Statistical experiments for Statistical experiments for monitoring trees outside the forestmonitoring trees outside the forest

P. Corona*, M.V. P. Corona*, M.V. ChiriacòChiriacò*, L. *, L. FattoriniFattorini**, F. **, F. BaffettaBaffetta****

* University of * University of TusciaTuscia, Italy , Italy

** University of Siena, Italy** University of Siena, Italy

TOF play a basic role in biodiversity conservation, contribution to scenic beauty, wind erosion mitigation (windbreaks), agricultural crop protection, provision of wood and non‐wood products, bio‐energy and living space for animals and plants as well as carbon storage functions

TOF fall outside the domain of forests as defined by FAO

during the FAO Expert Consultation on Global Forest Resources Assessment 2000 held in Kotka (Finland) in the 1996, the importance of trees outside the forest (TOF) and the need for detailed information about those objects were underlined for the first time

TREES OUTSIDE the FORESTTREES OUTSIDE the FOREST

forest inventories are currently requested to broaden their scopes to include TOF Kleinn, 2000 - On large area inventory and assessment of trees outside forests. Unasylva

TOF are all those TOF are all those forest trees or group of forest trees or group of trees within rural and trees within rural and

urbanized areas (i.e. urbanized areas (i.e. not classified under not classified under

forest land use)forest land use)

ToF

A/ Forestry setting: small « Forest/Other Wooded Land » (<0.5 ha)

B/ Rural setting

C/ Urban setting

Independentfrom farms

On farmsTrees in dense stands

Trees along roads

Trees scattered

Trees in linear features

Trees along streetsTrees along waterways

Trees along waterways

Trees in public parks

Trees on boundariesTrees inside private gardens

Independent from houses

Dependent from houses

ToF

Dense stands

Scattered trees

Linear features

Trees along street / road (urban or rural setting)

Trees along waterway (urban or rural setting) Trees on boundary (farm or urban setting)

Trees in hedgerow (farm setting)

« forestry » land (< 0.5 ha) farm land (tree crops, orchards, agroforests, etc)

urban setting (public parks, large private gardens)

Trees on contour (farm setting)

« forestry » land (< 0.5 ha)

farm land (orchards, agroforestry parkland, trees in fields or pasture)

urban setting (small private gardens)

FAO, 2010 Thematic Study on Trees Outside Forest (TOF). Inception Workshop

summary:

the economic, social and environmental importance of TOF is expected to dramatically increase in the current context of change

TOF are considered as a distinct item in the preparation of the Global Forest Resource Assessment 2015

one of the major reasons of the lack of information on TOF is the methodological challenges posed by the assessment: this issue is a priority for more detailed investigation

few published studies have been devoted to sampling strategies suitable for surveying TOF

examples of surveys performed by means of probabilistic sampling are due to Holmgren et al. (1994) in Kenya and Tokola (2006) in India: both these surveys adopt classical two‐phase sampling schemes, with aerial photos used in the first phase and field measurements in the second

alternatively, Corona and Fattorini (2007) propose a two‐stage scheme for surveying tree rows in a region of Central Italy

all the published surveys were performed for specific assessments, whileno attempt has been made to achieve well‐behaved estimators of TOF in the framework of multipurpose large‐scale inventories

Corona P, Fattorini L (2006) The assessment of tree row attributes by stratified two-stage sampling. Eur J For Res

Holmgren P, Masakha EJ, Sjöholm H (1994) Not all African land is being degraded: a recent survey of trees on farms in Kenya reveals rapidly increasing forest resources. Ambio

Tokola T (2006) Asia. In: Kangas A, Maltamo M (eds.). Forest Inventory. Methodology and Applications. Springer, Dordrecht

the purpose of this talk is to methodologically the purpose of this talk is to methodologically outline selected sampling exercises (with study outline selected sampling exercises (with study cases from Italy) to support largecases from Italy) to support large--scale forest scale forest inventories with the embedded assessment of inventories with the embedded assessment of

this kind of objectsthis kind of objects

first‐phase sampling points useful for identifying non‐traditional objects like TOFare among those assigned to non‐forest landby remotely sensed imagery, and then set aside from the subsequent (field) survey phases

large‐scale forest inventories usually adopt multi‐phase design‐based strategies

As concerns first‐phase sampling scheme, note that:

to avoid uneven coverage of the study area, stratified sampling schemes are usually exploited in large‐scale forest inventories; this is normally accomplished by partitioning the study region into quadrats or other regular shape (e.g. exagons) statistical units of the same size; then sampling points are randomly or systematically assigned to each unit

if the points are randomly and independently located within eachstatistical unit, the scheme is usually referred to as tesselated stratified sampling (TSS) (or unaligned systematic sampling)

when sampling points are distributed at a fixed spatial lag, starting from a randomly drawn origin, the scheme is referred to as aligned systematic sampling (ASS)

