# An Alternative Approach to Using LiDAR Remote Sensing Data to Predict Stem Diameter Distributions across a Temperate Forest Landscape

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## Abstract

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^{2}≥ 0.75, and another 23% had 0.5 ≤ R

^{2}< 0.75). The predicted frequency of larger stems was much better than that of small stems (8 ≤ x < 11 cm diameter), particularly small conifers. We used the predicted SDDs to calculate aboveground carbon density (ACD; RMSE = 21.4 Mg C/ha), quadratic mean diameter (RMSE = 3.64 cm), basal area (RMSE = 6.99 m

^{2}/ha) and stem number (RMSE = 272 stems/ha). The accuracy of our predictions compared favorably with previous studies that have generally been undertaken in simpler conifer-dominated forest types. We demonstrate the utility of our results to spatial forest management planning by mapping SDDs, the proportion of broadleaves, and ACD at a 0.25 ha resolution.

## 1. Introduction

^{2}versus point densities as low as 0.5 pulses/m

^{2}for ABA; [26]), it is less effective at capturing broadleaf crowns than coniferous ones [27], it does not detect all trees beneath the canopy [22], and it carries high computational demands. Some of these issues can be addressed through hybrid approaches that combine ITD with ABA [28]. For example, the distribution of stem diameters can be extrapolated from an ITD model to encompass smaller stems [29], or diameters can be calibrated to match an ABA-predicted stand metric [30].

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Aerial Input Data

^{2}.

_{AERIAL}) in combination with the ground-based plot data to develop a binary ecosite classification: sugar maple dominated stands and mixed stands.

_{GROUND}). We then developed rule-based conditions for ecosites (forest types) that best separated the calibration plots along the first and second PCA axes. We defined the sugar maple ecosite to be where SPECIES

_{AERIAL}contained low proportions of balsam fir, white spruce and eastern hemlock (<20%) and a high proportion of broadleaves (>60%), which were predominantly sugar maple. Areas that did not meet the conditions for the sugar maple ecosite were defined to belong to the mixture ecosite. This ecosite definition separated the majority of plots on the first two PCA axes based on their species composition as a proportion of both stem numbers and basal area (Figure 2).

_{LIDAR}for each 0.25-ha LiDAR tile was defined by the proportion of first returns falling into a set of 1-m height bins. We also calculated gap fraction and top canopy height as LiDAR summary metrics for each tile. Gap fraction (GF) was calculated as the proportion of first returns recorded below 2 m [20]. Top canopy height (TCH) was calculated by averaging the maximum height recorded from each complete 1 m

^{2}section of the tile, excluding gaps where all returns were below 2 m. The tile-level data that we used to predict stem diameter distributions (SDD

_{PREDICT}) therefore consisted of HD

_{LIDAR}, TCH, GF and ecosite.

#### 2.3. Ground Plot Data

_{C}for conifers and QMD

_{B}for broadleaves), aboveground carbon density (ACD, in Mg C/ha), basal area (BA, in m

^{2}/ha; BA

_{C}for conifers and BA

_{B}for broadleaves), and the density of stems with dbh ≥ 8 cm (N, in ha

^{−1}; N

_{C}for conifers and N

_{B}for broadleaves) for each plot. For ACD, individual stem biomass was calculated from dbh using Canadian biomass equations [40,41], then summed to obtain the total plot biomass. Biomass was halved to give the carbon content [42]. The biomass equations used to estimate ACD from field data were species-specific, but generic allometries for conifers and broadleaves were used in calculating ACD from SDD

_{PREDICT}. These attributes were variously used for model priors and to measure model performance, as detailed below.

#### 2.4. Overview of the Modelling Approach

_{LIDAR}). Briefly, our approach estimates four parameters (described in the following section) for each tile that specify a SDD that, when converted into a predicted distribution of LiDAR first return heights (HD

_{PREDICT}) using the allometry-based algorithm described in [32], most closely approximates HD

_{LIDAR}for a given tile. The allometry-based algorithm uses relationships between stem diameter, tree height, crown diameter, and crown taper to estimate the canopy height profile for a given SDD, then uses correction factors that account for crown overlap, crown permeability, and gaps to compute a corresponding HD

_{PREDICT}. The resulting SDD is used to calculate a suite of structural attributes (BA, N, QMD, and ACD) for each LiDAR tile across the full landscape (an overview of the approach used is depicted in Figure 3).

