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## Materials and Methods

**Study area**

The study was conducted at the Montmorency Forest which is a teaching and research forest facility of the Laval University. The forest is located in central Quebec, Canada (47.3°N, 71.1°W), about 70 km north of Quebec City (Figure 1.1). The study site covered an area of 66 km2. Elevation at the site ranges from approximately 460 to 1040 m above sea level. The average annual precipitation is 1589 mm, and the average annual temperature is 0.3°C (Environnement Canada, 2016). The site lies on the Laurentian Plateau and is part of the balsam fir/white birch domain. Balsam fir (Abies balsamea [L.] Miller), black spruce (Picea mariana [Miller] BSP and white spruce (Pinus glauca [Moench] Voss) are the dominant conifers found in the forest. White birch (Betula papyrifera Marshall) and trembling aspen (Populus tremuloides) are also common (Bélanger, 2001).

Figure 1.1 Locations of the Montmorency Research Forest study site and the field plots over a lidar DTM

**Field plot data**

The Montmorency Forest has a network of circular permanent plots (radius = 11.28 m, area = 400 m2). The permanent plots are re-measured on a five-year cycle. The plots center position were georeferenced using a dual-frequency antenna GPS (Trimble Yuma, accuracy ~ 30 cm) and were not post-processed. A sample of 119 plots was selected following these criteria: 1) the dominant species was balsam fir, 2) the last cut was before 1996 and 3) the dominant height was at least 7 m. The merchantable trees with a diameter at breast height (dbh) above 9 cm were measured using a diameter tape. Tree height was measured on eight sample trees per plot using a Vertex hypsometer. A height-diameter relationship was adjusted for each plot (nonlinear mixed effects model) and then used to calculate the height of the other trees. The volume of each merchantable tree (MV) was computed using height and diameter based equations as described in Fortin et al. (2007). The total MV was computed as the sum of the individual MVs. We used field data that were collected between 2007 and 2014. Twenty-four field plots had been initially inventoried in 2011, the same year as the lidar survey. Attributes of the other plots were estimated by interpolating the field measurements between two dates of inventory. Table 1.1 presents a summary of the field data.

### Lidar data

The airborne laser data were acquired in August 2011 with an Optech ALTM 3100 discrete return sensor. The average flying altitude was 1000 m above ground. The incident pulse repetition rate was of 100 kHz with a 50% strip overlap. The scanning angle ranged from 0 to 24° relative to the nadir. The sensor could record up to four measurements per pulse. Each field plot was covered by two to four lidar strips. The average above-ground return density was 6.4 point m-2. Information for each return was recorded in .las files (version 1.0). The returns classified as ground (by the vendor) were interpolated to construct a digital terrain model using the LAStools software (Isenburg, 2016). Non-ground returns were normalized by measuring the elevation difference with the underlying terrain model.

**Data analysis**

The data analysis is organized in six sections: section 1) describes the nonlinear generalized least squares regression, section 2) provides an analysis of the spatial distribution of returns using an example field plot, section 3) calculates the lidar variables from the point clouds, section 4) describes the merchantable volume modelling, section 5) analyses the residual variance, and section 6) provides the model validation.

The spatial distribution pattern group of covariates (SP), which will be described later, have been included in this model either in the fixed part through the f(.) function or in the random part through the g (.) function.

** Analysis of the spatial distribution of returns**

The return density, return spacing and scanning angle play an important role in the spatial distribution of returns. To provide a visual example of this, we have extracted a sample of the 2d distribution of all the returns of a given field plot. The plot had been covered by three lidar strips (#8, #9 and #10) during the survey (Figure 1.2). The strips had an average return density of 2.15, 2.19 and 2.04 point m-2 respectively, an average return spacing of 6.8, 6.8 and 7.0 dm and an average scanning angle of 16, 1.6, and 9.4°. The returns were plotted in a 2d geographic grid and a line segment was drawn between consecutive returns to show the scanning sequence. Strips #8 and #10 had a heterogeneous spatial distribution of returns. Local clumps of returns alternated with local gaps. Conversely, strip #9 had a more homogeneous spatial distribution of returns throughout the plot. When the three strips were pooled together at the plot level, the average return spacing drastically decreased (4.0 dm). However, local gaps were still observable within the plot.

