DET FORSTLIGE FORSØGSVÆSEN I DANMARK
THE DANISH FOREST EXPERIMENT STATION STATION DE RECHERCHES FORESTIÉRES DE DANEMARK
DAS FORSTLICHE VERSUCHSWESEN IN DÄNEMARK
BERETNINGER UDGIVNE VED DEN FORSTLIGE FORSØGSKOMMISSION
R E P O R T S - R A P P O R T S - BERI CHTE
HÆF T E 2 ISSN 0 3 6 7 - 2 1 7 4
I N D H O L D
P. H o l t e n - A n d e r s e n : Danish Yield Tables in the Past Century. (Produk- tionsoversigter gennem et århundrede i dansk skovbrug). S. 71-145. (Beretning nr. 356).
KØBENHAVN
T R Y K T I KANDRUPS BOGTRYKKERI
1989 .
W achstum ju n g e r D ouglasien (Pseudotsuga m en ziesii M irb. Franco) und K üsten
tannen (A bies grandis Dougl. Lindley) in A bhängigkeit von der N ährstoffver- sorgung. (D rought R esistance, W ater R elations and G row th o f Douglas F ir and G ra n d F ir in R elation to N u trien t Supply). S. 1. - N r. 328. H. H o lste n e r- Jørgensen og G . L au ritsen : K obberm angel hos sitkagran på M eilgaard. (C opper Deficiency in Sitka Spruce at M eilgaard). S. 83. - N r. 329. H. A. H enriksen og K.
Sanojca: Bøgeunderskovs betydning for udvikling a f vanris på stilkeg (Quercus robur L.). (T he Influence o f an U nderstorey o f Beech upon th e D evelopm ent o f E picorm ic B ranches o f O ak (Q uercus robur L.). S. 93. - H . 2: N r. 330. N. E le rs K och: Skovenes friluftsfunktion i D anm ark. III. del. A nvendelsen a f skovene, lokalt betragtet. (Forest R ecreation in D enm ark. P art III: T h e Use o f th e Forests Considered Locally). S. 121. - H . 3: N r. 331. J. Bo L arsen, B. G ade L arsen og H.
K . K r o m a n n : Abies n o rd m an n ian a provenienser til pyntegrønt og juletræ er. (Pro
venances o f A bies nord m an n ian a for G reenery- and C hristm astree P roduction). S.
363. - N r. 332. B ent T. C hristensen and P er Å. M alm ros: Spatial V ariability o f Litterfall and Soil O rganic M atter in a Beech Stand. (Den rum lige variation i løv
fald og i jordens organiske stofindhold i en bøgebevoksning). S. 383. - N r. 333. H.
H o lste n er-Jø rg e n sen og T. M e l l e r g a a r d : G ræ sning i skove. (G razing in Forests).
S. 397.
Bd. XXXX, H . 1: N r. 334. Erik Holmsgaard: O prensning, vandrisdannelse m. v. i en 25-års periode på en u rø rt og to tyndede parceller i et hugstforsøg i bøg. (Self- Pruning, F orm ation o f E picorm ic Shoots etc. during 25 Y ears in a N on-T h in n ed an d two T h in n ed Plots o f a T h inning E xperim ent in Beech). S. 1-51. - N r. 335. P.
Moltesen, T. Lynge Madsen og P. O. Olesen: Planteafstandens betydning for rød
granens tørstofproduktion og vedkvalitet. (The Influence o f Spacing on th e P roduc
tion o f D ry M atter and W ood Q uality o f N orw ay Spruce). S. 5 3-76. - N r. 336. H.
Holstener-Jørgensen og E. Stope: Et gødskningsforsøg i C ham æ cyparis law soniana på A alholm skovdistrikt. (A Fertilizing E xperim ent in C ham æ cyparis law soniana at the A alholm Forest D istrict). S. 77-84. - N r. 337. H. Holstener-Jørgensen og Jørgen Lundberg: Et faktorielt gødningsforsøg i 40-årig rødgran i H overdal p la n tage, U lborg statsskovdistrikt. (A Factorial Fertilizing E xperim ent in 40-Y ear-01d N orw ay Spruce in H overdal P lantation, U lborg State Forest D istrict). S. 8 5-94. - N r. 338. Søren Fl. Madsen: O verensstem m ende stam m eside- og vedm assefunktio- n er for fem forskellige nåletræ arter. (C om patible T ree T ap e r and V olum e F uncti
ons for five D ifferent Conifers). S. 9 5-140. - H . 2: N r. 339. T. Lynge Madsen, P.
