Draft - tspace.library.utoronto.ca · Draft Updatingplotstoimprovesmall-areaestimates 27 1 Introduction 28 Due to the international agreements on climate change, countries now need

Draft

Updating plots to improve the precision of small-area estimates: the example of the Lorraine region, France

Journal: Canadian Journal of Forest Research

Manuscript ID cjfr-2019-0405.R2

Manuscript Type: Article

Date Submitted by the Author: 11-Mar-2020

Complete List of Authors: Fortin, Mathieu; Canadian Forest Service, Canadian Wood Fibre Centre

Keyword: hybrid inference, multiple imputation, Bayesian method, population dynamics model, growth model

Is the invited manuscript for consideration in a Special

Issue? :Not applicable (regular submission)

https://mc06.manuscriptcentral.com/cjfr-pubs

Canadian Journal of Forest Research

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Updating plots to improve the precision of small-area1

estimates: the example of the Lorraine region, France2

Mathieu Fortin3

Canadian Wood Fibre Centre, Canadian Forest Service, Natural Resources4

Canada, 580 Booth Street, Ottawa, ON K1A 0E4, Canada, E-mail:5

[email protected]

March 19, 20207

1

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Updating plots to improve small-area estimates

Abstract8

The sampling intensity of national forest inventory is usually low. Forest dynamics9

models can be used to update plots from past inventory campaigns to enhance the10

precision of the estimate on smaller areas. By doing this, however, the inference relies11

not only on the sampling design but also on the model.12

In this study, the contribution of the model predictions to the variance of the en-13

hanced small-area estimates was assessed through a case study. The French national14

forest inventory (NFI) provided different annual campaigns for a particular region and15

department of France. Three past campaigns were updated using a forest dynamics16

model, and estimates of the standing volumes were obtained through two methods: a17

modified multiple imputation and the Bayesian method.18

The updating greatly increased the precision of the estimate and the gain was similar19

between the two methods. The sampling-related variance represented the largest share20

of the total variance in all cases. This study suggests that plot updating provides more21

precise estimates as long as (i) the forest dynamics model exhibits no systematic lack22

of fit and it was fitted to a large dataset and (ii) the sampling-related variance clearly23

outweighs the model-related variance.24

Keywords: hybrid inference; multiple imputation, forest dynamics model; growth25

model; Bayesian method26

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1 Introduction27

Due to the international agreements on climate change, countries now need to assess the28

state of their forests annually (McRoberts et al., 2009; Vidal et al., 2016). This need has29

induced a fundamental change in the way national forest inventories (NFI) are conducted,30

and many countries have adopted what is called a continuous forest inventory to produce31

annual estimates (McRoberts, 2001; Alderweireld et al., 2016; Hernández et al., 2016; Tomter,32

2016). Continuous forest inventories generally rely on design-based estimators and they cover33

the whole national forested area annually. This is obviously demanding in terms of resources;34

therefore the sampling intensity, i.e. the ratio between the sample size and the population35

size, remains very low. For instance, the sampling intensity in Wallonia, Belgium, is as low36

as 0.2%, but it is still one of the highest in Europe (Alderweireld et al., 2016).37

This low sampling intensity has little impact on the precision of national-level estimates38

because the sample size remains large. In estimating the forest characteristics over smaller39

areas, the sampling intensity remains roughly the same, but the sample size decreases dras-40

tically leading to imprecise regional or local estimates. This problem is included within the41

general topic of small-area estimation (Rao and Molina, 2015). One way to improve the42

precision of these regional estimates is to update the plots of past inventory campaigns using43

a forest dynamics model. Forest dynamics models include different types of models from the44

traditional stand-level models to the individual-based models (Shifley et al., 2017) and they45

are commonly referred to as growth models in forestry.46

Plot updating to improve the precision of the current estimates has interested biometri-47

cians for decades (Kangas, 1991; Bokalo et al., 1996; Shortt and Burkhart, 1996; Van Deusen,48

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1996, 1997; McRoberts, 2000, 2001; Xie et al., 2011). Although the approach seems very ap-49

pealing, it has a major consequence on the inferential framework. At the national level, the50

inferential framework of continuous NFI is based on the sampling design. However, when51

some plots are updated using a forest dynamics model, the inference is based on both the52

design and the model (Kangas et al., 2019). This framework is known as hybrid, a term53

coined by Fattorini (2012) and Corona et al. (2014). Other expressions such as Mandallaz’s54

(2013) pseudo-synthetic estimators also fit in this hybrid inferential framework. A notable55

characteristic of the hybrid inferential framework is that uncertainty arises not only from the56

sampling design but also from the model predictions. The model-related variance, which is57

often overlooked, is known to increase along with growth projection length (Melo et al., 2018,58

2019). The gain in precision as induced by the updated plots could be partially compensated59

for by the increasing model-related variance.60

In a more general perspective, updating plots from past inventory campaigns to enhance61

the precision of the estimates of current inventories is a special case of data assimilation62

(Lahoz et al., 2010). This approach assumes that updated past information can be merged63

with current information, which is a fundamental distinction with updated forest inventories,64

which solely rely on past information (e.g., Anttila, 2002; Haara and Leskinen, 2009). Dif-65

ferent statistical approaches, such as Bayes’ theorem and the Kalman filter (Kalman, 1960),66

have been tested for merging past and current information (e.g., Ehlers et al., 2013; Nyström67

et al., 2015). However, the estimation has never been formally addressed through a hybrid68

inferential framework and the contribution of the model to the uncertainty of the estimates69

remains to be quantified.70

The objective of this study was to estimate the gain of precision when plots of past71

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inventory campaigns are updated and integrated into the current inventory campaign. The72

French NFI was used as a case study. More precisely, the Meurthe-et-Moselle department73

and the Lorraine region in eastern France were selected to showcase the approach. The plots74

from three past inventory campaigns were updated and integrated into the current inventory75

under a hybrid inferential framework. The reduction of the variance as well as the model76

contribution to the variance of the estimates was assessed.77

The estimated variances obtained under the hybrid inferential framework were compared78

with those obtained with another method commonly used to merge past and current infor-79

mation, namely through the Bayesian method. In the former, the plots of past inventory80

campaigns are updated and then merged in the current inventory campaign. In the latter,81

it is assumed that updated plots and current plots provide different estimates that can be82

merged using Bayes’ theorem.83

2 Methods84

2.1 Hybrid inferential framework85

McRoberts et al. (2016) reported four key features that formally describe the context of86

hybrid inference: (i) a model that predicts the variable of interest using some auxiliary87

information, (ii) a probability sample of auxiliary information, (iii) a point estimator such as88

the mean or the total and (iv) an estimator of the variance that can account for the variances89

of the model and the probabilistic sampling design. The first two features are addressed when90

using a forest dynamics model to update the individual plots of a past inventory campaign.91

