V, -. TAPER-BASED SYSTEM w r3 FOR ESTIMATING STEM V,

U LzI VOLUMES OF UPLAND OAKS

by Donald E. Hilt

FOREST SERVICE RESEARCH PAPER NE-458 1980

,=OREST SERVICE, U.S. DEPARTMENT OF AGRICULTURE NORTHEASTERN FOREST EXPERIMENT STATION

370 REED ROAD, BROOMALL, PA 19008

DONALD E. HILT received a bachelor's degree in forestry from Iowa State University in 1969, and a master of science degree in forestry from Oregon State Universitgr in 1975. He joined the Northeastern Forest Ex- periment Station in 1975 as a statistician in the biometries group at Upper Darby, Pennsylvania. In 1976, he transferred to Delaware, Ohio, where he is a research forester specializing in studies of growth and yield of managed upland oaks.

MANUSCRIPT RECEIVED FOR PUBLICATION 2 JANUARY 1980

A taper-based system for estimating stem volumes is developed for Central States upland oaks. Inside bark diameters up the stem are pre- dicted as a function of dbhib, total height, and powers and relative height. A Fortran IV computer program, OAKVOL, is used to predict cubic and board-foot volumes to any desired merchantable top dib. Volumes of test trees were predicted with acceptable accuracy. Other uses for the taper-based system are discussed.

N UMEROUS VOLUME TABLES for up- land oaks have been constructed in past

years. While most are satisfactory for the intended use, few provide the Qexibility available with a taper-based vstem for estimating stem volumes. Users often desire predicted volumes to merchantable top diameters that differ from the published tables. A taper-based system allows the user to calculate volumes to any desired top diameter. Volume in any portion of the stem may also be calculated.

Some of the older volume tables con- structed by the alignment chart method cannot be computerized because there is no prediction equation. Other volume tables based on regression equations sometimes fail to produce consistent trends between cubic volumes to various top dimeters, or between board-foot and cubic-foot volumes.

L.) Dbh,, ranged from 2.4 to 25.5 inches and total height from 21 to 116 feet. A sys- tematic sample of 334 trees was selected to develop the taper equation. The remaining 84 trees were used to validate the model (test data set one).

Dbh,, was recorded for each standing tree. The trees were then felled and bucked into 6 to 20 sections. Height above ground and diameter inside bark (dib) were mea- sured on each section. Dbh, was measured at the section point 4.5 feet above ground.

An additional 124 oak trees from West Virginia and Virginia were used as a second test data set. Each tree was bucked into nine sections and the dib and height above ground were recorded for each section. Dbh,, ranged from 6.0 to 21.9 inches and total height from 48 to 104 feet.

Logical and consistent trends result when a taper-based system is used. Taper equation

This paper presents a taper-based system for estimating cubic and board-foot volumes of upland oaks. Diameter breast height in- side bark (dbh,) and total tree height are required to estimate stem volumes with the taper-based system. Diameter breast height outside bark (dbh,,), measured in the field, can be converted to dbh, using appropriate bark-thickness factors for each oak species. Total height may be measured in the field or estimated with a local height-diameter curve.

The taper equation was constructed from tree measurements by using a modification of the taper model proposed by Bruce e t al. (1968). The modified form of the equation is

DATA where

Felled-tree measurements were collected = d2/D2 on 418 upland oak trees located in Ohio, = (H-h)/(H-4.5) Kentucky, Missouri, Indiana, and Illinois. d = dib at measurement point (inches) Oak species were white oak (&. alba L.),

= dbhib (inches) chestnut oak (Q. prinus L.), black oak (&. veluntinn Lam. j, scarlet oak (Q. coccinea = total tree height (feet) Muenchh.), and northern red oak (Q. rubra h = height at measurement point (feet)

and the 0's and y's are constants to be deter- mined. The equation satisfies the logical con- ditions that d equals zero when h equals H, and d equals D when h equals 4.5 feet. I used dbhib in the equation because bark thickness varies according to oak species. Dbh,, can be converted to dbh, using ap- propriate bark-thickness factors for each species.

