# An Inventory-Based Regeneration Biomass Model to Initialize Landscape Scale Simulation Scenarios

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data

#### 2.1.1. German National Forest Inventory (NFI)

#### 2.1.2. Bavarian State Forest Inventory (BSFI)

^{2}that encloses several smaller concentric inventory circles. Only trees above a certain threshold DBH (typically 30 cm) are recorded on the whole plot area. Trees with a DBH < 30 cm and ≥ 11 cm are measured within an 80 to 125 m

^{2}circle. The smallest class of trees with a DBH < 11 cm, including regeneration trees, is surveyed on a 25 m

^{2}circle [25]. The minimum height of regeneration trees to be sampled in the BSFI is 0.2 m, the same as in the NFI. Trees up to a height of 1.3 m are recorded at a higher-class width, i.e., 0.1 m. Trees of a DBH > 0 and < 7 cm are recorded per DBH levels 1.5, 2.5, 3.5, 4.5, 5.5, and 6.5 cm. The beech-dominated BSFI data set used for model calibration and evaluation was selected by the same criteria as the NFI data. That set of plots with beech-dominated overstory will be named “BSFI data” in the following (Table 2).

#### 2.2. Data Preparation

_{0}and d

_{0}are tree height and DBH as measured, respectively. Thus, HD can obtain any value between 0 and 1. For a tree of any diameter, HD = 0.5 indicates an average, i.e., normal height to diameter ratio. HD > 0.5 marks an above average tall tree and HD < 0.5 a stout one. That characteristic, hence, aims to indicate the extent to which a tree had been inhibited in diameter growth through neighbor competition in the past.

_{ij}> 0 considered). The Species Profile Index in its standard form is based on three stand height layers whose upper borders are H

_{max}× 1, H

_{max}× 0.8, and H

_{max}× 0.5, where H

_{max}is the maximum tree height in the stand of interest. SPI, thus, might indicate whether biomass is concentrated into exactly one tree class or shared among several ones within the stand being considered.

#### 2.3. Models

_{1}, …, β

_{5}). In contrast, the effect of Dq is modeled with a nonlinear spline-based smoother [31], which is indicated by the symbol s(Dq). It accounts for a hypothesized nonlinear relation between Dq and regeneration stock due to a likely minimum of light transmission between the phase of adolescence and intense harvest.

_{i}is a cluster specific random effect and ε

_{ij}represents i.i.d. errors. The BSFI-based model comprises two group effects (Equation (6)):

_{i}is the random effect related to the forest management unit and b

_{ij}is the stand-related one. ε

_{ijk}represents the i.i.d. errors.

_{i}~ N(0, σ

_{1}

^{2}) and ε

_{ij}~ N(0, σ

_{2}

^{2}). For fitting Equation (4b), we correspondingly assumed b

_{i}~ N(0, σ

_{3}

^{2}), b

_{ij}~ N(0, σ

_{4}

^{2}), and ε

_{ijk}~ N(0, σ

_{5}

^{2}). In order to facilitate the reproduction and application of the deterministic model part, we complemented both Equations (4a) and (4b) with an approximation that refrained from the smoothing spline s(Dq). To that end, we replaced s(Dq) with a small set of polynomials. Therefore, we predicted regeneration biomass based on the observed Dq values and the mean of each remainder predictor. Then, we fit one polynomial to the data of predicted biomass over Dq within each of several Dq intervals.

_{1}, u

_{2}, v

_{1}, and v

_{2}are regression parameters; and index l stands for an observation. Both ε

_{1}~ N(0, σ

_{6}

^{2}) and ε

_{2}~ N(0, σ

_{7}

^{2}) are the i.i.d. errors. In order to describe both trends, we divided the range of predicted biomass values into 40 intervals, each representing a quantile width of 2.5%. Within each of them, we fitted one gamma probability density function to the enclosed relative residuals. That way, we obtained a set of 40 distribution functions and, concomitantly, a data set of 40 estimated values for both parameters, shape and rate. For describing the biomass-related trend of a parameter considered—i.e., either shape or rate—we associated each parameter value to the corresponding interval’s predicted biomass median and applied a linear regression (Equations (9a) and (9b)) to the resulting point set. Thus, for any predicted biomass and its accompanying residual distribution, we estimated the corresponding parameters, shape and rate. For fitting the density functions, we used the statistical software R and the function fitdist from package fitdistrplus with moment matching estimation [35].

#### 2.4. Evaluation

## 3. Results

^{−1}at a Dq of 30 cm vs. a value of 5 t ha

^{−1}at Dq values larger than 40 cm (Figure 1a and Figure 2a).

