15 April 2014 Science Briefs

SNAMP Pub #20: Estimating tree growth from complex forest monitoring data

Article Title: Estimating tree growth from complex forest monitoring data

Authors: Eitzel M. V., J. Battles, J. Knape, R. York, P. de Valpine

Research Highlights:

  • Gaining a better understanding of tree growth limitation has many applications for forest ecology and management goals.

  • Forest managers at Blodgett Forest Research Center have collected tree diameter at breast height information for forest plots since 1976. This dataset provides a great opportunity to learn about long-term growth limitation in forest trees.

  • Problems with using these datasets include measurement error and asynchronous measurement periods. Measurements from the same tree or field plot also have something in common and so are not statistically independent.

  • Sophisticated statistical models (hierarchical state-space models) can represent measurement error (in order to correct for it and get a more accurate growth measurement), uneven time intervals (to avoid making simplifications about constant growth between measurements), and non-independent measurements (in order to satisfy statistical assumptions and account for different sources of variation).

  • Our hierarchical state-space model incorporates these features and allows us to understand how annual tree growth depends on factors like tree size, competition with other trees, and light, water, and nutrient availability.

  • In unmanaged stands at Blodgett, white fir growth is most strongly limited by tree size (small trees grow more slowly) and tree competition (trees grow more slowly on crowded plots), as well as additional individual variation between trees.

Understanding forest tree growth, especially as it depends on tree size, is important for many applications including modeling changes in tree populations, evaluating forest management goals, and estimating biomass and carbon sequestration. Tree growth rates partly determine tree mortality rates, are an important component of dendrochronology, and estimation of growth rates from field data is needed to run forest simulators.

Factors that limit tree growth include tree size, competition with neighbors, and resource limitation (water, light, and nutrients), and competition and resource limitation may affect smaller trees more strongly than larger trees (statistically, an interaction between size and these variables). Forest inventories provide a rich data source to study growth limitation, but typically these datasets are complex in a number of ways.

Most statistical models use the standard assumption that data points are independent of each other (have nothing in common), but in a forest inventory, the same tree may be measured multiple times, or trees from the same field plot may be more similar to each other than trees from a different plot. We need to use a hierarchical statistical model with random effects to represent the random differences due to being a particular tree or being from a particular plot. Otherwise, growth rates might be dominated by particularly productive plots or particularly fast-growing individuals. In addition, leaving out random effects results in pseudoreplication – a common modeling error which may render conclusions invalid.

In addition, diameter censuses (a common forest monitoring data type) are intended to be conducted over regular time intervals but often intervals are uneven. Even if censuses are conducted regularly in one forest or study, when data from different locations are combined, this problem of mismatched intervals becomes inevitable. Diameter measurements are prone to measurement error when a diameter tape slips or bark has been lost during the measurement interval, often giving rise to unrealistic shrinkage. Typical ways of correcting for this problem can bias growth estimates (for example, throwing out negative measurements without knowing which positive measurements are also errors and should be thrown out as well). A specific kind of hierarchical statistical model, called a state-space model, explicitly models missing data and observation error, allowing us to understand growth increment on an annual timescale despite the uneven time intervals and measurement error.

We used forest inventory data from Blodgett Forest Research Station (east of Georgetown, CA on the central western slope of the Sierra Nevada) to fit a hierarchical state-space model for annual growth of white fir (Abies concolor) in 0.04 ha (1/10 acre) plots. All of the trees in this study were from the reserve portions of the research station, with no management other than fire suppression (which is typical of large areas of the Sierra Nevada). Our model accounts for uneven time intervals, repeated measurements of the same trees, the effects of being in specific plots, and the effect of tree size, competition, and resource limitation on growth, as well as interactions of tree size with the other variables. We use readily available forest management data for these variables: diameter at breast height for tree size; plot basal area for competition; and annual climatic water deficit (a measure of drought), slope, elevation, insolation (amount of solar radiation), and soil type as variables to represent resource limitation. We fit the model using OpenBUGS, an open source software package which allows the flexible creation and fitting of complex statistical models.

The mean annual growth increment for white fir trees in reserve (unmanaged) portions of Blodgett is 0.182 inches (0.463 cm) per year, increasing 0.013 inch per year for each additional inch in tree diameter (all else being equal, larger trees grow faster). More crowded stands with higher basal area reduced individual tree growth overall, and this effect was more pronounced for larger trees (a significant interaction between size and basal area). See Figure, with dashed lines showing uncertainty. Trees on steeper slopes had slower growth than those on gentler slopes. Some soil types differ significantly from each other, with trees on Cohasset soils having higher growth rates than the other types. In addition, there is some evidence that water deficit explains some of the variation in our data.

Other resource variables (elevation, insolation) were not important at this site. While the climate variable is not significant, our inventories do not contain measurements for all years, and therefore the uncertainty in this parameter estimate is large. We were able to estimate the error in our diameter measurements at 0.111 to 0.149 cm (0.044 to 0.059 in, for two slightly different models). This error is quite low compared to other studies, but those studies included obvious mistakes in record-keeping as well as outliers, which we have removed in our dataset. We found that the effect of being in different plots in different locations was significant, both for overall annual growth increment and for trees of different sizes (an interaction between tree size and plot). We also found that individual trees consistently grew faster or slower than other trees of the same size in addition to growing faster when they were larger.


  1. We have created a model which appropriately represents a complex forest inventory dataset and were able to investigate the dependence of tree growth on tree size, competition, and resource supply, as well as interactions between size and other variables.

  2. Tree size, plot basal area, and the interaction between them strongly affect white fir growth at Blodgett in the reserve parts of the research station, which have seen no management action other than fire exclusion. Hill slope and soil type also affect growth, and there is some evidence for the effect of annual climatic water deficit.

  3. Even when we have included a plot measurement of basal area, the effect of being in a particular plot is important to tree growth. This implies that some kind of unmeasured environmental or competitive factor affects growth at or below the plot scale, and gives each plot a slightly different relationship between tree size and tree growth rate. Similarly, even when we have included tree size, certain trees have a growth advantage or disadvantage over their lifetime in the inventory. This individual advantage could be caused by a variety of factors, including the individual's genetics, a favorable site (or microsite when the tree was a seedling), or maternal effects (the parent tree had good resources but not necessarily good genetics).

  4. OpenBUGS is one of several tools for fitting such complex models, and there are a number of excellent textbooks on hierarchical modeling for a variety of applications. This published research paper includes an extensive set of appendices illustrating the technical details of designing, constructing, fitting, and evaluating these models, as well as supplements including the data and model files required to replicate the results. The appendices and supplements are intended to make these modeling tools more transparent and accessible to scientists and managers.

Full Reference:
Eitzel, M. V., J. Battles, R. York, J. Knape, P. de Valpine. 2013. "Estimating Tree Growth Models from Complex Forest Monitoring Data." Ecological Applications. 23:6, 1288-1296.

The full paper is available here.

For more information about the SNAMP project and the Fire and Forest Health team, please see the: Fire and Forest Health Team Website.

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