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The aim of this study was to evaluate top height (TH) estimates for Norway spruce stands calculated according to different computational methods, and to assess the effects of stand density and plot size on TH estimation accuracy. Field data were collected from twelve 1 ha research plots located in even-aged spruce stands. Conventional estimates were found to generally overstate TH. The accuracy of TH estimation was dependent on sample plot size. TH estimation error decreased rapidly with increasing sample plot area, but only up to a certain cut-off point. Errors in TH estimation were also related to local stand density, with low and very high density levels leading to decreased accuracy. The most reliable TH estimates were obtained using the U-estimator method, which is resistant to changes in sample plot size.

Measuring site productivity is critical for predicting forest growth and yield (

Spatial dependence among individual trees in a forest is typically positive and operates at a scale of microsite variation, but this is confounded by negative spatial dependence over small inter-tree distances caused by competition among immediate neighbors (

Estimation of TH by simply selecting a certain number of the thickest trees appropriate to the size of the sample plot is inherently biased. An unbiased procedure must follow the guidelines of

The main objective of this study was to evaluate the effect of sample plot size on differences between TH estimates calculated according to the conventional method and methods involving the division of research plots into 0.01 ha sub-plots, as well as to verify to what extent the proposal by

Field data were collected from twelve 1 ha research plots located in three even-aged spruce stands which were 58, 85, and 120 years old. The studied stands are situated in the Beskid Zywiecki Mountains, Western Carpathians (

In each of the stands, four 1 ha research plots (100 × 100 m each) were marked out for examination (see _{i}, _{i}) of every tree on the plots.

Tree coordinates were used to randomly select locations for square sample plots with area (

Next, a pseudo-random number generator was applied to make a random selection of coordinates for the center

Trees with coordinates _{i} and _{i} were counted in the sample plot if they met the following condition (

Random selection was repeated 200 times for each sample plot area.

The TH of trees in a randomly selected sample plot was calculated by the following four methods (

Conventional estimate (CE): TH for a sample plot with area

Adjusted largest trees (ALT): TH was calculated as the arithmetic mean of (1.6

U-estimator (UE): TH was calculated as a weighted mean for trees on the sample plot. The weight was defined as the frequency of occurrence of the thickest trees in a subset composed of

Sub-plot estimation (SUB): TH was calculated as the arithmetic mean of the height of the thickest trees, selected one from each non-empty 0.01 ha sub-plot; for example, if the area was 0.04 ha, then four trees were taken, one from each 0.01 ha sub-plot. The sample plots were divided into 0.01 ha transects (lanes), whose direction was consistent with the accepted system of rectangular coordinates. In order to reduce the effect of transect orientation on the results of the experiment, the transects were designated alternately in parallel to the

where

The variation of DBH and H was expressed by variance and the coefficient of variation.

The height determined using the SUB method was taken as the actual TH (_{act}) for the sample plot. Then, differences between the top heights (_{cal}) determined using the CE, ALT, and UE methods and the actual value were computed for each sample plot (

The value of systematic error (bias) was determined for the evaluated methods as the arithmetic mean of the deviations observed in subsequent repetitions for particular sample plot sizes (

Precision was expressed as the standard deviation of the observed errors (

The null hypothesis that the average values of the observed deviations equaled zero was verified using the

The calculated indices, characterizing the accuracy of the various methods, were then listed and analyzed by sample plot size.

The TH of each 1 ha research plot (_{1ha}) was calculated as the arithmetic mean of the heights of the thickest trees, selected one from each non-empty 0.01 ha (10 × 10 m) sub-plot. This TH was taken as a reference value. Subsequently, TH was estimated based on plots with area (

Absolute errors for individual estimates, as well as their mean values and standard deviations were calculated as measures of bias and precision.

Empirical error distributions obtained by random selection of various sample plot sizes and the application of four different calculation methods were compared with normal distribution using the Kolmogorov-Smirnov test at

Differences between the conventional (CE) and sub-plot (SUB) methods generally result from different selection of trees used for TH calculation. Therefore, these differences are discussed first, both in terms of DBH and H on all research plots. ^{-1}) among the research plots, and possibly also to the specific spatial distribution of the trees. These two factors may especially affect the number of trees selected for top height estimation using the compared methods. The conventional method, even in areas with low tree density and large irregularity in tree distribution, usually leads to the selection of a higher number of trees, typically equal to the size of the area expressed in ares, while in the case of small density the SUB method usually selects fewer trees as some 0.01 ha sub-plots are empty. This was true for all 1 ha research plots located in the oldest stand, but the effect was the most pronounced on research plot OS4. The mean height determined based on trees selected using the CE method was higher than that estimated by the SUB method (

The ALT and the UE methods have been developed in order to reduce the errors inherent in the CE method.

