InIIthe estimation of forest characteristics related to stem diameters at breast height from ALS based ITD was studied. In this case the observed markm(1)was the radius rof a crown, whose cross-sectional shape was assumed to be circular, and mark of interestm(2)was the stem diameter,d. The forest characteristics of interest were the stem diameter distribution, quadratic mean diameter and basal area.
Different forests have different relationships between the crown radii and stem diameters. This can result from spatial grouping of the data. For example, in II the training and target data consist of several square field plots that are spatially separate. To take this spatial grouping into account, the transformation g was de-fined as a mixed-effects model [48, 49]. Let djt be thet-percentile of stem diameter distribution on plotjand letrjtbe the corresponding percentile of the crown radius distribution with Horvitz–Thompson-like corrections. Then the general form of the model we want to fit is
djt=g(rjt,φj) +εjt, (6.3) where the parameter vector φj = (β,xj,bj) consists of fixed effects β that are common to all data, plot-specific covariates xj, and plot-specific random effects bj ∼ N(0,σ2D), independent from plot to plot. The covariance matrix σ2Dis un-known. The residual errors εjt ∼ N(0,δ2) are assumed to be independent for all data points with an unknown varianceδ2.
During model fitting, values for the plot-specific random effects are predicted.
However, these values are only available for plots with observations of the response variable, in other words for the training plots. These predicted values cannot be used for prediction outside the training data – the random effects can only be set to their expected value 0 and the model used for population-level prediction. This is why the plot-specific covariates can be useful: if a covariate that explains between-plot variation in the random effects can be found, the covariate could be added to the model to make plot-level prediction possible. Naturally, the potential covariates should be observed both in training data and the target areas where prediction is needed to be of any benefit to the prediction problem. Remote sensing based covariates are a natural choice.
Two different model types were considered forg: a quadratic polynomial form which was also considered in [47], and the Richards’ curve, also known as the gen-eralized logistic function. The two polynomial forms used were
g(rjt,φj) =β1rjt+β2r2jt+b1jrjt+b2jr2jt (6.4) and
g(rjt,φj) =β1rjt+β2r2jt+b1jrjt, (6.5) the first of which was used only for exploring the model fit, and the second simpli-fied form for exploring the prediction performance of the polynomial model. The
Richards’ curve was given by the formula g(rjt,φj) = Kj
(1+exp(Qj−Bjrjt))1v. (6.6) Here Kj, Qj, and Bj are parameters that contain fixed effects, random effects, and possible covariates, whereasvis a purely fixed effect. This form of parametrization was chosen by visually inspecting the plot-specific curves formed by the crown radii and stem diameters. Convergence problems in model fitting prohibited considering vas a plot-specific parameter, although preliminary analysis of fitting separate non-linear fixed-effects models using the Richards’ curve to every plot did show some variation invfrom plot to plot.
Whether polynomial model curve or the Richards’ curve was used, the models were first fitted without any plot-specific covariates. After this, the plot-specific co-variates with the highest absolute correlations with the predicted values of random effects were added to the models, and the models fitted again. During prediction, the random effects were set to their expected value, zero.
Let us now redefinerias a detected tree crown radius from a forest areajwhere prediction of forest characteristics related to stem diameters is needed. After an appropriate transformation g is found, the cumulative distribution function F of diameters can be estimated simply as
Fb(d) =∑g(ri,φj)≤dNbi
Nb = ∑g(ri,φj)≤dp(si)−1
∑ip(si)−1 . (6.7) The estimated basal area is
BAc =
∑
i
π(g(ri,φj)/2)2
p(si) , (6.8)
the clearest example of a Horvitz–Thompson-like estimate of a population total in this case, and the quadratic mean diameter is
QMD\ =
s∑iNbig(ri,φj)2
Nb , (6.9)
simply a square root of the weighted mean of the transformed stem diameters.
7 SUMMARY OF RESULTS
7.1 RESULTS OF I AND II
The methodologies described above for estimation of stem density from ALS-based ITD, presented inI, and estimation of characteristics related to stem diameters from ALS-based ITD, presented in II, were tested on a data set collected from a typical managed boreal forest area in eastern Finland (62◦31′N, 30◦ 10′ E). The dominant tree species on the study area is Scots pine (Pinus sylvestris L.), representing 73%
of total volume. Norway spruce (Picea abies [L.] H. Karst.) represents 16% of the volume, and deciduous trees the last 11%. The data consist of field measurements, collected from 79 square field plots, and ALS data. This data set has been used in several earlier studies, e.g. [47, 50–52].
The field measurements were carried out in May–June 2010. The field plots were placed subjectively to record species and size variation over the area. The plot side length was either 20, 25 or 30 metres. All trees with diameter at breast height ≥5 cm or height≥4 metres were mapped for locations, their diameter at breast height and height measured, and species registered. Statistics related to field measured forest characteristics can be found in Table 1 ofII.
The ALS data for the area were collected on 26 June 2009. Optech ALTM Gemini laser scanning system was used. Flying altitude was approximately 720 metres above ground level, the opening angle of scanner was 26◦and the pulse repetition frequency was 125 kHz. These parameters led to a nominal sampling density of 11.9 pulses·m−2.
The ALS data were interpreted with the ITD algorithm of Lähivaara et al. [23]
(described briefly in Section 3.3). This method requires training data; hence, the data set was split into two, a training set of 43 plots and a validation set of 36 plots. The validation set consists of plots that were directly below a flight line. The ITD data are available only for the plots in the validation set; hence, the Horvitz–Thompson-like estimation was performed in the 36 validation plots.
Results inIshowed that the Horvitz–Thompson-like estimator was able to achieve significant reductions of root mean square error (RMSE) and mean of errors (ME) for stem density estimates when compared to a reference estimate formed from the detected number of trees and ABA estimator. A Monte Carlo cross-validation ex-periment where the data were repeatedly randomly split in half to a training set for choosing an optimal αand validation set for estimation of the stand density using the chosenαconfirmed that the Horvitz–Thompson-like estimator performed well.
The results showed that there was a connection between the spatial structure of the forest plots and the estimation errors: on average, the clustered patterns resulted in underestimation and regular patterns in overestimation.
In II the best performing parameter value α from I was chosen as the ”true”
parameter value and the Horvitz–Thompson-like stem density estimation using this parameter as the basis to estimate stem diameter distribution, quadratic mean diam-eter and basal area. The Horvitz–Thompson-like estimator combined with either the polynomial transformation or Richards’ curve transformation from crown radii to
stem diameters performed generally well in all of the estimation tasks when com-pared to the other tested methods. The Richards’ curve transformation achieved good fitting results, showing that it is a viable choice for stem diameter distribution estimation. It did not perform well in prediction in the full data set, tested with a leave-one-out cross-validation experiment, however, the prediction performance in a more homogeneous subset of the data was comparable or better to the other tested methods. Polynomial transformation performed well also in the prediction experiment in the full data set.