United States

Department of

Agriculture

Forest Service

Southern &rest

Experiment Station

Research Note -

SO-275

August, 1981

Estimation of Total Tree Height from Renewable Resources

Evaluation Data

Charles E. Thomas

SUMMARY

Many ecological, biological, and genetic studies use

the measurement of total tree height. Until recently, the

Southern Forest Experiment Station’s inventory procedures through Renewable Resources Evaluation (RRE)

have not included total height measurements. This note

provides equations to estimate total height based on

other RRE measurements.

INTRODUCTION

Interest is growing in estimating whole tree volume and

biomass relationships for southern tree species (Clark

1978, Taras 1978, Hitchcock 1979):For example, four

southern universities’ are conducting research that uses

contemporary procedures for estimating individual tree

biomass. These studies are based on felled tree data.

Results of recent studies using these same procedures

indicate a strong relation between total tree biomass and

diameter-squared-height (d’h) (Clark 1978). Once these

relationships are established, it is possible to estimate

individual tree biomass directly from dbh and total height

without destructively sampling trees.

The RRE unit at the Southern Forest Experiment Station would like to estimate biomass by using its standard

inventory information. Unfortunately, measurements of

’ Auburn University, Louisiana Tech University, Mississippi State University, Stephen F. Austin University in cooperation with the Renewable Resources Evaluation program of Southern Forest Experiment

Station.

individual trees, while including diameter at breast height,

have not included total height. The merchantable height

and (since 1971) upper stem diameter outside bark at

merchantable height have been measured. A need for

good estimates of total height exists in RRE research.

A second consideration which requires some attention

is that past tree volume equations have generally been

restricted to specific top limits. Volume equations for various top limits often crossed illogically (Burkhart, 1980).

Future changes in the measurement of tree dimensions

should attempt to maintain compatability with past inventories while offering the flexibility to choose any level of

utilization of individual trees by species and/or other

criteria. Volume estimation should be more stable when

estimated total heights are included in predictive models.

It is possible to estimate total height from past inventory

measurements. Diameters and merchantable height permit better height estimates than dbh alone. Thus, a sample of trees from an inventory region can be measured as

usual, and include total height. Equations developed to

relate total height and standard variables on this sample

can then be used to produce reliable estimates of total

height for all trees. This estimated total height, along with

measured dbh, can improve whole bole volume and

biomass estimates considerably.

OBJECTIVE

The objective of this study was to estimate regression

coefficients for the computation of total tree height from

standard survey measurement variables. Both softwood

and hardwood species were included in the study populations.

Southern Forest Experiment Station/T-l0210 U.S. Postal Services Bldg., 701 Loyola Avenue, New Orleans, La. 70113

Forest Service, U.S. Department of Agriculture.

Serving Alabama, Arkansas, Louisiana, Mississippi, Eastern Oklahoma, Tennessee, Eastern Texas.
Figure l.- Thr Delta forest sunvy rrgion of Arkansas.

METHODS

A pilot study for the estimation of total height using

standard survey data was run in Units I and II of Arkansas

(fig. 1). These two units were located in the Mississippi

Delta region of the state and were considered to have

species composition and population characteristics that

would represent some of the estimation problems in the

Southern Forest Experiment Station area. Existing survey sample locations were revisited and total height was

measured on each tree. In addition, some temporary

plots were established by using standard procedures in

order to increase the size of the sample. On these supplementary plots, all pertinent plot and individual tree

measurements-including total height-were made.

A list of variables considered relevant to the current

problem is included (table 1). Several additional variables

such as site index and damage class were examined, but

later excluded as poor predictors. The sample was subdivided into species and species groups based on numbers

of observations, analysis of simple statistics on the samples, and standard RRE species groups.

Analysis was performed on individual species or species groups to determine the height prediction equation.

“Best” equations were chosen to maintain both F statistic

significance and R2 values as high as possible (table 5).

Charles E. Thomas is Research Forester, Southern Forest Experiment Station, Forest Service-USDA, New Orleans, La.

