J. For. 112(1):51-54 EXPLORING THE ROOTS http://dx.doi.org/10.5849/jof.13-065 Accurately Measuring the Height of (Real) Forest Trees Don C. Bragg uick and accurate tree height1 measurement has always been a Q goal of foresters. The techniques and technology to measure height were developed long ago—even the earliest textbooks on mensuration showcased hypsometers (e.g., Schlich 1895, Mlod- ziansky 1898, Schenck 1905, Graves 1906), and approaches to refine these sometimes remarkable tools appeared in the first issues of For- estry Quarterly, Proceedings of the Society of American Foresters, and the Journal of Forestry. For example, one such hypsometer based on the geometric principle of similar triangles (top of Figure 1) employed rotary mirrors to allow the user to simultaneously see the top and bottom of the tree in “proper parallax” (Tieman 1904). Other early hypsometers applied different approaches that used angles and dis- tance (e.g., Graves 1906, Detwiler 1915, Noyes 1916, Krauch 1918). Of these trigonometric hypsometers, those that calculated total tree height (HT) as a function of the tangent of the angles to the top (B2) and bottom (B1) of the tree and a baseline horizontal dis- tance (b) to the stem were most common (Figure 1). Because they are easy to apply and required only simple tech- nology, these approaches (hereafter, the similar triangles and tangent methods) have dominated tree height measurement. That is not to say the challenge of accurate tree height measurement was solved— many early foresters reported problems with getting consistent data Figure 1. Graphical representation of tree height measurement on in uneven terrain or dense understories or with the use of different sloping ground (the same mathematics apply for a level surface) types of hypsometers. Good measurement practices usually miti- using the similar triangles and tangent methods. Note that all gated these issues and became standard components of forester train- historic equations and figures referenced in this work follow the notation of these figures. ing programs. Others addressed these challenges by designing new techniques based on different trigonometric relationships. As an ex- ample, Haig (1925) proposed a slide rule solution that calculated tree Yet such corrections, although they improved the reliability of height using the sine law and slope distance from the observer to the indirect height measurement, failed to address other potential tree base (s1) sources of error. For these adjustments to work, the trees had to be ͑ ϩ ͒ truly vertical, i.e., the point at their base measured for the horizontal s1 sin B2 B1 HT ϭ distance was directly below the highest point at the top of the tree. As sin A 1 an example, if the subject tree happened to be leaning, the de- (see the trigonometric example in Figure 1 for the angles involved). termination of A1 in Haig’s correction is less straightforward and In very steep terrain, the ability to use a slope distance rather than required additional measurements. Hypsometers based on similar horizontal distance (b) made height determination much simpler. triangles could be used on leaning stems, but the geometry of this McArdle and Chapman (1927) applied a different trigonometric approach requires the user to exactly match the angle of lean, a solution for sloping ground when determining tree heights. Using a daunting and imprecise task in most forested settings. For tree height ϭ fixed slope distance (s1 100 ft) from the eye of the observer to the measuring devices that used the tangent method, adjusting for lean ϭ ϩ base of the tree, they proposed the following: HT 100 sin B1 also presented a serious challenge. Falconer (1931, p. 744) succinctly 100 cos B1 tan B2. addressed this quandary: Received August 19, 2013; accepted November 18, 2013. Affiliations: Don C. Bragg ([email protected]) is a research forester with the USDA Forest Service, Monticello, AR, and editor-in-chief of the Journal of Forestry. Acknowledgments: I recognize the contributions of the Native Tree Society for their efforts to develop and promote the sine method. Mike Shelton, Nancy Koerth, and Jim Guldin (all of the USDA Forest Service), Bob Leverett, and several anonymous reviewers all contributed to the development of this article. Journal of Forestry • January 2014 51 The incorrect location or choice of the dle the plumb lines and the third to identify His seminal Forest Science monograph pro- point to which the measurement for dis- tance is taken is the only common and con- the point on which these two vertical planes vided such an insight: in his Figure 3, tinued source of error in measuring the intersect). Unfortunately, such an approach Grosenbaugh (1963, p. 8) provided the fol- ϭ Љ height of leaning trees. The error is due, not required further training of field crews, sig- lowing equation