Tesselated Stratified Sampling Aligned Systematic Sampling

* in the next slides, R denotes the area covered by the quadrats overlaying the region of interest

ASS, which is based on a regular grid of points,  may be unsuitable in presence of some spatial regularity, leading to heavy losses of efficiency

TSS strategy produces population total estimators with a remarkableconvergence rate of variance which in addition is normally‐distributed forlarge samples (the variance convergence rate of the TSS estimator of total value isof order n−γ with 1<γ≤3, while the corresponding URS estimator solely possesses a variance convergence rate of order n−1)

Barabesi (2003) A Monte Carlo integration approach to Horvitz-Thompson estimation in replicated environmental designs. Metron

Barabesi & Franceschi (2011) Sampling properties of spatial total estimators under tessellation stratified designs. Environmetrics

it has been theoretically proven the superiority of the TSS over Uniform Random Samplingwhile the same cannot be demonstrated for ASS

the use of ad hoc sampling schemes for inventorying each of the target TOF parameters would likely be an efficient solution 

however, in order to save time and resources, it may be appealing to carry out the survey within the framework of large‐scale inventories

Reference standards

SmallSmall woodlotwoodlot (SWOF)(SWOF): tree groups witharea between 0.05 and 0.5 ha and mininum width of 20 m

TreeTree rowsrows (TROF)(TROF): tree rows with at leastthree trees, at least 20-m long and lessthan 20-m wide

ScatteredScattered treestrees (STOF)(STOF): single trees and tree groups not in rows and less than 0.05 ha wide

INFC, 2003

SWOF & TROF

Let U be a population of N small woodlots or tree rows, say Naa ,,1 K in a

delineated study area A of size ||A

Denote by || ja the size of unit j and by jy the value of a physical attribute

Suppose that the population total

∑∈

=Uj

jyT

and the population mean NTY /= are the interest quantities to be estimated

Baffetta F., Fattorini L., Corona P. (2011) Estimation of small woodlot and tree row attributes in large-scale forest inventories. Environmental and Ecological Statistics

1 ‐ a set of  N points is selected in accordance with a spatial sampling scheme (e.g. TSS), thus achieving a discrete population of points to be overlaid on VHR imagery  (as usually carried out in the first‐phase of multi‐phase large‐scale inventories)

2 ‐ SWOF and TROF containing at least one first‐phase point are included in the sample

3 ‐ sampled SWOF and TROF are delineated on the imagery and their size measured

4 ‐ complete tally of sampled SWOF and TROF can be carried out in the field 

SURVEY PROTOCOL

Denote by S the set of distinct units selected at least once by the n points and let

m be the cardinality of S

As proven by Baffetta et al. (2010), under the TSS scheme the quantity

∑∈

=Sj j

jn

yn

T || ||

~a

R

may be viewed as an approximation of both the HT and HH estimators and as

such it turns out to be approximately unbiased

A conservative estimator for the variance of nT~ is given by

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−⎟⎟⎠

⎞⎜⎜⎝

⎛

−= ∑

∈Sjn

j

jnY Tn

ynn

V || 2

2

22)(

~||)1(

1~a

R

while nYn VT )(~96.1~ ± provides a confidence interval with nominal coverage equal

to 95%.

