The Sine Method as a More Accurate Height Predictor for Hardwoods

Most hypsometers apply a mathematical technique that utilizes the tangent of angles and a horizontal distance to deliver the exact height of a tree under idealized circumstances. Unfortunately, these conditions are rarely met for hardwoods in the field. “new” predictor based on sine and slope distance and discussed here does not require the same assumptions for accurate height determination. Case studies using a sycamore (Platanus occidentalis L.), a water oak (Quercus nigra L.), and a southern red oak (Q. falcata Michx.) from southern rkansas are presented to emphasize the sensitivity of the tangent method to erroneous measurement procedures. When heights were measured properly and under favorable circumstances, the results obtained by the tangent and sine methods differed only by about 2 percent. Under more challenging conditions, however, errors ranged from 8 to 42 percent. These examples also highlight a number of distinct advantages of using the sine method, especially when exact tree height is required. INTRODUCTION Tree height is one of the most conventional attributes of forest mensuration. Equipment and methods specifically designed to enumerate the vertical dimension of individual trees have been available since the earliest years of forestry (for example, Noyes 1916; Schlich 1911). Few people have questioned the application of these techniques because they are based on fundamental geometric or trigonometric principles. Given textbook definitions of tree height measurement under idealized circumstances, there seemed little need for criticism. Unfortunately, the accuracy of height measurement has for too long been taken for granted. If carelessly applied, the conventional method of height determination is prone to significant errors. Even subtle violations of the assumptions of this technique (for example, an almost imperceptible lean in a tall tree) can produce noticeable departures from the exact height. In all fairness, it has only been in recent years that technology has caught up to the science behind tree height measurement, making it possible to control or eliminate this error ( lozan, W. 2004. Tree measuring guidelines of the Eastern Native Tree Society. Unpublished report. http://www.uark.edu/misc/ents/measure/tree_measuring_guidelines.htm. [Date accessed: ugust 20, 2005]. This paper briefly reviews the basic assumptions behind the traditional height measurement technique, including some that can lead to significant errors in height estimation. I will also describe a “new” estimator of height that uses a set of trigonometric relationships that is not sensitive to the same assumptions. Differences between the tangent and sine methods are illustrated in case studies of height measurements of hardwoods in southern rkansas. METHODS Basic Height Measuring Principles Mathematically speaking, hypsometers typically apply a technique that utilizes the tangents of angles and a horizontal distance to determine tree height. Figure 1 illustrates the basic principles of height determination. With accurate distance and angle measurements, tangent-based hypsometers determine total tree height (TanHT) as follows: TanHT = [tan( ) × b] + [tan( ′) × b] (1) 1 Research Forester, USD Forest Service, Southern Research Station, Monticello, R 71656.