Instrumental method for reducing error in compression-derived measurements of rainfall interception for individual trees

Abstract

This technical note presents an instrumental method for the precise and timely installation of mechanical displacement sensors to investigate stem compression and relaxation associated with whole-tree rainwater loading and evaporation, respectively. We developed this procedure in response to the conclusions of Friesen et al. (2008), which called for the development of a precision mounting method for strain sensors on inherently-irregular trunk cross-sections so that rainfall interception, storage and evaporation may be distinguished from other strain-related phenomena. To supply precise sensor installation locations, high-resolution trunk profiles are generated using the LaserBark^TM automated tree measurement system. These scans are utilized to approximate the location of neutral bending axes. A routine then instructs a mobile rangefinder along the cross-section to optically indicate exact positioning for strain sensors over the bending axes. As imprecise sensor placement linearly increases error and diminishes signal-to-noise ratio, this automated installation routine is designed to remove significant distortions created by wind throw, off-centre loading within unevenly-distributed canopies, and human error that can lead to erroneous measurements of rainfall interception.

Citation Van Stan, J. T. II, Jarvis, M. T., Levia, D. F. Jr & Friesen, J. (2011) Instrumental method for reducing error in compressionderived measurements of rainfall interception for individual trees. Hydrol. Sci. J. 56(6), 1061–1066.

Résumé

Cette note technique présente une méthode instrumentale pour l'installation précise et opportune de capteurs de déplacement mécanique pour étudier la compression et la détente de tige associées respectivement au chargement en eaux pluviales et à l'évaporation de l'ensemble de l'arbre. Nous avons développé cette procédure en réaction aux conclusions de Friesen et al., qui appellent à la mise au point d'une méthode de précision pour le montage des capteurs de contrainte sur les sections transversales des troncs afin que l'interception des précipitations, leur stockage et les processus d'évaporation puissent être distingués d'autres phénomènes liés à la contrainte. Pour fournir des emplacements précis d'installation des capteurs, des profils de troncs haute résolution sont générés en utilisant le système automatisé de mesure LaserBark™. Ces profils numérisés sont utilisés pour déterminer l'emplacement des axes de flexion neutre. Ensuite un programme ordonne à un télémètre mobile le long de la section transversale d'indiquer d'une manière optique le positionnement exact de capteurs de contrainte sur les axes de flexion. Comme un placement imprécis du capteur augmente les erreurs et diminue le rapport signal sur bruit linéairement, ce programme d'installation automatique est conçu pour éliminer les distorsions importantes créées par les chablis, le chargement excentrique au sein de canopées irrégulièrement réparties, ainsi que l'erreur humaine, qui peuvent conduire à des mesures erronées de l'interception des précipitations.

Key words:

• forest hydrology
• interception
• stem compression
• mechanical displacement sensor
• LaserBark™

Mots clefs:

• hydrologie forestière
• interception
• compression de tige
• capteur de déplacement mécanique
• LaserBark^TM

INTRODUCTION

Intercepted rainfall stored by, and evaporated from, canopy bark and foliar surfaces can diminish incident precipitation inputs beneath the forest canopy by as much as 50% (Muzylo et al. 2009), depending on species and season (Crockford and Richardson 2000, Levia and Frost 2003, Keim et al. 2006, Llorens and Domingo 2007). However, estimates of canopy water storage and evaporation are primarily based on indirect methods and model estimates which are capable of producing >30% error (Muzylo et al. 2009). Error is compounded further if these indirectly-derived interception components are falsely linked to net precipitation measurements by simply treating interception as a “black-box” process (Pollacco and Angulo-Jaramilo 2009), or when splash droplet evaporation is neglected (Murakami 2006, 2007, Dunkerley 2009). The identification of uniquely linked interception mechanisms (Pollacco and Angulo-Jaramilo 2009) and investigation of splash droplet evaporation requires direct canopy interception measurements at the intra-storm scale (Dunkerley 2009). Historically, direct measurements of water fluxes at varying temporal resolutions have been performed using weighing lysimeters (e.g. Fritschen et al. 1973, Edwards 1986, Storck et al. 2002). Although weighing lysimeters have provided valuable insights into the roles of vegetation in hydrological processes (e.g. Seyfried et al. 2001, Petrone et al. 2006, Stumpp et al. 2009), these investigations focus on the whole soil–plant–atmosphere system. Thereby processes such as canopy interception, bark interception and forest floor interception, as well as evaporation and transpiration are observed through one single measurement. To derive a more process-based understanding of plant-specific processes (e.g. canopy interception, evaporation of intercepted water and intra-storm canopy storage), and of soil or surface-specific parameters (e.g. soil evaporation and forest floor interception), it is necessary to observe the different processes as separately as possible. Weighing lysimeters are also expensive and require extensive construction (Sayler et al. 1984), which disturbs the surrounding environment.