## 4. Discussion

#### 4.1. The Study Responds to a Common Requirement for Model Initialization

#### 4.2. The Study Conceptualizes and Evaluates a Novel Biomass-Based Approach

#### 4.3. The Modelling Approach Exemplified Provides a Basis of Future Development

#### 4.4. Browsing Is a Likely Missing Indicator

#### 4.5. Most of the Random Effect Is Due to Within-Stand Variability

#### 4.6. Application of Such Model Has to Account for General Limitations of Any Inventory

#### 4.7. The Most Relevant Predictors Will Be Accessible through Remote Sensing

#### 4.8. Future Work Has to Explain Random Effects and Might Extend the Focus of Model Application

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

_{0}and d

_{0}are tree height and DBH as measured, respectively.

**Table A1.**Parameter values that result from fitting of Equation (A1) to the distribution of individual tree height–diameter data.

Distribution Characteristic Equation (A1) Was Fitted to | Parameter Value | |
---|---|---|

A | B | |

Mean | 5.142 | 0.437 |

2.5% quantile | 1.803 | 0.580 |

97.5% quantile | 7.563 | 0.420 |

## Appendix B

## Appendix C

_{1}, d

_{2}, and d

_{3}. Each of these approximations is exclusively valid within the corresponding interval of Dq (hence, the intercepts may notably differ).

**Table A2.**The parameters of the approximation to the deterministic part of the NFI-based model (see Section 2.3 and Equation (A3)) that substitutes s(Dq) by two polynomials; the intercept implies that of the polynomial approximation and thus depends on the interval of Dq.

Dq Interval | Intercept | Slope of | |||||||
---|---|---|---|---|---|---|---|---|---|

SDI | SPI | SI | Hmax | HD | d_{1} | d_{2} | d_{3} | ||

≤56.6 | 79,000 | −6.87 | −1498 | −1304 | 304 | −2927 | −2543 | 66.88 | −0.54 |

>56.6 | 52,630 | −6.87 | −1498 | −1304 | 304 | −2927 | −2.06 | −0.58 | 0 |

**Table A3.**The parameters of the approximation to the deterministic part of the BSFI-based model (see Section 2.3 and Equation (A3)) that substitutes s(Dq) by two polynomials; the intercept implies that of the polynomial approximation and thus depends on the interval of Dq.

Dq Interval | Intercept | Slope of | |||||||
---|---|---|---|---|---|---|---|---|---|