_{PREDICT}and related stand attributes to observations from the validation plots.

#### 2.5. Description of the Predictive Model

_{PREDICT}) [32]. Given an observed distribution of LiDAR first returns for a particular tile (HD

_{LIDAR}), our goal is to infer the Bayesian posterior distribution of SDD parameters, $\mathsf{\theta}\text{}=\left\{\mathrm{k},\mathsf{\lambda},\mathrm{N},{\mathrm{p}}_{\mathrm{B}},{\mathsf{\sigma}}_{\mathrm{L}}\right\}$, that could have generated the observed LiDAR returns. In addition to the four parameters described above, θ also includes a noise term (${\mathsf{\sigma}}_{\mathrm{L}}$) for the residual error between HD

_{LIDAR}and HD

_{PREDICT}. Using Bayes’ theorem:

_{LIDAR}and HD

_{PREDICT}for given values of θ, and (2) informative priors, which specify the probability of a given set of θ from related (but statistically independent) analyses of the LiDAR point cloud data, an aerial imagery-derived ecosite classification, and the set of calibration plots. (The normalization constant, which ensures the posterior distribution integrates to one, is ignored when using MCMC sampling.)

#### 2.6. Model Likelihood

_{LIDAR}given the HD

_{PREDICT}generated from θ is calculated from differences in the proportion of returns falling within each 1-m height interval ($\mathrm{h}$ where $\mathrm{h}=\mathrm{H}$ at the highest interval). Specifically, the likelihood for a given set of parameters ($\mathsf{\theta}$) is the product of the probability densities of each of these differences under a normal distribution, where f represents the normal probability density function with mean zero and standard deviation ${\mathsf{\sigma}}_{\mathrm{L}}$.

#### 2.7. Prior Distributions

_{S}and P

_{M}, we derived a prior probability distribution from the mean, variance, and covariance of SDD parameters and stand attributes across the set of calibration plots. The prior for P

_{S}also used tile-level estimates of TCH and GF to help predict stem density. Both components of the prior were specified separately for sugar maple and mixture ecosites.

_{S}. Using linear regression, we found that the total number of stems in a given tile was related to TCH and GF for each ecosite:

_{C}, BA

_{B}, QMD

_{C}, QMD

_{B}and TCH) that could be calculated for a given SDD, and specified their contribution to the joint prior distribution as:

_{M}is defined as a probability distribution for quantities that are calculated from the list of stems associated with given SDD parameters (somewhat analogous to defining priors on the predicted response from a regression-type model [44]). The parameter values for ${\mathrm{P}}_{\mathrm{M}}$ are listed in Table S1. It is important to note that since we estimate a separate set of SDD parameters for each individual tile, the fitted relationships that inform the priors only serve to constrain the SDD parameters and are not used to predict the SDD parameters directly, as in standard area-based approaches.

#### 2.8. Model Output and Performance

^{2}) as:

^{2}indicates that model predictions are an improvement over the simple mean number of stems, while R

^{2}= 1 indicates perfect agreement between the predicted and observed number of stems in each diameter bin. We also used the Reynolds error index ($\mathrm{e}$) [45] and root mean square error (RMSE) to compare our predictions for SDDs and stand attributes with previous studies:

## 3. Results

#### 3.1. SDD Predictions

^{2}≥ 0.75) in half of the 40 validation plots and moderately well (0.5 ≤ R

^{2}< 0.75) in another nine plots. The remaining 11 plots did not demonstrate particularly good SDD predictions (R

^{2}< 0.50), with six of these being worse than a simple mean. The mean Reynolds error index was 53.3 ± 24.7 across all plots (Table 2). Overall, sugar maple plots were predicted more accurately than mixture plots (19 of 24 vs. 10 of 16 plots having R

^{2}> 0.5), a finding also supported by the Reynolds error index.

^{2}and the number of stems or basal area (Pearson correlation coefficient = −0.05 and 0.12, respectively). Small-diameter stems were the most poorly predicted (see Figure S3), being generally over-predicted in the lower basal area plots and under-predicted in higher basal area plots. As an indication of the impact on predictive performance, if the smallest bin is excluded from the R

^{2}calculation (i.e., considering only stems with dbh ≥ 11 cm), the number of moderately well-fitting plots (R

^{2}≥ 0.5) increases from 29 to 34 (85%).

