1

Fig. 1. The difference between the predictions of the two volume models from the 1970s to the 2010s (right) by the National Forest Inventory of Finland (NFI) sampling region (left). Sampling regions 2 and 3 are divided into western (W) and eastern (E) parts.

2

Fig. 2. The plots selected for TLS data collection (all dots), scanned plots (green dots), felled trees (red dots).

Table 1. The minimum, maximum and mean of diameter at breast height (dbh), height (h) and volume (v) in the old and new volume modelling data by tree species.
old data new data
min max mean min max mean
pine dbh (cm) 0.90 50.60 20.23 3.90 49.70 20.90
h (m) 1.50 28.30 13.67 5.40 29.60 16.59
v (dm3) 0.40 1921 312 5.10 1947.7 340.20
spruce dbh (cm) 1.5 61.9 18.04 5.55 57.00 22.25
h (m) 1.8 32.7 13.82 4.50 32.70 18.19
v (dm3) 0.7 3796 264 7.40 2856.4 448.20
birch dbh (cm) 1.2 49.7 16.65 4.00 42.45 16.18
h (m) 2.4 29.5 15.33 4.50 29.80 16.20
v (dm3) 0.4 2026 229 4.30 1427.70 213.36
3

Fig. 3. The form height in old (0) and new (1) dataset for pine (upper) spruce (middle) and birch (lower) in the percentile diameter classes.

Table 2. The estimates of coefficients of the volume model 5 (61.1. in Laasasenaho 1982), their standard error and t-values for all species
full data model old model
coefficient standard error t-value coefficient
pine β0 –2.293114 0.0228019 –101.3275 –2.29450
β1 2.570253 0.007707629 333.4687 2.57025
δ0 0.2105365 0.05147734 4.089886
δ1 –0.03352298 0.01723158 –1.945438
spruce β0 –2.411956 0.02326293 –104.5267 –2.41218
β1 2.624621 0.008255297 317.9317 2.62463
δ0 0.06596944 0.06211427 1.062066
δ1 0.009008474 0.0206251 0.4367724
birch β0 –2.102088 0.03392367 –62.55514 –2.09787
β1 2.55162 0.01231355 207.2205 2.55058
δ0 0.1252633 0.06947937 1.802885
δ1 –0.03105756 0.02547401 –1.219186
Table 3. The residual standard error (RSE) and the coefficient of determination (Multiple R2) for old and new volume models (5) by species. The F-test describes the significance of the full model compared to the old model.
Tree species Full data Old data
RSE original RSE Multiple R2 F-test p-value RSE original RSE Multiple R2
Pine 19.51 19.45 0.9792 115.36 0 18.01 17.81 0.9833
Spruce 20.41 20.93 0.9811 48.768 0 19.91 20.77 0.9823
Birch 20.30 19.88 0.9792 6.1824 0.002 20.08 19.66 0.9812
4

Fig. 4. Relative residuals of model 5 in the dbh classes of the new data with λ = 0 and λ = 1 for pine (upper row), spruce (middle row) and birch (lower row).

Table 4. The estimates of coefficients of the volume model 6 (61.2. in Laasasenaho 1982), their standard error and t-values for all species
full data model old model
coefficient standard error t-value coefficient
pine β0 –5.392811 0.06715021 –80.54965 –5.39417
β1 3.480598 0.03040095 114.4898 3.48060
β2 –0.03198842 0.001596443 –20.03731 –0.039884
δ0 –0.002811802 0.1647923 –0.0170627
δ1 0.06637664 0.07175107 0.9250962
δ2 –0.005370931 0.003522453 –1.52477
spruce β0 –5.398914 0.06499235 –83.34407 –5.39934
β1 3.46464 0.03031531 114.2868 3.46468
β2 –0.02731815 0.001716997 –15.91042 –0.0273199
δ0 –0.4109449 0.2124345 –1.934454
δ1 0.2087533 0.08939057 2.335294
δ2 –0.00843131 0.004000025 –2.107814
birch β0 –5.327978 0.08839265 –60.48643 –5.41948
β1 3.546208 0.04242831 83.58118 3.57630
β2 –0.03882196 0.002573179 –15.08716 –0.0395855
δ0 –5.327978 0.08839265 –60.48643
δ1 3.546208 0.04242831 83.58118
δ2 –0.03882196 0.002573179 –15.08716
Table 5. The residual standard error (RSE) and the coefficient of determination (Multiple R2) for old and new volume models (6) by species. The F-test describes the significance of the full model compared to the old model.
Tree species Full data Old data
RSE original RSE Multiple R2 F-test p-value RSE original RSE Multiple R2
Pine 18.09 17.77 0.9807 69.078 0 17.30 17.04 0.9846
Spruce 19.04 18.45 0.9829 38.97 0 18.86 18.42 0.9841
Birch 19.27 18.54 0.9811 8.3378 0 18.89 18.08 0.9834
5

Fig. 5. The predicted tree volumes as a function of dbh in old data (light blue dots) and new data (blue dots) for pine (upper), spruce (middle) and birch (lower) using model 5 (A) and model 6 (B). Black curve is based on new data and grey on the old data.

6

Fig. 6. Relative residuals of model 6 in the dbh classes of the new data with λ = 0 and λ = 1 for pine (upper row), spruce (middle row) and birch (lower row).

