Appendix F Developing and using allometric models to estimate biomass
Within a specified forest stratum biomass carbon can be estimated using ground-based methods entailing an inventory of stem diameters and/or heights, and application of allometric models which relate above- and below-ground biomass to the inventory measurements. For a detailed treatment of important issues see Picard et al., (2012) and Chave et al., (2004). Stratification is a critical step in defining the appropriate domain in which an allometric model is developed and applied.
Allometric models for estimation of biomass have most commonly used stem diameter as the explanatory variable, with some also using tree heights, and to a lesser extent, canopy width and wood density. A growing number of researchers have shown that stem diameter can be an adequate biomass predictor at local or regional scales, with height or wood density, providing little improvement in the efficiency of allometric predictions of above-ground or below-ground biomass (e.g. Brown et al., 1989; Ketterings et al., 2001; Jenkins et al., 2003; Chave et al., 2005; Basuki et al., 2009; Xiang et al., 2011: Paul et al., 2014). This suggests that stem diameter accounts for common geometric, biomechanical and hydrodynamic principles that govern the transport of essential materials in trees (West et al., 1999; Enquist and Niklas, 2001). However, in some tropical forests, height and wood density have been shown to be important variables and their explanatory power should therefore be examined (e.g. Chave, 2005; Feldpausch et al., 2011 and 2012; Chave et al., 2014). Feldpausch et al., (2011 and 2012) showed that inclusion of tree height as an allometric factor can reduce error in estimates of tropical biomass and hence carbon stocks and emissions due to deforestation. Height at which diameters are measured often varies between forests based on the heights of the trees, shape of the stem and the average height at which they branch into multiple stems. As a general rule, when establishing an allometric model, the diameters should be measured at 130 cm height, or as high as possible but below the height at which the stem becomes multi-stemmed. This decreases measurement errors. Generally for shrub species, diameters can be measured at 10 cm height. When using an allometric model, trees should be measured at a height consistent with the data used to establish it.
Number of trees to harvest (sample) for deriving allometric models
Sampling error may be significant when selecting trees or shrubs for harvest to develop allometric models. In a global review of the use of allometrics based on stem diameter to determine the biomass of different tree species, Zapata-Cuartas et al., (2012) found that there was an exponential improvement in the precision in predictions of tree biomass with increasing sample size. Similar results were obtained by Roxburgh et al., (2015) who analysed above-ground biomass data from 23 species to quantify sampling errors associated with the development of allometrics. They found marked variability between allometrics in the number of individuals required to satisfy a given level of precision, with a range of 17-95 individuals to achieve biomass estimates with a standard deviation within 5% of the mean for the best performing stem diameter selection algorithm, and 25-166 individuals for the poorest. This variability arises from (a) uncertainty in the relationship between diameter and biomass across allometrics, and (b) differences between the diameter size-class distribution of individuals used to construct an allometric, and the diameter size-class distribution of the population to which the allometric is applied. For pan-tropical forests Chave et al., (2004) found an exponential decline in %CV with increased sample size, with %CV increasing above 10% when 20 trees or fewer were sampled.
Correcting for moisture content
Total above- or below-ground biomass is weighed fresh in the field. Sub-samples are used to determine the dry-weight equivalent. These need to be representative, so as to reduce errors in estimation of dry weight. Ideally, this sub-sampling would be based on each tree component (foliage, bark, twigs, large branches and stems etc.). As a minimum, selected trees should be divided into crown (all foliage and twigs less than about 5 mm diameter) and the remaining bole (stem and branches). The fresh weights of these two components are measured in the field, and then sub-samples (at least three of about 2-3 kg) taken of each component, weighed and transported back to the laboratory and dried (at 70oC) until the dry weights stabilise. For the bole samples, this could take several weeks. Using the average moisture content of sub-samples of each component, a weighted average whole-tree moisture content can be determined based on the relative contribution to total fresh weight of the individual components. For shrubs with no pronounced stem, a separate bole component is not required.
Recent work (Ximenes et al., 2006; Paul et al., 2014) in temperate forests showed moisture content varies more between sites than between species within sites. Within a site, there is evidence that moisture contents varied between growth-habits (e.g. trees compared to shrubs), but within a growth-habit at a given site, variability was just as high within as between species (Paul et al. 2014). Therefore, species-specific moisture content determinations appear to be unnecessary. Rather, average moisture contents can be derived for key genera and growth-habits within sites. Data are limited for tropical forests, so further testing should be conducted.
Selecting the form of an allometric model
The traditional power law allometric model is a simple power function. The linear equivalent of such a power functions is: ln(y) = a + b × ln(x), where y is the dependent variable (biomass, kg DM tree-1), x is the independent variable (stem diameter, cm), a is the intercept coefficient, and b is the scaling exponent. Parameters a and b are estimated using least squares regression.
The logarithmic transformation, in addition to linearizing the relationship, also corrects for heteroscedasticity. Regressions such as these produce unbiased estimates of log-biomass. However direct transformation back to the original scale will yield biased estimates of biomass. There are a number of alternative ways of calculating a bias correction. A common method is to multiply estimates by a correction factor based on the ratio of arithmetic sample mean and mean of the back-transformed predicted values from the regression as described by Snowdon (1991).
