T he aim o f this p a p e r is, 1) to categorize the overw helm ing a m o u n t o f m eth o d s used in the co n stru ctio n o f yield tables in D en m ark , a n d 2) to u n d ertak e a critical analysis o f these m ethods.
T he p u rp o se is to clarify th e m erits o f the individual m ethods th ro u g h classification and evaluation.
In section 2.0, th e stru c tu re o f a yield tab le is presented. A yield tab le m ay be divided in to 3 parts: the grow th section , the treatm ent section an d the y ie ld section. This clear distinction is used th ro u g h o u t th e article.
Yield tables with one site class
In section 3.0, m eth o d s fo r constructing y ie ld tables with one site class are described. F ig .l gi- vesian outline o f th e n o rm al co n stru ctio n pro ced u re as it has developed u p till th e present tim e.
It is show n how a yield ta b le is derived fro m 6 basic factors:
fro m the g ro w th section: height developm ent: H 3, a n d cum ulative volum e p ro d u ctio n : P, fro m th e tre a tm e n t section: developm ent o f basal area: G3, a n d th in n in g intervals: C, fro m th e yield section: diam eter o f rem aining cro p : D s, a n d the q u o tien t: Q = (D 2/ D 3),
describing th e ra tio o f diam eter o f thinnings a n d d iam eter o f re
m aining cro p .
T otal yield: P = L IV, is th e fo u n d a tio n o f th e g row th section, a n d th e re fo re also the fu n d am en tal basis o f th e yield ta b le as a whole. A ccordingly it is the fa c to r we wish to determ ine w ith the greatest accuracy.
T o tal yield can only be d eterm ined directly, if th e basic d a ta o riginate from long-term p ro d u ctio n experim ents o r p e rm an en t local sam ple plots. Very o ften th e only d a ta available are fro m o b serv atio n stan d s, te m p o rary sam ple p lo ts, field surveys o r o th e r yield tables (section 3.2). H o w these d a ta can be used to reconstruct to ta l yield as well as height developm ent is des
cribed in sections 3 .3 .2 an d 3.3.3. T he p u rp o se o f these sections is to give an outline o f th e re
c o n stru ctio n m eth o d s. F o r fu rth e r details th e interested read er m ust refer to th e literatu re cited.
T ab le 1 sum m arizes these sections. A ccordingly, it show s a resum é o f m ethods used fo r d e term in in g to tal yield in all pub lish ed D anish yield tables.
O n th e basis o f th e resum é in ta b le 1, som e m ain outlines in the developm ent o f co n stru ctio n m ethods are traced . 4 perio d s ch aracterizing the em ployed sm oothing techniques can be id en ti
fied:
- 1880: g rap h ical sm oothing 1880- 1930: m ath em atical sm oothing 1930 - 1980: g rap h ical sm oothing 1980 - : m ath em atical sm oothing
It is observed th a t th e prom ising developm ent o f m ath em atical m ethods fo r constructing yield tables th a t was in itiated a t th e end o f last cen tu ry , w as d iscontinued w hen the graphical m ethods were in tro d u c e d . Several D anish a u th o rs show ed prom ising advances in the m a th e m atical fo rm u la tio n o f diam eter d istrib u tio n s alread y in th e 1880’s. This developm ent w as a necessary prerequisite fo r co n stru ctin g diam eter-class-w ise yield tables.
T he developm ent w as d isco n tin u ed w hen the g rap h ical m eth o d grad u ally replaced the m ath em atical m eth o d in the 1930’s. It is n o t until to d a y , a century later, th a t th e developm ent to w ard s diam eter-class-w ise yield tables has been resum ed.
Yield tables with several site classes
In section 4.0, th e descrip tio n is extended to cover th e co n stru ctio n o f y ie ld tables with several site classes.