Friesen et al. (2008) demonstrated that direct measurements of canopy water loadings can be obtained inexpensively at high temporal resolution (∼5 seconds) and sensitivity (<5 kg) using mechanical displacement sensors affixed to the tree bole with a minimally-invasive superstructure. Mechanical displacement sensors respond to the compression and relaxation of a trunk axial section from changes in canopy aboveground biomass, where axial loads creating trunk compression at the storm-event scale are largely attributed to the increased weight of intercepted rainwater within the canopy and, conversely, relaxations from the compressed state represent intra-storm evaporation of intercepted rainfall (Friesen et al. 2008). However, wind and off-centre loading within the canopy (Fig. 1) can produce bending anomalies many orders of magnitude greater than compressive forces of the same strength, regardless of tree height and diameter (Table 1).

Fig. 1 Physical forces acting on a tree within a storm event. Wind and the weight of intercepted rainfall upon unevenly-distributed canopies will produce bending- and torque-related anomalies within stem compression data.

Table 1  Ratios of bending stress to compressive stress (as measured 2 m from the tree base) under modest wind loads (500 N) and maximum rainfall interception* for trees of increasing diameter and canopy height. These calculations conservatively assume vertically-uniform diameter

Diameter (m)	Height (m)
	5	10	15	20
0.25	384	1024	1664	2304
0.50	48	128	208	288
0.75	14	38	62	85
* The interception load was calculated using the maximum value reported in the literature (∼50kg) (Huang et al. 2005, Zhang et al. 2006, Muzylo et al. 2009).

Assuming wood within tree boles is a linear elastic material that behaves orthotropically (FPL 1999, Lyons et al. 2002), measurement distortions from wind and off-centre loading can be removed if the sensors are orthogonally aligned along axes of no longitudinal stresses or strains (neutral bending axes) (Gere and Timoshenko 2004). Precision of sensor placement is critical, as measurement error will increase linearly with distance from a neutral bending axis or orthogonal alignment in any linearly elastic material. The installation procedure outlined in Friesen et al. (2008) hand-aligns mechanical displacement sensors, enhancing the potential for human error. Recognizing this shortcoming, Friesen et al. (2008) suggested the need for a precision mounting method which could quickly and accurately determine orthogonal, neutral bending axes and the placement of sensors. The objective of this technical note is to describe an instrumental method addressing this concern using high-resolution cross-sectional scans from the newly-developed LaserBark^TM automated tree measurement system (Van Stan et al. 2010) to: (1) estimate the location of orthogonal neutral bending axes for asymmetrical trees; and (2) optimize mechanical displacement sensor placement. The theoretical method outlined within this technical note represents an important development, as hydrologists and foresters currently utilize many forms of strain technology (e.g. Guitard and Castera 1995, Peltola 1996, Yoshida and Okuyama 2002, Moore and Maguire 2005, Murphy et al. 2005, Friesen et al. 2008, James and Kane 2008) without this level of precision placement.

METHODOLOGICAL SOLUTION TO DETERMINE SENSOR PLACEMENT

To approximate neutral axes of bending for trees of non-idealized cross-section and shape requires, at a minimum, high-resolution cross-sections over the installation area (Gere and Timoshenko 2004). The most recent embodiment of the Van Stan et al. (2010) bark microrelief instrument reads 15 data points per second from a sub-millimetre accurate laser rangefinder to produce high-resolution (≥10 radii measured per degree) two-dimensional stem profiles (interested readers are referred to Van Stan et al. 2010 for further details). Once a scan is complete, the cross-section is broken into triangles, formed by two adjacent measurement points and the origin. Neutral bending axes are then derived from the centroids and areas of the triangles using:

(1)

(2)

where (x_ci ,y_ci ) is the centroid and A_i is the area of any triangle enclosed by the points (x_i ,y_i ) and (x_{i +1},y_{i +1}), and the origin (or centre of cross-section). With the centroid and the area determined for each triangle, the centroid of the entire irregular stem profile can be computed as:

(3)

(4)

where (x_c ,y_c ) is the total centroid, A_t is the total area, and n is the total number of points enclosing the tree cross-section. In the case of a constant elastic modulus (E), neutral axes of bending will pass through the centroid of the cross-section (Gere and Timoshenko 2004) (Fig. 2(a)). This assumption may hold true for hardwoods and some softwoods, as several studies have found statistically insignificant (<5%) differences in E between fresh and dried samples of both heartwood and sapwood (Gillette 1914, FPL 1999, Berthier et al. 2001, Passialis and Adamopoulus 2002). In this scenario, finding a pair of neutral axes is as simple as choosing any two perpendicular lines which pass through the area centroid of the enclosed section (Gere and Timoshenko 2004) (Fig. 2(a)).

