Stem biomass is reconstructed from samples based on the fact that oven-dry biomass is a function of stem volume and basic density (Eq. 3.1)

BM = V ■ R (3.1)

Here BM is the oven-dry biomass (kg), V is the stem volume (m3), and R is the basic density defined as dry weight divided by green volume (kg/m3).

Volume calculation, although a long time established part of forest mensuration, is not always trivial in biomass studies because unlike in stem volume determination the non-merchantable branch volume matters in biomass studies. The predicament is that there is no standardised definition for a twig as opposed to a branch. Nor is it always clear in broad-leaved species that tend to grow multi-stemmed or with forked stems what is to be labelled as stem and what as branch. As indicated in Fig. 3.3 a solution is, for example, to select the largest branch as a stem and define all other parts as branches. It must, however, be noted that this practice is not compatible with most taper and volume models that are based on volume calculated from the sum of all stems of a merchantable size and thus are based on a virtual, single stemmed tree. With redefining all other forks of a stem as big branches the merchantable stem volume and the derived stem biomass will be underestimated compared to estimations based on available volume functions and basic density. Thus a method compatible to stem volume determination, where the basal area of all merchantable stems and branches is added up to a defined cut-off diameter might be more useful.

A large variety of sampling methods were tested for biomass sampling such as systematic sampling in absolute or relative heights or more recently randomised branch sampling techniques (Jessen 1955; Valentine and Hilton 1977; Valentine

et al. 1984; Gregoire et al. 1995; Gaffrey and Saborowski 1999; Saborowski and Gaffrey 1999; de Gier 2003), which have been proven to improve accuracy for a set sampling effort. Experience shows that the choice of the sampling method is not in the first place driven by aspects of the sampling theory, but rather by the level of sophistication and experience of the sampling team. In developing countries computers for field work are often unavailable. Field teams for biomass sampling are frequently recruited from an inexperienced body of persons rather than from scientific personnel or experienced field technicians. Therefore highly sophisticated sampling designs that rely on computer software in the field are often not feasible. Thus, often the principle “the simpler the better” pays off. A gain in efficiency as a result of the use of e. g. randomised sampling techniques can be easily negated by the incorrect application of the sampling method by inexperienced field crews, while straightforward systematic sampling methods (e. g. samples taken every 3 m) have a better chance to be implemented correctly. However, a decrease in sampling efficiency is to be taken into account.

The volume calculation of stem sections between measuring points is typically based on Smalians’s formula (van Laar and Akqa 2007) or alternatively on the geometric formula of a frustum of a cone (Eq. 3.2).

V = (R2 + Rr + r2) (3.2)

Here V is the section volume (m3), R is the bigger end radius of the stem section and r the smaller top end radius. In case no cut-off diameter was defined, the volume of the stem tip can be determined as a cone by setting the top end radius r to zero.

The initially introduced Eq. 3.1 implies that stem volume and basic density calculation are equally important for the successful determination of stem biomass. Trees are known to vary considerably in wood density within the stem in radial and longitudinal direction and also between trees and sites. Thus, pre-information on longitudinal density gradients in the species of interest should be used wherever possible to adjust the sampling design of discs beforehand accordingly. To base biomass upscaling merely on mean literature values of basic density is a very crude approach that might, due to density variations within and between trees lead to seriously biased estimates.

