1 Introduction

In the Swiss NFI (NFI), wood volume and changes in wood volume are estimated based on the stem volume of individual trees using various models. Many of the models applied currently were described by Kaufmann (2001), and this earlier text is thus referred to repeatedly below. However, some of them have subsequently been completed or adjusted based on methodological developments, and some of the descriptions provided by Kaufmann (2001) are not comprehensive. Therefore, this chapter mainly updates descriptions published previously, but also recapitulates fundamental information when necessary for readability and completeness.

Stem volume models estimate single stem volume using three measured tree dimensions: diameter at breast height (d1.3), upper diameter at 7 m height (d7) and tree height (h). The stem volume models currently implemented are the same as those described by Kaufmann (2001). Section 12.2 therefore provides complementary information and a summary of the essential information, such as equations and coefficients. Tariff models are the most general and estimate single stem volume as a function of d1.3 as the only measured tree dimension. The current tariff models have been slightly modified from the descriptions given by Kaufmann (2001). Section 12.3 provides updated descriptions.

In addition to stem volumes, volumes of large and small branches are estimated, mainly as inputs for biomass estimation. The ratio of the volume of large branches to stem volume over bark is estimated using a logit model described by Kaufmann (2001). The corresponding model for the ratio of small branches is very similar but has not been published previously. These two models and their estimated coefficients are described in Sect. 12.4.

As the NFI is based on permanent inventory plots, changes in resources in general can be estimated as the difference between consecutive NFIs. In particular changes in standing volume such as growth, harvest and mortality can be estimated directly as the difference in tree volume. However, in some cases growth models are needed. The models and coefficients for basal area increment and volume increment are described in Sect. 12.5.

2 Stem Volume Models

The stem volume models map d1.3, d7 and tree height to total stem volume over bark of individual trees, including the tree top and the aboveground part of the stump but excluding branches. They are able to explain nearly all variance and are more precise than the tariff models (Sect. 12.3). However, d7 and tree height are time-consuming measurements and are therefore only assessed on a subsample, namely the tariff trees.

The method for selecting tariff trees is explained in Sect. 2.3.4.5. The estimated volume of the tariff trees is used for two purposes. First, it is used as the dependent variable when fitting tariff models (Sect. 12.3). Second, it is used to improve stem volume predictions with the tariff models, which use d1.3 as the only explanatory variable measured in the field (Sect. 12.3). For the tariff trees, stem volume can be estimated using both the more precise stem volume model and the generally applicable tariff model. The difference between the two models is scaled up (by the inverse of the selection probability) and added to the tariff volumes to correct for the difference to the stem volume. Assuming that the stem volume is known and is unbiased for the tariff trees, this procedure results in unbiased estimates of standing and total volume even in small sampling units, i.e. for rare tree species or small regions (Mandallaz 1991, 1997). A full description of this two-stage estimation procedure and the reasons to adopt it were presented by Kaufmann (2001, page 176 ff.).

The stem volume functions of the NFI were developed in 1991. Nine different functions were fitted: seven were developed for the main tree species Norway spruce (Picea abies), silver fir (Abies alba), pine (Pinus sylvestris and P. nigra), European larch (Larix decidua), Douglas fir (Pseudotsuga menziesii), European beech (Fagus sylvatica) and oak (Quercus spp.), and the remaining two were developed for coniferous and broadleaved species (Kaufmann 2001; page 162 ff.). For the sake of completeness, the equations and the coefficients (Table 12.1) are presented here.

