3.3 Reconstruction o f basic d ata
3.3.3 Reconstruction o f total yield
A summary o f the individual methods o f reconstruction shown in table 1 is given in this sec
tion. F or further details it is necessary to refer to the cited literature.
A fter each summary, a short evaluation o f the reconstruction m ethod is given. The purpose is to give a brief evaluation o f the suitability o f the m ethod for reconstructing basic data, when the aim is the construction o f a yield table. When interpreting these evaluations, it is im portant to bear in mind th at the cited publications, in which the methods are described and used, fre
quently have had multipurpose goals. Producing a yield table has only been the main goal in some o f the publications.
As the following sections are not necessary for understanding the article as a whole, they are presented in small print.
Group I
The feature common to all 14 publications shown in table 1, group 1, is that total yield and its distribution over tim e has been derived from long-term production experiments or local sample plots.
Therefore, the construction o f these yield tables follows the general procedure outlined in section 3.1.
However, the G ram/O pperm ann method deviates considerably from this general procedure. The method prevailed at The Danish Forest Experiment Station until the beginning o f the 1930’s, and is descri
bed under group II (see p. 93-94).
Group I I
The feature common to all publications shown in table 1, group II, is that total yield has been derived from observation stands, tem porary sample plots or field surveys. The distribution o f total yield over time is de
term ined in a num ber o f ways.
G roup II is divided into 5 subgroups.
Subgroup I
In subgroup I, total yield for each sample plot is determined as P = V3 + ZV2. V3 is m easured in the sample plot, whereas thinning volumes: V2, are reconstructed. The individual m ethods in the subgroup differ ac
cording to the m ethod used for reconstructing the thinning volumes. The distribution over tim e o f total yield (i.e. the shape o f the volume-increment curve) is reconstructed with the help o f stem analyses.
Registration o f thinnings - stem analysis, 1.1
This 1st m eth o d uses a com bination o f registration o f thinnings and stem analysis. The m ethod has m ainly been used by Løvengreen (1935, 1949, 1951a, 1951b).
The following main points characterize the m ethod (see Løvengreen, 1949):
1) Volume factors, including V3, are m easured in sample plots situated in observation stands.
2) Thinning volumes: V2, and time o f thinning is determined from the accounts. Thinnings are conver
ted from commercial volume to total volume by the use o f local solid-content factors, in com bination with knowledge regarding minim um diam eter o f small end used at the time o f logging. If the sample plot does not cover an entire observation stand, the appropriate thinning volume is calculated as a proportion o f the total.
3) On every sample plot 5-7 trees are selected for comprehensive stem analysis. The mean diam eter o f the sample trees should lie near mean diam eter o f the stand as a whole, or possibly slightly below, due to the earlier status o f the present mean trees as dom inants or co-dom inants (see section 3.3.2). The average stem volume o f the sample trees (under bark) is calculated for each year throughout their grow th. It is then pos
sible to calculate the annual stem increment for the average tree: iv, and stem-increment percentage: pv.
4) Løvengreen further assumes th at the increment percentage for total volume: P v, is the same as the stem-increment percentage calculated for the average sample tree: pv. This assum ption is further discussed by Løvengreen (1935, p .585-586) - (refer also to the critical analysis o f these assum ptions by Nielsen, 1949, p.251-252; M øller & Nielsen, 1952, p. 108-110).
From the m easured volume in the remaining crop (para. 1) and the above calculated volume-increment percentages (P v = pv), the volume o f the remaining crop can be reconstructed backwards year by year until the previous thinning: V3. To this volume the calculated thinning volume: V2 (from p ara. 2), is added, thus giving the volume before thinning: V,. W ith the value o f V, as a new starting p oint, the value o f the rem ai
ning crop is reconstructed backwards yet another period. This process is repeated until the tim e o f first thinning is reached. In this way the development o f volumes before thinning, in thinning and after thin
ning (V|, V2, V3) is obtained as well as the volume increment (see also subgroup 2, where L øvengreen’s m ethod is compared with that o f West-Nielsen (1949, 1951)).
5) The total yield o f each sample plot is thus determined by the present volume o f remaining crop (para.