Note that the p's can be estimated by linear model procedures for given values of the y's, I used a simplified version of equa- tion ( I ) ,

Cubic volumes h

A form factor (F) representing the ratio of cubic volume inside bark from tip to a given height (h*) on the stem, to a cylinder of diameter D and height (H - 4.51, can be computed by integrating (3) :

n

y = x312 + X Pi (x312 - xyi) (2) - 7.6098 (z4) H

i= 1 + 9.041 ( lom5) (z5i2 ) DH

to determine reasonable values for the 7's. (5)

Different values of the y's were substituted - 5.651 (10") (z4) DH into (2) by trial and error, and the resulting equation was fitted to the data with step- + 1.28405 (lo-') (z512) D wise regression procedures (maximum R2 improvement method) after each substitu- - 1.0355 (z") D tion. The values yl = 3 and y2 = 30 suggested by Bruce proved to be satisfactory. Given - 8.48 (lo-') (z"~) DH these values for the y7s, stepwise regression procedures were then, used to estimate the + 6.84 (z3') DH p's in equation (1). The resulting model was where

Inside bark diameters on the stem can be estimated from (3) as

Equation (4) represents the average stem profile for trees of specified D and H. The calculated R2 for the_taper model based on residuals of the form (d - d) was 0.93. Typical curves shown in Figure I illustrate the reason- able taper curves, including butt flare, gen- erated by equations (3) and (4) for upland oaks.

A

Predicted cubic volume inside bark (V) from tip to h* is then calculated as

Cubic volume of the stem from tip to a 1-foot stump (TOTVOL) was predicted with equations (5) and (6) using h* = 1.0. Mer- chantable cubic volumes from a 1.0-foot stump to merchantable top dib were pre- dicted by setting h* equal to the height to merchantable top dib, then subtracting the resulting volume from TOTVOL. A stump volume equal to the volume of a cylin- der I foot long and diameter equal to the predicted dib at 1 foot was added to TOTVOL to arrive at the predicted total stem cubic volume (CVTS).

Figure 1.-Stem profiles as estimated by equa- tions (3) and (4). Constant dbh ib ldbh .b ratio = 0.91 used.

90

Dbhab

\ * * * * * * * * 21NGHES

80 \ ------ 10 INCHES

\ - 15 INCHES

RADIUS, ib (inches)

"Actual" cubic volumes of trees were calculated by summing the volumes of mea- sured sections. The volume of each section was computed as a frustrum of a cone. Pre- dicted volumes were then compared to the actual volumes for CVTS and cubic volume to a 4-inch top (CV4) (Table 1). The mea- sured dbhib was used to predict volumes for the mode data set. Dbh, was calculated as 0.91 x dbhob for the two test data sets to simulate actual use of the equations because bark thickness is not usually mea- sured in the field. The average percentage deviations and differences between total volumes were considered acceptable for the model data set, especially since the com- putation of actual volumes is certainly not exact. Excellent results were also obtained for the first test data set. The standard deviation of the precentage differences was approximately 10 percent in most cases. This value indicates that for a given tree, the CVTS or CV4: can be estimated with a + 10 percent error. Total volume for all trees is estimated with more accuracy.

Predicted volumes were consistently low for the second test data set. Part of the reason for low estimates may be due to the constant dbhib/dbh,, ratio of 0.91 that

Table 1 .-Corngarison of predicted and "actual" volumes

Average Standard deviation of Total Total Percentage

Data seta gf percentage trees percentage "actual" predicted difference, differencesc differences volume volume totals

Model : CVTS CV4 BF8

Test 1 : CVTS CV4 BF8

Test 2: CVTS CV4 BFR

aCVTS = cubic volume total stem, ib; CV4 = cubic volume t o 4-inch top, ib; BF8 = International 1/4-inch rule board-foot volume t o 8-inch top.

b~~~~ analyzed for all trees; CV4 and BF8 for trees 11.6 inches dbhOb and larger.