In section 4.3, it is show n how a yield tab le w ith several site classes m ay be co n stru cte d by th e help o f grow th fu n ctio n s, treatm en t fu n ctio n s an d y ie ld fu n ctio n s. T hese functions specify th e basic facto rs fro m each o f the 3 sections o f th e yield tab le as dep en d en t variables o f one o r m o re o th er basic facto rs (m ost freq u en tly height, possibly supplem ented by age). In this w ay, th e g ro w th -, treatm en t- a n d yield fu n ctio n s determ ine th e relatio n sh ip betw een the basic facto rs an d site class. T he 6 facto rs a n d 3 fu n ctio n s are related in th e follow ing way:
Section o f table Function F actor E xam ple
G ro w th section G ro w th function H ,P H 3 = f(T ,S I) , P = f ( H 3) T reatm en t section T reatm en t function G ,C C = f(T) , G 3 = f(H 3) Yield section Yield fu n ctio n D ,Q Q = f(T ) , D 3 = f(H 3)
In th e above » exam ple«-colum n, generally used fu n ctio n al fo rm u la tio n s o f th e 6 basic facto rs are show n. T he fu n ctio n s fo r P , G 3 an d D 3 are the ones com m only referred to as g row th-,
treat-130
m ent- an d yield fu n ctio n s in th is paper. W hen the 6 fu n ctio n al relationships are determ in ed , all rem aining volum e facto rs in the yield tab le m ay be derived fo r each site class, as described for y ie ld tables with one site class (section 3.1). T h u s, a to ta l yield table w ith several site classes is
p ro d u ced .
T ab le 2 gives a resum é o f the m ethods used in th e co n stru ctio n o f all D anish yield tables w ith several site classes. Special em phasis is laid o n classifying the g row th-, tre a tm e n t- a n d yield functions used in th e c o n stru ctio n o f th e individual tables.
F o r a detailed analysis o f 6 o f the m o re im p o rta n t D anish yield tab les, a special visual m eth o d o f analysis is in tro d u ced in section 4.2, i.e. th e graphical m atrix . A yield tab le is in reality a surface in a m u ltidim ensional space. In the graphical m atrix this m ultidim ensional space is system atically p ro jected o n to tw o-dim ensional surfaces. T he m atrix quickly reveals w hether th ere are inconsistencies in the tab les, as well as revealing w h at m ethods are used in their co n stru ctio n .
The 6 analyzed yield tab les are show n in fig .3-8.
F ro m tab le 2 an d these m atrices, it is seen th a t th e m ost com m only used grow th fu n ctio n is the Eichhorn rule (P = f(H ), cell (3,2) in the m atrix ). A ccordingly, section 4.3.1 gives a n exten
sive review o f th e developm ent o f the rule th ro u g h o u t this century, as well as th e use o f th e rule in D anish p u b licatio n s, covering the general to p ic o f g row th and yield. T he m ost com m only used treatm en t fu n ctio n is th e d eterm in atio n o f a h eigh t/volu m e-cu rve (V = f(H ), cell (6,2)) or a height/basal-area-curve (G = f(H ), cell (5,2)). F inally, a com m only used y ie ld fu n ctio n is the d eterm in atio n o f a.factor-curve, describing the D g/ H g-curve (D g = f(H g), cell (8,2)).
A s well as show ing th e use o f gro w th -, tre a tm e n t- an d yield functions in specific cells, seve
ral o th er cells in the grap h ical m atrix co n tain in fo rm a tio n o f fu rth e r im p o rtan ce. T he m o re im p o rta n t o f these are d escribed in section 4.4. In section 4 .4 .1 , the use o f relative distan ce o f trees as a tre a tm e n t fu n ctio n (fig. 14-15), is analyzed fo r th e ab ove-m entioned 6 m a jo r yield tables. F u rth e r, in section 4 .4 .3 , th e fo rm facto rs used in th e yield tab les are tested ag ain st fac
to rs derived fro m m o re recently p ro d u ced fo rm -fa c to r functions (fig. 16-19). F inally, the m ean-tree ta riffs in th e yield tables are analyzed in section 4.4 .4 (fig .20).
