# Reconstructing Multivariate Climate Using A Mechanistic Tree Ring Model

Post by John Tipton, statistics graduate student with Mevin Hooten at Colorado State University

Statistical Challenges of Paleoclimate Reconstructions
The ability to reconstruct paleoclimate from proxy data is important for understanding how climate has changed in the past and to allow exploration into how changing climate influences ecological processes. Statistical reconstructions of paleoclimate have unique challenges because proxy data are noisy, indirect observations of climate. Thus, any statistical model must address the following challenges: change of temporal support, sparse data, and the prediction of unobserved climate variables. When reconstructing climate from tree ring widths, the change of temporal support arises because our climate data are monthly average temperature and log total precipitation, whereas tree ring growth is measured on an annual scale. Therefore, any statistical model must account for this temporal misalignment. To overcome estimation issues common in sparse data scenarios, many previous reconstructions used linear statistical methods with constraints to regress the tree ring widths onto climate. For a multivariate climate reconstruction (e.g., temperature and precipitation), predicting paleoclimate using linear regression requires the inversion of a many to one functional that has potentially infinite solutions. Thus, multivariate climate reconstructions from univariate tree ring width time series are not commonly performed.

Mechanistic Models – A Promising Alternative
There is a need for rigorous statistical multivariate climate reconstructions, and hence, we developed an alternative to using linear statistical methods (Tipton et. al., In Press), using a mechanistic, biologically motivated model that “grows” tree rings to approximate the true growth process. By growing tree ring widths on a monthly time step, the mechanistic model aligns the monthly climate data with annual tree ring width data. Extending the mechanistic model to allow each tree species to have differential response to climate provides strong constraints on possible climate scenarios, ameliorating the difficulties that arise from having too many unknowns. We perform Bayesian inference to generate a probabilistic reconstruction that allows visual exploration of uncertainties. In contrast, many paleoclimate reconstructions generate point estimates that are not probabilistic in nature. The probabilistic reconstructions provide auxillary information that can be used to determine at what time periods the reconstruction is informative. Unfortunately, the use of a mechanistic growth model comes at a computational cost, thus we fit our model using Markov Chain Monte Carlo with compiled C++ code to increase computation speed.

Reconstruction of T and P – at the Same Time!
Our motivating goal was to generate a reconstruction of spatially explicit climate (temperature and precipitation) in the Northeastern United States that can be used to better understand how vegetation patterns have changed due to both climate change and direct human activity. Our work focuses on the Hudson Valley of New York, although in future work this model framework could be extended to the entire Northeastern United States. We focus on the Hudson Valley because there has been previous efforts to reconstruct the Palmer Drought Severity Index (PDSI), a combination of temperature and precipitation, which we can directly compare to our reconstruction, exploring the benefits and costs of different modeling frameworks. Figure 1 shows our joint temperature and precipitation reconstruction with the darkness of the shaded areas proportional to the probabilistic model predictions. For comparison, the black line in the log precipitation plot represents the previous centered and scaled PDSI reconstruction. Interestingly, there is little learning about temperature from our model (although the uncertainties are reasonable) while the log precipitation reconstruction is highly correlated (r=0.72) with the preexisting PDSI reconstruction. This result is in line with ecological expectations – drought in the Hudson Valley is strongly associated with precipitation (Pederson et al., 2015). When comparing our reconstruction to the previous efforts, our method has the added benefit of providing uncertainty estimates that illuminate where in time the reconstruction is informative without relying on statistically improper scoring rules like RE and CE commonly used in the paleoclimate literature. The use of proper scoring rules for assessing predictive ability is vital, because improper scoring rules can lead to incorrect inference about predictive skill.

Figure 1. Plot of probabilistic reconstruction of temperature and log precipitation using a mechanistic tree ring growth model. The reconstructions are shaded according to posterior predictive probabilities with the dotted lines giving the 95% credible interval. The solid black line in the log precipitation plot is a centered and scaled reconstruction of PDSI using the same data. The black lines at the far right of each reconstruction are the observational records.

What Did We Learn?
Reconstructing climate based on ecological proxies is tricky! Our simulations showed that the model we developed can reconstruct multivariate climate with great skill when tree growth responds to both temperature and precipitation. However, in the Hudson Valley, and many other temperate regions, trees respond mainly to precipitation and the bulk of tree growth occurs in a very limited range of temperatures. Thus, while the reconstruction of precipitation in these regions is both accurate and precise, the reconstruction of temperature is inherently more uncertain. The main benefit of a fully rigorous approach for obtaining the reconstructions is that the proper uncertainty can then be factored in to our scientific understanding of the climate process as well as accounted for in other modeling efforts (e.g., ecosystem computer models that depend on historic climate reconstructions).

References
Pederson, N., A.W. D’Amato, J.M. Dyer, D.R. Foster, D. Goldblum, J.L. Hart, A.E. Hessl, L.R. Iverson, S.T. Jackson, and D. Martin-Benito. (2015). Climate remains an important driver of post-European vegetation change in the eastern United States. Global Change Biology, 21(6), 2105-2110.

Tipton, J.R., M.B. Hooten, N. Pederson, M.P. Tingley, and D. Bishop. (In Press).
Reconstruction of late Holocene climate based on tree growth and mechanistic hierarchical models. Environmetrics.