A METHOD TO SET UP THE PHASES OF THE TREE GROWTH IN THICKNESS BASED ON DENDROCHRONOLOGY OF AN ENGLISH OAK

Elżbieta Gołąbek, Andrzej Tukiendorf

ABSTRACT

In the paper we proposed a methodology to distinguish the phases of the tree growth in thickness of an English oak in the past century based on the ring chronology. We analyzed northern, eastern, southern and western tree ring increments using both the statistical modeling and a taxonomic method. Each step of the methodological procedure was described in the paper and the results were displayed graphically in line plots via a dendrogram and set in a table. To familiarize the Reader with the chosen methodology, some general notes over the approaches were added to the text. In the analysis, three phases of the tree thickness growth in the examined monumental English oak in the 20^th century were set up statistically. The average-annual increments for these phases were calculated and plotted. The appropriateness of this technique for a practical use in dendrochronology was concluded finally and its wider application suggested.

Key words: growth in thickness, dendrochronology, Bayesian modeling, taxonomy, growth phase.

INTRODUCTION

Each tree during its lifetime has fluctuations in its growth in thickness. To the main factors that could have influence on this process we include aging effect, health status, biosocial site, climate, biotope conditions, and air pollution [7]. Hence, data concerning the annual growth measurements comprise the information on a tree itself as well as its nearby environment. The analysis of such growth changes is the main goal of the dendrochronology [12]. Levels and dynamics of the thickness increments are measures that may be useful to assess historical phases of a tree development and/or its health status which, in turn, can be used to more complex ecological studies. New dendrochronologic analyses, however, require modern statistical approaches. Among them, Bayesian modeling and taxonomy can be powerful and effective tools to resolve contemporary problems in dendrology.

In the presented study, the phases of the thickness growth in the recent epoch were established. Based on the time series data on growth increments in a monumental English oak, modern statistical approaches were proposed to resolve such a dendrochronological problem. To familiarize the Reader with the chosen methodology, some general notes over the statistical tools were added to the text. The results were graphically displayed and set in a table. Some concluding remarks about the approach were given in the final part of the paper.

DATA

In the study information on the growth in thickness referred to sets of time series of the ring chronology of northern, eastern, southern and western (NESW) wood samples bored from a trunk of the analyzed English oak (Quercus robur) on the height of 130 cm, using a 40 cm Pressler’s drill [6] (the gaps after drilling were protected against infection). The data comprised the measurements of the oak’s increments in the years 1901-2000 and were gauged optically using an increment core measuring instrument.

The tree was estimated to be 298 years old at the time of measurement (2001). Following the dendrological classification [6], its present health status was evaluated as poor. By the decision of the Provincial Nature Conservator, this English oak has been included in the list of monumental trees (registration no. 314/1). The tree is located in a communal park in Rogów Opolski, Opole Province, Poland. The annual NESW ring increments for the years 1901-2000 are presented in Figure 1.

From the line plots shown in Figure 1, fairly wavy changes of the oak’s growth in thickness in the NESW directions throughout the past century can be ascertained. This supports for periodical trends of the trunk increments in thickness with time.

METHODS

The chosen statistical methodology can be divided into two procedural parts.

(I) Firstly, the procedure relies on the Bayesian statistical modeling of the analyzed time series. In this study, after [2], for the succeeding 20^th century’s years

t_i = 1901, …, 2000, for i = 1, …, 100,

the following linear periodical perturbation model for each series was adopted

y_i ~ Normal(m_i, t)

m_i = d₁ + d₂t_i + d₃ cos(d₄t_i) + d₅ sin(d₄t_i)

where y_i is the growth in thickness, m _i and t – distribution parameters (i.e. expected values and a variance component respectively), whilst d₁, …, 5 are the unknown regression coefficients (to be estimated).

In the Bayesian approach, in addition to specifying the model for the observed data (usually in the form of a probability distribution – likelihood), a vector of unknown parameters (random quantity) is added as well, having its own prior distribution and a vector of hierarchical hyperparameters. Inference concerning these unknown parameters is then based on their posterior distribution expressed in the convenient shorthand as proportional to the likelihood times the prior (in prior-to-posterior analysis, the prior always informs the posterior) [1].

The computation was performed in WinBUGS v. 1.4 [10] based on a simulation technique known as Markov Chain Monte Carlo (MCMC) [8]. The basic philosophy behind MCMC is an iterative simulation of Markov chain whose equilibrium distribution is the desired distribution. Instead of calculating exact estimates, this technique generates a stream of simulated values for each quantity of interest [8].

In this study, to achieve the convergence, three parallel chains were run and the first 1,000 samples of each were discarded as a burn-in while the following 10,000 production cycles were used to estimate each quantity of interest. An equilibrium state of streams of values was established via an examination of within chain autocorrelation and a comparison of the results of the chains started with overdispersed initial values, including the use of the Gelman-Rubin statistic available within the software (see the web site www.mrc-bsu.cam.ac.uk/bugs for details). The posterior means of the NESW thickness growths were exposed graphically in line plots.

