Fusing tree-ring and forest inventory data to infer influences on tree growth

Better understanding and prediction of tree growth is important because of the many ecosystem services provided by forests and the uncertainty surrounding how forests will respond to anthropogenic climate change. With the ultimate goal of improving models of forest dynamics, here we construct a statistical model that combines complementary data sources – tree-ring and forest inventory data. A Bayesian hierarchical model is used to gain inference on the effects of many factors on tree growth – individual tree size, climate, biophysical conditions, stand-level competitive environment, tree-level canopy status, and forest management treatments – using both diameter at breast height (DBH) and tree-ring data. The model consists of two multiple regression models, one each for the two data sources, linked via a constant of proportionality between coefficients that are found in parallel in the two regressions. The model was applied to a dataset developed at a single, well-studied site in the Jemez Mountains of north-central New Mexico, U. S. A. Inferences from the model included positive effects of seasonal precipitation, wetness index, and height ratio, and negative effects of seasonal temperature, southerly aspect and radiation, and plot basal area. Climatic effects inferred by the model compared well to results from a dendroclimatic analysis. Combining the two data sources did not lead to higher predictive accuracy (using the leave-one-out information criterion, LOOIC), either when there was a large number of increment cores (129) or under a reduced data scenario of 15 increment cores. However, there was a clear advantage, in terms of parameter estimates, to the use of both data sources under the reduced data scenario: DBH remeasurement data for ~500 trees substantially reduced uncertainty about non-climate fixed effects on radial increments. We discuss the kinds of research questions that might be addressed when the high-resolution information on climate effects contained in tree rings are combined with the rich metadata on tree- and stand-level conditions found in forest inventories, including carbon accounting and projection of tree growth and forest dynamics under future climate scenarios.

and a process model. However, their process model did not include fixed effects: tree growth 99 was modeled as a function of year random effects and individual random effects. Individual tree 100 growth is known to be influenced by many factors, including first and foremost tree size, 101 followed by a suite of other factors varying at either the tree-level or the stand-(or site-) level, 102 such as stand basal area, various aspects of climate in multiple months or seasons, disturbances, 103 substrate, etc. (Cook and Briffa 1990, Paine et al. 2011, Bowman et al. 2012). In other words, the 104 problem of understanding individual tree growth is a complex one. 105 Here we develop a hierarchical Bayesian model that can infer the influence of many 106 factors on individual tree growth, relying on a combination of tree-ring and forest inventory-type 107 (diameter remeasurement) data at once. Hierarchical Bayesian modeling is appropriate for the 108 task because it allows one to formally combine information from multiple data sources, while 109 accommodating complexity and scale (Clark 2005). In this application specifically, it allows us 110 to tailor a statistical model to the structure of the data, including the different levels (tree-vs. 111 stand-) at which predictors are available, and the nature of the relationship between the two types 112 of observations. Further, hierarchical Bayesian modeling allows us to examine variation in radial 113 and diameter increments in terms of conditional probabilitiese.g., the influence of climate 114 conditional upon tree size. We develop this model at a single exceptionally well-studied site, 115 where much is known about the history of disturbance, forest management, climate limitations, 116 phenology, substrate, etc. This well-studied site provides the opportunity to unravel a complex 117 multiple regression model problem with confidence. We test hypotheses about the factors 118 influencing individual tree growth using a multiple regression approach, with several main 119 effects, expected nonlinearities, and potential interactions among factors, to evaluate the 120 7 hypotheses that growth is influenced by tree size, climate, biophysical setting, and stand 121 conditions. surrounding landscape for most of the 20 th century, which represents a significant deviation from 130 the historical regime of high frequency-low intensity fire (Falk 2006, Falk et al. 2007). An 8×9 131 grid of 0.25-ha 50×50 m plots, centered 500 m apart, was established in 1999 and 2000 132 (described in Falk 2004, Farris et al. 2013). Fifteen of these plots were included in the present 133 study, ranging in elevation from 2520-2590 m, with forest types ranging from essentially pure 134 Pinus ponderosa to dry mixed conifer stands. These plots vary with respect to substrate, slope, 135 aspect (Table 1), and other variables. Twentieth century fire suppression led to overly dense 136 forest, particularly on pumice-derived soils, such that a combined thinning (2006) and prescribed 137 fire (2012) treatment was applied to restore forest structure. Our sample included 3 plots that 138 experienced these treatments, one on each of the three substrate types present (tuff, pumice, and 139 alluvium; Table 1). The site is characterized by cool winters and warm summers (mean January using the program COFECHA (Holmes 1983 actual vapor pressure (in hectopascals). All covariate data were centered and scaled (to a mean of 181 zero and a standard deviation of 1.0), so that the magnitude of their effects could be compared.

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Dendroclimatic Analysis 183 We conducted a dendroclimatic analysis of the tree-ring data, for comparison against the 184 Bayesian multiple regression approach. In tree-ring studies, low-frequency variation in the 185 absolute value of radial increments are removed (i.e., the long-term reduction in radial increment 186 that results from increasing diameter over time), a procedure known as "detrending" (Speer 187 2010). We used a cubic smoothing spline with a 50% frequency cutoff response at 10 years to 188 detrend the tree-level measurement series (function detrend, R package dplR, Bunn 2008Bunn , 2010.

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This retained annual to decadal growth variability in the data while removing longer-term regressions together included two tree-level variables (tree size and height ratio) and five plot-208 level variables (radiation, slope, aspect, wetness index, and plot basal area; Table 2). In the 209 following, we detail each regression in turn, and then the connection between them.

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The tree-ring submodel was a multivariate normal Gaussian process model, which substrate that is known to vary among the plots (Table 1). Individual random effects ( . ), 224 normally distributed with a prior mean of zero and variance σ.i 2 , were also included. Residual .
Where the parameter η is within-year residual variation, and the parameter ρ is the rate at which 240 covariance (among years) decays.

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The diameter increment (d.incijkt) submodel assumes a normally-distributed response, 242 with the same tree-level and plot-level fixed and random effects as above, with the exception that 243 thinning+prescribed fire (thin.firej) was included as a plot-level indicator variable (Table 2) In this second model, we tested the interaction between plot basal area and climate, i.e., 297 two forms of stress, competition and climate, that are expected to exacerbate one another, such 298 that individual tree growth should be more greatly reduced by climate stress (high temperature, 299 low precipitation) if a tree is in a high density stand than a low density stand. We also tested 300 interactions between climate and height ratio (i.e., tree status in the canopy, relative to 301 neighbors), and between climate and tree size (basal area at the time of the first census).

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Model comparison 303 To evaluate whether a model that combines tree-ring and DBH data together outperforms  (Figures 3 and 4). the tree-ring data when the uncoupled model was used on a reduced tree-ring data set ( Figure   468 4b), but when the estimates of effects from the two data sources were coupled to one another, the 469 posterior distributions of non-climate effects were forced to resemble one another (tree-ring vs. 470 23 DBH data, Figure 4c), thus the coupled model was able to "borrow strength" from the rich DBH 471 data, and reduce uncertainty about shared fixed effects substantially (Figure 4c compared to 4b).

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The performance of our Bayesian fusion model under a data scenario where strength can be 473 borrowed across tree-ring vs. forest inventory data with respect to climate effects is a target of 474 further investigation.

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In this analysis, we treated tree size and proxies for competition (plot basal area, height 476 ratio) as invariant, when in fact we know that they evolve over time. In spite of this, we found   DBH columns indicate (X) which variables enter in the two submodels, respectively, and the 693 column "coupled" indicates (with an α) those effects whose estimates were coupled to one 694 another across the two submodels via a constant of proportionality.