SDI | SPI | SI | Hmax | HD | d_{1} | d_{2} | d_{3} | ||

≤35 | 50,530 | 40,074 | −4.59 | −1194 | -- | 255 | 1338 | −19.06 | 0.082 |

>35 | −25,973 | 40,074 | −4.59 | −1194 | -- | 255 | −5522 | 206 | −2.55 |

## References

- Biber, P.; Borges, J.; Moshammer, R.; Barreiro, S.; Botequim, B.; Brodrechtová, Y.; Brukas, V.; Chirici, G.; Cordero-Debets, R.; Corrigan, E.; et al. How Sensitive Are Ecosystem Services in European Forest Landscapes to Silvicultural Treatment? Forests
**2015**, 6, 1666–1695. [Google Scholar] [CrossRef] - Jonsson, B.; Jacobsson, J.; Kallur, H. The Forest Management Planning Package Theory and Application. Stud. For. Suec.
**1993**, 189, 1–56. [Google Scholar] - Pott, M.; Fabrika, M. An Information System for the Evaluation and Spatial Analysis of Forest Inventory Data. Forstwiss. Cent.
**2002**, 121 (Suppl. 1), 80–88. [Google Scholar] - Pretzsch, H. Forest Dynamics, Growth and Yield; Springer: Berlin/Heidelberg, Germany, 2009; ISBN 978-3-540-88306-7. [Google Scholar]
- Canadian Forest Inventory Committee. Canada’s National Forest Inventory Ground Sampling Guidelines: Specifications for Ongoing Measurement; Natural Resources Canada, Canadian Forest Service, Pacific Forestry Centre: Victoria, BC, Canada, 2008.
- Polley, H. Survey Instructions for the 3rd National Forest Inventory (2011–2012) 2nd Revised Version, May 2011 with 4. Corrigendum (21.03.2014); Bundesministerium für Ernährung, Landwirtschaft und Verbraucherschutz Ref. 535; Federal Ministry of Food, Agriculture, and Consumer Protection: Bonn, Germany, 2011. [Google Scholar]
- Hunziker, U.; Brang, P. Microsite Patterns of Conifer Seedling Establishment and Growth in a Mixed Stand in the Southern Alps. For. Ecol. Manag.
**2005**, 210, 67–79. [Google Scholar] [CrossRef] - Abdullahi, S.; Kugler, F.; Pretzsch, H. Prediction of stem volume in complex temperate forest stands using TanDEM-X SAR data. Remote Sens. Environ.
**2016**, 174, 197–211. [Google Scholar] [CrossRef] - Abdullahi, S.; Schardt, M.; Pretzsch, H. An unsupervised two-stage clustering approach for forest structure classification based on X-band InSAR data. A case study in complex temperate forest stands. Int. J. Appl. Earth Obs. Geoinf.
**2017**, 57, 36–48. [Google Scholar] [CrossRef] - Kayitakire, F.; Hamel, C.; Defourny, P. Retrieving forest structure variables based on image texture analysis and IKONOS-2 imagery. Remote Sens. Environ.
**2006**, 102, 390–401. [Google Scholar] [CrossRef] - Schweiger, J.; Sterba, H. A Model Describing Natural Regeneration Recruitment of Norway Spruce (Picea abies (L.) Karst.) in Austria. For. Ecol. Manag.
**1997**, 97, 107–118. [Google Scholar] [CrossRef] - Tremer, N.; Schmidt, M.; Hansen, J. Estimating the Structure of Natural Regeneration Based on Inventory Data. Allg. Forst Jagdztg.
**2005**, 176, 1–13. [Google Scholar] - Kolo, H.; Ankerst, D.; Knoke, T. Predicting Natural Forest Regeneration: A Statistical Model Based on Inventory Data. Eur. J. For. Res.
**2017**, 136, 923–938. [Google Scholar] [CrossRef] - Lutze, M.; Ades, P.; Campbell, R. Spatial Distribution of Regeneration in Mixed-Species Forests of Victoria. Aust. For.
**2004**, 67, 172–183. [Google Scholar] [CrossRef] - Gravel, D.; Beaudet, M.; Messier, C. Partitioning the Factors of Spatial Variation in Regeneration Density of Shade-Tolerant Tree Species. Ecology
**2008**, 89, 2879–2888. [Google Scholar] [CrossRef] [PubMed] - Pretzsch, H.; Biber, P.; Ďurský, J. The Single Tree-Based Stand Simulator SILVA: Construction, Application and Evaluation. For. Ecol. Manag.
**2002**, 162, 3–21. [Google Scholar] [CrossRef] - Seidl, R.; Lexer, M.J.; Jager, D.; Honninger, K. Evaluating the Accuracy and Generality of a Hybrid Patch Model. Tree Physiol.
