CROSS-COUNTRY T R A N S P O R T D I S T A K C E S A N D R O A D ATETS 6 5 Varden p 5 terrangtransportliorrelitionen har beraknats empiriskt bade p 5 kartor och vid en liombination a v falt- och kartarbete.
Korrelrtionen har befunnits vara orsakad a v foljande forh5llanden:
- FrBn vissa delar a v e t t transportomr5de in5ste ofta terrangtransporten ske till annan uiig an den narmast belagna.
- Virket transporteras i regel fram till sarskilda lastplatser vid vag.
- Den npraktiskao korvagen avviker fr5n f5gelvagen mellan pilastningsplat- sen i terrangen och lastplatsen vid vag.
\'id en undersokning, villien utforts p i kronoparken M o t e n som en kom- bination a v miitningar i falt och p 5 liartor, har terrangtransportkorrelitionens korrelation med det geometriska tilltransportavst5ndet w r i t rngcket w a g . Under denna forutsiittning bor mail for e t t aktuellt omr5de kunna ber"k a n a terrangtransportkorrelitionen med ledning a v relativt f & bestamningar och lBta det darvid erhBllna aritmetiska medelvardet galla for olika avst5nd. - D5 terriingtransportkorrelitionen beraknats enbart p 5 kartor har daremot en tydlig tendens till avtagande varde vid langre a v s t i n d erh5llits.
I e t t sarskilt avsnitt har beslirivits och exemplifierats hur vagnats- och terrangtransportkorrelrtionerna infores i den t p p a v kalkyler rorande vag- natets t a t h e t son1 redovisats a v Sundberg.
Avhandlingens andra del, som behandlar vaglangdsbestainningar, inleds med statistiska uppgifter rorande det svenska vagnatets omfattning vid olilia tidpunkter. Darefter har redogjorts for en metodik a t t bestiimma e t t vagnats langd genoin stickprovsrnassig insanlling a v data. Denna bygger p 5 en a v M a f i r m anvisad tillampning a v eBrr/fons n8lproblems.
Metodiken har f5tt sin hittills viktigaste anvandning slid rilisskogstaxerin- gens vaginventering, soin foretagits 5ren 1956-1963. Darvid har en regist- rering skett a v antalet skarningar inellan vagar och sidorna p 5 de s. k. taxe- ringstrakterna. RIed hansyn till vagens heskaffenhet m. m. i anslutning till skarningspunkten ~ n e d traktsidan har en klassificering utfbrts.
\'aginventeringen har speciellt syftat till a t t bestainma langden a v de vagar, soin paverkar tilltransportavst5ndet. For den totala langden a v dessa vagar har uppgifter ej funnits tillgangliga.
Fr5n inventeringarna 5ren 1956-1060, lrallad vaginventeringen 1056-60, har bl. a. redovisats uppgifter om langden a v allmanna och enskilda permanenta vagar a v betydelse for tilltransporten. Gppgifterna har givits dels fordelade pB riksskogstaxeringens regioner I-V och dels fordelade p i agarkategorier inom en nordlig och en sydlig regiongrupp. I den nordliga gruppen har dessutoin an- givits langden asr ointertid med traktor eller lastbil farbara s. lr. motorbas- vagar. - )led hjalp a v arealuppgifter har aven vagtatheten i meter vag per hektar beralinats. De s& beraknade vagtiitheterna har tillsarninans med an- tagna varden p5 vagnats- och terrangtransportkorrektionerna legat till grund for en berglining a v nledeltilltrailsportavst5nd inom olilia agargrupper och redovisningsomr5den.
I arbetets sista avsnitt har redogjorts for uppskattningen a v vaglangden inom s. k , sm8omrBden i Jamtlands lan. DBrvid har samma metodik, i form a v linjetaxering pB kartor, tillampats soin vid rilrsslrogstaxeringens vaginven- tering. Redovisningen har gjorts relativt omfattande for a t t liunna t j a n a till ledning vid andra uppskattningar a v liknande karaktar.
Appendix
A Method of Estimating the Total Length of Roads by Means of a Line Survey
b y BERTIL MATERN
The needle problem, t h e classical problem in geometric probabilities can be described as follows: A system of equidistant parallel lines is drawn on a sheet of paper. A needle of a length (a) less t h a n t h e distance (b) between two neighbouring lines in t h e system is placed a t random on t h e paper. M7hat is t h e probability t h a t t h e needle intersects t h e lines?
According t o t h e French naturalist Buffon, t h e probability is
Buffon published this solution in 1777 b u t stated t h e problem already 1733 (see Castelnuouo, 1925).
