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Volumetrics

Warning

This library is under development, none of the presented solutions are available for download.

Process forest inventory data. Adjust volumetric equations and taper functions for later use.

Class Parameters

Volumetrics(df, tree_identifier, tree_height, tree_dbh, tree_bark,
            segment_height, segment_diameter, tree_bark)
Parameters Description
df The dataframe containing the cubage data.
tree_identifier The name of the column that contains the unique identifiers of the trees.
tree_height The name of the column that contains the total heights of the trees (meters).
tree_dbh The name of the column containing the diameter at breast height (DBH) values of the trees (centimeters).
tree_bark (Optional) The name of the column containing the bark thickness values of the trees (centimeters). If tree_bark == None return only volumes with bark on 'get_volumes()' method.
segment_height (Optional) The name of the column containing the heights of the cubed segments of the trees (meters). Required for fit_taper_functions() method.
segment_diameter (Optional) The name of the column containing the diameters of the cubed segments of the trees (centimeters). Required for fit_taper_functions() method.
tree_bark (Optional) The name of the column containing tree bark (centimeters).

Class Functions

functions and parameters
  Volumetrics.get_volumes()  
  Volumetrics.fit_taper_functions(models, iterator, save_dir = None)#(1)!

  Volumetrics.fit_volumetric_functions(models, iterator, vol_column,
                                       save_dir = None)#(2)!

  Volumetrics.get_individual_diameter(hi, tree_height, tree_dbh)#(3)!

  Volumetrics.get_individual_taper_volume(tree_height, tree_dbh, stump=0.1)#(4)!

  Volumetrics.get_individual_volume(tree_height, tree_dbh)#(5)!

  1. models = (Optional) List of models to be fitted! If models == None uses all available models.
    iterator = (Optional) A column name string. Defines wich column will be used as a iterator. Could be a farm name, plot name, code or any unique identification tag.
    save_dir = (Optional) A directory to save the fitted function parameters and ann model.
  2. models = (Optional) List of models to be fitted! If models == None uses all available models.
    iterator = (Optional) A column name string. Defines wich column will be used as a iterator. Could be a farm name, plot name, code or any unique identification tag.
    vol_column = (Optional) A column name to the volume values. If vol_column == None, it uses de volumes obtained by the get_volumes() method to fit the volumetric functions.
    save_dir = (Optional) A directory to save the fitted function parameters and ann model.
  3. hi = Fraction of height from which you want to obtain the diameter (meters).
    tree_height = Total height of the tree (meters).
    tree_dbh = Diameter at breast height (DBH) value of the tree (centimeters).
  4. tree_height = Total height of the tree (meters).
    tree_dbh = Diameter at breast height (DBH) of the tree (centimeters).
    stump = Stump height (meters). By default, uses stump = 0.1.
  5. tree_height = Total height of the tree (meters).
    tree_dbh = Diameter at breast height (DBH) of the tree (centimeters).
Methods Description
.get_volumes() Returns the volume of each cubed segment and the total volume of each tree separated by tree_identifier. If tree_bark == None, it returns only the volume with bark; otherwise, it returns the volume with and without bark.
.fit_taper_functions() Fits the available taper function models. Saves a .json file with the coefficients for each fitted model and a .pkl file for the fitted ANN's.
.get_individual_diameter() Returns a pandas dataframe with diameter at a given hi height of the tree for each fitted taper model.
.get_individual_taper_volume() Returns a pandas DataFrame with the estimated volume for the provided height and diameter at breast height for each fitted taper function. It uses the integration of taper functions to obtain the result.
.get_individual_volume() Returns a pandas DataFrame with the estimated volume for the provided height and diameter at breast height for each fitted volumetric function.

Example Usage

Consider a volume dataset composed of 50 Eucalyptus trees, in which diameters were measured at heights of 0.1 meter, 0.6 meter, 1.30 meters, and, from that point onward, at 2-meter intervals along the stem.

Download example file.

First 5 rows of the file:

Fazenda arvore_n dap (cm) altura_total (m) seção (m) diametro_c_casca (cm) diametro_s_casca (cm) casca (cm)
Fazenda 1 1 24,8 28,0 0,1 30,05 26,51 1,74
Fazenda 1 1 24,8 28,0 0,6 26,55 23,15 1,96
Fazenda 1 1 24,8 28,0 1,3 24,75 21,93 1,94
Fazenda 1 1 24,8 28,0 2,0 23,85 21,48 1,90
Fazenda 1 1 24,8 28,0 4,0 21,40 19,30 1,11

taper_functions_example.py
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from fptools.Volumetrics import Volumetrics#(1)!

import pandas as pd#(2)!

