^{1}

^{*}

^{1}

^{2}

^{3}

^{4}

Reproduction is permitted for noncommercial purpose.

Airborne laser scanning (ALS) is an active remote sensing technique that uses the time-of-flight measurement principle to capture the three-dimensional structure of the earth's surface with pulsed lasers that transmit nanosecond-long laser pulses with a high pulse repetition frequency. Over forested areas most of the laser pulses are reflected by the leaves and branches of the trees, but a certain fraction of the laser pulses reaches the forest floor through small gaps in the canopy. Thus it is possible to reconstruct both the three-dimensional structure of the forest canopy and the terrain surface. For the retrieval of quantitative forest parameters such as stem volume or biomass it is necessary to use models that combine ALS with inventory data. One approach is to use multiplicative regression models that are trained with local inventory data. This method has been widely applied over boreal forest regions, but so far little experience exists with applying this method for mapping alpine forest. In this study the transferability of this approach to a 128 km^{2} large mountainous region in Vorarlberg, Austria, was evaluated. For the calibration of the model, inventory data as operationally collected by Austrian foresters were used. Despite these inventory data are based on variable sample plot sizes, they could be used for mapping stem volume for the entire alpine study area. The coefficient of determination R^{2} was 0.85 and the root mean square error (RMSE) 90.9 m^{3}ha^{−1} (relative error of 21.4%) which is comparable to results of ALS studies conducted over topographically less complex environments. Due to the increasing availability, ALS data could become an operational part of Austrian's forest inventories.

Airborne laser scanning (ALS), also referred to as Light Detection and Ranging (LiDAR), is an active remote sensing technique that has found widespread use in topographic mapping [^{2}) of small-footprint (< 1 m) laser pulses on the ground. A small part of the pulse energy is typically scattered backwards towards the sensor where a photodiode registers the time of arrival and, optionally the intensity of the echoes [

Over vegetated terrain most of the laser pulses are reflected by the leaves and branches of the vegetation canopy [

By subtracting the terrain height from the vertical coordinate of the 3D point cloud, a virtual representation of the vegetation canopy is obtained. The significant potential of this 3D canopy representation for forestry applications, including estimation of canopy heights, stand volume, basal area or above-ground biomass has been pointed out repeatedly [

Much research on the use of ALS for forestry applications has so far focused on boreal forests. In Scandinavia the operational use of ALS data within forest inventories has become a reality [

The aim of this study has been to investigate the transferability of this multiplicative regression model for mapping of alpine forests. A 128 km^{2} mountainous area in the western part of Austria was selected as study area. In Austria, ALS data covering large alpine areas have only become available recently. Despite the complex mountainous topography, the ALS derived terrain and canopy height models are of good quality as the investigation by Hollaus et al. [

The study area is situated in the southern part of the federal state of Vorarlberg, Austria (^{2} of the Montafon region. The complex alpine landscape of the study area has high relief energy, whereas the elevations range between 800 and 2,900 m. The land cover is characterized by coniferous and mixed forests, shrubs, meadows, and sparsely settled areas in the valley floors. According to the Stand Montafon Forstfonds^{3}ha^{−1}, which is high compared to 325 m^{3}ha^{−1 2} representing the average Austrian stem volume. Two thirds of the forests are located above 1,000 m sea level, whereas the timberline is at about 1,950 m. As a result of the mountainous terrain 6% of the forests grow on a slope inclination of more than 45°, and 61% between 30° and 45°. Furthermore, 28% of the forests grow on slopes between 15° and 30°, and only 5% on slopes less than 15°. Approximately 80% of the forested area is managed forest with a protection function (protection forest with yield), 10% is managed forest (commercial forest), and the rest is unmanaged protection forest (protection forest without yield) on extreme sites or at the timberline. In alpine regions protection forests are of great importance to protect inhabited areas, roads and railway lines from natural hazards such as rockfalls and snow avalanches. About half of the forests in the study area are managed by the forest administration Stand Montafon Forstfonds, which is the largest forest owner in Vorarlberg. They operate a precise forest inventory, which provides the field data used in this study. The forested areas managed by the Stand Montafon Forstfonds are regularly distributed over the entire test site and therefore, this forest inventory data are representative for the entire study area. Within the study area the average forest stand size is approximately 3.5 ha and is equal to the average area of private owned forests in Austria.

