Forestry

An International Journal of Forest Research

Forestry 2021; 1–11, doi:10.1093/forestry/cpab023

Xiao Dai1 , Mark J. Ducey2 , Haozhou Wang1 , Ting-Ru Yang1 , Yung-Han Hsu1 , Jae Ogilvie1 and John A. Kershaw Jr1
1

Faculty of Forestry and Environmental Management, University of New Brunswick, Fredericton, NB, Canada
2
Department of Natural Resources and the Environment University of New Hampshire, Durham, NH, USA*Corresponding author: Tel: +1(506)453-4933; E-mail: xdai1@unb.ca

Received 28 January 2021
Efficient subsampling designs reduce forest inventory costs by focusing sampling efforts on more variable forest
attributes. Sector subsampling is an efficient and accurate alternative to big basal area factor (big BAF) sampling
to estimate the mean basal area to biomass ratio. In this study, we apply sector subsampling of sphericalimages to estimate aboveground biomass and compare our image-based estimates with field data collectedfrom three early spacing trials on western Newfoundland Island in eastern Canada. The results show that sector
subsampling of spherical images produced increased sampling errors of 0.3–3.4 per cent with only about 60 trees
measured across 30 spherical images compared with about 4000 trees measured in the field. Photo-derived
basal area was underestimated because of occluded trees; however, we implemented an additional level ofsubsampling, collecting field-based basal area counts, to correct for bias due to occluded trees. We applied
Bruce’s formula for standard error estimation to our three-level hierarchical subsampling scheme and showedthat Bruce’s formula is generalizable to any dimension of hierarchical subsampling. Spherical images are easilyand quickly captured in the field using a consumer-grade 360◦ camera and sector subsampling, including all
individual tree measurements, were obtained using a custom-developed python software package. The system
is an efficient and accurate photo-based alternative to field-based big BAF subsampling.

Introduction
Aboveground biomass (AGB) plays a vital role in global climatechange mitigation and ecosystem dynamics (Brown, 1997; Metteet al., 2002; Le Toan et al., 2011) and can help in monitoring emissions of CO2 resulting from land use and land coverchanges (Sales et al., 2007). To restore, enhance and manage
forest resources and create a sustainable environment (Bartuska,
2006), forest attributes must be efficiently monitored (Brown,
1999; Pearson et al., 2007; Chen et al., 2019). Direct measurement of biomass requires complete harvest of sample plots anddrying and weighing of the different tree components (Kershaw
et al., 2016, pp. 153–154). This process is destructive, timeconsuming and costly and is generally limited to a few researchstudies rather than used operationally. Accurate and repeatable estimates often are obtained from allometric equations
applied to individual tree measurements (e.g. diameter at breast
height (DBH) and height (HT)) and expanded to per unit area
(Brown, 2002; Lu et al., 2016); however, even this process istime-consuming and may result in large errors because relationships between species, forest ages, site conditions and equations
must be considered (Telenius and Verwijst, 1995; Lu, 2006; Yanget al., 2017). Various indirect methods such as regression models(Baskerville, 1972; Brown et al., 1989; Usoltsev and Hoffmann,
1997; Montès et al., 2000), hemispherical photography (Clark andMurphy, 2011) and remote sensing (Armstrong, 1993; Lu, 2006;
Zolkos et al., 2013; Lu et al., 2016) are used for AGB estimation.
However, there is no standard for determining the best estimation methods for biomass because various data sources andprediction approaches are frequently applied (Fassnacht et al.,2014), even though they can give widely varying results (e.g.
MacLean et al., 2014). The most effective variables, equation
forms and estimation approaches are not clear (Lu et al., 2016)and have not, to our knowledge, been systematically studied.
Remote sensing is an important tool for landscape levelbiomass estimation (Lu, 2006). Light Detection and Ranging(LiDAR), either airborne (ALS) or terrestrial (TLS), scanning has
shown promise for biomass estimation (Goetz et al., 2009;
Gleason and Im, 2011; Hayashi et al., 2015) and has become
an almost ubiquitous tool in forest inventory (Dubayah andDrake, 2000; Dassot et al., 2011; Hayashi et al., 2015). ALS has
the capability of covering large landscapes and providing highresolution ground and canopy surface models (Gaveau and Hill,
2003; Räsänen et al., 2014; Wilkes et al., 2015; Erfanifard et al.,2018), while TLS has the capability of estimating understory
vegetation parameters (Hilker et al., 2010; Hopkinson et al.,2013), calibrating ALS estimates with auxiliary variables (Greaves
et al., 2017) and providing forest structural information (Duceyand Astrup, 2013; Astrup et al., 2014). However, LiDAR is not© Institute of Chartered Foresters, 2021. All rights reserved. For Permissions, please e-mail: journals.permissions@oup.com