2011

If 1=jy for each Nj ,,1K= , the population total coincides with the population

size N (the abundance), then

∑∈

=Sj j

n nN ||

||1~a

R

which provides an approximately unbiased estimator of N, with variance estimator

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−

= ∑∈Sj

nj

nN Nnnn

V || 22

22)(

~||

1)1(

1~a

R

with a nominal 95% confidence interval nNn VN )(~96.1~ ±

If || jjy a= for each Nj ,,1K= , the population total coincides with the total area

of the population units (the coverage, denoted by A), then

nmA ||

nR

=~

with variance estimator

)1()(~

2

22

)( −−

=nn

mnmV ||nAR

with a nominal 95% confidence interval nAn VA )(~96.1~

±

In order to estimate the population mean NTY /= , a very natural procedure is to

estimate T and N as well and then take the ratio

∑∑∈∈

==SS j jj j

j

n

nn

yNTY

||1/

||~~~

aa

with variance estimator

2

2

22

)( ||

~

)1(~~ ∑

∈ ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −−

=Sj j

nj

nnY

YynnN

V ||a

R

If || jjy a= for each Nj ,,1K= , the estimator of the average size of the units

reduces to

||

1/~

∑∈

=Sj j

n mAa

while the variance estimator reduces to

2

2

22

)( ||

~1

)1(~~ ∑

∈ ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=

Sj j

n

nnA

AnnN

V ||a

R

Simulation study for estimating abundance and coverage

a region of size 3600 km2 (side 60 km) was taken as the study area

within the area, a population of N = 10000 woodlots (density of 2.8 woodlots per 100 ha; unitary sizes less than 0.5 ha with expected size of about 0.2 ha and a positively skew distribution) was settled according to three different spatial patterns

within the area, a population of N = 30000 tree rows (density of 8.3 tree row per 100 ha; expected unitary size of about 0.1 ha and a positively skew distribution) was settled according to three different spatial patterns

to simulate TSS, the study area was partitioned into n = 3600 quadrats of size 1 km2

for each spatial pattern, the first and second order inclusion probabilities of each woodlot or tree row were computed by considering the intersections of the unit with the quadratspartitioning the study area

the sum of the first-order inclusion probabilities provided the expectation for the size m of the samples arising from the TSS

10000 samples were randomly generated; each sample was generated by randomly selecting a point in each quadrat, including in the sample all the SWOF/TROF containing at least one of these 3600 points

the knowledge of the first- and second-order probability allowed for the exact computation of the relative bias (RB) and of the relative root mean squared error (RRMSE) of the estimator of abundance and coverage based on HT criterion

the estimator of the average size, being a ratio of estimators, cannot be handled analytically; in these cases, RB and RRMSE were determined empirically on the basis of their Monte Carlo distributions

the Monte Carlo distributions were also used to determine the expectations of the RSE estimators (ERSEE) and the actual coverage of the 95%confidence intervals (AC95), empirically determined as the percentage of times the simulated intervals contained the true parameter value

∑∈

=Sj j

jyt

πˆ ; hj

N

jh hj

hjjhN

jj

j

jHT yyyT ∑∑

=>=

−+

−=

11

21)ˆ(V

πππππ

ππ

simulation results (SWOF)

Parameter UNIFORM CLUSTERED TRENDED RB(%)* 0 0 0 RB(%) -0.11 0.17 0.07 RRMSE(%)* 23.51 (23.51) 23.45 (23.45) 23.42 (23.42) RRMSE(%) 23.51 23.39 23.23 ERSEE(%) 23.95 23.91 23.97

Abundance

AC95(%) 93.6 93.79 94.02 RB(%)* 0 0 0 RB(%) -0.04 0.11 -0.05 RRMSE(%)* 22.65 (22.65) 22.59 (22.59) 22.56 (22.56) RRMSE(%) 22.61 22.50 22.50 ERSEE(%) 23.16 23.12 23.17

Coverage

AC95(%) 94.01 94.21 94.3 RB(%) 0.5 0.35 0.26 RRMSE(%) 6.4 (6.38) 6.45 (6.44) 6.46 (6.46) ERSEE(%) 5.98 5.95 5.97

Average size

AC95% 91.97 91.83 91.59 Expected sample size* Expected sample size

19.34 19.34

19.36 19.38

19.31 19.30

Single-quadrat woodlots: in the population (%)

expected in the sample (%) 91.86 91.71

91.58 91.30

91.72 91.61

Values in brackets refer to the HT estimator

(*) values determined analytically

simulation results (TROF)

Values in brackets refer to the HT estimator; (*) values determined analytically

Parameter UNIFORM CLUSTER TREND RB(%)* 0 0 0 RB(%) -0.06 0.07 -0.04 RRMSE(%)* 18.82 (18.82) 18.6 (18.60) 18.72 (18.72) RRMSE(%) 19.06 18.47 18.78 ERSEE(%) 19.05 19.04 19.03

