The unbiasedness of a generalized mirage boundary correction method for Monte Carlo integration estimators of volume
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Canadian Journal of Forest Research. 44(7): 810-819.
The typical "double counting" application of the mirage method of boundary correction cannot be applied to sampling systems such as critical height sampling (CHS) that are based on a Monte Carlo sample of a tree (or debris) attribute because the critical height (or other random attribute) sampled from a mirage point is generally not equal to the critical height measured from the original sample point. A generalization of the mirage method is proposed for CHS and related techniques in which new samples of critical heights or other dimensions are obtained from mirage points outside the tract boundary. This is necessary because, in the case of CHS, the critical height actually depends on the distance between the tree and a randomly located sample point. Other spatially referenced individual tree attribute or coarse woody debris (CWD) estimators that use Monte Carlo integration with importance sampling have been developed in which the tree or CWD attribute estimate also depends on the distance between the tree and the sample point. The proposed modified mirage method is shown to be design unbiased. The proof includes general application to Monte Carlo integration estimators for objects such as CWD sampled from points.
Lynch, Thomas B.; Gove, Jeffrey H. 2014. The unbiasedness of a generalized mirage boundary correction method for Monte Carlo integration estimators of volume. Canadian Journal of Forest Research. 44(7): 810-819. https://doi.org/10.1139/cjfr-2014-0031.