Accurate tree volume and biomass estimations are central in forestry, since their information is used for foresters when deciding amongst forest management options or when conducting research. Both volume and biomass are well correlated and thus estimated based on basal area measures, which is defined as the cross-sectional area at breast height (

The two most common instruments used to measure cross-sectional area are the tape and the caliper, but ignore eventual concavities in the cross-section (

Commercial plantations of valuable tree species such as walnut, cherry and poplar are good cases where foresters want to have accurate volume measurements. Plantations are established in their final density allowing a free growth of individual trees which are followed during stand development. In this study, we have focused in poplar plantations, a representative example of such cases. Poplar plantations cover approximately 900,000 hectares throughout Europe (

The main objective of this study was to compare cross sectional estimations based on diameter calculations obtained by girth or caliper measures. We also analyzed which variables (age, height from ground level, out-of-roundness and size of the cross-section) were correlated higher error rates, and how they influenced volume predictions.

Measurements were taken in a poplar plantation (_{o}). The cross-sectional area ranged from 0.53 to 1,741.41 cm2, the number of rings from the pith (R) ranged from 1 to 17 and log height position within the tree (H) ranged from 0.1 to 20.9 m above ground level. In order to evaluate the effect of tree size section shape, we decided to analyze the number of rings from the pith, rather than the number of rings from the bark.

On each digital cross-sectional image, both the circumference and the diameter of the maximum and minimum axis of the bole were measured for each annual ring (_{i}=(π/4)•D2) was then used to estimate each cross-sectional area. The three alternative estimators of diameter (AED) were applied to each of the 5,408 cross sections: [1] girth, [2] arithmetic mean diameter and [3] geometric mean diameter assuming an elliptical shape, making a total of 16,224 estimations (

For each AED (i=1 to 3), R (j=1 to 17) and H (k=1 to 9 from ground to top), the cross-sectional area percent differences (e_{ijk}) between measured values (A_{o}) and estimated values (A_{ijk}), relative to the measured value (A_{o}), were calculated in order to assess bias, precision and accuracy. The reference measurements were assumed to be unbiased. Bias (b) refers to trueness, and standard deviation (s) refers to precision. Since “n” is the number of trees analyzed for the same Alternative Estimator of Diameter (i), annual Ring (j) and Log Height (k), these expressions may be summarized as follows:

To account for differences in estimated cross-sectional area (e_{ijk}), a multilevel mixed-effect model was used because data were organized into clusters (observations from each log were clustered longitudinally according to each tree). General expression for the multilevel linear mixed model proposed was:

Where _{ijk}) taken from _{v} Log Height (H) within _{u} tree;

The “cross-sectional image” factor nested to the “tree” factor was considered random, so the objective was to estimate variance components using the intraclass correlation coefficient (_{o}) and out-of-roundness (OOR) were considered as covariates in the model. Multilevel mixed effects are usually very complicated numerical problems, and as a result convergence issues commonly arise. If there are many subjects with few observations estimation and convergence problems could result during an analysis. For this reason we discard the possibility of adding a tree random effect to account for all the observations from the same tree. Due to the hierarchical nature of the sample, generalized mixed models were adjusted with the GLIMMIX procedure in SAS/STAT version 9.2 statistical software (

Random effects were highly significant. The variance component (VC) value estimate for the random effect of “cross-sectional image” nested to the “tree” ranged from 60.3% to 77.7%, with an average value of 69.9%. The selected covariates (OOR and A_{o}) were very significant (p < 0.0001) at all Log Heights analyzed, except the measured cross-sectional areas at 2.7 m and 18.3 m above ground level (_{[1]} = –2.399; b_{[2]} = –1.046; b_{[3]} =–0.557) and among the seventeen rings analyzed by age (R_{maximum} = –0.930 to R_{minimum} = –6.258). However, in most cases, no significant differences were detected in the interaction (R x AED). Regardless of the AED, the worst estimates were obtained in the outer two rings, which were only present in the disks close to ground level. We also found that the worst estimate of Log Height was obtained at stump level (b_{[1] }= –4.024; b_{[2] }= –2.486; b_{[3]}= –1.980).