Subsequently, the third and fourth features are addressed, namely the need for point and92

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variance estimators that consider both the model and the design.93

Analytical point and variance estimators have been developed by some authors in the94

context of hybrid inference (e.g., Ståhl et al., 2011, 2016; Fortin et al., 2016). However,95

in many circumstances, the complexity of the model does not allow for the use of these96

analytical estimators because they rely on a Taylor series and the model derivatives cannot97

be calculated. Forest dynamics models working at the tree level (e.g., Stage, 1973; Pretzsch98

et al., 2002) are a typical example of these complex models. They are usually based on short99

growth steps so that the predictions are reinserted into the model as many times as necessary100

to obtain projections for longer time intervals. Although these models can be differentiated101

over a single growth step, the differentiation over several growth steps becomes intractable102

and, consequently, the analytical hybrid estimators cannot be applied (Fortin et al., 2018).103

Bootstrap estimators are an alternative to analytical estimators when the model or the104

way it is used are too complex to produce derivatives. Such bootstrap hybrid estimators were105

used by McRoberts and Westfall (2016) and McRoberts et al. (2016) following a multiple-106

imputation strategy as suggested by Rubin (1987). The imputation technique consists of107

imputing values for those missing in a sample using a procedure that may take the form of a108

model. In the context of plot updating, this means using a forest dynamics model to produce109

volume estimates that are substituted for unobserved current volumes in the plots of past110

inventory campaigns. Design-based point and variance estimators are then applied to the111

sample including imputed values.112

To account for the uncertainty in the imputation, the procedure can be repeated many113

times, hence the multiple-imputation denomination, with some random deviates, whose dis-114

tribution reflects the uncertainty, added to the imputation at each run. This method is, in115

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fact, a bootstrap and, more precisely, a parametric bootstrap (Efron and Tibshirani, 1993,116

p. 53) when the random deviates are drawn from parametric distributions to reflect the model117

residual variation, the random effects and the uncertainty in the parameter estimates. If b118

denotes the run and B the total number of runs, the multiple-imputation point and variance119

estimators are (Rubin, 1987, p. 76):120

µ =B∑b=1

µbB

(1)

V(µ) =1 + 1/B

B − 1

B∑b=1

(µb − µ)2 +B∑b=1

Vd(µb)

B(2)

where µb is the design-based estimate of the mean calculated from sample b, and Vd(µb) is121

the design-based estimate of the variance based on sample b, that is,122

Vd(µb) =

∑ni=1 (yb,i − µb)2

n(n− 1)(3)

where yb,i is the observed or imputed value in plot i of sample b. The first term on the right-123

hand side of Eq. 2 is the part of the variance that arises from the imputation procedure,124

whereas the last term represents the average designed-based variance estimate.125

This multiple-imputation technique has been used in forestry by many authors (Reams126

and McCollum, 2000; McRoberts and Westfall, 2014, 2016; Magnussen et al., 2017). However,127

some simulation studies evidenced that the variance estimator shown in Eq. 2 tends to over-128

estimate the true variance (Fay, 1992; Fortin et al., 2018). The major part of this variance129

bias comes from the model residual variation (Fortin et al., 2018). When using a model for130

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imputation, the relative bias in the variance estimator can be as large as 2(1−R2)/R2 where131

R2 is the coefficient of determination of the model (see Supplementary Material 3 in Fortin132

et al., 2018). In other words, a model whose R2 = 0.95 could result in a 10.5% overestimation133

of the variance in some contexts.134

Fortin et al. (2018) proposed and tested a corrected parametric bootstrap estimator based135

on multiple imputation, which can be expressed as136

V(µ) =

∑Bb=1 (µb − µ)2

B+ 2Vd(µy)−

∑Bb=1 Vd(µb)

B(4)

where137

Vd(µy) =

∑ni=1

(yi −

∑ni=1

yin

)2

n(n− 1)(5)

with yi =∑B

b=1yb,iB. Note that the variance estimated in Eq. 5 is, in fact, the design-based138

variance estimator (Eq. 3) applied to the average values yi.139

This parametric bootstrap variance estimator was supported by theoretical developments140

as well as different simulation studies, which showed that it is asymptotically unbiased (Fortin141

et al., 2018). In a more global perspective, this variance estimator is, in fact, similar in essence142

to that of Pfeffermann and Tiller (2005).143

Fortin et al. (2018) identified term Vd(µy) as an estimator of the sampling-related variance144

component. Subtracting this term from the right-hand side of Eq. 4 yields an estimator of145

the model-related variance component (Vm(µ)), namely146

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Vm(µ) = V(µ)− Vd(µy)

=

∑Bb=1 (µb − µ)2

B+ Vd(µy)−

∑Bb=1 Vd(µb)

B(6)

Like Rubin’s multiple-imputation variance estimator (Eq. 2), this corrected variance esti-147

mator shows a logical behaviour. If a census of the population is carried out, the design-based148

variance estimates Vd(µy) and Vd(µb) are both equal to 0 and the estimate of the total vari-149

ance reduces to∑Bb=1(µb−µ)2

B, a component that arises from the model only. If the model150

predictions are perfect, then term∑Bb=1(µb−µ)2

B= 0 and Vd(µb) = Vd(µy) so that the corrected151

variance estimator reduces to a classical design-based variance estimator. For further details152

on the mathematical developments behind this estimator, see Fortin et al. (2018) and the153

supplementary data in Melo et al. (2019).154

In practice, the point and variance estimators shown in Eqs. 1 and 4 require that the155

model implement a consistent stochastic approach. More precisely, random deviates must be156

generated to account for the different stochastic components of the model. Typically, those157

are (i) the uncertainty in the parameter estimates; (ii) plot, growth step or tree random158

effects; and (iii) the residual variation in each submodel that composes the model. Moreover,159

these random deviates must follow a hierarchical structure. For each bootstrap run, the160

random deviates in the parameter estimates are drawn only once. Likewise, a plot random161

effect is drawn only once for a particular plot. The growth step random effects are drawn162

only once for all the trees of a particular plot during a given growth step. Finally, the163

deviates accounting for the residual variation are those that are drawn at each occurrence164