Consider a chain consisting of N links, each one with length a. The chain is placed on t h e paper so t h a t t h e position of each individual link in relation t o t h e line system is determined b y a random procedure of t h e same nature as t h a t governing t h e position of t h e needle in t h e Buffon problem. The expected number of intersections between t h e chain and t h e line system is t h e n
2 L
(1)
-.-
n b
where L = A'. a = t h e length of t h e chain. The same result is obtained if instead the chain is initially placed on t h e paper and t h e position of t h e line system then is determined by random translation a n d rotation in relation t o fixed coordinate axes.
A simple argument (presented b y Barbier in 1860) can explain why t h e factor 2/n occurs in formula (1). We consider the limiting case obtained when t h e links of t h e chain are made smaller a n d smaller i n such a manner t h a t t h e chain is finally transformed t o a circle with a diameter equal t o t h e distance between t h e lines (b). The circumference of t h e circle is L = n b, a n d t h e circle c u t s t h e lines in two points, irrespective of position. For- mula ( I ) also gives t h e value 2:
Consider t h e n a n area with a network of roads with t h e total length L. If a system of parallel equidistant lines ( a t distance b a p a r t ) is placed according t o a randomly chosen direction, t h e expected number of intersections with t h e road net is 2 L l x . b as in formula (I). Conversely, let n be t h e number of intersections between t h e line system a n d t h e net- work of roads, a n d form t h e expression
The expected value of this expression then equals L, or using E for mathematical expectation:
E 0') = L
If t h e total length of roads is unknown, we t h u s have a possibility t o estimate it b y com- puting Y.
\Ye can also write:
CROSS-COCNTRY T R A S S P O R T DISTASCES A S D ROAD NETS 6 9
\?here t is t h e total length of t h e line system per unit area. ( I n t h e case now discussed t = lib). The formula E ( Y )
-
L also applies in t h e case of two line systems a t right angles. I t is also applicable t o t h e t r a c t sides of t h e third S a t i o n a l Forest Survey vhicli form a system of fragments from two line systems a t right angles. I n these cases we only have t o substitute for t in formula (3) t h e value which corresponcls t o the form of survey concerned.The formula E (Y) = L applies if the direction of the line system is determined crt r a n d o m . In other words, t h e average value of I' for a very great number of systems with different directions equals L. For a n individual system, however, Y m a y considerably deviate from L even if t h e lines of the system are spaced very closely. This situation may occur if the road net is located in such a way t h a t most of t h e roads run in t h e same direction. Ilow- ever, t h e bias is considerably reduced if we have two syslems a t right angles, or fragments of two systems of this Bind. This statement can be proved b y formulas b u t i t is perhaps b e t t e r to illustrate i t b y some examples. The examples have been obtained by placing line systems on maps a n d b y counting t h e number of intersections with some net of railways or roads.
Example
Province of Jaintland
Railways. . . .
Main roads. . . .
Province of Goteborg a n d Bohus
Railways. . . .
Isle of eland
Railways . . . Province of RIalnlollus
Railxays . . . Total . . .
Estimated length in kilometres Length
according t o curvc-
meter lim
4 s s h o ~ v n by t h e examples from t h e province of Goteborg and Bohos, and t h e isle of o l a n d , a n one-sidecl orientation of t h e roads m a y lead t o rather biased estimates if only one line system is uscd. No essential bias in estimating t h e road length has been encountered with two line systems a t right angles, neither in thc examples she\\-n above, nor in other investigations not reported here.
If t h e surveyed line length in a n area is small, large s a m p l i n g errors m a y occur in t h e estimate of t h e total length of roads. This statement can he supported by a n e x a n ~ p l e . On a map of Ostergotland, 15 different systems of survey tracts wcre laid out, each system having a n extent equal Lo t h a t of the annual set of survey tracts in t h e third National Forest Survey. The numher of intersectious b e t ~ ~ e e n t h e different systems and t h e public roads shown on the map amounted t o (the values arranged according t o magnitude): 40, 42, 42, 43, 46, 47, 50, 50,57, 57, 58, 58, 60, 62, 69. Using these observations, v e t h e n find for t h e relative standard error of t h e total length of t h e roads the estimate 17 per cent. This standard error applies to t h e estimate of road-length based on one year's tracts.
Postscript 1964. The investigations reported above were carried o u t in 1956. The author has briefly commented on t h e problem in a paper published in 1959. As mentioned in t h a t paper, the theory of geometric probabilities has been applied t o t h e estimation of t h e average fiber length ( B a c k m a n 1934, K i l p p e r 1939, K a n e 1956, Iiallrnes & Corte 1960) in a way analogous t o t h e procedure described above.
For a more general discussion t h e reader is referred t o a n interesting paper b y Steinhatrs (1954) and t h e recent monograph K e n d u l l & J l o r a n (1963).
[la (2) and line
in Both
Formula (2) and line system in Both