In Silviculturalist/forester: Forest Science and Forestry

knitr::opts_chunk$set(
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Vorläufige Fichten-Ertragstafel für Bayern

von E. Assmann und F. Franz

In this vignette solutions for the double-logarithmic parabolic functions used by F. Franz and E. Assmann for their interim yield tables for Norway Spruce in Bavaria are presented.

Original source:

Assmann, E. & Franz, F. 1965. Vorläufige Fichten-Ertragstafel für Bayern. Forstwissenschaftliches Centralblatt 84(1/2):13-43. Available online (2022-04-06): \url{https://www.waldwachstum.wzw.tum.de/fileadmin/publications/Assmann_1965_Vorlaeufige_Fichten-Ertragstafel.pdf}

Definitions: $A$ : age at breast height (1.3 m)

$h_0$ : dominant height , Bestandesoberhöhe.

$HO$ : Site index at age 100, Oberhöhenbonität im Bezugsalter 100.

$h_m$ : \~Lorey's mean height, Bestandesmittelhöhe (mittlere höhe des bestandes-grundflächenmittelstammes)

$G_{max}$ : Natural Basal Area, Natürliche Grundfläche

$n.B.G.$ : Natural stocking level, Natürliche bestockungsgrad.

$n.B.G._{opt}$ : Stocking level of maximum volume growth (Quotient of n.B.G.), bestockungsgrad der optimale volumenzuwächse (quotient von n.B.G.).

$d_m$: Quadratic Mean Diameter, Durchmesser.

$z_v$: Volume increment at optimal stocking level, volumenzuwachs bei optimaler grundflächenh.

Equation 1.

Development of dominant height $h_0$ over age:

$$ log_{10}h_0 = a_0 + a_1\cdot log_{10}A + a_2\cdot log_{10}^2A\ \quad \ a_1 = a_{10} + a_{11} \cdot e^{lnHO} + a_{12} \cdot e^{ln^2HO}\ \quad \ a_2 = a_{20} + a_{21} \cdot e^{lnHO} + a_{22} \cdot e^{ln^2HO}\ \quad \ \Downarrow \ \quad \ a_0 = log_{10}HO - a_1\cdot log_{10}100 - a_2\cdot log_{10}^2 100\ \quad \ h_0 = 10^{a_0+a_1\cdot log_{10}A + a_2\cdot log_{10}^2A} $$

ggplot()+
  scale_x_continuous(limits=c(20,120),breaks=c(seq(20,120,10)))+
  xlab('Age')+
  ylab('Dominant height, m')+
  geom_function(fun=\(x) Franz_Assmann_1965_height_trajectory_Germany_Bavaria_Spruce(16,100,x),n=600)+
  geom_function(fun=\(x) Franz_Assmann_1965_height_trajectory_Germany_Bavaria_Spruce(20,100,x),n=600)+
  geom_function(fun=\(x) Franz_Assmann_1965_height_trajectory_Germany_Bavaria_Spruce(24,100,x),n=600)+
  geom_function(fun=\(x) Franz_Assmann_1965_height_trajectory_Germany_Bavaria_Spruce(28,100,x),n=600)+
  geom_function(fun=\(x) Franz_Assmann_1965_height_trajectory_Germany_Bavaria_Spruce(32,100,x),n=600)+
  geom_function(fun=\(x) Franz_Assmann_1965_height_trajectory_Germany_Bavaria_Spruce(36,100,x),n=600)+
  geom_function(fun=\(x) Franz_Assmann_1965_height_trajectory_Germany_Bavaria_Spruce(40,100,x),n=600)+
  theme_classic()

Equation 2.

Difference $h_0-h_m$ over age:

$$ ln(h_0-h_m) = b_0+b_1\cdot lnA+b_2\cdot ln^2A\ \quad \ (h_0-h_m){EXTREMUM} = b{00}+b_{01}\cdot HO\ \quad \ A_{EXTREMUM} = b_{10} + b_{11}\cdot HO\ \quad \ (h_0-h_m){A=20} = b{20}+b_{21}*HO\ \quad \ \Downarrow \ \quad \ b_2 = \frac{(ln(h_0-h_m){EXTR.}-ln(h_0-h_m){A=20})}{-2\cdot lnA_{EXTR.} \cdot (ln20-lnA_{EXTR.}) + ln^220-ln^2A_{EXTR.}}\ \quad \ b1 = -2b_2\cdot lnA_{EXTR.}\ \quad \ b_0 = ln(h_0-h_m){EXTR.} + 2\cdot b_2\cdot lnA{EXTR.}\cdot ln20 - b_2\cdot ln^220\ \quad \ ln(h_0-h_m) = b_0+b_1\cdot lnA+ b_2\cdot ln^2A $$