#### 3.2. Stand Attributes

^{2}/ha). Weibull shape and scale parameters were generally over-predicted (RMSE: shape = 0.504 and scale = 7.87). Total number of stems in sugar maple plots was often over-predicted, whilst in mixture plots stem numbers tended to be under-predicted (RMSE = 68.0). Overall, sugar maple plots were better predicted than the mixture plots (BA RMSE: 4.84 vs. 9.32 m

^{2}/ha; N RMSE: 62.4 vs. 75.6). Partitioning basal area and the number of stems into broadleaf and conifer components revealed that broadleaves were estimated more accurately than conifers (BA RMSE: broadleaves = 5.93 m

^{2}/ha, conifers = 7.39 m

^{2}/ha; N RMSE: broadleaves = 38.6, conifers = 67.3). The errors were spatially uniform in the validation plots across the landscape.

#### 3.3. Applying the Model Across the Landscape

## 4. Discussion

#### 4.1. Assessing SDD Predictions

^{2}> 0.5) for 85% of the validation plots, with sugar maple plots predicted slightly more accurately than mixture plots (Figure 4). The smallest stems (8 ≤ dbh < 11 cm) were the least well predicted, as has also been reported in a comparison of parametric and non-parametric ABAs [20]. The smallest trees are most likely problematic because they are not captured well by LiDAR pulse data [48], particularly when considering first returns from low density data [49]. Issues also arise from the stems falling just below the measurement threshold of 8 cm, but which occupy the same space as those falling just above the threshold.

#### 4.2. Assessing Predictions of Stand Attributes

^{2}/ha for basal area and 3.64 cm for quadratic mean diameter fell within their recorded ranges (2.9–18.8 m

^{2}/ha and 2.1–4.6 cm, respectively), whilst our RMSE of 272 (scaled to per hectare) for the total number of stems was slightly better than their results (299–1030 stems/ha).

^{2}/ha). This is contrary to the findings of Peuhkurinen et al. (2008) [18] who, in their study aimed at predicting species-specific diameter distributions, predicted the volume of deciduous species with only half the accuracy of conifers in boreal forest plots. Broadleaves were minor species in their study and dominant in ours, and estimation methods are expected to best fit the dominant species. Sugar maple plots were also generally better predicted than mixture plots, which follows from their higher proportion of broadleaf stems. Conifers may be captured less accurately by our model because the reflective properties of coniferous crowns are difficult to capture mechanistically [56]. We partly accounted for this by incorporating a crown permeation term in the algorithm that predicted return heights, but crown permeability would likely increase further if this algorithm were re-tuned to coniferous stand data.

#### 4.3. Implementing the Model across the Landscape

## 5. Conclusions

_{LIDAR}, TCH, GF and an ecosite classification (alternatively, a forest could be classified entirely as sugar maple or a mixture). Further refinements might use new plot data to re-parameterize the model priors and stem number relationship, or to include other known predictive relationships, to better constrain parameter estimates for a new area. If a suitable network of plots already exists (as is common where LiDAR is used for forest inventories [11]), then a final step would be to re-parameterize the allometry-based algorithm for translating plot inventory data to a corresponding HD

_{LIDAR}. It would be particularly useful to assess whether parameters change significantly and whether this improves predictions compared with the current parameterization.

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Locations of 154 calibration and validation ground plots in Haliburton Forest and Wildlife Reserve in Ontario, Canada. Black dots represent calibration plots and light grey dots represent validation plots.