The ALT estimates of TH in two spruce stands (YS and MS) were similar to those obtained by the UE method, although in most cases there were some statistically significant differences between the two. In turn, in the oldest stand (OS), the ALT estimates were considerably lower than the UE ones. Therefore, in the case of the ALT estimates for YS and MS, the reduction in the absolute value of systematic error with respect to the CE method was similar to that of the UE estimates. However, the average ALT error reduction was lower than that achieved by the UE method, the former amounting to 70-75% for 0.02-0.05 ha sample plots and 75-80% for larger plots. The ALT method underestimated TH for all sample plot sizes on the research plots OS1 and OS3, but it was still more accurate than the CE method. While in the case of OS2 and OS4 the ALT method also underestimated TH for all sample plot sizes, the absolute error values were higher than those produced by the CE method (

TH estimates obtained by the SUB, ALT, UE, and CE methods were compared with the TH_{1ha} calculated for the set of the thickest trees selected one from each 0.01 ha (10 × 10 m) sub-plot.

In YS and MS, TH estimates generated by the SUB, ALT, and UE methods, irrespective of sample plot size, were very similar to each other and only slightly differed from TH_{1ha}. Among these methods, the smallest average bias (calculated for all sample plot sizes) was found for the UE method (0.031 m for YS and -0.004 m for MS), followed by ALT (0.040 m for YS and 0.050 m for MS), and SUB (-0.068 m for YS and -0.051 m for MS). The SUB and UE estimates of TH for OS1 and OS3 were similar, while the ALT method provided much lower results. In this case, just as in the younger stands, UE estimation was the most accurate method. In contrast, in OS2 and OS4, TH estimates produced by different methods were highly divergent (

It should be emphasized that in most cases the UE method yielded not only the least biased, but also the most precise TH estimates. SUB and ALT error variability for all sample plot sizes was on average higher by about 4% and 20%, respectively.

Error variation in TH estimation was found to be correlated with sample plot size. Initially, error variation fell sharply with increasing area, but after reaching a certain cut-off value, increasing plot area no longer caused a corresponding drop in error variation (

Error variation in TH estimation was also affected by the number of trees on the sample plot (

TH estimation errors were correlated with

The conventional method of estimating TH based on 100 thickest trees per 1 ha has the fundamental disadvantage of being sensitive to sample plot size (

The ALT and UE methods proposed by ^{-1} and 139 trees ha^{-1}. On those research plots, both UE and ALT estimates exhibited higher systematic errors than the CE results. This is due to the low density of trees and therefore different intensities of tree selection (percentage of “top height trees” selected on the sample plot) by these methods (

Previous studies examining the influence of sample plot size on TH estimation were usually based on data from small sample plots (less than 0.2 ha), established within a grid of permanent sample plots (

The accuracy of TH estimation was also influenced by the local density of trees. Unbiased height estimates of the studied spruce stands were obtained at

Three issues are important when assessing site productivity based on stand height: sample plot size, definition of “top height trees”, and the method of selecting those trees. As far as sample plot size is concerned, the most important aspect is to select the best method of TH estimation, which would ensure consistent results regardless of plot area. The widely used conventional approach in which TH is calculated for 100 thickest trees per hectare without taking into account their spatial distribution is not sufficiently precise and leads to divergent results depending on sample plot size. Since TH is typically employed to estimate site productivity, its determination as the mean height of the thickest trees in the stand, selected one from each 0.01 ha sub-plot, appears to be the most reliable solution as those trees are representative of the site conditions over the entire area of the stand. An additional argument for this solution is the spatial diversification of micro-site conditions generally observed in forest ecosystems (

The accuracy of TH estimates in the studied Norway spruce stands largely depended on the estimation method, sample plot size, and stand density. In all the methods analyzed (SUB, CE, ALT, and UE), TH estimates were influenced by sample plot size, with the strongest and weakest effects observed for CE and UE, respectively. Estimation results were also significantly affected by stand density. In the young- and medium-aged stands, characterized by relatively high tree density, the best results were obtained using the U-estimator and the adjusted largest trees method, while conventional estimation generally overstated TH. However, both the UE and ALT methods produced questionable TH results with a negative bias in stands with low tree density. Those two methods were found suitable for stands with tree density larger than approx. 160 per ha.

TH estimation precision was dependent on plot size. Initially, error variation fell rapidly with increasing plot area, but only up to a certain cut-off point. This suggests that an optimum sample plot size with relatively high accuracy can be obtained at reasonable labor intensity. In the analyzed stands, that size was determined to be approx. 0.05-0.10 ha in terms of area and 20-40 in terms of number of trees, as revealed by analysis of standard errors in TH estimation.

TH estimation errors were also linked to low and very high levels of local stand density. In general, relatively small errors were observed for

This paper was written as part of research project BZ/ZBiPL/14-17 “Actual and potential site productivity for main forest forming tree species in Poland” and statutory research (DS-3418/ZBiPL/15) carried out at the Department of Biometry and Forest Productivity, University of Agriculture in Krakow, Poland.

Location of the research area and the distribution of trees on individual 1 ha research plots.

