Vladimir L. Gavrikov

Abstract In this study, a geometric model of a growing forest stand has been explored. The basic relationships considered link stand volume and stand density, diameter at breast height (DBH), mean DBH and mean height. The model provides simple formulas connecting the exponents of all the relationships. Application of the formulas to real forestry data provided a high level of predictions of an exponent from two others measured through regressions from empirical data. The Pinus sylvestris L. data were of a static nature, a collection of individual stands, while the Pseudotsuga menziesii (Mirb.) Franco data were dynamic,representing forest stand development over time. The ability of the model to predict exponents in the empirical data implies, on the one hand, a substantial level of similarity between the model and the forestry data. And, on the other hand, the model gives an example in which parameters of one relationship may be linked to parameters of another.Supposedly this kind of ‘relationship between relationships’may be observed in forest stands undergoing active growth and competition-induced self-thinning.

Keywords Relationship between relationships · Selfthinning · Forestry data · Allometry · Scots pine · Douglasfir

Introduction

Forest mensuration is a highly developed field of measurement and research in forest science. During century-long studies, many variables have been introduced that have described dimensions, stand density, production and other variants of forest stands. Over one hundred years ago, especially when yield tables had become standard, it was realized that these variables are linked by various relationships(Frothingham 1914). Not only does mean stem diameter correlate (positively) with mean stem height, total stand volume and numerous other variables are density-dependent,i.e., vary with stand density. Density-dependent relationships were later explored in explicit mathematical form as functions of stand density (Reineke 1933; Hilmi 1955), and interrelationships among tree dimensions were analyzed in allometric studies (Mohler et al. 1978; Henry and Aarssen 1999; Vanclay 2010).

Studies of self-thinning “l(fā)aws” brought about an important direction to research. A highly debated question was whether the exponent, (the slope in log form), of the selfthinning line remained constant across higher plant species.An analysis of the self-thinning theories may be found in Vanclay and Sands (2009). Weller (1987, 1989) analyzed how the self-thinning exponent related to the exponent in the allometric relationship, linking plant dimensions as well as the exponent in biomass-area relationships, and a systematic relation between the former and the latter has been found. It is important to note that the ‘relationship between relationships’ concept was, in fact, put into focus by this research.Later, attention was drawn again to the fact that relationships of the sort ‘dimension-dimension’ or ‘dimension-density’may be linked to one another through their parameters. Inoue(2004, 2009) suggested an analysis relating the two following relationships:

where S stands for mean stem surface area, H for mean height, B d for biomass density and ε and η are exponents,i.e., parameters of the relationships. In the above expressions (1), biomass density is def ined as B d = S/2πRH, R being a mean stem radius. Inoue (2004, 2009) suggested that when the total stem surface area approached its maximum and became independent of stand density, the expression ε + η ≈ 1/2 was valid. The latter expression is also a manifestation of ‘relationships between relationships’ and it has an instrumental value. That is, if we know ε in (1) and possess some additional information, (regarding the total stem surface area), then we can predict η.

Gavrikov (2014, 2017, 2018) explored a simple geometric model of a forest stand and suggested that ‘relationships between relationships’ can be found between R(N) and H(R),N being stand density. The relations may be presented in the power functions form:

where A and B are exponents of the relationships, both derived through the geometric model that contained a common term λ. That is,Theoretically,under the supposition that total stem surface area is constant,independent of stand density, A and B may be directly calculated from one another. Through the use of a wide collection of published data, it has been shown that the model is,indeed, workable, i.e., the exponents A and B can be predicted from one another and for empirical data as well. If the total stem surface area was dependent on variations of stand density, a generalization of the model is possible, making the relationship between A and B still operable.

With regards to trees, studies involving a consideration of stem surface area are biologically sound for obvious anatomical reasons. It should be acknowledged however, that surface stem area as such attracts less attention in forest science than stem mass. A possible reason is that researchers of the self-thinning theory may expand their view to herbaceous plants as well, for which stem surface is less relevant than biomass. In any case, it is important to apply the geometric model of forest stands to traditional forestry data that contain stand volume measurements.