Table 6. The estimates of coefficients of the volume model 7 (61.3. in Laasasenaho 1982), their standard error and t-values for all species.
full data model old model
coefficient standard error t-value coefficient
pine β0 –3.383341 0.04207626 –80.45867 –3.32176
β1 2.015408 0.01445914 139.3864 2.01395
β2 2.144271 0.05597404 38.30831 2.07025
β3 –1.121108 0.04888704 –22.93263 –1.07209
β4 –0.004035521 0.0006782147 –5.950211 –0.003273
δ0 0.5831496 0.1432678 4.070347
δ1 –0.1778505 0.03040952 –5.848513
δ2 –1.1444 0.2984041 –3.835067
δ3 1.141898 0.2689587 4.245627
δ4 0.002406294 0.001429594 1.683201
spruce β0 –3.782985 0.04199002 –90.15069 –3.77543
β1 1.912913 0.01687974 113.326 1.91505
β2 2.833072 0.04900075 57.81691 2.82541
β3 –1.538206 0.04188057 –36.7284 –1.53547
β4 –0.008581622 0.0008074753 –10.62772 –0.0085726
δ0 0.03531895 0.1722487 0.2050462
δ1 –0.0461456 0.04193549 –1.100395
δ2 0.1403905 0.3540179 0.3965632
δ3 –0.1161943 0.3197521 –0.3633886
δ4 0.0003250218 0.001790375 0.1815384
birch β0 –4.553696 0.08843668 –51.52675 –4.49213
β1 2.122157 0.02553022 83.12333 2.10253
β2 4.096565 0.1526368 26.83864 3.98519
β3 –2.767257 0.1362035 –20.31707 –2.65900
β4 –0.01485222 0.001453626 –10.21736 –0.0140970
δ0 0.4878564 0.2303936 2.117491
δ1 –0.2301959 0.04762161 –4.833855
δ2 –0.5090479 0.5184651 –0.9818365
δ3 0.4926584 0.4648618 1.059795
δ4 0.01209118 0.002811613 4.300443
Table 7. The residual standard error (RSE) and the coefficient of determination (Multiple R2) for old and new volume models (7) by species. The F-test describes the significance of the full model compared to the old model.
Tree species Full data Old data
RSE original RSE Multiple R2 F-test p-value RSE original RSE Multiple R2
Pine 6.42 6.42 0.9975 32.933 0 6.82 6.81 0.9976
Spruce 7.00 6.97 0.9977 7.7947 0 7.46 7.42 0.9975
Birch 7.96 7.88 0.9967 10.531 0 8.13 7.98 0.9969
7

Fig. 7. The predicted tree volumes using model 7 in the old data (light blue) and new data (blue) as a function of dbh class (A) and h class (B) for pine (upper row), spruce (middle row) and birch (lower row).

8

Fig. 8. Relative residuals of model 7 in the dbh classes of the new data with λ = 0 and λ = 1 for pine (upper row), spruce (middle row) and birch (lower row).

Table 8. The coefficients, their standard errors and t-values of the new volume model (8) for all species.
full data model
coefficient standard error t-value
pine β0 –4.447527 0.02631802 –169.0749
β1 2.362339 0.01531425 154.2575
β2 –0.01054031 0.0006214953 –16.9596
β3 0.8566598 0.007997866 107.111
δ0 –0.05472306 0.06228864 –0.87854
δ1 –0.1009865 0.03211809 –3.144226
δ2 –0.002319712 0.001344036 –1.72593
δ3 0.1566015 0.01423715 10.9995
spruce β0 –4.162928 0.03039662 –137.0555
β1 2.083863 0.01977891 105.3578
β2 –0.00527749 0.0007558304 –6.982374
β3 1.040583 0.0114666 90.74904
δ0 –0.207592 0.09298762 –2.232469
δ1 0.007377705 0.04845477 0.1522596
δ2 –0.005278365 0.001738939 –3.035395
δ3 0.1034346 0.02262765 4.57116
birch β0 –5.12979 0.04217144 –121.7211
β1 2.509713 0.02722397 92.18763
β2 –0.02113233 0.001285776 –16.43546
β3 0.9830402 0.01676183 58.64755
δ0 0.2926902 0.09660794 3.02967
δ1 –0.2129817 0.05494384 –3.876353
δ2 0.009456602 0.002737932 3.453922
δ3 0.06331147 0.02997322 2.112268
Table 9. The residual standard error (RSE) and the coefficient of determination (Multiple R2) for old and new volume models (8) by species. The F-test describes the significance of the full model compared to the old model.
Tree species Full data
RSE original RSE Multiple R2 F-test p-value
Pine 6.62 6.67 0.9974 41.096 0
Spruce 7.88 8.13 0.9970 12.158 0
Birch 8.22 8.19 0.9965 9.1126 0
9

Fig. 9. The predicted tree volumes with the new model 8 in the old data (light blue) and new data (blue) as a function of dbh class (A) and h class (B) for pine (upper row), spruce (middle row) and birch (lower row).

10

Fig. 10. Relative residuals of model 8 in the dbh classes of the new data with λ = 0 and λ = 1 for pine (upper row), spruce (middle row) and birch (lower row).