Distance sampling

Estimating abundance

Abundance estimation is often accomplished using this equation: \[ \hat{N} = \frac{n}{\hat{p}} \]

  • \(N\) is abundance (population size)
  • \(n\) is the number of individuals detected
  • \(\hat{p}\) is an estimate of detection probability: The probability of detecting an individual


Most methods differ in how they estimate \(p\)

Estimating abundance

Challenges:

  • Detection probability is rarely constant
  • It is a function of:
    • Age
    • Sex
    • Habitat
    • Distance

Distance Sampling

Basic idea:

  • \(p_i\) is the probability of detecting individual \(i\) at distance \(x_i\).
  • Since every individual in the population has a different detection probability, we replace \(p\) with the average detection probability (\(\bar{p}\))
  • The equation is now:

\[ \hat{N} = \frac{n}{\hat{\bar{p}}} \]

Advantages of distance sampling:

  • Population size can be estimated from a single survey
  • Explicit link between population size and density

Design

  • Randomly place line transects or points throughout the study area
  • Record the distance to each animal detected

Line transects

Point transects

Line transects

Line transects

In line transect sampling, it is common to record the radial distance and bearing, rather than the perpendicular distance.

However, the analysis must be conducted on the perpendicular distance data.

Real design

Mongolian gazelle (Procapra gutturosa)

Line transects

Animal Distance (\(x\))
1 4.4
2 25.3
3 41.8
4 3.1
5 78.5
\(\vdots\) \(\vdots\)
\(n\) 4.4

Estimating average detection probability

Fit a detection function, \(\mathrm{g}(x)\), to the data

Assume individuals are “uniformly” distributed with respect to the transect (valid under random sampling): \(p(x) = 1/B\) where \(B\) is the transect half-width.

\(\bar{p}\) is then the average detection function, weighted by \(p(x)\).

\[\bar{p} = \int_0^B g(x)p(x) \mathrm{d} x\].

Computing average detection probability


Shiny App

Assumptions

  1. Animals on the line are detected with certainty (i.e., \(p=1\) when distance=0).
  2. Animals do not move during the survey
  3. Distance is measured accurately
  4. Transect lines are placed randomly with respect to the animals
    • This ensures that individuals will be uniformly distributed with respect to the transect
  5. Detections (of individuals or groups) are independent of one another

Point transects

Point transects

Differences between point vs line transects


We still use the same type of detection function (half-normal, hazard, etc…).


We don’t expect a “flat” histogram of distances when \(\bar{p}=1\). Why?


More individuals occur far from the observer than close to the observer because area increases with distance.

Point Transects

When average detection probability is 1, the histogram of detected distances will increase with area because there are more individuals far from the point.

Point Transects

If average detection probability is less than 1, the histogram will increase and then decline.

Assignment



Read Chapter 10 – Capture-Mark-Recapture