The selection process of representative trees for a stand has to consider the number of trees that should be sampled and how they should be selected from the population.

The actual number of sample trees is determined by the level of precision required and the variability of the resource, with the principle that model bias should be entirely avoided as stated by van Laar and Theron (2004). There is always a trade­off between accuracy and efficiency of the sampling procedure. Higher accuracy results in a higher cost and time investment. Improving the standard error of the mean by 50 % requires a four times higher sampling number, since the standard error of the mean is inversely proportional to the square root of the sample size (Eq. 2.2). This well-known fact and the effort involved in biomass studies are a strong motivation for choosing a sampling number close to the theoretical minimum number, which is still representative of the population.

Sy = P (2.2)

where

Sy is the standard error of the estimate of the arithmetic mean of a population s is the estimate of the standard deviation of the population n is the number of samples

The standard error of the mean, Sy, is then multiplied by the appropriate z-score, obtained from the Normal-(z)-Distribution, e. g., 1.96 for a 95 % confidence level, to obtain the margin of error (ME). The margin of error defines the confidence interval, when subtracted from or added to the estimate, E.

The minimum sampling percentage for simple random sampling from large populations depends on the variance in the population and the accepted error probability of the outcome, defined by the chosen confidence limits (Eq. 2.3).

z ■ s [1]

E

where

N is the minimum number of samples to be representative of the population E is the margin of error, the maximum accepted difference between estimated (x)

and true mean (д) of the population. z is the z-score obtained from a Normal-(z)-Distribution s is the estimate of the standard deviation of the population

Usually the true variance is unknown, thus the sample variance has to replace the population variance when calculating the t-statistic or the z-score, as indicated in Eq. 2.4 (van Laar an Akga 2007).

As stated by van Laar and Theron (2004) “There is no universally accepted criterion for the size of the acceptable margin of error, which is a function of the population variance, sample size and the statistical risk of exceeding the margin of error.” The margin of error for volume calculations in forest inventory data should preferably not exceed 3-6 % of the mean (Smith et al. 2004; Bundeswaldinventur 2008 cited by Waggoner 2009). For biomass and carbon inventories sometimes higher error margins, ranging from 10-15 %, are accepted for the estimation (van Laar and Theron 2004; Hollinger 2008).

However, the minimum sampling number of trees has to be determined itera­tively. Stauffer (1982) suggested an iterative procedure and expressed the margin of error as a percentage of the mean, which requires the substitution of the standard deviation by the coefficient of variation, according to Eq. 2.4. An iterative determination of the sample size with increasing n, converges after a few iterations (for an example see van Laar and Akga 2007, p. 258).

where

n is the number of samples ta is the t-statistic at n-1 degrees of freedom s% is the coefficient of variance and E% the relative margin of error.

If variance information on biomass is not readily available, which might be typical, other methods can be applied like regression-based sampling. This method

is based on the use of an auxiliary variable, which is easier and cheaper to measure and which is ideally linearly correlated to biomass. For example, tree basal area (BA) lends itself as an auxiliary variable for biomass sampling, since it is highly correlated to tree biomass and easy to determine from stem diameter measurements, which can be conducted in high numbers. The process is then a two-phase sampling procedure, where first the population mean of the BA is estimated and then a sub­sample is drawn for biomass sampling. However, Cunia (1990) raised doubts about the unbiased nature of the sample if anything other than diameter at breast height (DBH) or height were used as auxiliary variables.

Once the minimum tree number for sampling has been determined, the selection of trees which represent the stand is a substantial task. It requires prior knowledge about the representation of the independent variables that will be used for modelling. Tree height and DBH were chosen as independent variables in this example. A DBH-height-plot produced from a preceding terrestrial inventory of the stand or the region where the function should be applied, is a good help to choose the sample trees. A way to obtain robust regression estimates, which can be applied in similar stands, is to divide the classes of DBH and height and sample in each class in a stratified sampling procedure, so that the selected sample trees cover the full range of DBH-height relations. This can be done visually, based on equidistant classes as indicated in Fig. 2.2, or alternatively, in a bivariate binning procedure via a
classification based on quantiles for the DBH and the height. If too few DBH-height pairs are available, a selection based on DBH-classes presents a feasible alternative.

Van Laar and Akqa (2007) refer to Attiwil and Ovington (1968), who compared different sampling methods for the estimation of biomass from an area and found that the regression-based method (N = 20), performed better than the estimation of the biomass from the mean DBH tree (N = 5) sampling of four trees in five DBH classes (N = 20) and regression-derived mean DBH tree multiplied by the tree num­ber. The other sampling methods produced biased results in the range of —8.5, —3.1, and —11.6 %. These results and the criticism of Cunia (1990) on importance sam­pling may serve as valid arguments to base tree selection on a regression approach that covers the spectrum of DBH-height relations in the population. This should also result in more generic models that are easier to adapt to other sites later on.

There are two different approaches to estimate biomass for forest areas. Both include a forest inventory component based on a sampling procedure. The first approach is based on dry mass/area relations, where upscaling to the stand level is done per sample plot first and then the area of sample plots is used as a ratio estimator to upscale biomass from the total sampled area to the total stand area. The second approach is based on single tree equations (Chap. 3) and relies on stem frequencies in diameter classes, as produced in the inventory. For each mean tree per diameter class, the biomass is upscaled with a ratio estimator relating the sampled tree numbers to the population tree numbers.

The following example illustrates the difference of the application of an estab­lished allometric biomass prediction model for Pinus radiata (van Laar and van Lill 1978) to estimate the total biomass based on different selections of trees from the population. The assumption that the model produces unbiased estimates for the population should fit the purpose here. The model is applied to estimate dry mass for a pine stand of one hectare in size, which was fully mapped for DBH and tree heights (Ackerman et al. 2013) to illustrate the influence of tree selection on the outcome of a model prediction. The application of an allometric equation to all individual trees serves as a reference. This is tested against the application of the biomass model, based on the diameter of the quadratic mean diameter tree (dq) and based on average dq trees in seven diameter classes. The diameter distribution of the stand is shown in Fig. 2.3.

The application of the functions for all 318 individual trees of that stand resulted in an estimated total stand dry mass of 424.99 Mg ha_1 (tons). The estimate based on the mean quadratic diameter was clearly negatively biased at 422.87 Mg ha_1 (—2.12 Mg ha_1), while the diameter class-based approach produced a nearly unbiased dry mass estimate of 424.88 Mg ha_1 (—0.11 Mg ha_1). It is obvious that the estimate of biomass for a full stand based on the dq does not take into account the nonlinear relation between diameter and biomass, and thus underestimates the contribution of the bigger diameter trees in the stand, while the estimate with diameter classes mitigated that effect to an acceptable degree. This effect can be expected to be much stronger for a more skewed diameter distribution.

A multitude of inventory techniques developed for volume estimation in forests can be easily adapted for biomass inventories. The same variables that are used for biomass estimation, namely DBH and height, are typically used to estimate volume. The interested reader might therefore consider some standard volumes on forest inventory, such as Kangas and Maltamo (2006), van Laar and Akga (2007), Mandallaz (2008), for detailed information.