^{1}

^{*}

^{1}

Forest inventories were originally designed for the assessment of timber stocks over large areas. The large datasets gathered by these programs are becoming of increasing interest in other applications, particularly in ecosystem modeling. With inventory designs based on sampling proportional to size (angle-count plots) users should be cautious of using data pertaining to individual plots, as the plot-wise data is a statistical estimate rather than a true measurement. Estimates of per-plot basal area are mathematically unbiased, but the individual precision is extremely poor. Resampling of inventory datasets using multiple basal area factors can improve the precision of the estimates on single plots, thus providing better data for potential end users. Following two simulation studies to demonstrate our method we apply it to the sampling points of the Austrian National Forest Inventory, and show how the improved estimates of basal area give rise to more realistic estimates of basal area increment on individual points, reducing variance through the smoothing of extreme estimates. Our method will be useful in studies where angle count inventory data pertaining to individual plots is used to assess the precision of models or remote sensing methods.

Basal area is a key descriptor of a forest stand, and is often used to estimate other forest attributes such as biodiversity indices (

When inventory data is used for its designed purpose (the assessment of mean basal area or volume over large areas) the imprecision of individual plot data is not relevant, as (presumably) the sample design of the inventory uses sufficient plots to reduce standard errors to useful limits. Recently however large-scale inventory data is being used in carbon budget studies (

Sophisticated forest models usually have intensive data requirements, and NFI data is potentially an extremely valuable resource. Many models can be run in a point-based, scale indeterminate fashion to avoid the assumption that single-point data is fully representative of an area (

If stand basal area is used as an independent variable in any application, its improved estimation can only lead to better models and clearer understanding of its impact on stand density dependent attributes of forested ecosystems. Although inventory purists may argue that this is a misuse of data collected for a single specific purpose, the scope, comprehensiveness, reliability and high collection cost of NFI datasets demands that efforts should be directed at how to validly use this data in other applications.

A fixed-area inventory plot (“sampling proportional to area”) comprises an exhaustive sample of all trees of above some defined size, in a predefined area. In the context of the broader inventory it is a single sample, and while it should never be considered as being representative of the surrounding region it is nevertheless a complete record of the target population that exists within the plot boundaries. Nested fixed area plots are similar, with the constraint that in the larger plots the target population is restricted to the larger trees. An angle-count plot is different, in that trees are selected as being in or out of the sample based on the relationship between their diameter at breast height (^{2}/4R^{2}

In metric units, ^{2} ha^{-1}, thus the number of trees counted “in” the sample multiplied by

National Forest Inventories are commonly designed as systematic and/or cluster sampled grids, and (in central Europe) the points are generally too far apart to be considered as being in the same stand. For their design purpose of estimating national-scale forest attributes this is fine, but for stand-scale basal area assessment the angle-count method (as implemented in NFIs) represents a serious disadvantage. We show in this paper however that it is possible to develop a more useful estimation of basal area on single plots, through resampling the original dataset. While an angle-count based estimate from a single point will not approach the precision of a single fixed-area measurement within the area measured (which is of course the true value inside that area), the improvements gained through the methods presented in this study may improve precision enough that they become more useful in future applications.

The objective of this study is to improve the precision of angle count estimates on individual points through resampling the original database using multiple basal area factors. We first demonstrate the principle through two simulation studies:

to establish that the new estimates of basal area have an improved root mean square error on plots where true basal area is known, and

to show how our new estimates of basal area are more useful for model evaluation than the original estimates on plots where the true basal area is not known.

We then apply our method to data of the Austrian National Forest Inventory, showing substantial improvements in estimates at the individual plot level.

In this example we simulate four regular stands of trees, on square spacing, with densities of 324, 400, 784 and 1936 stems per hectare (as might be found in a timber plantation). All trees are identical and have a beginning

In the second simulation we establish 1000 stands of randomly located trees, of undefined extent. All trees are identical and increase from 11 to 30cm

As the extent of each stand is undefined, the “true’ value of the basal area surrounding each sample point cannot be determined. Assuming that we know however that there is no regeneration or mortality and the rate of

From each 20 year time series we extract 17 groups of four consecutive years. The first two years of each of these are used to estimate

The modern “permanent plot” Austrian National Forest Inventory made its first measurements in 1981, following two previous national inventories conducted with a temporary plot design. Inventory measurements covered the periods 1981-1985, 1986-1990, 1992-1996, 2000-2002 and 2007-2009. The inventory is organized into tracts each of 4 points on a 200 m square. 5600 such tracts are arranged in a square grid pattern across the country, including over areas that are not currently forested. Inventory field methods are fully described by

Because the Austrian NFI records tree diameters and locations, we may resample the available dataset to determine which trees would have been counted as “in” if BAFs greater than 4.0 had been used. In this study we apply BAFs from 4.0 to 8.0 in steps of 0.1, and thus obtain 41 different (but in theory, equally unbiased) estimates of basal area surrounding each inventory point in each period.