'Percentage difference = [(predicted volume - actual volume)/actual volume] x 100.

was applied. The main reason was because the stem form of the trees from West Virginia and Virginia was much better than the stem form of the trees used to build the taper equation. The average of the stem form factors predicted with equation (5) using h* = 1.0 was 0.472, while the actual stem form averaged 0.509. If the sample of 124 trees provide a good representation of all trees found in West Virginia and Virginia, then the taper equation constructed in this paper for Central States upland oaks would not be applicable and the construction of an appropriate taper system for this geographical region is recommended. If the sannple of well-formed trees occurred by chance, as the case may be occasionally for Central States upland oaks, then the predicted volumes can be adjusted by using methods proposed by Gevorkiantz (1955).

Cubic volumes to various merchantable top diameters are shown in Figure 2 for a

tree 80 feet tall. The volumes generated from equations (5) and (6) f o m smooth and logical progressions with increasing dbhOb Trends were similar for other height classes.

International 114-inch rule board-foot volumes were calculated for all trees 11.6 inches dbh,, . and larger. Total board-foot volume (BF8) in the tree was detemined by summing the board-foot volumes of all logs in the tree to an 8-inch top. Allowance was made for a 1.0-foot stump and 0.3 feet of trim for each 16-foot log. Fractional 16-foot log lengths at the top of the tree were calculated by 4-foot sections, including fractional length on the remaining 4-foot section. Predicted board-foot volumes were computed using scaling diameters estimated with equations (3) and (4). Actual board-

Figure 2.-Cubic volumes to El-, 6-,I-inch tops, and total stem for 80-foot total height tree.

I CUBIC VOLUME /'

foot volumes were computed using measured or interpolated scaling diameters.

A comparison of actual and predicted board-foot volumes is s h o w in Table I. Average percenbge deviations and differences between total values were acceptable for the model data set and the first test data set. The standard deviation of the percentage differences, 19.1 in both cases, is larger than it is for cubic volumes. This larger value should be expected because estimation of board-foot volumes in trees requires the estimation of several scaling diameters up the stem. Considering the variability that occurs in many upland oak stems, the results are excellent.

Estimates for the second test data set are low for board-foot volumes, just as they were for cubic volumes. Predicted board-foot volumes may also be adjusted according to methods proposed by Gevorkiantz (1955).

I3oard-footlcubic-foot ratios were plotted

for various heights and diameters (Figure 3). The ratios follow smooth and logical trends.

Program OAKVOL

The taper-based volume equations pre- sented in this paper are too complex to be solved easily with a desk calculator. A Fortran IV computer program titled OAKVOL (Ap- pendix) was written to provide the flexibility desired when using the taper-based system. The four subroutines provide the necessary programing logic for applying the taper- based system to a variety of uses. Explana- tion of each subroutine may be found in the program listing. The main program was mitten to allow the user to specify any number of fixed merchantable top diameters for both cubic and board-foot volumes. Modification of the main program, and possibly the subroutines, is necessary depend- ing on the intended use.

Figure 3.-International 114-inch board-footltotal stem cubic volume ratios.

. . . . . - - - - _ _ - ---

HEIGHT (feet)

110 . - . . - . 100 ---- ---

80 ---

10 12 14 16 18 20 22 24 DbhOb( inches)

Table 2.-Sample total cubic-foot volume table ' Diameter breast Total height ( f ee t )

high, ob (inches) 20 30 40 50 60 70 80 90 100 110 120

aCubic volume of entire stem, less bark. Includes 1.0-foot stump volume. Constant dbhib/dbhOb ratio = 0.91 used.

Table 3.-Sample board-foot volume table ' Diameter breast Total height (feet)

high,ob (inches) 50 60 70 80 90 100 110 120

aInternational 114-inch rule board-foot volumes to 8-inch top dib. Allowance made for 1.0-foot stump, 0.3 feet trim for each 16-foot log, and fractional length on upper log. Constant dbhib/dbhob ratio -: 0.91 used.

6

Even though a large number of computa- tions are necessary to calculate volumes using OAKVOL, the cost is very reasonable. In a sample run, CVTS was calculated for 33,600 trees and BF8 for 11,628 trees in 38 seconds on an AMDAHL 470V/6 com- puter- Computing could be simplified by fitting regression equations to explain rela- tionships such as those shown in Figure 2 and 3. However, most users have access to high-speed computers, and should prefer the flexibility available with OAKVOL.