The upscaling from sample discs typically contains a measurement component where basic density is determined at disc level and a modelling component based on the estimation of fresh weight/dry weight ratios or a regression approach to obtain information for the entire stem. In general, biomass should be provided as dry mass. Fresh mass is site and species specific (Marden et al. 1975; Kokkola 1993) and also subject to a substantial intra-annual and inter-tree tree variation (Kokkola 1993). In addition, the harvesting technique (debarking) and weather conditions after felling can modify the fresh weight (Adams 1971; Kokkola 1993) rendering it a suboptimal variable for tree biomass characterisation. A comparative quantification of biomass is only possible when an equilibrium in moisture content is reached that can be determined in accordance with scientific lab standards. The rich body of publications on biomass prove that there is a considerable variation in the definition of dry mass between studies that varies from about 40 to 105 °C. In nutrient studies with nitrogen content in phytomass as a response variable the drying temperature is usually limited to a maximum of 60-65 °C to avoid nitrogen loss as a result of volatility of some chemical components. Cones are often dried below 40 °C, for germination to remain possible. Oven-dry weight of wood refers per convention in wood science to drying to a state of constant weight at about 103 ± 2 °C (see e. g. DIN EN13183-1). Using drying temperatures of 70 °C and below, for example to maintain the volatile nitrogen parts for further nutrient analysis, leads to higher biomass values of a magnitude of 2-3 % in wood (Forrest 1969; Barney et al. 1978). Seifert and Muller-Starck (2009) reported similar weight changes for Norway spruce cones. The cone weight was 84 % (dried at 38 °C), 80 % (at 60 °C) and 78 % (at 105 °C) in proportion to the fresh weight. Evidence shows that it is paramount that differences in drying temperature are taken into account when pooling data and also when comparing established functions. Furthermore the drying regime

should be provided in publications for a reliable comparison of biomass studies. However, the establishment of further drying series at different temperatures seems to be warranted to obtain transfer functions for the different tree components, which would facilitate a conversion of biomass at different drying temperatures and would thus facilitate a comparison of results obtained in different biomass studies.

The most common principle in density determination of sample discs is the Archimedean principle of water displacement (American Society for Testing and Materials 1999). A basic physical principle is used that relates the buoyant force of a body directly to its volume. The method is simple and implies establishing of weight of a sample in air and in water with a scale. The density of water is 1 g/cm3 and that of air is negligible, which results in Eq. 3.3 for the final density (Gerthsen 1997, p. 97).

Wwater p sample Pwater) * V ^ (3 3)

Wair (psample ~ pair) * V psample

Here W is weight (g), p is density (g/cm3) and V is volume (cm3).

The weight in water is determined by full immersion of the wood sample in a water basin, which is placed on top of a scale. The scale reading in gram after immersion equals the volume of the sample in cm3. It is important that the sample is fully saturated with water (to constant weight), so that no additional water is taken up into the sample during the displacement measurement.

Based on individual disc density a longitudinal and partially a radial density variation can be taken into account, if a regression is applied that models basic density as a function of height in the stem. This procedure also averages out measurement errors to a certain degree (Fig. 3.4).

Based on the assumption that tree stems are rotationally symmetric along their longitudinal axis, basic density of a stem can be obtained by mathematical integration of the density regression function multiplied with the cross-sectional area along the tree height from the lower end to the upper end of the stem section (Eq. 3.4).

hu

j A(x)p(x)dx (3.4)

hi

Here hu and hl represent the height of the top and bottom end of the section, A(x) is the cross-sectional area of the stem at height x between hu and hl and p(x) is the basic density at the same height determined by a regression function.

A mathematically simpler, but still sufficiently accurate solution can be achieved by determining the height of the centroid (centre of gravity) of each section if enough sections are taken. The centroid of the frustum of a cone is calculated according to Weisstein (2013), based on Eshbach (1975), Harris and Stocker (1998) and Kern and Bland (1948) as indicated in Eq. 3.5.

i R2 C 2Rr C 3r2 4 (R2 C Rr C r2/

where z is the height of the center of gravity, R is the radius of the bottom end of the stem section (m), r is the radius at the top end (m), and l is the length of the stem section.

This height of the centroid z is then recalculated into the absolute height at the stem of that tree and then used to obtain a basic density value from the regression of density over height, as a representative value for the section. The obtained basic density is finally multiplied with the section volume. Using the regression example illustrated in Fig. 3.4 and a log of 3 m length and R = 0.40 m, r = 0.36 m, one would come to a height of the centroid (z) within the section of 1.44 m. If we assume that this log was the second 3 m section of the stem we have to add the stump height (0.3 m) and the first log (3.0 m) to z, resulting in an absolute height of the centroid that is 4.74 m. For this stem section an average basic density of 417 kg m“3 would be determined according to the tree-specific regression obtained in Fig. 3.4. Using this calculation template section by section will result in a reliable approximation of stem biomass.