Moltesen og P. O. Olesen: G ødsknings indflydelse på rødgrans rum tæ thed og tø r
stofproduktion. (Effect o f Fertilization on the Basic D ensity and P roduction o f D ry M atter o f N orw ay Spruce). S. 141-171. - N r. 340. J. Bo Larsen: Økofysiologiske og morfologiske undersøgelser a f forskellige Abies procera provenienser m ed hen
syn til deres egnethed til pyntegrøntproduktion. (Ecophysiological M orphological Investigations o f D ifferent Abies P rocera - Provenances in R elation to G reenery
DANISH YIELD TABLES IN THE PAST CENTURY
PRODUKTIONSOVERSIGTER GENNEM ET Å R H U N D R ED E I DANSK SKOVBRUG
B Y
P. HOLTEN-ANDERSEN
Holten-Andersen, P. Danish Yield Tables in the P ast Century (Produktionsoversigter gennem et Å rhundrede i Dansk Skovbrug). Accepted sept. 9, 1988. Forsti. Forsøgsv. Danm ., vol. 42, rep. 356, p. 71-145, 1989.
All Danish yield tables produced over the last century are analyzed and classified according to the methodology used in construction o f the tables.
More recent yield tables are all divided into 3 main sections showing growth-, treatment- and yield figures. It is shown how a yield table with a single site class, containing the above-mentio
ned 3 sections, can be constructed when 6 basic factors are known. The 6 basic factors are: to tal yield (P), height o f remaining crop (H3), basal area o f remaining crop (G3), diameter o f re
maining crop (D3), the ratio between diameter o f thinnings and diameter o f remaining crop called the thinning quotient (Q = D2/D 3) and thinning intervals (C).
When the construction o f a yield table is not based on data from perm anent sample plots, fi
gures for total yield (P) will be lacking and must accordingly be reconstructed. As total yield is considered to be the m ost im portant o f the 6 basic factors in a yield table, an extensive review of methods used for determining cumulative volume production in all Danish yield tables is gi
ven.
It is further shown how yield tables with several site classes are constructed. Through the use o f growth functions, treatm ent fu n c tio n s and yield functio n s, a total yield table with several site classes may be derived. The 3 types o f functions determine the relationship between the ba
sic factors and site class.
Finally, the concept o f »the graphical matrix« is introduced as a quick visual method o f ana
lyzing, am ongst other things, the growth-, treatm ent- and yield functions used in constructing the yield table.
Key words: Yield tables, methodology, classification, reconstruction o f total yield, graphical matrix.
The article is also available in a Danish version (see H olten-Andersen, 1989).
