Enhancing the Estimation of Stem-Size Distributions for Unimodal and Bimodal Stands in a Boreal Mixedwood Forest with Airborne Laser Scanning Data

Stem size distribution (SSD), which describes tree frequencies in diameter classes within an area, has a variety of direct and indirect applications that are critical for forest management. In this study, we evaluated which structural characteristics derived from Airborne Laser Scanning (ALS) data were best able to differentiate between unimodal and bimodal stands in a managed boreal mixedwood forest in Alberta, Canada. We then used wall-to-wall ALS data to predict (for 20 m-by-20 m grid cells) the parameters of a Weibull SSD in unimodal cells, and a Finite Mixture Model (FMM) in bimodal cells. The resulting SSDs were evaluated for their fit to ground plot-measured SSDs using an Error Index (EI). We found that the variance of ALS return heights was the best metric for differentiating between unimodal and bimodal stands, with a classification accuracy of 77%. Parameters of both the Weibull and FMM distributions were accurately predicted (r2 ~0.5, Root Mean Square Error (RMSE) ~30%), and that differentiating for modality prior to estimating SSD improved the accuracy of estimates (EI of 49.13 with differentiation versus 51.31 without differentiation). Unique to our presented approach is the stratification by SSD modality prior to the modelling of distributions. To achieve this, we apply a threshold to an ALS metric that allows SSD modality to be distinguished for each cell at the landscape level, and this a priori information is then used to ensure that the appropriate distribution is modelled. Our approach is parsimonious and efficient, enabling improved accuracy in SSD estimation across diverse landscapes when ALS data is the sole data source.


Introduction
Forest professionals manage forests using detailed forest inventories based on measurements and models characterizing tree dimensions, timber volume, and stand composition [1].A critical component of forest inventories is the stem size distribution (SSD), which represents the relative frequency of tree diameters on a given area (e.g.plot or stand; [2]).SSDs are versatile and are useful for a range of timber production and ecological monitoring purposes.For example, the SSD can be used directly to describe stand attributes such as structure, age, and volume [3][4][5], or used as inputs to models that can describe product quality [6], forecast growth, or provide information for planning and management considerations [7].To support these information needs, there is increasing interest Forests 2018, 9, 95 2 of 14 in the capacity to estimate stand-level SSDs over large areas, and to understand patterns of their variance-particularly across the diverse species and age gradients that exist in mixedwood stands.Mixedwood stands are the most common habitat in the boreal forest, which makes up almost 2 billion hectares globally-28% of which is in Canada, representing 78% of the nation's total forested area [8].
Currently, SSDs are usually acquired using field measurements where tree diameters are measured on sample plots and classified into diameter classes.The class sizes used can vary depending on the plot size, stand characteristics, or the application; however, 2-cm diameter classes are most frequently used [9].SSDs of homogenous stands are typically unimodal, while two or more maxima are often observed in heterogeneous stands with more complex structure.Various statistics, such as the bimodality coefficient or Hartigan's dip statistic [10], can be used to assess multimodality of SSD.However, it can also be characterized using the ecological characteristics of the stand, including if stands are multilayered [11], have high variance in diameters [12], ages [13], or number of species [14], or by more complex metrics such as the ratio of stem density to top height (D/H; [15]).As stands mature, they do not follow a linear pattern of structural development, making the prediction of mature stand structures difficult [16].Heterogeneity, typically associated with multimodal stands, results from ecological legacies and disturbance histories specific to individual areas [17].
In addition to size class frequencies, SSDs can be summarized using a variety of statistical models or probability density functions (PDFs).The most common PDF for characterizing SSDs is the Weibull distribution because of its flexibility with a limited number (two) of parameters to predict or impute [18].Fitting PDF parameters to measured diameter at breast height (DBH) values relies on optimization techniques such as maximum likelihood estimation (MLE; [19]).Despite the flexibility of the Weibull distribution, it is limited to the characterization of stands with unimodal SSD [20].If a SSD of a stand is not unimodal, it should be characterized by a more complex distribution or use a nonparametric estimation method such as a Finite Mixture Model (FMM; [15]) or k-Nearest Neighbors (k-NN) [21].
For complete characterization of forest structure, the SSDs of all stands in the area of interest need to be estimated or measured.SSDs derived from field measurements are spatially constrained, time consuming, and expensive to acquire.Hence, field-based estimates of SSDs alone cannot provide the large-area spatial coverage required in a forest management context.A common inventory approach to address the limitation in spatial coverage is to use air-photo interpretation, which can provide complete spatial coverage of an area.However, this approach is limited to the scale of the aerial photography and the expertise of the interpreter, and as a result it is difficult to provide the detailed tree-level information required for a SSD.Eid et al. [22], for a spruce/pine forest in Norway, determined that photo interpretation provided poorer estimates of stand inventory attributes, such as basal area, height, and number of trees, than estimation by other remote sensing methods.They found, for example, deviations of 20% for photo-interpreted height, compared to deviations of 12% for laser scanning estimates.Inaccurate photo-interpreted estimates were projected to have more than three times the loss in the value of a stand when compared to Airborne Laser Scanning (ALS)-based estimates.
These limitations in spatial coverage and accuracy, coupled with increasing financial pressures and needs for highly detailed data, are resulting in the increased use of active remote sensing technologies for informing forest inventories.ALS has been incorporated into forest inventories because it provides very detailed information on forest structure over large spatial extents [23].It has been demonstrated [24] that attributes such as dominant height, mean diameter, stem number, basal area, and volume can be adequately estimated using ALS data in combination with a sample of ground plots [25,26].ALS also has been used to characterize SSDs in unimodal stands requiring relatively simple estimation techniques [27] and in multimodal stands with more complex procedures [21].However, less has been done to estimate SSDs of boreal mixedwood forests, which can have both simple and irregular distributions in neighboring stands, making them more difficult to predict [14], and estimation of stand modality with ALS has been limited to vertical forest structure (e.g.[28,29]).In the case of forests varying between unimodal and multimodal SSDs, the thresholds for applying Forests 2018, 9, 95 3 of 14 different fitting techniques is also poorly understood.Landscape-scale evaluation of modality in predicting SSD in mixedwood stands can provide valuable insights into the complexity of these forests.
In this paper, we evaluate the efficacy of ALS metrics to first differentiate plots with unimodal and multimodal SSDs and then to predict parameters of those distributions.Using both ground plot measurements and ALS estimates of plot structural characteristics (e.g., height or age), we investigate the capacity of ALS metrics to distinguish between areas with single or multimodal distributions and then apply ALS to predict SSDs with the distribution most appropriate to the characterized distribution type.We then compare ALS parameter estimates to fitted parameters and ground measured SSDs.Finally, we discuss the applications of this methodology and explain how estimates could be predicted across an entire area of interest.