In section 4 .5, the d ifferen ce betw een em pirical an d p rognostic yield tables is described. It is stressed th a t all yield tables are , to som e extent, p rognostic.
5 .2 D a n is h s u m m a r y
A rtiklens m ål er, 1) a t stru k tu re re den overvæ ldende m æ n g d e a f m eto d er der er a n v en d t ved u d fo rm n in g en a f p ro d u k tio n so v ersig ter i D a n m a rk , og 2) a t foretage en kritisk analyse a f disse m eto d er. D et er sigtet a t bringe klarhed over m eto d ern e gennem en k lassifikation og en v u rd e
ring.
I afsn it 2 .0 gennem gås de 3 g rundlæ ggende dele en p ro d u k tio n so v ersig t kan opdeles i: til
væ kstoversigten, behandlingsoversigten og u dbytteoversigten . D enne sk arp e opdeling an v en des som g ru n d lag fo r artik len s videre gennem gang.
Een enkelt pro d u k tio n so versig t
A fsn it 3.0 gennem går m eto d ern e til frem stilling a f een en kelt p ro du ktion soversigt. F ig u r 1 gi
ver en oversigt over d en generelle frem stillingsprocedure, således som den h a r udviklet sig frem til idag. D et vises hvorledes en p ro d u k tio n so v ersig t afledes a f 6 g ru n d fa k to re r:
fra tilv æ k sto v ersig ten : h ø jdeudviklingen: H 3, og to ta lp ro d u k tio n e n : P ,
fra behandlingsoversigten: grund flad eu d v ik lin g en : G3, og tyndingsm ellem rum m ene: C, fra udbytteo v ersig ten : diam eteren i blivende b estand: D 3, og kvotienten: Q = (D 2/ D 3), der
beskriver udhugningens diam eter i forhold til bestandens diam eter.
D en ak k u m u lered e v ed m assep ro d u k tio n : P = £ I V, er tilvæ kstoversigtens, og derm ed hele p ro d u k tio n so v ersig ten s, a fg ø ren d e fu n d am en t. D et er den størrelse i p ro d u k tio n so v ersig ten , vi ø n sk er bestem t m ed stø rst sik k erh ed . T o ta lp ro d u k tio n e n kan k un bestem m es d irek te hvis g ru n d m aterialet b estår a f d a ta fra langsigtede p ro d u k tio n sfo rsø g eller d istrik tsp rø v eflad er.
O fte h a r vi kun d a ta fra k o n tro lafd e lin g er, eengangsprøveflader, d istrik tstak satio n eller an d re p ro d u k tio n so v ersig ter til råd ig h ed (afsnit 3.2). H vorledes såd an n e d a ta udnyttes til rekon stru ktion a f b åde to ta lp ro d u k tio n e n og højdeudviklingen gennem gås i afsn itten e 3 .3 .2 og 3 .3.3. F o rm ålet m ed disse afsn it er a t give et overblik over rek o n stru k tio n sm eto d ern e. D en in
teresserede læ ser m å h erefter gå videre i den citerede litte ra tu r.
I tabel 1 sam m en fa ttes disse afsn it, idet tabellen giver en oversigt over to ta lp ro d u k tio n e n s bestem m else fo r sam tlige publicerede d an sk e p ro d u k tio n so v ersig ter.
P å b a g g ru n d a f oversigten i tab el 1 spores i a fsn it 3.4 nogle hovedudviklingslinier i frem stil- lingsm etodikken. B lan d t an d et k an identificeres 4 perio d er, der k arak teriserer den anvendte u d jæ v n in g sm eto d ik :
- 1880: g rafisk u d jæ v n in g 1880 - 1930: m atem atisk u d jæ v n in g 1930 - 1980: g rafisk u d jæ v n in g 1980 - : m atem atisk u d jæ v n in g
D et k o n statere s, a t den lovende m atem atiske udvikling, d er var igang in d e n fo r u darbejdelse a f p ro d u k tio n so v ersig ter i slutningen a f forrige årh u n d re d e , blev b ru d t ved indførelse a f den g rafiske m etode. B lan d t an d et v ar flere d an sk e fo rfa tte re langt frem m e m ed den m atem atisk e form ulering a f d iam eterfo rd elin g er allerede i 1880’erne. D enne udvikling var den nødvendige fo rlø b er fo r egentlige diam eterklassevise p ro d u k tio n so v ersig ter.
