Abstract: Equations were developed to estimate components of above-ground woody bio- mass, as a function of diameter, height, spacing, and age for two hybrid poplar clones in western Washington. Independent and harmonized fitting techniques are compared. With the small sample sizes that are unavoidable in such experiments, harmonized equations provided more useful and consistent estimates of biomass increment than did independent equations by age and spacing. They were also better suited to interpolation and extrapola- tion of long term trends of biomass increment than those based on the independent fits.

Keywords: Biomass equations, Populus, cottonwood, spacing, stand density, yields.

Considerable interest has developed in short-rotation intensive culture of poplar and other species for energy and fiber products. Research to date has provided productive clones and successful establishment methods (Heilman and others 1991). Important questions remain about spacing, rotation age, and interrelationships with growth rate as affected by site, cultural treatment and genetic (clonal) interactions with site and cultural treatment. Answers to these questions and management decisions will vary with objectives but must be based on an understanding of growth patterns of trees and stands. Availability of equations to estimate biomass components based on more easily measured tree characteristics, such as diameter and height, are prerequisite to such understanding.

Many past attempts to characterize and understand tree and stand growth have involved applications of general allometric equations to estimate growth responses to treatment, including allocations of biomass among tree components, without examining differences in allometry among treatments. This has occurred in forestry applications and is especially prevalent in ecology and biomass research. Moreover, most available equations were developed independently for each year or growth period; each equation contains some unexplained variation and is influenced by sampling inadequacies, annual climatic fluctuations, and measurement errors. All of these influences become part of estimated changes and/or increments because such changes and increments are determined by differences between estimates from the independent equations.

Previous studies have developed biomass estimates across treatments and spacing but have not examined differences in allometry among treatments. Blankenhorn and others (1986) used diameter and height as predictor variables for separate equations for each treatment and year. Other recent examples are Dolan (1984), who used log-log equations for each spacing and year to estimate biomass with logarithms of diameter and height as predictor variables for the logarithm of biomass; Auclair and Bouvarel (1992), who estimated biomass for coppiced poplar using height of tallest shoot and number of dominant shoots as predictor variables in a nonlinear equation; and Blackwell and others (1992), who used the logarithm of diameter as a predictor of the logarithm of biomass for lodgepole pine (Pinus contorta var. latifolia Engelm.). Baldwin (1987) presents a summary of biomass prediction equations for planted southern pines, but there is no reference to modeling density (spacing) or age effects. The same is true of Clark's (1987) summary of biomass equations for southern softwood and hardwood species. Several other papers on biomass studies and equations outside the United States also do not mention spacing or age effects (Auclair 1987; Kasile 1987; Pelz 1987; Singh 1987).

A particular problem arises in experiments that involve estimates of biomass and biomass increment over a series of years. Relationships are expected to change over time. Develop- ment of equations to estimate biomass components at each measurement date requires destructive sampling. Yet, extensive sampling at each measurement is not possible without destroying the experiment. Consequently, equations for estimating biomass components from readily measurable variables such as diameter and height must be developed from very small samples. Regressions fit to very small samples can be expected to be relatively inaccurate and may not produce consistent estimates for successive measurement dates.

Harmonization and constrained fitting techniques have had considerable use in forestry, particularly in estimation of height and stem volume. Growth estimates obtained by subtracting two successive independent estimates of height (or volume derived from these heights) are often erratic and sometimes impossible, because of the combination of small height samples and unavoidable errors in height measurement. Techniques have been developed for smoothing these trends over time that provide consistent estimates and reasonable interpolations and extrapolations for ages where measurements are not available. Curtis (1967) states that harmonized curves provide more reasonable estimates of change over time and a more consistent basis for estimating volumes and increments than independently fitted curves. Omule and MacDonald (1991) said essentially the same thing when they said that curves may cross illogically when fitted independently across measurement periods. Hyink and others (1988), Omule and MacDonald (1991), and Flewelling and DeJong (1994) developed procedures for fitting constrained, harmonized height/diameter curves for use with repeatedly measured plot data. Similar problems exist in biomass studies resulting from small sample size and measurement errors.

This paper reports one of the first attempts to develop individual prediction equations by clone and to harmonize curves over age and spacing as a means of overcoming the unavoidable limitations of small samples. It outlines procedures used for biomass sampling, development of harmonized equations, and their use to facilitate understanding of patterns and trends in above-ground woody biomass increment across time and among treatments. The results of independent fit and harmonized fit estimates are also contrasted.

