# Model-Based Estimation of Forest Inventory Attributes Using Lidar: A Comparison of the Area-Based and Semi-Individual Tree Crown Approaches

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials

#### 2.1. Study Area

#### 2.2. Field Data

#### 2.3. Lidar Data Acquisition and Processing

## 3. Methods

#### 3.1. Constructing Population Units

#### 3.1.1. Grid Cells

#### 3.1.2. Segments

#### 3.2. Unit-Level Model

#### 3.3. Target Parameters

#### 3.4. Predictions for Target Parameters

#### 3.5. Model Selection

#### 3.6. Mean Squared Error Estimators

## 4. Results

#### 4.1. Selected Models

#### 4.2. Estimation for the Study Region

#### 4.3. Estimation for Stands

## 5. Discussion

#### 5.1. Contribution of Error Components

#### 5.2. Peculiarities of a Segment Population

#### 5.3. Implications for Forest Management Inventories

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Target Parameters and Their Components | |||

Notation | Description | Notation | Description |

${\mu}_{\alpha}$ | $\mathrm{The}\text{}\mathrm{target}\text{}\mathrm{parameter}\text{}\mathrm{for}\text{}\mathrm{an}\text{}\mathrm{area}\text{}\mathrm{indexed}\text{}\mathrm{by}\text{}\alpha $$\text{}(\tau $$\text{}\mathrm{for}\text{}\mathrm{the}\text{}\mathrm{study}\text{}\mathrm{region},\text{}i$ for stand). | ${h}_{ij}$ | $\mathrm{The}\text{}\mathrm{area}\text{}\mathrm{occupied}\text{}\mathrm{by}\text{}\mathrm{the}\text{}j$th population unit in the $i$th area in hectares. |

${h}_{i\cdot}$ | $\mathrm{The}\text{}\mathrm{sum}\text{}\mathrm{of}\text{}\mathrm{the}\text{}\mathrm{areas}\text{}\mathrm{of}\text{}\mathrm{the}\text{}\mathrm{population}\text{}\mathrm{in}\text{}\mathrm{the}\text{}i$th area in hectares. | ${h}_{\cdot \cdot}$ | The sum of the areas of all population units in the entire study region in hectares. |

Models and Their Components | |||

Notation | Description | Notation | Description |

$\mathit{y}$ | A vector of observable quantities of the response variable for all population units. | $\mathit{e}$ | A vector of residuals. |

$\mathit{X}$ | A design matrix of lidar covariates and an intercept for all population units. | $\mathit{\beta}$ | A vector of regression coefficients. |

$\mathit{Z}$ | A matrix that assigns population units to areas. | $\mathit{G}$ | The variance-covariance matrix of $\mathit{v}$. |

$\mathit{v}$ | A vector of realized random effects. | $\mathit{R}$ | The variance-covariance matrix of $\mathit{e}$. |

${\sigma}_{e}^{2}$ | The residual variance. | ${\sigma}_{v}^{2}$ | The random-effect variance. |

${N}_{i}$ | $\mathrm{The}\text{}\mathrm{number}\text{}\mathrm{of}\text{}\mathrm{population}\text{}\mathrm{units}\text{}\mathrm{in}\text{}\mathrm{stand}\text{}i$. | ${n}_{i}$ | $\mathrm{The}\text{}\mathrm{sample}\text{}\mathrm{size}\text{}\mathrm{in}\text{}\mathrm{stand}\text{}i$, i.e., the number of field plots (ABA) or the number of segments (s-ITC) |

Results Assessment Measures | |||

Notation | Description | Notation | Description |

${\widehat{RMSE}}_{\alpha}$ | The model-based root mean squared error for the predicted target parameter. | ${\delta}_{RMSE,\alpha}$ | The relative change between s-ITC and ABA model-based root mean squared errors. |

$C{I}_{\alpha}$ | The approximate confidence interval for the predicted target parameter. | $\widehat{C{V}_{\alpha}}$ | The estimated coefficient of variation of the predicted target parameter. |

