### Study site

This study was conducted in the Noonan Research Forest (NRF, N 45°59′12′′, W 66°25′15′′), a 1531-ha forest located 30 km northwest of Fredericton, New Brunswick, Canada. Since 1985, the forest is managed by The University of New Brunswick. In 2002, a total of 296 stands were delineated on the NRF, of which 38 stands were considered non- forested stands and 19 stands were recently harvested. Hardwood, mixed and softwood stands compose the remaining 239 stands. These stands all originated following land clearing and fire in the 1920s.

A permanent 100 m by 100 m inventory grid was overlaid on the NRF in a north-south, east-west orientation (Fig. 1). In this study we used random selection within stand polygons, and 3 grid intersections per stand were randomly selected. In stands with 3 or less grid intersections, all grid intersections within that stand were selected. A total of 705 sample points was selected (approximately 1 point per 2 ha), and all data were collected in the summer and early fall of 2002 using a 2 M basal area factor angle gauge (i.e., each tally tree represented 2 m^{2}∙ha^{− 1} of basal area).

Diameter at breast height (DBH, nearest 0.1 cm), distance from sample point to tree center (DIST, 0.1 m), and species were recorded for each count “in” tree. Total height (TOTHT, nearest 0.1 m) was measured on a subsample of up to three trees per species per plot. For each species the diameter distributions were divided into thirds and one tree selected from each third. Species-specific height-diameter equations were derived from the subsampled heights and total height for all trees estimated. A total of 8518 trees were tallied and 4572 trees were selected for height measurement.

The BAF at which trees become borderline (maxBAF) for all trees was calculated using measured DBH and DIST:

$$ \mathrm{maxBAF}={\left(\frac{\mathrm{DBH}}{2\cdot \mathrm{DIST}}\right)}^2 $$

(1)

and used in the sample simulations to quickly determine if trees were “in” or “out” for a given BAF (see Yang et al. (2017) for more details).

### Carbon estimation

To estimate aboveground Carbon (C) storage, aboveground biomass (AGB) needs to be estimated first using allometric equations (Elias and Potvin 2003). Using Lambert et al.'s (2005) Canadian biomass equations, aboveground biomass was estimated by four components/pools (stem wood, stem bark, leaf, and branches) for each tree. Total biomass was calculated by summing across the four components. Total C and C by component (kg) were estimated using the species-specific conversion factors from Lamlom and Savidge (2003) for each tree. The carbon to basal area ratio (kg∙m^{− 2}) for total, stem wood, stem bark, branch, and leaf, were calculated by dividing the estimated C component by tree basal area:

$$ {\mathrm{X}\mathrm{CBAR}}_i=\frac{C_{\mathrm{X},i}}{{\mathrm{BA}}_i}=\frac{C_{\mathrm{X},i}}{0.00007854\times {\mathrm{DBH}}_i^2} $$

(2)

where XCBAR_{i} = the carbon to basal area ratio of the X carbon component (total, stem wood, stem bark, leaf or branches) for the *i*th sample tree (kg·m^{-2}); *C*_{X,i} = estimated X carbon component (total, stem wood, stem bark, leaf or branches) for the *i*th sample tree (kg); BA_{i} = the basal area of the *i*th sample tree (m^{2}); DBH_{i} = the measured diameter at breast height of the *i*th sample tree (cm).

### Big BAF sample simulation

The 705 plots collected in the original complete inventory of NRF were used here to simulate smaller sample sizes. Sample plots were selected using simple random sampling, and 250 replicate sample simulations were generated with different fixed sample sizes ranging from 20 to 100 in steps of 10 and from 100 to 200 in steps of 50. Count BAFs (cBAF) ranged from 2 to 30 M in steps of 2 and measure BAFs (mBAF) ranged from 4 to 100 M in steps of 4. Only combinations where mBAF __>__ cBAF were used and all allowable combinations of cBAF × mBAF were simulated on each sample point selected in each replicate sample simulation. Count and measure trees were determined using maxBAF, if maxBAF was greater than or equal to cBAF, then the tree was considered “in” for that sample point and cBAF. Likewise, measure trees were those trees that had a maxBAF that was greater than or equal to a given mBAF.

### Data analyses

To explore the influence of cBAF and mBAF on sampling outcomes on total C and different C components, trends were evaluated across the range of cBAFs and mBAFs. For each sample combination (cBAF × mBAF × sample replicate), average basal area per hectare (m^{2}∙ha^{− 1}) was estimated using:

$$ \overline{\mathrm{BA}}=\frac{\sum \limits_{i=1}^N\left({g}_i\times \mathrm{cBAF}\right)}{N} $$

(3)

where \( \overline{\mathrm{BA}} \) is the average basal area (m^{2}∙ha^{− 1}); *g*_{i} is the number of count trees on the *i*th sample point; cBAF is the count BAF; *N* is the number of sample plots. The mean XCBAR was calculated by averaging the individual tree XCBARs across all measure trees in the sample (mean ratio estimator):

$$ \overline{\mathrm{XC}\mathrm{BAR}}=\frac{\sum_{m=1}^M{\mathrm{XC}\mathrm{BAR}}_m}{M}=\left(\frac{1}{M}\right)\sum \limits_{m=1}^M\left(\frac{{\mathrm{XC}}_m}{{\mathrm{BA}}_m}\right)=\left(\frac{1}{M}\right)\sum \limits_{m=1}^M\left(\frac{{\mathrm{XC}}_m}{0.00007854\times {\mathrm{DBH}}_m^2}\right) $$