Sample volume tables generated with OAKVOL for CVTS and BF8 are shown in Table 2 and 3. A constant dbh, /dbh,, ratio of 0.91 was used to determine dbh,. In practice, a variable ratio according to the oak species should be used.

"stem" into a finite number of sections (100 should be adequate), then calculate the surface area for each section with the formula for a truncated cone.

Volume tables based on the number of logs to variable top diameters, often desired for timber cruising, can also be constructed using the taper-based system. Field measure- ments would require dbh,, , total height, and number of 16-foot logs to the variable merchantable top dib. Total height could be estimated using a local height-diameter relationship. After converting the number of logs to a merchantable height, volumes could be computed using a modified version of OAKVOL.

The taper-based system developed in this paper should be useful to many foresters working in the upland oak timber type. High-speed computers permit inexpensive applications of the system.

The inherent flexibility of a taper-based system allows compatible cubic and board- foot volumes to be calculated to any desired top diameter. Modification of OAKVOL L IT E RATU RE C ITE D will dlow users to apply the system to a variety of other uses. Percentage distribu- Bruce, David, Robert 0. Curtis, and Caryanne tions of volumes within trees, often desired Vancoevering. for biomass and whole-tree chipping studies, 1968. Development of a system of taper and

can b e conveniently computed with a taper- volume tables for red alder. For Sei. 14:339-350. Gevorkiantz, S.R.

based Stem surface areas, ib, can also 1955. Composite volume tables for timber and be calculated- The easiest method for c o m ~ u t - their application in the Lake States. U.S. Dept. ing surface areas would be to segment the Agric. For. Serv. Tech. Bull. 1104. 51 p.

C 313 WR I I E I Cs, 2 j DGb..IOR, H i , ( T,I..J\.4lLS ( I ? ? I = 1, CIVTDPS 3 , I BFVQl.,.,S I i , I = I, BFTOPS ) 2 FORMAT(T1038F8.Zi

GO TQ 5 200 STBP

END

SUBRQLfT I P4E E3 I SECT i f)@f..: I g , i-j-r, I"\ITOFS? Ti7!IsD I B , T'CIF'I.-I'TS ! C C --.... TW I S SI.JRROU.1-I P\lE USES A B I SECT I QP.f IILGBR 1 Tk4M I I TEROIT I 'v'%

C APFRtlACk.4 ) 7-0 DETERM I NEI Ti. 1E l....Er.IGl'1,-E-; f TOF'i-.ITS 1 FROM T I I2 Ta IkfE C DESIRED MERC+-/Ap..!iAgL_E: I NC-jljDK @ARI.; ClIAMETEQS { TOPDf g j; .-.-- C

I MP I., 1: C 1 1" fi EkL*% ( &$ .... 1-f , Cj ... Z :i REAI .-*8 "["QFD f E3 t 1 i2 ) TaFt..;-l-S ! 1 0 1 k=O. (3 E=I--fT--l . I3 D=O. 0 ZMDIE3-Q. O DO 10 I=I ,NTOFS h i = :E I F i T O F D I B ( I :I .L.'T. .< ) I ?f;[3 7-0 $5

= I...t-r -.. . t: CAL ... L D I D i t>Bi-.IfEp t.-I"f', P. E::)) 1FfD.L.E. TOPDIBi; 1: 1 j G 0 Ti3 GO DO 20 J= l ? 50 2 M I fip "I" = ( A .+. [3 j / 2 CAL-L. E31fBiDB!-iIB, 1.-I.?, Zt'!IEI"T7^ :ZPii:!I:E) j Df FFzTCjPD'j'B i I j ..-ZM!::;iIB IF([&&,@S<DIf::'F> .i'./". JJJ, 'Tilj 45 'J3V ( /"J I F:'Fm i - Tm a, i i;l;Cf TiIj 50 &+= Z[y I [:q" GO T t l 20