0.2 CONTENTS
Page
0.1 A b strac t... 72
0.2 C ontents... 73
0.3 Symbols and term inology... 74
1.0 In troduction... 79
2.0 The structure o f yield tables... 80
3.0 Yield tables with one site class... ... 81
3.1 General construction procedure... 81
3.2 Basic d a ta ... 84
3.3 Reconstruction o f basic d ata... 84
3.3.1 Table 1... 84
3.3.2 Reconstruction o f height developm ent... 85
3.3.3 Reconstruction o f total yield... 86
3.4 Main outline o f the developm ent... 95
4.0 Yield tables with several site classes... 97
4.1 Table 2 ... 97
4.2 The graphical m atrix... 98
4.3 Growth-, treatm ent-and yield functions... 106
4.3.1 Growth functions... 107
4.3.2 Treatm ent functions... 117
4.3.3 Yield functions... 119
4.3.4 The master-table principle... 120
4.4 Some specific cells o f the m atrix... 120
4.4.1 Characterizing thinning treatm ent... 120
4.4.2 The basal-area/volum e-curve... 123
4.4.3 Form factors... 124
4.4.4 Mean-tree ta riff... 126
4.5 Empirical and prognostic yield tables... 127
5.0 Sum m ary... 128
5.1 English sum m ary... 128
5.2 Danish sum m ary... 130
6.0 Acknowledgem ents... 133
7.0 References... 133
M ain tables... 142
0.3 SYMBOLS AND TERM INOLOGY
A b b r e v i a t i o n T e x t U n it
T Age from seed year
H Mean height (Hg or similar) m
D Diameter (Dg or similar) cm
N Stems trees/ha
G Basal area m V ha
V Volume m3/h a
F Form factor over bark
P Cumulative volume production = m3/h a
total increment = £ I vb0_, =
total yield = V3 + ZV2
A Cumulative thinnings = I V 20_t m V ha
C Thinning intervals, given years
as a vector o f ages
R Relative distance o f trees = average spacing as a %
percentage o f mean height
Q Thinning quotient = D2/D 3 = ratio o f diameter o f m ean-basal-area tree in thinnings and after thinning
IN Intensity = G*F
SI Site index = site class = site quality class
Iv A nnual increment m3/h a * år
P v Increment -ratio or -percent for volume Pg Increment -ratio or -percent for basal area P h Increment -ratio or -percent for height P d Increment -ratio or -percent for diameter P f Increment -ratio or -percent for form factor
1 Index referring to before thinning, e.g. H,
2 Index referring to thinnings, e.g. H 2
3 Index referring to after thinning, e.g. H 3
4 Index referring to between thinnings, e.g. H 4
t Index referring to a given point in time, e.g. H„ where t denotes age from seed
t l Index referring to the beginning o f a period.
May be combined with the above index, e.g. N3,,, denoting stem number after thinning, at time tl
t2 Index referring to the end o f a period
In the following text capital letters refer to stand level, small letters to tree level. Further
more, standard IUFRO -notation is used (IUFRO, 1965).
75 VARIABELFORKORTELSER
F o r k o r t e l s e T e k s t E n h e d
T Alder fra frø år
H M iddelhøjde (Hg eller lign.) m
D Diam eter (Dg eller lign.) cm
N Stam tal stk /h a
G G rundflade m2/h a
V Vedmasse m3/h a
F U ægte form tal
P Akkum uleret produktion = m3/h a
total tilvækst = EIvb0-t =
total udbytte = V3 + ZV2
A Akkumulerede tyndinger = I V 20_, m3/h a
C Udhugningsmellemrum, angivet år
ved en vektor af aldre
R Relativ træ afstand %
Q Kvotienten = D2/D 3
IN Intensitet = G*F
Bon Bonitet
SI Bonitet
Iv Årlig løbende massetilvækst - m 3/h a * år
P v Tilvæ kstbrøk eller -procent for vedmasse P g Tilvæ kstbrøk eller -procent for grundflade P h Tilvæ kstbrøk eller -procent for højde P d Tilvæ kstbrøk eller -procent for diameter P f Tilvæ kstbrøk eller -procent for form tal
1 Index refererende til før hugst, f.eks. H,
2 Index refererende til udhugning, f.eks. H 2
3 Index refererende til efter hugst, f.eks. H 3 4 Index refererende til mellem hugst, f.eks. H 4
t Index refererende til et givet tidspunkt, f.eks. H (, hvor t er alderen fra frø
tl Index refererende til begyndelsen af en periode.
Kan kombineres med ovenstående index, f.eks. N 3 „, der angiver stam tal efter hugst, tidspunkt tl
t2 Index refererende til slutningen a f en periode.
I efterfølgende tekst refererer store bogstaver til bevoksningsniveau, små bogstaver til en- kelttræniveau. Iøvrigt anvendes IUFRO -notation (IU FRO , 1965).
TERM INOLOGY
Many technical concepts are not covered by a single, precise word in two or more languages.