2
Not all individual species or groups of interest (e.g.

swamp chestnut oak, and select white oak) were represented in sufficient numbers to allow estimation of total

height on the species by itself. Dummy variate methods

were used to “separate” individual species in some

groups.

Analysis of species groups 3 and 6 indicated a significant increase in the R2 value when dummy variates representing individual species within the group were introduced into models. For example, regression model 6

showed an increase in the R2 value with the addition of a

dummy variable for swamp chestnut oak. To apply the

equation coefficients, data is entered for each variable as

usual. A 7 is entered in the equation with the coefficient

- 9.70 if the species is swamp chestnut oak. If not, a 0 is

entered. Species and species groups are listed in table 2.

Basic sample statistics are presented in table 3.

RESULTS

Regression equations were run for each species or

species group. Merchantable bole length is the single

most important independent variable in all species and

diameter at the top of the bole is usually next in importance. Diameter at breast height (dbh) and the square of

pole top diameter contributed significantly to many of the

regressions. Coefficients are presented in table 4.

Use of these equations results in a marked improvement in prediction of height over the use of dbh alone. A

typical R2 for dbh is 0.5 while regressions for selected

variable rapidly approached 0.9 in many of the regressions. Standard errors of the estimates (root mean

squared error) also improved notably. In table 5, R2

values for simple linear regression equations based on

dbh alone are included to indicate the improvement

obtained by using multiple regressions.

CONCLUSIONS

The estimating equations should be of value in the

estimation of total height from past inventories. Based on

the height, estimates of above-ground bole volume and

biomass may be obtained using existing biomass equation (Clark 1978, et al.). Total height of trees can be

estimated using dbh, improved estimates can be

obtained using merchantable height and diameter outside bark. Height regressions incorporating standard

RRE variables provide a marked increase in the R2 over

the use of dbh alone.

The use of these equations is limited to Units I and II in

Eastern Arkansas at present. Continuing research will

expand the species list and the geographical applicability. The survey is currently measuring total heights on

trees in Tennessee.

Table 1 .-List of variables*

Variables in the regression model

Y = b, + b, xc,) + + b,, x (,,) + e are:

X(1) = double bark thickness (at breast height) (inches)

X(2) = Diameter breast height (inches)

X(3) = bole length to merchantable top (4.0 inches dob or merchantable limits) (feet)

X(4) = saw log top diameter outside bark (dob inches)

X(5) = saw log mid dob (diameter at % sawlog height) (inches)

X(6) = pole top dob (inches)

X(7) = pole mid debt (inches)

X(8) = crown ratio (crown length as a decimal proportion of total

height)

X(9) = stump dob (inches)

X(1 0) = transformation (1)

X(1 1) = transformation (2)

Y = total height (feet)

transformation (1) = X(3) - 4.5/X(2) + X(6)/(1 -X(6))/X(2)

transformation (2) = X(6) l * 2, pole top dob squared.

All variables are as measured in RRE surveys and reference to exact

measurement technique can be found in the Survey Field Manual for

Arkansas.

*Variables included in one or more regression equation. Variables were

each tested for significant contribution to the regression.

+Dob at h = X(3)/2 for pole timber; at h =(X(3) + saw log length)/2 for

saw timber trees.

Table P.-Species and species groups

Survey I.D.

1.110 Shortleaf pine

2.400,401 Hickories

3.531,541,544 Beech and Ash

4.611 Sweet gum

5.691,693 Tupelos

6. 802,825 Select White Oaks

7. 813,833,834 Select Red Oaks

8.835 Post Oak

9. 812,822,831,837 Other Oaks

10.971,972 Elms

Pinus echinata

Catya spp.

Fagus sp., Fraxinus spp.

tiquidambar styraciflua

Nyssa spp.

Quercus alba, Q. prinus

Quercus rubra, Q. falcata

Q. shumardii

Quercus stellata

Quercus spp.

Ulmus spp.

