Thinking outside the square: Evidence that plot shape and layout in forest inventories can bias estimates of stand metrics

Plot‐based data collection is an important component of quantitative ecological research and is widely used. Some of the most extensive plot‐networks can be found in country‐wide forest inventories, which provide critical information about the state of forest ecosystems. While sampling designs for forest inventories have been well studied, plot design and installation has received less attention. The New Zealand National Forest Inventory of natural forest uses a nested plot design with a 0.126 ha circular plot superimposed concentrically on a 0.04 ha square plot. Stems ≥ 60 cm diameter at breast height (DBH) are measured in the circular plot while stems ≥ 2.5 DBH are measured in the square plot. Stem density of ≥60 cm DBH stems measured in the circular plots were compared with those from square plots. Stem densities estimated from square plot measurements were 23.7% higher than those estimated from circular plot measurements in the 2002–2007 inventory, and 18.4% higher in the 2009–2014 inventory. The main cause of this discrepancy appears to be due to the placement of plot boundaries during establishment of square plots. This effect may have resulted from a subconscious tendency of field teams to include large trees inside plots when laying out these boundaries. It is concluded that estimates from the circular plots are unlikely to be biased while those from the square plots are positively biased. This study highlights the critical importance of plot design and plot placement in forest inventories to ensure that estimates of stand attributes are unbiased. Especially on undulating or uneven terrain, methods of determining whether trees are inside or outside plot boundaries of circular plots are likely to be more accurate than those typically used for square or rectangular plots.