**Figure 2.**Separation of calibration plots according to an ecosite classification defined from aerial imagery. The biplot shows the first two axes of a Principal Components Analysis (PCA), and the percentage of variation explained by each axis (in parentheses). The PCA was performed on the main species in each calibration plot as a proportion of (

**a**) the total stem number and (

**b**) the total basal area; the shape and colour of the points denote the ecosite classification for these plots.

**Figure 3.**Overview of the Markov chain Monte Carlo approach used for estimating the parameters required to predict stem diameter distributions (SDDs). In the top panel, the HD

_{LIDAR-SDD}model includes five parameters: four to generate SDD

_{PREDICT}(from which the allometry-based algorithm produces the accompanying HD

_{PREDICT}), and one for the residual error. The likelihood quantifies the goodness-of-fit between HD

_{PREDICT}and the LiDAR tile inputted into the model (HD

_{LIDAR}). The prior reflects the probability of the current set of parameters based on conditions found across a set of calibration plots within the study area combined with a LiDAR regression and ecosite classification. The prior and likelihood together inform the selection of the next set of parameters using the Metropolis-Hastings criterion. This process is iterated many times to obtain a chain of parameter samples from the posterior distribution. The bottom panel describes the five parameters being estimated: $\mathrm{k},\mathsf{\lambda},\text{}\mathrm{N},{\text{}\mathrm{p}}_{\mathrm{B}},{\text{}\mathsf{\sigma}}_{\mathrm{L}}$

**Figure 4.**Stem diameter distributions for the 40 validation plots. Histograms show the measured stems, points show the predicted number of stems within each bin, and the solid line shows the Weibull distribution fitted to the data. The plots are ordered from low to high total stem numbers; note the changing y-axis.

**Figure 5.**Predictions of stand attributes versus the measured values in validation plots for (

**a**) Weibull shape parameter for the SDD, (

**b**) Weibull scale parameter for the SDD, (

**c**) quadratic mean diameter, (

**d**) aboveground carbon density, (

**e**) total basal area and (

**f**) total number of stems. Solid lines denote 1:1 relationships. The shape and colour of the points denote the ecosite classification of the plot.

**Figure 6.**Mapped predictions of stand characteristics across the forest landscape. Purple regions correspond with (

**a**) low ACD and (

**b**) low broadleaf density, and green regions correspond with (

**a**) high ACD and (

**b**) high broadleaf density. In (

**c**) red regions correspond with a “reverse J” distribution and yellow regions with a unimodal distribution when predicted SDDs are reduced to the single best-fit axis of variation depicted in the inset figure (also see Figure S2). White regions represent non-forested areas, such as lakes, or land not under the same ownership as the forest.

Structural Metric | Sugar Maple Ecosite | Mixture Ecosite |
---|---|---|

Stem density | 145 ± 42 | 202 ± 76 |

Basal area | 21.8 ± 5.3 | 28.2 ± 6.4 |

Quadratic mean diameter | 21.6 ± 3.1 | 22.7 ± 3.1 |

Proportion broadleaves | 0.89 ± 0.14 | 0.52 ± 0.20 |

Top canopy height (from LiDAR) | 14.7 ± 3.2 | 13.6 ± 2.4 |

**Table 2.**Reynolds error index values for all validation plots and for sugar maple and mixture ecosites separately. Lower values indicate a closer match between the predicted and observed stem distributions.

Plots | No. Plots | Mean ± s.d. | Minimum | Maximum |
---|---|---|---|---|

All validation plots | 40 | 53.3 ± 24.7 | 16.8 | 131.8 |

Sugar maple ecosite | 24 | 52.5 ± 20.5 | 22.0 | 95.2 |

Mixture ecosite | 16 | 54.5 ± 30.0 | 16.8 | 131.8 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Spriggs, R.A.; Coomes, D.A.; Jones, T.A.; Caspersen, J.P.; Vanderwel, M.C. An Alternative Approach to Using LiDAR Remote Sensing Data to Predict Stem Diameter Distributions across a Temperate Forest Landscape. *Remote Sens.* **2017**, *9*, 944.
https://doi.org/10.3390/rs9090944

**AMA Style**

Spriggs RA, Coomes DA, Jones TA, Caspersen JP, Vanderwel MC. An Alternative Approach to Using LiDAR Remote Sensing Data to Predict Stem Diameter Distributions across a Temperate Forest Landscape. *Remote Sensing*. 2017; 9(9):944.
https://doi.org/10.3390/rs9090944

**Chicago/Turabian Style**

Spriggs, Rebecca A., David A. Coomes, Trevor A. Jones, John P. Caspersen, and Mark C. Vanderwel. 2017. "An Alternative Approach to Using LiDAR Remote Sensing Data to Predict Stem Diameter Distributions across a Temperate Forest Landscape" *Remote Sensing* 9, no. 9: 944.
https://doi.org/10.3390/rs9090944