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of the submodel. Note that these deviates can also follow a covariance structure. All these165

stochastic components contribute to the model-related variances.166

The variance estimator in Eq. 4 does not require an assumption regarding the distribution167

of the point estimate; however, such a distribution is needed to define the bounds of confidence168

intervals. Under fair conditions, that is, if the model does not exhibit strong curvature as per169

Bates and Watts (1988, Ch. 7) and the sampling is random, it can reasonably be assumed170

that the central limit theorem (Casella and Berger, 2002, p. 236) still applies and that the171

point estimate follows a normal distribution.172

When updating plots from past inventories and merging them into the current inventory,173

the model only applies to the plots of past inventories so that the yb,i does not vary across the174

bootstrap runs for the plots of the current inventory. The assertion is correct if the variable175

of interest is truly observed in the plots of the current inventory. It is incorrect when dealing176

with the standing commercial volume of tree species because this variable is almost never177

truly observed. Instead, it is predicted using submodels of height–diameter relationships178

and volume, which should ideally be the same as those used in the forest dynamics model.179

Consequently, the yb,i of past plots will vary across the bootstrap runs due the submodels of180

height–diameter relationships and volume plus the other components of the forest dynamics181

model, whereas the yb,i of current plots will only be affected by the submodels of height–182

diameter relationships and volume.183

This framework is flexible enough to accommodate different contexts that go beyond data184

assimilation. For instance, if the plots of past inventories are left aside, the variance estimator185

in Eq. 4 still applies and its model-related component accounts for the uncertainty arising186

from the submodels of height–diameter relationships and volume. Only updating plots of187

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past inventories without combining them with any current information is still an option and188

it results in a simple updated inventory as per Anttila (2002) and Haara and Leskinen (2009).189

2.2 Bayesian method190

A Bayesian approach can be used in conjunction with a hybrid inferential framework. If191

each inventory campaign provides an estimate of the population parameter of interest, these192

estimates can be combined using Bayes’ theorem (Casella and Berger, 2002, p. 23) to obtain a193

posterior distribution of the estimate. Other approaches for combining estimates exist, such194

as the mixed estimator (see Kangas, 1991), but the Bayesian approach is more flexible in195

the sense that it does not assume that the estimates are independent and that the posterior196

distribution is normal.197

The different estimates from the individual inventory campaigns are obtained using the198

point and variance estimators shown in Eq. 1 and 4. The fundamental difference between the199

Bayesian method and the one described in the previous section is that the plots from past200

inventory campaigns are updated but not directly combined with the plots of the current201

inventory campaign. Instead, they produce an estimate of the population parameter of202

interest, and both sources of information are merged through the estimates and not through203

the plots (Fig. 1).204

(Insert Figure 1 here)205

Bayes’ theorem can be expressed under a general form as (Howson and Urbach, 2006,206

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p. 21)207

P(h | e) =P(e | h)P(h)

P(e)=

P(e | h)P(h)∫P(e | h)P(h)dh

(7)

where P(·) is the probability of observing the argument; e is the evidence, what we know; and208

h is the hypothesis, what we want to test, respectively. P(h | e) is the posterior distribution209

of h given the evidence e, whereas P(h) is known as the prior distribution of h.210

In the context of continuous variables, the probabilities P are replaced by probability211

density functions (f):212

f(µ | µp) =f(µp | µ)f(µ)∫f(µp | µ)f(µ)dµ

(8)

where f(µ | µp) is the posterior distribution of µ given the estimates obtained from the213

different inventory campaigns µp, f(µp | µ) is the joint density of observing the estimates µp214

if µ was the true mean and f(µ) is the prior of µ, that is, the density without any knowledge215

of the inventory campaigns. Note that the notation in Eq. 8 follows the vector notation in216

McCulloch et al. (2008, p. 23).217

The knowledge about the prior distribution of µ is limited: the mean standing volume is218

necessarily positive and likely smaller than a particular limit which could be set to 400 m3ha−1219

according to field experience. This limited information leads to the use of a noninformative220

prior (see Box and Tiao, 1973, p. 27), so that f(µ) = c for any µ within the range ]0, 400] or221

0 otherwise.222

The expression of Bayes’ theorem shown in Eq. 8 might seem unusual to some biometri-223

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cians who prefer a sequential approach (see Box and Tiao, 1973, p. 11) so that224

f(µ | µp,2, µp,1) =f(µp,2 | µp,1, µ)f(µ|µp,1)∫f(µp,2 | µp,1, µ)f(µ|µp,1)dµ

(9)

where µp,1 and µp,2 are the estimates from a first and a second inventory campaign. The225

function f(µ|µp,1) is, in fact, the posterior of µ given the estimate from a first inventory226

campaign µp,1, which serves as prior to the posterior f(µ|µp,1, µp,2). This sequential approach227

can be repeated each time new information is made available.228

Using the general product rule (Korn and Korn, 2000, p. 589), it can be shown that229

the expression in Eq. 8 and the sequential approach are, in fact, equivalent (see Appendix).230

However, in this study, the first was preferred over the latter because it is more tractable231

when the estimates from the different inventory campaigns are correlated. The sequential232

approach can also accommodate these correlations but at the cost of additional developments.233

The density f(µp | µ) is expected to follow a multivariate normal distribution with234

variances estimated using the variance estimator shown in Eq. 4. However, the estimates235

are not entirely independent since they rely on the same model. It can be assumed that the236

covariance between the estimates is equal to237

∀k 6= k′, COV(µp,k, µp,k′) =

√Vm(µp,k)Vm(µp,k′) (10)

where k is the inventory campaign index, so that k = 1, 2, . . . , K and Vm(µp,k) is the estimated238

model-related variance of the estimate of the kth inventory campaign (Eq. 6). Note that239

Vm(µp,k) is only the estimated model-related variance and it is necessarily smaller than the240

estimated variance shown in Eq. 4.241

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Under the assumption of a multivariate normal distribution, the joint density becomes242

f(µp | µ) =e−(µp−µ)

