# Estimation of Changes of Forest Structural Attributes at Three Different Spatial Aggregation Levels in Northern California using Multitemporal LiDAR

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^{2}

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## Abstract

**:**

## 1. Introduction

^{2}and 900 m

^{2}. A direct application of predictive models will render predictions for grid units of size too small to be considered of interest for reporting in forest inventories. Areas of interest (AOIs) (i.e., the areas for which estimates are needed) are typically geographic units that can vary in size depending on the particular application. For worldwide inventories or inventories over continents or countries, AOIs are typically administrative or political units such as countries or municipalities. In forest management applications, AOIs are typically stands, compartments or even complete forests or landscapes. All these AOIs require spatial aggregation of grid units. However, validations of predictive models in the ABA literature are typically performed using global metrics of model fit, such as the sample-based root mean square error or bias, that provide average measures of uncertainty for predictions made for pixels or plots. These measures of uncertainty derived from the model fitting stage do not directly translate into measures of uncertainty for predictions for AOIs composed of multiple pixels (i.e., countries, municipalities, forests, stands, etc.). In addition, even when considering single pixels, they are not AOI-specific, as they only provide an average value, across the entire population, of the error that can be expected using a given model.

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Sampling Design and Field Data

^{2}) were measured in the field during the summer of 2009 and then remeasured during the summer of 2016. All field plots were located on nodes of the 100 m by 100 m grid of monumented markers at BMEF. Coordinates of the makers were determined using traverse methods and survey grade GPS observations and have an accuracy of 15 cm or better (see [23]). For each of the 26 stands selected for sampling, a node of the BMEF 100 m grid was randomly selected and used as a starting node for a 282 m by 282 m grid formed by selecting every other plot of the 100 m grid moving in the diagonal directions. Field measurements were taken on the nodes of the 282 m by 282 m grid (see Figure 1).

^{2}ha

^{−1}to 25.25 m

^{2}ha

^{−1}. For the remaining 20% of the area, approximately one quarter was not thinned while the other three quarters were thinned to a residual BA that ranged from 6.89 m

^{2}ha

^{−1}to 13.77 m

^{2}ha

^{−1}. Fresh weight of total extractions for the 427.40 hectares subject to thinning was 11,009.38 Mg of logs and 23,164.32 Mg of chipped material.

#### 2.3. LiDAR Data Acquisitions

^{2}covering the entire BMEF. The cell size matched the field plot size and each cell of the grid was considered a population unit, equivalent to the field plots. Predictors and their corresponding acronyms used in further sections are summarized in Table A1.

#### 2.4. AOIs, Target Parameter and Overview of Modelling Strategies

#### 2.5. $\delta $-Modeling Method

#### 2.5.1. Model $\delta $-modeling Method

#### 2.5.2. Target Parameter $\delta $-modeling Method

#### 2.5.3. Model Selection and Estimator $\delta $-modeling Method

#### 2.5.4. MSE Estimators for the $\delta $-modeling Method

#### 2.6. $y$-Modeling Method

#### 2.6.1. Model $y$-modeling Method

#### 2.6.2. Target Parameter $y$-modeling Method

#### 2.6.3. Estimator $y$-modeling Method, and Estimator of the MSE

#### 2.7. Comparison of Methods

#### 2.7.1. General Accuracy Assessment

#### 2.7.2. AOI-specific Comparisons.

#### 2.7.3. Extrapolation to Thinned Stands

## 3. Results

#### 3.1. Selected Models $\delta $-modeling Method and $y$-modeling Method

^{2}), and the exponents of the error variance function were very close to those obtained in [4,17,27] for V, and in [27] for B. For BA, variance of model errors was a function of the percentage of first returns above two meters (PcFstAbv2). Based on the results of the likelihood ratio tests, that for all variables resulted in p-values larger than 0.87, simplified models were selected and used for prediction.

#### 3.2. General Accuracy Assessment and Comparison of Methods

#### 3.3. AOI-Specific Estimates

#### 3.3.1. Entire Study Area

#### 3.3.2. Stands

^{3}ha

^{−1}year

^{−1}for V, of 0.02 to 0.15 m

^{2}ha

^{−1}year

^{−1}for BA and of 0.10 to 0.80 Mg ha

^{−1}year

^{−1}for B. However, for B and V, the $RMS{E}_{y}$ tended to be smaller than $RMS{E}_{\delta}$ while negligible differences between methods were observed for BA (Figure 3 and Figure 4).

#### 3.3.3. Pixel-level

^{3}ha

^{−1}year

^{−1}and 3.30 m

^{3}ha

^{−1}year

^{−1}for V, 0.39 m

^{2}ha

^{−1}year

^{−1}and 0.38 m

^{2}ha

^{−1}year

^{−1}for BA, and 1.67 Mg ha

^{−1}year

^{−1}and 1.65 Mg ha

^{−1}year

^{−1}for B. Mean and median values of $RMS{E}_{y}$ were 2.49 m

^{3}ha

^{−1}year

^{−1}and 2.20 m

^{3}ha

^{−1}year

^{−1}for V, 0.48 m

^{2}ha

^{−1}year

^{−1}and 0.48 m

^{2}ha

^{−1}year

^{−1}for BA and 1.89 Mg ha

^{−1}year

^{−1}and 1.76 Mg ha

^{−1}year

^{−1}for B (Table 5 and Figure A1). Predictions from the $\delta $-modeling method tend to be smoother than predictions from the $y$-modeling method. For all variables, the proportion of pixel-level predictions using the $\delta $-modeling method within the range of values observed for the field plots, was always 99.84% or larger (Figure A2). Considering that these results were obtained in the presence of thinned stands and the relatively small fraction of the forest that was sampled, obtaining less than 0.16% of the predictions outside of the measurement range seems to be a clear sign of over smoothing (see Appendix B).