To help facilitate understanding the text for Danish as well as English readers, a short des
cription a n d /o r translation o f a num ber o f technical terms used in this text is given below. The list is not organized alphabetically, but grouped according to subject, in the order in which the terms are used in the following text:
W o r d o r te r m D e s c r i p t io n D a n is h e q u i v a l e n t Volume components M ensurational characteristics o f remaining crop
and thinnings (H, N, G, V, D, F)
Vedmasse faktorer
Growth Total increase in volume (or another variable) over a period
Vækst
Increment Rate o f growth Tilvækst
Yield Volume removed as thinnings or by clear felling U dbytte
P roduction experiment Grow th and yield experiment Produktionsforsøg
Local sample plot A perm anent experiment established locally by a forest district
Distriktsprøveflade Observation stand Large and uniform production stand where
all commercial thinnings are registered
K ontrolafdeling T em porary sample plot P lot laid out tem porarily with a specific survey
in mind
Eengangsprøveflade
Field survey D istriktstaksation
G row th function A m athematical function (or graph), defining a factor o f the growth section o f a yield table as dependent variable o f other factor(s) (often height and sometimes also age). A commonly used growth function is the Eichhorn rule:
P = II„ = f(H 3)
Vækstlov
Treatm ent function A m athematical function (or graph), defining a factor o f the treatm ent section o f a yield table as dependent variable o f other factor(s) (often height and sometimes also age). A commonly used treatm ent function is the height/basal-area- curve: G 3 = f(H 3)
Behandlingsprincip
Yield function A m athematical function (or graph), defining a factor o f the yield section o f a yield table as dependent variable o f other factor(s) (often height and sometimes also age). A commonly used yield function is the factor-curve: D3 = f(H 3)
U dbyttefunktion
77
W o r d o r te r m D e s c r i p t i o n D a n is h e q u iv a l e n t Yield table with
a single site class
Produktionsoversigt med een bonitet Yield table with
several site classes
Bonitets vis produktionsoversigt Site class In this paper a site class denotes a specific growth
potential (i.e. a single figure) as well as all the growth and yield figures that apply to the site class in question
Bonitet
C urrent site class Aktuel bonitet
Initial spacing Density o f stocking a t time o f establishment Planteafstand
Square spacing Kvadratforband
Solid-content factor Fastmassetal
M inimum diam eter o f small end
Aflægningsgrænse Smoothing Graphic or m athem atical/statistic smoothing Grafisk eller
m atem atisk/statistisk udjævning
Increm ent period Period between two successive thinnings Tilvækstperiode Region Area exhibiting a certain uniform ity regarding
growth conditions. The definition does not exclude large variations in site class
Vækstom råde
M anagem ent class Species grouped together in a m anagem ent unit Driftsklasse
1.0 INTRODU CTION
»There are few countries in which foresters have so excelled in the construction o f yield tables, as in Denmark« (Møller, 1929, p .249).
Yield tables form the basis o f all long-term planning in forestry. They constitute the link be
tween silviculture and economics. Many have contributed with new yield tables but few have brought order in the profusion o f methods.
Yield tables are multidimensional. They belong to a world often analyzed by the aid of mathematics. Few foresters are m athem aticians. Even fewer can visualize a multidimensional sphere.
The purpose o f this article is to classify and analyze methods used in the construction o f Da
nish yield tables over the past century. It is the intention to cast light on the obscure, and to give a visual insight into a multidimensional sphere.
The classification o f Danish yield tables, and the methods behind their construction, is shown in tables 1-2 and fig .l.
Examples o f a m ethod for visual analysis are shown in fig.3-8, i.e. the graphical m atrix. In the graphical matrix the multidimensional sphere o f a yield table is projected systematically onto two-dimensional surfaces. The graphical m atrix is well suited for tracing the methods used in the construction o f a yield table with several site classes, especially when these methods are sparsely docum ented by the author. The matrix also reveals inconsistensies in the finished product - it casts light upon the obscure.
The graphical m atrix places the observer in the same position as the cave dwellers o f Platon, who could only com prehend a projection o f reality (in this case the multidimensional space) in the form o f moving shadows on the cave wall (the two dimensional surface), cast by rays o f sun shining through the opening o f the cave (Platon, 385-380, p .279-315). Likewise, the matrix enables the observer to view a multidimensional sphere in two-dimensional simplifications, as projections on paper.
The article is based solely on yield tables published in Danish periodicals. Foreign references are only included to the extent that they add further inform ation to the described methods de
veloped and used in Denmark.
Right until the end o f last century most foresters considered the yield from thinnings to be o f no practical or economic significance. A table giving the development for the remaining crop only, was considered by many to be a fully adequate yield table. These tables are not included as acceptable yield tables in this work.