Table 12.1 Coefficients of the stem volume models used in the NFI
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Norway spruce (Picea abies):

$$ V={b}_0+{b}_1{d}_7^2h+{b}_2{d}_{1.3}^2+{b}_3{d}_7^3+{b}_4h $$
(12.1)

Silver fir (Abies alba):

$$ V={b}_0+{b}_1{d}_7^2h+{b}_2{d}_{1.3}+{b}_3{d}_{1.3}^2+{b}_4{d}_{1.3}^3h+{b}_5{h}^4 $$
(12.2)

Pine (Pinus sylvestris and P. nigra):

$$ V={b}_0+{b}_1{d}_7^2h+{b}_2{d}_{1.3}+{b}_3{d}_{1.3}^2+{b}_4{d}_{1.3}^3h $$
(12.3)

European larch (Larix decidua):

$$ V={b}_0+{b}_1{d}_7^2h+{b}_2{d}_{1.3}^2+{b}_3{h}^2 $$
(12.4)

Douglas fir (Pseudotsuga menziesii):

$$ V={b}_0+{b}_1{d}_7^2h+{b}_2{d}_{1.3}^2+{b}_3{d}_{1.3}^3+{b}_4{d}_{1.3}^2{h}^2 $$
(12.5)

European beech (Fagus sylvatica):

$$ V={b}_0+{b}_1{d}_7^2h+{b}_2{d}_{1.3}^2+{b}_3{d}_7^3+{b}_4{d}_{1.3}^3h $$
(12.6)

Oak (Quercus spp.):

$$ V={b}_0+{b}_1{d}_7^2h+{b}_2{d}_{1.3}^2+{b}_3{d}_{1.3}^3+{b}_4{d}_{1.3}^3h $$
(12.7)

Coniferous (all species):

$$ V={b}_0+{b}_1{d}_7^2h+{b}_2{d}_{1.3}^2+{b}_3{d}_7^2+{b}_4{d}_7^3+{b}_5{d}_{1.3}{h}^3 $$
(12.8)

Broadleaved (all species):

$$ V={b}_0+{b}_1{d}_7^2h+{b}_2{d}_{1.3}+{b}_3{d}_{1.3}^2+{b}_4{d}_7^2 $$
(12.9)

where V is the stem volume over bark (in m3), d1.3 is the diameter at breast height (in m), d7 is the upper diameter measured at 7 m height (in m), and h is the total tree height (in m).

As the stem volume models are fundamental to the NFI, this section includes information about their derivation and the data used to fit them.

Model fitting was based on a data set including long-term growth and yield data from approximately 38,000 tree stems from the permanent plot network of the Experimental Forest Management (EFM) trials conducted at WSL. These stems were measured (lying) in 2-m sections as the trees were harvested, between 1888 and 1974. Thus, their precise volume is known. Table 12.2 gives an overview of the number of available stem measurements for each species and the regional distribution of the measured stems in Switzerland.

Table 12.2 Number of measured stems from the Experimental Forest Management data set for each species corresponding to a fitted model, along with the regional distribution of the trees used for measurements
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The NFI tariff sample trees and the trees from the EFM trials were assessed for very different objectives: the NFI aimed to infer the state of forests representatively over the whole of Switzerland and its production regions, whilst the EFM experiments aimed to answer various research questions (e.g. Zell 2018; Peck et al. 2014) but are not representative of Switzerland’s forests. In fact, the tree population from the EFM trials is clearly different from that of the corresponding species in Switzerland’s forests: trees of the regions Plateau and Pre-Alps are generally over-represented in the EFM data set, whereas trees of the Alps region are under-represented.

The two data sets actually represent tree populations with different characteristics. For example, the stem forms of EFM and NFI sample trees are clearly different for all species and species groups. More specifically, the slenderness ratio (h/d1.3) of trees measured in EFM plots is systematically larger than that of NFI sample trees, as shown for Norway spruce in Fig. 12.1

Fig. 12.1
figure 1

Slenderness ratio (h/d1.3) per diameter class of Norway spruce (Picea abies) sample trees measured in the NFI (measurements on standing trees) and as part of the EFM Project (measurements on lying trees) over all Switzerland

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Stem volume models are assumed to be valid for all of Switzerland, but this is not so the tariff models, which respect regional differences. Therefore, the stem forms were shown for the regions with respect to their fitting data set (i.e. the EFM data) compared to their prediction data set (i.e. the NFI data). The stem form of trees in the Swiss Plateau is similar to that of EFM sample trees, whereas trees in the Jura and the Pre-Alps show a lower slenderness ratio compared to trees from the EFM trials. This difference even increases in the regions Alps and Southern Alps. Figures 12.2 and 12.3 show examples of these differences for Norway spruce and European beech, respectively.