#### 3.4. Extrapolation to Thinned Stands

^{2}ha

^{−1}to −1.88 m

^{2}ha

^{−1}, which seems to be a very small change in basal area. Estimated changes in BA using the $y$-modeling method ranged from −3.58 m

^{2}ha

^{−1}to −7.01 m

^{2}ha

^{−1}. An advantage of the $y$-modeling method is that it allows obtaining the values of the structural attributes at a given point in time. Using the $y$-model we estimated BA for the thinned stands for 2015. For those stands where thinning prescriptions dictated leaving a residual BA of 17.22 m

^{2}ha

^{−1}to 25.25 m

^{2}ha

^{−1}, estimated BA for 2015 ranged from 19.87 m

^{2}ha

^{−1}to 26.22 m

^{2}ha

^{−1}, which is in accordance with the thinning prescriptions. For the remaining area subject to thinning the estimated BA for 2015 was 17.64 m2 ha

^{−1}, while the prescriptions dictated leaving a residual BA ranging from 6.89 m

^{2}ha

^{−1}to 13.77 m

^{2}ha

^{−1}in 75% of the area and leaving the remaining area untouched. In general, the estimated BA for 2015 are consistent with the prescriptions, which indicates that the $y$-modeling method produces reasonable estimates of BA when extrapolating to the thinned stands. In summary, for the estimation of changes, biases derived from extrapolation seemed to be of larger magnitude for the $\delta $-modeling method although they were also present for the $y$-modeling method.

## 4. Discusion

#### 4.1. General Accuracy Assessment and Comparison of Methods.

^{3}ha

^{−1}year

^{−1}when using the $\delta $-method and 3.76 m

^{3}ha

^{−1}year

^{−1}when using the $y$-method. These values are slightly smaller than the $mRMSE$ obtained by Poudel et al. [8] using the $\delta $-method (4.74 m

^{3}ha

^{−1}year

^{−1}) and two lidar acquisitions separated in time by five years. For B, $mRMSE$ using the $\delta $-method and the $y$-method were, respectively, 1.72 Mg ha

^{−1}year

^{−1}and 1.94 Mg ha

^{−1}year

^{−1}. These values were very close to those reported by Poudel et al. [8] using the $\delta $-method (1.88 Mg ha

^{-1}year

^{-1}) and worse than those reported by Temesgen et al. (1.25 Mg ha

^{−1}year

^{−1}and 1.63 Mg ha

^{−1}year

^{−1}), also using two LiDAR acquisitions separated in time by five years. Values of $RMSE$ for BA were similar to those obtained by Næsset and Gobakken [20] in coniferous forest in Norway, using the $y$-method with log-transformed models and two LiDAR acquisitions that were two years apart from each other. In relative terms, for V and B, the values that we obtained for $mRRMSE$ were considerably larger than those obtained by Poudel et al. [8]. These differences are due to the fact that observed growth rates in Poudel et al [8] are much higher than we observed at BMEF.

#### 4.2. AOI-Specific Estimates

#### 4.2.1. Entire Study Area

^{−1}year

^{−1}. These differences seem to be due to multiple factors such as differences between study areas, changes in live biomass versus changes in standing biomass, time between LiDAR acquisitions and field plot sizes etc. Further investigation is needed to test if the model-based estimators studied here and the GREG estimators in [10] have a similar performance when used under the same conditions.

#### 4.2.2. Stands

#### 4.2.3. Pixel-level

#### 4.3. Advantages of Modeling Alternatives

## 5. Conclusions

- The change of structural attributes and LiDAR auxiliary information are weakly correlated. This weak correlation seems to more evident in BMEF than in previous studies because of the slower growth in the study area and the relatively short lapse of time between LiDAR acquisitions, which indicates that for future studies in similar areas it might be necessary to increase the time lags between LiDAR flights.
- In general, the $\delta $-modeling method was found to be a slightly more accurate alternative to obtain estimates of change for the whole study area; however, the $y$-modeling method was able to produce better estimates at the stand level. In addition, the $y$-modeling method method also seemed to be less prone to extrapolation problems. This indicates that field campaigns for the $\delta $-modeling method have to be carefully designed while the $y$-modeling method might be less sensitive to certain bias problems.
- Despite the weak correlations with the changes in structural attributes, LiDAR auxiliary information allows obtaining estimates of growth for stands that improve over those derived using only field information.
- The large uncertainty observed for pixel-level predictions indicated that high-resolution maps of growth, generated using LiDAR auxiliary information in similar conditions, should be taken as approximated products.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Sets of candidate predictors used in the study. Predictors included in the models to predict structural changes are highlighted with a boldface font. HiD, LoD and RNA represent the high diversity, low diversity and research natural areas respectively.

