Where \(d_i\) is the number of detections at site \(i\), \(k\) is the (potential) number of individuals at site \(i\), \(J_i\) is the total number of visits to site \(i\), \(p_k\) is the probability of detecting at least one individual at site \(i\), provided that there are \(k\) individuals present and \(\lambda\) is the mean number of individuals across sites. Theoretically, it would be necessary to consider any number of birds per site and therefore the summation should go to infinity; in practice however, we may set an upper limit \(K\) to what is a reasonable expected maximum number of individuals at a given site \citep*{Royle2003}. In this case, we set an upper limit of five birds per pentad. This limit was somewhat arbitrary and based on past experience on conducting Black Harrier surveys.
The likelihood of the detection history in all sites is then