80
[See e.g. G yldenfeldt, 1881, p.284 (beech); G yldenfeldt, 1883, p .23 (beech); Steen, 1887, p . 102-103,136- 137,170-171,214-215,235-240 (beech); Müller, 1889, p .243 (oak); Oppermann, 1891, p .103-106 (general comments)].
Furtherm ore, works that do not describe the methods o f construction but only present the final result, are not included.
[See e.g. Hansen, 1877, p .27-43 (various species); M üller, 1889, p .260 (silver fir); M uus, 1931, p .341 (beech). Also belonging to this category are 8 fine yield tables for Norway spruce, produced by Opper
m ann & P rytz and described in M üller (1889, p .218-221, 253-256). The tables are based on data published by Oppermann & P rytz (1892), and give figures for 4 localities, supplemented by 2 alternative thinning treatm ents. Likewise, 4 yield tables for beech, oak, ash and Norway spruce, published by Oppermann (1896, p .232,242,249,255), belong to this category. U nfortunately, the m ethod o f construction is not de
scribed in any o f these publications].
In production experiments, volume com ponents for the individual plots are m ost frequently presented in a table showing the development o f the remaining crop and thinnings. These tab
les are called yield tables by some authors. Unless such data have undergone further sm ooth
ing, the resulting tables are not considered as yield tables and are consequently not included in this paper. The article only deals with the so-called »mean-tree tables«, i.e. yield tables based on the simplified condition that all trees in a stand are equal to the m ean-basal-area tree. All Danish yield tables so far published are based on this condition.
2.0 TH E STRUCTURE OF YIELD TABLES
A yield table is normally divided into 3 parts giving growth-, treatm ent- and yield figures (see Jørgensen & Andersen, 1959, p .485-489; Johnston, Grayson & Bradley, 1967, p .349).
All recent Danish yield tables are divided into the following 3 sections:
I : Growth - presents the biological production potential, II : Treatment - presents a standard treatm ent programme,
III: Yield - presents the combined effect o f I and II, in the form o f variables describing the physical yield.
The growth section gives cumulative height- and volume growth as a function o f stand age.
These variables constitute the foundation o f the whole yield table. Number o f stems, basal area and volume o f remaining crop are variables strongly influenced by silvicultural treatm ent.
The progression with age o f these variables therefore describes the treatment.
The yield section describes the physical yield, again as a function o f age. The most im portant variables are thinning volumes and diameter o f remaining crop. These variables are the main factors determining economic yield.
Descriptions o f quality, which is an important factor in determining economic yield, are lack
ing however. Such descriptions are not included in any Danish yield tables, partly because qua
lity is difficult to quantify and varies greatly with locality, and partly because quality is deter
mined by m arket forces and therefore varies with time.
3.0 YIELD TABLES W ITH ONE SITE CLASS This section describes how the methods o f constructing a yield table with one site class have de
veloped up till today. In section 4.0, the description is extended to yield tables with several site classes.
3.1 G e n e r a l c o n s t r u c t i o n p r o c e d u r e
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 .
82
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.
84
3 .2 B a s ic d a ta
Basic data are divided into 3 categories in the following (see table 1).
Group / consists first and foremost o f long-term, controlled production experiments on per
m anent sample plots. The group also includes perm anent local sam ple p lo ts established by forest districts.
Group I I consists o f observation stands (i.e. norm al production stands where commercial thinnings are registered), temporary sam ple p lots and data from fie ld surveys o f forest dis
tricts. Observation stands were previously often used for thinning control and in checking long-term planning targets (see e.g. Biilmann, 1943). For this purpose a num ber o f large and uniform production stands, evenly distributed am ong management classes (species grouped together in a m anagement unit) and age classes, were selected. In these stands all commercial volumes removed through thinning were accurately registered. The commercial volumes were converted to total volume with the use o f local solid-content factors.
The main difference between observation stands and local sample plots is that volume o f the remaining crop is measured at each thinning on local plots, whereas it is rarely measured in the observation stands (usually only in connection with field surveys).