Description Auxiliary Variables Sets 1, 2 and 3 | Acronym | Description Auxiliary Variables Set 4 | Acronym | ||
---|---|---|---|---|---|

Set 1 Year: 2009 | Set 2 Year: 2015 | Set 3, Difference 2015-2009 | Set 4 | ||

Minimum, maximum, mean, mode, standard deviation, variance, coefficient of variation and interquartile range of the distribution of heights of the point cloud. | Elev_min_{09} | Elev_min_{15} | $\delta $Elev_min_{15-09} | Incoming solar radiation | Solar_radiation |

Elev_max_{09} | Elev_max_{15} | $\delta $Elev_max_{15-09} | |||

Elev_mean_{09} | Elev_mean_{15} | $\delta $Elev_mean_{15-09} | Structural diversity, factor with three levels HiD, LoD and RNA. Coded using two dummy variables. RNA reference level. | HiD | |

Elev_mean^{2}_{09} | Elev_mean^{2}_{15} | $\delta $Elev_mean^{2}_{15-09} | |||

Elev_mode_{09} | Elev_mode_{15} | $\delta $Elev_mode_{15-09} | |||

Elev_stddv_{09} | Elev_stddv_{15} | $\delta $Elev_stddv_{15-09} | LoD | ||

Elev_var_{09} | Elev_var_{15} | $\mathit{\delta}$Elev_var_{15-09} | |||

Elev_CV_{09} | Elev_CV_{15} | $\delta $Elev_CV_{15-09} | Presence absence of prescribed fires. Coded using a dummy variable taking value 1 for stands where prescribed fires are applied and 0 otherwise. | Burned | |

Elev_IQ_{09} | Elev_IQ_{15} | $\delta $Elev_IQ_{15-09} | |||

Elev_AAD_{09} | Elev_AAD_{15} | $\delta $Elev_AAD_{15-09} | |||

Elev_MADmed_{09} | Elev_MADmed_{15} | $\delta $Elev_MADmed_{15-09} | |||

Elev_MADmod_{09} | Elev_MADmod_{15} | $\delta $Elev_MADmod_{15-09} | |||

Percentiles of the distribution of heights of the point cloud. | Elev_P01_{09} | Elev_P01_{15} | $\delta $Elev_P01_{15-09} | ||

Elev_P05_{09} | Elev_P05_{15} | $\delta $Elev_P05_{15-09} | |||

Elev_P10_{09} | Elev_P10_{15} | $\delta $Elev_P10_{15-09} | |||

Elev_P20_{09} | Elev_P20_{15} | $\delta $Elev_P20_{15-09} | |||

Elev_P30_{09} | Elev_P30_{15} | $\delta $Elev_P30_{15-09} | |||

Elev_P40_{09} | Elev_P40_{15} | $\delta $Elev_P40_{15-09} | |||

Elev_P50_{09} | Elev_P50_{15} | $\mathit{\delta}$Elev_P50_{15-09} | |||

Elev_P60_{09} | Elev_P60_{15} | $\delta $Elev_P60_{15-09} | |||

Elev_P70_{09} | Elev_P70_{15} | $\delta $Elev_P70_{15-09} | |||

Elev_P75_{09} | Elev_P75_{15} | $\delta $Elev_P75_{15-09} | |||

Elev_P80_{09} | Elev_P80_{15} | $\delta $Elev_P80_{15-09} | |||

Elev_P90_{09} | Elev_P90_{15} | $\delta $Elev_P90_{15-09} | |||

Elev_P95_{09} | Elev_P95_{15} | $\delta $Elev_P95_{15-09} | |||

Elev_P99_{09} | Elev_P99_{15} | $\delta $Elev_P99_{15-09} | |||

Canopy relief ratio | CRR_{09} | CRR_{15} | $\mathit{\delta}$CRR_{15-09} | ||

Percentage of first (Fst) and all (All) returns above 2 m | PcFstAbv2_{09} | PcFstAbv2_{15} | $\mathit{\delta}$PcFstAbv2_{15-09} | ||

PcAllAbv2_{09} | PcAllAbv2_{15} | $\mathit{\delta}$PcAllAbv2_{15-09} | |||

Ratio all returns above 2 m to first returns | AllAbv2Fst_{09} | AllAbv2Fst_{15} | $\delta $AllAbv2Fst_{15-09} | ||

Percentage of first returns above the mean and mode | PcFstAbvMean_{09} | PcFstAbvMean_{15} | $\delta $PcFstAbvMean_{15-09} | ||

PcFstAbvMode_{09} | PcFstAbvMode_{15} | $\delta $PcFstAbvMode_{15-09} | |||

Percentage of all returns above the mean and mode | PcAllAbvMean_{09} | PcAllAbvMean_{15} | $\delta $PcAllAbvMean_{15-09} | ||

PcAllAbvMode_{09} | PcAllAbvMode_{15} | $\delta $PcAllAbvMode_{15-09} | |||

Ratio of all returns above the mean and mode to number of first returns | AllAbvMeanFst_{09} | AllAbvMeanFst_{15} | $\delta $AllAbvMeanFst_{15-09} | ||

AllAbvModeFst_{09} | AllAbvModeFst_{15} | $\delta $AllAbvModeFst_{15-09} | |||

Proportion of points in the height intervals [0,0.5), [0.5,1), [1,2), [2,4), [4,8) and [8,16) meters. | Prop0_05_{09} | Prop0_05_{15} | $\delta $Prop0_05_{15-09} | ||

