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Forest managers need accurate biomass equations to plan thinning for fuel reduction or energy production. Estimates of carbon sequestration also rely upon such equations. The current allometric equations for ponderosa pine (
Tree biomass estimates influence our understanding of both carbon sequestration [
Therefore, not only are the accurate equations needed to quantify the past biomass or carbon storage [
Estimates of aboveground biomass often rely on allometric equations for individual trees with coefficients varying by species. These functions may reflect total aboveground biomass, or some component thereof. Bole volume is important because a large proportion of biomass is found in the main stem. Nonetheless, branch and foliage biomass are important from the standpoint of fire behavior [
Ponderosa pine (
These equations are often simple expressions relating treelevel biomass to some expression of tree size [
Biomass and carbon may be estimated with equations matched to species and geographic region of interest. If such equations are not available however, any equation for the species may be used [
Of particular concern for ponderosa pine is the potential for density management to influence the dynamics of biomass and its distribution. For example, an increase of foliage growth may follow mechanical treatment (thinning) with the duration depending on thinning intensity [
As an example of thinning effects on surface fuel accumulation through needle fall in California pine forests, trees thinned to 15 m^{2} ha^{−1} produced 2000 kg ha^{−1} while stands at 38 m^{2} ha^{−1} produced 3650 kg ha^{−1} after four years [
The purpose of this study is to develop allometric relationships for trees across an array of conditions in ponderosa pine stands in northeastern California to ascertain if stand variability within the forest is a contributing factor in aboveground biomass estimation. In addition we graphically compared published equations to those fitted to evaluate the sitespecific applicability of published equations.
The study was conducted at Blacks Mountain Experimental Forest (BMEF) in northeast California. The Experimental Forest (40°40' N, 121°10' W) is located approximately 35 km northeast of Mount Lassen and elevation ranges from 1700 to 2100 m. Soils are classed as Typic Argixerolls with mesic soil temperatures at lower elevations. Annual precipitation averages 460 mm falling primarily as snow between October and May. Stands at Blacks Mountain are dominated by ponderosa pine with occasional white fir (
We sampled stands from three separate thinning treatments within Blacks Mountain. These stands thinned from below two (T_{2}), eight (T_{8}), and ten (T_{10}) years prior to sampling for biomass. In addition, trees were sampled from five different unthinned (U) areas of Blacks Mountain. A total of 82 trees with targeted ages 120 years or younger were initially sampled, and of these, three were subsequently discarded due to exhibition of characteristics of oldgrowth trees, and ages exceeding 135 years. All stands are similar in respect to species composition and primary age cohort (approximately 100 years). All the thinnings at BMEF were designed to remove trees from smaller diameter classes, while also favoring the retention of pine and minimizing ladder fuels and creating a uniform open canopy condition.
Trees were selected to give an approximately equal representation across a range of diameters from 12 to 52 cm. The means of sampled trees varied from 30.1 to 33.3 cm (
Sampled trees were felled and sectioned in 2007 during late spring and summer. Disks were removed from the bole at the stump (0.3 m), breast height (1.37 m), half way to the base of the live crown, and then at five equally spaced points within the crown, beginning with the base of the live crown. The dimensions (diameter of two axes inside and outside bark and disk thickness at the end of each axis) of each sampled disk and length of each section were recorded in the field.
Mean breast height (1.37 m) diameter (
Treatment  Sample Size 



U  21  32.8  17.3  0.54  –  66 
T_{2}  29  33.3  15.2  0.59  66  16 
T_{8}  21  33.3  14.8  0.62  64  17 
T_{10}  8  30.1  14.8  0.61  42  18 
The crown was divided into five equal sections for branch sampling and diameter measurement (
Trees crowns were broken into five sections and portions of the bole were also removed at the stump (0.3 m), breast height and midcrown.
The samples for bole wood, branches and foliage were all weighed separately after being ovendried at 80 °C for two weeks to ensure a stable weight. Total bole biomass was estimated for each tree section from the average of density at either end of each section and the dimensions of that section.
Equations for branch wood biomass (
The number of years of foliage retained per branch was evaluated for each section of the tree and each of the four thinning treatments. We evaluated years of foliage retention in a linear model as a function of section identifier.
Branch and bole weights in this analysis all include bark. The foliage weight analyzed excluded the current growing season’s needle production and thus represented weight at the start of the growing season. Branch diameters ranged from 7 to 83 mm, ovendry foliage weight ranged from 5 to 3322 g (
Minimum, mean and maximum for branch diameter (mm) and weight (g) by crown position for weightsampled branches.
Section  Branch  Wood  Foliage  

