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## ESTIMATING MODEL- AND SAMPLING-RELATED UNCERTAINTY IN LARGE-AREA GROWTH PREDICTIONS

The aforementioned context is one of hybrid inference. The term hybrid was coined by Corona et al. (2014) and refers to the fact that the inferential process relies on both a model and a probability design (McRoberts et al., 2016). It arises when the variable of interest is not measured or not measurable, such as future forest growth, and when the explanatory variables are only available from samples and not for the whole population (Fortin et al., 2016). Some authors have already applied hybrid estimators in the context of large-area estimation of volume, biomass and carbon (e.g., Healey et al., 2012; McRoberts et al., 2016; Saarela et al., 2015; Ståhl et al., 2011, 2016). When working with complex models such as tree-level growth models, current analytic hybrid estimators, i.e., those based on algebra (e.g., Ståhl et al., 2011), can rarely be applied. An alternative consists in using bootstrap hybrid estimators (Fortin et al., 2018).

Quantifying sampling and model errors of large-area growth predictions is essen-tial since it can help us identify which issues need to be addressed in order to reduce the uncertainty of the predictions. Previous studies on large-area estimates of vol-ume and biomass have shown that the major source of uncertainty originated from the sampling (Breidenbach et al., 2014; McRoberts and Westfall, 2014; Ståhl et al., 2014). However, the use of growth models in the context of hybrid inference adds a temporal variability. Kangas (1999) reported that uncertainty increased along with projection length. Thus, the contribution of both sampling and models to prediction uncertainty may also change along the projection length.To the best of our knowl-edge, this temporal variability has not been addressed, with the notable exception of Condés and McRoberts (2017) who worked on short-term predictions. Moreover, tree-level growth models are complex when compared to stand-level models since they include many sub-models. This system of sub-models follows the dynamic of individual tree development through a temporal and spatial frame (Pretzch et al., 2008). Because mortality and recruitment are highly stochastic (Sheil and May, 1996), it could be anticipated that they would contribute more to the total un-certainty. Identifying which one of the sub-models contributes more uncertainty could help to improve models, understand forest dynamics and reduce prediction uncertainty. As far as we know, this has not been addressed yet.

The aim of this study was to estimate uncertainty in growth predictions at the level of a large area. To do this, we worked on two hypotheses: (i) over long simu-lation periods, i.e., a 100-year prediction, model uncertainty becomes greater than sampling uncertainty; (ii) among the model components, mortality is the major contributor to prediction uncertainty, then followed by recruitment. To confirm or invalidate these two hypotheses, we aimed at decomposing the total prediction variance into a model and sampling component that was made possible by using a bootstrap hybrid estimator. Secondly, through a variance decomposition approach, the prediction uncertainty was also decomposed into sub-models (i.e., growth, mor-tality and recruitment) in order to assess which one contributed the most to the variance of large-area growth predictions. We worked with the ARTEMIS tree-level growth model (Fortin and Langevin, 2012), which was used to generate large-area predictions for the Bas-Saint-Laurent region in Quebec, Canada.

**Material and methods**

**Growth model**

he 2009 version of the ARTEMIS distance-independent tree-level growth model was built, fitted and evaluated using data from the network of permanent plots of Quebec’s provincial forest inventory. This network consists of 12,570 randomly located sample plots that were established in Quebec’s commercial forests and that has been measured since the 1970s. ARTEMIS takes the vast majority of the forest types in Quebec into account (Fortin and Langevin, 2010, 2012).

The model consists of four dynamic and two static sub-models (Fig. 2.1). The dynamic parts are those typical of population dynamic models: a mortality sub-model, a diameter increment sub-model for survivor trees, and two sub-models that predict the number and the diameter at breast height (DBH, 1.3 m in height) of the recruits, respectively. The static sub-models predict tree height and commercial volume. All sub-models are of the linear or generalized linear type. The model is based on 10-year growth intervals. Longer predictions are obtained through an iterative procedure, the result of the previous interval being re-inserted in the model. Readers are referred to Fortin and Langevin (2010, 2012) for further details on the model.

ARTEMIS growth predictions are based on a wide range of explanatory variables that can be retrieved from the compilation of forest inventories in Quebec: tree species, harvest occurrence (yes/no), spruce budworm defoliation (yes/no), stem density (tree ha−1), basal area (m2ha−1) and forest type. ARTEMIS also makes use of climatic variables, such as the 1971-2000 mean annual precipitation (mm) and mean annual temperature (°C). These climate variables can be estimated using the BioSIM application (Régnière et al., 2010).

ARTEMIS can predict forest growth either in a deterministic or stochastic fash-ion (Fortin and Langevin, 2010), depending on the user’s decision. Stochastic pre-dictions rely on the Monte Carlo technique (Rubinstein and Kroese, 2007) and they can be either fully or partially stochastic. The full stochastic mode assumes that all the sub-models are stochastic. In contrast, the partial stochastic mode makes it possible to disable the stochasticity in the selected sub-models. Predictions are generated at tree level, and the plot-level outcome can be obtained by aggregating tree-level predictions. The model was also designed to handle many plots at the same time since the usual input data were expected to come from forest inventories. ARTEMIS has been used for diﬀerent applications, from simple productivity assess-ment to the comparison of diﬀerent silviculture options (e.g., Fortin, 2014; Fortin and Langevin, 2012; Laliberté et al., 2016).

