A stand-level growth and yield model for thinned and unthinned managed Norway spruce forests in Norway

Figures & data

Figure 1. Distribution of Norway spruce sample plots across Norway.

Table 1. Descriptive statistics of thinned and unthinned Norway spruce data.

Treatment	Growth periods		A	S	N	G	V	GTQ	HT	ALT	LAT
Unthinned	311	mean	41	16	2510	37.3	298			209	61
		SD	13	3	860	15.3	196			140	2.7
		min	18	11	777	6.1	21			20	58
		max	92	21	5021	79.8	893			530	67
1 Thinning	255	mean	50	16	1221	36.1	332	0.71	12.4	212	60
		SD	12	3	400	12.4	198	0.11	2.3	146	2.5
		min	25	11	600	6.3	26	0.47	8.5	20	58
		max	68	21	2640	67.4	917	0.92	21.7	450	64
2 Thinnings	187	mean	38	15	1796	25.4	149	0.79	11.4	260	61
1st thin		SD	8	2	312	9.2	82	0.08	1.8	132	2.3
		min	25	11	900	8	27	0.5	8.7	40	58
		max	61	20	2500	46.8	410	0.91	17.2	450	64
2nd thin	246	mean	55	15	1043	35.7	331	0.75	15.6
		SD	11	2	232	10.1	171	0.12	2
		min	31	11	460	11.8	72	0.35	11.8
		max	78	20	1700	60.3	812	0.92	21.1

Notes: A = stand age from planting (years); S = site index (m) at base age 40 years; N = number of trees (ha⁻¹); G = basal area (m² ha⁻¹); V = stand volume (m³ ha⁻¹); G_TQ = basal area thinning quotient (G after thinning divided by G before thinning); and HT = dominant stand height (m) at thinning, ALT = altitude (m); LAT = latitude.

Figure 2. Description of the model application in the new stand-level model. Numbers in brackets corresponds to the equations presented in the text.

Table 2. Definitions of variables and their abbreviations used in this work.

Abbreviation	Definition of Variable
A1,2	Stand Age (years from planting)
V1,2	Volume (m^3 ha^−1)
G1,2	Basal area (m^2 ha^−1)
H1,2	Dominant stand height (m), mean height of 100 thickest trees per hectare
N1,2	Number of trees per hectare
GA	Basal area after thinning (m^2 ha^−1)
GB	Basal area before thinning (m^2 ha^−1)
HT	Dominant stand height at thinning (m)
S	Site Index at base age 40 years (m)
NA	Trees per hectare after thinning
NB	Trees per hectare before thinning
QMD1,2	Quadratic mean diameter (cm)
CAI	Current annual volume increment (m^3 ha^−1 yr^−1)
MAI	Mean annual volume increment (m^3 ha^−1 yr^−1)

Note: All subscripts of 1 or 2 denote stand values at the measurement period at times 1 or 2, respectively.

Table 3. Stand data for four selected unthinned Norway spruce plots.

ID	S	A	N	G	H	QMD	V
1	11	46	2977	31.3	12.2	11.6	155.9
2	14	34	3090	26.2	12.2	10.4	125.9
3	17	25	2038	22.7	11.8	11.9	109.6
4	20	22	2710	24.6	10.7	10.8	115.9

Note: A = stand age from planting (years); S = site index (m) at base age 40 years; N = number of trees (ha⁻¹); G = basal area (m² ha⁻¹); H = stand dominant height (m), QMD = quadratic mean diameter (cm); and V = stand volume (m³ ha⁻¹).

Table 4. Parameter estimates (standard errors) and fit statistics for Equations (1–5).

	Equation – component
	1 – V2	2 – G2	3 – N2	4 – H2	5 – NA/NB
${\text{β}}_1$	0.24961	4.77696	−1.0085	0.01605	−1.93267
	(0.01097)	(0.02191)	(0.2619)	(0.00142)	(0.02303)
${\text{β}}_2$	1.15036	0.30957	0.03675	0.61208	1.92953
	(0.01594)	(0.02651)	(0.00792)	(0.17427)	(0.02439)
${\text{β}}_3$	1.01153	−0.1479	3.76228	4.43722
	(0.01224)	(0.0175)	(0.7144)	(0.54254)
${\text{β}}_4$	2.320398		2.5541
	(0.59361)		(0.4084)
${\text{β}}_5$			−1.0097
			(0.2631)
	All Data
E	0.325	−0.012	−1.78	0.019	−0.00044
MAE	11.8	1.33	73.3	0.499	0.0436
RMSE	16.6	1.8	138.1	0.644	0.0533
R^2	0.987	0.963	0.965	0.976	0.846
	Cross Validation
E	0.312	−0.0945	−1.23	−0.026	−0.00043
MAE	11.6	1.32	73.5	0.472	0.0441
RMSE	16	1.83	158	0.627	0.0542
R^2	0.988	0.962	0.949	0.976	0.845

Figure 3. Observed versus predicted values for future Volume (V₂), basal area (G₂), trees per hectare (N₂), and dominant stand height (H₂) based on predictions from Equations 1–4 with parameter values from Table 2. Residuals are based on direct predictions from observations (circles) and, in the case of G₂ and V₂, include predictions from the necessary model components (triangles).

Figure 4. Comparison of site-index curves between the new Equation (4) and the equations of Tveite (1977) and Sharma et al. (2011).

Figure 5. Development in number of trees per hectare for different site indices (m, base age 40 years) and different thinning intensities based on a common starting value of 2500 trees per hectare. The basal area thinning quotient (G_TQ) is the ratio of basal area before and after thinning where a value of 1 indicates the unthinned stand condition.

Figure 6. Comparison of residuals for predicted future trees per hectare (N₂) between the new survival function, Equation (3), and the survival function of Eid and Øyen (2003).

Figure 7. Comparison of residuals for future quadratic mean diameter (QMD2, cm) between the function of Blingsmo (1984) and the implied QMD2 calculated from future basal area and trees per hectare as predicted from Equations (2) and (3), respectively.

Figure 8. Comparison of residuals for future volume (V₂, m³ ha⁻¹) as predicted from Næsset and Tveite (1999) and from Equation (1).

Figure 9. Development in future trees per hectare and basal area for four Norway spruce stands (Table 3) representing different site indices (S-11, 14, 17, and 20 m) with three management scenarios including: Unthinned, one-thinning removing 25% of the basal area at a dominant stand height of 12 m, and two-thinnings removing 25% of the basal area at dominant stand heights of 12 and 16 m.

Figure 10. Maximum size-density relationships for four Norway spruce stands (Table 3) representing different site indices (S-11, 14, 17, and 20 m) with three management scenarios including: Unthinned, one-thinning removing 25% of the basal area at a dominant stand height of 12 m, and two-thinnings removing 25% of the basal area at dominant stand heights of 12 and 16 m.

Figure 11. Development of standing volume and net volume (standing + removed volume) for four Norway spruce stands (Table 3) representing different site indices (S-11, 14, 17, and 20 m) with three management scenarios including: Unthinned, one-thinning removing 25% of the basal area at a dominant stand height of 12 m, and two-thinnings removing 25% of the basal area at dominant stand heights of 12 and 16 m.

Figure 12. Comparison of current annual volume increment (CAI) and mean annual volume increment (MAI) stands of different site indices under three management scenarios: Unthinned, one-thinning removing 25% of the basal area at a dominant stand height of 12 m, and two-thinnings removing 25% of the basal area at dominant stand heights of 12 and 16 m.

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