T Ω−1

(µp−µ)2√

(2π)K | Ω |(11)

where Ω is the estimated variance-covariance matrix of µp whose diagonal elements are the243

estimated hybrid variances (Eq. 4), and off-diagonal elements are the covariances based on244

the estimated model-related variances (Eq. 10), | Ω | is the determinant of the same matrix245

and K is the number of inventory campaigns or, equivalently, the length of vector µp.246

2.3 Case study: Stands of oak, beech and hornbeam in eastern247

France248

Sessile oak (Quercus petraea Liebl.), pedunculate oak (Quercus robur L.), European beech249

(Fagus sylvatica L.) and European hornbeam (Carpinus betulus L.) are four emblematic250

species of northern France. They often grow in mixed stands and together account for more251

than a third of the total standing commercial volume in metropolitan France (IGN, 2014).252

In 2010, the nationwide harvest of these four species was estimated at 12.4 million m3, which253

represented more than 60% of the total harvest of broadleaved species (AGRESTE, 2012).254

The French NFI is a continuous inventory based on an annual two-phase sampling scheme255

(Mandallaz, 2008, p. 79), also known as double sampling (Gregoire and Valentine, 2008,256

p. 149). The first phase follows a systematic grid design that covers the whole metropolitan257

territory. The main grid has a sampling intensity of one node every 1 km2. One tenth of258

the nodes is systematically selected and successively sampled every year. This results in an259

annual systematic design with a lower sampling intensity of one node every 10 km2. The260

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successive annual grids are meant to overlap and to keep a systematic grid design after a261

five-year cycle. This system is described by some authors as one of interpenetrating panels262

(e.g., McRoberts, 2001). Every year, the land use at each node of the annual grid is observed263

using aerial photographs. During the second phase, a subsample of the nodes that fall into264

forested areas is randomly selected for ground measurements. A field plot is established265

at each node of this subsample. Each plot is composed of four concentric circular areas in266

which different tree- or plot-level attributes are recorded. A vegetation survey is also carried267

out following the Braun-Blanquet method (Wikum and Shanholtzer, 1978). The inventory268

protocol is described in IGN (2016).269

We focused on the Lorraine region where the oak–beech–hornbeam forest type is common.270

French administrative regions are divided into departments. To illustrate the effect of the271

area, we also selected the Meurthe-et-Moselle department of the Lorraine region to be part of272

the case study. The 2015 inventory campaign was chosen as the one for which an estimate of273

the standing commercial volume was needed. Three past inventory campaigns were selected274

for the plot updating. The inventory campaigns of 2005 and 2010 were left aside because275

the annual systematic grids followed a five-year cycle and the updated plots would have been276

very close to those of 2015. By selecting the inventory campaigns of 2006, 2009 and 2012,277

the set-up of the case study was less subject to spatial correlations.278

The plots that fell into the oak–beech–hornbeam forest type within the Meurthe-et-279

Moselle department and the Lorraine region were identified using the vegetation survey. The280

plots were kept only if the cover of oak, beech or hornbeam was at least 25% and the cover281

of coniferous species did not exceed 5%. It was assumed that these two criteria were enough282

to define the forest type and that the area covered by this forest type remained constant283

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from 2006 to 2015 in both the department and the region. Approximately 60% and 50% of284

the plots met these criteria at the department and the region levels, respectively. Given that285

the forested areas of the department and the region were estimated at 1600 and 8820 km2286

(IGN, 2014), respectively, the area occupied by the oak–beech–hornbeam forest type could287

be roughly estimated at 960 km2 in the Meurthe-et-Moselle department and at 4410 km2 in288

the whole Lorraine region. Table 1 provides a summary of the dataset. Fig. 2 shows the289

location of the plots. Note that most of the southeastern part of the region showed no plot290

of the oak–beech–hornbeam forest type. This is, in fact, the Vosges mountain range where291

coniferous stands dominate the landscape.292

(Insert Table 1 here)293


MATHILDE is a forest dynamics model that makes it possible to forecast the development295

of sample plots located in stands of oak, beech and hornbeam (Fortin and Manso, 2016).296

The model applies at the tree level and its structure is shown in Fig. 3. It consists of297

several submodels that respectively predict the individual mortality, the diameter increment298

of survivors (Manso et al., 2015a,b) and the recruitment of new trees. The mortality submodel299

includes dummy variables that account for the occurrence of droughts and windstorms so that300

the model can take into account the effect of these disturbances (Manso et al., 2015b).301


Since its construction in 2015, the model has been complemented by a plot- and a tree-303

level harvest occurrence submodel (Manso et al., 2018; Fortin et al., 2019a), a submodel304

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of height–diameter relationships (Fortin et al., 2019b) and a submodel of commercial vol-305

ume (unpublished). In addition to traditional forest metrics, the model takes into account306

stumpage prices in the prediction of harvest occurrence as well as the mean temperature and307

precipitation of the growing season in the prediction of tree height and diameter increment.308

The model was designed to work in a full stochastic fashion using the Monte Carlo tech-309

nique. Random deviates are drawn to account for the uncertainty in the parameter estimates;310

the plot, growth step and tree random effects; and the residual variation of each submodel311

that composes the model. Some submodels have a binary outcome such as the harvest oc-312

currence submodels and the mortality submodel. The random deviate that accounts for the313

residual variation is then drawn from a uniform distribution [0,1]. The event is assumed to314

occur when the deviate is smaller than the predicted probability of occurrence.315

In its stochastic mode, MATHILDE follows the hierarchical structure mentioned in Sec-316

tion 2.1. It produces growth realizations of each sample plot. The usual growth step is five317

years, but the model allows for growth steps ranging from three to six years. Given that the318

selected inventory campaigns were three years apart, the growth step was set to three years.319

Longer growth projections were obtained by reinserting the predictions into the model as320

often as needed. The deviates were simulated following the hierarchical structure and they321

propagated through the successive growth steps.322

The point and variance estimators shown in Eqs. 1 and 4 were used in conjunction with323

MATHILDE to update the plots from the 2006, 2009 and 2012 inventory campaigns to 2015.324

The stumpage prices and the mean temperature and precipitation of the growing season were325

set to those observed from 2006 to 2015. A major windstorm occurred in 2009 and it was326

accounted for in the 2006–2009 growth step when updating the plots of the 2006 campaign.327