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Biomass estimates derived from sector subsampling of 360◦ sphericalimagesForestry2

sample point. On spherical images, these trees are often very
distorted, or it is very difficult to clearly identify the tree tip
which make tree height harder to measure accurately (Wanget al., 2021). An effective subsampling protocol for sphericalimage sampling requires an alternative measure-tree selection
process.
Dai et al. (2021, in press) showed that sector subsampling
is a viable alternative measure-tree selection method to big
BAF sampling with differences in mean values averaging lessthan 1 per cent of the means obtaining using big BAF sampling
and nearly equivalent standard errors for a given measure-tree
subsample intensity (i.e. number of trees). Sector sampling (Ilesand Smith, 2006; Smith et al., 2008; Smith and Iles, 2012) uses
sectors of a circle to define sample plots. Originally designed
to efficiently sample small or irregular forest areas (Iles andSmith, 2006; Smith et al., 2008; Smith and Iles, 2012), Dai et al.
(2021, in press) applied sectors as a means to select subsamplesof trees for detailed measurement. As formulated by Dai et al.
(2021, in press), sector subsampling uses a small angle gauge
and horizontal point sampling to select count trees to estimate
BA and then a randomly oriented sector to select a subsampleof trees to measure. Two variants of sector subsampling weredeveloped. The first method, termed SectorIN, used a randomly
oriented sector to select a subsample of the ‘in’ trees selectedusing the small angle gauge. All ‘in’ trees that fell within the
sector were selected as measure trees. The second method,termed SectorDST, used a randomly oriented sector to select a
subsample trees within a predefined distance of plot centre formeasurement. In this case, measure-trees could be either ‘in’trees or other trees that fell within the predefined sector. Like big
BAF sampling, the measure trees are used to estimate the ratioof the tree attribute of interest to tree basal area, and the mean
ratio is multiplied by average BA to estimate the per ha averageof the attribute of interest.
In this paper, we combine photo point sampling (Stewart
et al., 2004; Fastie, 2010; Dick, 2012; Wang et al., 2020) withsector subsampling (Dai et al. 2021, in press) to develop an
efficient method for sample-derived estimation of area-based
AGB using spherical images. The specific objectives of this study
were: (1) apply sector subsample selection to PBA plots obtained
from spherical images; (2) develop a hierarchical approach tosubsampling from photos that includes correction for occluded
trees and (3) generalize Bruce’s formula (Goodman, 1960) for
multiple subsample levels.

Methods
Study sites
In this study, data from western Newfoundland Island (NL),
Canada, were used. These data came from three early spacing trials established in the early 1980s by the government of
Newfoundland and Labrador in cooperation with the Canadian
Forest Service (Donnelly et al., 1986) (Figure 1). Balsam fir (Abies
balsamea L.) was the dominant species with minor componentsof black spruce (Picea mariana (Mill.) Britton, Sterns and Poggenb.)
and white birch (Betula papyrifera Marshall). There were five spacing treatments: control/no spacing (S00), 1.2-m spacing (S12),
1.8-m spacing (S18), 2.4-m spacing (S24) and 3.0-m spacing

