# Temporal Dynamics of Root Reinforcement in European Spruce Forests

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Workflow and Methodological Approaches

#### 2.2. Data Sources

#### 2.3. Stem Diameter Growth Modelling

#### 2.4. Estimation of Mean Stand Age

#### 2.5. Upscaling of Root Reinforcement

#### 2.6. Model for the Temporal Dynamics of Root Reinforcement after Disturbances

## 3. Results

#### 3.1. Modeling of Growth Rate

#### 3.2. Calculation of Root Reinforcement

#### 3.3. Modelling Root Reinforcement Dynamic

## 4. Discussion

#### 4.1. Comparison of the Two Growth Rate Models

#### 4.2. Correlation between Mean DBH and Stand Age

#### 4.3. Upscaling of Root Reinforcement

#### 4.4. Root Reinforcement Dynamics

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DBH | Diameter on Breast Height |

NFI | National Forestry Inventory |

NLM | Nonlinear Model |

LM | Linear Model |

QMD | Quadratic Mean Diameter |

RBMw | Root Bundle Model Weibull |

RR | Root Reinforcement |

SDI | Stand Density Index |

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**Figure 1.**Flowchart showing the datasets and modeling steps done to quantify and model the temporal dynamics of root reinforcement in spruce forests.

**Figure 2.**Sensitivity analysis of the linear growth rate model. The data are presented in 2 cm DBH classes from 0.12 m up to 0.76 m. For each parameter, model predictions using median values for predictors and their 90th percentile interval are shown: (

**a**) SDI, (

**b**) altitude, (

**c**) slope, (

**d**) aspect, (

**e**) all variables. The yellow line shows the percent frequency of data included in each DBH class.

**Figure 3.**Comparison of the root reinforcement as function of the DBH for Swiss NFI data (n = 7191) and European data (n = 2640).

**Figure 4.**Root reinforcement dynamics over time, according to the model after Weibull. The data are presented in 10-year classes from 30 up to 250 years. The yellow line shows the distribution of data frequency in percentage.

**Figure 5.**Predicted values of RR using the Weibull RR-model compared with the calculated values based on Swiss NFI. The numbers on the top show the sample size of each box. The red line shows the 1:1 fitting.

**Figure 6.**Sensitivity analysis of the different variables on the RR dynamics over time. The altitude has the highest influence on the RR.

**Figure 7.**Comparison of the increments of the basal mean stem (dg) of spruce according to the Swiss yield table [52] with integration of the BHD increments resulting from the Swiss NFI data. The red dots stand for the subalpine altitudinal stage, blue stand for high montane, light green upper/lower montane, and dark green stands for the submontane/colline altitudinal stage.

Coefficient | Value | Std Error | Error t-Value | p-Value |
---|---|---|---|---|

Slope x1 | $-2.819\times {10}^{-4}$ | $2.928\times {10}^{-5}$ | −9.631 | <0.001 |

Altitude x2 | $-1.674\times {10}^{-5}$ | $2.025\times {10}^{-6}$ | −8.267 | <0.001 |

Aspect x3 | $1.062\times {10}^{-5}$ | $1.413\times {10}^{-5}$ | 0.751 | 0.452 |

SDI x4 | $-4.993\times {10}^{-5}$ | $2.329\times {10}^{-6}$ | −21.438 | <0.001 |

Slope y1 | $6.637\times {10}^{-5}$ | $7.763\times {10}^{-6}$ | 8.550 | <0.001 |

Altitude y2 | $3.865\times {10}^{-6}$ | $4.579\times {10}^{-7}$ | 9.502 | <0.001 |

Aspect y3 | $4.974\times {10}^{-7}$ | $3.156\times {10}^{-6}$ | −0.771 | 0.441 |

SDI y4 | $1.031\times {10}^{-5}$ | $5.279\times {10}^{-7}$ | 19.523 | <0.001 |

c_1 | $1.120\times {10}^{-1}$ | $2.822\times {10}^{-3}$ | 39.694 | <0.001 |

c_2 | $-2.528\times {10}^{-2}$ | $7.494\times {10}^{-4}$ | −33.740 | <0.001 |

Coefficient | Value | Std Error | Error t-Value | p-Value |
---|---|---|---|---|

Slope x1 | $-2.033\times {10}^{-3}$ | $2.695\times {10}^{-4}$ | −7.543 | <$2\times {10}^{-16}$ |

