## Introduction

Distance sampling is a common class of methods used to estimate abundance of wildlife populations (Buckland, Goudie & Borchers 1998). In conventional distance sampling (CDS), a region is sampled from randomly placed lines (line transect sampling) or points (point transect sampling) with *n* detected animals counted and their respective distances, *y*_{i,} *i* = 1, … *n*, to the traversed line or point recorded. The distribution of these distances is then used to estimate the decline in an animal's detection probability as a function of increasing distance from the observer. The fitted detection function then facilitates estimation of animal abundance or density in the surveyed region, and the precision thereof (Buckland *et al*. 2001).

One of the strengths of CDS is that the random placement of transects in the study area (i.e. design-based surveys) supports two assumptions. First, the surveyed area is assumed to be a random (i.e. unbiased) sample of the larger study area (between-transect scale), allowing extrapolation of density estimates from the former to the latter. Second, the perpendicular distance of animal locations to the survey transects is assumed to be uniform (within-transect scale). We use π(*y*) to denote the probability density distribution of an independent and identically distributed random variable *y* and π(*y*_{i}) to denote the probability density at a particular distance *y*_{i.} If π(*y*) is uniform, then any drop-off in the expected number of animals detected at greater distances is due to declining detectability, rather than changes in animal density.

Both assumptions above may not hold for distance sampling-type data collected from non-randomly located transects or paths. Violations of the assumption that the surveyed area is not a random sample of the study area can be accounted for by modelling population density as a spatially explicit function of habitat covariates (Hedley & Buckland 2004; Williams, Hedley & Hammond 2006; Johnson, Laake & Ver Hoef 2010). Without transect randomisation, however, accounting for bias caused by violations of the assumption that π(*y*) is uniform can pose a challenge (Johnson, Laake & Ver Hoef 2010; Marques *et al*. 2010). Violation of the uniformity assumption will commonly occur in surveys conducted from easily navigable permanent paths that either indirectly (e.g. via association with habitat variables) or directly (e.g. road avoidance) affect animal behaviour. Without knowing how a species' utilisation varies with distance from transects, results will be biased (Johnson, Laake & Ver Hoef 2010); although see Marques *et al*. (2010) for an alternative approach to disentangling detectability from π(*y*). Therefore, from an analytical point of view, most CDS literature holds that randomisation of transect location is indeed necessary (Buckland *et al*. 2001, 2004).

Opportunistic surveys – i.e. those in which detections are recorded while observers are performing other tasks – have some important advantages over design-based surveys, suggesting a need for methods that can account for the bias associated with non-uniform π(*y*). These advantages include: (1) the relatively low-cost of acquiring distance data from paths already being traversed for other reasons (Williams, Hedley & Hammond 2006; Kiszka *et al*. 2007) or from easily navigable paths (Walsh & White 1999), (2) the relative ease of collecting opportunistic data long-term (Kiszka *et al*. 2007; Himes Boor & Small 2012), and (3) avoidance of the problem that, when studying extremely rare or elusive animals, resource limitations may prevent feasibly sized design-based studies from detecting enough animals for informative inference.

Opportunistic surveys may also be preferable when counting rare events, such as sighting rare or elusive animals, or finding carcasses of even common animals. In the latter case, carcasses may be detectable for such short durations that an unfeasibly large design-based survey would be necessary to detect enough carcasses for adequate precision. Yet opportunistically sighted carcasses are often recorded in long-term data sets, often along with cause of death, thereby enhancing our understanding of a species' mortality dynamics. In this manuscript, we extend CDS through the development of new methods to incorporate estimates of non-uniform π(*y*) from auxiliary Global Positioning System (GPS) movement data in distance sampling estimators. In addition, we also extend CDS to incorporate data on rates at which scavengers dispose of carcasses into estimates of mortality rates. If carcasses are quickly removed from the environment (i.e. by decay or consumption), a smaller proportion of carcasses will be detected. Consequently, researchers have paid much attention to carcass removal rates when estimating wildlife mortality due to wind farms (Smallwood *et al*. 2010), roads (Santos, Carvalho & Mira 2011), pesticides (Rivera-Milán, Zaccagnini & Canavelli 2009) and power lines (Ponce *et al*. 2010). Using the ‘multiplier’ approach (Buckland *et al*. 2001; Buckland *et al*. 2004), one can then estimate the mortality rate by dividing the estimated carcass abundance (from distance sampling analysis) by the estimated duration for which a carcass is detectable (equivalent to multiplying by the estimated removal rate), taking care to incorporate variance due to the latter in the former (Plumptre 2000).

The above ‘multiplier’ method, however, is invalid when carcasses have multiple chances to be detected, but can only be detected once – i.e. during opportunistic surveillance when multiple trips may be made past a carcass, but communication amongst researchers ensures no double sampling. In such situations, detections are conditional on previous non-detection. The probability of detecting a carcass on one of the several trips is a function of the probability the carcass was available for detection at each trip (and hence on the removal rate), making the detection probability of each carcass a nonlinear function of the number of trips past that carcass, the interval between trips and the removal rate. Thus, our second extension to CDS is the explicit inclusion of detection and removal as competing processes within distance sampling estimators.

Finally, in some systems ‘removal rates’, which implicitly assume carcass removal to be a discrete event, may not be the relevant concept. For instance, detection of large terrestrial mammal carcasses occurs either by detection of the carcass itself or via detection of various scavenger species, each of which may be more or less detectable depending on size and capability for flight (e.g. large numbers of vultures in flight can be seen at great distances). Thus, carcass availability for detection depends on the probability of scavenger (i.e. sighting cue) presence as a function of time since carcass production (i.e. death). These probabilities will differ between scavenger species based on their abundance, search efficiency and niche partitioning (Hunter, Durant & Caro 2007). Hence, rather than modelling a ‘removal’ process, we model the sighting cue process itself. Specifically, we estimate π(*c*|*t*): the probability each sighting cue, *c*, is available as a function of time since death, *t*, from motion-sensor camera trap data on scavenger activity at carcasses.

In summary, we address several gaps in methods used to estimate cumulative mortality incidence from opportunistic surveillance data. As an example of the methods developed in this study, we estimate cumulative mortality during outbreaks of seasonally endemic anthrax in the plains zebra (*Equus quagga*) of Etosha National Park (ENP), Namibia. Using a hierarchical modelling framework (Royle & Dorazio 2008), we model carcass production, sighting cue availability and detection as concurrent dynamic processes. Our analysis explicitly accounts for surveillance effort by estimating mortality rates within surveilled space-time windows. We use bootstrap methods to incorporate error associated with estimation of π(*y*) and π(*c*|*t*) into the final incidence estimate. We first present a technical section extending distance sampling methods, as motivated above. We then introduce the ENP study system, focusing on the observational and ecological processes that play a role in producing the passive surveillance data to be analysed. We continue by using simulated carcass data to assess the accuracy and precision of the developed estimator, before applying it to estimate cumulative mortality in the surveilled region during an anthrax outbreak in ENP in 2010. Finally, we conclude with a discussion of the utility of these methods as well as suggest future directions.