingley, M.P., Evans, M.N. et al. Probabilistic reconstructions of local temperature and soil moisture from treering data with potentially timevarying climatic response. Clim Dyn 44, 791–806 (2015). https://doi.org/10.1007/s003820142139z
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Keywords
 Bayesian hierarchical modeling
 Biological–statistical modeling
 Multiproxy paleoclimate reconstruction
 Treering width
 Timevarying climatepaleodata relationship