There is some evidence that power-law models fail for very large trees, with over-estimates of biomass being common when DBH is >50 cm (Niklas, 1995, Chambers et al., 2001; Chave et al., 2005; Fatemi et al., 2011) due to greater damage, decay and senescence as trees mature. In such cases, non-linear models, or weighted-combined models, should be explored as an alternative to traditional power-law allometric models, with additional explanatory variables such as tree height being included (Brown et al., 1989; Parresol 1999; Bi et al., 2004; Ketterings et al., 2001).
Performance of allometric models
To evaluate model efficiency of allometric models, statistics used are based on those recommended in a review by Parresol (1999), the most important being the Fit Index, otherwise known as model efficiency (Soares et al., 1995). Efficiencies of >0.70 are regarded as reasonable predictors of biomass, but ideally the efficiency should be > 0.9.
Model EF is related to the ratio of the total sum of squares to the residuals sum of squares.
where Oi are the observed values, Pi are the predicted values, and Ō is the mean of the observed data. A positive value indicates that the simulated values describe the trend in the measured data better than the mean of the observations, with a value of 1 indicating a perfect fit. A negative value indicates that the simulated values describe the data less well than a mean of the observations. The percentage coefficient of variation (CV) can also calculated for each model fit.
and N is the number of observations, and p is the number of parameters used in the model.
Generalised (generic) allometric models
For native forests which may contain many different species, it is impractical to develop allometric models for each species at each monitoring site. Generic allometric models may be derived using biomass data from a given species, or growth-habit, across a number of different sites within a specified region, or domain.
Appropriate domain of generic allometric models
Recent studies in woodlands (Williams et al., 2005), eucalypt forests (Montague et al., 2005) and mixed-species plantings (Paul et al., 2013) have shown that although site-species differences were significant, the amount of variation accounted for by these site-species factors was small, thereby supporting the use of generalised allometrics which had slightly less accuracy, but much greater certainty. Several authors have proposed such generalised allometric models for large-scale application for a range of tree or shrub species (e.g. Pastor et al., 1984 (north-east USA); Zianis and Mencuccini 2003 (northern Greece); Jenkins et al., 2003 (USA); Williams et al., 2005 (northern Australia); Montagu et al., 2005 (eastern Australia); Muukkonen 2007 (Europe); Dietze et al., 2008 (south-eastern USA); Xiang et al., 2011 (China); Vieilledent et al., 2012 (Madagascar); Kuyah et al., 2012a (Kenya)).
Generic allometric models should not be applied outside their appropriate domain, given that significant variations in factors such as topography, hydrology and soil nutrient availability may result in systematic biases (Clark & Clark, 2000; Clark, 2005). For this reason, generalised allometrics which have entailed the use of larger pan-continental datasets (Cannell 1984; Brown et al., 1989; Brown, 1997; Chave et al., 2005; Zapata-Cuartas et al., 2012) need to be applied with caution. Verification at fine-scale of these pan-continental generalised allometrics have often failed (e.g. Basuki et al., 2009; Vieilledent et al., 2012). Madgwick et al., (1991) found that for the eucalypt genera, allometrics developed in one country may not be accurate for the same life-forms growing in other countries.
In order to avoid serious error and bias, the allometric model should not be applied to trees (or other vegetation) outside of the diameter range of the samples used to construct the allometric model.
Categorisation (species versus growth-habit) of generic allometric models
There is clear evidence that above-ground biomass allometry of shrubs differs greatly from that of trees (Keith et al., 2000; Bi, et al., 2004; Paul et al., 2013). Differences in allometry are less significant within these growth-habit categories. Recently, Paul et al., (2016) showed that cost effective prediction of biomass across a wide range of stands in Australia is possible using generic allometric models based on only five plant functional types. In addition to species and life-form, climate is also an important factor influencing allometric models for above-ground biomass. Mean annual rainfall can be a major factor (Brown et al., 1989; Sternberg and Shoshany, 2001; Drake et al., 2003; Chave et al., 2005; De Walt and Chave, 2004).
Development of allometrics for below-ground biomass has generally entailed development of generic rather than site-and-species specific relationships due to limited available data on root biomass (Barton and Montagu, 2006; Ouimet et al., 2008; Peichl and Arain, 2007; Xiang et al., 2011; Paul et al., 2014).
Testing of allometric models
Allometric models should always be tested by comparing with direct measurements of above- and below-ground biomass across the domain region of interest. Examples include: northern hardwood forests in New Hampshire, USA (Arthur et al., 2001), mixed-species found within the Sonoran Desert (Búquez and Martínex-Yrízar, 2011), pure stands of Poplar or Norway Spruce (Pérez-Cruzado et al., 2015) and mixed-species plantings across Australia (Paul et al., 2014).
For direct measurement of above-ground biomass, either a sample of individual trees encompassing the full range of sizes found in the forest in which the allometric is to be applied, or whole plots of about 20m x 20 m (but probably larger in rainforest) are harvested and weighed. Within these plots, sub-plots are selected for root excavation. In forests where stocking is too low (<500 stems/ha) to make whole plot root excavation efficient, roots are excavated around individual trees or shrubs with excavation boundary varied according to the size and distance to neighbouring trees (Picard et al., 2012). Required depth of excavation depends on the depth of tap roots. Previous work suggests that 2 m depth is sufficient (Mokany et al., 2006; Paul et al., 2013). Schenk and Jackson (2002) concluded that globally 50% of all roots are within the upper 0.3 m while 95% of all roots are within the upper 2 m of the soil profile. The majority of root mass is in the coarse (> 2mm) fraction, so that roots finer than this can be ignored where the objective is to measure total tree biomass.
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