##
General Idea
Section* *

The detectability of the objects usually becomes smaller as the distance from the transect line increases. However, there may be some narrow strip along the line in which detectability is virtually perfect. Therefore, we can use the observations within the narrow strip to do the estimation.

### Notations

- \(y_i\) - number of objects observed from the ith transect
- \(n\) - number of transects selected
- \(\tau\) - total number of objects in the study region
- \(A\) - area of the study region
- \(D=\tau /A\) - density of the objects -namely, number of objects per unit area
- \(L\) - length of the transect
- \(w_0\) - maximum distance from the line to which detectability is assumed perfect
- \(2w_0\) - width of the strip
- \(2w_{0}L\) - area of the strip
- \(y_0\) - number of objects detected within the narrow strip

#### Formulas for Estimation

- Density Estimation
\(\hat{D}=\frac{y_0}{2w_{0}L}\)

- Estimation of Total Number of Objects in the Study Region
\(\hat{\tau}=A\hat{D}=\frac{Ay_0}{2w_{0}L} \)

**Note!**

There are several ways to choose \(w_0\), the perfect detection distance. We will illustrate one method in the example later.

Burnham et al. (1980, p. 33) suggest that the data should include at least 40 detections to provide a reliable estimation. We will have a smaller number just for illustrative purposes.

##
Example 13-1:
Section* *

(

Reference: text p. 231)On a line transect of length L=100 meters, a total of *y *= 18 birds were detected at the following distances (in meters) from the transect line:

0, 0, 1, 3, 7, 11, 11, 12, 15,15, 18, 19, 21, 23, 28, 33, 34, 44

Please estimate the density of birds in the study region.

How to choose \(w_0\)?

From the data, we know that

5 birds were seen within 10 meters

7 birds were seen between 10 and 20 meters

3 birds were seen between 20 and 30 meters

2 birds were seen between 30 and 40 meters

1 bird was seen between 40 and 50 meters

We plot the above information in a histogram.

We can see from the above histogram that the relative frequency of observing the birds drops off sharply (from 7 to 3) after 20 meters from the transect line. Thus we choose \(w_0\) = 20.

Given \(w_0\) = 20, we know that \(y_0\) = 12

\(\hat{D} =\dfrac{y_0}{2w_{0}L} =\dfrac{12}{2(20)(100)}=0.003\)

So, the density estimate is 0.003 birds per square meter or 30 birds per hectare.

##
Limitation of Narrow-Strip Method
Section* *

- Not all observations obtained are used.
- The determination of the width of the narrow strip seems somewhat subjective.
- The detectability may, in fact, decrease smoothly with distance so that the narrow strip with perfect detectability really has a width of zero.

(We will introduce other line transect sampling methods in the following sections.)