**2005**, 25, 939–951. [Google Scholar] [CrossRef] [PubMed] - Fischer, R.; Bohn, F.; Dantas de Paula, M.; Dislich, C.; Groeneveld, J.; Gutiérrez, A.G.; Kazmierczak, M.; Knapp, N.; Lehmann, S.; Paulick, S.; et al. Lessons Learned from Applying a Forest Gap Model to Understand Ecosystem and Carbon Dynamics of Complex Tropical Forests. Ecol. Model.
**2016**, 326, 124–133. [Google Scholar] [CrossRef][Green Version] - Kupferschmid, A.D.; Heiri, C.; Huber, M.; Fehr, M.; Frei, M.; Gmür, P.; Imesch, N.; Zinggeler, J.; Brang, P.; Clivaz, J.C.; et al. Einfluss Wildlebender Huftiere Auf Die Waldverjüngung: Ein Überblick Für Die Schweiz. Schweiz. Z. Forst.
**2015**, 166, 420–431. [Google Scholar] [CrossRef] - Clark, D.A.; Brown, S.; Kicklighter, D.W.; Chambers, J.Q.; Thomlinson, J.R.; Ni, J. Measuring Net Primary Production in Forest Concepts and Field Methods. Ecol. Appl.
**2001**, 11, 356–370. [Google Scholar] [CrossRef] - Thünen Institute, Germany. German National Forest Inventory (BWI) Results Database. Available online: https://bwi.info (accessed on 1 September 2015).
- Kändler, G.; Bösch, B. Überprüfung und Neukonzeption einer Biomassefunktion. Abschlussbericht 2b; Abt. Biometrie und Informatik Wonnhaldestraße 4 79100; Forstliche Versuchs- und Forschungsanstalt Baden-Württemberg: Freiburg, Germany, 2013. [Google Scholar]
- Röhling, S.; Dunger, K.; Kändler, G.; Klatt, S.; Riedel, T.; Stümer, W.; Brötz, J. Comparison of Calculation Methods for Estimating Annual Carbon Stock Change in German Forests under Forest Management in the German Greenhouse Gas Inventory. Carbon Balance Manag.
**2016**, 11, 12. [Google Scholar] [CrossRef] [PubMed] - Marklund, L.G. Biomass Functions for Norway Spruce (Picea abies (L.) Karst) in Sweden; SLU, Swedish University of Agricultural Sciences, Department of Forest Survey: Uppsala, Sweden, 1987. [Google Scholar]
- Neufanger, M.; Faltl, W.; Schelhaas, C. Anleitung zur Durchführung von Betriebsinventuren in den Bayerischen Staatsforsten. FE AA 010 Durchführung von Betriebsinventuren; Bayerische Staatsforsten AöR: Munich, Germany, 2012. [Google Scholar]
- Kozlowski, T.T. Physiological Ecology of Natural Regeneration of Harvested and Disturbed Forest Stands: Implications for Forest Management. For. Ecol. Manag.
**2002**, 158, 195–221. [Google Scholar] [CrossRef] - Pardos, M.; Montero, G.; Cañellas, I.; Ruiz del Castillo, J. Ecophysiology of Natural Regeneration of Forest Stands in Spain. For. Syst.
**2005**, 14, 434–445. [Google Scholar] [CrossRef] - Reineke, L.H. Perfecting a stand-density index for even-aged forests. J. Agric. Res.
**1933**, 46, 627–638. [Google Scholar] - Pretzsch, H.; Biber, P.; Uhl, E.; Dauber, E. Long-Term Stand Dynamics of Managed Spruce-Fir-Beech Mountain Forests in Central Europe: Structure, Productivity and Regeneration Success. Forestry
**2015**, 88, 407–428. [Google Scholar] [CrossRef] - Thünen Institute, Germany. Digital Map of Forest Ecological Regions (Wuchsgebiete/Wuchsbezirke). Available online: https://gdi.thuenen.de/wo/wgwb/ (accessed on 1 September 2015).
- Wood, S.N. Generalized Additive Models: An Introduction with R, 2nd ed.; Chapman and Hall/CRC Press; Taylor Francis Inc.: New York, NY, USA, 2017; ISBN 978-1-49-872833-1. [Google Scholar]
- Fahrmeir, L.; Kneib, T.; Lang, S.; Marx, B. Extensions of the Classical Linear Model. In Regression; Springer: Berlin/Heidelberg, Germany, 2013; pp. 177–267. ISBN 978-3-642-34332-2. [Google Scholar]
- R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2016; Available online: https://www.R-project.org/ (accessed on 24 April 2016).
- Wood, S.; Scheipl, F. Gamm4: Generalized Additive Mixed Models Using ‘mgcv’ and ‘lme4’. R Package Version 0.2-5. Available online: https://CRAN.R-project.org/package=gamm4 (accessed on 25 November 2017).
- Delignette-Muller, M.L.; Dutang, C. fitdistrplus: An R Package for Fitting Distributions. J. Stat. Softw.