The ALS data were acquired in the framework of a commercial terrain mapping project during two flight-campaigns. The first flight took place on December 10, 2002 (snow-free, leaf-off) and covers the lower altitudes, the second on July 19, 2003 and covers the higher altitudes (snow-free, leaf-on) as shown in ^{2} and 2.7 points per m^{2}, respectively. For both flights the flying heights above ground range between ∽650 m and ∽2,000 m, whereas the average flying height was 1,000 m. For the winter flight an Airborne Laser Terrain Mapper (ALTM i1225) and for the summer flight an ALTM 2050 was used. The beam divergence for both systems is 0.3 mrad, which results in a mean footprint size of 0.3 m for the average flying height. The maximum scan angle was ±20 deg. During both flight-campaigns 650 millions laser scanner points were acquired, including first- and last-echoes. Further information about the used data sets can be found in Wagner et al. [

The forest inventory data are provided by the forest administration Stand Montafon Forstfonds, which operates a precise inventory since 1988 for approx. 65 km^{2} forests in the southern part of Vorarlberg, in the so-called Montafon region. The design of this inventory is based on permanent sample plots distributed in a regularly 350 m grid (

For all selected trees, the tree species was determined and the diameter at breast height (

For each sample plot one tree height (_{0} is a variable coefficient, _{1} is a constant coefficient, and _{0} depends on the site quality and is calculated by means of the measured _{0} for each sample plot all the missing tree heights within the plot are calculated. The expected standard deviation of the estimated heights ranges between 1.3 and 1.9 m [

For the calibration of the ALS data the stem volumes per unit area are used as ground reference quantities. The estimation of stem volume is based on regional volume equations of the Bavarian forest inventory, as these reflect the growth pattern of spruce in the province of Vorarlberg best. These volume equations are based on a so-called form height concept, which means that the conical shape of the stems is transformed to a cylinder. The diameter of the cylinder corresponds to the _{ij}_{stem,fi}) per ha can be calculated with the following formula:
_{stem,fi} is the stem volume in m^{3}ha^{−1}, _{stem,fi} corresponds to the commercial useable timber volume and is therefore not equal to the real biological available stem volume within the forest. In contrast to the commercial useable timber volume, the biological one includes additional small trees (

The pre-processing of the ALS data includes the georeferencing, the generation of topographic models (e.g. digital terrain and surface model), and the derivation of canopy heights. The georeferencing of the ALS data is based on the method developed by Kager [

The georeferenced 3D point clouds are used to calculate the digital terrain model (DTM), digital surface model (DSM), and the canopy heights. The last-echo points were used for generating the DTM using a hierarchic robust filtering technique described in Briese et al. [

Although the forest inventory is based on permanent sample plots the accuracies of the available coordinates of the sample plot centers vary in the range of several meters. The inaccuracies can be explained by the used surveying instruments (i.e. compass in combination with an ultrasonic range instrument), which were applied for the positioning. About two tens of reference measurements with differential GPS have shown that the average position error is approximately 10 m, which is quite large compared to a 20 m diameter of the sample plots used for the ALS data analyses. To overcome this problem manual co-registration of the forest inventory data to the ALS data has been carried out. For this purpose positions of single trees are needed, which were measured during the forest inventory field campaign. For all sampled trees the polar coordinates relative to the sample plot center were measured with a compass in combination with an ultrasonic range instrument. The expected accuracies of the measured polar coordinates are in the range of 0.3 to 0.5 m. As shown in

In this way the positions of 103 of the 143 available sample plots could be clearly co-registered to the ALS data. For the further analyses the calculated _{stem,fi} of these clearly co-registered sample plots are used as reference data. Within the 103 sample plots 925 trees were measured, whereas the