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without its shortcomings. Occlusion of all or parts of some trees
is a frequent issue with both ALS and TLS (Hilker et al., 2010;
Ducey and Astrup, 2013; White et al., 2016) and is a major sourceof uncertainty in LiDAR-assisted forest inventories (Ayrey et al.,2019). Reliance on model-assisted predictions based on data thatare not probabilistic samples is another source of uncertainty(Yang et al., 2019). While ALS data are becoming increasingly
freely available, terrestrial scanners remain very expensive andboth data sources require field data for calibration, extensive
postprocessing, model fitting and prediction verification. In many
respects, LiDAR trades off costs associated with field work withcosts associated with equipment and office work.
Close-range digital photogrammetry is a cost-effective alternative to TLS (Stewart et al., 2004; Perng et al., 2018; Lu et al.,2019; Wang, 2019). Several researchers have demonstrated thathigh-resolution panoramic imagery is an accurate tool for collecting basic tree and forest data (Dick et al., 2010; Fastie, 2010;
Lu et al., 2019;Wang, 2019 ; Wang et al., 2020). Horizontal point
sampling (or angle count sampling (Bitterlich, 1984; Iles, 2003;
Stewart et al., 2004)) is easily implemented on 360◦ panoramic
images (Fastie, 2010; Dick, 2012). The angle required for a given
basal area factor is expressed in terms of pixels, and a pixel-based‘gauge’ is moved across the image, and trees appearing larger
than the gauge are counted as ‘in’ trees (DeCourt, 1956; Stewart
et al., 2004; Fastie, 2010; Dick, 2012; Wang et al., 2020). However,occluded (hidden or partially hidden by closer trees) trees area problem with photo-based angle count sampling resulting inundercounts of ‘in’ trees and, as a result, photo basal area (PBA) is
underestimated (Stewart et al., 2004; Dick, 2012; Wang, 2019). In
addition to PBA estimates, Perng et al. (2018) and Lu et al. (2019)
demonstrated how individual tree diameters and heights can beobtained from stereographic 360◦ /180◦ hemispherical images,
and Wang et al. (2021) extended this idea to spherical images.
The newer consumer-grade 360◦ spherical cameras makephoto-based angle count sampling even easier because image
stitching is done onboard the camera and the two fixed fisheye lenses minimize alignment errors (Wang, 2019; Wang et al.,2020). Dai (2021) compared biomass estimation models basedon PBA derived from spherical images obtained using a Ricoh
Theta S 360◦ camera (Ricoh Imaging Company, LTD, 2016) tomodels based on common TLS metrics. Root mean square errors(rMSEs) for models based on TLS metrics ranged from 20 to 33 percent, while rMSEs for models derived from PBA estimates ranged
from 17 to 21 per cent. Given the low cost, portable size and fieldefficiency, the spherical camera offers much promise as a forest
inventory tool (Wang et al., 2020; Dai et al. 2021, in press).
Model development requires calibration for every new application. Efficient sample-derived estimation may be an effective
alternative to model estimates (Yang et al., 2019). Big basal areafactor (BAF) sampling is a widely used subsampling design thatutilizes a small angle gauge to count ‘in’ trees and estimate basal
area per ha (BA; m2 ha−1 ) for each sample point; and a larger
angle gauge to select trees to measure (Iles, 2003; Marshallet al., 2004; Yang et al., 2017). The ratio of the tree attribute ofinterest to individual tree BA (XBAR) calculated from the measuretrees and the mean BA are used to calculate the per unit area
estimates of the attribute of interest. However, implementing
big BAF sampling on spherical photos is challenging, since big
BAF sampling tends to select trees that are very close to theBiomass estimates(S30). The treatments were arranged in a randomized complete
block design with 3 blocks per trial (45 sample units were usedin this study). Each treatment was applied to a 0.25-ha area
(50 m × 50 m), and a circular permanent sample plot (PSP) wasestablished near the centre of each 0.25-ha area. The PSP size
varied such that there were ∼100 trees per plot at the time ofestablishment. Only the most recent measurements for each trial
were used in this study (2013 for Pasadena and Cormack; 2017
for Roddickton).

Field biomass estimation
Individual tree biomass was estimated using the Canadian
National Biomass equations (Lambert et al., 2005). We used
Eq. 3 from Table 4 in Lambert et al. (2005) which included
both DBH (cm (BH = 1.3 m above ground)) and total height (HT;m). Total tree biomass (BMi ; kg) was obtained by summing theseparate component biomass estimates (wood, bark, branches
and foliage). Field biomass per ha (FBM; tonnes·ha−1 ) for eachsample point was estimated by summing the individual tree
biomass estimates multiplied by the plot expansion factor anddividing by 1000 kg per tonne

FBM =

P
EF × ni=1 BMi
,
1000

(1)

where n = the number of field measure trees on each PSP.

