# Individual-Based Allometric Equations Accurately Measure Carbon Storage and Sequestration in Shrublands

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## Abstract

**:**

^{–1}year

^{–1}) exceeded the margin of error produced by the simulated sources of uncertainty. This demonstrates that, even when the major sources of uncertainty were accounted for, we were able to detect relatively modest gains in shrubland C storage.

## 1. Introduction

#### 1.1. Potential Sources of Uncertainty

#### 1.2. Aims and Objectives

## 2. Experimental Section

#### 2.1. Study Sites

#### 2.2. Field Sampling

#### 2.3. Biomass Sampling

#### 2.4. Using Allometric Equations to Calculate Carbon

^{2}), Height is maximum shrub height (m), and a and b are fitted coefficients. Dry weight was converted to carbon storage using a multiplication factor of 0.5, which is standard in New Zealand [25]. This function form is analogous to applying a logarithmic link function in a generalized linear modelling (GLM) framework. Using a logarithmic link function has the advantage over ordinary linear least squares regression (OLS) in that it reduces bias in fitted dry weight. In particular, using this type of function prevents negative fitted values for small shrubs, since the result of the exponential transformation is always non-negative. Previous generalized linear modelling (GLM) analyses on the same harvested shrub dataset demonstrated that there was no AIC support for including site- or species-specific coefficients in the allometric equation [26].

#### 2.5. Simulating Allometric Uncertainty

^{4}permutations to generate a bootstrapped mean and 95% confidence interval (CI) for regression coefficients and predicted-dry-weight values. We stored the coefficient values for each permutation to estimate the effect of allometric uncertainty on C storage estimate error (see section “Simulating Uncertainty in C Storage Estimates” below).

#### 2.6. Simulating Measurement Errors

#### 2.7. Simulating Uncertainty in C Storage Estimates

**Figure 1.**Schematic diagram for simulation of independent effects of the three sources of uncertainty studied—harvest and plot measurement error and bootstrapped allometry uncertainty—on carbon (C) storage estimate errors.

^{6}simulations in total). When combining harvest measurement error and bootstrapped allometric uncertainty (simulation type 5) for each harvest error simulation we generated 1000 bootstrapped values for the linear coefficient and intercept, and then combined these coefficient values with the observed measurements for shrubs in plots to estimate C storage (again giving 10

^{6}simulations in total). When combining bootstrapped allometric uncertainty and plot measurement errors (simulation type 6) for each bootstrapped estimate of allometric coefficients we ran 1000 plot measurement error simulations. When combining all three sources of uncertainty (simulation type 7), we used 1000 harvest measurement error simulations. For each harvest error simulation, we generated 1000 bootstrapped values for the allometric coefficients and then ran 1000 plot measurement error simulations (giving 10

^{9}simulations in total).

**Figure 2.**Schematic diagram for simulation of interactive effects between the three sources of uncertainty studied—harvest and plot measurement error and bootstrapped allometry uncertainty—on carbon (C) storage estimate errors.

## 3. Results

**Figure 3.**Allometric relationship between harvested shrub dry weight and shrub total basal area multiplied by shrub height (

**a**). This curve was obtained by fitting the equation: Dry weight = exp(a

**•**ln(basal area

**•**height) + b). Solid line in (

**a**) shows the curve fitted on the entire harvested shrub dataset with a = 1.0215 and b = 6.1512. Dotted lines show the upper and lower bounds of the 95% confidence interval for predicted values under bootstrapping. Confidence interval breadths for predicted values are expressed in terms of carbon (C) storage in (

**b**).

^{–1}or 42% of observed C storage). However, in all simulations the degree of uncertainty varied considerably amongst individual plots (Figure 4, Figure 5 and Figure 6). This variation was strongly related to the “observed” C storage of plots. Plots with large observed C storage tended to have greater uncertainty in C storage (i.e., larger confidence interval breadth for simulated values) expressed in absolute terms (Figure 7a). By contrast, plots with low C storage tended to have high uncertainty in percentage terms (Figure 7b). The greatest uncertainty in estimates of mean C storage across plots was observed for the three-way interaction including all sources of uncertainty (Figure 8), with the 95% CI for mean C estimates spanning over 2 Mg ha

^{–1}or 28% of observed mean C.