Table 3.--Basic statistics for species

Dbh Height

Species or Number of Standard Standard

species group’ observations Mean deviation Mean deviation

1 2 2 11.2 3.6

2 58 12.1 5.5

3 5 0 13.9 6.8

4 6 4 i i .a 4.8

5 31 13.9 6.1

6 86 12.6 5.7

7 5 0 la.4 7.6

a 5 9 12.1 5.9

9 132 12.4 5.8

10 38 10.4 5.4

‘Species or species group as designated in table 2.

54.5 11.5

64.5 17.5

70.0 la.0

66.0 la.5

60.0 16.0

62.0 16.5

78.0 17.0

53.0 12.0

63.5 15.0

54.5 16.6

3
LITERATURE CITED

Burkhart, H.E.

1980. Cubic foot volume of loblolly pine to any merchantable top limit. South. J. Applied For.

4(2):7-9.

Clark, A.

1978. Total tree and its utilization in the Southern

United States., For. Prod. J. 28(10):4752.

Hitchcock, H.C. III.

1979. Converting traditional CFI data into biomass

values: A case study. Invited paper presented at the

1979 Forest Inventory Workshop SAF-IUFRO, Ft.

Collins, Co.

Hitchcock, H.C. III. and J.P. McDonnell.

1979. Biomass measurement: A synthesis of the literature. Invited paper presented at the 1979 Forest Inventory Workshop SAF-IUFRO, Ft. Collins, Co.

Taras, M.A.

1978. Biomass of Southern pines and their product

yields. In: C.W. McMillan (Ed.) Complete utilization of

Southern Pines. For. Prod. Res. Sot. Madison, Wise.

p. 15-23.

Taras, M.A. and D.R. Phillips.

1978. Above ground biomass of slash pine in a natural

sawtimber stand in Southern Alabama. U.S. Dep.

Agric. For. Serv. Res. Pap. SE-188 13 p., Southeast.

For. Exp. Stn. Ashville, N.C.

Table 4.--Regression coefficients

Species’ bo b, bz b3 ba bs bo b be bg ho b1 1 b(d)

Softwood

1 11.1 . . . . . . . . . 0.810 . . . . 5.94 -1.763 ,.,.,., . . . . ,.,.... . . . . . . . . . . . .

Hardwood

2

3

4

5

6

7

8

9

10

(a) beech

16.7 . . . . . . 1.049

51.6 ‘-21’2 1.612 0.508

22.0 1.007 0.558

17.1 . . . . . . -1.022 0.906

25.0 . . . . . . -0.634

10.2 . . 0.932

20.3 . . -0.301 0.755

23.9 -10.2 -0.123 0.986

29.9 . . . . -0.414 0.869

(b) swamp chestnut oak

‘Species or species groups as designated in table 2.

. . . .

-0.185

.,....

. . . .

,.....

0.237

0.088

. . . . . 7.09 - 1.927

. . .

. . . . 1.96 . . .

. . . . . .

.,.... 2.60

. . . 3.86

. . . 1.91

‘0.892

5.04

2.26 .

. .

. . . .

5.22

.

.

3.34

Table 5.-Regression statistics

R2

W’ F+ Dbh

Species Equation Regression SEE* alone

1 0.91 59.6 18,3 3.82 0.68

2 0.87 74.4 52,5 6.36 0.65

3 0.85 47.3 44,5 7.59 0.55

4 0.67 39.5 60,3 11.15 0.60

5 0.75 19.7 26,4 8.48 0.44

6 0.84 93.2 69,5 7.62 0.61

7 0.81 63.7 46,3 7.63 0.32

8 0.86 80.1 54,4 4.81 0.54

9 0.74 60.7 125,6 8.35 0.23

10 0.85 37.2 345 7.33 0.74

*Coefficient of multiple determination.

+F value and degrees of freedom for regression, P = 0.05.

*Standard error of the estimate.

,.... . . . . . -0.173 . .

. . . . . . . . 0.027 - 15.8(a)

.,.., ..,.... . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

0.657 0.429 9.70(b)

. . . . -0.050

,.... . . . . . . . . . .

4.16 -0.038 . . . . . .

. . . . . . . .

*U.S. GOVERNMENT PRINTING OFFICE: 1981-771-081/4