Methods

This study was conducted as an adjunct to an experiment established in 1986 in cooperation with the Washington State Department of Natural Resources in Olympia, Washington. The experiment was a factorial design with two poplar clones and three spacings arranged in three blocks. One clone, D-01, was a Populus hybrid (taxonomic identity unknown, but suspected to be either P. trichocarpa × P. nigra or P. trichocarpa × P. angustifolia) developed originally at the University of Idaho and subsequently selected from a Canadian planting by Dula's Nursery of Canby, Oregon (Dula 1984). The other clone, 11-11, was a P. trichocarpa × P. deltoides hybrid developed and tested by the University of Washington and Washington State University (Heilman and Stettler 1985). The clones were established as pure plantings at square spacings of 0.5, 1.0, and 2.0 m (corresponding to 40,000, 10,000, and 2,500 trees per hectare respectively). Size of plots varied with spacing such that each plot had 100 interior measurement trees and a buffer approximately one-half as wide as the projected height of the trees at five years (original anticipated time of harvest). Each buffer had a minimum width of three rows of trees spaced at the measurement plot spacing. Clone-spacing treatments were assigned randomly within each block. Yields were estimated by applying the oven-dry biomass component equations developed during this study to diameter and height measurements taken on the 100 trees on each plot.

Field procedures

Trees were selected in the interior buffer rows of each plot for destructive sampling. Two trees were selected for each 1.0-m plot for year one because it was assumed that inter-tree competition (thus differences among spacings) were negligible and six trees per treatment (two per clone and spacing combination per block) were selected for years two through five and year seven to represent the distribution of diameters in each treatment. Trees were not sampled during the sixth year. A total of 192 trees were sampled. Sampling was done after the cessation of growth and bud set. The sample trees were felled, measured for diameter at 0.3 and 1.3 m and total height, and separated into stem and branch components. Each component was weighed in the field to obtain fresh weights. A subsample of each component was obtained from each tree; the subsamples were pooled by treatment and oven-dried to a constant weight at 105o C. The dry weight/fresh weight ratios so obtained were used to estimate dry weights of the components for individual sample trees.

Office procedures

Independent fits—Stem and branch biomass equations were developed independently for each clone and spacing combination at the end of each growing season. Equations were fit using stepwise linear regression procedures and the Statistical Analysis System, SAS/STAT (SAS Institute Inc. 1987) of the following form:

Ln(Y)=f[Ln(D), Ln(H), Ln(D)Ln(H)]

where:

• Ln(Y)=natural logarithm of the component to be estimated, such as stem dry weight or branch dry weight
• Ln(D)=natural logarithm of diameter at 0.3 meters above ground, and
• Ln(H)=natural logarithm of total tree height,

Harmonized fits—After several years' data were available, a harmonized model of the following form was also fit:

Ln(Y)=f[Ln(D), Ln(H), S, Ln(S), A, Ln(A)]

where:

• Ln(Y)=natural logarithm of the component to be estimated such as stem dry weight or branch dry weight,
• Ln(D)=natural logarithm of diameter at 0.3 meters above ground,
• Ln(H)=natural logarithm of total tree height,
• S=spacing (0.5, 1.0, or 2.0 m),
• Ln(S)=natural logarithm of spacing,
• A=age (years since establishment), and
• Ln(A)=natural logarithm of age.

Spacing was considered a continuous variable; the actual value of 0.5 was used for the 0.5-meter spacing, 1.0 was used for the 1.0-meter spacing, and 2.0 was used for the 2.0- meter spacing.

The full model is represented by each of the above predictor variables in linear combination and all possible cross products of those single variables with the restriction that S and Ln(S) could not appear together and A and Ln(A) could not appear together in the same cross product. A single variable or cross product was allowed to enter the equation at each step if its coefficient was significant at the 0.05 probability level and was removed from the equation if its coefficient was not significant at the 0.05 probability level. Only variables with coefficients significant at a probability level less than 0.05 were allowed to remain in the final equations (Draper and Smith 1981).

Residuals analysis—Residuals were examined using Cook's distance to evaluate an observation's influence on the regression coefficients, and outliers were tested by using the Bonferroni t-test (Weisberg 1980). Three data points out of 378 were classified as outliers and were rejected from the data set at the 0.05 probability level. Standardized residuals of the rejected data points ranged from -3.9 to 5.8. Before rejection, these data points were carefully examined for possible reasons why they were different from other data points; no biological reason was found and the sampled material was not available for re-examination. Therefore, it was assumed that some type of measurement or recording error had occurred.