The multiplicative regression model used for estimating stem volume [_{stem,fi} in m^{3}ha^{-1}; _{0,f}, _{10,f}, …, _{90,f} are percentiles of the first-echo laser canopy heights for 0%, 10%, …, 90% in m; _{0,1}, _{10,1}, …, _{90,1} are percentiles of the last-echo laser canopy heights for 0%, 10%, …, 90% in m; _{mean,f}, _{mean,1} are mean values of the first- and last-echo canopy heights in m; _{cv,f}, _{cv,1} are coefficients of variation of the first- and last-echo canopy heights in percent. As suggested by Næsset [_{0,f}, _{1,f}, …, _{9,f} are cumulative canopy densities of first-echo laser points for the fraction no. 0, 1, …, 9; _{0,1}, _{1,l}, …, _{9,1} are cumulative canopy densities of last-echo laser points for the fraction no. 0, 1, …, 9. The 10 fractions are of equal height and are calculated by dividing the difference between the highest and the lowest (2 m) canopy height by 10. For each fraction no. 0, 1, …, 9 (> 2 m) the proportion of laser hits above the fraction limits to the total number of laser hits were calculated for both, first- and last-echo points. The model parameter of

For the development of the final model stepwise multiple regression analyses is used. Starting with the regression equation,

For the multiple regression analysis the multiplicative model (

_{stem,fi,}_{i}^{3}ha^{−1}; _{stem,}_{i}^{3}ha^{−1}, retransformed back to the original untransformed scale without bias correction; and _{stem,}_{i}

Due to the variable sizes of the forest inventory sample plots the appropriate sample plot size to extract the ALS data is not known in advance. Therefore, several diameters are used and evaluated. The sample plot size, which leads to the highest accuracy, is used to analyze the effects of varying ALS properties.

As mentioned in section 2.2, the study area is covered by two ALS data sets with different point densities and acquisition times. The winter ALS data cover 92 and the summer ALS data cover 64 from the 103 forest inventory sample plots, which could clearly be co-registered. Within the overlapping area of the two data sets 52 forest inventory sample plots are available, which are used to analyses the effects of varying acquisition times. As the summer ALS data have a point density of 2.7 p/m^{2} a thinning of 66% is applied to reduce the density to those of the winter data (0.9 p/m^{2}). Based on the sorted acquisition time the thinning is done by a systematic removal of points.

To understand the impact of different ALS point densities the original and the thinned data are analyzed for each acquisition time separately. Thus, the results can easily compared within each flight campaign as the flying height, the local incidence angles, the acquisition times, the sensor characteristics, and the used sample plots are similar for the original and the thinned data set.

For the validation of the calibrated models a cross-validation procedure is used, where for each step one observation is excluded for the calibration of the model. Since the model is fitted

For the determination of the appropriate sample plot size five different diameters (18 m, 20 m, 22 m, 24 m, and 26 m) were used. All co-registered sample plots (103) were used for the multiple regression analysis. The final models and statistical parameters of the derived results are shown in

A sample plot size with a diameter of 24 m leads to the highest coefficient of determination (R^{2} = 0.84) and to the lowest root mean square error (RMSE = 96.8 m^{3}ha^{−1} corresponding to a relative error of 22.9%). The final model consists of three independent variables including _{0,f}, _{30,l} and _{6,f}. The residuals derived from the cross-validation vary between -254.7 and 204.1 m^{3}ha^{-1} and have a standard deviation of 100.0 m^{3}ha^{−1}.

The analysis of the effects of different point densities was done for all (103), the winter (92), the summer (64), and the 52 forest inventory sample plots located within the overlapping area of the two data sets separately. Based on the original and on the thinned ALS data sets a circular sample plot with a diameter of 24 m was used. The thinning of the ALS data sets leads to a minor decrease of R^{2} from 0.84 to 0.82 using all 103 FI sample plots and from 0.81 to 0.77 using only the 92 plots covered by the winter ALS data. For the 64 sample plots covered by the summer data no change of the R^{2} (R^{2}=0.82) could be observed by thinning the point density of 66%. As it is summarized in ^{3}ha^{—1} for all plots, from 110.4 to 117.5 m^{3}ha^{−1} for the winter plots, and from 97.9 to 99.9 m^{3}ha^{−1} for the summer plots. Also the standard deviations of the residuals increase from 100.0 to 106.7 m^{3}ha^{−1} for all plots, from 114.8 to 122.7 m^{3}ha^{−1} for the winter plots, and from 110.7 to 111.3 m^{3}ha^{−1} for the summer plots.