precision. Most NFIs use fixed-area plots to estimate stem density (number of stems per hectare) by counting the trees that fall within plots and dividing by the plot area (Tomppo, Gschwantner, Lawrence, & McRoberts, 2010). Usually all stems above a specified minimum diameter are counted. It is important that plots in large-scale forest inventories and ecological studies are established in a manner which provides unbiased estimates of forest parameters such as stem density and basal area. Curtis and Marshall (2005) note that imprecisely surveyed plots are a frequent cause of errors in the calculation of plot area and corresponding values of stand statistics.
Although there is no consensus concerning plot shape, in most cases circular plots are preferred over square or rectangular plots as they require only a single control point at the plot centre compared with the four corner points required for square or rectangular plots.
In addition, circular plots have the shortest plot perimeter for a given plot area meaning that they require fewer decisions regarding inclusion or exclusion of trees close to the plot boundary (McRoberts, Tomppo, & Czaplewski, 2015). However, only a few studies have tested the effect of plot shape on stand descriptors. Comas, Mateu, and Delicado (2011) looked specifically at the effect of plot shape and the number of plots on the sample variance of stem numbers per unit area and found dependencies indicating that plot shape might be critical for accurate sampling. Condit et al. (1996) tested plot size in regard to tree-diversity in three tropical rainforests and showed that species-area curves (species numbers as a function of plot size) had different forms for stems of different diameters, and showed that in tropical forests 20 × 20 m plots are inadequate to estimate large-tree species composition. Laurance, Ferreira, Merona, and Hutchings (1998) studied the effect of large square (100 × 100 m) and rectangular plots (40 × 250 m) on estimates of tree diversity and composition in a tropical rainforest.
They did not find any significant differences in tree species diversity while the size of their plots met the requirement in regard to the minimum sampling area for phytosociological studies (Mueller-Dombois & Ellenberg, 1974), informing current efforts to establish NFI's in tropical and subtropical countries (Saket, Branthomme, & Piazza, 2010). Plot size and shape and their influence on stand structural variables such as stand density are touched on in forest inventory textbooks (Husch, Beers, & Kershaw, 2003;Spurr, 1952) highlighting the importance of the correct plot size to include a representative number of trees accounting for the forest heterogeneity.
A review of 41 countries with NFIs reveals that more than 76% use circular plots to measure the largest tree size class either in a cluster (43%) or a single plot design (56%; see Supporting Information).
In cluster-plot designs, circular large-tree subplots average 0.06 ha while the full cluster area sampled for large trees averages 0.29 ha per cluster (permanent plots). In single-plot designs the average area for large-tree circular plots is 0.07 ha, only marginally bigger than the average size of subplots in cluster-plot designs. Only 15% of NFIs use rectangular or square plots. These are mostly from South American countries including Brazil, Ecuador and Peru, which have deployed plot forms based on the Food and Agriculture Organisation of the United Nations, National Forest Monitoring and Assessment design (Saket et al., 2010;Vidal, Alberdi, Hernández, & Redmond, 2016).
The rectangular plots used in these countries are large in area with an average plot-size of 0.36 ha. Smaller square plots are used only in China which uses a plot-size of 0.06 ha. In the past, New Zealand has used even smaller square plots of 0.04 ha although its current NFI only uses these smaller square plots for stems < 60 cm diameter at breast height (DBH).
No study known to the authors has closely examined practical difficulties associated with the setting up of circular versus square or rectangular plots that could influence the estimation of area-based variables in large-scale forest inventories or ecological studies.
However, Daubenmire (1959) highlighted the practical limitations of overly elongated plots. The limitations mentioned were the difficulty of laying out such plots, and the high proportion of plants close to plot boundaries requiring decisions as to whether the plant is inside or outside the plot.
While New Zealand, like most other countries, uses circular plots in its NFI for planted forests (Beets et al., 2011(Beets et al., , 2012Herries et al., 2017) surveys and ecological studies of New Zealand's indigenous forests have since the 1970s made extensive use of a 20 × 20 m square plot design (Allen, 1979(Allen, , 1993. This plot design was therefore chosen as the base unit for New Zealand's inventory of natural forest, which consists of land classified as forest in 1990 excluding planted forest, and is comprised predominantly of tall indigenous forest (Payton, Newell, & Beets, 2004). The use of different plot shapes between strata (planted vs. natural) was accepted in order to maintain compatibility with previous datasets (Coomes, Allen, Scott, Goulding, & Beets, 2002;Payton et al., 2008), often an important consideration when updating current forest inventories designs (Tomppo et al., 2010). It was also considered useful to utilise the analytical and data storage systems that had been developed for analysing plant composition and change (Allen, 1979(Allen, , 1993Hurst & Allen, 2007). This was despite some concern about the desirability of deriving estimates (e.g. of carbon and stand volume) using plot designs developed under a theoretical background of vegetation ecology/phytosociology (Mueller-Dombois & Ellenberg, 1974) rather than forest mensuration (Husch et al., 2003).
Data collected in natural forests using the 20 × 20 m plot design have been used recently to classify the indigenous forests in New Zealand (Wiser, Hurst, Wright, & Allen, 2011). This intended use is reflected in the design and field manual that describes the plot establishment and the data collection methods. The focus of placing plots in "homogenous" areas of vegetation and site conditions (Hurst & Allen, 2007) is common in such qualitative phytosociological studies (Mueller-Dombois & Ellenberg, 1974), but can result in less than stringent operating procedures for laying out plot boundaries as described in Hurst and Allen (2007) or Payton et al. (2004).
Because there was concern that 20 × 20 m plots were too small to sample the large trees often found in New Zealand's natural forest, it was decided to use a nested plot design in the natural forest NFI, consisting of a 20 m radius plot for stems ≥ 60 cm, superimposed on a 20 × 20 m square plot for stems ≥ 2.5 DBH (Figure 1).
The use of square plots as sampling units introduces the challenge of laying out plot boundaries in a manner that ensures plot size can be precisely determined and vegetation is sampled in an unbiased fashion. Goulding and Lawrence (1992) advocate establishing a square plot by initially setting up a diagonal line from the start corner to the furthest plot-corner, then establishing the other two corners, and stretching lines between corners to determine which trees are inside or outside the plot. However, such a procedure is difficult to apply in uneven terrain and in natural forest with large trees or dense vegetation, and the methods used to establish 20 × 20 m plots in New Zealand's natural forest NFI differ from this approach in a number of ways. Briefly, each nested plot is laid out using the following procedure (Payton et al., 2004): 1. Use a GPS unit to navigate to within 30 m of the plot location.
Then use the compass bearing and 30 m tape from the GPS location to establish point P (Figure 1).

2.
The perimeter line P-A is laid out on a compass bearing at right angles to the slope following the predominant contour of the slope, or if the terrain is flat on a northerly bearing (magnetic 0°).

3.
The perimeter line P-M is laid out at right angles to the P-A line.

4.
The open ends of the plot are then connected using two 20 m tapes to form a 20 × 20 m square.

5.
Once the outer boundary of the 20 × 20 m plot is laid out, the inner subplots are segregated by laying tapes across the plot at 5 m intervals along each boundary, resulting in a grid with 5 × 5 m subplots. Some concern about the plot design used in New Zealand's natural forest NFI began after analysis of data from the first inventory carried out between 2002 and 2007 revealed the estimate of stem density of large diameter trees based on trees measured in the 20 m radius circular plots was lower than the estimate based on trees measured in the 20 × 20 m square plots. At first it was believed this discrepancy could have been caused by field teams overlooking some trees near the perimeters of circular plots. Great care was therefore taken during the second inventory carried out during 2009-2014 to ensure that all trees ≥ 60 cm DBH within the circular plots were identified and measured. However, analysis of data from the second inventory revealed that the stocking discrepancy remained. Suspicion then fell on the methodology used to lay out plot boundaries of 20 × 20 m square plots. Careful examination of the protocols used suggested there could be scope for field teams when establishing boundary lines to subconsciously adjust them so as to include stems close to the plot boundary.
The objective of this study was to test the hypothesis that stem density of large trees is overestimated in the 20 × 20 m square plots used in New Zealand's natural NFI. A secondary objective was to test the hypothesis that large trees are over-represented close to plot boundaries in the 20 × 20 m plots.