Study Area
The study area is an actively managed boreal mixedwood forest near Lesser Slave Lake in Alberta, Canada (Figure 1).The area is approximately 700,000 ha in size.Ten tree species were present, with white spruce (Picea glauca), black spruce (Picea mariana), trembling aspen (Populus tremuloides), and lodgepole pine (Pinus contorta) as the most common.The study area covers three natural subregions of Alberta-Central Mixedwood, Lower Foothills, and Upper Foothills.These regions receive approximately 600 mm of annual precipitation and have mean summer and winter temperatures of 20 • C and −21 • C, respectively [30].Common disturbances include timber harvesting and fires, which are slightly less common than in other forested regions in Alberta.Ecology varies from mesic to wetland areas and features some of the most productive timber-producing areas in Alberta [28].
Forests 2018, 9, 95 3 of 15 [14], and estimation of stand modality with ALS has been limited to vertical forest structure (e.g.[28,29]).In the case of forests varying between unimodal and multimodal SSDs, the thresholds for applying different fitting techniques is also poorly understood.Landscape-scale evaluation of modality in predicting SSD in mixedwood stands can provide valuable insights into the complexity of these forests.
In this paper, we evaluate the efficacy of ALS metrics to first differentiate plots with unimodal and multimodal SSDs and then to predict parameters of those distributions.Using both ground plot measurements and ALS estimates of plot structural characteristics (e.g., height or age), we investigate the capacity of ALS metrics to distinguish between areas with single or multimodal distributions and then apply ALS to predict SSDs with the distribution most appropriate to the characterized distribution type.We then compare ALS parameter estimates to fitted parameters and ground measured SSDs.Finally, we discuss the applications of this methodology and explain how estimates could be predicted across an entire area of interest.

