The methodology related to TLS-based ITD, presented inIIIwas tested with simu-lated field plots, generated by several different marked point processes, and field plots generated from actual mapped field data. The reason for using process-generated data was to compare the estimator performance in different types of spa-tial patterns, with full control over the process parameters. The reason for using the field data was to test the performance in spatial patterns that exist in the real world.
Six different types of processes were used: Poisson process, non-overlapping discs process, Gibbs hard core process, Log-Gaussian Cox process, and a plantation-like process where the trees are located in a grid. Two different non-overlapping discs processes were simulated: one where the non-overlapping discs were the tree stem discs, another one where the non-overlapping discs had ten times the radii of the stem discs. Three different Gibbs hard core processes were used, with hard core distances 1, 1.5 and 2 metres. Five different variants of the Log-Gaussian Cox process were simulated with varying parameter values controlling the sizes of the clusters produced by the process. Four different diameter distributions were used. All of the simulated circular plots had radius 10 metres. The processes were simulated in larger windows to take possible edge-effect into account. Only plots where the plot centre point was not covered by a stem disc were accepted. 10000 plots were simulated from the Poisson process; for other processes, 2000 plots were simulated for each process variant.
The field data consist of 111 mapped square field plots with side length 30 metres placed to an area of approximately 43000 hectares in eastern Finland, measured in 2017. The plots were sampled from a systematic network using information on stand maturity and dominant tree species. All trees with diameter at breast height ≥ 5 cm were mapped for locations, their diameter at breast height and height measured, and species registered. The field measurements are documented in detail in [53].
For every plot, the central square with side length 10 metres was covered with a triangular grid of points, every point being at least 1 metre apart from the other points. These points were considered as centre points for circular sample plots of radius 10 metres. If a tree stem disc covered a plot centre point, the stem disc was removed from the plot. This procedure produced 13986 sample plots, 126 from each original mapped field plot.
The Horvitz–Thompson-like estimator was compared to two previously pre-sented estimators, namely those prepre-sented in [54] and [45], in plot-level stem density and basal area estimation. The Horvitz–Thompson-like estimator produced mostly lower or comparable RMSE and ME values to the other estimators. Exceptions were the clustered data sets produced by the Log-Gaussian Cox process; however, none of the estimators perform well in the clustered data sets. Approximate confidence
in-tervals for the Horvitz–Thompson-like estimates were also calculated; it was shown that the empirical coverage probabilities of these intervals – the proportion of true values covered by the interval – were close to nominal confidence-levels in the Pois-son process data and data sets that did not deviate too much from CSR. Especially in the case of the field data set the intervals were mostly conservative, having larger empirical coverage probabilities than the nominal levels.
8 DISCUSSION AND CONCLUSIONS
This dissertation studied the estimation of forest characteristics from remote sens-ing data with Horvitz–Thompson-like estimators based on stochastic geometry. Two cases were presented: estimation from airborne laser scanning data interpreted with an individual tree detection algorithm presented in [23], and estimation from indi-vidual tree detection material from terrestrial laser scanning data. These cases were studied inIandIII, respectively, and concentrated on estimation problems related to forest characteristics that can be observed from the remote sensing material, in other words estimation of population totals related to those tree characteristics that are modelled by ITD. The case of estimating unobserved forest characteristics, stud-ied inII, was also presented.
There are several other methods for estimation of population totals from ITD, although they are more plentiful in the TLS case. Mehtätalo [44], a huge inspiration for the author, presented a method for airborne ITD, based on the expected surface area covered by a Boolean model of discs and a detectability-weighted crown radius distribution. This method requires the radius distribution to be specified, for exam-ple as a Weibull distribution. Additionally, the method assumes that a tree is not detected if the centre point of the crown is under the crown of a larger tree (α=0).
Generalization to other detection conditions is not effortless, as the expected surface area covered by an erosion of a Boolean model is not known; on the other hand, a dilated Boolean model is still a Boolean model, with a shifted radius distribution.
Mehtätalo [44] also presented a method based on the Horvitz–Thompson estimator and a multinomial distribution of crown areas. The distribution assumption in this case is very flexible and does not actually make assumptions about the shape of the distribution. The assumptions made lead to a formula for detectabilities where the detectability of a tree jdepends on the detectabilities of the treesi < j. Note that Mehtätalo [44] uses the same sequential assumption that is used in this thesis: trees are ordered from largest to smallest, and only the previous trees in this sequence affect the detection of the current tree. However, the methodology presented here differs from the first method of Mehtätalo by requiring no distributional assump-tions on the marks of the point process of forest, and by enabling the effortless in-clusion of erosion-based detection conditions to the estimator. Whereas the second method of Mehtätalo uses the estimated detectabilities iteratively, the detectabilities of different trees do not depend on each other on the constructions presented here.
Mehtätalo [44] also notes that this iterative dependence might lead to error propa-gation. These two methods are the only other existing methods for estimation from airborne ITD that the author is aware of.