U dviklingen blev im idlertid sa t i stå, d a den g rafiske m eto d e i 1930’erne in d to g den d o m in e
rende p o sitio n . F ø rst n u , 100 å r senere, er udviklingen m o d diam eterklassevise p ro d u k tio n s
oversigter ved a t blive genoptaget.
B onitetsvise pro d u k tio n so versig ter
I afsn it 4.0 udvides beskrivelsen h erefter til a t o m fa tte bon itetsvise produ ktion soversigter.
D et vises i a fsn it 4 .3 , hvorledes et helt sæ t bonitetsvise pro d u k tio n so v ersig ter k an frem stilles ved h jæ lp a f væ k stlo ve, behandlingsprincipper og u dbyttefu n ktion er. D isse fu n k tio n e r angi
ver g ru n d fa k to re rn e fra pro d u k tio n so v ersig ten s 3 dele som afh æ n g ig e v ariable a f een eller fle
re a n d re g ru n d fa k to re r (oftest h ø jd e i blivende bestand evt. suppleret m ed alder). D e fo rm u le
rede v æ kstlove, beh an d lin g sp rin cip p er og u d b y tte fu n k tio n e r fastlæ gger p å denne m åde sam
132
m enhæ ngen m ellem g ru n d fa k to re rn e og b o n itet. D e 6 fa k to re r og 3 relatio n er h ø rer indbyrdes sam m en p å følgende m åde:
D el a f oversigt R elation F aktor E ksem pel
T ilvæ kstoversigten V æ kstlov H ,P H 3 = f( T ,B o n ), P = f(H 3) B ehandlingsoversigten B ehandlingsprincip G ,C C = f (T ) >g3 = f(H 3) U dbytteoversigten U d b y tte fu n k tio n D ,Q Q = f (T ) >d3 = f(H 3)
I »eksem pel«-kolonnen er a n fø rt generelt anv en d te ud try k fo r de 6 g ru n d fa k to re r. F u n k tio nerne fo r P , G 3 og D 3 er dem der alm indeligvis betegnes som en v æ k stlov, et b eh an d lin g sp rin cip eller en u d b y tte fu n k tio n i denne artik el. N år de 6 fu n k tio n er fo r g ru n d fa k to re rn e er fo rm u leret, kan alle resterende ved m assefak to rer afledes fo r hver b o n itet som beskrevet fo r een en
k elt p ro d u k tio n so versig t (afsn it 3.1). E n ko m p let bonitetsvis p ro d u k tio n so v ersig t er derm ed frem stillet.
T abel 2 giver en oversigt over m eto d er an v en d t til frem stilling a f sam tlige danske b o n itetsv i
se p ro d u k tio n so v ersig ter. D er lægges sæ rlig v æ g t på a t klassificere hvilke væ kstlove, b e h a n d lingsprincipper og u d b y tte fu n k tio n e r, der er an v en d t ved u d fo rm n in g en a f de enkelte p ro d u k tionsoversigter.
Til en videregående analyse a f 6 a f de vigtigste danske p ro d u k tio n so v ersig ter p ræ sen teres i a fsn it 4.2 en sæ rlig visuel an alysem etode - den grafiske m atrix. E n p ro d u k tio n so v ersig t ud g ø r i virkeligheden en flade i et m u ltid im en sio n alt ru m . I den g rafiske m atrix projiceres d ette m u lti
dim ensionale ru m system atisk p å to-dim ensionale flader. M atrixen giver således et hu rtig t overblik over, om der er in d re sam m en h æ n g i pro d u k tio n so v ersig ten , sam t over hvilke m eto der der er an v en d t til frem stilling a f pro d u k tio n so v ersig ten .