513 B=Zp[IDPT 20 C13P~4TII'~ftlE 40 TOP/.+T'E; I - 1 ...l'T'...- 1 6

GO TC? 1. <3: 45 . - Tapt[f-S(I)=:ZMID!:"'y

SWROUTINE CUVQLCDBHIB,WT,NTQFSZTOPHTS,CUVOtS) C C ....-.... THIS SUBROUTINE CALCULATES CUBIC VOLUMES (CUVOLS> TO

C DESIRED MERCHANTABLE INSIDE BARK DIAMETERS. C TOPWTS = DISTANCE FRQM T I P TO MEECI-fANTMLE HEIGt-IT -.-- C

IMPL IC IT REAL*€3 (A-H,O-Z> REAL*8 TOPHTS(lO),CUVOLS(lO) L=O

5 DO 50 I = l , NTOPS K = I IF(TOPHfS( I ) .LT, ,0QI iGO 70 55 IF(L.EQ. 1 )GO TO I 0 Z=TOFHTS i I 1 / i WT-4-5)

10 xl=z**a.s Xt3..Z*.tt4 x3=z**31 X4=XI*HT X5=X2*HT XG=X4*DBHIB

X7=X5*aatiIs XB=XI*DBHIB X9=X3~~DBHI8 x 1 O= Xss*l..+l- F=. 4*Xl F=F+ i %.21757D-.S ) * X 4 F=F- ( 7. &oc38D-. 4 j 4t-:<y;

F=F+ ( '3, Q41[)--5) *XE: F -F .- (5 . E,ft;lC> .... 5 ).*jI:-y

F=F+ ( : iw28405D-.3) *Xz F=F- 6 1 0355f,I.-..4 ) 35:<13 F=F-- ( 8. .480.-.5 ) =E"X[Et FzF; ( 84fi.-.(5 ) -PkX 1 (J

I F < 1-. EQ. 2 )GO TO E:O CJ,Ja[_s ( f ) = * Qt'J5454 1 5<-ag31.-4 f @*[)@I-..( g* i' i-iT'-.-L!+ * 5 *F

50 Ct)NTLi'.ltJE GO TCf 5-7

55 CIJiQOLS(K)=O,O 57 x = (HT-.j. a Q ) / (i-{T-.."i., 5 )

L= 1 GO TO 5

E,Q ~af-l,)f~l-= , 1 cj-%ggt..r I s3t.f-isr.d 1 s-5 ( i...i'r .... L+ a ..* f:: . 'j 3-c::

C C .... -. .... CAtCULA-rE 1 , 9 FCIjjJT S'TLjMF! v[']I_-UplE ... --. ....

STUMFt..;: {-.IT .... 1 , Q

CALL- f""IB ( D@b.iIBp /--i.rp STUl~~~I . - -~ , Si-l_iPli""[l!) STLIMPV=. . 005.454 J 5.ztfi~;-r[-[MFI:Ji3-S~tjj~fpf:3

C e ..., -. .... CAl.-t"'L!LATE CIJa f C +,*!Ot_tfPiES I:fiE~~<~i.../~NTA~L-E: TOf7 f,? J: AM%')"ER 5 -. -. .-.

nn 70 I= J , NTOPS CUfq-JLS ( I ) =l"Q.Tval-.-.ctff\,Iftff<; ( .s 1 f T O ~ ~ . ~ T S i I j ~ ' 7 , , QQ 1 ;I c ijij[-~-s ( 1 :! = ~a-r.vo~_- .+. s.j-upttf:! v

70 CONTINUE FtE?"UW N EN0

SUBROUTINE BFVOL C DBHIB, kiT, J'.ITUt7E;, BFDIBS, TOFtf-iTS, BFVOLS 1 I* L

C --- TH I S SUBROUT I b4E CAl-CUtkTES I NT ' L 1 /4.- I NCt.1 RLILE BC3ARD

C FOOT VtlLUMES ( BFVOLS ) TCI DES I RED MERCC.iANTAEi-E I f4S I DE BARC( C @ IFiMETEf?S ( BFD I BS ) . TOF"f.:f'S = D I ST At4CE FF<DM T I P TQ MEF.;"CF~AN~..~~B[-E C HEIGHT .---- C

IMPLICIT EEAL*S CA-1.4, O-.Z i REAL*E TOW-iTSi SE31 ,BFVOLSi 10!, BFDIBSt S O ) ,LI-J<,EI~ LCtGDIB=O. O DO '30 I = 1, NTOPS TOFHTS f I i =I-iT~-.T0Fl-i'T S i I ) H=l.O BFVOLS ( I > =O.

