Assumption 1

Random Target Distribution

Targets are equally present at all distances

To correct for declining detection probabilities, we must assume something about the distribution of targets in the study area. Otherwise, we have basis to infer missed targets at any distance. In almost all distance-analysis studies, we assume targets are randomly distributed relative to survey paths or point. If true, perpendicular distances between routes and targets will be evenly distributed between 0 and w under perfect detection (Figure 4). If this assumption holds, and we see a decline in the number of observations far away from routes, we can infer that detection declines far from the observers, and not that fewer targets exist far from the observers. In short, we can infer that we missed some far-away targets if this assumption is true.

Under this assumption and if detection declines with distance, large distances will be artificially under-represented relative to short distances (Figure 5). This assumed fact allows us to estimate the decline in detection probabilities because the difference between a constant number of detections for every distance and the observed number of detections at every distance is an estimate of detection probability decline. In other words, if this assumption is true we can estimate detection decline as the difference between a uniform distance histogram and the realized distance histogram (Figure 5). This difference between the theoretical and observed histograms is the key to (classical) distance sampling analysis.

uniCaption = paste("Figure 2:",
  "Targets (left: dots) randomly distributed relative", 
  "to the transect (left: black line).  Assuming perfect detection in a", 
  "strip 15 m on both sides, observers detected all targets",
  "(left: dashed red lines) and observe a uniform distribution", 
  "of off-transect distances")

distPlot("uniform")  

uniCaption = paste("Figure 3:",
  "Targets (left: dots) as in Figure 2 randomly distributed relative", 
  "to the transect (left: black line).",
  "Assuming detection declines with distance,", 
  "observers detected more targest near the transect than",
  "farther away",
  "(left: dashed red lines).",
  "This situation results in a non-uniform distance distribution (right).")

distPlot("halfnorm")  

uniCaption = paste("Figure 4:",
  "Theoretical (black) and observed (red) distance distributions.", 
  "The proportion of missed targets is the area of the grey polygon.")
  
layout( c(1) )
par(mar = c(5.1, 3.1, .5, .5))

plot(c(0, 16), c(0, 1.1), type = "n", 
   xlab = "Distance off transect",
   ylab = "Relative probability of detection")
   
lines(c(0, 15), c(1,1))
lines(c(15,15), c(0, 1))
xD <- seq(0, 50, length= 100)
yD <- rnorm(xD, 0, 3.5) / rnorm(0, 0, 3.5)
lines(xD, yD, col = "red")

xP <- c(xD, rev(xD))
yP <- c(yD, seq(15, 0, length = 100))
polygon(xP, yP, col = "grey")
text(14, 0.85, "Missed targets", adj = 1)