In the case of TLS, several estimators have been studied. Olofsson and Ols-son [54] presented a method where the detected forest characteristics are weighted by the surface area of the sample plot visible from the scanning location – for ex-ample, the detected number of trees would be divided by the visible plot area, not the whole plot area. The logic is simple: the trees were detected because they were located in the visible parts of the plot, and not the nonvisible parts. They also presented three detection conditions: full (α = 1), centre point (α = 0) and any
(α= −1) visibility, previously discussed in Section 5.2. However, these conditions were not taken into account when calculating the visible area. Kuronen et al. [45]
built upon this methodology and took into account the visibility conditions on the visible area, leading to a tree-specific weights based on morphologically transformed visible areas. The construction is quite similar to the one presented in this thesis but does not utilize any sequentiality in the detection process. Other proposed methods utilize gap probabilities of a Poisson forest [33] and traditional distance sampling methods [30, 55]. None of the above-mentioned approaches have been shown to be unbiased or have unbiased variance estimators under given, known model assump-tions.
Here, a unified methodology for Horvitz–Thompson-like estimation both from ALS-ITD and TLS-ITD was presented. In both cases, the estimator can improve re-sults derived straight from the ITD detected trees significantly. For example, inIthe RMSE of stem density was reduced by 54 per cent and the ME shifted significantly closer to zero. Similarly, in III the RMSE reductions for stem density range from 57 to 64 per cent in the simulated sample plots and from 71 to 78 per cent in the Poisson process simulated plots, depending on the detection condition used. These examples show that the Horvitz–Thompson-like estimators can be very useful for remote sensing supported forest inventory. However, the results and studies pre-sented here should not be considered a finalized product, as improvements can still be made.
There are several ways of moving forward and improving the studies presented here. First of all, a large scale simulation study, similar to the one presented inIII, for the airborne remote sensing case could be beneficial to understand what effects different deviations from the model assumptions could have on the estimation re-sults. For the terrestrial caseIIIshowed that the estimator was unbiased for Poisson process, and that the estimator produced underestimation in clustered processes and overestimation in regular processes. A connection between the magnitudes of bias and divergence from the Poisson process assumption was also found. I did show similar properties for real world data, however, the number of forest plots studied, 36, was quite small. On the other hand,IIIconsidered only simulated for-est plots – although some simulated from field measured data – and not actual data derived from terrestrial laser scanning. Testing the methodology using actual ITD derived from TLS is an important step for showing the operational performance of the estimator.
Both presented special cases of detectability constructions include a tuning-parameter α. The value of this parameter is not known. Hence, a way of fitting the value in a training set is needed. Two different situations are possible. First, the training set is a collection ofcalibrationplots in a forest area over which estimation is needed. Second possibility is that the training set is a spatially separate forest area, and the estimation is performed in a new forest area. In the second case the differing forest conditions and data collection parameters can provide additional difficulty. It should also be noted that because there is only one parameter, the fitting of it will capture all sorts of effects not necessarily related to the detection condition; for ex-ample, a training data set that is regular will produce different fitted value of α when compared to clustered training data: the fitted value would want to correct the overestimation present in regular data and underestimation present in clustered data.
The cases presented in this dissertation were simplified to include only 2D tree objects. However, many ITD algorithms, such as Lähivaara et al. [23], produce 3D
tree objects. Hence, constructing detectabilities that utilize the full 3D information produced by the ITD algorithms is an important way forward, and could provide estimators that produce more accurate estimates than the 2D variants.
Although complete spatial randomness and modelling the location of a tree as a single random point uniformly distributed in the study area are viable starting points, they are quite naive assumptions. Trees do interact, and this interaction can affect the point patterns they form. Hence, using a different probability model as a basis for the detectability could prove useful. A sequential point process, where the distribution of the the current point is affected only by the points that came before it (e.g. [56] or [57]) would be the most natural candidate to be used with the sequential detection structure presented here.
The results relating to the Horvitz–Thompson-like estimator presented here – un-biasedness, variance of the estimator – rely on the absence of measurement errors.
In reality, both the marks of interest and the marks that affect detectability can con-tain errors – not to mention the measured tree locations. It could be possible to take the properties and formulas given above as the expected value and variance condi-tioned on the marks, and use the laws of total expectation and variance to derive the unconditional formulas – expected value and variance that take measurement errors into consideration – to study the behaviour of the estimator in this case. This nat-urally relies on ITD algorithms being able to produce information on measurement errors. Additionally, even if information on measurement errors is available, the detectabilities are quite complicated morphological transformations of sets derived from the marks. This might greatly complicate the derivation of formulas under measurement errors. Bayesian ITD methods could bypass these complications with a computationally demanding solution: if the method produces a sample of several ITD fits on an estimation area, hence representing the uncertainty in measurements through that sample, the Horvitz–Thompson-like estimator could be used for every ITD fit in the sample. This would produce a distribution of estimates that quantifies the uncertainty related to the measurement errors.
All in all, the studies presented inI–III show that Horvitz–Thompson-like esti-mators based on stochastic geometry are viable options for estimating forest char-acteristics from remote sensing data. Several individual tree detection algorithms exist and are the norm in interpreting terrestrial laser scanning data. Hence the problem of estimating forest characteristics from the partially observed tree collec-tions produced by these algorithms is, and will be, an important one to solve for the operational use of remote sensing in forestry. However, the studiesI–IIImerely scratch the surface, and there is lot to be done in the future. I hope that my efforts can offer a useful starting point.
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