Prop05_1_{09} | Prop05_1_{15} | $\delta $Prop05_1_{15-09} | |||

Prop1_2_{09} | Prop1_2_{15} | $\delta $Prop1_2_{15-09} | |||

Prop2_4_{09} | Prop2_4_{15} | $\delta $Prop2_4_{15-09} | |||

Prop4_8_{09} | Prop4_8_{15} | $\delta $Prop4_8_{15-09} | |||

Prop8_16_{09} | Prop8_16_{15} | $\delta $Prop8_16_{15-09} |

## Appendix B

**Figure A1.**Maps of change in V, BA and B and corresponding pixel-level RMSE maps for the $\delta $-modeling method.

**Figure A2.**Comparison pixel-level predictions for V, BA and B using the $\delta $-modeling method and $y$-modeling method predictions for the unsampled stands subject to thinnings are in red. The range of V, BA and B observed in the sample is indicated by the grey ribbons. The proportions, ${P}_{\delta}$ and ${P}_{y}$, of predictions within the range of values observed in the sample, and the correlation between predictions from both methods are indicated in the upper left corner. The proportion of pixels in the thinned stands where the $\delta $-modeling method and $y$-modeling method predict losses (i.e., $P({\widehat{\delta}}_{i,\delta}<0)$ and $P({\widehat{\delta}}_{i,y}<0)$) are indicated on the lower left quadrant of the figure.

## Appendix C

**Figure A3.**Indexes of extrapolation. Average of Mesgaran’s novelty index relative to the mean, ${\overline{NT2}}_{mean}$, for the sampled and not thinned stands (dark blue), unsampled stands not thinned (green) and unsampled and thinned stands (red). The value of this index for the field plots (light blue) provides the baseline value (i.e., the value observed for the sample of field plots).

**Figure A4.**Comparison of density functions for the predictors in the models used to estimate changes in Basal Area using the $\delta $-modeling method and $y$-modeling method in field plots (light blue), sampled and not thinned stands (dark blue), unsampled and not thinned stands (light blue) and unsampled and thinned stands (red). For each group the area of overlap, AO, with the density function for the field plots (green) is provided for each predictor.

**Figure A5.**Comparison of density functions for the predictors in the models used to estimate changes in Biomass using the $\delta $-modeling method and $y$-modeling method in field plots (light blue), sampled and not thinned stands (dark blue), unsampled and not thinned stands (light blue) and unsampled and thinned stands (red). For each group the area of overlap, AO, with the density function for the field plots (green) is provided for each predictor.

