Data generated by any photo-id project in which the risk of missing a match is not negligible will include multiple encounter histories from some of the animals. Over five sampling occasions, an individual (in our case study, a female grey seal) may, for example, generate history 01001 and 00100 if the image taken on the third sampling occasion is not matched to either of those taken on the second or fifth. That might be because it was identified only from the left in samples two and five and only from the right in sample three, or because although all images were from the same side, the standardised scores between the third image and the other two failed to exceed the threshold used.
The risk of multiple encounter histories precludes the use of the usual multinomial model for presence/absence capture history frequencies. Inference in our approach is based on maximising the likelihood of encounter history frequencies where each history specifies the sides from which the seal was photographed during each season. For example, 0L00L represents a seal that we know was photographed from the left at least once during seasons 2 and 5, whereas 0L00B represents a seal that we know was photographed at least once from the left during season 2 and from both sides during season 5. We calculate, according to the demographic parameter values at that stage and allowing for the measured pairwise FRR, the expected frequency of each possible encounter history. Any history consisting of only L or R encounters is possible, plus any mix of L and R provided there is at least one B, for example, L0R0B. A seal identified from both sides in the same breeding season only as a result of matches to a B encounter in a different season generates two histories (for example, L0R0B and 00L0B, if there were separate right and left side encounters with the same seal during season three).
To calculate the expected frequency of a given history, we use the following notation, where subscript i refers to the season number from season 1 to n.
FS and LS, the first and last encounter in the history, for example, 2 and 5 in 0L00L.
nB, the number of B encounters in the history, for example, one such in L0R0B.
nL and nR, the number of seasons in which the seal was identified from the left and the number in which it was identified from the right, for example, two and two in L0R0B, and three and one in L0L0B.
, the number of seals joining the local population as a result of immigration or recruitment between seasons i and i − 1. represents the size of the local population at season 1; thus, the number of females that had their pup within the PA in that year plus recruited females that failed to pup but would have used the PA.
φi , annual apparent survival (thus including emigration) from season i − 1 to i. For convenience, φn+1 is set to 0.
Plocal, the probability that a seal in the local population is available to be photographed during a season, assumed to be independent from season to season and the same in each season.
PL,i, the probability an available seal is photographed from the left at least once during season i, varying from season to season as a result of variation in sampling effort.
PR,i, the probability an available seal is photographed from the right at least once during season i, varying from season to season as a result of variation in sampling effort.
τi, the probability that the seal turns when it is encountered so that both sides can be photographed and linked to the same ID at that time.
Pmiss, the estimated pairwise FRR.
, φi, Plocal, PL,i, PR,i and τi are all free parameters used to maximise the likelihood; we regard Plocal, PL,i , PR,i and τi as nuisance parameters and , φi as parameters of interest as estimating initial population size, apparent survival and recruitment/immigration. In the likelihood function, Pmiss is assumed to be a known without error.
To calculate the probabilities of events L, R and B as functions of the nuisance parameters, we assume left and right encounters with an available individual occur as Poisson processes at rates varying with side and from season to season. Then, the probabilities of events L, R and B equal Plocal , Plocal and Plocal , where ∼PL,i and ∼PR,i represent 1 − PL,i and 1 − PR,i (see Appendix S1).
Multiplying those probabilities for the encounters in a given history gives the probability that those encounters would be recorded for a local seal alive at least over that period (i.e. from FS to LS). However, to generate the given history, all those encounters must also be recognised as being with the same seal. The algorithm used to calculate the probability of that happening is described in Appendix S1. Finally, we need to multiply by the probability that it is not recorded in each season for which it was in the local population but for which the encounter type in the given history was 0, allowing for the risk that it could be photographed in that season but not recognised. Assume the risk of failing to recognise an individual reduces exponentially with the number of images available. Then, the probability of not recording the seal at a time i when it was part of the local population equals
T1 is the probability that the seal is photographed from the left only and recognised, T2 that it is photographed from the right only and recognised, and T3 that it is photographed from both sides and recognised, whether or not both sides are linked to the same ID at that time. Here, we assume that only a single photograph representing the left or right side of an individual is retained in a given season; if each individual is represented by a number of photographs within a season to aid identification between seasons, then the definition of Pmiss needs to be modified to the risk of failing to match between randomly selected sets of photographs of that size rather than between single photographs.
To generate a given history, a seal must arrive in the local population by season FS and survive and remain in the population until at least season LS. Let Pr (histh, a, d) represent the probability of observed history h calculated as described above for a seal that arrives in the local population in season a and stays till season d; a and d determine the number of seasons preceding FS and following LS in which the seal was not seen or not recognised. So, to calculate the expected frequency, we sum over the possible arrival and departure times, multiplying Pr (histh, a, d) by the number of seals arriving at those arrival times and their probability of survival over exactly that period:
where denotes expectation, and fh represents the observed frequency of history h.
To derive the likelihood, we assume the observed frequencies are independently distributed as Poisson variates:
and the first summation is over all observed histories, whereas the second is over all possible histories.