Temporary sample p lo ts and fie ld surveys only provide inform ation for remaining crop at the time of measurement. The difference between inform ation from tem porary sample plots and field surveys, is usually that tem porary sample plots are m ost frequently selected for a more specific study, e.g. the construction o f a yield table.
Finally, G roup I II consists o f inform ation from other yield tables, the basis o f which will na
turally be d ata from group I or II.
3 .3 R e c o n s t r u c t i o n o f b a s ic d a ta
For com pilation o f the growth section o f a yield table, data from group I will always be prefer
red. The precise measurements from long-term experiments provide exact figures for both height and total yield as a function o f age.
If basic data from group I are lacking, it is necessary to resort to alternative solutions. With basic data from group II, the growth section must to a certain extent be based on reconstructed data.
The methods o f reconstruction for the development o f height and total yield are classified in table 1, and elaborated in sections 3.3.2 and 3.3.3. As they only describe the alternatives used when sufficient basic data are lacking, the m ajor p art o f the two sections are presented in small print.
3 . 3 . 1 T a b le I
Table 1 gives a resumé o f the reconstruction methods (refer to p. 142-143, where a fold-out version o f table 1 is found).
The principal key to the table is column 1 and 2. Together they present a classification o f the methods used in the estim ation o f total yield, by a division into main groups (A,B,C), groups (I,I I,III), subgroups (1,2,3,4,5) and finally methods within each subgroup.
Column 1 distinguishes between 3 main groups, A, B and C:
A) Total yield and its distribution over tim e is known for each sample plot. In this case the basic data appear as complete time-series of volume com ponents. Accordingly there is no need for any reconstruction.
B) Total yield and its distribution over time is unknown for each sample plot, but can be re
constructed. Thus, d ata for each sample plot appear in the same form as data from main group A.
C) Total yield and its distribution over time is unknown, and cannot be reconstructed for each sample plot. D ata belonging to this group cannot be brought to the same level as d a
ta from main group A, and it is necessary to accept that total yield is reconstructed by in
direct methods for the yield table as a whole.
The trend from main group A over B to C therefore shows a continuous development, in which the decisive factor: total yield, is determined with increasing uncertainty, and through increasingly indirect m ethods.
As the m ajority o f all yield tables are based on data from more than one main group, co
lumn 4 provides a summary o f the combined basic data. Column 2 only provides the group o f data that has been used for determining total yield.
Finally, column 6 shows the reference for each yield table and references describing the m ethods in general.
3 . 3 . 2 R e c o n s t r u c t i o n o f h e i g h t d e v e l o p m e n t
The following section is not essential for the article as a whole, and is therefore presented in small print.
Height development is often reconstructed on the basis o f more extensive data than those used for estim a
ting total yield. However, frequently the same classification applies to reconstruction o f height develop
ment as that applying to total yield (see section 3.3.1).
In classifying the methods o f reconstruction, the following 4 main groups are used (symbols from table 1, column 1-2):
A I
When basic data belong to g roup I, height development will be measured and known for each sample plot, and there is accordingly no need for any reconstruction.
B II
If basic data belong to group II, height development will n ot be measured for each sample plot. One possi
bility is therefore, to reconstruct the height developm ent f o r each sam ple p lo t. This will normally be the m ethod applied when basic d ata are derived from tem porary sam ple plots. The m ethod used for recon
struction is stem analyses carried o ut on a num ber o f mean trees in the sample plot (see e.g. Prytz, 1889, p.
84-86; Møller, 1951, p. 190-193). (Examples for conifers are: Picea abies: Løvengreen, 1951b, p .356-361;
Picea sitchensis: H enriksen, 1958b, p .52-53,70; Abies nobilis: Elingård-Larsen & Jensen, 1985, p .255; P i
cea om orika: M øller M adsen, 1989. Examples for broadleaves are: Quercus robur: Løvengreen, 1949, p . 113-114; Acer pseudoplatanus: K jølby, 1958, p .46).