## References

- Næsset, E. Predicting forest stand characteristics with airborne scanning laser using a practical two-stage procedure and field data. Remote. Sens. Environ.
**2002**, 80, 88–99. [Google Scholar] [CrossRef] - Andersen, H.-E.; McGaughey, R.J.; Reutebuch, S.E. Estimating forest canopy fuel parameters using LIDAR data. Remote. Sens. Environ.
**2005**, 94, 441–449. [Google Scholar] [CrossRef] - González-Ferreiro, E.; Diéguez-Aranda, U.; Miranda, D. Estimation of stand variables in Pinus radiata D. Don plantations using different LiDAR pulse densities. For. Int. J. For. Res.
**2012**, 85, 281–292. [Google Scholar] [CrossRef] - Mauro, F.; Molina, I.; García-Abril, A.; Valbuena, R.; Ayuga-Téllez, E. Remote sensing estimates and measures of uncertainty for forest variables at different aggregation levels. Environmetrics
**2016**, 27, 225–238. [Google Scholar] [CrossRef] - Valbuena, R.; Packalen, P.; Mehtätalo, L.; García-Abril, A.; Maltamo, M. Characterizing forest structural types and shelterwood dynamics from Lorenz-based indicators predicted by airborne laser scanning. Can. J. For. Res.
**2013**, 43, 1063–1074. [Google Scholar] [CrossRef] - Eggleston, H.S.; Buendia, L.; Miwa, K.; Ngara, T.; Tanabe, K. IPCC Guidelines for National Greenhouse Gas Inventories, Volume 4: Agriculture, Forestry and Other Land Use; Institute for Global Environmental Strategies: Hayama, Japan, 2006; Volume 4, ISBN 4-88788-032-4. [Google Scholar]
- Babcock, C.; Finley, A.O.; Bradford, J.B.; Kolka, R.; Birdsey, R.; Ryan, M.G. LiDAR based prediction of forest biomass using hierarchical models with spatially varying coefficients. Remote. Sens. Environ.
**2015**, 169, 113–127. [Google Scholar] [CrossRef][Green Version] - Poudel, K.P.; Flewelling, J.W.; Temesgen, H. Predicting Volume and Biomass Change from Multi-Temporal Lidar Sampling and Remeasured Field Inventory Data in Panther Creek Watershed, Oregon, USA. Forests
**2018**, 9, 28. [Google Scholar] [CrossRef] - Temesgen, H.; Strunk, J.; Andersen, H.-E.; Flewelling, J. Evaluating different models to predict biomass increment from multi-temporal lidar sampling and remeasured field inventory data in south-central Alaska. Math. Comput. For. Nat.-Resour. Sci. (MCFNS)
**2015**, 7, 66–80. [Google Scholar] - McRoberts, R.E.; Næsset, E.; Gobakken, T.; Chirici, G.; Condes, S.; Hou, Z.; Saarela, S.; Chen, Q.; Stahl, G.; Walters, B.F. Assessing components of the model-based mean square error estimator for remote sensing assisted forest applications. Can. J. For. Res.
**2018**, 48, 642–649. [Google Scholar] [CrossRef] - Næsset, E.; Gobakken, T.; Solberg, S.; Gregoire, T.G.; Nelson, R.; Ståhl, G.; Weydahl, D. Model-assisted regional forest biomass estimation using LiDAR and InSAR as auxiliary data: A case study from a boreal forest area. Remote. Sens. Environ.
**2011**, 115, 3599–3614. [Google Scholar] [CrossRef] - Massey, A.; Mandallaz, D. Design-based regression estimation of net change for forest inventories. Can. J. For. Res.
**2015**, 45, 1775–1784. [Google Scholar] [CrossRef] - Rao, J.N.K.; Molina, I. Introduction. In Small Area Estimation, 2nd ed.; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2015; pp. 1–8. ISBN 978-1-118-73585-5. [Google Scholar]
- Breidenbach, J.; Astrup, R. Small area estimation of forest attributes in the Norwegian National Forest Inventory. Eur. J. For. Res.
**2012**, 131, 1255–1267. [Google Scholar] [CrossRef] - Mauro, F.; Monleon, V.J.; Temesgen, H.; Ford, K.R. Analysis of area level and unit level models for small area estimation in forest inventories assisted with LiDAR auxiliary information. PLoS ONE
**2017**, 12, e0189401. [Google Scholar] [CrossRef] [PubMed] - Goerndt, M.E.; Monleon, V.J.; Temesgen, H. Small-Area Estimation of County-Level Forest Attributes Using Ground Data and Remote Sensed Auxiliary Information. For. Sci.
**2013**, 59, 536–548. [Google Scholar] [CrossRef] - Breidenbach, J.; Magnussen, S.; Rahlf, J.; Astrup, R. Unit-level and area-level small area estimation under heteroscedasticity using digital aerial photogrammetry data. Remote. Sens. Environ.
**2018**, 212, 199–211. [Google Scholar] [CrossRef] - Magnussen, S.; Næsset, E.; Gobakken, T. LiDAR-supported estimation of change in forest biomass with time-invariant regression models. Can. J. For. Res.
**2015**, 45, 1514–1523. [Google Scholar] [CrossRef] - Næsset, E.; Bollandsås, O.M.; Gobakken, T.; Gregoire, T.G.; Ståhl, G. Model-assisted estimation of change in forest biomass over an 11year period in a sample survey supported by airborne LiDAR: A case study with post-stratification to provide “activity data”. Remote. Sens. Environ.
**2013**, 128, 299–314. [Google Scholar] [CrossRef][Green Version] - Nasset, E.; Gobakken, T. Estimating forest growth using canopy metrics derived from airborne laser scanner data. Remote. Sens. Environ.
**2005**, 96, 453–465. [Google Scholar] [CrossRef] - Ritchie, M.W. Multi-scale reference conditions in an interior pine-dominated landscape in northeastern California. Ecol. Manag.
**2016**, 378, 233–243. [Google Scholar] [CrossRef] - Adams, M.B.; Loughry, L.H.; Plaugher, L.L. Experimental Forests and Ranges of the USDA Forest Service; United States Department of Agriculture, Forest Service, Northeastern Research Station: Newton Square, PA, USA, 2008; p. 191.
- Oliver, W.W. Ecological Research at the Blacks Mountain Experimental Forest in Northeastern California; United States Department of Agriculture, Forest Service, Pacific Southwest Research Station: Albany, CA, USA, 2000; p. 73.
- Wing, B.M.; Ritchie, M.W.; Boston, K.; Cohen, W.B.; Olsen, M.J. Individual snag detection using neighborhood attribute filtered airborne lidar data. Remote. Sens. Environ.
**2015**, 163, 165–179. [Google Scholar] [CrossRef] - Hudak, A.T.; Strand, E.K.; Vierling, L.A.; Byrne, J.C.; Eitel, J.U.; Martinuzzi, S.; Falkowski, M.J. Quantifying aboveground forest carbon pools and fluxes from repeat LiDAR surveys. Remote. Sens. Environ.
**2012**, 123, 25–40. [Google Scholar] [CrossRef] - Area Solar Radiation—Help | ArcGIS Desktop. Available online: http://desktop.arcgis.com/en/arcmap/10.6/tools/spatial-analyst-toolbox/area-solar-radiation.htm (accessed on 8 April 2019).
- Mauro, F.; Monleon, V.; Temesgen, H.; Ruiz, L. Analysis of spatial correlation in predictive models of forest variables that use LiDAR auxiliary information. Can. J. For. Res.
**2017**, 47, 788–799. [Google Scholar] [CrossRef][Green Version] - Rao, J.N.K.; Molina, I. Basic Unit Level Model. In Small Area Estimation, 2nd ed.; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2015; pp. 173–234. ISBN 978-1-118-73585-5. [Google Scholar]
- Rao, J.; Molina, I. Empirical Best Linear Unbiased Prediction (EBLUP): Theory. In Small Area Estimation, 2nd ed.; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2015; pp. 97–122. [Google Scholar]
- R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2018. [Google Scholar]
- Pinheiro, J.; Bates, D.; DebRoy, S.; Sarkar, D.; R Core Team. nlme: Linear and Nonlinear Mixed Effects Models. R package version 3.1-137. Available online: https://cran.r-project.org/web/packages/nlme/index.html (accessed on 15 April 2019).
- Datta, G.S.; Lahiri, P. A unified measure of uncertainty of estimated best linear unbiased predictors in small area estimation problems. Stat. Sin.
**2000**, 10, 613–628. [Google Scholar] - Silverman, B.W. Density Estimation for Statistics and Data Analysis; Chapman and Hall: Boca Raton, FL, USA, 1986. [Google Scholar]
- Mesgaran, M.B.; Cousens, R.D.; Webber, B.L. Here be dragons: A tool for quantifying novelty due to covariate range and correlation change when projecting species distribution models. Divers. Distrib.
**2014**, 20, 1147–1159. [Google Scholar] [CrossRef] - Bollandsås, O.M.; Gregoire, T.G.; Næsset, E.; Øyen, B.-H. Detection of biomass change in a Norwegian mountain forest area using small footprint airborne laser scanner data. Stat. Methods Appl.
**2013**, 22, 113–129. [Google Scholar] [CrossRef] - Fekety, P.A.; Falkowski, M.J.; Hudak, A.T. Temporal transferability of LiDAR-based imputation of forest inventory attributes. Can. J. For. Res.
**2014**, 45, 422–435. [Google Scholar] [CrossRef] - Ozdemir, I.; Donoghue, D.N. Modelling tree size diversity from airborne laser scanning using canopy height models with image texture measures. For. Ecol. Manag.
**2013**, 295, 28–37. [Google Scholar] [CrossRef]