  1. Import Volumetrics class.
  2. Import pandas for data manipulation.

taper_functions_example.py
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dados = pd.read_excel(r'C:\seu\diretório\exemplo_volumetrics.xlsx')#(1)!

vol = Volumetrics(df=dados, tree_identifier='arvore_n',
                  tree_height='altura_total (m)',
                  tree_dbh ='dap (cm)',
                  segment_height='seção (m)',
                  segment_diameter= 'diametro_c_casca (cm)',
                  tree_bark='casca (cm)')#(2)!

calculated_volumes_df = vol.get_volumes(iterator="Fazenda")#(3)!
calculated_volumes_df.to_excel("exemplo_volumetrics_with_vol.xlsx",
                               index=False)#(4)!

metrics_taper = vol.fit_taper_functions(save_dir="seu/diretório/para/salvar",iterator="Fazenda")#(5)!

individual_diameter = vol.get_individual_diameter(10, 31, 30)#(6)!

taper_volume = vol.get_individual_taper_volume(35, 32, stump=0.15)#(7)!

metrics_vol = vol.fit_volumetric_functions(save_dir=output_path, iterator="Fazenda")#(8)!

individual_volume = vol.get_individual_volume(tree_height=21.8, tree_dbh=19)#(9)!

  1. Load your .xlsx file.
  2. Create the variable vol containing the Volumetrics class.
  3. Calculate the volumes for each tree and segment, grouped by iterator, and save the results in the variable calculated_volumes_df. The original DataFrame will be returned with the following additional columns: segment_vol_with_bark, segment_vol_without_bark, tree_vol_with_bark, tree_vol_without_bark, bark_factor, and mean_bark_factor.
  4. Save the DataFrame with the calculated volumes as exemplo_volumetrics_with_vol.xlsx.
  5. Fit taper functions and artificial neural networks for each iterator, saving the performance metrics in the variable metrics_taper. This will create a .json file with the model coefficients and .pkl files for the trained ANN models, storing all generated files in save_dir.
  6. Get the diameter at 10 meters for a tree with a total height of 31 meters and a diameter at breast height (DBH) of 30 centimeters using the fitted models, saving it in the variable individual_diameter.
  7. Get the volumes calculated by integrating the taper functions for a tree with a height of 35 meters and DBH of 32 centimeters, considering a stump height of 0.15 meters, and save it in the variable taper_volume.
  8. Fit all volumetric functions for each iterator and save the results in save_dir, storing the metrics in the variable metrics_vol. Since no vol_column was provided, it will use the volume obtained through the get_volumes method to fit the models.
  9. Calculate the volume of a tree with a height of 21.8 meters and a DBH of 19 centimeters using all volumetric functions and save it in the variable individual_volume.

Output

Tables

calculated_volumes_df(1)

  1. Initial DataFrame with the following additional columns: segment_vol_with_bark, segment_vol_without_bark, tree_vol_with_bark, tree_vol_without_bark, bark_factor, mean_bark_factor
Fazenda arvore_n dap (cm) altura_total (m) seção (m) diametro_c_casca (cm) diametro_s_casca (cm) casca (cm) segment_vol_with_bark segment_vol_without_bark tree_vol_with_bark tree_vol_without_bark bark_factor mean_bark_factor
Fazenda 1 1 24,84076433 28 0,1 30,05 26,51 1,74 0,007092165 0,005544635 0,559879936 0,387477996 0,692073373 0,682430303
Fazenda 1 1 24,84076433 28 0,6 26,55 23,15 1,96 0,031450877 0,023341421 0,559879936 0,387477996 0,692073373 0,682430303
Fazenda 1 1 24,84076433 28 1,3 24,75 21,93 1,94 0,036171179 0,026055822 0,559879936 0,387477996 0,692073373 0,682430303
Fazenda 1 1 24,84076433 28 2,0 23,85 21,48 1,90 0,032463883 0,02310445 0,559879936 0,387477996 0,692073373 0,682430303
Fazenda 1 1 24,84076433 28 4,0 21,40 19,30 1,11 0,080407591 0,065402308 0,559879936 0,387477996 0,692073373 0,682430303
Fazenda 1 1 24,84076433 28 6,0 20,50 19,17 1,81 0,068942643 0,047175553 0,559879936 0,387477996 0,692073373 0,682430303