Abundance

AC95(%) 94.03 94.55 93.86 RB(%)* 0 0 0 RB(%) -0.09 0.09 0.02 RRMSE(%)* 18.15 (18.15) 17.92 (17.92) 18.05 (18.05) RRMSE(%) 18.36 17.87 18.03 ERSEE(%) 18.41 18.39 18.39

Coverage

AC95(%) 93.29 93.71 93.17 RB(%) 0.24 0.27 0.33 RRMSE(%) 5.05 (5.04) 4.97 (4.97) 5.05 (5.04) ERSEE(%) 4.81 4.83 4.8

Average size

AC95(%) 92.73 93.32 92.75 RB(%)* 0 0 0 RB(%) -0.08 0.12 -0.04 RRMSE(%)* 18.48 (18.48) 18.26 (18.26) 18.39 (18.39) RRMSE(%) 18.68 18.15 18.36 ERSEE(%) 19.41 19.35 19.38

Total length

AC95(%) 94.21 94.64 94.04 RB(%) 0.12 0.18 0.14 RRMSE(%) 3.57 (3.57) 3.58 (3.57) 3.54 (3.54) ERSEE(%) 3.41 3.43 3.42

Average length

AC95(%) 93.02 92.73 93.04 Expected sample size* Expected sample size

30.07 30.05

30.03 30.06

30.07 30.08

Single-quadrat tree rows: in the population (%)

expected in the sample (%) 89.15 88.84

89.23 88.66

89.55 89.26

the relative bias of developed estimators is negligible in all situations and always smaller than 0.5%

the RRMSEs are practically identical to those achieved by the HT estimator

the RSE estimators turn out to be conservative estimators of RRMSEs when population totals are under estimation (abundance, coverage and total length);  slight underestimation takes place for population averages (average size and length). 

the actual coverage of confidence intervals always results smaller than the nominal coverage of 95 %, even if in no case it turns out to be smaller than 91% (these discrepancies are mainly due to the tails of the actual distributions which turn out to be slightly heavier than those of the normal density)

the estimation of averages gives rise to RRMSEs of about 3‐6% while the estimation of totals turns out to be less efficient with RRMSEs of about 18‐24% (these accuracy levels seem however to be quite satisfactory if compared with the expected sampling fraction which turns out to be invariably smaller than 0.002)

in presence of clustered and trended spatial patters the use of URS will lead to much higher standard errors than those achieved in the case of random or uniform patterns: on the other hand the use of TSS, ensuring an even coverage of the study area, carries to standard errors similar for all the spatial patterns

STOF STOF inventoryinventory →→ multiphasemultiphase samplingsampling schemescheme

in the third phase, a fraction of points is selected from each two‐phase sample by means of SRSWOR

each point selected in the third phase is visited on the ground and dendrometric attributes of each STOF within the large plot centred at the point is quantified

from each stratum, a fraction of the first‐phase points is drawn by SRSWOR

for each point selected in the second phase the number of STOF lying within a large plot centred at the point is recorded by means of the VHR imagery

in the first phase a set of  N points is selected onto the study area in accordance with a spatial sampling scheme (e.g. TSS), thus achieving a discrete population of points 

the first‐phase points are spatially overlaid on VHR imagery and classified into land‐cover classes (strata) by visual interpretation (e.g.: agricultural land, artificial areas, wetland, water, woodland, other land) 

In the first phase a set of N points is selected onto the study area in accordance with a

spatial sampling scheme (e.g. TSS), thus achieving a discrete population of points

{ }Npp ,,K1=P

P is partitioned into L strata, say LPP ,,1 K of sizes 1,,1 >LNN K and a sample of

points, say ll UQ ⊂ , of size 2≥ln is selected from each stratum Ll ,,1K= by means of

SRSWOR

The double-expansion estimator is

∑=

=L

lllTwT

122ˆˆ

where

NNw Ll /=

∑∈

=li

il

l Tn

TQ

ˆˆ 12

NiyaRT

ijji ,,1,ˆ K== ∑

∈S

and its variance estimator

⎭⎬⎫

⎩⎨⎧

−+−−

= ∑∑==

L

lll

L

l l

lTllT TTw

nsNw

NV

1

222

1

222

2 )()1(1

1

where

∑∈

−−

=li

lil

lT TTn

sQ

22

22 1

1 )ˆ(

estimatingestimating the the numbernumberof STOF (of STOF (secondsecond--

phasephase))