**2015**, 64, 1–34. Available online: http://www.jstatsoft.org/v64/i04/ (accessed on 24 April 2016). [CrossRef] - Akaike, H. Prediction and Entropy. In A Celebration of Statistics; Atkinson, A.C., Fienberg, S.E., Eds.; Springer: New York, NY, USA, 1985; pp. 1–24. [Google Scholar]
- Marsaglia, G.; Tsang, W.W.; Wang, J. Evaluating Kolmogorov’s distribution. J. Stat. Softw.
**2003**, 8. Available online: http://www.jstatsoft.org/v08/i18/ (accessed on 24 April 2016). [CrossRef] - Becker, R.A.; Chambers, J.M.; Wilks, A.R. The New S Language; Chapman & Hall/CRC: London, UK, 1988; ISBN 978-0-534-09192-7. [Google Scholar]
- Efron, B. Bootstrap Methods: Another Look at the Jackknife. Ann. Stat.
**1979**, 7, 1–26. [Google Scholar] [CrossRef] - Price, D.T.; Zimmermann, N.E.; van der Meer, P.J.; Lexer, M.J.; Leadley, P.; Jorritsma, I.T.M.; Schaber, J.; Clark, D.F.; Lasch, P.; McNulty, S.; et al. Regeneration in Gap Models: Priority Issues for Studying Forest Responses To Climate Change. Clim. Chang.
**2001**, 51, 475–508. [Google Scholar] [CrossRef] - Ferguson, D.E.; Carlson, C.E. Predicting Regeneration Establishment with the Prognosis Model; No. 467, 1-U54; Forest Service: Ogden, UT, USA, 1993. [Google Scholar]
- Landsberg, J.J.; Waring, R.H. A Generalised Model of Forest Productivity Using Simplified Concepts of Radiation-Use Efficiency, Carbon Balance and Partitioning. For. Ecol. Manag.
**1997**, 95, 209–228. [Google Scholar] [CrossRef] - Barna, M. The Effects of Cutting Regimes on Natural Regeneration in Submountain Beech Forests: Species Diversity and Abundance. J. For. Sci.
**2008**, 54, 533–544. [Google Scholar] [CrossRef] - Foster, J.R.; Reiners, W.A. Size Distribution and Expansion of Canopy Gaps in a Northern Appalachian Spruce-Fir Forest. Vegetatio
**1986**, 68, 109–114. [Google Scholar] - Asner, G.P.; Kellner, J.R.; Kennedy-Bowdoin, T.; Knapp, D.E.; Anderson, C.; Martin, R.E. Forest Canopy Gap Distributions in the Southern Peruvian Amazon. PLoS ONE
**2013**, 8, e60875. [Google Scholar] [CrossRef] [PubMed] - Ammer, C. Impact of Ungulates on Structure and Dynamics of Natural Regeneration of Mixed Mountain Forests in the Bavarian Alps. For. Ecol. Manag.
**1996**, 88, 43–53. [Google Scholar] [CrossRef] - Motta, R. Impact of Wild Ungulates on Forest Regeneration and Tree Composition of Mountain Forests in the Western Italian Alps. For. Ecol. Manag.
**1996**, 88, 93–98. [Google Scholar] [CrossRef] - Boulanger, V.; Baltzinger, C.; Saïd, S.; Ballon, P.; Picard, J.F.; Dupouey, J.L. Ranking Temperate Woody Species along a Gradient of Browsing by Deer. For. Ecol. Manag.
**2009**, 258, 1397–1406. [Google Scholar] [CrossRef] - Golser, M.; Hasenauer, H. Predicting Juvenile Tree Height Growth in Uneven-Aged Mixed Species Stands in Austria. For. Ecol. Manag.
**1997**, 97, 133–146. [Google Scholar] [CrossRef] - Lindberg, E.; Holmgren, J. Individual Tree Crown Methods for 3D Data from Remote Sensing. Curr. For. Rep.
**2017**, 3, 19–31. [Google Scholar] [CrossRef] - Jucker, T.; Caspersen, J.; Chave, J.; Antin, C.; Barbier, N.; Bongers, F.; Dalponte, M.; van Ewijk, K.Y.; Forrester, D.I.; Haeni, M.; et al. Allometric Equations for Integrating Remote Sensing Imagery into Forest Monitoring Programmes. Glob. Chang. Biol.
**2017**, 23, 177–190. [Google Scholar] [CrossRef] [PubMed] - Hyyppä, J.; Hyyppä, H.; Leckie, D.; Gougeon, F.; Yu, X.; Maltamo, M. Review of Methods of Small-footprint Airborne Laser Scanning for Extracting Forest Inventory Data in Boreal Forests. Int. J. Remote Sens.
**2008**, 29, 1339–1366. [Google Scholar] [CrossRef] - Hothorn, T.; Müller, J. Large-Scale Reduction of Ungulate Browsing by Managed Sport Hunting. For. Ecol. Manag.
**2010**, 260, 1416–1423. [Google Scholar] [CrossRef] - Didion, M.; Kupferschmid, A.D.; Bugmann, H. Long-Term Effects of Ungulate Browsing on Forest Composition and Structure. For. Ecol. Manag.
**2009**, 258, 44–55. [Google Scholar] [CrossRef]