Sector subsampling of spherical images
A Ricoh Theta S 360◦ camera (Ricoh Imaging Company, LTD,
2016) was used to obtain spherical images of the NL spacing
trial PSPs. We obtained images in three locations on eachPSP. The images were obtained at half the plot radius ofeach PSP at azimuths of 0◦ (360◦ ), 120◦ and 240◦ . At each
image acquisition location, spherical images were obtained atheights of 1.6 and 2.6 m using a tripod-stabilized height pole
(Wang et al., 2021).
In Dai et al. (2021, in press), sector subsampling intensity wasdefined in terms of the angular percentage of the 360◦ azimuth
subsampled (Figure 2a). The software developed by Wang et al.
(2021) was modified to randomly select a sector of a specified
intensity by randomly generating an azimuth and superimposing
parallel vertical lines on the cylindrically projected images basedon sector intensity (Figure 2b) with the inclusion region centredon the randomly generated azimuth (in cylindrical projections,
fixed horizontal angles are of fixed image width; thus the verticalparallel lines project the specified sector angle onto the cylindrical image; Figure 2b).
In our first simulation experiment, a modified SectorDST sampling method (Dai et al. 2021, in press) was implemented on eachphoto pair at each image acquisition location within each PSP(three locations per PSP were used in this study) across the threespacing trials to select measure-trees to determine the mean
biomass to basal area ratio (BBAR; kg·m2 ). PBA (m2 ha−1 ) was
estimated from the spherical images (1.6-m image heights) using
the software (available from: https://github.com/HowcanoeWa
ng/Panorama2BasalArea) developed by Wang et al. (2020) and a
2 M BAF (i.e. each count tree represents 2 m2 ha−1 of basal area).
Mean PBA for each PSP was estimated using
PBA =

Pp

i=1 PBAip=Pp

i=1 BAF × Counti

p

,

(2)

where p = the number of image acquisition locations (three in the
study).

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Figure 1 Location of Newfoundland in Atlantic Canada and locations of three early spacing trials on Newfoundland Island.

Forestry

The SectorDST selection method, as developed in Dai et al.
(2021, in press), selects measure-trees within a randomly oriented sector within a predetermined distance from the sample
point (Figure 2a). For the spherical images, tree occlusion is an
issue, so instead of selecting all trees within a given distance, wechose all trees within a sector that were clearly visible (Figure 2b).
All visible trees within the sector were measured for DBH (PDBH)
and HT (PHT) on the spherical image pairs using the stereographic
methods described by Wang et al. (2021) as implemented in themodified software (available from: https://github.com/Howcanoe
Wang/Spherical2TreeAttributes).

4

Photo tree basal area (PTBA) and photo tree biomass (PTBM)
were calculated using the same methods described above for thefield estimates (assuming all trees were balsam fir for biomass
estimation) and mean BBAR (BBAR) was calculated using a ratioof means approach (Dai et al. 2021, in press)

Pm
Pm
PTBMi
i=1 PTBMi,=
BBAR = Pi=1
Pm
m
PTBA
0.00007854
× (PDBHi )2
i
i=1
i=1

(3)

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Figure 2 SectorDST subsampling as viewed: (a) from a map perspective of a photo sample plot and (b) as viewed from a cylindrically projected sphericalimage. (BAF count trees are identified by horizontal white bars; measure trees are those trees within the two vertical black lines defined by sector
angle ‘a’).

Biomass estimates

where m = the number of measure-trees. Mean photo biomass
per ha (PBM, tonnes·ha−1 ) was then estimated using
BBAR × PBA
.
1000

(4)

Percent standard error for (PBM) was estimated using Bruce’s
formula (Goodman, 1960; Marshall et al., 2004)

%se PBM =

r


2

2
%se PBA + %se BBAR ,(5)

where %se() = the standard error as a percent of the mean.
Although Bruce’s formula assumes independence of the error
components, simulations show that it performs well in comparison with
more
complicated formulations (Gove et al., 2020). For%se PBA , we used the formula for standard error (se) undersimple random sampling (Zar, 1999)v
uP
u PBA2 − P PBA2 /k
,
se = t
k k−1

(6)



where k = the number of sample plots. For %se BBAR , we used

the formulation provided in Kershaw et al. (2016 p. 348, Eq. 10.53)

%se BBAR
v
u
u
=t

2

BBAR
m (m−1)

! P

PTBM2

PTBM

2

+

P

PTBA2
2

PTBA

−

2

P

PTBM · PTBAPTBM · PTBA

!