Baffetta F., Fattorini L., Corona P. (2011) Assessing the attributes of scattered trees outside the forest by a multi-phase sampling strategy. Forestry

a sample of points ml (of size 2≥lm ) is selected from second-phase sample of each

stratum by means of SRSWOR

in this case, the triple-expansion estimator is

∑=

=L

lllTwT

133ˆˆ

where

∑∈

=li

il

l Tm

TG

ˆ1ˆ3

with variance estimator

⎭⎬⎫

⎩⎨⎧

−+−−

= ∑∑==

L

lll

L

l l

lTllT TTw

msNw

NV

1

233

1

232

3 )ˆˆ()1(1

1

where

∑∈

−−

=li

lil

lT TTm

sG

23

23 )ˆˆ(

11

estimatingestimating the total of the total of a STOF a STOF attributeattribute

((thirdthird--phasephase))

in order to estimate the population mean XTY /= , a very natural procedure is to take

the ratio

23ˆ/ˆˆ XTY =

as the estimator of Y where 2X̂ is the abundance

By using the first-order Taylor series approximation, this estimator can be proven to be

approximately unbiased with an approximately conservative variance estimator

42

22

23

32

3322

232

ˆˆ

ˆˆ

2ˆ XVT

XCT

XVV XXTT

Y +−=

where 22 XV is the estimator of abundance variance and XTC3 denotes the three-phase

estimator for the covariance between 2X̂ and 2T̂ , which is given by

⎭⎬⎫

⎩⎨⎧

−−+−−−

−= ∑ ∑ ∑

= ∈ =

L

l i

L

llllili

ll

llXT

l

TTXXwTXXmn

NwN

C1 1

332223 )ˆˆ)(ˆˆ(ˆ)ˆˆ(111

11

G

estimatingestimating the the meanmean of of a STOF a STOF attributeattribute

((thirdthird--phasephase))

Simulation study for STOF inventory

a quadrat of size 625 km2 (side 25 km) was taken as the study area

since the target areas of real STOF surveys are mostly constituted by agricultural land, then 72 % contiguous quadrats located in the lower part of the study region were presumed to belong to Stratum 1 (agricultural land), while the remaining (28 %) were presumed to belong to Stratum 2 (other land)

overall densities of D = 50, 100, 150 STOF per ha were presumed within the whole study area and their were settled according to three different spatial patterns

for each generated STOF , an AGTB value (in ton) was generated from a log-normal distribution with expectation and variance both equal to 0.3 (coefficient of variation 180%)

STOF population generated from an aggregated spatial pattern with overall

density of 50 (trees x 100 ha), Agricultural

land density (white) of 65 and Other land

density (grey) of 11

STOF population generated from a

random spatial pattern with overall density of

50 (trees x 100 ha), Agricultural land density (white) of 65 and Other

land density (grey) of 11

to simulate TSS, the study area was partitioned into n = 625 quadrats of size 1 km2 (strata 1 and 2 were invariably constituted by 450 and 175 points)

50000 samples were randomly generated; each sample was generated by randomly selecting a point in each quadrat, including in the sample all the forests containing at least one of these 3600 points

from each stratum a fraction of 5% and 10% of points were selected by SRSWOR to build the second-phase sample; in the third phase, a fraction of 50% of second-phase points was again selected by SRSWOR)

the sampling simulation provided the Monte Carlo distributions of the abundance estimates, total biomass estimates and average biomass estimates as well as the Monte Carlo distributions of the corresponding RSE estimates

the relative bias (RB) and the relative root mean squared error (RRMSE) of the estimators were empirically determined from the Monte Carlo distributions

the Monte Carlo distributions of the RSE estimates were used to determine the expectation of the RSE estimators (ERSEE) obtained from the variance estimators to be compared with the corresponding RRMSEs

Performance of Performance of the twothe two--phase phase estimator of estimator of abundanceabundance and and of its variance of its variance estimatorestimator(RB = relative bias; (RB = relative bias; RRMSE = relative RRMSE = relative root mean squared root mean squared error; ERSEE = error; ERSEE = expectation of the expectation of the RSE estimators)RSE estimators)

