ShadeMotion: tree shade patterns in coffee and cocoa agroforestry systems

Abstract

Shade trees in coffee and cocoa agroforestry systems provide valuable livelihoods and other ecosystem services. Unfortunately, most shade canopies are sub-optimally designed and managed and the reasons behind this sub-optimality are poorly known. There is evidence, however, indicating that farmers and extension agents lack both the knowledge and access to user-friendly tools to optimally design their shade canopies. To fill this gap, we developed ShadeMotion, a simple to parameterize, yet powerful software, capable of calculating the spatial and temporal distribution of the shade cast by trees on a plot (horizontal or tilted) anywhere on Earth. Trees may be planted in any spatial arrangement, their population density can change according to planting, thinning, mortality or harvest, tree crowns may take any of eight possible, regular, geometric shapes, vary in density and monthly leaf fall patterns, and may be pollarded or not. Shade patterns may be calculated for one instant, 1 year or less, or for the entire life cycle of a plantation; in the latter case, tree growth data must be added as input. Shade can be measured at ground level or at any height above the ground. In this article, we: (1) describe the key features of the software, and (2) simulate the spatial and temporal shade patterns of a traditional agroforestry system of Coffea arabica cv. Caturra, Erythrina poeppigiana and Cordia alliodora in Costa Rica. ShadeMotion can realistically model farm scenarios and may be used in the participatory design of agroforestry systems in farmers field schools and in classrooms.

Appendix 1. Analytic expressions of tree crown shadows

Horizontal terrains

Following values are common to crown shapes in horizontal terrains.

$$X = (x - x_{0} )cos\left( Z \right) - (y - y_{0} )sen\left( Z \right)$$

$$Y = (x - x_{0} )sen\left( Z \right) + (y - y_{0} )cos\left( Z \right)$$

For crowns with system SRA located at the center of the crown: spheres and ellipses:

$$x_{0} = x_{a} + (T + h - m)cotan\left( {elev} \right)sen\left( Z \right)$$

$$y_{0} = y_{a} + \, (T + h - m)cotan\left( {elev} \right)cos\left( Z \right)$$

$$Z = azim + 180 - Y^{ + }$$

For crowns with system SRA located at the base of the crown: semi-spheres, semi-ellipses, cones, inverted cones, cyllinders and umbrellas:

$$x_{0} = x_{a} + (T - m)cotan\left( {elev} \right)sen\left( Z \right)$$

$$y_{0} = y_{a} + (T - m)cotan\left( {elev} \right)cos\left( Z \right)$$

$$Z = azim + 180 - Y^{ + }$$

With x_a, y_a being the Cartesian coordinates where the tree is planted in the terrain (SRU Cartesian coordinate System), azim is the suns azimuth, and Y ⁺ is the angle (clockwise) between the Y-axis and the north, a is half-diameter of the tree crown, H is the height of the crown, h is half crown height, and (x₀, y₀) are the Cartesian coordinates of the origin of the SRB reference system in SRU. A grid cell with coordinates X, Y is shaded by a tree crown if it satisfies at least one of the systems of inequalities describing a tree crown shadow.

Spheres

$$X^{2} + \left( {Y\;sen\left( {elev} \right)} \right)^{2} \le a^{2}$$

Ellipsoids

$$\frac{{X^{2} }}{{a^{2} }} + \frac{{Y^{2} }}{{\left( {h\;cotan\left( {elev} \right)} \right)^{2} + a^{2} }}$$

Semi-spheres

$$\begin{aligned} & \left\{ {\begin{array}{*{20}l} {X^{2} + \left( {Y\;sen\left( {elev} \right)} \right)^{2} \le a^{2} } \hfill \\ {Y \ge 0} \hfill \\ \end{array} } \right. \\ & X^{2} + Y^{2} \le a^{2} \\ \end{aligned}$$

Semi-ellipsoids

$$\begin{gathered} \left\{ {\begin{array}{*{20}l} {\frac{{X^{2} }}{{a^{2} }} + \frac{{Y^{2} }}{{\left( {H*cotan\left( {elev} \right)} \right)^{2} }} \le 1} \hfill \\ {Y \ge 0} \hfill \\ \end{array} } \right. \hfill \\ X^{2} + Y^{2} \le 0 \hfill \\ \end{gathered}$$

Cones

$$\begin{aligned} & \left\{ {\begin{array}{*{20}l} {Y \le \frac{{H\;cotan\left( {elev} \right)}}{a}X + H\;cotan\left( {elev} \right)} \hfill \\ {Y \le - \frac{{H\;cotan\left( {elev} \right)}}{a}X + H\;cotan\left( {elev} \right)} \hfill \\ {Y \ge 0} \hfill \\ \end{array} } \right. \\ & X^{2} + Y^{2} \le a^{2} \\ \end{aligned}$$

Inverted cones

$$\begin{aligned} & \left\{ {\begin{array}{*{20}l} {Y \ge \frac{{H\;cotan\left( {elev} \right)}}{a}X} \hfill \\ {Y \ge - \frac{{H\;cotan\left( {elev} \right)}}{a}X} \hfill \\ {Y \le H\;cotan\left( {elev} \right)} \hfill \\ \end{array} } \right. \\ & X^{2} + \left( {Y - H\;cotan\left( {elev} \right)} \right)^{2} \le a^{2} \\ \end{aligned}$$

Cylinders

$$\begin{aligned} & \left\{ {\begin{array}{*{20}l} {Y \le H\;cotan\left( {elev} \right)} \hfill \\ {Y \ge 0} \hfill \\ {X \le a} \hfill \\ {X \ge - a} \hfill \\ \end{array} } \right. \\ & X^{2} + Y^{2} \le a^{2} \\ & X^{2} + \left( {Y - H\;cotan\left( {elev} \right)} \right)^{2} \le a^{2} \\ \end{aligned}$$

Umbrellas (cap of sphere)

An umbrella can be modeled by a flattened semi-ellipsoid that looks like a circle when viewed from above.

Tilted terrains

Inequalities presented for all crown shapes in horizontal terrains are valid in tilted terrains with the following modifications:

$$X = \left( {x \, - x_{0} } \right)cos\left( Z \right) - \left( {y - y_{0} } \right)sen\left( Z \right)cos\left( \sigma \right)$$

$$Y = \left( {x - x_{0} } \right)sen\left( Z \right) + \left( {y - y_{0} } \right)cos\left( Z \right)cos\left( \sigma \right)$$

For crowns with system SRA located at the center of the crown: spheres and ellipses:

$$x_{0} = x_{a} + \left( {T + h - m} \right)cotan\left( {elev} \right)sen\left( Z \right)$$

$$y_{0} = y_{a} + \, \left( {T + h - m} \right)cotan\left( {elev} \right)cos\left( Z \right)$$

$$Z = azim + 180 - \delta$$

For crowns with system SRA located at the base of the crown: semi-spheres, semi-ellipses, cones, inverted cones, cyllinders and umbrellas:

$$x_{0} = x_{a} + \left( {T + - m} \right)cotan\left( {elev} \right)sen\left( Z \right)$$

$$y_{0} = y_{a} + \left( {T + - m} \right)cotan\left( {elev} \right)cos\left( Z \right)$$

$$Z = azim + 180 - \delta$$

where σ is the angle of maximum slope of the terrain and δ the angle of orientation of the positive Y⁺ axis, which is aligned along the maximum slope of the terrain.