## Appendix B

Predictor Name | Description |
---|---|

p_1, p_10, p_20, p_25, p_30, p_40, p_50, p_60, p_70, p_75, p_80, p_90, p_95, p_99 | The percentile of the z-dimension indicated by the trailing number. For example, p_95 describes the elevation at which 95% of the lidar points fall below. |

max_z | The maximum z value. |

min_z | The minimum z value. |

mean_z | The mean z value. |

stddev_z | The standard deviation of the z values. |

var_z | The variance of the z values. |

mean_z_sq | The square of the mean z value. |

vol_cov | The product of the mean z value and the pct_r_1_above_2 metric. |

pct_all_above_2 | The proportion of all returns above 2 m. |

pct_all_above_mean | The proportion of all returns above the mean z value. |

pct_r_1_above_2 | The proportion of first returns above 2 m. |

pct_r_1_above_mean | The proportion of all returns above 2 m. |

r_1, r_2, r_3, r_4 | The number of returns indicated by the trailing number. For example, r_1 indicates the number of first returns. |

area | The area of the population unit (only included for s-ITC models) |

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**Figure 1.**Vicinity map of the Panther Creek watershed situated in the northwest of Oregon, USA. Forest stands are colored according to omitted, sampled and unsampled. Field plots are indicated in blue. Universal Transverse Mercator (UTM) Zone 10 N coordinates are given at the margins of the figure, with UTM grid lines demarcated in gray.

**Figure 2.**Canopy height model (blue to green background) displayed underneath delineated segments (black lines) and grid cells (white lines). Two example field plots are shown (red circles) with included segments shaded in orange. The green line in the center represents a stand delineation.

**Figure 3.**Pearson’s standardized residuals of ABA (green) and s-ITC models (orange) for the four forest attributes.

**Figure 4.**Predictions of area-level parameters for VOL, BA, DEN, and QMD for s-ITC models and ABA models. Whether or not the stand had at least one field plot is indicated by the color of the point. The confidence intervals are shown as horizontal bars for s-ITC and vertical bars for ABA.

**Figure 5.**The ratio of the stand-specific sample size of segments to field plots plotted against ${\delta}_{RMSE,i}$, for VOL, BA, DEN, and QMD in sampled stands. A simple linear regression (red) was fit to each attribute to demonstrate the trend, with slopes and intercepts reported for each attribute.

**Figure 6.**Error components for segment and cell models for VOL, BA, DEN, and QMD in sampled stands. Component rows are ranked by the ratio of the number of segments to the number of field plots (as in Figure 5) in ascending order.

**Figure 7.**The error component ${g}_{1i}$ for all sampled stands in the study area plotted against the stand-specific sample size ${n}_{i}$. Note that ${g}_{1i}$ for unsampled stands, not displayed in this figure, was equal to the respective random effect variance estimate (Table 4).

**Figure 8.**The error component ${g}_{2i}$ for all stands in the study area plotted against the stand-level prediction ${\widehat{\mu}}_{i}$. The prediction of the study region mean, ${\widehat{\mu}}_{\tau}$, is shown for the ABA as a dashed vertical line as a reference.

**Table 1.**Distribution of the number of stands by stand-specific sample size. A given column indicates the number of stands that contain the number of field plots indicated in the top row. For example, the first column indicates there are 94 stands with 0 field plots.

Number of Field Plots | 0 | 1 | 2 | 3 | 4 | 5 | 7 |

Number of Stands | 94 | 18 | 6 | 4 | 1 | 5 | 1 |

Term | Definition |
---|---|

Grid cell | A square area 0.08 hectares in size. The population unit for the ABA. |

Population | The set of all geographical units, either grid cells for the ABA or segments for the s-ITC approach, used in the analysis. |

Segment | An irregular polygon of varying size produced by a segmentation procedure. The population unit for the s-ITC approach. |

Stand | An area of homogeneous forest structure used as a small area of interest. If a stand contains at least one field plot it is considered “sampled”, if it does not it is considered “unsampled”. Stands are indexed by $i$. |