Overall STOF density

per km2

STOF

density

in

Stratum

1

STOF

density

in

Stratum

2

Spatial

pattern

Second-

phase

sample

size

RB RRMSE ERSEE

31 -0.36% 14% 14% RAND

63 -0.22% 107% 10%

31 -0.14% 18% 18 %

50 50

AGGR

63 -0.20% 12% 13%

31 -0.28% 15% 15% RAND

63 -0.22% 10% 10%

31 -0.13% 21% 20%

50

65 11

AGGR

63 -0.19% 13% 13%

31 -0.35% 10% 10% RAND

63 -0.25% 7% 7%

31 -0.18% 12% 12%

100 100

AGGR

63 -0.21% 9% 9%

31 -0.21% 10% 10% RAND

63 -0.22% 7% 7%

31 -0.13% 13% 13%

100

130 23

AGGR

63 -0.12% 9% 9%

31 -0.38% 8% 8% RAND

63 -0.32% 6% 6%

31 -0.12% 10% 10%

150 150

AGGR

63 -0.21% 7% 7%

31 -0.27% 8% 8% RAND

63 -0.23% 6% 6%

30 -0.13% 10% 10%

150

195 34

AGGR

63 -0.26% 7% 7%

Performance of Performance of the threethe three--phase phase estimator of estimator of total total AGTBAGTB and of its and of its variance variance estimatorestimator(RB = relative bias; (RB = relative bias; RRMSE = relative RRMSE = relative root mean squared root mean squared error; ERSEE = error; ERSEE = expectation of the expectation of the RSE estimators)RSE estimators)

Overall STOF density

per km2

STOF

density

in

Stratum

1

STOF

density

in

Stratum

2

Spatial

pattern

Second-

phase

sample

size

RB RRMSE ERSEE

31 -0.48% 39% 34% RAND

63 -0.21% 27% 25%

31 -0.27% 43% 38%

50 50

AGGR

63 -0.26% 30% 29%

31 -0.26% 39% 34% RAND

63 -0.24% 28% 26%

31 -0.40% 45% 39%

50

65 11

AGGR

63 -0.17% 31% 28%

31 -0.43% 28% 25% RAND

63 -0.23% 19% 18%

31 -0.18% 30% 28%

100 100

AGGR

63 -0.22% 28% 20%

31 -0.30% 27% 25% RAND

63 -0.24% 19% 19%

31 -0.34% 29% 27%

100

130 23

AGGR

63 -0.09% 21% 20%

31 -0.42% 23% 21% RAND

63 -0.42% 16% 15%

31 -0.01% 24% 23%

150 150

AGGR

63 -0.16% 17% 16%

31 -0.19% 22% 21% RAND

63 -0.23% 16% 15%

31 -0.11% 24% 223%

150

195 34

AGGR

63 -0.22% 17% 17%

Performance of Performance of the threethe three--phase phase estimator of estimator of average AGTBaverage AGTBand of its and of its variance variance estimatorestimator(RB = relative bias; (RB = relative bias; RRMSE = relative RRMSE = relative root mean squared root mean squared error; ERSEE = error; ERSEE = expectation of the expectation of the RSE estimators)RSE estimators)

Overall STOF density per km2

STOF

density

in

Stratum

1

STOF

density

in

Stratum

2

Spatial

pattern

Second-

phase

sample

size

RB RRMSE ERSEE

31 -0.12% 37% 31% RAND

63 0.00% 25% 23%

31 -0.16% 40% 33%

50 50

AGGR

63 -0.07% 28% 25%

31 0.00% 37% 31% RAND

63 0.00% 26% 23%

31 -0.27% 40% 34%

50

65 11

AGGR

63 0.02% 28% 25%

31 -0.10% 26% 23% RAND

63 0.03% 18% 17%

31 0.01% 27% 25%

100 100

AGGR

63 -0.01% 19% 18%

31 -0.10% 27% 23% RAND

63 -0.01% 18% 17%

31 -0.15% 27% 24%

100

130 23

AGGR

63 0.04% 19% 18%

31 -0.02% 22% 20% RAND

63 -0.11% 15% 14%

31 0.13% 22% 20%

150 150

AGGR

63 0.05% 15% 15%

31 0.09% 21% 19% RAND

63 0.01% 15% 14%

31 0.03% 22% 20%

150

195 34

AGGR

63 0.04% 16% 15%

the relative bias of the estimators  is invariably negative owing to edge effects but negligible, being always smaller than 0.5 %

the RSE estimator turns out to be a conservative estimator of RRMSEs when abundance is the target parameter

underestimation of variance and consequently of RSE takes place for the three‐phase estimators of totale and average AGTB; underestimation reduces drastically as the plot sampling fraction of the second‐phase increases from 5 to 10%