ggplot()+
xlim(c(20,120))+
ylab(latex2exp::TeX("$h_0 - h_m$"))+
xlab('Age')+
geom_function(fun=\(x) Franz_Assmann_1965_height_trajectory_Germany_Bavaria_Spruce(dominant_height = 40,age = 100,age2 = x)- Franz_Assmann_1965_Lorey_height_trajectory_Germany_Bavaria_Spruce(SIH100 = 40,age = x,YieldLevel = 1))+
geom_function(fun=\(x) Franz_Assmann_1965_height_trajectory_Germany_Bavaria_Spruce(dominant_height = 40,age = 100,age2 = x)- Franz_Assmann_1965_Lorey_height_trajectory_Germany_Bavaria_Spruce(SIH100 = 40,age = x,YieldLevel = 2),linetype=2)+
geom_function(fun=\(x) Franz_Assmann_1965_height_trajectory_Germany_Bavaria_Spruce(dominant_height = 40,age = 100,age2 = x)- Franz_Assmann_1965_Lorey_height_trajectory_Germany_Bavaria_Spruce(SIH100 = 40,age = x,YieldLevel = 3),linetype=3)+
  theme_classic()

Equation 3.

$G_{max}$ over age:

$$ lnG_{max} = c_0 + c_1\cdot lnA + c_2\cdot ln^2A\ \quad \ {lnG_{max}}{EXTR.} = c{00}+c_{01}\cdot lnHO\ \quad \ lnA_{EXTR.} = c_{10}+c_{11}\cdot lnHO \ \quad \ \frac{1}{c_2} = c_{20}+c_{21}\cdot lnHO\ \quad \ \Downarrow \ \quad \ c_1 = -2\cdot c_2 \cdot lnA_{EXTR.}\ \quad \ c_0 = {lnG_{max}}{EXTR.} - c_1\cdot lnA{EXTR.} - c_2\cdot ln^2A_{EXTR.}\ \quad \ lnG_{max} = c_0 + c_1\cdot lnA + c_2\cdot ln^2A $$

ggplot()+
  xlim(c(20,120))+
  xlab('Age')+
  ylab(latex2exp::TeX('$G_{max}/ha$'))+
  geom_function(fun=\(x) Franz_Assmann_1965_maximumBasalArea_Germany_Bavaria_Spruce(40,x,1))+
  geom_function(fun=\(x) Franz_Assmann_1965_maximumBasalArea_Germany_Bavaria_Spruce(40,x,2))+
  geom_function(fun=\(x) Franz_Assmann_1965_maximumBasalArea_Germany_Bavaria_Spruce(40,x,3))+
  geom_function(fun=\(x) Franz_Assmann_1965_maximumBasalArea_Germany_Bavaria_Spruce(30,x,1),linetype=2)+
  geom_function(fun=\(x) Franz_Assmann_1965_maximumBasalArea_Germany_Bavaria_Spruce(30,x,2),linetype=2)+
  geom_function(fun=\(x) Franz_Assmann_1965_maximumBasalArea_Germany_Bavaria_Spruce(30,x,3),linetype=2)+
  theme_classic()

Equation 4.

$n.B.G._{opt}$ over age:

$$ ln(1-n.B.G.) = d_0+d_1\cdot lnA + d_2\cdot ln^2A\ \quad \ ln(1-n.B.G.){EXTR.} = d{00} + d_{01}\cdot lnHO\ \quad \ lnA_{EXTR.} = d_{10}+d_{11}\cdot lnHO\ \quad \ d_2 = d_{20}+d_{21}\cdot lnHO\ \quad \ \Downarrow \ \quad \ d_1 = -2\cdot d_2\cdot lnA_{EXTR.}\ \quad \ d_0 = (d_{00} + d_{01}\cdot lnHO)+d_2\cdot lnA_{EXTR.}\ \quad \ n.B.G._{opt} = 1-e^{d_0 + d_1\cdot lnA + d_2\cdot ln^2A} $$