,

Pj
FBAi
,
FPBAR = Pji=1
i=1 PBAi(8)

where FPBAR was the mean of FBA to PBA ratio, j = the numberof field plots used for correcting PBA (5, 10, 15, 20 in this study),
and the corrected biomass estimate (CBM) became
CBM = FPBAR · PBM = FPBAR · PBA · BBAR .

(9)

We further used an extension of Bruce’s formula to include thisthird source of sampling error%se CBM =

r




2
2
2
%se FPBAR + %se BBAR + %se PBA ,
(10)



where %se FPBAR was estimated using Eq. 4. Derivation of Eq.

10 is given in the Supplemental Materials (Derivation of CBM
Error). To assess the validity of this extension, we used coveragebased on nominal 95 per cent confidence intervals and thecorrelations between the sources of errors. All simulations wererepeated 100 times for estimation comparisons and 1000 times
for assessing coverage of nominal confidence intervals.

(7)

where PTBM and PTBM were the individual measure-tree biomass
estimates and mean measure-tree biomass estimates; PTBA and
PTBA were the individual measure-tree stem basal areas (estimated from PDBH measures triangulated from the 1.6- and 2.6m images) and mean measure-tree stem basal area and BBAR
was the mean biomass to basal area ratio of the measure-trees
from Eq. 3.
The SectorDST sampling procedure was implemented on eachspherical image pair at each image acquisition point on eachspacing PSP on each replicate across the three spacing trials (3trials × 5 treatments × 3 blocks × 3 locations = 135 image acquisition points). All analyses were conducted in the R Statistical
Language (R Development Core Team, 2019).

Occluded tree correction
One sampling issue with using PBA is tree occlusion (Dick, 2012).
On photo plots, it is not possible to move from the sample point
to check for occluded (hidden) trees. Occluded trees result inunder–counting of ‘in’ trees and negative biases in BA estimates.
To explore the potential of using a subsample of in-field tree
counts to correct for PBA bias, we implemented a simulation

Results
Using a sector intensity of 2 per cent (7.2◦ of the full 360◦ ), 163
photo measure-trees were selected across the 135 image acquisition points (three points per PSP) over the 45 spacing trial PSPs.
BBAR averaged 3214 kg·m−2 with a standard error of 30.7 kg·m−2 .
PBA averaged 34.3 m2 ha−1 with a standard error of 1.8 m2 ha−1 .
The resulting mean biomass was 110.3 kg·ha−1 with a standard
error of 5.4 kg·ha−1 (Table 1). PBM was underestimated relative
to FBM on all but two of the 45 spacing trial plots (Figure 3).
While underestimation of biomass was evident in the sector
subsampling, the LOWESS smoothing line (Cleveland, 1981) indicated a relatively strong linear relationship that could be corrected using ratio estimation (Figure 3). Based on 100 repeatedsimulation samples using 1 image acquisition point per spacing PSP and a random subsample of FBA to correct PBA, theunderestimation associated with PBA was efficiently corrected(Figure 3; Table 1). As few as 5 field plots were sufficient to correct
the PBM underestimation that resulted from occluded trees in
the PBA estimates (Table 1). With 10 field samples, standard
errors (Table 1) were comparable to those obtained using theregression estimates of BM developed in Dai (2021, pp. 24 &27), and with 20 subsamples, the standard errors were nearly5

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PBM =study using the spherical image results from the previous section
and a subsample of field basal area (FBA) counts. We explored
field BA: PBA correction (field to photo basal area ratio, FPBAR) by
randomly selecting 5, 10, 15 or 20 plots to obtain FBA measures.
A second ratio of means estimator was obtained to correct PBA
using the subsampled field BAs

Forestry

SourceCorrection ratio

Sample size

Meana

Field measured
TLS prediction
PBA prediction
Sector subsampling
Uncorrected
5
10
15
20

1.44
{0.93, 2.35}
1.42{1.03, 1.86}
1.40{1.06, 1.66}
1.40{1.18, 1.60}

Biomass estimate
Standard error (%)

20.1
13.4
9.3
7.1

Meanb

Standard error (%)

148.9
148.8
149.3

7.3
17.2
19.9

102.9
148.5
{95.5, 241.7}
146.4{105.8, 191.4}
144{108.8, 171.2}
143.9{121.9, 163.5}

5.4
30.1
19.7
13.2
10.3

a Correction ratio is unitless since it is field BA (m2 ha−1 ) divided by photo BA (m2 ha−1 ). b Biomass is in tonnes·ha−1 .