To study the effects of different acquisition times the accuracies of the winter and the thinned summer data are compared. Only those 52 sample plots are used which are covered by the winter and the summer data simultaneously. The achieved R^{2} is higher for the thinned summer data than for the winter data and are 0.85 and 0.79 respectively. For the RMSE and the standard deviation of the residuals a decrease from the winter to the thinned summer data can be observed. The RMSE decreases from 108.2 to 93.3 m^{3}ha^{−1} (^{3}ha^{−1} as shown in

The results of this study indicate that the multiplicative stem volume model can successfully be applied for a mountainous forest in Austria. The derived coefficient of determination (R^{2} = 0.84) is comparable with those derived from boreal forests (R^{2} = 0.83 – 0.97) reported in Næsset [^{3}ha^{-1} (17.5 to 22.5%) which is comparable to 96.8 m^{3}ha^{−1} (22.9%) derived in the current study. As shown in

For the operational application of this model it has to be considered that the variables derived from first- and last-echoes are known to change depending on different ALS system parameters. For example, Næsset [

Concerning the effects of varying ALS point densities this study shows that for each ALS data set an individual model can successfully be fitted as summarized in ^{3}ha^{−1}) with decreasing point densities from 0.9 to 0.3 p/m^{2} is slightly higher than those (ΔRMSE = 2.4 m^{3}ha^{−1}) from 2.7 to 0.9 p/m^{2} (

Concerning the different acquisition times the results derived from the winter and the thinned summer data were compared. While similar R^{2}s were derived, the differences of the RMSEs, and the standard deviation of the residuals derived from the cross-validation are relatively large and are 14.9 m^{3}ha^{−1} and 16.0 m^{3}ha^{-1} respectively as can be seen in

Due to geometric discrepancies between the forest inventory plots and the ALS data a manual co-registration was applied. About two-thirds of the available sample plots could clearly be co-registered to the ALS data. The relative high number of remaining sample plots (

Even though the sample plot areas of the forest inventory data were variable, the statistically calculated stem volumes in m^{3}ha^{−1} could be used for the calibration of the stem volume models. As summarized in

A critical point of the used forest inventory data is the use of tree height (

The used forest inventory data do not include trees with a ^{3}ha^{−1}) the lack of these trees has a minor influence on the calculated stem volumes. Apart from the estimation of the commercial useable timber volume, which is represented by the forest inventory data, the estimation of the real biological available stem volume should consider also trees with a

The technique of airborne laser scanning has reached the maturity to be of use for large-scale mapping of structural forest parameters in alpine environments. By combing the ALS derived 3D point cloud with forest inventory data in the multiplicative regression model approach used by Næsset [^{2} large area. The validation showed that R^{2} is 0.84 and RMSE 96.8 m^{3}ha^{−1} (22.9%), which is comparable to results derived for boreal forests. The good results obtained in this study offer an operational perspective for the use of ALS data in the Austrian forest inventory, considering that both the ALS and inventory data were collected as part of routine data collection activities. In that respect it is also important that the accuracy of the method decreased only slightly when the laser point density on the ground was decreased from 2.7 to 0.3 points per m^{2}.

Future research will address the question if and how this method can be transferred to other forest ecosystems in Austria using data from the national forest inventory. The Austrian national forest inventory is carried out regularly with a time interval of six to eight years. Today, more than 170 attributes are assessed, which provide information on quantity, quality and trends of the Austrian forests [

Also, we plan to investigate the benefits of ALS systems that record the echo intensity or the complete echo waveform [

We would like to thank the Landesvermessungsamt Feldkirch for granting the use of the ALS data and the forest administration “Stand Montafon Forstfonds” for providing the forest inventory data. Furthermore, we are grateful to our colleagues C. Eberhöfer and H. Kager for the pre-processing of the ALS data. Parts of this study were funded by the Austrian Federal Ministry of Agriculture, Forestry, Environment and Water Management (BMLFUW) in the framework of the project ÖWI-Regio.