**Figure 1.**Profiles of regeneration biomass as predicted by the deterministic model part based on the NFI (German National Forest Inventory, Equation (4a)). Each profile is presented over one predictor with, at the same time, the remainder predictors at their mean value (abbreviation H-D: Height-Diameter); profiles ordered by the relevance of the referring predictor based on the AIC criterion ((

**a**–

**f**), see Section 2.4); dotted lines refer to the confidence interval; each predictor shown within its 95% interval; the stochastic part (Figure 3a,b) covers the residual distribution.

**Figure 2.**Profiles of regeneration biomass as predicted by the deterministic model part based on the BSFI (Bavarian State Forest Inventory, Equation (4b)). Each profile is presented over one predictor with, at the same time, the remainder predictors at their mean value (abbreviation H-D: Height-Diameter); profiles ordered by the relevance of the referring predictor based on the AIC criterion ((

**a**–

**e**), see Section 2.4); dotted lines refer to the confidence interval; each predictor shown within its 95% interval; the stochastic part (Figure 3c,d) covers the residual distribution (profiles based on 5881 BSFI plots used for model calibration).

**Figure 3.**Plausibility test of the theoretical probability distribution that provides the basis for the stochastic part of the model (Equation (8)); diagrams (

**a**,

**b**) are based on the NFI, i.e., German National Forest Inventory; (

**c**,

**d**) are based on the BSFI, i.e., Bavarian State Forest Inventory. That distribution of relative residuals (R in Equation (8), per-plot regeneration biomass to predicted regeneration biomass) is presented here from within a center range of the data (see Section 2.4) where the three main predictors of the deterministic model part (Figure 1), Dq, SDI, and Top Height, lie within their interquartile range each; both (

**a**) (resp. (

**c**)) and (

**b**) (resp. (

**d**)) compare the theoretical distribution to the empirical one; diagrams (

**a**) and (

**c**): probability density of the relative residual; the bars refer to the empirical density, and the line refers to the density obtained from a fitted gamma probability density function; diagrams (

**b**) and (

**d**): the QQ-plot of the quantile based on the fitted probability density function (Theoretical Quantile) over the quantile based on the measured data (Empirical Quantile); all shown within the 95% quantile of the relative residual ((

**a**), (

**b**) based on 1353 NFI plots, (

**c**), (

**d**) based on 1069 BSFI plots used for model calibration).

**Figure 4.**Diagrams (

**a**,

**b**) show the parameters α and β of the stochastic model part (Equation (8)) based on the NFI (German National Forest Inventory), as represented by the trend line of shape α (diagram (

**a**)) and rate β (diagram (

**b**)) over the predicted regeneration biomass (Equations (9a) and (9b), parameterized equation shown above corresponding figure); the stochastic model part considers the scattering of the relative residuals of the deterministic part (Figure 1); diagram (

**c**) is the result of the corresponding plausibility test of the stochastic model part with a QQ-Plot that compares 7784 quantiles Q of the modelled relative residual vs. the empirical one up to Q = 95% (relative residual is R in Equation (8), per-plot regeneration biomass to predicted regeneration biomass); diagrams (

**d**–

**f**) present the corresponding results based on the data of the BSFI (Bavarian State Forest Inventory, (

**d**,

**e**) based on 5881 plots used for model calibration; (

**f**) based on 6073 points spared from calibration for evaluation).

**Figure 5.**Diagrams (

**a**,

**b**): Mean of measured regeneration biomass over mean of predicted regeneration biomass per stratum obtained from the BSFI (Bavarian State Forest Inventory); (

**a**) if strata have been formed based on the diameter class (Table 3); (

**b**) if strata have both been formed based on diameter class and Forest Management Unit: the higher data spread on that scale level points to an as yet stochastic factor to be captured; in diagram (

**a**), the numbers 4 to 60 denote diameter classes [cm] of 4 ± 4, 12 ± 3, 20 ± 5, 30 ± 5, 40 ± 5, 50 ± 5, and 55 to 80; in diagram (

**b**), due to the sample size required per stratum the data focus on diameter classes 30, 40, and 50 (71% of all 11,954 BSFI plots); diagrams (

**c**), (

**d**) show the according result of the cumulative frequency (Cum. Freq.) of mean biomass; the cumulative frequency serves to indicate the quality of the modelled distribution characteristics (all based on 6073 inventory plots spared from model calibration for evaluation).

**Table 1.**Value characteristics of the base data set from the German national forest inventory NFI (n = 7823) used to analyze the influence of the overstory on the biomass of the regeneration fraction. Columns q 2.5 and q 97.5 show the 2.5% and 97.5% quantile, respectively. Column n denotes the number of data records used to calculate that range, as well as the mean of the variable considered. Column Ori. (origin) indicates whether the data were native (N) or derived by own computation (D). Further abbreviations are Lr. (layer), BAF (basal area factor), ER (ecoregion), and EL (elevation).