Altitude x2 | $-1.301\times {10}^{-4}$ | $1.783\times {10}^{-5}$ | −7.299 | <$2\times {10}^{-16}$ |

Aspect x3 | $-5.059\times {10}^{-5}$ | $1.246\times {10}^{-4}$ | −0.406 | 0.6848 |

SDI x4 | $-4.255\times {10}^{-4}$ | $2.086\times {10}^{-5}$ | −20.400 | <0.001 |

Slope y1 | $1.671\times {10}^{-5}$ | $7.129\times {10}^{-6}$ | 2.343 | 0.0191 |

Altitude y2 | $1.164\times {10}^{-6}$ | $4.754\times {10}^{-7}$ | 2.449 | 0.0144 |

Aspect y3 | $8.503\times {10}^{-7}$ | $3.223\times {10}^{-6}$ | 0.264 | 0.7919 |

SDI y4 | $6.143\times {10}^{-6}$ | $5.461\times {10}^{-7}$ | 11.249 | <0.001 |

c$\_1$ | $9.049\times {10}^{-1}$ | $2.388\times {10}^{-2}$ | 37.885 | <0.001 |

c$\_2$ | $5.751\times {10}^{-3}$ | $6.393\times {10}^{-4}$ | −8.996 | <0.001 |

Coefficient | Value | Std Error | Error t-Value | p-Value |
---|---|---|---|---|

Slope x1 | $-2.035\times {10}^{-3}$ | $2.694\times {10}^{-4}$ | −7.553 | $4.70\times {10}^{-14}$ |

Altitude x2 | $-1.306\times {10}^{-4}$ | $1.780\times {10}^{-5}$ | −7.339 | $2.34\times {10}^{-13}$ |

SDI x4 | $-4.257\times {10}^{-4}$ | $2.084\times {10}^{-5}$ | −20.429 | <0.001 |

Slope y1 | $1.678\times {10}^{-5}$ | $7.127\times {10}^{-6}$ | 2.355 | 0.0186 |

Altitude y2 | $1.170\times {10}^{-6}$ | $4.747\times {10}^{-7}$ | 2.466 | 0.0137 |

SDI y4 | $6.147\times {10}^{-6}$ | $5.453\times {10}^{-7}$ | 11.272 | <0.001 |

c_1 | $9.014\times {10}^{-1}$ | $2.231\times {10}^{-2}$ | 40.411 | <0.001 |

c_2 | $-5.691\times {10}^{-3}$ | $6.019\times {10}^{-4}$ | −9.455 | <0.001 |

**Table 4.**Coefficient of the RR~age model. The RMSE has a value of 1.738 kN/m on 7192 degrees of freedom.

Coefficient | Value | Std Error | Error t-Value | p-Value |
---|---|---|---|---|

$q1$ | −0.06986 | 0.0027 | −25.98 | <0.001 |

$q2$ | −0.0085 | 0.0002 | −39.38 | <0.001 |

lambda | 150.04 | 2.7191 | 55.18 | <0.001 |

k | 2.017 | 0.0293 | 68.79 | <0.001 |

s_1 | 27.87 | 0.5932 | 46.99 | <0.001 |

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**MDPI and ACS Style**

Flepp, G.; Robyr, R.; Scotti, R.; Giadrossich, F.; Conedera, M.; Vacchiano, G.; Fischer, C.; Ammann, P.; May, D.; Schwarz, M.
Temporal Dynamics of Root Reinforcement in European Spruce Forests. *Forests* **2021**, *12*, 815.
https://doi.org/10.3390/f12060815

**AMA Style**

Flepp G, Robyr R, Scotti R, Giadrossich F, Conedera M, Vacchiano G, Fischer C, Ammann P, May D, Schwarz M.
Temporal Dynamics of Root Reinforcement in European Spruce Forests. *Forests*. 2021; 12(6):815.
https://doi.org/10.3390/f12060815

**Chicago/Turabian Style**

Flepp, Gianluca, Roger Robyr, Roberto Scotti, Filippo Giadrossich, Marco Conedera, Giorgio Vacchiano, Christoph Fischer, Peter Ammann, Dominik May, and Massimiliano Schwarz.
2021. "Temporal Dynamics of Root Reinforcement in European Spruce Forests" *Forests* 12, no. 6: 815.
https://doi.org/10.3390/f12060815