1) and the sum o f the registered thinning volumes (para. 2). The distribution over time o f this total yield (i.e. the shape o f the volume-increment curve) is determined by the stem analysis (para. 3 and 4), (see N iel
sen, 1949, p .249).
Evaluation: L øvengreen’s m ethod is exceedingly tim e consuming, as it requires comprehensive stem analyses o f a num ber o f trees in each sample plot. It is however a thoroughly tested and satisfactory m ethod. Inaccuracies may result from the assum ption P , = pv, as well as from the problem regarding the earlier social status o f the present mean trees o f the sample plot.
S tum p measurements - stem analysis, 1.2
The 2nd m ethod makes use o f a com bination o f stum p m easurements and stem analysis. The m ethod is mainly developed by H enriksen (1958b).
It is characterized by the following m ain points (H enriksen, 1958b, p .52-53, 58-65, 67-71):
1) Volume factors, including V3, are m easured in each sample plot.
2) Basal area o f stumps o f earlier thinned trees are m easured. A single caliper m easurement per stum p is
D el forstlige Forsøgsvæ sen. X L II. H. 2. 27. ju n i 1989.
sufficient. As far as possible the stumps should be grouped according to the time o f thinning, e.g. last thin
ning, previous thinning, earlier thinnings (see para. 4). As it is often difficult to determine the age o f the stum ps, it is seldom possible to make a m ore precise grouping.
3) W ithin each sample plot a relationship between stum p diameter and basal area at breast height is established from m easurements on trees in the remaining crop. A relationship comm on to all the m easured sample plots may alternatively be established or deducted from general stemline functions.
The basal area and m ean-basal-area diam eter can now be established for each o f the thinning groups.
4) It is further necessary to determine height in order to calculate thinning volumes, as form factors are derived from a table or a function. Thinning heights ought to be determined from diam eter/height-regres- sions for the individual thinnings. These regressions are however unknown. Thinning heights are therefore determined by entering the calculated thinning diam eter into the m ean-diam eter/m ean-height relation
ship, established from the analysed sample trees. However, this m ethod introduces a biased estimate.
The m ean-diam eter/m ean-height-curve will approxim ately correspond to the Dg/ H g-curve o f the stand (referred to as the factor-curve in the following, see Henriksen, 1952, p. 179-180). General experience shows us that the slope o f the factor-curve will always be steeper than the slope o f the correct diameter/height- regression. The height o f the thinnings will therefore be estimated too low. This discrepancy will be more pronounced the more the diam eter o f thinnings differ from the mean diam eter o f the remaining crop, i.e.
in the case o f light thinnings.
The fewer the groups into which total basal area o f thinnings is divided (para. 2), the greater will be the discrepancy in height determ ination. This will be especially pronounced when m any small and early thin
ned trees result in a low m ean diam eter o f the accumulated thinnings (Henriksen, 1958b, p .58-65). The purpose o f the above-m entioned grouping o f stumps into 3 thinning phases (para. 2) is to minimize the bias associated with the determ ination o f thinning height.
5) By following the above-m entioned procedure in para. 2-4 the total thinning volume is determined:
ZV2, but we still do not know the distribution o f these thinnings over time.
The total thinning volume can be allocated to each year o f thinning, either in proportion to the thinning registrations for the entire stand in which the sample plot is situated, or in proportion to thinning volumes derived from an existing yield table (H enriksen 1958b, p .70).
6) Total yield can now be determined as the sum o f the volume o f remaining crop m easured in the sam
ple plot, and the reconstructed thinning volumes: P = V3 + I V 2. The total yield so determined is distributed over time by the aid o f stem analysis, as described in para. 3-5, m ethod 1.1.
Evaluation: The described procedure will normally lead to an underestim ate o f thinning volumes due to decayed stum ps and too low estimates o f thinning height (cf. para. 4) (Henriksen, 1958b, p .58-65). The discrepancy has been examined by H enriksen and found to be within a margin o f about -10% o f the true thinning volumes. Accordingly, the errors resulting from this m ethod are as a whole considered to be grea
ter than those o f m ethod 1.1.