**Figure 1.**Study area location map, delineated stands and field plots, and detailed diagram showing the light detection and ranging LiDAR field plots grid over the permanent Blacks Mountains Experimental Forest (BMEF) grid of permanent makers.

**Figure 2.**Estimates of V, BA and B change for the sampled stands of Blacks Mountains Experimental Forest. LiDAR-derived estimates using the $\delta $-modeling method are indicated by blue dots, LiDAR-derived estimates obtained using the $y$-modeling method are indicated with red dots and field-based estimates are indicated using black.

**Figure 3.**Estimates of V, BA and B change for the unsampled stands of Blacks Mountains Experimental Forest. LiDAR-derived estimates using the $\delta $-modeling method are indicated by blue dots and LiDAR-derived estimates obtained using the $y$-modeling method are indicated with red dots. Thinned stands are to the left and non-thinned stands to the right.

**Figure 4.**Values of $RMS{E}_{\delta}$ (blue), $RMS{E}_{y}$ (red) and $S{E}_{f}$ (black) for the stand-level estimates of V, BA and B.

**Figure 5.**Indexes of extrapolation. Average of Mesgaran’s novelty index [35], $\overline{NT2}$, for the sampled and not thinned stands (dark blue), unsampled stands not thinned (green) and unsampled and thinned stands (red). The value of this index for the field plots (light blue) provides the baseline value (i.e., the value observed for the sample of field plots).

**Figure 6.**Comparison of density functions for the predictors in the models used to estimate changes in Volume using the $\delta $-modeling method and $y$-modeling method in field plots (light blue), sampled and not thinned stands (dark blue), unsampled and not thinned stands (light blue) and unsampled and thinned stands (red). For each group the area of overlap, AO, with the density function for the field plots (green) is provided for each predictor.

**Table 1.**Minimum (Min), mean (Mean), standard deviation (Sd), and maximum (Max) of the plot-level values for 2009, 2015 and yearly increments for the period 2009–2015. Values of volume V, basal area BA and biomass B are expressed on a per-hectare basis.

Variable (Units) | Period | Min | Mean | Sd | Max |
---|---|---|---|---|---|

V(m^{3} ha^{−1}) | 2009 | 19.87 | 166.93 | 119.66 | 619.43 |

BA(m^{2} ha^{−1}) | 3.81 | 23.43 | 12.02 | 66.54 | |

B(Mg ha^{−1}) | 8.31 | 83.65 | 61.55 | 323.30 | |

V(m^{3} ha^{−1}) | 2015 | 17.20 | 175.52 | 117.04 | 644.30 |

BA(m^{2} ha^{−1}) | 3.42 | 25.45 | 12.01 | 67.47 | |

B(Mg ha^{−1}) | 8.34 | 89.38 | 60.29 | 335.03 | |

V(m^{3} ha^{−1}year^{−1}) | Increment 2009–2015 | −10.89 | 1.43 | 3.88 | 11.19 |

BA(m^{2} ha^{−1}year^{−1}) | −0.91 | 0.34 | 0.45 | 1.74 | |

B(Mg ha^{−1}year^{−1}) | −5.81 | 0.95 | 1.97 | 5.99 |

**Table 2.**Summary models for the $\delta $-modeling method. Model coefficients, standard errors of the model coefficients, variance parameters and general metrics for accuracy assessment are provided. Predictor acronyms are explained in Table A1. Coef is the value of the coefficient and Std.Error its corresponding standard error. V indicates volume, BA indicates basal area and B indicates biomass.

