A comprehensive yield table is constructed by determining the development, in time, o f the fol
lowing volume components:
H |, N |, G |, V], D,, F, (before thinning) H 2, N2, G2, V2, D2i F2 (thinnings) H 3, Ns, G,, V3, Dj, F3 (after thinning)
In fig.l the general production procedure is illustrated. To the left in the diagram the volume com ponents we wish to determine are shown for: before thinning, thinnings and after thin
ning. A t the top, the 4 main stages in the construction procedure are shown.
FIX ED DERIVED
STAGE 1 STAGE 2 STAGE 3 STAGE 4
A F T E R Hj h3
T H IN N IN G Nj N3(D3,G3)
o 3 g3
v 3 - V3(H3,G 3,F3)
D, d3 \
Table F3 Fj(D3,H 3) \
Total yield p \
Auxiliary Q = (D2/ D 3) \
factors c \
T H IN N IN G S H 2 A H 2(D2iG2,V2,F2)
n2 N2(N3,N.) \
g2 ---— g2(d2,n2) \
V2 V2(C ,P) \
D2 d2(Q ,d3) \
Table F2 F 2(H 2,D2,G2,V2) a
B EFO RE H | H I(D 1,G „ V „ F 1)
TH IN N IN G N, n,(C ,n3)
G, - G , ( G 3,G 2)
V, V ^ Vj. Vj)
D, D ^ N , ^ ,)
Table F, F 1(H „ D „ G 1,V i)
F ig u r e 1. The general construction procedure o f a yield table. The arrow s show the overall structure o f the calculations. The position o f the individual volume factors on an x-axis (from left to right in the figu
re), shows the order in which they are calculated from fixed or previously derived factors, shown in p aren
thesis following the factor.
F ig u r I. D en generelle frem stillingsprocedure f o r en produktionsoversigt. Pilene angiver hovedbereg- ningsgangen. D e enkelte vedmassefaktorers placering p å en x-akse (gående fr a venstre m o d højre i figuren) angiver den ræ kkefølge, ved hvilken de beregnes a ffastlagte eller tidligere afledte fa ktorer, der vises i p a rentes efter fa kto ren .
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In stage 1 the 6 basic factors o f the yield table are determined as a function o f age, through statistic or graphic smoothing o f the basic data: total yield (P), height o f remaining crop (H3), basal area o f remaining crop (G3) and diameter o f remaining crop (D3). Furtherm ore, determ i
nation o f the thinning quotient (Q = D 2/D 3) and thinning intervals (C) is required.
When stage 1 is com pleted, all remaining volume factors can be derived. In stage 2, volume factors for the remaining crop are determined. In stage 3, volume factors for thinnings are de
rived from the basic factors and the factors calculated in stage 2. Finally, volume factors for before thinning are derived partly in stage 3 and ultimately in stage 4.
In the following text the construction procedure is described in detail. As this description is not necessary for the understanding o f the remaining text, it is presented in small print.
Form factors
In fig. 1, it is assumed th at form factors are derived from form -factor tables or functions. They m ay also be obtained by smoothing form -factor measurements available in the basic m aterial. This procedure was used for all yield tables constructed before general form -factor functions were available from the middle o f this century. In the smoothing process, height or diam eter are most frequently used as independent variables (e.g. M øller, 1933, p .540), rarely both height and diam eter (e.g. Andersen, 1950, p .332-336; Henriksen, 1957, p .321-324).
The form -factor functions known today for the m ain tree species (Fog & Jensen, 1952; Olsen, 1976;
Madsen, 1987) are based upon extensive d ata, which means th at deriving local form -factor functions rather than using general functions will lead to a poorer result. Best use o f local form -factor measurements is made by calculating a local form -factor level, either totally or for each age class (for further reference, see H enriksen, 1952, p . 147-150).
Growth section
The whole basis o f a yield table is the growth section, which is derived by smoothing height o f remaining crop: H 3, and total yield: P , as a function o f age.
The most im portant factor in the yield table is total yield. This is stressed by Oppermann as early as the beginning o f this century (Opperm ann, 1905, p. 124-125; Oppermann, 1914, p .342).
As an alternative to the statistic or graphic smoothing o f accumulated total production: P = I I , , one may sm ooth the differential coefficient o f P , i.e. the function for annual increment: Iv (see e.g. Henriksen, 1957, p .327-329; Henriksen, 1958b, p .25-26; Kjølby, 1958, p .50-53). The advantage o f this procedure is to ensure th at the form o f the annual increment function is in accordance with general knowledge regarding the increase, culm ination and decrease o f the curve.
Treatment- and Yield- sections
Treatm ent can be described in two ways, either by determining the volume o f thinnings or by determining the volume o f remaining crop. Most frequently treatm ent is established by determining the volume o f re
maining crop: V3, or basal area o f remaining crop: G3.