(4)

where \( \overline{XCBAR} \) is the carbon: basal area ratio of the *X*th C component for the *m*th measure tree (kg∙m^{− 2}); *M* = the total number of measure trees in the sample; XC_{m} = the content of the *X*th C component in the *m*th measure tree (kg); BA_{m} = the basal area of the *m*th measure tree (m^{2}); and DBH_{m} = the DBH of the *m*th measure tree (cm). Average C per ha (kg∙ha^{− 1}) by C component is then obtained by multiplying each \( \overline{XCBAR} \) by \( \overline{BA} \):

$$ \overline{\mathrm{XC}}=\overline{\mathrm{XC}\mathrm{BAR}}\times \overline{\mathrm{BA}} $$

(5)

Bruce’s formula (Goodman 1960; Yang et al. 2017) was used to estimate percent sampling error for each C component for each sample combination:

$$ se\%\left(\overline{XC}\right)=\sqrt{se{\%}^2(Counts)+ se{\%}^2(XCBAR)} $$

(6)

where,

$$ \mathrm{se}\%\left(\overline{Y}\right)=100\left(\frac{S_{\overline{Y}}}{\overline{Y}}\right)=100\left(\frac{S}{\sqrt{n}}\right)\kern0.2em \left(\frac{1}{\overline{Y}}\right)=\frac{\mathrm{CV}\left(\overline{Y}\right)}{\sqrt{n}} $$

(7)

where \( \overline{Y} \) is either Counts or XCBAR, \( {S}_{\overline{Y}} \) is the associated standard error, *S* is standard deviation; \( \mathrm{CV}\left(\overline{Y}\right) \) = coefficient of variation; and *n* = sample size (Note that while \( \mathrm{se}\left(\overline{\mathrm{BA}}\right) \) > se(Counts), se % (Counts) = \( \mathrm{se}\%\left(\overline{\mathrm{BA}}\right) \) because Count and BA differ only by the constant BAF, which appears in both the numerator and denominator of eq. 7).

The “true” or “best” value of \( \overline{XC} \) (denoted as \( \widehat{XC} \)) was assumed to be the normal horizontal point sample average based on cBAF = 2 and all trees measured (i.e., mBAF = 2) across the 705 sample points. \( \widehat{XC} \) and its associated standard error (\( se\%\left(\widehat{XC}\right) \)) were compared to the estimates of \( \overline{XC} \) and se % (XC) by cBAF and mBAF using beanplots (Kampstra 2008) and other graphical methods.

### Sample size and sample cost estimation

Using the sampling results, a nonlinear mixed effects model was fit to predict \( se\%\left(\overline{XC}\right) \) as a function of sample size (Yang et al. 2017):

$$ \mathrm{se}\%\left(\overline{\mathrm{XC}}\right)={b}_0\left(\frac{1}{n^{b_1}}\right) $$

(8)

with random effects for cBAF and mBAF nested within cBAF fitted for both *b*_{0} and *b*_{1}. Using the random effects, the variance contributions associated with cBAF and mBAF were estimated for each C component. The minimum samples sizes required for 10% error were then estimated for each C component over the range of cBAF and mBAF.

Yang et al. (2017) outlined methods for determining costs associated with big BAF sampling and their methods were used here to estimate inventory costs associated with the various C components and combinations of cBAF and mBAF. Yang et al.’s (2017) cost equation was:

$$ {\mathrm{Cost}}_{\mathrm{Total}}={\mathrm{Cost}}_{\mathrm{Overhead}}+n\left({\mathrm{Cost}}_{\mathrm{Est}}+k\left(\frac{\overline{\mathrm{BA}}}{\mathrm{cBAF}}\right)\right)+r\times n\left(\frac{\overline{\mathrm{BA}}}{\mathrm{mBAF}}\right)+w\sqrt{n\times A} $$

(9)

where Cost_{Overhead} = the costs of planning the forest inventory; Cost_{Est} = the costs associated with plot establishment, *k* = the costs of a count tree determined, species identified and DBH measured; *r* = costs to measure tree height, *w* = travel costs between sample points (dollars per m travelled). Assuming a two-person crew-day costs $650.00 Canadian, and using the time-motion values reported by Yang et al. (2017), the coefficients in eq. 9 were as follows: Cost_{Overhead} = $1300.00 (assumed 2 days for planning); Cost_{Est} = $2.70; *k* = $0.765; *r* = $4.41; and *w* = $ 0.0945. Using these values and the minimum sample sizes estimated from Eq. 8, total inventory costs were calculated. Design effects, defined as the relative efficiency of sample design 1 versus sample design 2 (Särndal et al. 1992), were calculated as the ratio of costs of the two sample designs:

$$ {DE}_{1,2}=\frac{{\mathrm{Cost}}_1}{{\mathrm{Cost}}_2} $$

(10)

If DE < 1, then sample design 1 is more efficient than sample design 2. In this study, sample design 1 and 2 are various cBAF and mBAF combinations.