5 0 I-.f=H+. 3 I F C H. GE. TOFf-i'T:5 I I ) > GCi T'Q 5x3 t-,l=l-4+ I f . I F iH , .GT , ' fOPk i7 'S i I j ?GO T'U 55 Z=HT-.kt CALL.. DIB!DBI-.IIB7 I-il", Z p LlZiGfrIB i BFvOLS ( I ) =E;F'$CftS ( I 1 .+. . 7 '3G%.~._D{~~ j ,I: Bdi.[...<]<;D 1 a .... 1 , 3-$5?$t_-JGD 'J .... L . 23 GO TO 50

55; kt=t-:-..lE,. A = TOP 1.j1-S < 1 ) .... 1.1 1 LQG4=Aif4, i-OGf?I@=gF-CjIE;S ( I :)

IFC IL..0%4. EQ. O:kGfi 1-Ci El5 DO EtO J = 1 , 1L"OGri- BF{f("Jt-S( i )=BF:'vjJ[.-S( 1 > .+.. I<~'~~"i._fiZi::>~:B;Ti .,- r",:;-T ;.-i.;.; :.,,I .I. [,> ::i .... . E1,42+tj [ii-'' 'i 1 E3 -- " .t I....

LO~~DiB=t_Clt;DIB.t.. 5 E L I CCSNT I f4CJE E,5 FR&,C= ( 4 .-.. {F1-{"JAT{ 'Ji,6G4>+$4 ' 8 ) ,:!4. * .

BFr?;O[ Ej( I )=gF:'f,-)f-J1, C : ' " ) .;., = j,::lj'3.l.'i-'Ji.'f ~ B . " . " . 3 i ' ' . ~ ] : ~ . - Ex&2.::$i-i-[GC1;B )-3IZRt.;C; - L - ..> '.. .I. \.7 ..J *\ L... i.f t.? .. . -. '3C3 (3 0 NT I f'.II_IE

R ET'I..!RN EN0

VOLUMES GENERATED FOR F I V E SAMPLE TREES:

TOTAL CV4 CVTS BF8 DBHoB H E I G H T BF6

ACKNOWLEDGEMENTS

The author thanks Fred Berry, North- eastern Forest Experiment Station, and Will Carmean, North Central Forest Experiment Station, for use of their felled-tree measure- ments of Central States upland oaks. Thanks also to Jeff Martin, Northeastern Forest Experiment Station, for supplying the test data set from Virginia and iVest Virginia.

Headquarters of the Northerastern Forest Experiment Station are in Broomall, Pa. Field lahratories and research units are maintained at:

@ Amberst, Massachusetts, in cooperation with the University of Mwachusetts. Bel-tsvi%Be, Mary land.

@ Berea, Kentucky, in cooperation with Berea College. 0 BurlinBon, Vermont, in cooperation with the University of

Vermont. @ &laware, Ohio,

Durhm, New Hampshire, ;ln cooperation with the University of New Hmpshire.

@ Hamden, Connecticut, in cooperation with Yale University. Kingston, Pennsylvania.

@ Morgantown, West Virginia, in coopration with West Virginia Univemity, Morgantown.

@ Orono, Maine, in cooperation with the University of Maine, Brono.

@ Parscbns, West Virginia. @ Prhcebon, West Virginia. 0 Syracw, New York, in cooperation with the State University of

New York College of Environrnental Sciences and Forestry at Syracuse University, Syracuse.

0 University Park, Pennsylvania, in cooperation with the Pennsylvania State University.

@ Warren, Pennsylvania.