Study Area
The study area is an actively managed boreal mixedwood forest near Lesser Slave Lake in Alberta, Canada (Figure 1).The area is approximately 700,000 ha in size.Ten tree species were present, with white spruce (Picea glauca), black spruce (Picea mariana), trembling aspen (Populus tremuloides), and lodgepole pine (Pinus contorta) as the most common.The study area covers three natural subregions of Alberta-Central Mixedwood, Lower Foothills, and Upper Foothills.These regions receive approximately 600 mm of annual precipitation and have mean summer and winter temperatures of 20 °C and −21 °C, respectively [30].Common disturbances include timber harvesting and fires, which are slightly less common than in other forested regions in Alberta.Ecology varies from mesic to wetland areas and features some of the most productive timber-producing areas in Alberta [28].

Ground Plot Data
The permanent sample plots (PSPs) employed (Table 1; Figure 1) were a component of the over 650 PSPs established in Alberta since 1960 [31].The PSPs were established to help understand the stand dynamics in all forest types in Alberta.Measurements included tree height, height to living

Ground Plot Data
The permanent sample plots (PSPs) employed (Table 1; Figure 1) were a component of the over 650 PSPs established in Alberta since 1960 [31].The PSPs were established to help understand the stand dynamics in all forest types in Alberta.Measurements included tree height, height to living crown, DBH, species, and crown class for all trees on the plot >7 cm DBH.In addition, cores were taken from trees outside of a sample plot to assess the age structure of the stand.The PSPs are fixed-radius plots with a radius of 11.28 m (400 m 2 ).Measurements in the study area were taken between 2006 and 2007.Only the PSPs that had 20 or more trees (n = 71) were used in the modeling, in order to be effective in predicting parameters for SSD functions that were representative of ground measured SSDs [32].The bimodality coefficient [33] is a statistical approach to assessing a stand's modality.It has also been used in botany [34] and psychology [10] and provides a measure from 0 (perfect unimodality) to 1 (perfect bimodality).A critical value of 5/9 (~0.5556) is used to distinguish bimodal (>5/9) from unimodal (<5/9) distributions [10].We applied the bimodality coefficient to 71 PSPs and identified 23 (32%) as bimodal and 48 as unimodal with respect to SSD.

ALS Data and Metrics
ALS data were acquired for the study area between 2006 and 2008, with most coverage occurring in 2007.The data were collected with an Optech ALTM 3100 sensor flying at a height of either 1250 m or 1400 m, and scan frequencies of 30 Hz and 33 Hz depending on the year flown (Table 2).The result was complete coverage of the study area with an average point density of 1.5 returns/m 2 .ALS data processing began with separating point clouds into ground and non-ground returns based on adaptive Triangulated Irregular Network (TIN) models [35].Next, point clouds were normalized to heights above the ground surface before being clipped to the extent of sample plots.Metrics describing the vertical distribution of returns in each plot were calculated based on normalized ALS point clouds using FUSION [36] as well as the statistical software R [37] with the lidR package [38].A suite of these metrics was selected to fully characterize the height, variability, cover, and structure of the area corresponding to each PSP (Table 3).ALS metrics were separated into categories of similar types: height (e.g., height of 50th percentile), cover (e.g., percent above 2 m), variance (e.g., standard deviation), and canopy structure (e.g., rumple).Filling ratio (FR) is a proportion of filled voxels under the canopy [40].Voxels are volumetric pixels-cubic bins of a pre-defined size (e.g., 1 m × 1 m × 1 m), which, when stacked, cover the entire three-dimensional extent of the ALS point cloud [44].
where V VEG is the volume of vegetation (represented as the volume of voxels with returns), H MAX ij is the maximum voxel height for ij, and H G ij is the ground height of point ij (a value of 0 in normalized point clouds).

Analysis Approach
Figure 2 summarizes the workflow applied.The measured trees within each ground plot were first combined into 2-cm diameter classes and classified as either unimodal or bimodal using the bimodality coefficient.Then, various stand characteristics, measured on plots and predicted using ALS, were assessed for their ability to identify the plots as either unimodal or bimodal.Once the best ALS metric for identification was determined, it was used to categorize plots for estimation of SSD parameters by ALS.Field-based SSDs on classified plots were used as response data for prediction with ALS metrics.Each of these steps is described in further detail below.