The aims of this study were: (1) to reformulate the geometrical model of Gavrikov (2014) in the case of volume data; and, (2) to examine how well the self-thinning and allometric exponents predict each other in real forestry data.

Materials and methods

Model formulation

Initially the geometrical model presented a view of a forest stand as a population of equal cones packed densely into a horizontal space (Gavrikov 2014). Under such conditions,the population may be described by the following, where V denotes the total volume of the population, N stand density,R radius of the cone base and H height of the cones.

The relation (4) follows from the condition of dense packing so that a decrease of stand density must be accompanied with an increase of radial dimensions. The exponent γ ≠ 1 gives the model more flexibility and corrects an unnatural consequence, namely, a constancy of the total basal area(Gavrikov 2014).

Another part of the model is the introduction of a relationship between total stand volumes and stand density, a reflection of the self-thinning concept. The model itself does not introduce a relationship between total stand mass and stand density, only a relationship between mean linear dimension and stand density (4). However, the former relationship may be borrowed from the widely known self-thinning concept which says that, the relationship is (a) non-linear and (b)may be presented in a power function form. Therefore, suppose that the total stand volume may be related to stand density as:

where V, N and α are the total volume, stand density and exponent of the relation, respectively.

The relations (3) and (5) through?H ?N ∝ Nαcombine to give a relationship of H to R:

where H, R and α are height, cone radius and exponent of the relation, respectively.

The exponent in (6) will be further denoted as:

where γ and α are exponents of the relations (4) and (5),respectively.

The exponents in (4), (5) and (6) are all functionally interrelated with each other. The interrelationship means that the exponents may be found from each other because they contain the same parameters. If, for example, γ and α are known,then β may be directly calculated or predicted.

Data used

In order to test how the above approach works with real forestry data, a number of datasets were taken from public sources. The first is a database of Usoltsev (2010) who collected approximately 10,000 forest stand descriptions over the Eurasian boreal geographical space. From this database, all possible datasets for Scots pine (Pinus sylvestris L.) stands were taken. A dataset is usually a number of stand descriptions published by one author, and the dataset contains basic forestry information: mean diameter at breast height (DBH), mean height, stand density and volume. Forest stands belonging to one dataset have always close growth conditions, which is reflected in a Russian site quality class.This is a traditional Russian system of taking into account growth conditions varying from I to V, I being the best and V the poorest conditions. The number of forest stands in each dataset was a variable and depended on the availability of the data. The number ranged from 4 to 12, being mostly 6-7 forest stands in a dataset.

The datasets were first inspected as to how stand volume relates to stand density. This is important because the model implies that an actively growing forest stand is considered in which competitive mortality plays an important role. This is why stand volume should increase when stand density decreases.

Altogether 46 Scots pine datasets were found in the database of Usoltsev (2010). Three showed that lower stand density corresponded to lower stand volume. These three datasets were therefore discarded, and the remaining 43 datasets were taken for subsequent treatment.

Another data source was publications on growth of Douglas-f ir (Pseudotsuga menziesii (Mirb.) Franco) stands within a levels-of-growing-stock cooperative study by the USDA Forest Service since the early 1960s. The publications (Marshall and Curtis 2001; King et al. 2002; Curtis and Marshall 2009a, b) contain abundant information on the growth and development of Douglas-f ir. For the purposes of this study,parameters of stem radius (half dbh), mean height, and stand density were taken. Altogether, five datasets or forest stands were available from these publications. The Douglas-f ir data concerned only control plots on which no management was applied throughout the time of observations.

A brief description of the datasets may be found in the electronic Supplement, Tables S1 and S2.

Data treatment

This consisted of an estimation of the exponents of the relationships (4), (5) and (6). According to the model, volumes were first found in (3). The use of calculated stand volume V c instead of stand volumes given in the data sources (tables)V t is discussed in the “ Discussion” section.

All the data, N, R, H and V c were first log 10 -transformed and the exponents then found as slopes μ of correspondent linear functions of the form logY=k+μlogX , Y and X being dependent and independent variables, correspondingly,and k standing for an intercept. For example, the γ value was found from the regression logR=k+(?γ∕2)logN [see Eq. (4)]. The parameter estimation was done with Statistica 6 software, module ‘Non-linear estimation’, the loss function being ordinary least squares.