| Data
The New Zealand natural forest inventory, which is now in its third cycle of measurement, was implemented as part of the larger NFI

| Calculations
Areas of all 20 m radius plots in the inventory are 0.1257 ha because the radii are measured horizontally and plot areas are therefore not affected by terrain unevenness or slope. However, in uneven terrain, the approach used to establish the 20 × 20 m plots results in nonsquare plot shapes of variable area. Tapes also often have to be laid around boundary trees introducing problems when calculating area-based estimates such as basal area or stand density. The true horizontal areas of the 20 × 20 m plots are generally less than their nominal 0.04 ha because the boundary is laid out with tapes that follow the terrain surface. To allow the horizontal area to be calculated, the horizontal lengths and bearings of the four sides of each plot are measured using the Vertex IV (Haglofs, 2016). To calculate the horizontal area of each 20 × 20 m plot, we used Bretschneider's formula for the area of a convex quadrilateral: where a, b, c and d are the lengths of the four sides, s = (a + b + c + d)/2, and A and C are the internal angles of the quadrilateral for either of the two opposite corners. In a few cases due to uneven terrain, a plot side was measured in multiple sections, each with a separate length and bearing. In such cases, the horizontal distance and bearing of the straight line between plot corners was calculated from the separate section measurements, and used as the boundary line for that side of the plot.
We compared estimates of stem density (stems per ha) of ≥60 cm DBH stems in the 20 m radius circular plots with those obtained from the 20 × 20 m square plots. As is standard practice in forest inventories, we used ratio estimators to calculate stem densities. If N i is the number of stems with DBH ≥ 60 cm in plot i and A i (ha) is its area, then the ratio estimator of stem density D (stems per ha) is: We calculated bootstrap 95% confidence intervals of the difference in stem density estimates between circular and square plots.
These were calculated using the bias-corrected and accelerated method (Efron, 1987) and were performed using the SAS macro %BOOTCI. Under this procedure, bootstrap samples were obtained by sampling with replacement from the list of 676 grid points in the inventory, with each bootstrap sample of size 676. Ratio estimates of stem density were calculated from the stem counts and plot areas for both the circular and square plots for each sample, and the difference between the two estimates calculated for each sample. The empirical distributions of differences between the two estimates were then used to estimate 95% confidence intervals of the difference and perform two-tailed bootstrap tests of the null hypothesis of no difference between the two estimates using 20,000 bootstrap samples. Differences with p-value < 0.05 were taken to be statistically significant. This procedure was applied for live stems, standing-dead stems, and all stems, with separate analyses for the We obtained bootstrap 95% confidence intervals and tests of significance of differences in density of ≥60 cm DBH stems between perimeter and central subplots, and between these and circular plots.
To test whether sampling issues might also apply to smaller stems, we calculated ratio estimates in perimeter and central subplots by tests. Finally, we repeated the above analyses comparing perimeter and central subplots, but using the percentage of stems within each DBH size class as the dependent variable. These were calculated as follows. If n i is the number of stems in a size class and N i is the total number of stems with DBH ≥ 10 cm within a particular grouping of subplots of plot i, then the percentage of stems for the size class in that subplot grouping was calculated using 100 × Σ i n i /Σ i N i . In contrast, the number of live stems in the square plots decreased slightly from 1,010 to 986 stems. For standing-dead stems, the number in the circular plots increased slightly from 715 to 726, and decreased slightly in the square plots from 222 to 214.