(as expected) the presence of clustered spatial patterns causes reductions of efficiency in all the cases and for all the estimators; these reductions are always moderate owing to the use of TSS, which in the first phase ensures even coverage of the study area, but also owing to the stratifications performed in the subsequent phases which provide balanced samples with respect to the variation in STOF densities over the study area (more detrimental effects of aggregation might be expected if plots were selected completely at random over the area)

CASE STUDYCASE STUDY

the protocol was tested in various landscapes over the protocol was tested in various landscapes over Italy, for a total inventoried land of 3533 kmItaly, for a total inventoried land of 3533 km22

ID  Region – Landscape unit  Superficie (km2) 

1  Friuli – floodplain  936 

2  Veneto – carbonatic hills  330 

3  Toscana – intermountain plain  297 

4  Campania – sandstone mountains  1345 

5  Sicilia – clay hills  625 

CORONA P., CHIRIACÒ M.V, SALVATI R, MARCHETTI M, LASSERRE B, FERRARI B (2009). Proposta metodologica per l’inventario su vasta scala degli alberi fuori foresta. L'ITALIA FORESTALE E MONTANA

ProtocolProtocol

in the first phase a grid of 1 km x 1 km was overlaid onto the study areas and TSS was applied

the first‐phase points were overlaid on VHR imagery and classified into land‐cover classes (strata) by visual interpretation (e.g.: agricultural land, artificial areas, wetland, water, woodland, other land) 

SWOF and TROF containing at least one first‐phase point were included in the sample; they were delineated on screen and their size measured, and then complete talling in the field was carried out

from each stratum, a fraction of 10% of the first‐phase points was drawn by SRSWOR

for each point selected in the second phase the number of STOF lying within the plot with radius of 100 meters centred at the point was recorded by means of the VHR imagery

in the third phase, a fraction of 20% of points is selected from two‐phase sample by means of SRSWOR

all the third‐phase points are visited on the ground and dendrometric attributes of each STOF within the plot with radius of 100 meters centred at the point is quantified

TOF proves to stock TOF proves to stock 121 Mg C km121 Mg C km‐‐22, on , on average, with an average, with an 

annual sequestration annual sequestration of of 4 Mg C km4 Mg C km‐‐22

expanding these expanding these figures to overall Italy, figures to overall Italy, the estimated  the estimated  CC‐‐stock stock is about 30 Mtis about 30 Mt, with , with 

an an annual Cannual C‐‐sequestration of about sequestration of about 

1 M t1 M t

ID  Number  Area  Wood volume  AG carbon stock 

  (N km‐2)  (ha km‐2)  (m3 km‐2)  (Mg km‐2) 

1  1,8  0,43  74,9  34,2 

2  2,7  0,61  14,0  6,5 

3  0  0  0  0 

4  5,1  0,74  51,0  25,7 

5  2,8  0,32  12,1  6,3 

ID  Number  Lenght  Wood volume  AG carbon stock 

   (N km‐2)  (m km‐2)  (m3 km‐2)  (Mg km‐2) 

1  37,8  3821  278,2  126,4 

2  6,4  945  109,0  49,3 

3  4,5  679  169,0  81,8 

4  18,3  1573  145,9  71,5 

5  10,7  227  6,7  3,0 

ID  Number  Wood volume  AG carbon stock 

   (N km‐2)  (m3 km‐2)  (Mg km‐2) 