**Figure 4.**Mean (SimMeanC) and lower and upper 95% confidence interval limits (SimHighBound and SimLowBound) for carbon (C) estimate uncertainty simulations using the basal area approach. Simulated sources of uncertainty were diameter and height measurement errors for harvested shrubs (

**HM**), bootstrapped allometry uncertainty (

**AU**), and diameter and height measurement errors for shrubs on plots (

**PM**). The trend lines indicate a y = x curve.

**Figure 5.**Mean 95% confidence interval breadth (i.e., difference between 2.5th and 97.5th percentile) for simulated C storage of individual plots using the basal area approach. Confidence interval breadths are expressed either in absolute terms (

**a**) or as a percentage of “observed” plot C estimates (

**b**). The error bars show standard deviation (taken across plots). Codes are as follows: HM, harvest measurement error; PM, plot measurement error; AU, bootstrapped uncertainty in the allometry.

**Figure 6.**Mean (SimMeanC) and lower and upper 95% confidence interval limits (SimHighBound and SimLowBound) for the effect of interactions between different sources of error on carbon (C) estimate uncertainty simulations using the basal area approach. Simulated sources of uncertainty were diameter and height measurement errors for harvested shrubs (

**HM**), bootstrapped allometry uncertainty (

**AU**), and diameter and height measurement errors for shrubs on plots (

**PM**). The trend lines indicate a y = x curve.

**Figure 7.**95% confidence interval (CI) breadth for simulated C storage estimates of individual plots expressed in absolute terms (

**a**), or as a percentage of “observed” C storage (

**b**). The confidence intervals were taken from the simulations including all three error sources–harvest shrub measurement error, plot shrub measurement error and bootstrapped allometry uncertainty (i.e., simulation type 7).

**Figure 8.**Mean and 95% confidence interval breadth (i.e., difference between 2.5th and 97.5th percentile) for simulated mean C storage taken across plots using the basal area approach (

**a**); Confidence interval breadths expressed either in absolute terms (

**b**) or as a percentage of “observed” mean C across plots (

**c**). Codes are as follows: HM, harvest measurement error; PM, plot measurement error; AU, bootstrapped uncertainty in the allometric equation.

**Figure 9.**Mean simulated carbon (C) storage estimates for individual plots in the 2012 measurement vs. that for 2008 (

**a**). The line indicates a y = x curve. 2008 SimUpperBound is the upper bound of 95% confidence interval for simulated C estimates of individual plots in the 2008 measurement. The confidence intervals in (

**a**) were taken from the simulations including all three error sources: harvest shrub measurement error, plot shrub measurement error, and bootstrapped allometry uncertainty (i.e., simulation type 7); (

**b**) Mean and standard error of C sequestration across all plots and those in either the d’Urville or Oxford site. Mean 2012−Mean 2008 refers to C sequestration estimates using the mean of the simulated values (for each plot) in both 2012 and 2008, while Mean 2012−Upper 2008 refers to estimates derived by subtracting the upper bound of 95% confidence interval for each plot in 2008 from the mean of simulated values in 2012. This provides a test of whether or not we can detect C sequestration beyond the margin of error introduced by the uncertainty sources we simulated. Significance of paired t-tests for differences in C storage between 2008 and 2012 are as follows: * p < 0.05; ** p < 0.01; *** p < 0.001.

## 4. Discussion

^{–1}year

^{–1}). Considerable time and effort may be required for basal diameter measurements when stem densities are high. However, this might be offset by the requirement of fewer plots relative to more rapid, but less precise methods (e.g., [27]).

_{2}e (± s.e. 2907) between 2008 and 2012. Obviously, the economic viability of our survey method would depend heavily on carbon prices. With a unit price of US $20 per Mg CO

_{2}e our method could be applied for less than 10% of the money gained from carbon credits. Further, it may be possible that a reduction in sampling intensity would still allow us to detect changes in C storage with reasonable accuracy. Indeed, the sampling intensity we have used to estimate C sequestration (one plot per 37 ha) is much greater than that used in New Zealand’s national carbon monitoring system (one plot per 6400 ha).

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Mason, N.W.H.; Beets, P.N.; Payton, I.; Burrows, L.; Holdaway, R.J.; Carswell, F.E.
Individual-Based Allometric Equations Accurately Measure Carbon Storage and Sequestration in Shrublands. *Forests* **2014**, *5*, 309-324.
https://doi.org/10.3390/f5020309

**AMA Style**

Mason NWH, Beets PN, Payton I, Burrows L, Holdaway RJ, Carswell FE.
Individual-Based Allometric Equations Accurately Measure Carbon Storage and Sequestration in Shrublands. *Forests*. 2014; 5(2):309-324.
https://doi.org/10.3390/f5020309

**Chicago/Turabian Style**

Mason, Norman W.H., Peter N. Beets, Ian Payton, Larry Burrows, Robert J. Holdaway, and Fiona E. Carswell.
2014. "Individual-Based Allometric Equations Accurately Measure Carbon Storage and Sequestration in Shrublands" *Forests* 5, no. 2: 309-324.
https://doi.org/10.3390/f5020309