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Two different sets of simulations were conducted. In the first set, the inventory campaigns328

of 2006, 2009, 2012 and 2015 were successively used to produce a single estimate of the 2015329

standing volume. For the 2015 campaign, this was, in fact, the traditional method. For the330

inventory campaigns of 2006, 2009 and 2012, the approach was one of updated inventory331

(UI) as per Anttila (2002) where the past information is simply updated but not merged332

with that of the current inventory.333

Data assimilation (DA) was performed in the second set of simulations. The information334

of past inventory campaigns was updated and combined with current information under a335

hybrid inferential framework. The resulting estimates were then compared to those obtained336

using the Bayesian method. Note that the estimates of the first set of simulations, that is, the337

estimates from each inventory campaign, were those used in the Bayesian method (Fig. 1).338

3 Results and discussion339

The estimates of the mean standing volume per hectare were similar across the different340

inventory campaigns and the methods (Table 2). They ranged from 191 to 206 m3ha−1 and341

from 200 to 215 m3ha−1 at the department and the region levels, respectively. The estimated342

variances obtained from the individual inventory campaigns were of similar magnitude. Re-343

gardless of the method and the level, updating plots from past inventories and combining the344

information with that of the current inventory contributed to decrease the variance of the345

estimates. The estimated variance was roughly inversely proportional to the sample size for346

both methods. This trend is shown for the hybrid inferential framework in Figure 4. For in-347

stance, compared to the 2015 inventory campaign, the estimates based on the 2009 and 2015348

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campaigns had approximately twice the number of plots, but their estimated variances were349

half those of 2015 at both the department and the region levels (Fig. 4). The estimates based350

on all the available campaigns relied on a sample size four times larger and their estimated351

variances were approximately one fourth those of 2015. The smaller estimated variances352

resulted in narrower confidence intervals which were again very similar between the methods353

(Table 2).354



The contribution of the sampling and the model to the total variance is shown in Table 3.357

At both the department and the region levels, the sampling represented the largest share.358

Except for the 2006 inventory campaign, the estimated model-related variances remained359

relatively small, ranging from 0.6 to 8.8 m6ha−2 and from 0.5 to 8.3 m6ha−2 at the department360

and the region levels, respectively. In proportion, the model-related variance represented361

between 0.2 and 1.9% of the total variance at the department level when the 2006 inventory362

campaign was not included in the estimation. Whenever this was the case, the model-related363

variance accounted for 4.2–8.5% of the total variance.364

At the region level, the model had a greater relative contribution to the total variance365

mainly due to fact that the sampling-related variance was smaller. Without the 2006 in-366

ventory campaign, the model-related variances ranged from 0.5 to 9.9%. When the 2006367

inventory campaign was included in the estimation process, the model-related variances ac-368

counted for 14.9–25.6% of the total variance.369

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The results of this study show that updated inventories, that is, using only information371

from past inventories, can lead to estimates with a precision comparable to that of the372

current inventory (Table 2). Similar results were also reported by Anttila (2002). Combining373

updated past inventories with current inventory information leads to more precise estimates374

of the standing volume. This result is consistent with those of past studies (Kangas, 1991;375

Bokalo et al., 1996; McRoberts and Lessard, 1999; Ehlers et al., 2013). In fact, the updated376

plots increase the sample size and therefore, they contribute to decrease the sampling-related377

variance (Table 3). Because the model-related variance was much smaller than that induced378

by the sampling, the total variance was approximately inversely proportional to the sample379

size, as it would have been with a pure design-based variance estimator, that is, V(µ) = σ2/n.380

Fortin et al. (2016) showed that the model-related variance mainly stems from the vari-381

ances of the parameter estimates and not so much from the residual variation. A decisive382

factor that can explain the small model-related variance in this case study is the size of the383

datasets to which the different components of MATHILDE were fitted. For all components,384

the fit was based on thousands of observations and, consequently, the estimated variances of385

the parameter estimates were rather small. The context of the case study, namely a large386

sampling-related variance coupled with a small model-related variance, matches the condi-387

tions in which the largest gain in precision can be expected according to Ehlers et al. (2013).388

If the sampling-related variance had been smaller, there would have been less improvement.389

This would be the case if the sample size of the current inventory were much greater than it390

was in the case study. For instance, if the population had been the stands of oak, beech and391

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hornbeam all over the metropolitan territory, the 2015 inventory campaign would have had392

a sample size of 2150 plots. This would have led to a smaller sampling-related variance and,393

consequently, the model-related variance would have been proportionally more important. In394

such a context, adding plots updated from past inventory campaigns would necessarily have395

a smaller impact on the total variance of the estimate.396

Both the sampling- and the model-related variances of large-area estimates are known397

to be affected by the projection length: the sampling-related variance tends to decrease398

asymptotically, whereas the model-related variance increases along with the projection length399

(Melo et al., 2018). Even when the model-related variance is initially small, it can account for400

a greater part of the variance than the sampling especially when natural disturbances come401

into play (e.g., Melo et al., 2019). As per our hypothesis, it was expected that the model-402

related variance would increase with the projection length, and the gain in precision induced403

by the greater sample size would be partially compensated for. This was not the case in404

any of our combinations of campaigns regardless of the level. In fact, the longest projection405

occurred for the 2006 campaign, and the nine-year period separating that campaign from406

2015 represents a short-term projection compared to those of Melo et al. (2018). This short-407

term projection can partly explain why the model-related variance remained relatively small408

compared to the variance induced by the sampling.409

Contrastingly, Melo et al. (2019) found that the model-related variance could sharply410

increase and largely exceed the sampling-related variance in short-term projections when411

natural disturbances were taken into account. There is a fundamental difference between412

the context of this study and that of Melo et al. (2019). In plot updating, the occurrence of413

large-scale natural disturbances is known. When making large-area projections as in Melo414

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et al. (2019), the occurrence of large-scale disturbances is unknown and it must be predicted.415