Lr. | Variable | Symbol | Basis | Level | Ori. | n | Mean | q 2.5 | q 97.5 | Unit |
---|---|---|---|---|---|---|---|---|---|---|

Overstory | tree diameter | D_{t} | -- | tree | N | 52,404 | 42 | 10 | 80 | cm |

tree height | H_{t} | -- | tree | N | 52,404 | 28 | 11 | 40 | m | |

tree number/ha | N_{t} | D_{t} BAF | tree | N | 52,404 | 76 | 7 | 520 | ha^{−1} | |

Stand Density Idx. | SDI | N_{t}, D_{t} | plot | D | 7784 | 516 | 90 | 1 099 | -- | |

Spec. Profile Idx. | SPI | H_{t} | plot | D | 7784 | 0.49 | 0 | 1.26 | -- | |

H-D Characteristic | HD | H_{t}, D_{t} | plot | D | 7784 | 0.58 | 0.05 | 0.97 | -- | |

Top height | Hmax | H_{t} | plot | D | 7784 | 30 | 13 | 42 | m | |

Site Index | SI | ER, EL | plot | D | 7784 | 36.58 | 35.37 | 37.33 | m | |

Quad. Mean Dia. | Dq | N_{t}, D_{t} | plot | D | 7784 | 36 | 11 | 67 | cm | |

Regeneration | tree height | -- | -- | tree | N | 22,808 | 3.69 | 0 | 9 | m |

tree diameter | -- | -- | tree | N | 22,808 | 0.7 | 0 | 5.5 | cm | |

tree biomass | b_{c} | -- | tree | N | 22,808 | 0.74 | 0 | 8.5 | kg | |

tree number/ha | n_{c} | -- | tree | N | 22,808 | 5253 | 0 | 31,831 | ha^{−1} | |

biomass/ha | -- | b_{c}, n_{c} | plot | D | 7784 | 6.0 | 0 | 46 | t ha^{−1} | |

tree number/ha | -- | n_{c} | plot | D | 7784 | 15,392 | 0 | 98,218 | ha^{−1} |

**Table 2.**Value characteristics of the base data set from the Bavarian State Forest inventory BSFI (n = 11,954) used to analyze the influence of the overstory on the biomass of the regeneration fraction. Columns q 2.5 and q 97.5 show the 2.5% and 97.5% quantile, respectively. Column n denotes the number of data records used to calculate that range, as well as the mean of the variable considered. Column Ori. (origin) indicates whether the data were native (N) or derived by own computation (D). Further abbreviations are Lr. (layer), BAF (basal area factor), ER (ecoregion), and EL (elevation).

Lr. | Variable | Symbol | Basis | Level | Ori. | n | Mean | q 2.5 | q 97.5 | Unit |
---|---|---|---|---|---|---|---|---|---|---|

Overstory | tree diameter | D_{t} | -- | tree | N | 96,429 | 34 | 8.5 | 65 | cm |

tree height | H_{t} | -- | tree | N | 96,429 | 25 | 9.3 | 37 | m | |

tree number/ha | N_{t} | D_{t} BAF | tree | N | 96,429 | 73 | 15 | 404 | ha^{−1} | |

Stand Density Idx. | SDI | N_{t}, D_{t} | plot | D | 11,954 | 519 | 74 | 1080 | -- | |

Spec. Profile Idx. | SPI | H_{t} | plot | D | 11,954 | 0.6 | 0 | 1.4 | -- | |

H-D Characteristic | HD | H_{t}, D_{t} | plot | D | 11,954 | 0.54 | 0.03 | 0.95 | -- | |

Top height | Hmax | H_{t} | plot | D | 11,954 | 29 | 10 | 40 | m | |

Site Index | SI | ER, EL | plot | D | 11,954 | 35.4 | 36.25 | 37.33 | m | |

Quad. Mean Dia. | Dq | N_{t}, D_{t} | plot | D | 11,954 | 32 | 9.7 | 59 | cm | |

Regeneration | tree height | -- | -- | tree | N | 42,342 | 2.18 | 0.2 | 9.2 | m |

tree diameter | -- | -- | tree | N | 42,342 | 1.4 | 0 | 6.5 | cm | |

tree biomass | b_{c} | -- | tree | N | 42,342 | 1.4 | 0 | 12 | kg | |

tree number/ha | n_{c} | -- | tree | N | 42,342 | 2503 | 321 | 16,040 | ha^{−1} | |

biomass/ha | -- | b_{c}, n_{c} | plot | D | 11,954 | 4.3 | 0 | 30 | t ha^{−1} | |

tree number/ha | -- | n_{c} | plot | D | 11,954 | 7667 | 0 | 47,705 | ha^{−1} |

**Table 3.**Classification levels of the average diameter used for stratification with pct of plots in the BSFI (Bavarian State Forest Inventory) data (see Section 2.1.2).

Class | Diameter [cm] ^{1} | Pct of Plots |
---|---|---|

4 | 0 to 8 | 3 |

12 | 8 to 15 | 6 |

20 | 15 to 25 | 15 |

30 | 25 to 35 | 28 |

40 | 35 to 45 | 28 |

50 | 45 to 55 | 15 |

60 | 55 and more | 5 |

^{1}Lower limit excluded.