ggplot()+
scale_x_continuous(limits=c(20,120))+
  ylim(c(0.7,1))+
  xlab('Age')+
  ylab(latex2exp::TeX("$n.B.G._{opt}"))+
  geom_function(fun=\(x) Franz_Assmann_1965_optimal_BA_Germany_Bavaria_Spruce(SIH100 = 40,age = x,YieldLevel = 3),linetype=3)+
  geom_function(fun=\(x) Franz_Assmann_1965_optimal_BA_Germany_Bavaria_Spruce(SIH100 = 30,age = x,YieldLevel = 3),linetype=3)+
  geom_function(fun=\(x) Franz_Assmann_1965_optimal_BA_Germany_Bavaria_Spruce(SIH100 = 40,age = x,YieldLevel = 2),linetype=2)+
  geom_function(fun=\(x) Franz_Assmann_1965_optimal_BA_Germany_Bavaria_Spruce(SIH100 = 30,age = x,YieldLevel = 2),linetype=2)+
  geom_function(fun=\(x) Franz_Assmann_1965_optimal_BA_Germany_Bavaria_Spruce(SIH100 = 40,age = x,YieldLevel = 1),linetype=1)+
  geom_function(fun=\(x) Franz_Assmann_1965_optimal_BA_Germany_Bavaria_Spruce(SIH100 = 30,age = x,YieldLevel = 1),linetype=1)+
  theme_classic()

Equation 5.

$d_m$ over age:

$$ d_m = e_0+e_1\cdot A\ \quad \ {d_m}{A=100,n.B.G.=1.0} = e{00} + e_{01}\cdot lnHO\ \quad \ ln{d_m}{A=100} = e{10} + e_{11}\cdot ln(1.1-n.B.G.)\ \quad \ lnA_{d_m=0} = e_{20} + e_{21}\cdot lnHO\ \quad \ \Downarrow \ \quad \ e_{10} = ln(e_{00}+e_{01}\cdot lnHO)-e_{11}\cdot ln0.1\ \quad \ d_{100} = e^{e_{10}+e_{11}\cdot ln(1.1-n.B.G.)}\ \quad \ A_{d_m=0} = e^{e_{20}+e_{21}\cdot lnHO}\ \quad \ slope = \frac{d_{100}-0}{100-A_{d_m=0}}\ \quad \ intercept = d_{100}-slope100\ \quad \ d_m = intercept+slopeA $$

ggplot()+
  xlab('Age')+
  ylab(latex2exp::TeX("$d_m\t,cm$"))+
xlim(c(0,120))+
geom_function(fun=(\(x) Franz_Assmann_1965_diameter_age_Germany_Bavaria_Spruce(x,40,YieldLevel = 2)))+
geom_function(fun=(\(x) Franz_Assmann_1965_diameter_age_Germany_Bavaria_Spruce(x,40,YieldLevel = 2,nBG = 0.79)),linetype=2)+
geom_function(fun=(\(x) Franz_Assmann_1965_diameter_age_Germany_Bavaria_Spruce(x,40,YieldLevel = 2,nBG = 0.94)),linetype=3)+
geom_function(fun=(\(x) Franz_Assmann_1965_diameter_age_Germany_Bavaria_Spruce(x,30,YieldLevel = 2)))+
geom_function(fun=(\(x) Franz_Assmann_1965_diameter_age_Germany_Bavaria_Spruce(x,30,YieldLevel = 2,nBG = 0.92)),linetype=2)+
geom_function(fun=(\(x) Franz_Assmann_1965_diameter_age_Germany_Bavaria_Spruce(x,30,YieldLevel = 2,nBG = 0.98)),linetype=3)+
  theme_classic()

Equation 6.

$z_v$ over age:

$$ lnz_v=f_0+f_1\cdot lnA+f_2\cdot ln^2A\ \quad \ {lnz_v}{EXTR.} = f{00} + f_{01}\cdot lnHO\ \quad \ lnA_{EXTR.} = f_{10}+f_{11}\cdot lnHO\ \quad \ f_2=f_{20}+f_{21}\cdot lnHO\ \quad \ \Downarrow \ \quad \ f_1 = -2\cdot f_2\cdot lnA_{Extr.}\ \quad \ f_0 = (f_{00} + f_{01}\cdot lnHO)-f_1\cdot lnA_{EXTR.}-f_2\cdot lnA_{EXTR.}\cdot lnA_{EXTR.}\ \quad \ lnz_v=f_0+f_1\cdot lnA+f_2\cdot ln^2A $$

ggplot()+
  xlim(c(20,120))+
  ylim(c(10,30))+
  ylab('Current Annual Increment VfmS')+
  xlab('Age')+
  geom_function(fun=\(x) Franz_Assmann_1965_CAI_optimal_BA_Germany_Bavaria_Spruce(x,40,YieldLevel = 1))+
  geom_function(fun=\(x) Franz_Assmann_1965_CAI_optimal_BA_Germany_Bavaria_Spruce(x,40,YieldLevel = 2))+
  geom_function(fun=\(x) Franz_Assmann_1965_CAI_optimal_BA_Germany_Bavaria_Spruce(x,40,YieldLevel = 3))+
  theme_classic()

Silviculturalist/forester documentation built on April 20, 2024, 2:13 p.m.