Fig. 12.2
figure 2

Slenderness ratio (h/d1.3) per diameter class of Norway spruce (Picea abies) sample trees measured in the NFI (measurements on standing trees) in the regions Plateau (a), Pre-Alps (b) and Alps (c) and as part of the EFM Project (measurements on lying trees) over all Switzerland

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Fig. 12.3
figure 3

Slenderness ratio (h/d1.3) per diameter class of European beech (Fagus sylvatica) sample trees measured in the NFI (measurements on standing trees) in the regions Plateau (a), Jura (b) and Southern Alps (c) and as part of the EFM Project (measurements on lying trees) over all Switzerland

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To compensate for these differences and to complete the EFM data set over the full range of slenderness ratios, approximately 500 additional trees were sampled in 1990 and 1992. They were selected with the criteria of a low slenderness ratio and a large diameter at breast height (spruce: 199, fir: 11, larch: 89, pine: 20, beech: 174 and oak: 10). The diameters at 10%, 30%, 50%, 70% and 90% of the total tree height, as well as the diameter at breast height (d1.3), the upper diameter at 7 m height (d7) and the tree height (h), were measured on these standing trees. The stem volumes were then calculated using a method similar to that described in Sect. 13.2. Cubic interpolation splines were fitted between the measured diameters to achieve continuous stem profiles (Sect. 13.2 step 2), and the stem volumes were then calculated with rotational integrals (Kaufmann 1993, unpublished internal report; Sect. 13.2 step 7). This data set was then merged with the EFM data set to fit the stem volume functions.

Another difference between the NFI and EFM sample trees is the diameter distribution. In the EFM data set, small diameters are over-represented while middle and large diameters are under-represented compared to the NFI tariff sample trees (see Fig. 12.4 for examples with Norway spruce and European beech). To increase the comparability of the data, only a subsample of the full EFM data set was used to fit the models: stems <12 cm in breast height diameter (d1.3) were not considered and stems 12–35 cm in d1.3 were selected randomly and proportionally to their d1.3. Stems with diameters >35 cm were all included.

Fig. 12.4
figure 4

Densities of the diameter distributions of Norway spruce (left) and European beech (right) sample trees measured in the NFI and in the EFM Project

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Coefficients of the Eqs. 12.1, 12.2, 12.3, 12.4, 12.5, 12.6, 12.7, 12.8 and 12.9 were estimated with weighted linear regressions. The increasing variance over predicted values (heteroscedasticity) was respected by weighting the residuals with the term \( 1/\left({d}_7^3h\right) \), i.e. assuming that the variance is proportional to \( \left({d}_7^3h\right) \). To reduce bias, a grouped jackknife (Efron and Stein 1981) was applied instead of using only one estimator. The data were randomly split into 20 groups. An estimator was calculated for each group, as well as for the whole data set.

This fitting methodology has the following advantages. First, the data set used to fit the coefficients is closer to observations made in the NFI, owing to subsampling and complementing of the data with extreme forms. Second, by weighting the residuals, the inherent increasing variance is incorporated into the prediction. Third, a smaller bias in the estimated coefficients can be assumed, owing to the grouped jackknife. Fourth, as the models may result in negative predictions, a simple imputation for these trees was established using geometrically reasonable volume estimates. Fifth, the standardised residuals are unbiased over d7/d1.3 and h/d1.3, as illustrated in Fig. 12.5.

Fig. 12.5
figure 5

Residual analysis of the stem volume functions for Norway spruce (left) and European beech (right)

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To analyse the predictive behaviour of the models using the NFI data, a sample from the EFM data (the source of information for volumes) was selected proportionally to the three-dimensional density of d1.3, h and d7 in the NFI. The stem volume models were applied to this sample data set and residuals were calculated. As the residuals are representative of the NFI data, they could be used to show over- or underestimation of the stem volume models. Figure 12.5 shows the results of this analysis for examples with Norway spruce and European beech. For the largest trees (with an estimated volume >5 m3), an overestimation of about 5–10% was found, whereas the form factor and the slenderness showed better residual distributions.