The use of the log-transformed data was as follows. For some datasets, a direct non-linear regression by the function Y = k· X μ produced a confluent result because of too large a value of k. Instead, the linear regression oflog-transformed was always stable, producing reasonable estimated values of the parameters. However, if the non-linear regression worked normally for the data, then the estimated exponent μ values did not differ numerically from slope μ values received by the linear regression oflog-transformed data.

Results

The results of parameter estimations are summarized in Tables S3 and S4 (see the electronic Supplement). In the H(R) section, a comparison of parameters estimated by regression vs. those calculated from formula is given.The estimated parameter values are denoted as βrthat was received by the above regression procedure. At the same time, the value β c is calculated from α and γ through expression (7) suggested by the model.

As seen from Table S3, the deviation of β c from βr,which can be estimated as (βc? βr)/β c , is small for most of the cases. The deviation was less than 1% in 30 out of 43 datasets (～ 70% of Scots pine datasets). For the Douglas-f ir datasets, four of five datasets had the same small deviation,and the Skykomish dataset was only a 1.5% deviation.

Fig. 1 Values of β r and β c for Scots pine datasets plotted against each other. The solid line denotes β r = β c

Fig. 2 Values of β r and β c for Douglas-f ir datasets plotted against each other. The solid line denotes β r = β c

Graphically, the close correspondence of β r and β c is shown in Figs. 1 and 2, in which the values are plotted against each other.

By analogy with the allometric exponent given in relationship (6), any of the other three exponents, for example,the self-thinning exponent given in relationship (5), can be predicted. Formula (7) can be easily converted into:

The calculated values with Eq. (8) of α c were plotted against αrreceived through regression (Figs. 3 and 4). The figures show the close correspondence of the exponents,which is also reflected in the deviation measure (α c ? α r )/α c . As found from the data of Table S3, the deviation was less than 1% in 31 cases out of 43 (～ 72%).

To compare the above results from the regression of V c (N), Figs. 5 and 6 illustrate what happens if V t (N) is used instead of V c (N). It is obvious that the estimations of α c deviate substantially from αrshowing smaller α r values than predicted. The same picture is seen for β r versus β c values, with β c deviating substantially from predicted β r = β c .

Fig. 3 Values of α r and α c for Scots pine datasets plotted against each other. The solid line denotes α r = α c

Fig. 4 Values of α r and α c for Douglas-f ir datasets plotted against each other. The solid line denotes α r = α c

Fig. 5 Values of α r and α c for Douglas-f ir datasets plotted against each other. For the calculation of α r , the regression V t (N) was used, V t being the table values of volume stock taken from the published data.The solid line denotes α r = α c

Discussion

The most important result above is the surprisingly high accuracy with which an exponent can be predicted from two others with the help of simple formulas, Eq. (7),derived from the model. Attempts to create a model within which various exponents could be predicted from one another have been performed before. Weller (1987, 1989)suggested a decomposition of self-thinning exponent in the form:

Fig. 6 Values of β r and β c for Douglas-f ir datasets plotted against each other. For the calculation of β r , the regression V t (N) was used, V t being the table values of volume stock taken from the published data.The solid line denotes β r = β c

where φ stands for an exponent linking a radius of occupied area r and biomass m, r ∝ m φ , another allometric exponent θ links height h and biomass, h ∝ m θ and exponent δ links density of biomass in occupied space d and biomass, d ∝ m δ .Further, because φ is also linked to a self-thinning exponent ρ, the resultant formula for a subsequent analysis looked in Weller (1987) like:

where the prime in φ′ indicates that the value is not measured directly but calculated from the self-thinning exponent.Weller, therefore, compared the measured self-thinning exponents with those calculated from measured allometric exponents. As a rule, experimental data corresponded to the prediction that φ′ should show a negative relation to the allometric exponents, e.g., θ. However the graphs provided by Weller (1987) show a rather wide dispersion of the compared values regarding both theoretical and regression lines.