Fig. 2 LaserBark^TM tree scans of an irregularly-shaped Liriodendron tulipifera L. (yellow poplar) tree bole which show the location of neutral bending axes and placement of displacement sensors (x ₁, x ₂, y ₁ and y ₂) about: (a) a single cross-section where neutral bending axes intersect at the computed centroid for strain sensors; and (b) two cross-sections along a section of trunk for potentiometer-based mechanical displacement sensors (as per Friesen et al. 2008).

The heartwood of some tree species contains high levels of soluble “hot water” extractions in comparison to their sapwood (Grabner et al. 2005). These extracts have been shown to increase axial stiffness at the cellular level (Grabner et al. 2005). For these species, we propose an alternative method for computing neutral bending axes by meshing the cross-section with internal points of a regular spacing. The mesh can then be used to compute a Delaunay triangulation for the combined internal and external section points (Shewchuk 2002). Each element in the mesh is assigned a weighted E based on its distance from the section centroid, or alternatively, the distance to the closest exterior point of the section. The location of the neutral axis of bending parallel to the y-direction is then found by numerically solving the following equation:

(5)

where the integrated area of element j and E_j is the unique elastic modulus assigned to each element j for the total number of elements n.

After an acceptable set of neutral bending axes have been computed, the LaserBark^TM automated tree measurement system can optically indicate each sensor location on the physical tree surface for installation. This is done by instructing the carriage to move to a precise position along the support ring and powering-on the laser emitter at the exact location intended for sensor placement. In the case of a strain gauge transducer attached directly to the vascular tissue of the tree (as in Peltola et al. 2000, and Moore and Maguire 2005), the measurement of a single section is adequate. In the case of a linear potentiometer (as in Friesen et al. 2008), more than one section needs to be measured, marked, and then their properties combined through the methods used to solve non-prismatic beams (Gere and Timoshenko 2004) (Fig. 2(a)).

Once the sensors are installed, the distances of each to the neutral axes are used in conjunction with strain measurements to interpolate the non-shear stress at the neutral bending axes at any given time using:

(6)

where σ_c is the non-shear stress between the shear stresses of two orthogonally-aligned measurements σ₁ and σ₂ at the vertical and horizontal distances d ₁ and d ₂. Because we are measuring solely elastic deformations of assumedly orthotropic tree boles (Lyons et al. 2002), stress and strain are assumed to be linearly related. Thus, we can substitute Hooke's law for uniaxial stress to convert the above equation to strain:

(7)

where all stress (σ) terms become strain (ϵ) terms. These computations can be done for each set of axes and averaged to allow for an accurate measurement of non-shear strain regardless of wind direction.

Finally, compressive strain can be converted into a measurement of force through Hooke's law coupled with the area of the cross-section:

(8)

in which F is the compressive force measured from the interpolated strain (ϵ_c) over the total cross-sectional area (A_t ) with the elastic modulus (E). Determination of E may prove to be difficult as the strength of wood varies with tree age (Kokutse et al. 2004) and height (Tong et al. 2009). Friesen et al. (2008) used a calibration method to experimentally determine the value of E for each tree that was instrumented. This is an appropriate and robust method for in-situ determination of the E along axial strains, yet axial calibrations should be complemented with static pull tests to ascertain the relationship between shear forces and trunk bending moments (James and Kane 2008).

CONCLUSION

This instrumental method for precision mounting of the Friesen et al. (2008) mechanical displacement sensors along orthogonal neutral bending axes provides a precise, quick and repeatable installation procedure, enabling forest hydrologists to quantify the dynamics of individual canopy water loading (due to interception and storage) and unloading (due to evaporation) with confidence at high temporal resolutions. The theoretical framework outlined here is of great value in itself, as forest scientists currently use mechanical displacement and strain sensors for interception and wind throw research without precision installation techniques. As such, the methodological advance contained herein provides the forest hydrology community with a solution in the use of mechanical displacement sensors for the determination of whole tree canopy interception and intrastorm evaporation.

Acknowledgements

The authors would like to thank the Bucktoe Creek Preserve in Chester County, Pennsylvania, USA and the Office of Economic Innovation and Partnerships at the University of Delaware for supporting the instrument development and providing the use of their facilities as a beta-testing site for embodiments of the LaserBark^TM automated tree measurement system. We are also grateful for the translation services provided by Dr Hans-Jörg Busch of the Foreign Languages and Literatures Department at the University of Delaware.