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

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.

Appendix: Sampling algorithm

Posterior sampling is complicated by the many-to-one mapping from climate to tree-ring width inherent to the VS-Lite model. As a result, it is usually possible to construct many hot, dry moisture-limited climate histories as well as many cool, wet temperature-limited ones with the same likelihood from a given series of interannaul ring-width variations. The nonidentifiability between hot and dry solutions versus cool and wet solutions is related to the switching in the controls on tree-ring growth that can occur with large shifts in the low-frequency 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 (2-variable $\times$ 12-month $\times$ N-year) histories of climatic anomalies $\varvec{T}', \varvec{M}'$ will only be consistent with either a hot, dry climate and moisture-limited growth, or with a cool, wet climate and temperature-limited 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 time-scale 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 non-zero values of $\varDelta T$ and $\varDelta M$, and conditional on the parameters of VS-Lite controlling the growth response, a relatively long “burn-in” period is necessary to wander through the (2-variable $\times$ 12-month $\times$ N-year) 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 tree-ring width time series, while the climatic anomalies $T', M'$ mainly control the relative interannual variations in ring width index. By treating the ring-width 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

$$\begin{aligned}{}[\varvec{W}|\varvec{T}',\varvec{M}',\varDelta T,\varDelta M]&= [\varvec{W}',\bar{W},s^2_W|\varvec{T}',\varvec{M}',\varDelta T,\varDelta M] \\&\approx [\varvec{W}'|\varvec{T}',\varvec{M}'] [\bar{W},s^2_W|\varDelta T, \varDelta M]\end{aligned}$$

(16)

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 data-level 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 ring-width 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 monthly-resolved 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 right-hand side is given by our VS-Lite data-level model, and the second is given by the prior model (Eq. 9) for the climatic anomalies. We use a burn-in 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 high-performance computing (HPC) environment.

The sampler for the reconstructions based on real tree-ring 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 calibration-interval temperature, moisture, and ring-width 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 Tolwinski-Ward et al. (2013). The likelihoods in Steps 1 and 2 also depend, in the real-data reconstructions, on the sampled values of ($\varvec{\theta }_w, \varvec{W}_c,\varvec{T}_c,\varvec{M}_c$), which serve as inputs to VS-Lite. For inference conditioned on both tree-ring 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 del-index $\bar{I}_\delta$.

Table 2 Computing time for sampling

All sampling was coded and run in MATLAB®. The CPU time for each step and each experiment is given in the Table 2.