In other words, in the context of large-area projections, both the occurrence and the effects416

of the disturbances contribute to the model-related variance, whereas only the effects of the417

disturbances are accounted for in plot updating. This fundamental difference can also explain418

why the model-related variance remained small.419

The inability of many forest dynamics models to take into account disturbances definitely420

limits their use in plot updating. This applies not only to large-scale natural disturbances421

but also to harvesting (Haara and Leskinen, 2009; Ehlers et al., 2013). Some authors have422

suggested that harvested plots should be dealt with through a stratification before using the423

forest dynamics model (Van Deusen, 1996; Magnussen et al., 2017). However, it is not always424

known whether a particular plot has been harvested or not over the interval, and delineating425

such harvested areas on the map is subject to errors. Even if this information is available, it426

may happen that more than one disturbance occurred during the time interval, which could427

lead to a more complex stratification and the sample size issues that come with it.428

An alternative consists of integrating the harvesting into the model as suggested by429

Condés and McRoberts (2017). In MATHILDE, the harvesting is considered through a430

harvest occurrence submodel, which was much simpler than retrieving the plot statuses from431

GIS layers, assuming they would have been available. Because its mortality submodel also432

considers the effects of windstorms and droughts (Manso et al., 2015b), MATHILDE was433

adapted to the contexts of the different inventory campaigns. The integration of natural434

and human-made disturbances into forest dynamics models is crucial in forest management435

(Seidl et al., 2011). It can also facilitate data assimilation procedures by avoiding a complex436

stratification. There is no certainty that this results in more precise estimates though. The437

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comparison with a simple stratification whenever the data and the context make it possible438

remains to be investigated.439

The hybrid inferential framework yielded results that were very similar to those of the440

Bayesian method. Even though both methods require distributional assumptions, the Bayesian441

method is more demanding. The bootstrap hybrid variance estimator shown in Eq. 4 requires442

distributional assumptions for the stochastic components of the model. However, there is no443

need to assume a distribution for the estimates unless confidence limits are required. This is444

a fundamental difference with the Bayesian method where the distributions of the estimates445

are required to compute the posterior distribution.446

Another difference between the methods is that the hybrid inferential framework can easily447

be extended to the multivariate case. In fact, forest dynamics models that apply at the tree448

level can predict many variables of interest for each individual plot. This represents a major449

advantage over stand-level models and remote sensing information. In such a context, the450

yb,i, µ, µb, and µy in the estimators shown in Eqs. 1 and 4 all become vectors. The variance451

estimator produces an estimated variance-covariance matrix for the vector µ. Typically,452

the vector µ would contain some estimated species-specific volumes, basal areas and stem453

densities.454

The extension to the multivariate case with the Bayesian method is more complex. As455

shown in Eq. 11, the single estimates of the different inventory campaigns are assumed to456

be correlated. In the multivariate context, there could be correlations between the different457

variables of interests and between the different campaigns. For instance, the estimated basal458

area of the 2006 campaign could be correlated to the volume of the 2012 campaign, because459

these variables were predicted using the same model. This results in a variance-covariance460

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matrix of much larger dimensions than in the hybrid inferential framework. This remains to461

be investigated.462

Both methods rely on the assumption that the model exhibits no systematic lack of fit.463

This is a strong assumption. In practice, all models show a certain lack of fit. Differences464

between the context of the data used to fit the components of the model and that of the465

inventory campaigns will likely induce a lack of fit. These differences can be related to the466

sampling design, the inventory protocol or the time period. In MATHILDE, the submodels467

of harvest occurrence, recruitment and height–diameter relationships are not too much of a468

concern since they were all fitted to data that were collected in the context of the French469

NFI. However, the submodels of mortality, diameter increment and volume were fitted to data470

collected in other contexts, with larger plots and measurements carried out several decades471

ago in some cases (see Manso et al., 2015a,b). This potential lack of fit could not be assessed472

here because the plots of the different inventory campaigns were not remeasured over time.473

The variance estimator shown in Eq. 4 only addresses the precision of the point estimator.474

The mean square error (MSE) measures the quality of an estimator and it accounts for both475

the precision and the accuracy through the variance and the bias (Casella and Berger, 2002,476

p. 330). It is calculated as the average squared difference between the estimates and the true477

value: MSE = (B(µ))2 + V(µ), where B(µ) is the bias of the point estimator. Any lack of478

fit in the model can result in a bias in the point estimator. This bias induces an increase of479

the MSE or equivalently, a decrease in the quality of the point estimator. As pointed out by480

Gregoire (1998), the model-based inferential framework has been criticized for the fact that481

model misspecification can result in serious biases. The same criticism applies to the hybrid482

inferential framework.483

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In this case study, the estimates with the updated inventory approach were close to the484

estimate of the current inventory (Table 2). This suggested that there was no major lack of485

fit. In fact, major differences between the estimate of the current inventory campaign and486

those based on the updated inventory approach should be considered as a potential indicator487

that the model is not performing well in the context. A sensitivity analysis could also bring488

new insight into the impact of potential lacks of fit on the MSE of the point estimator.489

Another assumption is related to the variance estimator shown in Eq. 4. It assumes that490

the Monte Carlo technique provides a reliable estimate of the variance. There is no certainty491

that this is really the case, especially if some correlations were overlooked during the fitting492

of the model. Fortin et al. (2009) tested the estimated variances through the comparison493

of the nominal and empirical coverage of confidence intervals. The same test applied to494

MATHILDE showed that the variances were slightly underestimated. As a consequence, all495

the variances shown in Table 2 might be slightly underestimated as well.496

Some other estimators are less restrictive that those of the hybrid inferential framework.497

Model-assisted estimators do not make any assumption regarding the correctness of the model498

(Mandallaz, 2008, p. 109). Fischer and Saborowski (2019) developed a model-assisted esti-499

mator in the context of a continuous forest inventory based on a stratified sampling scheme500

(Gregoire and Valentine, 2008, p. 135). This was only possible because some plots of past501

campaigns had been remeasured in the current inventory campaign. When these remea-502

surements are available, it is definitely recommended to use these model-assisted estimators503

because they relax the assumption of model correctness. When the plots are not remea-504

sured over time, as in the French NFI, model-assisted estimators cannot be used and the two505

methods tested in this study remain viable options.506

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A common method in data assimilation is the Kalman (1960) filter and its extended507

version (Grewal and Andrews, 2001, p. 181). Ehlers et al. (2013) and Nyström et al. (2015)508

provided examples of data assimilation based on the extended Kalman filter which could not509

be applied in this case study because it requires the model to be differentiable. Tree-level510

models such as MATHILDE are complex and can hardly be differentiated. The bootstrap511

variance hybrid estimator used in this case study was developed in the context of complex512

models (Fortin et al., 2018). Based on bootstrap methods, it does not require any derivatives.513