Subgroup 2
In subgroup 2, 1, is reconstructed for each sample plot, and as a result total yield can be determined at any point in tim e as P = I I , . The distribution o f total increment over time (i.e the shape o f the increment curve) is reconstructed by the aid o f stem analysis.
Registration o f thinnings - stem analysis, 2.1
The only m ethod in this subgroup is described in references 20 and 21, table 1. The following m ain points characterize the m ethod (W est-Nielsen, 1951, p.227-241):
1) Volume factors, including V3, are measured in tem porary sample plots situated in observation stands.
2) Thinning volumes registered in the observation stands are converted from commercial volume to to tal volume, and distributed proportionately to the sample plot (see para. 2, m ethod 1.1).
3) From these volumes, stem num bers in the thinnings are derived (by a special procedure:
West-Niel-sen, 1951, p .233-234): N2. As the present num ber o f stems on the sample plot is known, it is possible to re
construct the stem num bers before and after thinning, backwards: N, and N3.
4) Stem analysis is completed for 6 sample trees in each plot. For every increment period (the time be
tween two successive thinnings) the volume increment o f the mean tree is calculated: r i v,].l2, as the average o f the volume increment for the 6 sample trees in the same period. The volume increment o f the sample plot in the increment period: E lv tl.12, is calculated as the volume increment o f the mean tree multiplied by the reconstructed stem num ber. As volume increment is reconstructed backwards, period by period, the volume-increment curve for the sample plot is finally obtained.
5) The m ethod o f West-Nielsen (1951) is a variant o f Løvengreen 's m ethod (1949, 1951a). The differen
ce is purely arithmetic, although West-Nielsen was probably not aware o f this. If we let index t l refer to the beginning of an increment period and index t2 to the end o f the period, the two m ethods are related as fol
lows:
In the aforem entioned increment period the increment ratio for the sample plot is:
where v:1 and v,2 are determined by stem analysis o f the sample trees.
W ith P v determined by (2), Løvengreen reconstructs the development o f standing volume over time as the relationship linking the volumes is:
In reconstructing the volume development o f the stand back in time, starting with the present value for standing volume: V,2, Løvengreen uses (3) in the following form:
As West-Nielsen is also reconstructing the volume development back in time, he uses (5) in the following form:
According to (5) the periodic increment: EIV t,_l2) is calculated as the increment o f the mean tree: I i v „.12, multiplied by the stem num ber. The m ethods o f West-Nielsen and Løvengreen are therefore, according to (3) and (5), totally identical.
Evaluation-. W est-Nielsen’s m ethod o f reconstructing N in (5) (see para. 3 and West-Nielsen, 1951, p .233-234) is unnecessarily complicated. Løvengreen ’s and West-Nielsen ’s m ethods b oth require the same basic inform ation. The detour via determ ination o f stem number in the latter m ethod adds an extra and unnecesssary source o f error. There is therefore no advantage in applying W est-Nielsen’s m ethod com pared with that o f Løvengreen (method 1.1).
In subgroup 3, total yield is reconstructed for each sample plot as P = V3 + I V 2. The distribution o f total yield back in time is, however, unknow n for each sample plot. As a complete time-series o f m ensurational data has not been reconstructed for each sample plot, it follows that we are now in the zone between m ain groups'B and C, table 1.
Stum p measurements, 3.1
The only method described uses stum p measurements without stem analysis. This m ethod is used by Elin- gård-Larsen & Jensen (1985) and by Jakobsen (1976). (It is not evident in the latter that thinning volumes are derived from stump measurements. This, however, was confirm ed by Jakobsen (1988)).
Pv = (Vt2-V „)/V tl
In L øvengreen’s m ethod it is assumed th at P v can be determined from the m ean tree as follows:
Pv = Pv = (vt2-vtl)/v ü
(1)
(2)
v t2 — v tl + E I v t l - t 2 — V,1 + V t l * ( ( v l 2 " v t l ) / v t l ) — V „ + V ,!*P v (3)
v „ = Vt2/(l + pv)
West-Nielsen, on the other hand, uses (3) as follows:
Vt2 = V,| + LIV n_12 = V„ + (Vt|/v,|)*(v[2-v,i) = V„ + N3 t,*£iv n.t2
(4)
(5)
(6)
Subgroup 3
90
1) Volume factors, including V3, are measured in a number o f tem porary sample plots.