Model | Predictor | Coef | Std. Error | ${\widehat{\mathit{\sigma}}}_{\mathit{\delta}\mathit{v}}^{2}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{\delta}\mathit{\epsilon}}^{2}$ | $\mathit{m}\mathit{R}\mathit{M}\mathit{S}\mathit{E}$ | $\mathit{m}\mathit{R}\mathit{R}\mathit{M}\mathit{S}\mathit{E}$ | $\mathit{m}\mathit{B}\mathit{i}\mathit{a}\mathit{s}$ | $\mathit{m}\mathit{R}\mathit{B}\mathit{i}\mathit{a}\mathit{s}$ |
---|---|---|---|---|---|---|---|---|---|

V(m^{3} ha^{−1} year^{−1}) | Intercept | 1.16 | 0.31 | 0.50 | 10.53 | 3.47 | 241.99% | −1.83 × 10^{−4} | −0.01% |

$\delta $Elev_P50_{15-09} | 1.33 | 0.27 | |||||||

$\delta $PcFstAbv2_{15-09} | 0.23 | 0.07 | |||||||

BA(m^{2} ha^{−1} year^{−1}) | Intercept | 0.31 | 0.12 | 0.01 | 0.14 | 0.39 | 116.30% | −8.2 × 10^{−4} | −0.25% |

$\delta $PcAllAbv2_{15-09} | 0.05 | 0.01 | |||||||

Elev_P75_{09} | −0.03 | 0.01 | |||||||

PcAllAbv2_{15-09} | 0.02 | <0.01 | |||||||

B(Mg ha^{−1} year^{−1}) | Intercept | 1.03 | 0.17 | 0.19 | 2.52 | 1.72 | 180.20% | −1.09 × 10^{−3} | −0.11% |

$\delta $Elev_var_{15-09} | 0.05 | 0.02 | |||||||

$\delta $Elev_P50_{15-09} | 1.03 | 0.20 | |||||||

$\delta $CRR_{15-09} | −16.67 | 6.58 |

**Table 3.**Summary models for the $y$-modeling method. Model coefficients, standard errors of the model coefficients, variance-covariance parameters and general metrics for accuracy assessment are provided. Covariate acronyms are explained in Table A1. Coef is the value of the coefficient and Std.Error its corresponding standard error. Coef is the value of the coefficient and Std.Error its corresponding standard error. V indicates volume, BA indicates basal area and B indicates biomass.

Model | Year | Covariate | Coef | Std.Error | ${\widehat{\mathit{\sigma}}}_{\mathit{y}\mathit{u}}^{2}$ | Kijt | ${\mathit{\omega}}_{\mathit{y}\mathit{t}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{y}\mathit{t}\mathit{\epsilon}}^{2}$ | ${\mathit{\rho}}_{\mathit{e}}$ | General Accuracy Metrics for Change Per Hectare and Year | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{m}\mathit{R}\mathit{M}\mathit{S}\mathit{E}$ | $\mathit{m}\mathit{R}\mathit{R}\mathit{M}\mathit{S}\mathit{E}$ | $\mathit{m}\mathit{B}\mathit{I}\mathit{A}\mathit{S}$ | $\mathit{m}\mathit{R}\mathit{B}\mathit{i}\mathit{a}\mathit{s}$ | ||||||||||

V(m^{3} ha^{−1}) | 2009 | Intercept | −19.09 | 10.36 | 640.29 | Elev_mean^{2}_{09} | 0.64 | 3.00 | 0.85 | 3.76 | 262.62% | 0.13 | 9.24% |

Elev_mean^{2}_{09} | 2.52 | 0.23 | |||||||||||

PcFstAbv2_{09} | 0.63 | 0.05 | |||||||||||

2015 | Intercept | 2.69 | 0.23 | Elev_mean^{2}_{15} | 0.61 | 4.17 | |||||||

Elev_mean^{2}_{15} | 0.69 | 0.05 | |||||||||||

PcFstAbv2_{15} | −26.30 | 11.10 | |||||||||||

BA(m^{2} ha^{−1}) | 2009 | Intercept | −0.22 | 1.57 | 7.42 | PcFstAbv2_{09} | 0.48 | 0.81 | 0.85 | 0.47 | 138.06% | 0.01 | 1.53% |

Elev_P10_{09} | −1.37 | 0.34 | |||||||||||

Elev_P30_{09} | 1.58 | 0.24 | |||||||||||

PcFstAbv2_{09} | 0.51 | 0.03 | |||||||||||

2015 | Intercept | −2.16 | 0.61 | PcFstAbv2_{15} | 0.45 | 1.12 | |||||||

Elev_P10_{15} | 2.56 | 0.51 | |||||||||||

Elev_P20_{15} | 0.57 | 0.03 | |||||||||||

PcFstAbv2_{15} | −0.97 | 1.72 | |||||||||||

B(Mg ha^{−1}) | 2009 | Intercept | −11.86 | 5.19 | 165.69 | Elev_mean^{2}_{09} | 0.71 | 0.39 | 0.85 | 1.94 | 203.69% | 0.08 | 8.60% |

Elev_mean^{2}_{09} | 1.19 | 0.11 | |||||||||||

PcFstAbv2_{09} | 0.34 | 0.02 | |||||||||||

2015 | Intercept | 1.31 | 0.12 | Elev_mean^{2}_{15} | 0.58 | 1.47 | |||||||

Elev_mean^{2}_{15} | 0.37 | 0.03 | |||||||||||

PcFstAbv2_{15} | −14.15 | 5.76 |

**Table 4.**Average increments of volume V, basal area BA and biomass B in the entire study area excluding the thinned stands (SA) and for the union of the sampled stands (SS). Estimates $(\widehat{\Delta})$, root mean square errors $(RMSE)$, coefficients of variation $(CV)$ and confidence intervals ($CI$) obtained using the $\delta $-modeling method and the $y$-modeling method are compared to estimates (${\widehat{\Delta}}_{f}$), standard errors ($S{E}_{f}$) coefficients of variation ($C{V}_{f}$), and confidence intervals ($C{I}_{f}$) using only the field information.