De 6 analyserede p ro d u k tio n so v ersig ter vises i figurerne 3-8.
D et ses a f tabel 2 og m atrix ern e, at d en hyppigst anvendte væ k stlo v er E ichhorns væ k stlo v (P = f(H ), celle (3,2) i m atrix en ). D er gives d e rfo r i a fsn it 4.3.1 en grundig gennem gang a f lo vens udvikling gennem d ette å rh u n d re d e , sam t anvendelsen a f loven i den danske p ro d u k tio n s- litte ra tu r. D et hyppigst an v en d te behandlingsprincip er fastlæ ggelse a f en h ø jd e/ved m a sse- kurve (V = f(H ), celle (6,2)) eller h ø jde/gru n dflade-ku rve (G = f(H ), celle (5,2)). Fastlæ ggelse a f fa k to rb a n en , der beskriver D g/H s-kurven (D g = f(H g), celle (8,2)), er endelig en o fte anvendt u d b yttefu n k tio n .
U dover a t vise anvendelsen a f v æ kstlove, b ehandlingsprincipper og u d b y tte fu n k tio n e r i en
kelte celler, in d eh o ld er den g rafiske m atrix yderligere en ræ k k e celler m ed afg ø ren d e in fo rm a tio n . De vigtigste supplerende celler gennem gås i a fsn it 4 . 4 . 1 afsn it 4.4.1 ses på de 6 analysere
de p ro d u k tio n so v ersig ters anvendelse a f relativ træ a fs ta n d som b ehandlingsprincip (figur 14- 15), og i a fsn it 4.4.3 p å a fp rø v n in g a f de enkelte pro d u k tio n so v ersig ters anv en d te fo rm ta l i fo rh o ld til nyere u d arb ejd ed e fo rm ta lsfu n k tio n e r (figur 16-19). E ndvidere analyseres p ro d u k - tionsoversigternes m id d e ltræ ta riffe r i afsn it 4 .4 .4 (figur 20).
I afsn it 4.5 gennem gås sluttelig forskellen mellem em piriske og p rognostiske p ro d u k tio n s
oversigter. D et und erstreg es, a t alle p ro d u k tio n so v ersig ter vil indeholde et stø rre eller m indre p ro g n o stisk elem ent.
6.0 A C K N O W L E D G E M E N T S
The presen t article is published w ithin the co n tex t o f the p ro ject »New Y ield T able fo r Beech in D en m ark « , p artly financed by th e D anish A g ricu ltu ral an d V eterinary R esearch C ouncil (p ro ject no. 13-3041). T he C ouncil has su p p o rted the research as well as th e publishing o f this a rtic le.
The concept o f th e graphical m atrix p resented in this p ap er was developed by the a u th o r in 1986, fo r th e p u rp o se o f analyzing th e interdependence o f stan d p ro p ertie s in d a ta form ing com plete tim e-series fro m th in n in g experim ents, w ith several th inning tre a tm e n ts. T he a u th o r atten d ed th e IU F R O conference o n » F orest G ro w th M odelling an d P red ictio n « (M inneapolis, august 1987), w here a sim ilar a p p ro a c h , a d a p te d to yield tables, w as p resented by d r. R o lfe Leary. I am in d eb ted to d r. L ea ry fo r his readiness to lend m e his d ra ft m an u scrip t (Leary, 1987), in w hich he ad o p ts the nam e o f »B akuzis-m atrix« fo r graphical m atrices sim ilar to the ones p resented in this paper.
7.0 R E F E R E N C E S
F or p u rposes o f cross-referencing, th e pages w here each p u blication is cited in this p a p e r are show n in square brack ets.
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