Logarithmic bias—Both the independently fit and the harmonized equations are logarith- mic in form, thus a correction for logarithmic bias was needed. Guidelines given by Flewelling and Pienaar (1981) were used to select an appropriate log bias correction fac- tor. Log bias correction factors were applied to each prediction equation which increased the estimated biomass by a small fixed percentage (less than 3% for woody biomass).

Application of tree level equations for stand estimates—The biomass equations were used with measured diameters and heights to estimate biomass of tree components for individual trees on the research plots. The individual tree estimates were summed over all trees on the plot and expanded to per hectare estimates. Biomass growth rates were estimated indirectly as differences between estimates for successive growth periods. Woody biomass was estimated as the sum of stem biomass and branch biomass.

Results and Discussion

Independent fits

Initially, biomass was modeled using independent annual fits to provide information on allometry of component values relative to diameter and height and immediate short term estimates of yield. Independently fit equations may theoretically provide “best” estimates of cumulative growth (or size) at any given time, but unfortunately, “best” or even “good” estimates of annual growth cannot be derived therefrom because of year-to-year inconsistencies and variation associated with sampling, field measurements, or shifts in predictor variables. Moreover, independently fit annual equations cannot be interpolated or extrapolated reliably for periods for which measurements are not available. Maximum points are not easily identifiable because of fluctuations and inconsistencies in estimated growth rates.

Each year the same independent model was fit, but the same predictor variables did not always enter. This is a common occurrence when using stepwise regression procedures. This is accentuated by the small size (six trees) of the available samples for each clone and spacing combination for each year. The difference in equation forms from year-to-year was probably a major contributor to the inconsistencies observed when the independent annual fits were used to estimate increments by subtraction. Forcing consistent independent variables for each year, however, would have resulted in the inclusion of insignificant variables and/or exclusion of significant variables. Table 1 shows the regression coefficients for the independently fit stem dry weight equations for both clones and number of observations, R2s, mean square errors (MSE), means for the transformed Y, and standard errors of estimate as a percent of the untransformed (geometric) mean for those fits by spacing and year. Variables entering the equations from year-to-year and spacing-to-spacing were most consistent in stem dry weight fits. Branch weights were much less consistent and fits were not as good as for stem dry weights.

Woody biomass—In the first three years the independently estimated increments (Fig. 1a) decreased with increased spacing for the D-01 clone, a result which was expected because the trees had not fully occupied the site. In the fourth year the trend of increment with spacing had reversed (i.e., growth increased with increased spacing); in the fifth year, however, the trend changed again with the 1-m spacing having the maximum increment. In years six and seven, the increments increased with increased spacing.

In the first year the trend in biomass increment for the 11-11 clone (Fig. 1b) decreased with increased spacing, but in years two and three the increment trend was mixed with the 1-m spacing having maximum increment. In year four the increment increased with increased spacing, but in year five an unexpected reversal occurred when the increment trend decreased with increased spacing. In years six and seven the trends again reversed and increased with increased spacing. In year six, the increment of the 0.5-m spacing was exceptionally small, especially when compared to the extremely large increment the year before.

Figure 1. Above-ground woody biomass (stem + branches) annual increments for independent fits (a and b) and harmonized fits (c and d) by clone, spacing, and year.

These inconsistent patterns in increment trends with spacing were disconcerting because they did not match expectations and current knowledge about general stand growth patterns. Once trees had fully occupied the site and biomass increments began to increase with spacing, the trend was expected to continue. Increment was expected to increase as spacing increased between trees. Current annual diameter and height increment trends (Fig. 2a, b, c, and d) did not show the inconsistent patterns observed for above-ground woody biomass (Fig. 1a and 1b), however. Moreover, no mortality occurred during the sixth and seventh growth periods to explain the inconsistencies in per hectare patterns and disparities with diameter and height growth trends. Subtracting the estimates from the independent equations was unreliable for estimating annual growth trends or patterns.

Figure 2. Annual height increments (a and b) and diameter increments (c and d) by clone, spacing, and year.

In year six, diameters and heights were measured but no trees were sampled for biomass; therefore, no equation could be developed. However, equations were available for years five and seven. An estimate of biomass (and corresponding increments) can be obtained from the independent estimates for the sixth growth period by either of two interpolation methods. The first is simple linear interpolation between year five and seven. This apportions increment, including mortality, equally between growth periods six and seven. The mortality assumption may or may not be correct. Differences in annual diameter and height increments are not accounted for. Figure 3a illustrates the results of using straight line interpolation for clone D-01 at the 2-m spacing. Notice that the trend for year five to year seven for the independent fit does not follow the trend lines for diameter and height increments, thereby washing out the effects of the drought in the early growing season in year six. Figure 3b illustrates the results of using straight line interpolation for clone 11-11 at the 2-m spacing.