In this study, the deviations of predicted values β c and α c from those measured through pairs γ and α r and β r and γ,correspondingly, are, for the most part, very small (Figs. 1 through 4). In this regard, there was no obvious difference between Scots pine and Douglas-f ir data. Douglas-f ir data are about def inite individual forest stands whose natural dynamics has been followed in the course of time. In contrast, the Scots pine datasets are arbitrary collections of forest stands. By selecting forest stands as study objects, the subjective choice of the authors may influence the resultant set of the published data. Inclusion of the data into larger databases may also be a matter of chance. It is, therefore,surprising that the approach developed-that is, a simple geometric model-is still workable for the substantial amount of data of this kind. In the literature, the use of simple models has been advocated (Vanclay 2010) and it appears that simple models may often be useful.

Also, for a few datasets, the estimation of α was scarcely significant (see Table S3, datasets #1, 29, 34, 35, 39, 41, 42).Nevertheless, for five of these seven cases, the accuracy of prediction of β c from γ and α r was 3% or less, which is quite good for the usual forestry studies.

As noted above, the model stand volume V c calculated through expression (3) was used in all the computations.An alternative might be the use of the stand volume V t taken from the same forestry data tables, as long as dbh and mean height were also taken from there. There are two points to note in this respect. First, practical attempts to use V t values from forestry tables were not successful in this study. The deviations of predicted exponents from the measured ones were too large (Figs. 5 and 6). The cause of the deviations is because the relationships V c (N) and V t (N) have substantially different slopes.

Second, the question of which of the slopes should be considered to be the ‘genuine’ one is open-because as it is well known from forest mensuration, volume values in forestry tables are not measured directly but calculated. The way the calculations are carried out is often not transparent compared to using simple formulas. The calculations are based on volume tables and/or regressions; in a sense, one has to trust the experience accumulated by forest mensuration researchers over decades.

The approach to calculate stand volume through expression (3) does not attempt to provide real forest stand volumes. Expression (3) is a part of the geometric model and is thought to provide a way to compute a sort of self-thinning exponent (α). However, the relationships V c (N) and V t (N)relate rather well to each other empirically. Figure 7 presents a selection of Scots pine datasets; the calculated and the estimated stand volumes are plotted against each other. Both volumes relate to each other linearly, even the slope, though not specially estimated, appears to be the same for all the datasets. Figure 8 presents the calculated vs. the estimated stand volumes for the Douglas-f ir datasets. In this case, the relationships seem to be slightly non-linear. Therefore, if the relation between the relationships V c (N) and V t (N) is known for a particular forest stand, they can be, if necessary, recalculated from one another.

Fig. 7 Relationship between volume of model stand V c calculated after (3) and stand volume V t taken from Scots pine forestry data tables for the same forest stands. Legend dataset number in brackets,names see Table S1: ● is Kurbanov-02 (#1), ○ is Gruk-79 (#2), + is Uspenski-87 (#4), ? is Mironenko-98 (#11), □ is Heinsdorf-90 (#20),▲ is Kozhevnikov-84 (#24)

Fig. 8 Relationship between volumes of model stand V c calculated after (3) and stand volume V t taken from Douglas-f ir forestry data tables for the same forest stands. Legend names see Table S2; □ Iron Creek, ? Hoskins, ● Skykomish, ○ Clemons, △ Rocky Brook

Conclusions

With forestry data combining growth, density, and selfthinning, there are two kinds of data. One is static and an arbitrary collection of stands of different ages. They may be sometimes considered as a reflection of the natural development of a forest stand. Another type of data is dynamic and a consecutive series of observations of stand development over time. In both cases, the geometric model under consideration is a workable instrument. The model shows how relationships, including self-thinning exponents and allometric exponents, may be linked to each other. The linkage has been demonstrated by the ability of model formulas to predict an exponent if two other exponents are known. The predictions were accurate for static Scots pine data and for dynamic Douglas-f ir data, which may manifest the work of‘relationship between relationships’ in the forestry data. It is possible that the recommended geometric model, in spite of its simplicity, is substantially similar with real forest stands.