**Table 4.**Coefficients of the regeneration biomass model based on the NFI (German National Forest Inventory, Equations (4a) and (5)). The variables are denoted as in Equations (4a) and (5), i.e., SDI (Stand Density Index), SPI (Species Profile Index), SI (Site Index), Hmax (maximum height), HD (Height Diameter Characteristic), Dq (Quadratic Mean Diameter); β

_{1}to β

_{5}are corresponding fixed effect coefficients; s(Dq) is a univariate penalized cubic regression spline [31] over Dq (Figure 1a); b is the NFI cluster specific random effect, and s

_{1}is its standard deviation; ε represents the i.i.d. errors and s

_{2}represents their standard deviation.

Intercept | Fixed Effect of | Random Effect | |||||||
---|---|---|---|---|---|---|---|---|---|

SDI | SPI | SI | Hmax | HD | s(Dq) | b_{i} | ε_{ij} | ||

Symbol ^{1} | β_{0} | β_{1} | β_{2} | β_{3} | β_{4} | β_{5} | -- | s_{1} | s_{2} |

Value | 74,344 | −6.87 | −1498 | −1304 | 304 | −2927 | -- | 5536 | 11,750 |

SE | 143 | 0.60 | 380 | 265 | 32 | 692 | -- | ||

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

^{1}In Equation (4a).

**Table 5.**Coefficients of the regeneration biomass model based on the BSFI (Bavarian State Forest inventory, Equations (4b) and (6)). Site Index (SI) was not significant as a predictor and removed from Equation (4b) (see Section 2.4); the variables are denoted as in Equations (4b) and (6), i.e., SDI (Stand Density Index), SPI (Species Profile Index), Hmax (maximum height), HD (Height Diameter Characteristic), Dq (Quadratic Mean Diameter); β

_{1}to β

_{5}are corresponding fixed effect coefficients; s(Dq) is a univariate penalized cubic regression spline [31] over Dq (Figure 2a); b

_{i}is the random effect specific for the forest management unit, and s

_{3}is its standard deviation; b

_{ij}is the stand specific random effect, and s

_{4}is its standard deviation; ε

_{ij}represents the i.i.d errors and s

_{5}represents their standard deviations.

Intercept | Fixed Effect of | Random Effect | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

SDI | SPI | SI | Hmax | HD | s(Dq) | b_{i} | b_{ij} | ε_{ijk} | ||

Symbol ^{1} | β_{0} | β_{1} | β_{2} | -- | β_{4} | β_{5} | -- | s_{3} | s_{4} | s_{5} |

Value | 40,074 | −4.59 | −1194 | -- | 255 | −4610 | -- | 1150 | 3667 | 7684 |

SE | 251 | 0.48 | 289 | -- | 26 | 571 | -- | |||

p-value | <0.0001 | <0.0001 | <0.0001 | -- | <0.0001 | <0.0001 | <0.0001 |

^{1}In Equation (4b).

**Table 6.**The increase of the AIC related to each predictor when it was removed as the only one from Equation (4a) and the nested model of Equation (4b) with minimum AIC (see Section 2.4). Abbreviations are NFI (German National Forest Inventory) and BSFI (Bavarian State Forest Inventory).

Model | Increase in AIC Related to Predictor | |||||
---|---|---|---|---|---|---|

SDI | SPI | SI | Hmax | HD | s(Dq) | |

NFI (Equation (4a)) | 96 | 14 | 10 | 127 | 10 | 229 |

BSFI (Equation (4b)) | 388 | 67 | -- | 406 | 253 | 2682 |

© 2018 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**

Poschenrieder, W.; Biber, P.; Pretzsch, H.
An Inventory-Based Regeneration Biomass Model to Initialize Landscape Scale Simulation Scenarios. *Forests* **2018**, *9*, 212.
https://doi.org/10.3390/f9040212

**AMA Style**

Poschenrieder W, Biber P, Pretzsch H.
An Inventory-Based Regeneration Biomass Model to Initialize Landscape Scale Simulation Scenarios. *Forests*. 2018; 9(4):212.
https://doi.org/10.3390/f9040212

**Chicago/Turabian Style**

Poschenrieder, Werner, Peter Biber, and Hans Pretzsch.
2018. "An Inventory-Based Regeneration Biomass Model to Initialize Landscape Scale Simulation Scenarios" *Forests* 9, no. 4: 212.
https://doi.org/10.3390/f9040212