| RE SULTS
Mean stem density for live stems ≥ 60 cm estimated from data from the 2002-2007 measurement cycle was 35.4 stems per ha in the circular plots and 43.8 stems per ha in the square plots, a difference of 8.4 stem per ha which was statistically significant (p < 0.0001, Table 1).
In the second measurement, the mean stem density estimates for live trees ≥ 60 cm DBH were 36.6 and 43.2 stems per ha, respectively, for circular and square plots, a difference of 6.6 stems per ha which was also statistically significant (p < 0.0001). Thus, densities of large live stems in the 20 × 20 m plots were 23.7% higher than those in 20 m radius circular plots for the 2002-2007 measurement cycle, and 18.0% higher for the 2009-2014 measurement cycle. There were also significant differences in stem density of standing-dead stems between circular and square plots in both measurement cycles (p < 0.012 and p < 0.044 respectively; Table 1).
When stems were remeasured during the 2009-2014 measurement cycle, it was noticed that a number of stems, mainly in the EXT subplot (outside the square but inside the circular plot) had been missed during the first measurement cycle, and correction for this omission may be the reason for the increase in stem density estimates for circular plots between the two measurement cycles. Therefore, subsequent analyses focus only on data from the 2009-2014 measurement cycle.
Examination of the numbers of ≥60 cm DBH stems in the sixteen 5 × 5 m subplots summed across the entire inventory shows that larger numbers of stems were sampled in subplots along sides P-A and P-M, and to a lesser extent in subplots along sides A-D and D-M compared with the centre four subplots (Figure 2). This suggests that when laying out these boundary lines, there could have been a subconscious tendency for field teams to vary the position of the line slightly to include large stems close to the plot boundary. We therefore next examined stem density estimates based on perimeter subplots and centre subplots, and for various groupings of the perimeter subplots (Table 2). This analysis showed that perimeter subplots had significantly higher stem densities than circular plots, but that estimates from centre subplots did not differ significantly from circular plot estimates. A detailed analysis of various groupings of perimeter subplots indicated that the highest stem density was recorded for the first subplot established in each 20 × 20 m square plot (subplot P), followed closely by subplots along the first two established perimeter lines, followed by subplots along the remaining perimeter lines (Table 2).
When the analysis was extended to compare stem density estimates from perimeter subplots with centre subplots by DBH size class, only the ≥60 cm DBH size class showed a significant difference in stem density between perimeter and centre subplots ( Table 3).

Comparison of the percentage of stems by DBH size class between
perimeter and centre subplots showed there was a higher percentage of ≥60 cm DBH stems in the perimeter subplots (Table 4).

Our dataset from the inventory of New Zealand's natural forests
is unique in providing a comparison of square and circular plots.
The discrepancy between estimates of large-tree stem densities  Herries et al. (2013)). The decision of whether a stem is inside or outside the plot boundary is made by determining whether the measured distance from the plot centre to the centre of the stem at ground level is greater or less than the plot radius, which can be achieved simply and accurately in most cases (Herries et al., 2017;Husch et al., 2003). Furthermore, the placement of plot centres which are located at the intersection of the 20 × 20 m plot diagonals is unlikely to be influenced by the local distribution of large stems.
In contrast, careful consideration of the protocols used to establish the 20 × 20 m plots reveals several steps where errors could occur. Firstly, the initial corner point (point P, Figure 1) is established 30 m along a bearing from a GPS location using a compass and tape.
However, even if bearings are predefined and accurate mensuration compasses are used, the accuracy of actual bearings will be much poorer than the nominal accuracy of the compass due to operator error. Audits of plots reveals that actual bearings commonly vary from predetermined bearings by up to 5º (field staff pers. comm.).
A ± 5º variation in bearing corresponds to a variation of ±2.6 m TA B L E 1 Estimates of stem density (stems per ha) for stems ≥ 60 cm DBH obtained from 20 m radius circular plots and 20 × 20 m square plots in New Zealand's natural forest NFI. Also shown is the difference between the two estimates with bootstrap 95% confidence interval, and p-values of bootstrap tests of the difference between the two estimates when determining the location of the P subplot, in practice a field crew can vary it by several metres, and a subconscious tendency to include rather than exclude large trees could account for the excess stem density in the P subplot compared with the centre subplots (Table 2). Our analysis also shows that there is a higher stem den- Portugal that have lower DBH thresholds in large-tree plots of similar or greater size than the New Zealand 20 m radius plot (see Table S1).  (Kent & Coker, 1992).
We note that large 100 × 100 m square plots are often used in tropical forest inventories (FAO, 1998). These are preferred to circular plots because in such large plots, trees near the plot boundary cannot be seen from the plot centre making it difficult to establish the plot boundary of a circular plot. In large plots, missed trees can be a significant source of error, and to assist in reducing this error, the plots are usually subdivided into subplots. Each large plot is effectively a cluster of contiguous smaller square subplots, typically of 20 × 20 m.
Edge effects between the subplots are of no consequence because the subplots are contiguous, and decisions regarding whether trees are inside or outside a plot boundary will only be an issue along the outer perimeter of the large plot. In such large plots, edge effects caused by the inclusion or exclusion of boundary trees will have a smaller influence on estimates of stand metrics than they do in small square plots such those used in the New Zealand NFI.
However, based on our study, apart from such large-plot surveys, we would recommend the use of circular plots in preference to square or rectangular plots. The plot centres of such circular plots

DATA ACCE SS I B I LIT Y
The third party data used for this study is held in the New Zealand