We triangulated neighboring returns into a 3d network to analyze the spacing distribution. This was done using the Delaunay triangulation algorithm of the ArcGIS software. The area of each triangle was calculated. Figure 1.3 shows the area distribution of triangulated returns for the previous example field plot. Strips #8 and #10 which had a heterogeneous pattern (see Figure 1.2) produced an asymmetric right-skewed area distribution. The areas ranged from 0.2 to 211.44 dm2 and 0.68 to 217.29 dm2 respectively. Conversely, strip #9, which had a homogeneous pattern, produced a symmetric leptokurtic area distribution. The areas ranged from 0.14 to 67.12 dm2. The heterogeneous pattern observed, when pooling all strips together, induced an asymmetric right-skewed mesokurtic area distribution. The number of small triangles increased. The areas ranged from 0.06 to 53.14 dm2. This example shows that an irregular distribution of returns persisted locally even with three overlapping strips. Strips were always pooled together in the following analyses.

#### Lidar variable generation

Lidar returns were clipped to the extent of each plot spatial boundary to extract the corresponding point cloud. Lidar variables were computed both at the strip and plot levels using the Fusion software (McGaughey, 2014). They were classified into three standard groups of explanatory variables: canopy height variables (CH: height percentiles, average, mode), canopy density variables (CD: percentage of first returns above or below a 2 m, 5 m or 7 m threshold) and canopy structural heterogeneity variables (CSH: standard deviation, variance, coefficient of variation, canopy surface roughness index). The canopy surface roughness index, hereafter called rumple index (Ri), was computed as the ratio of the canopy outer surface area to the ground surface area as measured by the lidar-derived canopy surface and digital terrain models in a 1 m x 1 m grid (Kane et al., 2010).

The fourth group of explanatory variables was computed to describe the spatial distribution pattern of returns (SP). Variables included in this group were: return density variable (computed as the number of above ground returns of a given plot divided by its area), area distribution variables (average, standard deviation, minimum, maximum, skewness and kurtosis of the area distribution of triangulated returns) and scanning angle variables (mean, mode, standard deviation, minimum, and maximum of the scanning angle distribution). Table 1.2 presents a summary of the lidar data for the study site.

**Results**

We found a very good correlation between the predicted and the field merchantable volumes even before the introduction of the SP variables (pseudo-R2 = 0.91, Figure 1.4 and Table 1.3). The best predictors were the average height of first returns (Mean_h1st), the percentage of first returns below 2 m (Per_1st_2m) and the rumple index (Ri). Their importance in the model was of 87%, 2%, and 1% respectively. The RSD was 28.0 m3 ha- 1 for the regression model. It was estimated to 29.0 m3 ha-1 after cross-validation. The best equation was: MV = exp (2.04) * Mean_h1st 1.33 * Per_1st_2m-0.08 * Ri 0.64 + ε (6)

Variable transformation was also used to account for the heteroscedasticity. We tested a square root, a cube root, and a logarithmic transformation given that the response and explanatory variables of the MV model were strictly positive (see Table 1.1 and 1.2), and that the three variables (MV, Per_1st_2m, and Ri) had a right-skewed distribution. The heteroscedasticity was best reduced with the logarithmic transformation. The p-value for the predicted MV parameter was improved, however, it remained significant (0.01). The model had the following form: log(MV) = 1.99 + 1.18 * log (Mean_h1st) – 0.08 * log (Per_1st_2m) + 0.95 * log (Ri) + ε

**Table of contents :**

Introduction

**Chapitre 1: Modelling the effect of the spatial pattern of airborne lidar returns on the prediction and the uncertainty of timber merchantable volume **

Résumé

Abstract

Introduction

Materials and Methods

Results

Discussion

Conclusion

**Chapitre 2: Lidar and multispectral imagery classifications of balsam fir tree status for accurate predictions of merchantable volume**

Résumé

Abstract

Introduction

Materials and Methods

Results

Discussion

Conclusion

**Chapitre 3: A generalized lidar-based model for predicting the merchantable volume of balsam fir of sites located along a bioclimatic gradient in Quebec, Canada **

Résumé

Abstract

Introduction

Materials and methods

Results

Discussion

Conclusion

**Conclusion générale **

**Bibliographie **