**Input dataset**

The input data we used to make large-area predictions also came from the provincial network of permanent plots of Quebec’s Ministry of Forests, Wildlife and Parks (MFWP). Since our focus was to generate regional-level growth predictions, we kept only the measurements from the Bas-Saint-Laurent region, which covers an area of 28,401 km2 and two vegetations zones: the northern temperate and the boreal zones (Poirier et al., 2013). The forest composition of the Bas-Saint-Laurent region made it possible to perform predictions for diﬀerent forest types, since it encompasses broadleaved, mixed and coniferous stands. Broadleaved and mixed forests are mainly composed of sugar maple (Acer saccharum Marsh.), yellow birch (Betula alleghaniensis Britton), balsam fir (Abies balsamea Mill.) and white spruce (Picea glauca Voss). Coniferous stands are dominated by balsam fir, white and black spruce (Picea mariana Britton) with a minor component of white birch (Betula papyrifera Marsh.) and trembling aspen (Populus tremuloides Michx.) (Poirier et al., 2013). Moreover, this region has been exploited for timber since the beginning of the 19th century, and for this reason, it is of historical importance for forestry in Quebec Boucher et al. (2009b). This first region-based screening resulted in 1,572 plot measurements that covered the period from 1975 to 2012. We chose the year for which we had the largest sample size, 2003, with a total of 393 plots.

The subsequent screening took the ecotype into account. We used the current ecological classification system used by the MFWP, which is based on the physical characteristics of the site, forest dynamics and its structural elements (Saucier et al., 2009, p. 186-205). Since there were too many ecotypes, we decided to keep three ecotypes that represented the diversity of forest stand composition and for which we had the largest sample sizes. We therefore worked on the following three ecotypes: sugar maple-yellow birch, balsam fir-white birch and balsam fir-white cedar. For convenience, we will refer to these three ecotypes as the broadleaved, mixed and coniferous ecotypes, respectively. The final dataset contained 188 plots.

In each of these 400-m2 plots, all trees with DBH equal to or greater than 9.1 cm were tagged for individual monitoring. All explanatory variables required by ARTEMIS were available from the compilation of the input dataset of Quebec’s MFWP. A summary of the dataset and the study area is provided in Table 2.1. The spatial distribution of the plots is shown in Fig. 2.2.

**Simulation framework**

We chose to quantify the variance of large-area growth predictions in terms of basal area. The simulation framework consisted of 100-year growth predictions running from 2003 to 2103 based on 10,000 Monte Carlo realizations. These predictions were generated for each one of the three aforementioned ecotypes and excluded all exogenous disturbances such as harvesting, pest outbreaks and fires. Additionally, we considered that the mean temperature and the mean precipitation would increase by 2°C and 5%, respectively, over the 21st century.

First, a large-area prediction was generated for each ecotype using the full stochastic mode, i.e., by considering the stochasticity of all ARTEMIS sub-models. The bootstrap variance estimator that is described in the next section made it pos-sible to split the total variance of large-area predictions into a sampling- and a model-related component.

We then generated a series of large-area predictions using the partial stochastic mode, with the aim of decomposing the total variance. Using the above framework, we alternately disabled the stochasticity of the mortality, diameter increment and recruitment sub-models, while keeping the other sub-models in a stochastic mode. The same bootstrap variance estimator was used.

Basal area predictions at the plot level were obtained by aggregating the tree-level predicted basal areas since they were produced by ARTEMIS. The predictions were run on the CAPSIS platform (Dufour-Kowalski et al., 2012).

**Table of contents :**

**1 Introduction **

1 Techniques and applications of growth models

2 Uncertainties in forest growth forecasts

2.1 Sources of uncertainties

2.2 Methods for estimating uncertainty

3 Objectives

4 Thesis context

4.1 Forest dynamics in Quebec

4.2 Quebec’s Forest Inventory

4.3 ARTEMIS-2009

4.4 Simulation framework

**2 Estimating model- and sampling-related uncertainty in large-area growth predictions**

1 Introduction

2 Material and methods

2.1 Growth model

2.2 Input dataset

2.3 Simulation framework

2.4 Hybrid estimator

3 Results & Discussion

4 Conclusions

**3 Using survival analysis to predict the harvesting of forest stands in Quebec, Canada **

1 Introduction

2 Material and methods

2.1 Dataset

2.2 Statistical development

2.3 Model evaluation

3 Results

4 Discussion

**4 The effect of natural and anthropogenic disturbances on the uncertainty of growth forecasts **

1 Introduction

2 Material and methods

2.1 ARTEMIS growth model

2.2 Uncertainty estimation

2.3 Study area and dataset

2.4 Forecasting

3 Results

4 Discussion

5 Conclusions

**5 Discussion and Perspectives **

1 Research problems

2 Framework

3 Estimation of uncertainties in large-area growth forecasts

4 Perspectives