86
The m ethod has a serious draw back, as trees th at are mean trees in a stand today will normally have been dom inants or codom inants earlier in the life o f the stand. Height development reconstructed by stem ana
lysis o f a num ber o f mean trees will therefore tend to show too sharp a rise in grow th in the early years, fol
lowed by a too rapid levelling o ff in later years. Henriksen (1958b, p .67-70) mentions the problem, but considers the defect to be o f m inor im portance. In this he is supported by Løvengreen (1949; 1951b, p .363- 364)
CI.I
Another possibility for group II data is not to reconstruct height developm ent f o r each sam ple plot, but f o r the yield table as a whole. This procedure is norm al when the basic data consist o f inform ation from field surveys, but it may also be used for tem porary sample plots.
W hen height inform ation from field surveys is used in the construction o f yield tables, one creates the il
lusion that heights from different stands and ages form a true time-series. It is accordingly necessary that uniform growth and treatm ent conditions prevail among the stands in which volume factors are measured.
Differences in site class between age classes, or improvement in site class through time, may result in incor
rect height development (see e.g. Andersen, 1984).
cm
Very often the height development in a yield table is constructed to conform with height curves from other yield tables. M øller’s yield tables for beech, oak and Norway spruce are nevertheless the only tables, where inform ation from other yield tables constitute the m ain basis. For the lower site classes this m aterial is sup
plemented, however, by inform ation from stem analyses (Møller, 1933, p.462-466).
3 . 3 . 3 R e c o n s t r u c t i o n o f t o t a l y i e l d
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.
3) It follows that total yield (P = V3 + I V 2) will be a highly derived figure. The determ ination o f thinning intervals: C, is decisive in the calculation o f thinning volumes, and consequently also for deriving total yield. If the thinning intervals are too long, com pared with the intervals inherent in the data used for estab
lishing the V,’- and V3-curves, then LV2 will be too small. If, on the other hand, the thinning intervals are too short, then I V 2 may be far too large. (When the individual thinning intervals approach 0, I V 2 a p p ro a
ches infinity).
P r y tz’ method, 5.2
P r y tz’ m ethod (for a further resumé, see M øller, 1951, p .262-266) resembles that o f M øller (1929) descri
bed above (m ethod 5.1).
P r y tz ’ m ethod differs however, in that all sm oothing is done by the use o f m athematical functions.
P rytz uses the general power-function Y = aX b, in logarithmic form: log(Y) = log(a)+'b*log(X ), for sm oothing a num ber o f the volume factors. The function is not sufficiently flexible to sm ooth all volume- factor relations. W here P r y tz ’ m ethod is used, it is therefore often seen th at some relations are sm oothed mathematically, while others are sm oothed graphically. The m ain points are as follows:
1) Sample plots are laid out in a num ber o f stands representing all stages between thinnings (i.e one year since thinning, two years since thinning...one year before thinning, before thinning, after thinning). In some o f the stands, trees must be m arked for thinning, so th at remaining crop and thinnings can be dis
tinguished.
Instead o f sm oothing volume before thinning: V,, and volume after thinning: V3 (as above, m ethod 5.1, para. 1), Prytz sm oothes the intensity before and after thinning: IN ,, IN 3.
Intensity is defined as the proportion o f available growing space occupied by the standing volume:
IN = (H *G *F)/(H * stand area) (19)
which, when stand area is 1 hectare, is equivalent to
IN = G*F (se e M øller, 1951, p . 18-19). (20)
From this it follows that volume is equal to intensity m ultiplied by height: V, = IN ,*H , and V3 = IN3*H 3.
P rytz assumes that IN, and IN3 are constant regardless o f age (Prytz, 1889, p .76-84; Møller, 1929, p .252-253; M øller, 1951, p. 19), which is equivalent to volume before and after thinning being a function o f height only. The intensities, IN , and IN 3, may however also be sm oothed as a function o f age, or possibly height (see Møller, 1929, p .263\Fabricius, 1919, p .347-348).
2) W ith thinning intervals: C, determined, thinning volumes are calculated as: V2 = IN ,*H r IN 3*H 3.
3) As before, total yield becomes a highly derived factor, calculated as P = V3 + £V 2.
Evaluation: The weakness inherent in the two above-described variants (m ethod 5.1-5.2) is that the whole basis o f the yield table - i.e. total yield - is a highly derived factor. Growth and treatm ent variability in the basic m aterial will lead to large standard deviations around the V,- and V3-curves. In addition, when d eter