Variable | Area | $\mathit{\delta}$-modeling Method | $\mathit{y}$-modeling Method | Field Only Estimates | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\widehat{\Delta}}_{\mathit{\delta}}$ | $\mathit{R}\mathit{M}\mathit{S}{\mathit{E}}_{\mathit{\delta}}$ | $\mathit{C}{\mathit{V}}_{\mathit{\delta}}$ | $\mathit{C}{\mathit{I}}_{\mathit{\delta}}$ | ${\widehat{\Delta}}_{\mathit{y}}$ | $\mathit{R}\mathit{M}\mathit{S}{\mathit{E}}_{\mathit{y}}$ | $\mathit{C}{\mathit{V}}_{\mathit{y}}$ | $\mathit{C}{\mathit{I}}_{\mathit{y}}$ | ${\widehat{\Delta}}_{\mathit{f}}$ | $\mathit{S}{\mathit{E}}_{\mathit{f}}$ | $\mathit{C}{\mathit{V}}_{\mathit{f}}$ | $\mathit{C}{\mathit{I}}_{\mathit{f}}$ | |||||

V(m^{3} ha^{−1} year^{−1}) | SS | 1.66 | 0.27 | 16.29% | 1.12 | 2.20 | 1.95 | 0.32 | 16.48% | 1.31 | 2.60 | 1.43 | 0.32 | 22.21% | 0.80 | 2.07 |

SA | 1.67 | 0.30 | 17.98% | 1.07 | 2.27 | 1.98 | 0.29 | 14.67% | 1.40 | 2.56 | ||||||

BA(m^{2} ha^{−1} year^{−1}) | SS | 0.36 | 0.03 | 8.68% | 0.30 | 0.42 | 0.37 | 0.04 | 9.93% | 0.30 | 0.45 | 0.34 | 0.04 | 10.87% | 0.26 | 0.41 |

SA | 0.42 | 0.04 | 8.41% | 0.35 | 0.49 | 0.44 | 0.04 | 9.61% | 0.35 | 0.52 | ||||||

B(Mg ha^{−1} year^{−1}) | SS | 1.07 | 0.13 | 12.35% | 0.81 | 1.34 | 1.24 | 0.17 | 13.61% | 0.90 | 1.57 | 0.95 | 0.16 | 16.89% | 0.63 | 1.28 |

SA | 1.15 | 0.16 | 13.66% | 0.83 | 1.46 | 1.29 | 0.15 | 11.83% | 0.98 | 1.59 |

**Table 5.**Minimum (Min), 5th percentile (p05), mean, median, 95th percentile (p95) and maximum (Max) of $RMS{E}_{\delta}$ (27) and $RMS{E}_{y}$ (28) for the pixels of the study area.

Variable | Method | Min | p05 | Mean | Median | p95 | Max |
---|---|---|---|---|---|---|---|

V(m^{3} ha^{−1} year^{−1}) | $\delta $-modeling method | 0.42 | 0.42 | 2.30 | 3.30 | 3.59 | 9.41 |

$y$-modeling method | 0.08 | 0.37 | 2.49 | 2.20 | 6.01 | 32.69 | |

BA(m^{2} ha^{−1} year^{−1}) | $\delta $-modeling method | 0.38 | 0.38 | 0.39 | 0.38 | 0.40 | 0.59 |

$y$-modeling method | 0.11 | 0.30 | 0.48 | 0.48 | 0.64 | 1.47 | |

B(Mg ha^{−1} year^{−1}) | $\delta $-modeling method | 1.62 | 1.63 | 1.67 | 1.65 | 1.76 | 4.57 |

$y$-modeling method | 0.47 | 1.10 | 1.89 | 1.76 | 3.09 | 10.45 |

© 2019 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**

Mauro, F.; Ritchie, M.; Wing, B.; Frank, B.; Monleon, V.; Temesgen, H.; Hudak, A. Estimation of Changes of Forest Structural Attributes at Three Different Spatial Aggregation Levels in Northern California using Multitemporal LiDAR. *Remote Sens.* **2019**, *11*, 923.
https://doi.org/10.3390/rs11080923

**AMA Style**

Mauro F, Ritchie M, Wing B, Frank B, Monleon V, Temesgen H, Hudak A. Estimation of Changes of Forest Structural Attributes at Three Different Spatial Aggregation Levels in Northern California using Multitemporal LiDAR. *Remote Sensing*. 2019; 11(8):923.
https://doi.org/10.3390/rs11080923

**Chicago/Turabian Style**

Mauro, Francisco, Martin Ritchie, Brian Wing, Bryce Frank, Vicente Monleon, Hailemariam Temesgen, and Andrew Hudak. 2019. "Estimation of Changes of Forest Structural Attributes at Three Different Spatial Aggregation Levels in Northern California using Multitemporal LiDAR" *Remote Sensing* 11, no. 8: 923.
https://doi.org/10.3390/rs11080923