Differentiation of Modality in Stem Size Distributions
Stands classified as multimodal are thought to be highly variable and structurally heterogeneous, and this heterogeneity has been quantified in different ways (Table 4).We examined how effective each of the characteristics identified in Table 4 were for differentiating between unimodal and bimodal SSD using both ground measurements and ALS-derived predictions of the five characteristics for differentiation.In an operational context, the use of a single ALS metric would be advantageous if it could be used as an effective heuristic to identify unimodal and bimodal grid cells, because a single metric would be available wall-to-wall (wherever ALS data is acquired), would be generated Forests 2018, 9, 95 6 of 14 as a standard preliminary processing step for an area-based approach (described below), and would require no ground samples or additional modelling.First, as a baseline, five ground-measured characteristics were assessed for their ability to discriminate unimodal and bimodal plots.Next, using an area-based approach, predictive models were developed to estimate each of the five ground-measured characteristics in Table 4 using ALS metrics (Table 3) as predictors and the ground-measured characteristics as response variables.Models were built using stepwise linear regression, and final models were selected based on combinations of up to three metrics (to avoid overfitting) that captured the most variation in the sample population.Predictor variables were selected so that those showing a strong correlation (r > 0.8; [46]) with each other or coming from the same category of descriptors were not included in the same model.The accuracy of area-based models for each of the aforementioned characteristics was assessed in terms of an adjusted r 2 and relative Root Mean Square Error (RMSE) for all sample plots.Finally, individual ALS metrics were used to differentiate SSD modality.First, as a baseline, five ground-measured characteristics were assessed for their ability to discriminate unimodal and bimodal plots.Next, using an area-based approach, predictive models were developed to estimate each of the five ground-measured characteristics in Table 4 using ALS metrics (Table 3) as predictors and the ground-measured characteristics as response variables.Models were built using stepwise linear regression, and final models were selected based on combinations of up to three metrics (to avoid overfitting) that captured the most variation in the sample population.
Forests 2018, 9, 95 7 of 14 Predictor variables were selected so that those showing a strong correlation (r > 0.8; [45]) with each other or coming from the same category of descriptors were not included in the same model.The accuracy of area-based models for each of the aforementioned characteristics was assessed in terms of an adjusted r 2 and relative Root Mean Square Error (RMSE) for all sample plots.Finally, individual ALS metrics were used to differentiate SSD modality.

Accuracy of Modality Differentiation
Modality differentiation was assessed by determining the overall accuracy of the classification that each measure produced.The classification accuracy was defined as the percentage of all PSPs that were correctly classified as either unimodal or bimodal.This was assessed for each characteristic and ALS metric, and allowed for consistent comparison between structural characteristics and individual metrics.A successful classification was defined as a statistically significant improvement from no classification (i.e. if we assume all plots are unimodal).The exact binomial test was used to assess the statistical difference between each classification method and the classification of all plots as unimodal [46].

Predictive Modeling of SSD Parameters Using ALS Metrics
Once a SSD of a plot was classified as either unimodal or bimodal using the most accurate ALS model, a structurally appropriate distribution function was fit to the ground-measured SSD (Figure 3).As complete data in the study area exists for trees > 7 cm DBH, a truncated Weibull distribution was used for describing the SSD in unimodal stands, with the truncation point set at 7 cm [47].While nonparametric imputations such as k-NN and Random Forest have been used to predict SSDs [48], these typically require large amounts of samples to be taken [49].Instead, a Finite Mixture Model (FMM), which applies k separate distributions to k components of data that are split at statistical breakpoints, where k is the number of modes in the data [14], was used for the bimodal distributions (i.e., k = 2).Similar to Thomas et al. [15], separate Weibull distributions were fit using MLE.Once the parameters of the appropriate distributions were estimated from the measured DBHs, the parameters were used to develop area-based models from the ALS metrics in order to estimate the PDF parameters and the SSD across the management area.assess the statistical difference between each classification method and the classification of all plots as unimodal [47].

Predictive Modeling of SSD Parameters Using ALS Metrics
Once a SSD of a plot was classified as either unimodal or bimodal using the most accurate ALS model, a structurally appropriate distribution function was fit to the ground-measured SSD (Figure 3).As complete data in the study area exists for trees > 7 cm DBH, a truncated Weibull distribution was used for describing the SSD in unimodal stands, with the truncation point set at 7 cm [48].While nonparametric imputations such as k-NN and Random Forest have been used to predict SSDs [49], these typically require large amounts of samples to be taken [50].Instead, a Finite Mixture Model (FMM), which applies k separate distributions to k components of data that are split at statistical breakpoints, where k is the number of modes in the data [14], was used for the bimodal distributions (i.e., k = 2).Similar to Thomas et al. [15], separate Weibull distributions were fit using MLE.Once the parameters of the appropriate distributions were estimated from the measured DBHs, the parameters were used to develop area-based models from the ALS metrics in order to estimate the PDF parameters and the SSD across the management area.