Figure 3 Corrected/uncorrected sector subsampled biomass estimates versus biomass estimated from field measurements under four different fieldBA correction sample sizes 5, 10, 15 and 20 for the Newfoundland (NL) spacing trials.

equivalent to those obtained for the complete field data fromthe 45 spacing plots with all trees measured for HT and DBH(Table 1). The 45 plots across the 3 spacing trials had 4181
trees measured with a standard error of 7.3 tonnes·ha−1 . In our
simulations presented in Figure 3 and Table 1, there were, on
average, 54 trees subsampled for photo measurement across6

the 45 randomly sampled spacing trial plots with an average
standard error of 10.3 tonnes·ha−1 .
The addition of subsampling field PSPs for BA estimation andratio correction added another source of error associated with
our estimate of biomass. As expressed in Eq. 10, we proposed
to expand Bruce’s formula by adding a third component to the

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Table 1 Means and associated percent standard errors for biomass (tonnes·ha−1 ) by study source (field measurements, terrestrial LiDAR scanning(TLS) and PBA regression-based predictions (Dai, 2021), and corrected/uncorrected sector subsampling estimates) and the associated correctedratios by PBA correction subsample size for the western Newfoundland (NL) spacing trials. Ranges are shown in brackets {} (The correction ratios were
determined by dividing FBA by PBA)

Biomass estimates

Factor%se PBA%se BBAR%se FPBAR%se CBMBM
Coveragecor(PBA vs. FPBAR)
cor(PBA vs. BBAR)
cor(BBAR vs. FPBAR)

Field sample sizea
5

10

15

20

7.08

7.08

7.03

7.07

1.97

1.96

1.95

1.99

20.19

14.39

12.19

10.56

21.66

16.24

14.25

12.90

157
94.7
0.16
0.00
0.05

156
98.5
0.23
0.05
0.01

155
99.5
0.30
0.04
0.04

155
99.9
0.46
0.08
0.05

a Number of field sample points used to correct PBA.

error formulation. Bruce’s method relies on independence of thecomponent errors (Goodman, 1960). As shown in Figure 3, correlations (r) between the error components were quite low. The
highest correlation was r = 0.16 (r2 = 0.03) for %se(PBA) versus%se(FPBAR), while the other two correlations were less than 0.10
(Figure 4). We expanded our number of simulations to 1000 andassessed coverage (number of 95 per cent confidence intervals
containing the true mean, as determined by the PSP measurements) by field sample size. For 5 field BA samples, coverage was
93.7 per cent, for 10 field BA samples, 98.6 per cent, for 15 field
BA samples, 99.5 per cent and for 20 field BA samples, 99.9 percent (Table 2).

Discussion
While selection of measure-trees did not produce biases, theunderestimation of basal area from the spherical images (PBA)
resulted in serious underestimation of biomass (Table 1, Figure 3).
Occlusion issues with panoramic photo sampling were identified
in several previous studies (Fastie, 2010; Dick, 2012; Lu et al.,2019). In the field, occluded trees can be identified and correctly
counted using a number of techniques including moving fromplot centre to actually measuring distances and DBHs to determine if trees are in or out of the count sample (Kershaw et al.,2016, pp. 372–374). With panoramic or spherical images, it is notpossible to move around and only visible trees can be counted
and/or measured. Our use of a third-stage subsample to collectfield plots to correct for occluded trees was an effective and efficient sampling procedure to correct PBA to account for occluded
trees (Figure 3; Table 1). Based on the resulting correction ratios
for field: photo BA (FPBAR, Table 4), about 70 per cent of the ‘in’trees were identified on the spherical images. This is consistent
with the results from Dick (2012) who reported 60–90 per centof trees correctly counted depending on the BAF used and standdensity. In this study, we used 3 image acquisition points offset
from plot centre and averaged across the three photo samples.
Wang (2019) found that this to be an effective strategy for