The precision of the estimates is assessed by examining the variance at each point of the basal area increment between periods. Increment, by definition, is the difference in basal area between two time periods, plus the basal area of any trees removed from the plot. Directly calculating increment in this manner (the “Difference” method) is however extremely imprecise, and as early as the 1950s

In an angle count inventory using a single BAF it is common for the data on a single point to suggest that no increment occurred;

The original data shows that four trees were in the sample on this plot in periods 3, 4 and 5. In periods 6 and 7, 10 trees were present. The interperiod increments suggested by this are thus 0, 0, 24 and 0 m^{2} ha^{-1}, which gives a mean of 6.0 m^{2} ha^{-1} period^{-1} and a plotwise variance of increment estimation of 144.0. Increments according to the multiple BAF method are 3.8, 5.3, 6.6 and 3.5 m^{2} ha^{-1} period^{-1}, with a mean of 4.8 m^{2} ha^{-1} and a plotwise variance of only 2.1. Although this is an extreme example of the possible improvements, it illustrates how we assess that improvement: if the variance in the plotwise increment estimates is reduced, then the basal area estimates are more precise.

On each of the four simulated regular stands of trees the multiple BAF estimate has a lower RMSE than the estimates using only a BAF of 4.0 (^{2} ha^{-1} (7.30% of the mean basal area of 68.96 m^{2} ha^{-1} throughout the simulated period) and the multiple BAF estimate has a mean RMSE of 2.39 m^{2} ha^{-1} (3.47%). Wilcoxon signed rank tests suggest all differences are significant at p < 0.004 except for the simulation with 324 stems per hectare, which is not significant (p = 0.570). The outliers and highest whisker extents seen in

Our second simulation assumes that the “true” basal area of each randomly distributed stand is unknown, as the extent of the stands is undefined. The test thus examines the ability of models constructed using basal area estimates across two measurement years to predict the estimates (made from the same point) in the following two years. For a stand with 784 stems per hectare, across 1000 trials models constructed with the BAF = 4.0 estimates had a mean root mean square difference of 134 stems per hectare, while the Multiple BAF method yielded a mean root mean square difference of 99 stems per hectare (

If the angle-count estimates of basal area were assumed to be “true measurements”, this would seem to imply that models constructed based on

The estimates using BAFs from 4.0 to 8.0 of mean national basal area across all periods are shown in

Basal areas calculated with the recorded

Increments calculated based on the resampled data are around 4% higher than that derived from the non-resampled data calculated either with the Difference or Starting Value measurements, which give almost identical results (

When broken down further to the level of individual plots and periods, the different methods give remarkably different estimates of increment (

The variance of the increment estimates on each plot were calculated across the five periods. The “mean plotwise variance” values in

The improvements in per-point basal area estimation we claim in this study cannot be directly proven, but better estimates of increment calculated with the Difference method must come from better estimates of basal area, as only those estimates of basal area are used to estimate the increment. It is not possible to assess how close the various estimates are to a “true” value, because no true value exists; basal area is a measure of density that on single angle count plots pertains to no definable area. If multiple samples are made within a defined area then the mean of those estimates may be assessed against a known true density, but this is not useful if the desired information is an estimate of basal area surrounding individual points.