Evaluation of SSD Parameters Using the Error Index (EI)
To examine the fit of the ALS-predicted SSD to the original measured tree stem DBHs, the Error Index (EI) proposed by Reynolds et al. [51] was used.The EI reports the sum of observed differences in each class as a proportion of the number of trees at a site.The EI is a frequently used method of evaluating SSD functions, as it allows for comparison across different fitting techniques (e.g., [52]) or PDFs (e.g., [32]).Two different EI calculations were used-one to compare fits of ground-measured distributions (EIG, Equation 2) and one to compare fits of ALS-estimated distributions (EIALS, Equation 3):

Evaluation of SSD Parameters Using the Error Index (EI)
To examine the fit of the ALS-predicted SSD to the original measured tree stem DBHs, the Error Index (EI) proposed by Reynolds et al. [50] was used.The EI reports the sum of observed differences in each class as a proportion of the number of trees at a site.The EI is a frequently used method of evaluating SSD functions, as it allows for comparison across different fitting techniques (e.g., [51]) or PDFs (e.g., [32]).Two different EI calculations were used-one to compare fits of ground-measured distributions (EI G , Equation 2) and one to compare fits of ALS-estimated distributions (EI ALS , Equation ( 3)): where f REFi is the measured frequency in DBH class i, f PDFi is the PDF-derived frequency in DBH class i, f ALSi is the estimated frequency in DBH class i, n REFi is the total number of measured stems in class i, and n ALS is the total number of estimated stems in class i. EI ALS was determined following Packalén and Maltamo [19], with n ALS determined using an area-based approach and the same variables used for the PDF parameter estimation.Both error indices range from 0, indicating a perfect fit, to 200, indicating non-overlapping distributions.

Differentiation of Modality in Stem Size Distributions
Most of the metrics and models were successful in differentiating plots into either unimodal or bimodal distributions (Table 5).SD DBH and the SD H had the highest overall classification accuracy (70.4% and 67.6%, respectively) of the ground-measured metrics, while the SD A and % dominant species had the lowest accuracy (both 59.2%).Characteristics related to tree size variability (SD H , SD DBH , D/H) were most accurately predicted by ALS (Adj.r 2 > 0.6), while as expected, characteristics not related to tree dimensions (SD A , % dominant species) had poorest predictions (Adj.r 2 < 0.2).In terms of their capacity to discriminate modality, the best ALS-predicted characteristic was SD DBH (74.7%), while the poorest was % dominant species (63.4%).Of the individual ALS metrics used to differentiate modality, the variance of ALS return heights was the most accurate (77.5%) and provided the most accurate differentiation overall.The variance of ALS return heights was therefore selected for differentiating modality for the remainder of the study.

Predictive Modeling of SSD Parameters Using ALS Metrics
The adjusted r 2 values for the predicted Weibull and FMM parameters were similar (0.5-0.6) with the exception of the unimodal shape parameter and the shape parameter for the second group of the FMM (0.3925 and 0.2019, respectively; Table 6).Although each of the metrics in Table 3 was used in at least one predictive model, some appeared more frequently than others.The most frequently used metrics were P95, Kurtosis, % All Returns Above 2 m, and the Filling Ratio.

Accuracy of Predicted Distributions
Using a mixture model on bimodal plots resulted in a higher accuracy than using a unimodal distribution on all plots for both ground-measured and ALS-predicted parameters (Table 7; Figure 4).For ground-measured plots, the mean EI G was 28.20 using mixture models and unimodal distributions when appropriate, while only using a unimodal distribution would have resulted in an EI G value of 31.24.Similarly, the mean EI ALS was 49.13 after differentiating modality, while predicting only a unimodal distribution on all plots would have resulted in a mean EI ALS value of 51.31.Plots deemed bimodal based on ground measurements and ALS predictions had higher mean error values than unimodal plots, with a difference of 9.01 between EI G values and 19.35 between EI ALS values.

