Determining maximum entropy in 3D remote sensing height distributions and using it to improve aboveground biomass modelling via stratification

McArthur's foliage height diversity (FHD) has been the gold standard in the determination of structural complexity of forests characterized by LiDAR vertical height profiles. It is based on Shannon's entropy index, which was originally designed to describe evenness in abundances among qualitative typologies, and thus the calculation of FHD involves subjective layering steps which are essentially unnatural to describe a continuous variable (X) such as height. In this contribution we aim to provide a mathematical framework for determining maximum entropy in 3D remote sensing datasets based on the Gini Coefficient of theoretical continuous distributions, intended to replace FHD as entropy measure in vertical profiles of LiDAR heights (1D, X), with extensions to variables expressing dimensions of higher order (2D or 3D, Z proportional to X2 or X3). Then we apply this framework to Boreal forests in Finland to describe landscape heterogeneity with the intention to improve the modelling of forest aboveground biomass (AGB), hypothesizing that LiDAR models of AGB should essentially be different in areas of differing structural characteristics. We carried out a pre-stratification of LiDAR data collected in 2012 using simple rules applied to the L-skewness (Lskew) and L-coefficient of variation of LiDAR echo heights (Lcv; equivalent to the Gini coefficient, GCH), determining a new threshold at GCH = 0.33 as a consequence of the newly developed mathematical proofs. We observed only moderate improvements in terms of model accuracies: RMSDs reduced from 41.7% to 38.9 or 37.0%. More remarkably, we identified critical differences in the metrics selected at each stratum, which is useful to understand what predictor variables are more important for estimating AGB at each area of a forest. We observed that higher LiDAR height percentiles are more relevant at open canopies and heterogeneous forests, whereas closed canopies in homogeneous forests obtain most accurate predictions from a combination of cover metrics and percentiles around the median. Without stratification, the overall model would neglect explained variability in the structural types of lower occurrence, and predictions from a model influenced by structural types of higher occurrence would be biased at those areas. These results are thus useful in terms of improving our understanding on the relationships underlying LiDAR-AGB models.

Automated summary

This study introduces a new mathematical approach by using the Gini coefficient to replace the traditional FHD method in describing the vertical complexity of LiDAR height profiles, with extensions to higher-dimensional variables. The research found differences in the accuracy of LiDAR models in different forest structural types, with implications for estimating forest aboveground biomass.