About the generated columns

  • segment_vol_with_bark: Stem segment with bark volume, calculated between two consecutive measurement sections.
  • segment_vol_without_bark: Stem segment without bark volume, calculated between two consecutive measurement sections.
  • tree_vol_with_bark: Total tree with bark volume, obtained by summing all segment volumes with bark.
  • tree_vol_without_bark: Total tree without bark volume, obtained by summing all segment volumes without bark.
  • bark_factor: Bark factor of the segment, calculated as the ratio between the volume with bark and the volume without bark for that segment.
  • mean_bark_factor: Average bark factor of the tree, calculated as the mean of the bark factors from all segments.

metrics_taper(1)

  1. DataFrame with evaluation metrics generated for each iterator and taper model and artificial neural network, including the score assigned to each model based on its performance.
iterator model MAE MAPE MSE RMSE R squared Explained Var Mean Error score
Fazenda 1 kozak 0,580666 5,718699 0,572843 0,756864 0,994621 0,994621 -0,000167 10
Fazenda 1 bi 0,635942 5,972947 0,643045 0,801901 0,993962 0,994016 -0,075560 9
Fazenda 1 ann 0,576319 5,651187 0,560707 0,748804 0,994735 0,994739 0,020658 8
Fazenda 1 schoepfer 0,769151 7,532993 1,033545 1,016634 0,990295 0,990356 -0,080168 7
Fazenda 1 johnson 1,078775 10,108434 1,978243 1,406500 0,981425 0,981571 -0,124507 6
Fazenda 1 matte 1,424262 14,718423 3,337336 1,826838 0,968664 0,969344 0,269282 5
Fazenda 2 ann 1,342497 7,238220 3,318417 1,821652 0,988023 0,988023 0,001891 10
Fazenda 2 kozak 1,350729 7,357823 3,468079 1,862278 0,987483 0,987483 -0,002186 9
Fazenda 2 bi 1,374535 7,880664 3,483492 1,866412 0,987427 0,987435 -0,045863 8
Fazenda 2 schoepfer 1,678529 8,785657 4,895865 2,212660 0,982329 0,982373 0,110280 7
Fazenda 2 matte 2,180025 13,928099 7,414827 2,723018 0,973238 0,974475 0,585444 6
Fazenda 2 johnson 2,216439 12,099970 9,165003 3,027376 0,966921 0,966956 0,097858 5

individual_diameter(1)

  1. DataFrame with the individual diameters estimated by the taper functions and artificial neural networks for each iterator and taper model, for a tree with a height of 31 meters, a DBH of 30 centimeters, at a height of 10 meters.
iterator model Predicted_diameter (cm)
Fazenda 1 schoepfer 22,75296935
Fazenda 1 bi 22,35954058
Fazenda 1 kozak 22,45146956
Fazenda 1 johnson 23,41945889
Fazenda 1 matte 22,83651908
Fazenda 1 ann 23,33443805
Fazenda 2 schoepfer 21,48912862
Fazenda 2 bi 23,08526645
Fazenda 2 kozak 22,40805692
Fazenda 2 johnson 22,46401882
Fazenda 2 matte 21,43830806
Fazenda 2 ann 23,71472482

taper_volume(1)

  1. DataFrame with the volumes estimated by integrating the taper functions and artificial neural networks for each iterator and taper model, for a tree with a height of 35 meters and a DBH of 32 centimeters.
iterator model Predicted_volume (m³)
Fazenda 1 schoepfer 1,166056704
Fazenda 1 bi 1,143061281
Fazenda 1 kozak 1,165284829
Fazenda 1 johnson 1,201758409
Fazenda 1 matte 1,150231088
Fazenda 1 ann 1,326654958
Fazenda 2 schoepfer 1,055148596
Fazenda 2 bi 1,179288353
Fazenda 2 kozak 1,147756782
Fazenda 2 johnson 1,108690858
Fazenda 2 matte 1,066598472
Fazenda 2 ann 1,178903371

metrics_vol(1)