Differentiation of Modality in Stem Size Distributions
The fact that neighboring stands in the boreal mixedwood forest can have vastly different structures requires flexible and spatially detailed approaches to SSD estimation.Field campaigns to measure SSD across large, diverse, and often remote areas are not feasible given constraints on time and resources.In addition, a priori knowledge of an appropriate PDF to use for fitting the SSD would be valuable when operating in structurally diverse areas.The methodology outlined in this study provides the ability to quickly and effectively characterize diverse forests over large spatial extents by providing detailed measures of the vertical distribution of vegetation over large areas using ALS data.
The ground-measured variables best able to differentiate bimodal distributions were those relating to tree sizes, such as SDH and SDDBH.This is consistent with estimating SSD, which represents the variation in tree DBH.If a stand has highly variable tree sizes, there will be a correspondingly large variance in DBHs and heights, likely reflecting in a multimodal SSD.Age variability and species mixtures provided less accurate differentiations, as maturing stand structure depends more on disturbance trends and ecological legacies than species or age differences [16,17].
Likewise, the most accurate ALS predictions of plot characteristics came from those related to tree size.This was expected, as ALS metrics best characterize physical structure and have more difficulty in estimating intrinsic characteristics such as the age or species of a tree.While attributes such as the percent dominant species and the SD of ages can be used to classify bimodal plots at the ground level, they are not predicted very accurately with ALS, which limits their use at an operational capacity.The variance of ALS heights performed the best in classifying bimodal plots and was chosen as the preferred determinant of modality in this study.This metric is commonly calculated as part of the standard suite of ALS metrics generated from software packages such as LAStools [53] and FUSION [37].Consequently, it is easily generated, accessible, and readily interpretable as a heuristic for distinguishing modality.While alternative approaches to modality characterization could include

Differentiation of Modality in Stem Size Distributions
The fact that neighboring stands in the boreal mixedwood forest can have vastly different structures requires flexible and spatially detailed approaches to SSD estimation.Field campaigns to measure SSD across large, diverse, and often remote areas are not feasible given constraints on time and resources.In addition, a priori knowledge of an appropriate PDF to use for fitting the SSD would be valuable when operating in structurally diverse areas.The methodology outlined in this study provides the ability to quickly and effectively characterize diverse forests over large spatial extents by providing detailed measures of the vertical distribution of vegetation over large areas using ALS data.
The ground-measured variables best able to differentiate bimodal distributions were those relating to tree sizes, such as SD H and SD DBH .This is consistent with estimating SSD, which represents the variation in tree DBH.If a stand has highly variable tree sizes, there will be a correspondingly large variance in DBHs and heights, likely reflecting in a multimodal SSD.Age variability and species mixtures provided less accurate differentiations, as maturing stand structure depends more on disturbance trends and ecological legacies than species or age differences [16,17].
Likewise, the most accurate ALS predictions of plot characteristics came from those related to tree size.This was expected, as ALS metrics best characterize physical structure and have more difficulty in estimating intrinsic characteristics such as the age or species of a tree.While attributes such as the percent dominant species and the SD of ages can be used to classify bimodal plots at the ground level, they are not predicted very accurately with ALS, which limits their use at an operational capacity.The variance of ALS heights performed the best in classifying bimodal plots and was chosen as the preferred determinant of modality in this study.This metric is commonly calculated as part of the standard suite of ALS metrics generated from software packages such as LAStools [52] and FUSION [36].Consequently, it is easily generated, accessible, and readily interpretable as a heuristic for distinguishing modality.While alternative approaches to modality characterization could include logistic regression or other modeling techniques, the parsimony, consistency, and transferability of a single metric makes our approach more applicable in other study areas and research.Further work could investigate more detailed ALS-based stratification of the study area to appropriately model other forest characteristics in addition to SSD.

Predictive Modeling of SSD Parameters Using ALS Metrics
The parameters of the Weibull and FMM distributions were predicted well using ALS-derived metrics.Thomas et al. [15] predicted Weibull and FMM parameters using ALS in a similar study area and achieved similar or slightly more accurate results for parameter predictions.However, their study first stratified by species and structural groups and predictive models included up to seven variables, whereas we used no more than three input variables in our models and did not stratify by species groups.Applying predictions across the landscape using the approach of Thomas et al. [15] would require the availability of reliable species information at the same spatial resolution as the ALS data, which can be difficult or expensive to acquire.Likewise, the current capacity of ALS does not allow for accurate and spatially detailed species characterization, making stratification by species often not feasible.
There was a large difference in prediction accuracy among parameters of the same model.Given the pulse density of our ALS data and the number of sample plots that were available, it was difficult to accurately predict all five FMM parameters, and these distributions had a wide range of resulting curves.In a few cases, the resulting distribution was one that imprecisely characterized the ground-measured SSD.However, ALS data with higher point densities should be able to capture more variation within sample plots and more accurately characterize bimodal plots.