2) The stum p measurements are carried out as described in subgroup 1, m ethod 1.2. In this way total thinning volume is reconstructed. Total yield is now determined as the sum o f the m easured volume o f standing crop: V3, and the thinning volumes determined by stum p measurements: I V ,. As stem analysis is not carried out, we do n ot know the distribution over time o f total yield o f each sample plot.
3) If the basic data originate from a region with uniform growth conditions, figures derived for total yield can be sm oothed over age as independent variable. If, on the other hand, considerable variation in site class is found within the region, it is advisable to use height as independent variable. This procedure is equivalent to using the Eichhorn rule (see section 4.3.1), and it is the approach used by Jakobsen (1976) and Elingård-Larsen & Jensen (1985).
4) Finally, the age/height-curve is reconstructed for the yield table. The distribution o f total yield over time, i.e. the form o f the increment curve, can now be derived for the yield table as a whole from the recon
structed heights and the height/total-yield-curve (para. 3).
E valuation: The above m ethod is the briefest o f the 4 so far described. It is relatively simple to use, as the laborious comprehensive stem analyses are n ot required. The m ethod is suitable in cases where only a small and incomplete am ount o f basic data are available. If, on the other hand, the purpose is to add basic data to already existing d ata from group I, then the m ethods described in subgroup 1 and 2 are more applicable.
In subgroup 4, annual increment is only reconstructed for one increment period: Iv ,U2l for each sample plot, whereas the total increment curve is reconstructed for the yield table as a whole. In table 1, subgroup 4 is therefore placed between m ain groups B and C. As already mentioned, it is also reasonable to place subgroup 3 here, although to a lesser degree.
Periodic increm ent - stem analysis, 4.1
The only m ethod described in this subgroup is used by K jølby (1958, p .31-58). Henriksen (1958b, p .53,71) has partly used the m ethod and refers to it briefly. The m ethod is characterized by the following main points:
1) Volume factors are measured in a num ber o f tem porary sample plots.
2) Height increment o f the sample plot is determined by m easuring height growth for the past 5 years on felled trees. For broadleaved species, the measurement can either be done by identifying the bud-scale scars in the bark, by counting the num ber o f annual rings on discs sampled at fixed heights, or by a combi
nation o f the two m ethods. For coniferous species, forming only one whirl each year, height growth is m o
re easily measured.
3) Diameter increment over the last 5 years is measured at breast height, either with an increment borer or by direct m easurements on sampled discs. Choosing a 5-year increment period is a comprom ise between two considerations. The periods must not be too short as this may cause great influence from climatic va
riations. O n the other hand the periods must not be too long, as this would cause distortion due to the n or
mal influence o f age.
4) If the periodic increment ratios for volume, basal area, height, diam eter and form factor are denoted:
P». P e. Ph. Pd and P f respectively, we get:
where V, H, G and F stand for the values at the beginning o f an increment period. From this it follows that:
Subgroup 4
V(l + P v) = H(1 + P h)*G(l + P g)*F(l +Pf) (7)
( 1 + P V) = (1 + P h) * ( l + P g)*(l + Pf) (8)
where
(1 + P g) = (1 + P d)2 (9)
which leads to
(1 + P V) = (1 +P„)»(1 + P „ )2*(1 + P f) (10)
It may be assumed that P f = 0. This will be approxim ately correct for older stands a n d /o r short increment periods. If smaller products, squares etc. are excluded in (10), the following approxim ation results:
P v = 2*Pd + P h (seeM øller, l9 5 l,p .2 5 -2 6 ;K jø lb y, 1958, p .50) (II) K jølby determines the periodic increment (ZIV [M2, t2-tl = 5 years) by entering the measured values o f P d and P h into (II), and multiplying the result with V. The periodic annual increment Iv 1M2 is thereafter deter
mined as (EIvtl.t2)/5.
[In general, the m easured increment ratios must be multiplied with the respective values for the start of the increment period (V,,, Dü , H„) in order to get the correct volume factors for the end o f the period.
