13.3 - Smooth-by-Eye Method

Motivation of the Smooth-by-Eye Method Section

Recall that in 17.2, one main limitation of the Narrow-Strip Method is that it assumes the detectability of objects is perfect within the strip. But, in reality, the detectability decreases as the distance to the transect line increases.

In this section, the main idea of the Smooth-by-Eye Method is to use the histogram for distance x from the transect line to approximate the density of the detectability function f (x). We will approximate f (x) as a decreasing function, which agrees with the situation in real life.

How to Approximate the Detectability Function f (x) Section

The first step is to construct a histogram for the distance x from the transect line.

The formula for the height of the histogram for a given distance x is

\( \hat{f}(x)=\dfrac{y_x}{y \times w_x}\)

  • \(y_x\) - number of observations in the interval containing x
  • \(y\) - total number of observations
  • \(w_x\) - interval width

Example 13-2: Section

( Reference: Section 17.2 page 231 of the text)

y = 18 (total number of observations)

  • 5 birds were seen within 10 meters
  • 7 birds were seen between 10 and 20 meters

Choose the interval width of the histogram to be 10 meters.


  • the height for the 1st interval is \(5/[18(10)] = 0.0028\)
  • the height for the 2nd interval is \(7/[18(10)] = 0.039\)


  • the height for the 3rd interval is 0.017
  • the height for the 4th interval is 0.011
  • the height for the 5th interval is 0.006

Knowing the heights of each interval, we got the following histogram:

Distance (Meters) from Transect Line Density of Detections Histogram of Distance from Transect 0 10 20 30 40 50 0.00 0.01 0.02 0.03 0.04 0.05 0.06

Notice that the smooth-by-eye curve intersects the vertical axis at 0.048

Notice from the histogram that the detectability actually increases a little bit in the second interval compared with that in the first interval. However, the true density of detectability decreases smoothly with distance. We regard the increase in the irregularities in the histogram due to random chance and the small number of observations. 

We still fit a smooth, decreasing curve irrespective of the irregularity. The curve is fitted by eyeballing and each person may have a slightly different fit.

According to the “height of histogram” formula given before, we know:

\(f_0=\dfrac{y_0}{y \times w_0} \)

Combining with the formula in 17.2: 

\(\hat{D}=\dfrac{y_0}{2w_{0}L} \)

We got a new formula to estimate the density:

\( \hat{D}=\dfrac{\hat{f}(0) \times y}{2L}\)

\(\hat{f}(0)=0.048\) (the smooth-by-eye curve intersects the vertical axis at 0.048)


\( \hat{D}=\dfrac{\hat{f}(0) \times y}{2L}=\dfrac{0.048 \times 18}{2 \times 100}=0.00432 \)

So, the estimated bird population density is 0.00432 birds per square meter or 43.2 birds per hectare.