Westfall & Laustsen (2006): Height-to-Diameter Model for Maine Tree Species

_{Predicted height vs. diameter at breast height (DBH) for a hypothetical poplar using the allometric model from Westfall and Laustsen (2006).}

Overview

This package provides predict_height_westfall, a function that predicts tree height (ft) at a given top diameter using the Chapman-Richards allometric model from Westfall and Laustsen (2006).

It supports 18 species groups in Maine and accepts diameter at breast height (DBH, inches), compacted crown ratio (percent), tree class, crown class, and an optional top diameter (inches).

Setting top_diam_in=0 (the default) returns total tree height; providing a non-zero top diameter returns the height to that stem diameter, enabling estimation of merchantable bole height.

All six parameters accept either a scalar or an array-like, so mixed-species, mixed-class stands can be predicted in a single call. When any parameter is array-like the inputs are broadcast together and a NumPy array is returned; otherwise a single float is returned.

Model description

Here, we show Equation (2) from Westfall and Laustsen (2006) modified to remove the error term and the random-effects parameters. The model is based on the Chapman-Richards growth equation (Richards 1959).

$$ \begin{aligned} H_{i} = & \left(\beta_0 D_{i} + \beta_1 CC_{1i} + \beta_2 CC_{2i} + \beta_3 CC_{3i}\right) \\ & \cdot \left(1 - \exp\left(-\beta_4 DBH_i\right)\right)^{\beta_5 CR_i + \beta_6 TC_i + \left(\frac{D_{i}}{DBH_i} + 0.01\right)^{\beta_7}} \end{aligned} $$

where:

$H_{i}$ = tree height (feet) of the $i^{th}$ tree at top diameter $D$.

$D_{i}$ = top-diameter (inches) within the $i^{th}$ tree.

$DBH_i$ = diameter at breast height (inches) of the $i^{th}$ tree.

$CC_{1i}$, $CC_{2i}$, $CC_{3i}$ = The crown class indicators of the $i^{th}$ tree. These are a set of one-hot encoded variables.

$$ CC_{ki} = \begin{cases} k = 1, ; = 1 & \text{intermediate, dead; 0 otherwise} \\ k = 2, ; = 1 & \text{dominant, codominant, open grown; 0 otherwise} \\ k = 3, ; = 1 & \text{overtopped; 0 otherwise} \end{cases} $$

$TC_i$ = The tree class of the $i^{th}$ tree. TC is an integer coded value between 1 and 3.

$$ TC_i = \begin{cases} 1 & \text{preferred} \\ 2 & \text{acceptable} \\ 3 & \text{rough/rotten cull, dead} \end{cases} $$

$CR_i$ = compacted crown ratio (%; integers 0 - 100) of the $i^{th}$ tree.

$\beta_0 \text{--} \beta_7$ = the fixed-effects population parameters.

predict_height_westfall

predict_height_westfall(
    species_group: int | array_like,
    dbh_in: float | array_like,
    ccr_pct: float | array_like,
    tree_class: str | array_like,
    crown_class: str | array_like,
    top_diam_in: float | array_like = 0.0,
    fia_spcd: int | array_like | None = None, # OPTIONAL
) -> float | numpy.ndarray

Predicts tree height (ft) at a specified top diameter using the Chapman-Richards allometric model.

Parameter	Type	Description
species_group	int or array	Species group number (1–18). Mutually exclusive with fia_spcd; exactly one must be provided.
dbh_in	float or array	Diameter at breast height (inches, > 0).
ccr_pct	float or array	Compacted crown ratio (percent, 0–100).
tree_class	str or array	One of "preferred", "acceptable", "rough", "rotten", or "dead".
crown_class	str or array	One of "dominant", "codominant", "intermediate", "overtopped", "open grown", or "dead".
top_diam_in	float or array	Top stem diameter (inches, >= 0) at which to predict height. Defaults to 0.0 for total tree height.
fia_spcd	int or array	FIA species code(s) (keyword-only). Converted to species group numbers before prediction. Mutually exclusive with species_group; exactly one must be provided.

Returns: Predicted height in feet as a float (scalar inputs) or numpy.ndarray (array inputs).

Species Groups

The species_group parameter accepts integers 1–18. The table below lists the species group number, the group name, the species associated with the group, and the FIA species codes (SPCD) that correspond to each of the species.