The examples we have presented in this paper show clear advantages of using a multiple BAF mean when examining individual plots. At first glance this seems counter-intuitive: the resampled values at higher BAFs are simply nested subsets of the original data, and thus seemingly should confer no increase in precision. In the context of the broader inventory this is of course true; the original (smallest) BAF gives the largest sample sizes and thus the greatest precision across the entirety of the sampled area. At the individual plot level however, the precision of the estimate is both indeterminate (because there is no defined area) and limited by the fact that the density estimate must be in integer multiples of the basal area factor. The new information gained from the resampling emerges from the knowledge that certain trees would be in the sample with a basal area factor of 4.0, but not in the sample with a BAF of some greater value. There is no issue of “pseudoreplicating” (

Regardless of the estimate variance in an individual angle count sample, a certain degree of error will almost certainly be present because we are estimating a continuous variable (the basal area) with a discrete function (the tree count). This is clearly shown in

Regardless of the basal area factor applied, the estimates are unbiased (statistically expected results are equal, even if the estimates themselves are not) so for all estimates _{1}) = E(G_{2}) = ... E(G_{n})_{1} ≠ G_{2}

_{EST}(G)

If the two basal area estimates are made with basal area factors _{1}_{2}

Although a general expression relating the covariance to the basal area factors cannot be determined, the results of this study show that this condition can be met.

It must be stressed that we do not claim to improve overall inventory estimates. In the case of

If we define an area and take several samples inside that area (with fixed plots or angle counts) we can estimate the parameters of the population, and test those estimates against the truth of the population (if we know it). Using multiple BAFs does not help with this; better results are gained with larger sample sizes (^{2} ha^{-1}. The whole forest thus has an average basal area density of 25 m^{2} ha^{-1}. Angle counts in each stand with a BAF of 4.0 might (hypothetically) suggest 12, 12, 32 and 44 m^{2} ha^{-1}, for an average of 25 and a sample variance of 249. Our multiple BAF method might give us estimates of 9, 15, 34 and 43 m^{2} ha^{-1}, which makes an average of 25.25 and a sample variance of 254. If we are interested in estimating the forest population density, then the best results come from the BAF=4.0. However, if our interest is in the individual stands, then we look at the root mean square error of the individual estimates. The RMSE of the BAF=4.0 estimates is 4.69, and the RMSE of the multiple BAF estimates is 3.57. So, in applications where we treat individual points as representing reality we should use the multiple BAF estimate. Wherever we treat individual points only as samples from a population we should use the original data.

Our second simulation shows the risks in assuming that individual angle-counts comprise individual “measurements” of basal area. The “model” we use for estimating stem density (

The lack of precision in per-plot basal area estimates from angle-count inventories reduces the utility of these datasets in ecological applications, but improvement over the raw data is possible. Basal area or attributes directly derived from this (

Considering the fact that permanent forest inventories based on angle count sampling theory have been established for repeated observations to assess forest changes over time (

This work is part of the project “Comparing satellite

Demonstration of improved basal area estimation using multiple BAFs. Histograms show the distribution of all estimates on a single NFI plot, using BAFs ranging from 4.0 to 8.0.

Root mean square errors of stand basal area estimation in regular stands. Boxes show the median and distribution of RMSEs using nine different sample points in forests of four different stem densities arranged in regular square grids. Open boxes are for estimates made with a basal area factor of 4.0, while solid green boxes are the RMSEs of the mean of 41 estimates made using basal area factors of 4.0 to 8.0 in increments of 0.1.

Comparison of the apparent error in models relying on estimates of basal area to estimate stem density. Boxes show the root mean square difference between models constructed using two consecutive years of data with those using the next two consecutive years.

Mean Austrian basal area, determined with basal area factors from 4.0 to 8.0.

Per-period increment estimates using three methods (Starting Value, Difference and MultiBAF mean). For graphical purposes the Difference method results have a minor random component added, true values are all in integer multiples of 4.0. Each data point represents one of four increment periods on one of 1915 NFI plots.

Example of basal area estimates made with various basal area factors in a regular plot of 784 stems per hectare. Data is drawn from Simulation 1, with a plot centre at coordinate (1, 1).

Summary details of basal area, increment and plotwise increment variance. (n/a): not applicable.

Parameter | NFI original | MultiBAF mean | Starting Value method |
---|---|---|---|

Mean basal area, periods 3 to 7 | 31.4 | 31.89 | n/a |

Mean increment per period | 4.37 | 4.54 | 4.34 |

Mean plotwise variance in increment | 22.68 | 11.03 | 5.47 |

Increment, periods 3 to 4 | 3.8 | 4.27 | 4.18 |

Increment, periods 4 to 5 | 4.02 | 4.35 | 4.29 |

Increment, periods 5 to 6 | 5.23 | 5.06 | 4.75 |

Increment, periods 6 to 7 | 4.39 | 4.48 | 4.11 |