Estimated differences (mean bias) in the mean diameters at breast height of trees selected according to the conventional and sub-plot methods, by sample plot size and research plot. (YS): young stand, plot 1-4; (MS): middle-age stand, plot 1-4; (OS): old stand, plot 1-4.

Estimated differences (mean bias) in the top heights determined using the conventional and sub-plot methods, by sample plot size and research plot. (YS): young stand, plot 1-4; (MS): middle-age stand, plot 1-4; (OS): old stand, plot 1-4).

Observed effects of plot size on differences between top height estimates made using the CE, ALT, and UE methods and the SUB method in 4 groups of research plots: group 1 (YS: young stand, plot 1-4), group 2 (MS: middle-age stand, plot 1-4), group 3 (OS: old stand, plot 1 and 3), and group 4 (OS: old stand, plot 2 and 4).

Observed effects of plot size and estimation method (CE, ALT, UE, SUB) on top height estimation error in 4 groups of research plots: group 1 (YS: young stand, plot 1-4), group 2 (MS: middle-age stand, plot 1-4), group 3 (OS: old stand, plot 1 and 3), and group 4 (OS: old stand, plot 2 and 4).

Dispersion of top height estimation errors using UE method, depending on the sample plot size.

Dispersion of top height estimation errors arising from the UE method, by the number of trees on the sample plot.

Top height estimation errors depending on stand density index (SDI) for the UE method.

Characteristics of the research plots. (^{-1}; (^{2} ha^{-1}); (%

Plot | Altitude(m a.s.l.) | N | G | % |
% |
---|---|---|---|---|---|

YS1 | 750-780 | 634 | 43.2 | 68.8 | 82.2 |

YS2 | 780-800 | 662 | 47.5 | 79.9 | 89.5 |

YS3 | 755-775 | 563 | 45.8 | 89.3 | 93.1 |

YS4 | 775-800 | 669 | 52.0 | 86.2 | 92.9 |

MS1 | 695-740 | 478 | 54.5 | 100.0 | 100.0 |

MS2 | 735-805 | 499 | 53.1 | 100.0 | 100.0 |

MS3 | 690-735 | 478 | 55.4 | 100.0 | 100.0 |

MS4 | 735-795 | 463 | 52.0 | 100.0 | 100.0 |

OS1 | 700-745 | 193 | 45.4 | 98.4 | 99.0 |

OS2 | 745-785 | 164 | 38.6 | 97.0 | 96.3 |

OS3 | 700-745 | 199 | 46.8 | 100.0 | 100.0 |

OS4 | 745-785 | 140 | 36.5 | 99.3 | 98.6 |

Characteristics of Norway spruces on individual research plots: number of spruces (

Variable | Plot | n | Mean | Min | Max | SD |
---|---|---|---|---|---|---|

DBH | YS1 | 436 | 31.59 | 15.0 | 57.1 | 6.39 |

YS2 | 530 | 31.27 | 13.8 | 52.7 | 6.62 | |

YS3 | 503 | 32.28 | 14.4 | 52.7 | 6.24 | |

YS4 | 577 | 31.89 | 13.0 | 51.2 | 6.94 | |

MS1 | 478 | 37.44 | 22.3 | 63.8 | 7.06 | |

MS2 | 499 | 36.17 | 21.6 | 56.8 | 6.88 | |

MS3 | 478 | 37.65 | 24.0 | 62.9 | 7.57 | |

MS4 | 463 | 37.11 | 22.2 | 59.5 | 7.30 | |

OS1 | 190 | 54.22 | 35.3 | 81.5 | 8.36 | |

OS2 | 159 | 53.87 | 31.0 | 76.7 | 8.59 | |

OS3 | 199 | 54.10 | 33.9 | 77.7 | 8.42 | |

OS4 | 139 | 56.59 | 26.1 | 84.0 | 9.87 | |

H | YS1 | 436 | 29.08 | 17.9 | 36.3 | 2.85 |

YS2 | 530 | 29.30 | 16.5 | 36.9 | 3.21 | |

YS3 | 503 | 29.89 | 13.7 | 37.0 | 3.01 | |

YS4 | 577 | 29.72 | 13.1 | 36.2 | 3.57 | |

MS1 | 478 | 36.22 | 24.8 | 44.9 | 3.39 | |

MS2 | 499 | 35.25 | 25.7 | 43.5 | 3.31 | |

MS3 | 478 | 36.27 | 25.8 | 44.8 | 3.51 | |

MS4 | 463 | 36.27 | 26.5 | 44.4 | 3.30 | |

OS1 | 190 | 40.82 | 30.1 | 48.4 | 3.34 | |

OS2 | 159 | 38.95 | 30.4 | 49.0 | 2.96 | |

OS3 | 199 | 41.32 | 33.6 | 48.1 | 3.13 | |

OS4 | 139 | 39.10 | 27.3 | 45.5 | 3.08 |