The location of the study area Montafon in the western part of the Austrian Alps. The left image shows the flight paths for the airborne laser scanner acquisition during the summer and the winter campaign, overlain over the shaded terrain model. The right image shows the location of the forest inventory plots overlain over the shaded terrain model with highlighted forested areas.

a) Co-registration of the FI sample plots to the ALS data. The center coordinates of each sample plot have been adapted, that the measured single tree positions fit best to the visually detectable tree positions in the ALS canopy height model and that the measured tree height fit best to the height extracted from the canopy height model. b) Distribution of distances of all 925 measured single trees from 103 FI sample plot centers.

Scatter plots of stem volumes from ALS data versus forest inventory data. The analyses were based on the 52 forest inventory sample plots covered by the winter and the summer ALS data.

Summary of forest inventory data. The calculations of the shown values are based on mean values per plot as well as on parameters measured on single trees. The statistics are calculated from all available forest inventory plots and only from those plots which could clearly be co-registered to the ALS data. The shown statistics of the winter (W), summer (S) and winter & summer (W&S) data are based on the co-registered sample plots.

Variable | Data | MIN | MAX | MEAN | SD |
---|---|---|---|---|---|

Diameter at breast height [cm] | All: 1,373 trees | 10.0 | 127.0 | 48.2 | 18.5 |

Co-registered: 925 trees | 10.0 | 127.0 | 47.6 | 18.8 | |

- W: 853 trees | 10.0 | 127.0 | 46.8 | 18.9 | |

- S: 559 trees | 10.0 | 127.0 | 51.1 | 18.8 | |

-W&S: 485 trees | 10.0 | 127.0 | 50.0 | 19.1 | |

| |||||

Tree height [m] | All: 1,373 trees | 5.4 | 43.8 | 27.7 | 6.8 |

Co-registered: 925 trees | 5.4 | 42.2 | 27.0 | 6.8 | |

- W: 853 trees | 5.4 | 42.2 | 27.1 | 7.0 | |

- S: 559 trees | 7.0 | 42.2 | 27.7 | 6.6 | |

-W&S: 485 trees | 7.0 | 42.2 | 28.1 | 6.8 | |

| |||||

Number of trees per plot | All: 143 plots | 1 | 29 | 9.8 | 5.4 |

Co-registered: 103 plots | 1 | 22 | 8.8 | 4.6 | |

- W: 92 plots | 1 | 22 | 9.3 | 4.6 | |

- S: 64 plots | 1 | 22 | 8.7 | 4.3 | |

-W&S: 52 plots | 1 | 22 | 9.3 | 4.4 | |

| |||||

Number of trees per ha | All: 143 plots | 8 | 1,876 | 414 | 392 |

Co-registered: 103 plots | 11 | 1,876 | 393 | 395 | |

- W: 92 plots | 11 | 1,876 | 429 | 406 | |

- S: 64 plots | 11 | 1,602 | 309 | 316 | |

-W&S: 52 plots | 18 | 1,602 | 352 | 335 | |

| |||||

Mean diameter at breast height per plot [cm] | All: 143 plots | 11.0 | 78.0 | 49.0 | 14.0 |

Co-registered: 103 plots | 11.0 | 77.3 | 48.5 | 14.1 | |

- W: 92 plots | 11.0 | 77.3 | 47.7 | 14.3 | |

- S: 64 plots | 21.0 | 77.3 | 51.9 | 13.6 | |

-W&S: 52 plots | 21.0 | 77.3 | 51.3 | 13.9 | |

| |||||

Mean tree height per plot [cm] | All: 143 plots | 6.0 | 42.1 | 26.9 | 6.5 |

Co-registered: 103 plots | 9.5 | 38.6 | 26.2 | 6.4 | |

- W: 92 plots | 9.5 | 38.6 | 26.6 | 6.5 | |

- S: 64 plots | 11.3 | 38.0 | 27.1 | 6.1 | |

-W&S: 52 plots | 11.3 | 38.0 | 27.7 | 6.2 | |

| |||||

Calculated stem volume [m^{3} ha^{-1}] |
All: 143 plots | 10.7 | 1,398.0 | 472.8 | 293.8 |

Co-registered: 103 plots | 15.7 | 1,137.7 | 423.4 | 239.0 | |

- W: 92 plots | 15.7 | 1,137.7 | 440.2 | 241.9 | |

- S: 64 plots | 23.0 | 1,137.7 | 415.9 | 230.0 | |

-W&S: 52 plots | 27.0 | 1,137.7 | 446,6 | 231.9 |

Evaluation of the appropriate sample plot size. Shown are the final models, the condition numbers (κ), the bias correction factors (RE), the coefficients of determinations (R^{2}), the root mean square errors (RMSE) between estimated and ground reference values and the results of the cross-validation (MIN, MAX, MEAN, and SD) for each sample plot size. The calculations were done using all (103) sample plots.