Stand-specific sample size | The sample size for a particular stand, denoted by ${n}_{i}$. For the area-based approach, this refers to the number of field plots in the stand. For the s-ITC this refers to the number of sample segments in the stand, i.e., those segments contained in the field plots (Figure 2). |

Study region | The set of 129 stands included in the analysis. |

**Table 3.**Sample means, standard deviations, and ranges for forest inventory attributes for the sample of field plots and the sample of segments. Note: for the mean and the standard deviations, a weighted mean was calculated for each plot using segment and plot areas as weights, respectively, and means and standard deviations of these scaled observations are reported in the table.

Attribute | Source | Mean | Std. Dev. | Minimum | Maximum |
---|---|---|---|---|---|

VOL $({\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$) | Field Plots | 601.3 | 389.4 | 3.3 | 1733.3 |

Segments | 542.6 | 340.8 | 0.0 | 1975.3 | |

BA $({\mathrm{m}}^{2}\mathrm{h}{\mathrm{a}}^{-1}$) | Field Plots | 48.7 | 23.7 | 1.8 | 102.0 |

Segments | 46.0 | 22.3 | 0.0 | 148.5 | |

DEN $(\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{s}\mathrm{h}{\mathrm{a}}^{-1}$) | Field Plots | 31.9 | 12.8 | 160.6 | 1519.9 |

Segments | 30.1 | 12.1 | 0.0 | 4019.9 | |

QMD $(\mathrm{c}\mathrm{m}$) | Field Plots | 659.3 | 277.2 | 5.3 | 69.1 |

Segments | 687.8 | 302.2 | 0.0 | 82.8 |

Attribute | Model | Predictor | Coefficient | Std. Error | $\mathit{\eta}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{v}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{e}}$ |
---|---|---|---|---|---|---|---|

VOL (${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$) | ABA | Intercept mean_z mean_z_sq | −8.09 10.50 0.83 | 12.90 4.02 0.15 | 0.5 | 0.00 ^{1} | 7.37 |

s-ITC | Intercept mean_z_sq | −20.27 1.25 | 16.66 0.04 | 0.5 | 61.65 | 8.23 | |

BA (${\mathrm{m}}^{2}\mathrm{h}{\mathrm{a}}^{-1}$) | ABA | Intercept P_60 | 0.74 1.98 | 1.50 0.08 | 0.5 | 0.00 ^{1} | 2.33 |

s-ITC | Intercept vol_cov | −2.24 2.50 | 1.69 0.08 | 0.5 | 5.7 | 3.53 | |

DEN ($\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{s}\mathrm{h}{\mathrm{a}}^{-1}$) | ABA | Intercept P_80 vol_cov | 935.42 −33.84 34.32 | 77.92 7.06 9.21 | 0.0 | 73.17 | 228.14 |

s-ITC | Intercept canopy_relief_ratio P_95 | 210.17 1301.40 −10.26 | 96.32 152.93 3.41 | 0.5 | 175.32 | 614.55 | |

QMD ($\mathrm{c}\mathrm{m}$) | ABA | Intercept canopy_relief_ratio P_60 | 16.79 −28.59 1.26 | 3.57 7.88 0.08 | 0.0 | 1.43 | 5.83 |

s-ITC | Intercept P_80 Pct_r_1_above_2m | 1.68 0.99 3.09 | 0.98 0.05 0.87 | 0.5 | 2.72 | 1.54 |

^{1}Indicates a value less than 0.001.

**Table 5.**Predictions of forest attributes and error components at the scale of the study region. Note that, for all models, ${g}_{1\tau}$ and ${g}_{3\tau}$ were < 0.01 and are not included in this table.