Some critical points of the stem volume functions are: (a) the model selection was not fully described; (b) biased predictions were obtained over the size of trees (see Fig. 12.5 for examples with Norway spruce and European beech); and (c) reproduction of the coefficients was not entirely possible, and discrepancies were large for some species (see Fig. 12.6 for examples with Norway spruce and European beech).

Fig. 12.6
figure 6

Distribution of 1000 refitted coefficients for Norway spruce (left) and European beech (right), based on the fitting procedure described in this section (subsampling proportional to d1.3, grouped jackknife). Continuous lines indicate the coefficients reported in the NFI and the dotted lines indicate coefficients based on all observations and jackknifing)

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Possible Further Developments

A re-analysis of the entire system for estimating volume is planned. Special attention will be given to unbiased predictions over the range of observations. Therefore, different prediction methods will be tested. The weighted residuals technique will be compared with a model-based approach in which all explanatory variables of the stem form functions are predicted (by simple or multiple imputation). Further, the stem volume functions will be analysed with respect to model selection, model formulation and heteroscedasticity. The overall aim will be to develop a prediction system capable of reproducing the observed (i.e. the representatively resampled) volumes.

3 Tariff Models

The stem volume models described in Sect. 12.2 can only be applied to trees with measured d1.3, d7 and h, namely the tariff trees (Sect. 2.3.4). However, for the majority of NFI sample trees only d1.3 is measured. For these trees, stem volume over bark, including the tree top and the aboveground part of the stump, is estimated with a tariff model that only depends on measured d1.3 and on stand and site attributes as auxiliary variables. A total of 30 different tariff models were fitted for different combinations of tree species and production regions (Table 12.3).

Table 12.3 Combinations of tree species and production regions along with the number of stems used to fit each tariff model
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The tariff models have largely remained the same since 1992, and the model descriptions and analysis documented by Kaufmann (2001; pages 166–169) are mostly still valid. This section therefore focuses on minor changes.

Tariff models were fitted based on the stem volume of the tariff sample trees (Sect. 12.2). The models that were described by Kaufmann (2001) had an independent set of coefficients for each inventory cycle. However, this caused artificial jumps in the volume prediction between inventories and therefore resulted in implausible estimations of growth in some cases, for example in small sampling units or for rare tree species. For this reason, the tariff models were refitted in a slightly different form so that all information from NFI1–3 was used simultaneously. The different inventories were considered through two additional coefficients (b8 and b9). Table 12.3 shows the number of observations used to fit each model. Stem volumes in NFI4 are currently estimated using coefficients fitted for NFI3.

$$ \hat{Y}=\mathit{\exp}\left({b}_0+{b}_1\mathit{\ln}\left({d}_{1.3}\right)+{b}_2{\mathit{\ln}}^4\left({d}_{1.3}\right)+\sum \limits_{i=3}^9{b}_i{B}_i\right) $$
(12.10)

where Ŷ is the tariff volume over bark fitted for each tariff number (201, 202, …, 230; Table 12.3). The index i corresponds to the additional single tree and sample-plot attributes (3,…, 9), and d1.3is the measured diameter at breast height. B3 to B9 are the following additional single tree and sample-plot attributes:

B 3 :

TMI: site quality expressed as the maximum of the total mean increment from stand establishment until the age of 50 years, in kg dry weight ha–1 year–1 (Sect. 15.5)

B 4 :

ddom: dominant diameter, i.e. mean diameter of the 100 thickest trees per hectare (derived from the diameters of the sample trees in the plot)

B 5 :

bifurcation of the stem (0 = no bifurcation, 1 = bifurcation) based on field observations

B 6 :

elevation (m a.s.l.), taken from the digital elevation model with a 25 m grid

B 7 :

stand layer to which the single tree belongs (0 = upper layer, 1 = understorey) based on field observations

  • B8 (inv2) and B9 (inv3) together indicate the inventory cycle(s) in which the tree was measured:

    • inv2=0 and inv3=0: tree measured in NFI1

    • inv2=1 and inv3=0: tree measured in NFI2

    • inv2=0 and inv3=1: tree measured in NFI3

The model coefficients b0 to b9 are presented in Table 12.7.