(II) Secondly, to distinguish types of the growth throughout the century, the cluster analysis [3] was conducted. In order to avoid excessively diminutive types of the thickness increments during these years, their taxonomic typology was performed using the modeled data only.

The sense of the cluster analysis is to group similar objects into homogenous clusters and to separate them from objects of other type under similarity/dissimilarity assumption [3, 9]. The objects are assumed to be similar when the differences/distances between them are relatively small. A ‘difference/distance’ term between objects is widely understood. In our analysis we used the Euclidean metric that has a ‘natural’ geometric interpretation [3, 9].

In the beginning of the typological procedure, the so-called ‘distance matrix’ of objects is built. In the presented study, we used Euclidean distances between the years. Founded on this matrix, a threshold distance value was established up to whose level the objects are still being similar to each other and above which one they are becoming dissimilar. Naturally, this discrimination can be performed through the ordering of distances between the objects, although some arbitrary thresholds may be assumed as well. This technique is usually displayed graphically in a line plot where the changes across steps can be easily observed.

Then, a taxonomic dendrogram [3, 9] of the linkage between the objects was produced. It is of note that this system consists of many U-shaped lines connecting objects in a hierarchical tree based upon a generated linkage function. The height of each U represents the distance between the two objects being connected and the objects can be contained in wider clusters with the size determined by the cutoff (threshold) value. Hereof, individual clusters (equivalent types) of objects can be distinguished that have relatively small variances inside the groups and big differentiations between the clusters [4, 5].

In our study, following this typological procedure, we identified phases of the thickness growth in the oak that were represented by taxonomical types of its growth years. The results were displayed graphically.

The cluster analysis is commonly provided in many statistical packages. The presented one was conducted in STATISTICA 5.0 software [11].

For the established phases of the thickness growth, the annual averages of the real and modeled data were calculated. For the particular geographical directions, the values were presented graphically.

RESULTS

The periodical models of the NESW growths in thickness of the analyzed English oak are presented in line plots in Figure 2.

From the models drawn in Figure 2, much clearer pictures of the oak’s thickness increments for NESW directions can be seen in comparison with the real data presentation (Figure 1). Across the analyzed century the most decreasing trends in growth in thickness can be observed for the northern direction. The eastern and the southern wave increments remained quite stable during this epoch but with different lengths. The west side of the trunk characterizes almost a linear trend of the growth with a slight decreasing tendency.

A line plot of increments of taxonomical (Euclidean) distances between the objects (years) is presented in Figure 3.

The line plot presented in Figure 3 testifies that up to the established 1.07 value of the linkage measure, the distances between the objects increase rather mildly, while above it, a dynamic increment of taxonomical distances is observed.

The taxonomical dendrogram of the objects together with their typology following the established threshold value is drawn in Figure 4.

Using the threshold of 1.07 as a cutoff value, three subsets of objects can be distinguished in the linkage dendrogram (see Figure 4). In consequence, three types of growth years – as objects (I, II and III) of 1901-1939, 1940-1969, and 1970-2000 were identified in the English oak’s lifetime, representing phases with the similar thickness growths in the NESW directions – Figure 5.

The lines plotted in Figure 5 show that in Phase I (Type III) the oak’s trunk had quite similar thickness growths in the NSW directions, while some distinct increments in the E side during this period. Phase II (Type II) characterized relatively equal growths in all the directions but generally less dynamic than in the preceding period. In Phase III (Type I), in turn, the NESW increments were the most dispersed in comparison with the earlier times. These features seemed to be predominant in characteristics of the phases.

The annual averages of the NESW thickness growths for the identified I, II, and III Phases both for the real and the modeled data are set in Table 1.

A calculated Pearson correlation r = 0.993 between the data and the model values set in Table 1 testifies a good fit of the thickness growths by the chosen periodical function. A graphical presentation of these values is given in Figure 6.

The plots in Figure 6 confirm a strong similarity between the data and the model. They show that a relatively strong decrease of the thickness growth in the northern side of the trunk took place in the analyzed century. Moreover, in both series, a decrease of the southern growth with age is observed but not as strong as in the northern side. The changes of the eastern and the western increments seemed to be fluctuating during this epoch.

DISCUSSION

The final remarks can be resolved into the following points:

• the raw measurements of the NESW growths in thickness in the analyzed monumental English oak provided evidence of their periodical changes throughout the 20^th century;

• the chosen Bayesian statistical modeling based on the trigonometric function satisfactorily fitted the data;

• the conducted taxonomical analysis of the trunk increments in these years revealed the consecutive phases of the tree growth determined by the directional differences;

• the chosen statistical methodology proved its appropriateness in such a dendrochronological analysis. It is suggested that the results can be used as a starting point in more advanced ecological studies concerning e.g. a wide range of environmental relationships.

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Andrzej Tukiendorf
Unit of Agricultural and Forest Technology
Technical University of Opole
ul. Mikołajczyka 5, 45-071 Opole
e-mail: antu@po.opole.pl

Responses to this article, comments are invited and should be submitted within three months of the publication of the article. If accepted for publication, they will be published in the chapter headed ‘Discussions’ in each series and hyperlinked to the article.