Attribute | ${\widehat{\mathit{\mu}}}_{\mathit{\tau}}$ | ${\mathit{g}}_{2\mathit{\tau}}$ | ${\widehat{\mathit{C}\mathit{V}}}_{\mathit{\tau}}$ | ${\widehat{\mathit{R}\mathit{M}\mathit{S}\mathit{E}}}_{\mathit{\tau}}$ | ${\mathit{\delta}}_{\mathit{R}\mathit{M}\mathit{S}\mathit{E},\mathit{\tau}}$ | ||||
---|---|---|---|---|---|---|---|---|---|

ABA | s-ITC | ABA | s-ITC | ABA | s-ITC | ABA | s-ITC | ||

VOL $({\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$) | 395.73 | 417.22 | 159.94 | 166.05 | 3.12% | 3.09% | 12.65 | 12.89 | 1.90% |

BA $({\mathrm{m}}^{2}\mathrm{h}{\mathrm{a}}^{-1}$) | 35.32 | 36.03 | 1.06 | 1.50 | 2.91% | 3.40% | 1.22 | 1.03 | −15.57% |

$\mathrm{DEN}\text{}(\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{s}\mathrm{h}{\mathrm{a}}^{-1}$) | 696.27 | 701.13 | 1225.02 | 1690.60 | 5.03% | 5.86% | 35.00 | 41.12 | 17.49% |

QMD $(\mathrm{c}\mathrm{m}$) | 25.37 | 24.97 | 0.77 | 0.36 | 3.47% | 2.39% | 0.87 | 0.60 | −31.03% |

**Table 6.**Medians of estimated mean squared errors for stands partitioned by sampled (S) and unsampled (U) stands.

Attribute | Sampled | $\mathbf{Median}\text{}{\widehat{\mathit{C}\mathit{V}}}_{\mathit{i}}$ | $\mathbf{Median}\text{}{\widehat{\mathit{R}\mathit{M}\mathit{S}\mathit{E}}}_{\mathit{i}}$ | $\mathbf{Median}\text{}{\mathit{\delta}}_{\mathit{R}\mathit{M}\mathit{S}\mathit{E},\mathit{i}}$ | ||
---|---|---|---|---|---|---|

ABA | s-ITC | ABA | s-ITC | |||

VOL (${\mathrm{m}}^{3}\mathrm{h}{\mathrm{a}}^{-1}$) | S | 2.3% | 4.9% | 9.7 | 29.9 | 208.2% |

U | 4.5% | 18.7% | 9.4 | 48.4 | 414.9% | |

BA (${\mathrm{m}}^{2}\mathrm{h}{\mathrm{a}}^{-1}$) | S | 2.3% | 6.1% | 1.1 | 2.5 | 127.3% |

U | 4.6% | 21.0% | 1.2 | 5.4 | 350.0% | |

DEN $(\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{s}\mathrm{h}{\mathrm{a}}^{-1}$) | S | 4.8% | 3.7% | 32.3 | 22.8 | −29.4% |

U | 12.1% | 26.4% | 89.4 | 183.0 | 104.7% | |

QMD ($\mathrm{c}\mathrm{m}$) | S | 3.5% | 5.8% | 1.2 | 1.7 | 41.7% |

U | 10.0% | 12.7% | 1.9 | 2.6 | 36.8% |

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**MDPI and ACS Style**

Frank, B.; Mauro, F.; Temesgen, H.
Model-Based Estimation of Forest Inventory Attributes Using Lidar: A Comparison of the Area-Based and Semi-Individual Tree Crown Approaches. *Remote Sens.* **2020**, *12*, 2525.
https://doi.org/10.3390/rs12162525

**AMA Style**

Frank B, Mauro F, Temesgen H.
Model-Based Estimation of Forest Inventory Attributes Using Lidar: A Comparison of the Area-Based and Semi-Individual Tree Crown Approaches. *Remote Sensing*. 2020; 12(16):2525.
https://doi.org/10.3390/rs12162525

**Chicago/Turabian Style**

Frank, Bryce, Francisco Mauro, and Hailemariam Temesgen.
2020. "Model-Based Estimation of Forest Inventory Attributes Using Lidar: A Comparison of the Area-Based and Semi-Individual Tree Crown Approaches" *Remote Sensing* 12, no. 16: 2525.
https://doi.org/10.3390/rs12162525