reducing occlusion bias but that was not the case here, eventhough some of the spherical images were in common.
As few as 5 field counts were sufficient to correct the underestimation; however, the resulting standard errors were quite large(Table 1). The percent standard errors associated with FPBAR
were, in general, quite large relative to the percent standard
errors for the other components (PBA and BBAR versus FPBAR;
Table 2). At 10 field samples, the resulting standard errors werecomparable to the standard errors obtained using model assisted
approaches (Dai 2021). With 20 subsamples, the standard errors
were only about double the full field sample data. The full fieldsamples required over 4000 trees be measured for DBH and HT,
our procedure only required about 60 trees be measured.
Subsampling with ratio estimation is, in general, a very efficient method for correcting bias, reducing sample sizes andfocusing sampling efforts on the level where variation is greatest(Iles, 2003, pp. 557; Yang et al., 2019; Hsu et al., 2020). In thisstudy, we used simple random selection of field subsamples.
Yang et al. (2019) showed that simple random sampling was asefficient and, in some cases, can be more efficient than variable
probability selection methods for LiDAR assisted ratio estimation
when sample sizes were small. Hsu et al. (2020) and Yang et al.
(2019) showed that list sampling was the most effective variable
probability selection approach for ratio estimation. List sampling
requires prior knowledge of the covariate for all sample units(Kershaw et al., 2016, pp. 353–356). Both Hsu et al. (2020) andYang et al. (2019) showed that the sampling with probability
proportional to prediction (3P) could also be effectively used.
Hsu (2019) developed methods for implementing 3P sampling
using spherical images. A variable probability approach most
likely would improve the results presented here.
Comparing our estimated standard serrors for single replicated simulated samples to the standard deviations of themeans across all samples, our extension of Bruce’s formula tothree dimensions appears reasonable. Coverage, based on 95 percent confidence intervals, seem to also support this extension.
Bruce’s formula relies on the independence of errors among thehierarchical components of the sample (Goodman, 1960). As

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Table 2 Comparison of percent standard errors from the sector-sample simulations for PBA (m2 ha−1 ), biomass to basal area ratio (BBAR; kg·m−2 ),field to photo BA ratio (FPBAR; unitless) and corrected biomass (CBM; tonnes·ha−1 ), coverage (number of 95% confidence intervals containing thetrue mean) and standard error correlations (r) by field PBA correction sample sizes for the western Newfoundland (NL) spacing trials

Forestry

shown in Figure 3, the assumptions of independence appear tobe met in this case, consistent with the observations of Goveet al. (2020) and Lynch et al. (2021, in press) that Bruce’s formulaprovides adequate confidence interval coverage in simulations
for ordinary big BAF sampling.

Conclusion
To make appropriate management decisions, foresters require
accurate and timely data (Kershaw et al., 2016, p. 3). An efficient
sample design can not only reduce costs and save time but also
provide accurate estimates (Lynch, 2017; Yang et al., 2017; Chenet al., 2019). In this study, we applied an alternative subsampling8

procedure using sectors to select measure-trees and coupledthis with a hierarchal sampling scheme that combines sphericalimage measurements with horizontal point sample counts toestimate biomass in three spacing trials in western Newfoundland Island. Our results show that this approach is efficient andaccurate and can be used to estimate biomass at a much lower
cost than more expensive technology such as terrestrial LiDAR.

Data availability statement
Plot-level biomass estimates and photo-sector data used in thisstudy are available at Dai and Kershaw (2021).

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Figure 4 Pairwise comparisons and correlations (r) between standard errors for PBAs, field:photo basal area ratios (FPBAR) and biomass:basal area
ratios (BBAR) for the Newfoundland (NL) spacing trials.

Biomass estimates

Acknowledgements

Conflict of interest statement
None declared.

Funding
Contribution Agreement with the Government of Newfoundlandand Labrador; Natural Sciences and Engineering ResearchCouncil of Canada, Discovery Grant Program (RGPIN04280);
New Brunswick Innovation Foundation Research Assistantship
Initiative (RAI2017–032).

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