However, a full stochastic implementation of the model is needed.514

4 Conclusions515

The following conclusions can be drawn from the results of this study:516

1. A forest dynamics model can be used to update the plots of past inventory campaigns517

in order to increase the precision of the estimate in the current inventory campaign.518

2. Bootstrap hybrid estimators based on multiple imputation yield estimates that are519

similar to those obtained with the Bayesian method.520

3. Given that forest dynamics models are usually fitted to large datasets and that the521

design-based variance of forest inventories is generally large for small areas, the updating522

of past inventories provides a major gain in precision.523

4. In this case study, updating information from an inventory campaign that took place524

nine years earlier still largely improved the precision of the estimate of the standing525

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commercial volume, and the variance related to the model remained a minor component526

of the total variance.527

5. Both the hybrid inferential framework and the Bayesian method assume that the model528

exhibits no systematic lack of fit, which is a strong assumption inherent to model-based529

estimators (Gregoire, 1998). Any systematic lack of fit can affect the quality of the point530

estimator.531

Acknowledgements532

The author wishes to thank Steen Magnussen (Canadian Forest Service), Ronald E. McRoberts533

(University of Minnesota and U.S.D.A. Forest Service), two anonymous reviewers and the534

Associate Editor for their comments on a preliminary version of this paper and Paula535

Irving (Canadian Forest Service) for her editing. The French NFI database is available536

from the website of the Institut national de l’information géographique et forestière (http:537

//www.ign.fr/). Some GIS layers that were used to produce Fig. 2 were downloaded from the538

GeoBoundaries dataset (www.wm.edu/as/data-science/researchlabs/geolab/get_data/539

geoboundaries), the European Environment Agency website (www.eea.europa.eu) and the540

Natural Earth website (www.naturalearthdata.com).541

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Appendix705

Let us consider Bayes’ theorem as expressed in Eq. 8 and set the number of inventory cam-706

paigns to two so that vector µp = (µp,1, µp,2)T . Using the general product rule (Korn and707

Korn, 2000, p. 589), the equation can be developped as follows708

f(µ | µp) =f(µp | µ)f(µ)∫f(µp | µ)f(µ)dµ

=f(µp,2 | µp,1, µ)f(µp,1 | µ)f(µ)∫f(µp,2 | µp,1, µ)f(µp,1 | µ)f(µ)dµ

=f(µp,2 | µp,1, µ)f(µp,1|µ)f(µ)

f(µp,1)∫f(µp,2 | µp,1, µ)f(µp,1|µ)f(µ)

f(µp,1)dµ

=f(µp,2 | µp,1, µ)f(µ | µp,1)∫f(µp,2 | µp,1, µ)f(µ | µp,1)dµ

which yields the sequential approach shown in Eq. 9. This development can easily be709

extended to a greater number of inventory campaigns. A similar development can also be710

found in Howson and Urbach (2006, p. 253).711

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Table 1: Summary of the inventory campaigns of 2006, 2009, 2012 and 2015 in the Meurthe-et-Moselle department and the Lorraine region, France. n is the number of plots.

Level and campaign n Minimum Median Mean MaximumMeurthe-et-Moselle department2006 campaignStem density (trees ha−1) 51 0 375 512 2476Basal area (m2ha−1) 0 20.8 20.1 43.32009 campaignStem density (trees ha−1) 35 0 422 569 2299Basal area (m2ha−1) 0 17.7 19.6 46.22012 campaignStem density (trees ha−1) 55 0 464 531 1837Basal area (m2ha−1) 0 20.0 20.2 46.72015 campaignStem density (trees ha−1) 46 42 463 605 1680Basal area (m2ha−1) 1.9 21.1 21.4 63.4

Lorraine region2006 campaignStem density (trees ha−1) 203 0 479 634 2476Basal area (m2ha−1) 0 22.0 21.8 60.72009 campaignStem density (trees ha−1) 170 0 497 688 2692Basal area (m2ha−1) 0 23.1 22.2 56.42012 campaignStem density (trees ha−1) 240 0 442 614 2564Basal area (m2ha−1) 0 21.1 21.3 54.12015 campaignStem density (trees ha−1) 180 0 519 666 2564Basal area (m2ha−1) 0 22.2 21.9 63.4

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Table 2: Estimates of the 2015 mean standing volume per hectare in mixed stands of oak, beech and hornbeam in the Meurthe-et-Moselle department and the Lorraine region, France, in function of the different methods and inventory campaigns. Confidenceintervals are based on a 0.95 probability.

Hybrid inferential framework Hybrid framework + Bayesian methodLevel and Approacha nb

Estimate of Confidence Estimated Estimate of Confidence Estimatedcampaigns the mean interval variance the mean interval varianceMeurthe-et-Moselle department2015 ST 46 206.2 [167.6, 244.7] 386.12012 UI 55 202.3 [168.3, 237.3] 309.82009 UI 35 191.2 [150.7, 231.6] 426.62006 UI 51 201.7 [166.5, 236.9] 322.12012 and 2015 DA 101 204.3 [178.7, 230.0] 171.7 204.3 [178.6, 230.1] 172.52009 and 2015 DA 81 199.5 [171.6, 227.4] 202.7 199.0 [171.1, 227.0] 203.12006 and 2015 DA 97 203.8 [177.8, 229.8] 175.7 203.7 [177.6, 229.8] 177.62009, 2012 and 2015 DA 136 200.7 [178.9, 222.4] 123.3 200.5 [178.8, 222.3] 123.42006, 2012 and 2015 DA 152 203.3 [182.6, 224.0] 111.9 203.4 [182.4, 224.5] 115.32006, 2009, 2012 and 2015 DA 187 201.2 [182.5, 219.9] 90.9 200.9 [182.1, 219.6] 91.7

Lorraine region2015 ST 180 215.0 [196.5, 233.4] 88.42012 UI 240 209.1 [192.9, 225.4] 68.92009 UI 170 212.0 [194.0, 230.0] 84.22006 UI 203 199.7 [181.3, 218.1] 88.12012 and 2015 DA 420 211.6 [199.4, 223.9] 39.0 211.7 [199.4, 224.0] 39.42009 and 2015 DA 350 213.5 [200.6, 226.4] 43.3 213.4 [200.4, 226.5] 44.12006 and 2015 DA 383 206.9 [193.8, 220.0] 44.4 207.3 [194.1, 220.6] 45.82009, 2012 and 2015 DA 590 211.9 [201.5, 222.2] 27.9 211.8 [201.3, 222.3] 28.52006, 2012 and 2015 DA 623 208.0 [197.3, 218.6] 29.4 208.3 [197.6, 219.0] 29.92006, 2009, 2012 and 2015 DA 793 208.7 [199.4, 218.1] 22.8 209.4 [199.7, 219.1] 24.5

aST: standard approach, that is, solely based on the current inventory campaign; UI: updated inventory approach, that is, based on pastinventory campaigns only; DA: data assimilation approach, that is, based on both past and current inventory campaignsbn: number of plots.