Accuracy of Predicted SSDs
The aforementioned discrepancies in prediction accuracy of SSD parameters likely compounded the errors of distributions fitted to ALS data.Thus, the mean EI ALS value for bimodal plots was slightly higher than that for unimodal plots.However, the methodology used in this study produced more accurate results in terms of both EI G and EI ALS than if all plots had been classified as unimodal.The difference in EI ALS values was relatively low for predicted distributions and slightly higher for measured distributions; this suggests that the accuracy of a predicted SSD decreases with decreasing parameter prediction accuracy.Thomas et al. [15] did not report EI values; however, a study by Tompalski et al. [42] reported similar EI ALS values to those reported herein when predicting SSD for unimodal distributions.Tompalski et al. [42] scaled EI values by 0.5 while we used 100 (Equation 3).When correcting for this difference in scale, our mean EI ALS values were slightly more accurate than those reported by Tompalski et al. [42], whose mean value was 71.6.

Model Application
Unless a stand is small or completely homogeneous, a single plot-level SSD will likely not be representative of SSD for the stand [53].Therefore, techniques for the aggregation of predictions from cell-level SSD are necessary to generate a stand-level SSD.One such approach involves summing predicted SSDs from each grid cell composing a delineated stand [54].More complex approaches involve multidimensional scaling [55], in which an estimator can be used for extrapolation to larger units, or segmenting areas into smaller units such as microstands, which are areas grouped by similar ALS-predicted attributes such as volume and height [56].If stand-level predictions are the desired result, a final aggregation step should be used to scale up from cell-level predictions; however, this was beyond the scope of the current study and SSD predictions remained at the cell level.

Conclusions
The structurally complex SSDs that exist in the boreal mixedwood forest should be fit with correspondingly complex distribution models.The difference in structures among stands requires detailed, landscape-level information to guide the fitting and modeling process.In order to meet the scope and detail needed for accurate forest management decisions, we used ALS as a means of differentiating and predicting SSD parameters in a boreal mixedwood forest.The differentiation step allowed us to fit structurally appropriate SSDs to respective stands and allowed for more robust characterizations of SSD than using a single model for the entire study area.For forest professionals who rely on detailed stand-level information, differentiating bimodal areas and their subsequent characterization by FMMs should provide insights into stand characteristics that would lead to more informed decisions and more accurate understanding of stand structure in complex habitats.

Figure 1 .
Figure 1.Map of the Slave Lake study area and sample plots (n = 71).

Figure 1 .
Figure 1.Map of the Slave Lake study area and sample plots (n = 71).

Figure 2 .
Figure 2. Workflow for differentiating bimodal plots and estimating stem size distribution (SSD) parameters with ALS.

Figure 2 .
Figure 2. Workflow for differentiating bimodal plots and estimating stem size distribution (SSD) parameters with ALS.

Figure 3 .
Figure 3. Weibull distributions (blue) best fit unimodal stands (left), while more complex distributions such as a Finite Mixture Model (green) best fit multimodal stands (right).

Figure 3 .
Figure 3. Weibull distributions (blue) best fit unimodal stands (left), while more complex distributions such as a Finite Mixture Model (green) best fit multimodal stands (right).

Figure 4 .
Figure 4. Examples of measured and predicted curves following correct and incorrect differentiation on four plots.

Figure 4 .
Figure 4. Examples of measured and predicted curves following correct and incorrect differentiation on four plots.

Table 3 .
ALS metrics used to predict probability density function (PDF) parameters.

Table 4 .
Published predictors of multimodal diameter distributions and how they are used to classify bimodal plots.DBH, diameter at breast height.

Table 5 .
Accuracy of bimodal plots classification using ground measurements, ALS predictions, and corresponding ALS metrics.indicates a significant improvement from no classification (i.e.assuming all plots are unimodal).SD A is the standard deviation of ages, D/H is the density divided by top height, SD H is the standard deviation of ages, and SD DBH is the standard deviation of DBH's. *

Table 6 .
Prediction accuracy of SSD parameters for unimodal (n = 48) and bimodal (n = 23) plots as differentiated by ALS.

Table 7 .
Measured (EI G ) and predicted (EI ALS ) values on sample plots, showing differences in unimodal and bimodal plots.