One can now choose either to determine the num ber o f stems in the remaining crop: N3, or diameter o f the remaining crop: D3. It is preferable to determine diam eter, as this is the m ost im portant factor in estab
lishing economic value o f the yield, and then let stem num ber be derived as shown in fig .l.
N3 is derived from G3 and D3. F3 is determined from a form -factor table or function and V3 is calculated as V3 = H 3*G3»F3. Thus all required volum e fa cto rs f o r the remaining crop have been determ ined.
The thinning intervals: C , are determined in accordance with the desired thinning régime, whereupon volumes o f thinnings: V2, and volume before thinning: V,, are derived from P and V3, as P = V3+ LV2 and V, = V2 + Vj. N2 and N, are also determined by the thinning intervals, as the num ber o f stems before thin
ning is equal to num ber o f stems after the previous thinning, and N2 = N r N3.
The diam eter o f thinnings is norm ally determined by sm oothing o f the thinning quotient Q = D2/ D 3.
With the thinning diam eter: D2, determined by the quotient D2/D 3 and D3 (D2 = D3*Q), G 2 is calculated from D2 and N2.
If a reliable form -factor table is available, with diam eter and height as independent variables, H2 and F 2 are finally determined simultaneously as the factors solving the equation V2 = H 2*G2»F2, in which V2 and G2, as well as the one independent variable for the determ ination o f form factor: D2, are know n. The value o f F2 is determined through a process o f iteration in the form -factor function (this m ethod is used by Elin- gård-Larsen & Jensen, 1985, p .254).
If a form -factor function with both diameter and height as independent variables is not available, H 2 is frequently determined by smoothing the ratio H 2/ H 3 or H 3-H2, equivalent to the procedure described for D2 (see e.g .O pperm ann, 1914, p. 346; Andersen, 1950, p. 353; Møller, 1951, p .262,272-273).
If, however, a reliable form -factor table, with both diam eter and height as independent variables,.is available, the above m ethod will result in an ambiguous determ ination o f H 2. This ambiguity m ay result in inconsistencies in the yield table, as some variables may be calculated in several ways, not necessarily lea
ding to the same result. The correct procedure for the determ ination o f H 2 is to solve the equation V2 = H 2*G2»F2, which ensures against ambiguity.
Thinning height can also be determined from thinning diam eter and the m ean-diam eter/m ean-height- regression o f the remaining crop (Dg/ H g-curve). This is not equivalent to using the diam eter/height-regres- sion o f the remaining crop (se e.g. Henriksen, 1958b, p .59,63). Systematically low thinning heights will normally result from the form er procedure.
Finally, H 2 can be derived from thinning diam eter, D2, and standard diam eter/height-regressions for re
maining crop, determined otherwise (see e.g. H enriksen, 1957, p .327). As the diam eter/height-regression of thinnings, resulting from norm al low thinning, will have a lower level than the diam eter/height-regres- sion o f remaining crop, thinning heights will systematically be determined too high through this m ethod.
Thus all volum e fa c to rs fo r thinnings have been determined.
The remaining volume factors for before thinning (H ,, G,, D ,, F,) can now be determined. Basal area before thinning is established as G, = G2 + G3, whereupon D, is calculated from N, and G ,. Hi and F, are determined (by iteration) as the factors solving the equation V, = H I*GI*F,) where V,, G, and the one inde
pendent variable for the determ ination o f form factor: Di, are known (see above regarding H 2 and F2).
Thereby all volum e fa cto rs in f i g . l have been determined.
Iterative adaption
5 o f the basic factors (stage 1, fig .l) are determined by smoothing o f volume factors derived directly from the basic data. C is derived likewise. The basic factors are thus determined strictly in accordance with the data (note however section 4.5 on the topic o f empirical and prognostic yield tables). The factors calcula
ted in stage 2, 3 and 4 are concurrently checked with the basic d ata during the process o f construction. If evident discrepancies appear, it is necessary to correct the sm oothing o f the 6 basic factors. In this way an iterative adaption is achieved. (Today this adaption can also be obtained by the use o f more advanced statis
tic smoothing techniques). It is, however, frequently seen that P and H 3 are unequivocally determined as a function o f age at the very start. As a result the adaption only affects G3, D3 and possibly Q = D2/D 3.
Conform ity
If the above-described general construction procedure is followed, all volume factors in the yield table will be unambiguously determ ined (i.e. each volume factor can only be determined in one way), whereby con
formity within the yield table is ensured.
However, many o f the methods described in various publications will result in am biguous determ ina
tion o f several volume factors. This may result in inconsistencies in the constructed yield table. In the past, when reliable form -factor tables did not exist, such inconsistencies were absorbed in the form factor, as this could not be determined accurately anyway.
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