Sample plot size | Parameter | |||||||
---|---|---|---|---|---|---|---|---|

κ | RE | R^{2} |
RMSE [m^{3}ha^{−1}] |
Cross-validation [m^{3}ha^{−1}] | ||||

MIN | MAX | MEAN | SD | |||||

Ø18 m | ln_{stem,fi} = 4.861837 - 0.419251 ln_{10,l} + 0.857076 ln_{6,f} - 0.123035 ln_{9,l} | |||||||

19.9 | 1.025544 | 0.84 |
0.354 |
-1.151 |
1.442 |
0.000 |
0.376 | |

0.83 |
101.4 |
-253.1 |
267.1 |
0.0 |
104.0 | |||

Ø20 m | ln_{stem,fi} = 3.178314 - 0.027323 ln_{2,f} + 0.262734 ln_{7,f} + 0.526734 ln_{6,l} | |||||||

29.0 | 1.039258 | 0.81 |
0.379 |
-1.612 |
1.397 |
0.001 |
0.410 | |

0.82 |
101.5 |
-264.9 |
291.0 |
-0.1 |
105.0 | |||

Ø22 m | ln_{stem,fi} = 3.090095 - 0.031669 ln_{2f} + 0.609494 ln_{6,f} + 0.204359 ln_{7,l} | |||||||

29.0 | 1.033626 | 0.83 |
0.361 |
-1.794 |
1.088 |
0.001 |
0.385 | |

0.83 |
99.1 |
-255.2 |
272.3 |
0.0 |
102.1 | |||

Ø24 m | ln_{stem,fi} = 4.662327 - 0.901615 ln_{0,f} + 0.523643 ln_{30,l} + 0.812855 ln_{6,f} | |||||||

27.2 | 1.017166 | 0.85 |
0.343 |
-1.773 |
0.936 |
-0.002 |
0.365 | |

0.84 |
96.8 |
-254.7 |
204.1 |
-0.3 |
100.0 | |||

Ø26 m | ln_{stem,fi} = 4.662327 - 0.901615 ln_{0,l} + 0.523643 ln_{40,l} + 0.812855 ln_{6,f} | |||||||

24.5 | 1.019927 | 0.84 |
0.355 |
-1.908 |
0.767 |
-0.001 |
0.375 | |

0.82 |
103.9 |
-338.4 |
282.1 |
-0.3 |
107.5 |

The values are calculated in the logarithmic scale. The quantities in the logarithmic scale are dimensionless.

The values are calculated in the arithmetic scale. The bias was corrected using a ratio estimator [

Effects of different ALS densities and acquisition times. Shown are the derived coefficients of determinations (R^{2}), the root mean square errors (RMSE) between estimated and ground reference values and the condition numbers (κ). Furthermore, the statistics of the residuals (MIN, MAX, MEAN, and SD) derived from the cross-validation are shown. The calculations were done using all (A), the winter (W), the summer (S), and the winter & summer (W&S) data corresponding to 103, 92, 64, and 52 sample plots. The calculations were done based on the original ALS points and on the thinned (percentage of thinning was 66%; data sets.

Time | κ | CF | R^{2} |
RMSE [m^{3}ha^{−1}] |
Cross-validation [m^{3}ha^{-1}] | |||||
---|---|---|---|---|---|---|---|---|---|---|

MIN | MAX | MEAN | SD | |||||||

Percentage of thinning: 0% | ||||||||||

A | ln_{stem,fi} = 4.662327 - 0.901615 ln_{0}_{,f} + 0.523643 ln_{30,l} + 0.812855 ln_{6,f} | |||||||||

27.2 | 1.017166 | 0.91 |
0.343 |
-1.773 |
0.936 |
-0.002 |
0.365 | |||

0.84 |
96.8 |
-254.7 |
204.1 |
-0.3 |
100.0 | |||||

W | ln_{stem,fi} = 5.621786 - 1.120663 ln_{0,f} + 0.553731 ln_{4,f} + 0.761191 ln_{6,f} | |||||||||