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Table 3: Estimated sampling-related and model-related variances of the point estimator ofthe 2015 mean standing volume per hectare in mixed stands of oak, beech and hornbeamin the Meurthe-et-Moselle department and the Lorraine region, France, under the hybridinferential framework.

Level and Sampling-related Model-relatedcampaigns variance varianceMeurthe-et-Moselle department2015 385.5 (99.8%) 0.6 (0.2%)2012 307.1 (99.1%) 2.7 (0.9%)2009 425.5 (99.7%) 1.1 (0.3%)2006 294.7 (91.5%) 27.4 (8.5%)2012 and 2015 169.8 (98.9%) 1.9 (1.1%)2009 and 2015 201.2 (99.3%) 1.4 (0.7%)2006 and 2015 166.9 (95.0%) 8.8 (5.0%)2009, 2012 and 2015 120.9 (98.1%) 2.3 (1.9%)2006, 2012 and 2015 107.1 (95.8%) 4.7 (4.2%)2006, 2009, 2012 and 2015 85.6 (94.1%) 5.3 (5.9%)

Lorraine region2015 88.0 (99.5%) 0.5 (0.5%)2012 65.4 (94.8%) 3.6 (5.2%)2009 75.8 (90.1%) 8.3 (9.9%)2006 65.5 (74.4%) 22.5 (25.6%)2012 and 2015 37.4 (95.9%) 1.6 (4.1%)2009 and 2015 41.1 (94.8%) 2.2 (5.2%)2006 and 2015 37.8 (85.1%) 6.6 (14.9%)2009, 2012 and 2015 25.3 (90.7%) 2.6 (9.3%)2006, 2012 and 2015 23.9 (81.5%) 5.4 (18.5%)2006, 2009, 2012 and 2015 18.2 (80.1%) 4.5 (19.9%)

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Figures713

Figure captions714

Figure 1 - Conceptual flowchart of the two methods used to integrate the information of past715

forest inventory campaigns into the current campaign.716

Figure 2 - Location of the ground plots of the 2006, 2009, 2012 and 2015 NFI campaigns in717

the Lorraine region (light grey) and the Meurthe-et-Moselle department (dark grey). Some718

GIS layers from the GeoBoundaries dataset (www.wm.edu/as/data-science/researchlabs/719

geolab/get_data/geoboundaries), the European Environment Agency website (www.eea.720

europa.eu) and the Natural Earth website (www.naturalearthdata.com) were used to pro-721

duce the map base. The plot coordinates were retrieved from the French NFI database,722

which is available from the website of the Institut national de l’information géographique et723

forestière (http://www.ign.fr/). The layers were compiled using QGIS 2.18.724

Figure 3 - Flowchart of MATHILDE for a single growth step. dt is the step length which can725

range from three to six years. In stochastic mode, random deviates for the uncertainty in the726

parameter estimates and the residual variation are drawn for all the submodels as indicated727

by the boxes. The letters a, b and c indicate that random deviates are also drawn to account728

for plot, growth step and tree random effects, respectively.729

Figure 4 - Relative decreases of the total variance in function of the relative increases of the730

sample size under the hybrid inferential framework. The point located at (1,1) represents the731

reference, that is a standard inventory based on the 2015 campaign only. The other labels732

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www.eea.europa.eu

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indicate which inventory campaign was updated and merged with the 2015 campaign.733

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t-3

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t-1

t

Update to t using the model



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using Bayes'

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t

t-3

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t-1

t




t

t

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b) Hybrid inferential framework combined with the Bayesian method

a) Hybrid inferential framework

Figure 1: Conceptual flowchart of the two methods used to integrate the information of pastforest inventory campaigns into the current campaign.

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Figure 2: Location of the ground plots of the 2006, 2009, 2012 and 2015 NFI campaigns inthe Lorraine region (light grey) and the Meurthe-et-Moselle department (dark grey). SomeGIS layers from the GeoBoundaries dataset (www.wm.edu/as/data-science/researchlabs/geolab/get_data/geoboundaries), the European Environment Agency website (www.eea.europa.eu) and the Natural Earth website (www.naturalearthdata.com) were used to pro-duce the map base. The plot coordinates were retrieved from the French NFI database,which is available from the website of the Institut national de l’information géographique etforestière (http://www.ign.fr/). The layers were compiled using QGIS 2.18.

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www.eea.europa.eu

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Predict harvest occurrence

a

b

b

a,c

b

a

Figure 3: Flowchart of MATHILDE for a single growth step. dt is the step length which canrange from three to six years. In stochastic mode, random deviates for the uncertainty in theparameter estimates and the residual variation are drawn for all the submodels as indicatedby the boxes. The letters a, b and c indicate that random deviates are also drawn to accountfor plot, growth step and tree random effects, respectively.

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2006

2009

2012

200620122012

2009

201220092006

2015 only

2006

2009

2012

20062012

20122009

201220092006

2015 only

b) Lorraine region

a) Meurthe-et-Moselledepartment

Figure 4: Relative decreases of the total variance in function of the relative increases of thesample size under the hybrid inferential framework. The point located at (1,1) represents thereference, that is a standard inventory based on the 2015 campaign only. The other labelsindicate which inventory campaign was updated and merged with the 2015 campaign.

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Draft - tspace.library.utoronto.ca · Draft Updatingplotstoimprovesmall-areaestimates 27 1 Introduction 28 Due to the international agreements on climate change, countries now need