When girth measurements were used, the cross-sectional areas were overestimated and the bias was usually higher than in the other two methods (

When we analyzed precision by the standard deviation, girth measurements provided more precise estimates (lowest standard deviation) in 77.9% of total cases. Cross-sections predicted from the geometric mean diameter were more precise than those predicted from the arithmetic mean diameter. In 83.7% of the cases studied, greater precision was obtained in the cross-section estimates when we used the geometric mean diameters than when the arithmetic mean was used. The arithmetic mean diameter was the least precise in 83.3% of the cases.

In terms of accuracy, the best results came from cross-section estimations based on the geometric mean diameter (68.2% of total cases). Accuracy results based on circumference or arithmetic mean diameter did not differ significantly. Circumference-based estimates were the most accurate in 20.1% of total cases, but the most inaccurate in 48.3% of the cases. Cross-section estimates based on arithmetic mean diameter were the most accurate option in only 11% of the cases.

_{i}) and out-of-roundness: when OOR increased, the estimated error increased proportionally. The correlations between _{[1]} = 0.765; r_{[2]} = 0.627 and r_{[3]} = 0.512. Estimations based on the geometric mean diameter resulted in a lower slope in the e_{i}-OOR relationship. _{o}). The covariate A_{o} seems to suggest a clear correlation between size and percent error: as cross-section increased, the error also tended to be higher. With circumference-based measurements, we also found that the error was clearly higher (r_{[1]} = 0.289) with increasing distance from the pith (i.e., with increased Ring Age or estimated cross-section diameter). When mean diameter was used, this trend disappeared and the Pearson correlation coefficients were no longer significant (r_{[2]} = 0.131; r_{[3]} = 0.138 ). Notably, the largest errors were found at stump level and in the central part of the tree due to irregular and asymmetric cross-sections resulting from branch insertions and pruning scars. Errors were smaller when we used geometric mean diameters and larger when we used circumference measurements. All cases presented non-significant Pearson correlation coefficients (r_{[1]} = 0.077; r_{[2]} = 0.009; r_{[3]} = 0.014).

In poplar, the three commonly used methods for estimating the cross-sectional area of trees overestimated basal area as shown by (_{[1]=}2.02). _{[3]}=2.44) than the arithmetic mean diameter (σ_{[2]}=2.73). In line with these results, we found that accuracy was greatest when we used the geometric mean diameters (rmse_{[3]}= 2.82), while the results based on circumference were least accurate (rmse_{[1]}= 3.24).

Out-of-roundness was the factor most correlated to estimation error in determining cross-sectional area. The mean OOR of 1.137 found in our poplar study was similar to means reported for other species, such as 1.07 for Norway spruce (

The cross-sectional area is one of the most important forest inventory variables, since it is widely used in growth and yield models at tree and stand level. In commercial plantations where timber has a high value, accurate measures are specially needed. For example, in poplar plantations, future merchantable volume predictions are sometimes made solely on the basis of current basal area (_{[1]} = –2.399; b_{[2]} = –1.046; b_{[3]} = –0.557), obtaining the ‘worst case’ growth and yield values: G_{[1]} = 26.44 m2·ha-1, G_{[2]} = 26.09 m2·ha-1, G_{[3]} = 25.97 m2·ha-1, V_{[1]} = 239.4 m3·ha-1, V_{[2]} = 235.3 m3·ha-1 and V_{[3]} = 233.8 m3·ha-1. Thus, errors due to volume overestimation were of 3.1%, 1.3% and 0.7%, respectively for girth… Error propagation has also been shown to be significant in the construction of taper equations and in data application to other models (e.g.

In conclusion, in order to avoid error propagation leading to erroneous estimates of other growth and yield variables, we recommend measuring the cross section from the geometric mean diameter including both the largest and smallest diameters of the section, especially on trees that are clearly elliptical. Caliper and girth measures could differ in the amount of time needed to take the measures in the field, which should be taken into account when evaluating the reported accuracy improvements.

We are grateful to the Diputación General de Aragón for providing support for this study. We also would like to thank Jonàs Oliva and Cinnamon Nolan, who read through the text, for their suggestions regarding the English.