  1. DataFrame with evaluation metrics generated for each iterator, volumetric model, and artificial neural network, including the score assigned to each model based on its performance.
iterator model MAE MAPE MSE RMSE R squared Explained Var Mean Error score
Fazenda 1 honner 0,028987 6,510184 0,001409 0,037535 0,998022 0,998143 -0,009291 10
Fazenda 1 ann 0,020789 5,362507 0,000713 0,026703 0,998999 0,998999 -0,000569 9
Fazenda 1 takata 0,027501 5,250705 0,001291 0,035934 0,998187 0,998214 -0,004318 8
Fazenda 1 spurr_log 0,026612 4,438766 0,001235 0,035137 0,998267 0,998277 -0,002731 7
Fazenda 1 schumacher_hall 0,027584 4,649301 0,001203 0,034683 0,998311 0,998321 -0,002636 6
Fazenda 1 meyer 0,025136 6,229777 0,001112 0,033346 0,998439 0,998439 0,000000 5
Fazenda 1 stoate 0,026301 6,530368 0,001158 0,034029 0,998374 0,998374 0,000000 4
Fazenda 1 spurr 0,026008 9,348743 0,001177 0,034306 0,998348 0,998348 0,000000 3
Fazenda 1 naslund 0,024859 11,898225 0,001020 0,031930 0,998569 0,998571 0,001263 2
Fazenda 1 ogaya 0,410646 151,1923 0,213968 0,462567 0,699650 0,719168 -0,117918 1
Fazenda 2 naslund 0,153069 6,603060 0,048230 0,219612 0,974528 0,974529 -0,001349 10
Fazenda 2 schumacher_hall 0,160824 9,781453 0,048410 0,220023 0,974433 0,974447 -0,005183 9
Fazenda 2 spurr_log 0,160540 9,932109 0,048446 0,220105 0,974414 0,974428 -0,005270 8
Fazenda 2 takata 0,159037 6,922203 0,049057 0,221488 0,974091 0,974091 -0,000386 7
Fazenda 2 ann 0,150892 5,508166 0,043498 0,208562 0,977027 0,977053 0,007015 6
Fazenda 2 meyer 0,155147 8,411204 0,047824 0,218687 0,974742 0,974742 0,000000 5
Fazenda 2 stoate 0,161432 14,918844 0,048986 0,221328 0,974129 0,974129 0,000000 4
Fazenda 2 spurr 0,181962 28,583026 0,055897 0,236425 0,970479 0,970479 0,000000 3
Fazenda 2 honner 0,189541 6,829942 0,062964 0,250927 0,966746 0,967397 0,035110 2
Fazenda 2 ogaya 0,430466 40,223791 0,284181 0,533086 0,849913 0,850523 0,034001 1

individual_volume(1)

  1. DataFrame with the volumes estimated by the volumetric functions and artificial neural networks for each iterator, for a tree with a height of 21.8 meters and a DBH of 19 centimeters.
iterator model Predicted_volume (m³)
Fazenda 1 spurr 0,241827831
Fazenda 1 schumacher_hall 0,248607044
Fazenda 1 honner 0,262977386
Fazenda 1 ogaya 0,860375777
Fazenda 1 naslund 0,245824632
Fazenda 1 takata 0,255820523
Fazenda 1 spurr_log 0,251481659
Fazenda 1 meyer 0,228396501
Fazenda 1 stoate 0,241410100
Fazenda 1 ann 0,271805848
Fazenda 2 spurr 0,421606899
Fazenda 2 schumacher_hall 0,332105882
Fazenda 2 honner 0,232727813
Fazenda 2 ogaya 0,939382897
Fazenda 2 naslund 0,324612548
Fazenda 2 takata 0,287273333
Fazenda 2 spurr_log 0,336234242
Fazenda 2 meyer 0,200621992
Fazenda 2 stoate 0,242861095
Fazenda 2 ann 0,284503571
.json Files

For each taper and volumetric function, .json files are generated containing the estimated coefficients of each model. An individual .json file is created for each iterator, named according to its respective identifier.

.pkl Files

Similarly, for each artificial neural network trained for tapering or volume prediction, a .pkl file is generated containing the trained model, named according to the respective iterator.

  • taper_model_ann_Fazenda 1.pkl
  • taper_model_ann_Fazenda 2.pkl
  • volumetric_ann_Fazenda 1.pkl
  • volumetric_ann_Fazenda 2.pkl

Although this type of file cannot be viewed directly, it stores the neural network parameters and internal configurations, which will be used later in the generation of assortments.