Depending on the tariff, some explanatory variables of the general model do not contribute substantially to improving the model. Variables with a p-value >0.05 were sequentially deleted using a backward model selection procedure. These deleted variables are marked with a hyphen in Table 12.7.

Discussion of the Tariff Models

The tariff models are used to predict single tree volumes, where the only measured biometric variable is d1.3. All other variables represent auxiliary information and reflect changes in growth conditions that influence tree height and stem form. In this sense, the tariff models use information beyond d1.3. However, it is unclear how precise volumes predicted by tariff models are compared to observed volumes. Since stem volumes are not measured in the NFI, such a comparison is not straight forward. Furthermore, the regression models ignore the inherent heteroscedasticity of the data, and thus the largest trees have the most impact on the coefficient estimations. The resulting p-values are potentially overoptimistic, which in turn influences model selection. Overall, the tariff models shown in Figs. 12.7 and 12.8 show unbiased behaviour over all Switzerland and only minor biases within some production regions.

Fig. 12.7
figure 7

Residual analysis of the tariff models for Norway spruce, shown for all regions together and for each region separately. The grey background is the two-dimensional density of observations (the darker the background the more observations) and the black line is a locally weighted regression spline

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Fig. 12.8
figure 8

Residual analysis of the tariff models for European beech, shown for all regions together and for each region separately. The grey background is the two-dimensional density of observations (the darker the background the more observations) and the black line is a locally weighted regression spline

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4 Volume Models for Large and Small Branches

The volumes of large branches (≥7 cm in diameter) and of twigs and small branches (<7 cm in diameter) are predicted as fractions of the stem volume (i.e. tariff volume (Sect. 12.3)).

$$ {branch\ volume}_i= stem\ volume\ast {pa}_i $$
(12.11)

where i indicates large or small branches and pai is volume of branches as a fraction of stem volume.

The proportion pai is estimated with a logit model:

$$ {pa}_i=\mathit{\exp}\left({lga}_i\right)/\left(1+\mathit{\exp}\left({lga}_i\right)\right) $$
(12.12)
$$ {lga}_i={b}_{0i}+{b}_{1i}{d}_{1.3}+{b}_{2i}{h}_1+{b}_{3i}{h}_2 $$
(12.13)

where b0i to b3i are species- and region-specific regression coefficients for large and small branches, d1.3 is the measured diameter at breast height, and h1 and h2 are indicator variables (used for spruce and beech only) for elevation (m a.s.l.) in combination with production region, as indicated in Table 12.4.

Table 12.4 Indicator variables for production region and elevation (m a.s.l.), used in models to predict the volume of branches for Norway spruce and European beech
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The models were fitted to data collected on the permanent plot network of the Experimental Forest Management (EFM) trials conducted at WSL (N = 14,712).

Species-specific logit regression models were fitted to large and small branches for spruce, fir, larch, pine and oak. For beech, three different models were fitted based on region (Tables 12.5 and 12.6). For Norway spruce, the proportion of large branches is negligibly small, so trees of this species are assumed to have no large branches at all. Development of the volume models for large branches was documented by Kaufmann (2001). A similar procedure was used to develop volume models for small branches.

Table 12.5 Coefficients of the volume models for large branches
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Table 12.6 Coefficients of the volume models for small branches
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5 Changes in Forest Resources

As the NFI is based on permanent plots, changes such as growth, harvest and mortality are estimated by re-measuring single trees in these plots. Single-tree estimates of gains and losses per year can be directly estimated as the difference in tree volume between subsequent NFIs. Additionally, specific growth models for basal area increment (BAI) and – in the case of tariff trees – for stem volume increment (SVI) are applied to estimate the growth of trees that were cut, died, or reached the calliper threshold between two inventories.