The second method of interpolation is to substitute the measured diameter and heights for year six into the year five and seven equations and then average the two estimates (substitution method). This procedure incorporates differences in mortality for the two growth periods plus differences in diameter and height increments. Figure 3c and 3d illustrates the results of using the substitution method for clone D-01 and 11-11, respectively, at the 2-m spacing. Intuition suggests that the substitution method should track the trends in diameter and height increment; however, in this case the results for clone 11-11 (Fig. 3d) are the opposite of those trends. In this instance the year five equation greatly overestimated the biomass when applied in year six, which included heights outside the range of those found in year five.

Figure 3. Above-ground woody biomass (stem + branches) mean tree annual increment trends as estimated from harmonized and independent fits at 2-m spacing using two different interpolation methods to estimate increment in year six for the independently fit biomass equations. Diameter and height increment trends are included for comparison

Harmonized fits

Because of inconsistencies between growth estimates from year-to-year when using the independent equations, especially year five increment in woody biomass trends across spacing, equations harmonized over age and spacing were fit to the combined data for each clone. The new harmonized equations resulted in more consistent estimates of annual yields and increments over age and spacing for each of the clones than did the independent equations. Table 2 shows the regression coefficients for the harmonized stem dry weight equations for both clones and number of observations, R2s, MSEs, means for the transformed Y, and standard errors of estimate as a percent of the untransformed (geometric) mean for those fits.

Harmonized fits across time require several successive measurements, and annual fluctuations in estimated growth rates, artifacts, and data collection biases tend to be reduced by the smoothing process. The harmonization process produces smoothed long- term trends in allometry and provides for interpolation for periods where samples were not taken. They also can be extrapolated more reliably than independent annual fits. Maximum points are more easily identified than with independent fits because of the smoother trends.

Woody biomass—Figures 1c and 1d illustrate the current annual above-ground woody increments per hectare using the harmonized prediction equations. The trends in annual increments decreased with increased spacing during the first two years and began to transition to increasing with increased spacing in year three. Clone D-01 remained in transition (Fig. 1c), increased from 0.5-m to 1-m, and decreased from 1-m to 2-m spacing, until year seven when the 2-m spacing increased above the 1-m spacing. The 11-11 clone was in transition (Fig. 1d) during years three and four and increased with increased spacing in year five and continued to show increased trends with increased spacing in years six and seven. The only apparent anomaly was in clone D-01 in year seven when the 1-m increment fell below the 0.5-m and there was a large increase in the 2-m spacing increment (Fig. 1c); however, this same pattern was also present in the measured diameter increments (Fig. 2c).

The harmonized fits proved to be superior to the independent fits when estimating biomass for year six when biomass was not sampled. Years four and six were both dry years and irrigation during the early spring was less than normal because of equipment failures, resulting in 50% less water input during the early growing season than other years. The diameter and height increment (Fig. 2) reflect this early water deficit, especially height increment for D-01. The interpolated values based on the independent fits (Fig. 1a, b) do not show a corresponding depression in the year-six growth rates, but the harmonized fits do reflect the reduced growth in a similar manner as the height and diameter increments illustrated in Figures 2a and 2b. The harmonized fit for clone D-01 shows the same depression in growth for the 2-m spacing as does height increment (Fig. 2a and 3a). Figure 3 illustrates that the harmonized fit can be used (interpolated) for measurements where no biomass samples were taken to estimate growth and that in fact the estimate reflects the real climatic effect which occurred as measured by the height and diameter increments for the same period.

Conclusions

The anomalies that appear in biomass estimates based on independently fit equations are in large part a consequence of very small samples used in fitting the biomass equations. This is an unavoidable limitation in experiments that require estimates of biomass over a series of years, because only small numbers of trees can be destructively sampled to determine tree biomass, without destroying the overall experiment.

Because of the intrinsic unreliability of biomass regressions derived by independent fits to very small samples, results will generally be more consistent and more readily interpretable if regressions are harmonized over age and spacing, using all data combined, rather than fit as independent equations for each year and spacing. If fewer than three years of data from one spacing are available, harmonization across age is not reasonable; however, harmonization across three or more spacings is possible with fewer years of data. Harmonized fits will also provide more accurate estimates for years lacking biomass measurements than can interpolations based on independent fits.