Field Sampling on a Fibonacci Spiral

The question of where to locate field samples is complex.  There are many things to consider.  This is an important consideration in planning field studies in ecology because this will partly determine the quality of the data and what it can tell you.  I will here discuss a few common options for laying out sampling sites and when a more complex plan using a Fibonacci spiral is warranted.

Commonly used schemes to locate samples in a study area include:  random, stratified random, rectangular grid, and radiating spokes.  These all have different strengths and weaknesses in the experimental design and analysis stages.  Factors include: ease of planning and implementation, assumptions of the analysis, data needs.  It is relatively easy to follow a rectangular grid in the field, especially if it runs parallel to roads and cardinal compass directions.  However, such a grid has its own inherent pattern that may filter the resulting patterns detected (Fortin & Dale, 2005).  Samples equally-spaced along transects radiating outwards from a point can provide information on a phenomenon around a focal point, but these become more diffuse with distance.  Neither of these sampling set-ups lead to a good distribution of inter-sample distances for examining spatial patterns with, for example, semi-variograms.

Fortin & Dale (2005) suggest overcoming these limitations by location samples using Fibonacci spirals.  This pattern leads to good coverage of an area, as with a rectangular grid, but maintains equal sample density regardless of distance from the center of the study area—the identified problem with radiating transects.  It may sound intimidating to locate samples along such spirals, but using polar coordinates of (angle, radius) rather than Cartesian (latitude, longitude) makes the location quick.  For implementation in the field, you simply use trigonometry to then convert the points back to (latitude, longitude) format.  This should result in data that are optimal for examination of spatial trends.  I will let you know how it works in practice at the end of the 2017 summer field season.

fibonacci

Figure 1:  400 sampling locations placed on a Fibonacci spiral.  Plot results from the R script below.  Points are originally ordinated with polar coordinates but returned as latitude — longitude (universal trasverse mercator coordinates in this case).

The R script below places a required number of points on a Fibonacci spiral (Figure 1) and then converts them to (latitude, longitude) format for input into your gps unit.  The code is very inelegant so that the steps are obvious.

#####  Fibonacci Spiral for Sampling  #####
#####          JDH, Feb. 2017         #####

tau <- (1 + sqrt(5))/2                                       ## The golden ratio.
try <- c(1:400)                                                  ## INSERT value sequence here.
theta <- try*(2*pi)/tau                                 ## Angle.
rad <- sqrt(try)                                                 ## Radius.

orig.x <-  508000                                                                ## INSERT utm center long.
orig.y <- 4475000                                                              ## INSERT utm center lat.

delta.x <- (cos(theta))*rad
delta.y <- (sin(theta))*rad

pts.x <- delta.x + orig.x
pts.y <- delta.y + orig.y

my.border <- 2                                                  ## INSERT width of extra white space.
plot(pts.x, pts.y, col=”darkgreen”, pch=16, cex=0.5,
                ylim=c(min(pts.y-my.border), max(pts.y+my.border)),
                xlim=c(min(pts.x-my.border), max(pts.x+my.border))
                )

Reference:
Fortin, M.-J. and M.R.T. Dale. 2005. Spatial Analysis: A Guide for Ecologists. Cambridge University Press. Pp. 248–254.
This book is an excellent, readable introduction to the topic.  

 

 

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