for “exact” height, H0 R to the actual chainage of the distance, but in nificantly increased the amount of time sin ␪Љ, where the height to the diameter mea- the location of the point to which the dis- tance is measured. The distance can be ac- needed to measure tree heights, especially in surement location (H0) was a function of a curately measured only when the point is heavy timber or rough terrain, and may have sloping baseline distance (RЉ) and angle be- located on the ground vertically below the necessitated adding staff for proper execu- tween the observer and the measurement lo- tip of the tree. tion. cation (␪Љ). He used sine rather than tangent In other words, because the tangent Such additional burdens did not endear at this point because the slope distance method projects height as a function of the themselves to forestry operations wanting to (whose measurement was possible using a horizontal distance from the observer to the streamline inventory procedures. Not sur- rangefinder-dendrometer) was required to apex of the crown, if a tree leans away from prisingly, then, less rigorous corrections for get the exact height above the ground for or toward the observer, the proper baseline tree lean and wide crowns became ingrained that point along the trunk. distance must be adjusted for accordingly. in mensuration textbooks and professional Grosenbaugh (1980, p. 204) later in- Think of how tree height measurement is curricula. The study of field-based tree troduced further measures to avoid den- often portrayed: the images are almost al- height measurement focused on optimizing drometry bias for leaning trees because ways straight, vertical, pointed crown coni- the efficiency of existing techniques (e.g., “…range, height, and (for optical forks) di- fers growing on level ground—an all-too- Barrett 1929, Morey 1931, Hunt 1959, ameter biases attributed to neglected lean to- often inadequate representation of “real” Bruce 1975, Rennie 1979, Long and Mohai ward or away from observer can be much trees in a more complicated forest environ- 1986, Williams et al. 1994) and the reintro- more serious and are not overcome by merely ment. duction and/or modification of largely un- tilting dendrometer and hypsometer to Hypsometers based on the mathemati- used tools (e.g., Curtis and Bruce 1968, match tree tilt in crosslevel.” This assertion cal principles of similar triangles and tan- Anunchin 1971, Buckner et al. 1977, Larsen questioned the use of certain ad hoc correc- gents incorporate assumptions that are only et al. 1987). Overall, most foresters were sat- tions for leaning trees. His use of spherical sometimes (rarely?) met in the field. Thick isfied with the degree of accuracy possible trigonometry yielded a more complete ver- undergrowth could be cleared and sloping from existing approaches (when the proper sion of the total tree height (Grosenbaugh Ϫ ground adjusted for, but tree lean and crown degree of fealty to adjusting for atypical trees 1980, p. 207), s2 sin B2 s1 sin B1, repre- irregularities proved much harder to ad- was observed) and that continuity in tech- senting the vertical difference between the dress. A fix for these was critical: Chapman nique over time trumped any possible im- crown high point (A) and the intersection of (1921, p. 246) noted that “(t)he error from pacts of systemic bias. To borrow a judicial the bole with the ground (C),where B1 and the measurement of broad-crown trees [un- phrase, it seems that height measurement B2 are the angles to the bottom and top of less corrected] is cumulative and tends to was considered “settled law.” the tree, respectively, and s1 and s2 are the over-estimate their heights.” To this end, However, in the latter half of the 20th corresponding “slant range” (the sloping Graves (1906) recognized the challenges of century, a new approach to tree height mea- baseline distance from his 1963 article) esti- measuring leaning and wide-crowned trees surement that could address the complexity mates from the observer to those points (Fig- and proposed corrections based on measur- of tree architecture appeared. Starting in ure 1). ing heights perpendicular to the lean and the 1950s, forest mensurationist Lewis R. For a truly vertical tree on level ground, moving further away to ensure that subordi- Grosenbaugh developed and refined den- this “sine method” yielded the exact same nate branches are not mistaken for the high- drometers capable of measuring tree heights, height as the tangent method (i.e., HT ϭ ͉b Ϫ ͉ ϭ ͉ Ϫ ͉ est point of the crown. Likewise, Krauch stem diameters, and bole volume under tan B2 b tan B1 s2 sin B2 s1 sin B1 ). (1918, 1922) suggested a number of adjust- many different conditions (e.g., Grosen- Figure 2 highlights the fact that for a leaning ments for the USDA Forest Service standard baugh 1954, 1963, 1980, 1981, 1991).