Use of allometric models for predicting the dbh and the stem dimensions Converting STRS observations of species, height and Dcrm into dbh   Kalliovirta and Tokola (2005) have published functions for predicting dbh and tree age using height, crown width and existing stand database information in Finland using NFI data. The functions are divided  for the different vegetation zones. In addition to the resuts: models and their parameters - the main findings in their article are: The relationship between the variables is almost linear, however the geographic division of the data and models reflects large-scale differences Sp and Dcrm result in dbh-estimation inaccuracy that is in the order of 13-15 % (in model RMSE) Having all three variables - Sp, Dcrm and height - improve the estimation accuracy of dbh to 7-12 % (in model RMSE) In practice this means that there is an upper limit of accuracy for STRS. It is approximately 7-12% for dbh in STRS with height, Dcrm and the species. In a stand, where the average diameter is 20 cm, and the diameters follow the normal distribution with a SD of 4 cm (20%, variance of 16 cm^2), STRS can explain at most 1 - 2^2/4^2 = 75% of the variation in dbh. In practice, there are measurement errors in Dcrm, height and species, which worsens the achievable accuracy. And if the measurements are biased, there will be bias in the model outputs (Fig 1). Fig. 1. An example of a dbh x height distribution in a birch stand. The dots represent the true, field measured values and the vectors point to the STRS estimates. Note that dbh was underestimated in most cases because Dcrm measurements made in images were underestimates. The error vectors are mostly horizontal, which reflects the height measurement accuracy. Allometric models are regression models and their use results in an averaging effect (Fig. 2). Fig. 2. An example, where STRS was done by applying the multi-scale TM approach for treetop positioning and height estimation and LiDAR -based crown modeling for Dcrm-estimation. The distributions show that STRS and allometric modeling were unable to reproduce the variation in dbh (SD of errors was 2.6 cm in dbh and 06 m in h). The field measurements range from 15 to 30 cm, but the STRS estimates are averaged in the 17 to 27-cm range. Regression models dbh = f(sp, Dcrm, h) do not produce the extreme values correctly, unless the fit (between the Y- and the X-vector) has been perfect and the X-variables have been measured without error. Estimation of stem dimensions using taper curves   The Sp, dbh and h estimates can be used (we will use)  for predicting a taper curve using for example Laasasenaho's (1983) polynomial functions, which are well known in Finland. They are allometric models also, and they have been estimated using a large data set of stem measurements in the 1960s and 1970s. If they are used for computing the stem volume, the model error is approximately 9% when the basic species-spesific polynom has been corrected with the dbh and height observations. This model inaccuracy can be divided into error variance between trees and between stands (Korhonen ####), and the errors of the model estimates can be considerably correlated among trees in one stand (i.e. model estimates are locally biased). This could reflect the stand-history and its effect on the allometry of trees in that stand. Fig 3. Basic taper curve (tapering relative to diameter at the 20% height) for pine, and points Y1, Y4 and Y7, which are used for calculating additive corrections (a third degree polynom with zero constant) to three of the eight coefficents of the taper curve, when dbh and h have been observed. Fig. 4. The taper curve can be used for computing volumes of logs, when the dbh (d) and height (h) are known.