Diameter  Weight  Weight  










1  13  41  70  33  1122  4241  16  403  1425 
2  8  36  83  4  1186  7477  5  437  3322 
3  7  34  80  3  891  6379  5  388  2507 
4  11  32  53  21  417  1536  18  281  964 
5  9  23  39  6  116  489  11  135  412 
Wood biomass (g) divided by volume (cm^{3}), was obtained for each weighed bole section and then averaged over the volume between the sections. Multiplying this value by the volume of each bole section yielded biomass per section. Total ovendry bole biomass was derived by summation over all sections.
Initial graphical analysis of trends among the trees from untreated areas of the forest indicated no trends across different untreated stands at Blacks Mountain. This was consistent with our observation of fairly uniform conditions within these areas across the entire Experimental Forest. For this reason, in all subsequent analyses we aggregated the untreated trees together into one treatment group (U).
The individual branch foliage biomass was fit to data pooled within a stand using weighted nonlinear regression:
The individual branch woody biomass was fit using weighted nonlinear regression:
These fitted regressions were then applied to each branch to obtain wood and foliage biomass for each branch and, through branch summation [
Using unconstrained iterative seemingly unrelated regression (SUR) [
In this model,
Heteroskedasticity for the three components of the model was addressed by transformation:
Various combinations of a reduced form were fit by removing standidentifier terms (indicator variables) and crown ratio from the model. A final model was selected after fitting the full model and fourteen selected reduced forms of the system. Model selection was guided by an informationcriteria approach [
We estimated the annual allocation of foliage by proportioning annual foliage for the terminal in each sampled branch. We then fit the current full year’s foliage as a function of the estimated total for the terminal. This gives a proportional estimator that was used to estimate the amount of current year’s foliage in any given tree. We tried models with slope adjustments for thinning, for section, for both section and thinning, and with no adjustments. We employed a nonlinear, mixedmodel with a random treelevel effect (ε_{s}) for the slope term to obtain the following predictive model for current annual branch foliage (
We graphically evaluated the number of years of needle retention for branches by treatment area and position in the tree. We then fit a tree level (linear) response for annual foliage biomass for individual trees to evaluate changes in litter fall within the Experimental Forest.
To evaluate the stand level impact of thinning on total foliage biomass and production rates, we applied the fitted model to a thinned stand sampled in this study (
At the treelevel, we graphically evaluated predictions from three published biomass models [
Fits for branchlevel foliage biomass (
Parameter estimates with standard errors for the branch foliage biomass model (1) and for the branch wood biomass model (2) by treatment, with weighted mean squared error.






Foliage  
k_{1}  0.1658 (0.033)  0.2048 (0.033)  0.1383 (0.030)  0.1654 (0.044) 
k_{2}  2.0136 (0.054)  2.0170 (0.044)  2.2260 (0.056)  2.0614 (0.075) 
1.61  1.32  1.91  1.14  

136  187  134  56 
Wood  
q_{1}  0.0319 (0.0043)  0.0221 (0.0029)  0.0161 (0.0029)  0.0209 (0.0043) 
q_{2}  −0.0181 (0.0033)  −0.0120 (0.0021)  −0.0075 (0.0019)  −0.0112 (0.0032) 
q_{3}  2.7674 (0.034)  2.8448 (0.034)  2.9366 (0.045)  2.8861 (0.053) 
0.741  0.813  1.151  0.654  

136  187  134  56 
Among the 15 models considered based on model 3, most could be eliminated from consideration due to high values (>10) of
The transformed root mean squared error estimates for model 5 were 30.3 for foliage, 85.6 for branches and 114.7 for bole. To obtain root mean squared error in grams, these values must be multiplied by
Parameter estimates for tree foliage biomass, branch biomass, and bole biomass from selected models based on models 5 and 6.
Model  Parameter  Model 5  S.E.  Model 6  S.E. 