Group	Species Group Name	Species Names	FIA SPCD
1	Miscellaneous softwood	Larch (introduced), Norway spruce, Jack pine, Red pine, Pitch pine, Pond pine, Scotch pine	6212, 91, 105, 125, 126, 128, 130
2	Tamarack (native)	Tamarack (native)	71
3	Eastern white pine	Eastern white pine	129
4	White spruce	White spruce	94
5	Black spruce	Black spruce	95
6	Red spruce	Red spruce	97
7	Balsam fir	Balsam fir	12
8	Eastern hemlock	Eastern hemlock	261
9	Northern white-cedar	Northern white-cedar	241
10	Sugar maple	Sugar maple	318
11	Ash	White ash, Black ash, Green ash	541, 543, 544
12	Poplars	Balsam poplar, Eastern cottonwood, Bigtooth aspen, Swamp cottonwood, Quaking aspen	741, 742, 743, 744, 746
13	Miscellaneous hardwood	Shagbark hickory, Black cherry, Scarlet oak, Northern red oak, Black oak, American basswood	407, 762, 806, 833, 837, 951
14	Yellow birch	Yellow birch	371
15	Paper birch	Paper birch	375
16	Other hardwood	Maple, Striped maple, Silver maple, Mountain maple, Norway maple, Ohio buckeye, Serviceberry, Sweet birch, Gray birch, American hornbeam, Butternut, Osage-orange, Apple, Eastern hophornbeam, Pin cherry, Chokecherry, White oak, Swamp white oak, Willow, Black willow, White willow, American mountain-ash, Elm, American elm	310, 315, 317, 319, 320, 331, 356, 372, 379, 391, 601, 641, 660, 701, 761, 763, 802, 804, 920, 922, 927, 935, 970, 972
17	Red maple	Red maple	316
18	American beech	American beech	531

Demonstration calculations

Westfall and Laustsen (2006) provided the following example of how to apply the equations using a poplar tree (species group = 12) with the following attributes:

• dbh = 15.5 in.
• Compacted crown ratio = 40 percent
• Tree class = acceptable (TC = 2)
• Crown class = codominant (CC₁ = 0; CC₂ = 1; CC₃ = 0)

We demonstrate that this implementation reproduces the same results.

1) Prediction of total height

When computing total height, you set the top-height diameter to 0.

$$\begin{aligned} H_{i0} &= \left(-4.2401(0) + 84.2529(0) + 91.5048(1) + 78.7788(0)\right) \\ &\quad \cdot \left(1 - \exp(-0.1023 \cdot 15.5)\right) \\ &\quad \cdot \exp\left(0.0054(40) + 0.0638(2) + \left(\tfrac{0}{15.5} + 0.01\right)^{0.1422}\right) \\ H_{i0} &= (91.5048) \cdot (0.7943) \cdot \exp\left(0.3436 + (0.01)^{0.1422}\right) \\ H_{i0} &= 75.0 \text{ ft} \end{aligned}$$

from westfall_2006 import predict_height_westfall

total_height = predict_height_westfall(
    species_group = 12,
    dbh_in = 15.5,
    ccr_pct = 40,
    tree_class = "acceptable",
    crown_class = "codominant",
    top_diam_in = 0.0,
)
# total_height => 75.0 ft

2) Prediction of bole height (4-in. top diameter)

Compute the top height of the 4-in. diameter bole.

$$\begin{aligned} H_{i4} &= \left(-4.2401(4) + 84.2529(0) + 91.5048(1) + 78.7788(0)\right) \\ &\quad \cdot \left(1 - \exp(-0.1023 \cdot 15.5)\right) \\ &\quad \cdot \exp\left(0.0054(40) + 0.0638(2) + \left(\tfrac{4}{15.5} + 0.01\right)^{0.1422}\right) \\ H_{i4} &= (74.5444) \cdot (0.7943) \cdot \exp\left(0.3436 + (0.2681)^{0.1422}\right) \\ H_{i4} &= 56.9 \text{ ft} \end{aligned}$$

bole_height = predict_height_westfall(
    species_group = 12,
    dbh_in = 15.5,
    ccr_pct = 40,
    tree_class = "acceptable",
    crown_class = "codominant",
    top_diam_in = 4.0,
)
# bole_height => 56.9 ft

3) Prediction of sawlog height (9-in. top diameter)

Compute the top height of the 9-in. diameter bole.

$$\begin{aligned} H_{i9} &= \left(-4.2401(9) + 84.2529(0) + 91.5048(1) + 78.7788(0)\right) \\ &\quad \cdot \left(1 - \exp(-0.1023 \cdot 15.5)\right) \\ &\quad \cdot \exp\left(0.0054(40) + 0.0638(2) + \left(\tfrac{9}{15.5} + 0.01\right)^{0.1422}\right) \\ H_{i9} &= (53.3439) \cdot (0.7943) \cdot \exp\left(0.3436 + (0.5906)^{0.1422}\right) \\ H_{i9} &= 39.8 \text{ ft} \end{aligned}$$

sawlog_height = predict_height_westfall(
    species_group = 12,
    dbh_in = 15.5,
    ccr_pct = 40,
    tree_class = "acceptable",
    crown_class = "codominant",
    top_diam_in = 9.0,
)
# sawlog_height => 39.8 ft

Installation

Install the latest release from PyPI:

pip install westfall-2006

Or clone the repository and install as an editable package:

git clone https://github.com/johnkilbride/westfall_height_to_diameter_2006.git
cd westfall_height_to_diameter_2006
pip install -e .

Citation

If you use this library, please cite Westfall and Laustsen (2006).

@article{westfall2006merchantable,
  title={A merchantable and total height model for tree species in Maine},
  author={Westfall, James A and Laustsen, Kenneth M},
  journal={Northern Journal of Applied Forestry},
  volume={23},
  number={4},
  pages={241--249},
  year={2006},
  publisher={Oxford University Press}
}