1  78,2  28,7  12,2 

2  199,6  29,6  13,0 

3  378,5  201,9  94,8 

4  135,8  66,6  34,2 

5  76,2  28,2  13,4 

SWOF

TROF

STOF

ESTIMATESESTIMATES (all the valuesare referred to land area)

both the interpretation of remotely‐sensed imagery and the fieldwork prove to be easier than in a classical forest inventory, 

as TOF recognition in the VHR imagery is obviously clearer, access in the field is more direct and measurement is simpler

in the case of a region of 100 km2 and under sampling intensity and conditions similar to those of the case studies, an expert 

photointerpreter requires half a day to perform first‐ and second‐phase steps using orthophotos while a team of two forest technicians requires roughly another one day to record the 

attributes in the field on the sampled TOF 

small patches such as woodlots are difficult to sample unbiasedly by plots because of edge effects (trees near to the edge of woodlots have less chance of being included in the sample than the inner trees)

while edge effects are negligible and can be ignored when the inclusion zones of trees are much smaller than the study region, when sampling small woodlots the inclusion zones and the study region are likely to have quite similar orders of magnitude (in these situations a relevant amount of negative bias may be induced by ignoring edge effects)

edge problems are well investigated in literature and several solutions have been proposed (e.g. Gregoire and Valentine, section 7.5), but any solution is likely to much elongate the field work with respect to the standard plot procedures

sectorsector samplingsampling

Gregoire TG, Valentine HT (2008). Sampling Strategies for Natural Resources and the Environment. Chapman & Hall

a sampling method referred to as sector sampling has been recently proposed and proves to be particularly useful for sampling small cluster of trees within irregularly shaped regions such as small woodlots

Iles K, Smith NJ (2006). A new type of sample plot that is particularly useful for sampling small clusters of objects. Forest Science

‐ a pivot point is purposively selected within the woodlot, usually in the middle of the woodlot

‐ a sector of fixed angle θ centred on a direction  randomly selected on 0‐2π is emanated from the pivot point and all the trees within the sector are sampled

as consequence, all trees in the woodlot have the same probability θ/2π of being included in the sample

Denote by jU the population of trees lying within woodlot j and let ly be the value of the interest

variable Y corresponding to the l-th tree in jU , in such a way that jt can be expressed as the

population total

∑∈

=jl

lj ytU

Sampling is performed in which a sample of trees, say jQ , is selected from jU in accordance with

SS and the interest variable is recorded for any jl Q∈

The HT estimator of jt based on a single sector turns out to be

∑∈

=jQl

lj ytθπ2ˆ

Without involving edge effects, this is an unbiased estimator for jt with variance

∑∑∈>∈

−+=jj lh

jlhhll

lj tyyyUU

22

22 22 θθπ

θπσ

where lhθ is the overlap of the angles of width θ constructed around the directions joining the l-th and h-th trees to the

pivot point

As customary in forest surveys, the sampling scheme is replicated jR times

In the case of SS, jR sectors can be randomly and independently emanated from the pivot point giving rise to jR

estimates jRj jtt ,,ˆ,,ˆ K1 , which constitute the realizations of jR iid random variables having expectation jt and

variance 2jσ

The arithmetic mean of the jR estimates

∑=

=jR

iji

jj t

Rt

1

1,

ˆˆ

constitutes an unbiased and consistent ( ∞→jR ) estimator of jt with variance jj R/2σ

The empirical variance of the jR estimates is

∑=

−−

=j

j

R

ijji

jt

ttR

V1

2,

2ˆ )ˆˆ(

11

and it constitutes an unbiased and consistent estimator of 2jσ

SECTOR SAMPLING can be effectively performed in the SECTOR SAMPLING can be effectively performed in the framework of largeframework of large--scale inventories of relatively small scale inventories of relatively small

sized objectssized objects

some hints for inventory planning from statistical exercises :

use designs with inclusion probabilities approximatively proportional to the sizes of objects of interest 

since most variability is usually due to the first phase of sampling (i.e. the selection of sample objects), only few within‐object sectors can be measured (usually, two)

adequate choice of pivot points in order to reduce as much as possible the sector size variability, then reducing the variability of total estimator among the selected sectors

adopt random systematic sampling to select within‐object sample, based on a partition of 0‐2π into parts of equal width and the random selection of one direction within each part (the scheme is proven to be unbiased and invariably more efficient than URS)

(as to field work, some tests in Central Italy showed that a team of two forest technicians required on average 30 minutes to tally two sectors per woodlot with             )°= 36θ

piermaria.corona@unitus.itpiermaria.corona@unitus.itDISAFRIDISAFRI

Department of Forest Environment and ResourcesDipartimento di Scienze dell’Ambiente Forestale e delle sue Risorse

not all that can be counted counts,not all that counts can be counted

(Einstein)