Effects of measurement error on Monte Carlo integration estimators of tree volume: critical height sampling and vertical Monte Carlo methods

The effects of measurement error on Monte Carlo (MC) integration estimators of individual-tree volume that sample upper-stem heights at randomly selected cross-sectional areas (termed vertical methods) were studied. These methods included critical height sampling (on an individual-tree basis), vertical importance sampling (VIS), and vertical control variate sampling (VCS). These estimators were unbiased in the presence of two error models: additive measurement error with mean zero and multiplicative measurement error with mean one. Exact mathematical expressions were derived for the variances of VIS and VCS that include additive components for sampling error and measurement error, which together comprise total variance. Previous studies of sampling error for MC integration estimators of tree volume were combined with estimates of upper-stem measurement error obtained from the mensurational literature to compute typical estimates of total standard errors for VIS and VCS. Through examples, it is shown that measurement error can substantially increase the total root mean square error of the volume estimate, especially for small trees.