Available taper models

  • schoepfer

    • \[ \operatorname{d_i} =dbh\left( b_0 + b_1 \left( \frac{h_i}{H} \right) + b_2 \left( \frac{h_i}{H} \right)^2 + b_3 \left( \frac{h_i}{H} \right)^3 + b_4 \left( \frac{h_i}{H} \right)^4 + b_5 \left( \frac{h_i}{H} \right)^5 \right) \]

  • bi

    • \[ \operatorname{d_i}=dbh\left[ \left( \frac{log\;sin \left( \frac{\pi}{2} \frac{h_i}{H} \right)} {log\;sin \left( \frac{\pi}{2} \frac{1.3}{H} \right)} \right) ^{\beta_0+\beta_1sin\left(\frac{\pi}{2}\frac{h_i}{H}\right)+\beta_2sin\left(\frac{3\pi}{2}\frac{h_i}{H}\right)+\beta_3sin\left(\frac{\pi}{2}\frac{h_i}{H}\right)/\frac{h_i}{H}+\beta_4dbh+\beta_5\frac{h_i}{H}\sqrt{dbh}+\beta_6\frac{h_i}{H}\sqrt{H}} \right] \]

  • kozak

    • \[ \operatorname{d_i} =b_0 \cdot (dbh^{b_1}) \cdot (h^{b_2}) \cdot \left(\frac{1 - \left(\frac{h_i}{h}\right)^{1/4}}{1 - \left(p^{1/3}\right)}\right)^{b_3 \cdot \left(\frac{h_i}{h}\right)^4 + b_4 \cdot \left(\frac{1}{e^{dbh/h}}\right) + b_5 \cdot \left(\frac{1 - \left(\frac{h_i}{h}\right)^{1/4}}{1 - \left(p^{1/3}\right)}\right)^{0.1} + b_6 \cdot \left(\frac{1}{dbh}\right) + b_7 \cdot \left(h^{1 - \left(\frac{h_i}{h}\right)^{1/3}}\right) + b_8 \cdot \left(\frac{1 - \left(\frac{h_i}{h}\right)^{1/4}}{1 - \left(p^{1/3}\right)}\right)} \]

  • johnson

    • \[ \operatorname{d_i} = dbh \cdot \left( b_0 \cdot \log\left( \frac{b_1 + \frac{(H - h_i)}{(H - 1.3)}}{b_2} \right) \right) \]

  • matte

    • \[ \operatorname{d_i} = dbh \cdot \left( b_0 \cdot \left( \frac{H - h_i}{H - 1.30} \right)^2 + b_1 \cdot \left( \frac{H - h_i}{H - 1.30} \right)^3 + b_2 \cdot \left( \frac{H - h_i}{H - 1.30} \right)^4 \right) \]

  • ann

    • Notation

    • \( β_n \): Fitted parameters
    • \( d_i \): Diameter (cm)
    • \( \text{dbh} \): Diameter at breast height (cm)
    • \( H \): Total height (m)
    • \( h_i \): Segment height (m)

    Artificial Neural Network

    When selecting the 'ann' model, 6 different structures of artificial neural networks will be tested. Only the result from 1 model will be returned. The model returned will be selected by the ranking function.
    For the 'ann' model, the module sklearn.neural_network.MLPRegressor is used.

    --- title: ANN Parameters --- classDiagram direction LR class MLPRegressor { Epochs: 3000 Activation: logistic Solver Mode: lbfgs Batch size: dynamic Learning rate init: 0.1 Learning rate mode: adaptive } class Model-0 { Hidden layer sizes: (15, 25, 20, 30, 10) } class Model-1 { Hidden layer sizes: (35, 10, 25, 35, 15) } class Model-2 { Hidden layer sizes: (25, 15, 30, 20) } class Model-3 { Hidden layer sizes: (15, 35, 45) } class Model-4 { Hidden layer sizes: (35, 10, 25, 35, 15) } class Model-5 { Hidden layer sizes: (35, 10, 25, 35, 15, 20, 15, 30) } MLPRegressor <|-- Model-0 MLPRegressor <|-- Model-1 MLPRegressor <|-- Model-2 MLPRegressor <|-- Model-3 MLPRegressor <|-- Model-4 MLPRegressor <|-- Model-5

    Available volumetric models

  • spurr

    • \[ \operatorname{V} = b_0 + b_1 \cdot (\text{dbh}^2) \cdot H \]