19.7 | 1.012892 | 0.84 |
0.351 |
-1.290 |
0.949 |
-0.004 |
0.379 | |||

0.81 |
110.4 |
-249.2 |
300.5 |
-0.4 |
114.8 | |||||

S | ln_{stem,fi} = 2.955974 + 0.102221 ln_{90,f} - 0.001293 ln_{3,l} + 0.729716 ln_{6,l} + 0.099115 ln_{cv,l} | |||||||||

21.7 | 1.046855 | 0.76 |
0.410 |
-2.128 |
0.719 |
0.001 |
0.445 | |||

0.82 |
97.9 |
-385.7 |
317.5 |
0.8 |
110.7 | |||||

W&S | W | ln_{stem,fi} = 4.824553 - 0.399229 ln_{0,f} + 0.094519 ln_{80,f} - 0.012827 ln_{2,f} + 0.696564 ln_{6,f} | ||||||||

20.3 | 1.021042 | 0.81 |
0.327 |
-1.097 |
1.134 |
0.006 |
0.384 | |||

0.79 |
108.2 |
-224.7 |
332.0 |
1.6 |
120.0 | |||||

S | ln_{stem,fi} = 3.892341 - 0.217090 ln_{0,f} + 0.047210 ln_{80,f} + 0.775024 ln_{6,f} | |||||||||

14.9 | 1.018094 | 0.89 |
0.248 |
-0.887 |
0.607 |
-0.002 |
0.273 | |||

0.85 |
90.9 |
-242.1 |
258.9 |
-0.6 |
97.2 | |||||

Percentage of thinning: 66% | ||||||||||

A | ln_{stem,fi} = 2.818110 - 0.074578 ln_{0,f} + 0.282504 ln_{7,f} + 0.606238 ln_{6,l} | |||||||||

24.5 | 1.026639 | 0.84 |
0.356 |
-1.7996 |
0.968 |
0.000 |
0.375 | |||

0.82 |
104.4 |
-268.5 |
311.9 |
-0.1 |
106.7 | |||||

W | ln_{stem,fi} = 2.960790 - 0.035852 ln_{2,l} + 0.610804 ln_{6,l} + 0.270893 ln_{8,l} | |||||||||

18.6 | 1.025183 | 0.80 |
0.387 |
-1.514 |
1.795 |
-0.001 |
0.411 | |||

0.77 |
117.5 |
-297.7 |
410.2 |
-0.2 |
122.7 | |||||

S | ln_{stem,fi} = 3.124630 + 0.129229 ln_{90,f} + 0.014617 ln_{3,l} + 0.682713 ln_{6,l} + 0.065367 ln_{cv,l} | |||||||||

21.7 | 1.048572 | 0.74 |
0.425 |
-2.092 |
1.051 |
0.000 |
0.460 | |||

0.82 |
99.9 |
-370.9 |
309.2 |
0.2 |
111.3 | |||||

W&S | W | ln_{stem,fi} = 3.472091 + 0.295300 ln_{0,f} - 0.510809 ln_{30,f} + 0.589993 ln_{70,f} - 0.098405 ln_{2,f} + 0.483818 ln_{6,f} | ||||||||

28.3 | 1.021042 | 0.80 |
0.333 |
-1.259 |
1.047 |
0.003 |
0.402 | |||

0.77 |
113.3 |
-304.0 |
317.0 |
0.0 |
127.1 | |||||

S | ln_{stem,fi} = 2.925126 + 0.250145 ln_{0,f} + 0.492056 ln_{80,f} - 0.148622 ln_{2,f} + 0.353557 ln_{5,f} | |||||||||

25.8 | 1.010784 | 0.85 |
0.293 |
-1.251 |
0.940 |
-0.011 |
0.363 | |||

0.85 |
93.3 |
-276.4 |
282.0 |
-2.4 |
104.0 |

The values are calculated in the logarithmic scale. The quantities in the logarithmic scale are dimensionless.

The values are calculated in the arithmetic scale. The bias was corrected using a ratio estimator [