foliage 

1.66802  0.230  2.29770  0.237 

0.20355  0.055  0.29450  0.065  

0.55652  0.054  0.65466  0.062  

0.10737  0.063  0.18726  0.074  

1.68907  0.232  –  –  

2.08727  0.055  2.17566  0.063  
branch 

−0.15537  0.419  0.69671  0.429 

−0.16062  0.054  −0.03770  0.061  

2.20682  0.332  –  –  

2.81810  0.101  2.92409  0.114  
bole 

−2.73498  0.316  −2.97351  0.344 

1.47812  0.062  1.44091  0.066  

1.34334  0.065  1.39323  0.069 
The greatest degree of correlation was between foliage and branch biomass (
Normal probability plots using transformed residuals for the selected system (5) of equations.
Since crown ratio may be problematic for applications where crowns are not observed, we also present a restricted (
The transformed root mean squared error estimates for model 6 were 34.1 for foliage, 94.7 for branches and 114.9 for bole. As with model 5, these must be multiplied by
The treelevel foliage weight model, indicates a greater amount of foliage among thinned areas for trees of a given size and crown ratio (
In contrast, branch biomass model was marginally lower among thinned stands (
We do not have a good explanation about this difference. However, the observed increase in foliage production may be influenced by sitetosite differences in site productivity. If this were so, care should be taken in application of any particular foliage biomass model to sites not within the productivity range reflected in the original modeling data.
Foliage retention for the sampled branches ranged from one to nine years and tended to increase slightly in the lower portions of the crown in thinned stands (
Annual foliage production expressed as a proportion of total foliage appears to be slightly lower for trees eight and ten years post thinning (
Box and whisker plots showing median, quartiles and the minimum and maximum years of needle retention by crown position, for branches sampled in unthinned and thinned areas by years since treatment.
Treelevel annual foliage biomass production (kg) as a function of total foliage estimate (kg) across treatments.
The application of the fitted models to an example stand at Blacks Mountain resulted in a pretreatment foliage estimate of 12.0 Mg ha^{−1} at 781 trees ha^{−1}, thinning reduced foliage weight (applying the indicator
Estimates of foliage weight (
Condition  Annual(Mg ha^{−1})  

PreThin ( 
53.5  18.8  12.0  29.4  4.0 
PostThin ( 
10.6  34.9  2.5  8.1  0.8 
Thin + 5 year ( 
11.9  37.1  3.9  9.1  1.0 
Thin + 5 year ( 
2.9  9.5  0.8 
A graphical analysis of the foliage biomass models illustrated the degree to which the ponderosa pine foliage biomass to diameter relationship varied within the Experimental Forest (
Foliage biomass at Blacks Mountain Experimental Forest (BMEF) with crown ratio (shaded from 0.45 to 0.85) and without crown ratio (dashed line) compared to three published functions (solid lines B, C, G) from [
Branch biomass in model 5 did vary across the experimental forest as well and crown ratio contributed to the model (
The presence of crown ratio in the branch and foliage models suggests a weakness in estimation methods that do not include this variable. Inclusion of crown ratio in the foliage model reduced the transformed mean squared error by 21% and in the branch model the crown ratio term reduced the transformed mean squared error by 18%.
Comparison of our fitted branch model to other published equations produced mixed results. For larger trees both [
Not surprisingly, we found little difference between published and fitted values for bole biomass (
Estimated unthinned branch biomass (
Estimated bole biomass over diameter, at BMEF (dashed line) compared to two published estimates (solid lines C, G) from [
We also considered the impact of foliage biomass with regard to the more recent applications of the component ratio method for biomass estimation [
We applied this function and graphically compared this application with our own observed component ratio (
The SUR fit presented for tree biomass is unconstrained for the estimation of total aboveground biomass [
Foliage component ratio (
A simultaneous set of allometric equations for foliage, branch and bole biomass were developed from trees in different stand conditions. The bole biomass appeared to be robust across the range of stand conditions sampled. Foliage biomass, on both branchlevel as a function of branch diameter and treelevel as a function of tree DBH and crown ratio, was greater among trees in thinned areas than those in untreated areas. However, trees of a given DBH in the thinned areas had slightly less branch biomass than those in thinned area. The net effect of this was an apparent increase in crown biomass for trees in thinned stands when compared with models from unthinned areas of the forest.
The allometric shift could have an impact on planning for forest restoration and fuels treatments. At the stand level we found the recovery in foliage biomass to be accelerated toward pretreatment levels at a greater rate than that implied for branch biomass. Because foliage is an important source of litter accumulation, managers should consider possible impacts of thinning on the dynamics of crown and surface fuel accumulation.
This project was funded by the Joint Fire Science Program (JFSP 63304).The authors wish to thank anonymous reviewers for helpful comments on the manuscript.
The authors declare no conflict of interest.