  • schumacher_hall

    • \[ \operatorname{V} = b_0 \cdot (\text{dbh}^{b_1}) \cdot (H^{b_2}) \]

  • honner

    • \[ \operatorname{V} = \frac{\text{dbh}^2}{b_1 \cdot \left(\frac{1}{H}\right)} \]

  • ogaya

    • \[ \operatorname{V} = \frac{\text{dbh}^2}{b_1 \cdot H} \]

  • stoate

    • \[ \operatorname{V} = b_0 + b_1 \cdot \text{dbh}^2 + b_2 \cdot \text{dbh}^2 \cdot H + b_3 \cdot H \]

  • naslund

    • \[ \operatorname{V} = b_1 \cdot \text{dbh}^2 + b_2 \cdot \text{dbh}^2 \cdot H + b_3 \cdot \text{dbh} \cdot H^2 + b_4 \cdot H^2 \]

  • takata

    • \[ \operatorname{V} = \frac{\text{dbh}^2 \cdot H}{b_0 + b_1 \cdot \text{dbh}} \]

  • spurr_log

    • \[ \operatorname{V} = \exp(b_0 + b_1 \cdot \log(\text{dbh}^2 \cdot H)) \]

  • meyer

    • \[ \operatorname{V} = b_0 + b_1 \cdot \text{dbh}^2 + b_2 \cdot \text{dbh} + b_3 \cdot \text{dbh} \cdot H + b_4 \cdot \text{dbh}^2 \cdot H \]

  • ann

    • Use the same ann models used for taper function.

    Notation

    • \( b_n \): Fitted parameters
    • \( V \): Estimated volume (m³)
    • \( \text{dbh} \): Diameter at breast height (cm)
    • \( H \): Total height (m)

    Ranking function

    To select the best-performing models and rank them accordingly, the following metrics are obtained:

    Métric name Structure
    Mean Absolute Error (MAE) \( MAE = \frac{1}{n} \sum_{i=1}^{n} \|y_i - \hat{y}_i\| \)
    Mean Absolute Percentage Error (MAPE) \( MAPE = \frac{100}{n} \sum_{i=1}^{n} \left\|\frac{y_i - \hat{y}_i}{y_i}\right\| \)
    Mean Squared Error (MSE) \( MSE = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2 \)
    Root Mean Squared Error (RMSE) \( RMSE = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2} \)
    R Squared (Coefficient of Determination) \( R^2 = 1 - \frac{\sum_{i=1}^{n} (y_i - \hat{y}_i)^2}{\sum_{i=1}^{n} (y_i - \bar{y})^2} \)
    Explained Variance (EV) \( EV = 1 - \frac{Var(y - \hat{y})}{Var(y)} \)
    Mean Error \( Mean\ Error = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i) \)
    Notation
    • \( y_i \): Observed value for the i-th observation
    • \( \hat{y}_i \): Estimated (predicted) value for the i-th observation
    • \( n \): Total number of observations
    • \( \bar{y} \): Mean of the observed values

    After obtaining the metrics for each tested model, the best model receives a score of 10, while the others receive scores of 9, 8, and so on.

    References

    BI, H. (2000). Trigonometric variable-form taper equations for Australian eucalypts. Forest Science, 46(3), 397–409.

    JOHNSON, T. (1911). Taxatariska undersökringar om skogsträdens form. Skgsvardsföreningens tiedskrifle. Häfte, Berlim, 9(10), 285–329.

    KOZAK, A. (2004). My last words on taper equations. The Forestry Chronicle, 80(4), 507–515.

    MATTE, L. (1949). The taper of coniferous species with special reference to Loblolly Pine. Forestry Chronicle, Mattawa, 25(1), 21–31.

    MEYER, H. A. (1940). A mathematical expression for height curves. Journal of Forestry, 38, 415–420. https://doi.org/10.1093/jof/38.5.415

    NÄSLUND, M. (1936). Skogsförsöksanstaltens gallringsförsök i tallskog. Meddelanden från Statens Skogsförsöksanstalt, Swedish Institute of Experimental Forestry, 29: 169.

    SCHÖEPFER, W. (1966). Automatisierung des Massen-, Sorten- und Wertberechnung stehender Waldbestände. Schriftenreihe Bad. Wurtt-Forstl.

    SCHUMACHER, F. X.; HALL, F. S. (1933). Logarithmic expression of timber tree volume. Journal of Agricultural Research, Washington, 47(9), 719–734.

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