5.1 Model for Basal Area Increment

For sample trees for which d1.3 is only measured once in two consecutive NFIs, BAI is estimated based on a regression model fitted to data from NFI1 and NFI2. This model is used to predict BAI for trees that were cut or died between two inventories, using the d1.3 value measured before this event took place. Analogously, the model is used to back-estimate BAI if d1.3 was only measured at the end of the inventory interval, such as for trees that reached the calliper threshold of 36 cm in the outer circle ( nongrowth trees; Sect. 2.9.2). No BAI is predicted for trees that reached the calliper threshold of 12 cm in the inner circle.

As done in the previous NFIs (Kaufmann 2001; pages 174–175), BAI was modelled as a function of several tree- and plot-specific variables separately for every tariff number (Table 12.3) according to Eq. 12.14 which is based on Teck and Hilt (1991) and Quicke et al. (1994):

$$ {\displaystyle \begin{array}{c}\hat{BAI}= AS\ \exp \Big({b}_0+\sum \limits_{i=1}^6{b}_i{B}_i+{b}_7\left( AS+1\right)\left(1-\exp \left({b}_8{d}_{1.3}\right)\right)\\ {}+{b}_9\left( AS-1\right)\left(1-\exp \left({b}_{10}{d}_{1.3}\right)\right)\Big)\end{array}} $$
(12.14)

where BAI is the basal area increment to be estimated (in m2 per 10 years); AS indicates whether BAI is predicted forwards (+1, for trees that were cut or died) or backwards (–1, for nongrowth trees); the index i corresponds to the additional single tree and sample-plot attributes (1,…, 6), and d1.3 is the measured diameter at breast height. The following additional attributes (B1–B6) are included:

B1:

stand basal area (m2 ha–1), calculated based on all standing trees in the plot that are living

B2:

basal area of larger trees (m2 ha–1), calculated based on all standing trees in the plot that are living and have a d1.3 greater than the target tree

B3:

TMI: site quality expressed as the maximum of the total mean increment from stand establishment until the age of 50 years, in kg dry weight ha–1 year–1 (Sect. 15.5)

B4:

elevation (m a.s.l.), taken from the digital elevation model with a 25 m grid

B5:

for even-aged forests: stand age (years), estimated according to a regression model based on tree-ring counts from stumps in the sample plots, as described in detail by Kaufmann (2001; pages 175–176)

for uneven-aged forests: dominant diameter ddom (cm), calculated as the mean diameter of the 100 thickest trees per hectare (derived from the diameters of the sample trees in the plot)

B6:

stand layer to which the single tree belongs (0 = upper layer, 1 = understorey), based on field observations

The coefficients b0 to b10 of the BAI model are presented in Table 12.8 for the case of even-aged forests and in Table 12.9 for the case of uneven-aged forests.

Based on the single measurement of d1.3, either at the beginning (trees that died or were cut) or at the end (nongrowth trees) of the inventory interval, and on the predicted BAI, the d1.3 that was not measured is estimated. Changes in individual-tree volumes are then calculated by applying the tariff models introduced in Sect. 12.3.

A similar BAI model is included in the forest development model MASSIMO (Chap. 17) but with some application-specific differences (e.g. growth boost). An alternative climate-sensitive BAI model was recently developed for MASSIMO (Rohner et al. 2017), but its potential future use in NAFIDAS for estimating changes in resources still needs to be evaluated.

5.2 Model for Stem Volume Increment

For the particular case of tariff trees (Sect. 2.3.4) with only one d1.3 measurement in consecutive NFIs, a regression model for stem volume increment is applied in order to enable application of the two-stage volume prediction described in Sect. 12.2 and by Kaufmann (2001; page 176 ff.). Stem volume increment over bark is modelled according to the same nonlinear equation as presented for BAI (Eq. 12.14) including the same explanatory variables. The coefficients b0to b10 of the stem volume increment model are presented in Table 12.10 for the case of even-aged forests and in Table 12.11 for the case of uneven-aged forests.