In the present case, D and H are known for the start (D„, H tl) as well as the end (Dt2, H t2) o f the incre
ment period, whereas V is only known for the end (V12). It is therefore possible to calculate P d and P h on the basis o f the periodic increment in relation to end-diameter or end-height (Pdt2, Pha)- W ith Pd and P h calcu
lated in this way, P v, calculated from (11) (Pva), m ust be multiplied with the volume for the end o f the pe
riod (Vt2).
However, as we know diam eter and height at the start o f the period, P d and P h can also, as norm ally, be calculated by dividing the periodic increment by these start values. In this case we get P dti and P htl. If the (unknown) volume at the start o f the period is denoted V„, and the (known) volume at the end Vt2, we get:
I I , „-,2 = V.2-V,, = V12-Vl2/(1 + P vtl) = Vl2»(Pvtl/ ( l + P vtl)) (12) T herefore, when P dü and P htl are calculated in this way, and Pvt, is derived from (11), we get the correct pe
riodic increment, cf. (12), by m ultiplying the end volume not with P vll, but with P v,i/(1 + P v,i).
It is n ot clear whether K jølby (1958) has allowed f o r this correction. M øller (1951, p .25-26) does not m ention these discrepancies. L a ck o f correction m ay easily lead to errors o f as m uch as 10%.
Furtherm ore, the following general relationships between increment ratios calculated from start and end values apply:
V,2 = Vtl»(l + P vll) and (13)
V„ = V,2* (l-P vt2) which leads to (14)
V,| = Vt2/(1 + P vtl) = Vt2« (l-P vl2) giving (15)
1-PVI2 = 1/(1 + P vtl) (16)
This leads to:
Pv„ = Pv,2/(1-Pv,2) and (17)
P v,2= P v „ /( 1 + P « i) (18)
The two form ulas (17-18) apply to P d, P h, P g, P f as well as P v].
5) After having calculated the periodic increment o f the sample plots (para. 2-4), these values are sm ooth
ed (graphically or statistically) for the yield table as a whole, resulting in a curve for the current annual increment (K jølby, 1958, p .50-53).
Evaluation: The m ethod used by K jølby is applicable when thinning volumes can neither be determined from thinning registrations nor from stump measurements. However, the risks o f error inherent in the m ethod are considerable. One problem is that all m easured increments refer to the same climatic 5-year pe
riod. If the climate during this period is atypical, it is certain that the estimates will be highly biased. The simplification from (10) to (II ) is o f less im portance (according to Møller, 1951, p .25-26). On the other hand, large errors may result from combining increment ratios with the wrong volume figures.
Subgroup 5 ■
In subgroup 5, we are working with basic data, where total yield is not known for each sample plot. Total yield is therefore reconstructed by various indirect methods for the yield table as a whole. As the volume development (the course o f V, and V,) is reconstructed, total yield is determined as P = V3 + I(V,-V3) = V3 + EV2. With d ata from stands o f different ages, an illusion o f a true time series is created (see section
92
3.3.2). From this it follows that total yield is reconstructed from volume factors which are all m easured in existing stands.
In the 1st m ethod, curves for volume o f standing crop before and after thinning are determined. Two re
ferences refer to this procedure. M øller (1929) describes a m ethod in which all sm oothing is done graphi
cally. P rytz (1889, 1891) describes the so-called P r y tz ’m ethod, where mathematical functions are used for smoothing.
M ø ller’s graphical method, 5.1
The m ethod described by M øller (1929) contains the following main points:
1) A num ber o f stands are m arked for thinning. Volume factors (H, D, N) are m easured and allocated to thinnings and remaining crop. O n the basis o f these data, sm oothed curves are drawn for volume before thinning: Vj, and volume after thinning: V3, (alternatively for volume before thinning: V,, and volume in thinnings: V2, see M øller, 1929, p .278).
2) W ith thinning intervals: C, equivalent to local practice, the volume o f each thinning: V2, can be calcu
lated as the difference between V, and V3 in the year o f thinning.
lated as the difference between V, and V3 in the year o f thinning.