Estimation of bird and bat mortality at wind-power farms with superpopulation models

Authors

  • Guillaume Péron,

    Corresponding author
    1. Patuxent Wildlife Research Center, U.S. Geological Survey, Laurel, MD, USA
    • Colorado Cooperative Fish and Wildlife Research Unit, Department of Fish, Wildlife, and Conservation Biology, 1484 Campus Delivery - Colorado State University, Fort Collins, CO, USA
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  • James E. Hines,

    1. Patuxent Wildlife Research Center, U.S. Geological Survey, Laurel, MD, USA
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  • James D. Nichols,

    1. Patuxent Wildlife Research Center, U.S. Geological Survey, Laurel, MD, USA
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  • William L. Kendall,

    1. USGS Colorado Cooperative Fish and Wildlife Research Unit, Department of Fish, Wildlife, and Conservation Biology, 1484 Campus Delivery – Colorado State University, Fort Collins, CO, USA
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  • Kimberly A. Peters,

    1. New Jersey Audubon Society, Cape May Bird Observatory, Cape May Court House, NJ, USA
    Current affiliation:
    1. Massachusetts Audubon Society, Lincoln, MA, USA
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  • David S. Mizrahi

    1. New Jersey Audubon Society, Cape May Bird Observatory, Cape May Court House, NJ, USA
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Correspondence author. E-mail: peron_guillaume@yahoo.fr

Summary

  1. Collision of birds and bats with turbines in utility-scale wind farms is an increasing cause of concern.
  2. Carcass counts conducted to quantify the ‘take’ of protected species need to be corrected for carcass persistence probability (removal by scavengers and decay) and detection probability (searcher efficiency). These probabilities may vary with time since death, because of intrinsic changes in carcass properties with age and of heterogeneity (preferential removal of easy-to-detect carcasses).
  3. In this article, we describe the use of superpopulation capture–recapture models to perform the required corrections to the raw count data. We review how to make such models age specific and how to combine trial experiments with carcass searches in order to accommodate the fact that carcasses are stationary (which affects the detection process).
  4. We derive information about optimal sampling design (proportion of the turbines to sample, number of sampling occasions, interval between sampling occasions) and use simulations to illustrate the expected precision of mortality estimates. We analyse data from a small wind farm in New Jersey, in which we find the estimated number of fatalities to be twice the number of carcasses found.
  5. Synthesis and applications. Our approach to estimation of wind farm mortality based on data from carcass surveys is flexible and can accommodate a range of different sampling designs and biological hypotheses. Resulting mortality estimates can be used (1) to quantify the required amount of compensation actions, (2) to inform mortality projections for proposed wind development sites and (3) to inform decisions about management of existing wind farms.

Introduction

The production of electricity from wind power is exponentially increasing world-wide (GWEC 2012). This renewable energy source does not come without environmental impacts, especially on wildlife and habitats (Drewitt & Langston 2006). In particular, we know that birds and bats are frequently killed by direct collision and barotrauma. One of the greatest concerns for stakeholders is that some of the affected species benefit from official protection. For example, in the United States, each and every death of a migratory bird caused by human activities is illegal in the absence of a permit (Migratory Bird Treaty Act of 1918). Golden and Bald eagles (Aquila chrysaetos and Hieraeetus leucocephala) are afforded additional protection under the Bald and Golden Eagle Protection Act. Incidental take permits can be obtained (e.g. USFWS 2011), but are often conditional on stakeholders monitoring, mitigating or compensating the take of protected species (or all of the above). Thus, one needs a reliable estimate of the number of animals killed.

The issue is not a formal and legal matter only. The numbers of fatalities reported at some locations (Arnett et al. 2008; Smallwood & Thelander 2008) raise, at least locally, serious concerns about the conservation status of impacted populations. Despite early reports that mortality at wind-power plants may on average be negligible compared to other sources of anthropogenic mortality (Erickson et al. 2002), more recent studies indicate that a deleterious effect is possible in some species, typically those for which even small changes in adult mortality have large consequences on population growth rate, and especially those that would not otherwise be impacted by anthropogenic mortality (Everaert & Stienen 2007; Martinez-Abrain et al. 2012). In some cases, most fatalities occurred at a small number of ‘problem turbines’ (Martinez-Abrain et al. 2012) or at particular times (e.g. weather events during migration nights: Kerns & Kerlinger 2004). Refining and quantifying our current understanding of factors responsible for variation in collision risk can thereby prove valuable, both to inform siting decisions for new wind farms and to fine-tune the operation of existing ones. Reliable estimates of the number of animals killed are necessary for this purpose too.

Our work was motivated by interactions with the New Jersey Audubon Society who undertook specific investigations of bird and bat mortality at a small wind farm (K.A. Peters, D.S. Mizrahi and Audubon Society, unpublished data). The present article includes an analysis of these data (‘Current study design and New Jersey application case’).

Raw carcass counts need to be corrected for (1) persistence probability (scavengers, decay, weather and tide may remove carcasses before surveys occur) and (2) detection probability (observers may miss some of the carcasses that are present during surveys) (Arnett et al. 2008; Smallwood et al. 2010; Huso 2011), in order to estimate the true number of fatalities. In the field of wildlife biology, capture–recapture protocols were introduced for this exact purpose of correcting animal count data for persistence and detection probabilities (Williams, Nichols & Conroy 2002). These models are flexible, and in particular, they can incorporate age-effects in persistence and detection probabilities. It is indeed expected that carcasses that are easy to detect for scavengers should be removed first, thereby inducing a variation in persistence and/or detection probabilities with time since death. Carcass age can also influence their intrinsic properties such that scavengers lose interest in old, dry or partly scavenged carcasses, and field crews have more difficulty finding them (K.A. Peters, D.S. Mizrahi and Audubon Society, unpublished data).

Our aim in this article is thus twofold. First, we review how ‘superpopulation’ capture–recapture models (Crosbie & Manly 1985; Schwarz & Arnason 1996) can be made age specific. We present two different methods: ‘multistate models’ and ‘regression models’. Then, we consider the fact that carcasses are stationary and field crews can record their positions. There is thereby little to no information about the detection process in the relocation data. To overcome this issue, field biologists often use trial experiments, in which externally obtained carcasses are planted and monitored (Smallwood et al. 2010; Barrientos et al. 2012; see also ‘Current study design and New Jersey application case’). We modify superpopulation models in order to simultaneously analyse data from carcass surveys and from trial experiments. Finally, an unaltered version of superpopulation models can also be fit if carcasses are marked upon initial detection and a multiple-observer sampling protocol is used to deal with detection probability. In the last section, we look for the optimal sampling design of this type of study.

Materials and methods

Multistate model

A first method to incorporate age-effects into superpopulation models is to define age-classes to characterize how long a carcass has been on the ground. Hereafter, we consider two such carcasses. The model is then based on a classes, or states, representing, for example, fresh and dry Markovian, stochastic transition between these two states (Hestbeck, Nichols & Malecki 1991). The fieldwork follows a standard multistate capture–recapture protocol with T sampling occasions (Hestbeck, Nichols & Malecki 1991). We adopt the maximum-likelihood approach and thus describe the model in terms of sufficient statistics (see below). Model parameters are presented in Table 1. An alternative and increasingly used approach is to describe capture–recapture models in terms of fundamental processes (e.g. Schofield & Barker 2008), which render their translation into Bayesian software (WinBUGS) programming code more straightforward (Bonner & Schwarz 2006; Royle 2009; Schofield & Barker 2009; Link & Barker 2010). Of interest is the superpopulation size NTot, defined as the number of distinct carcasses that are available during at least one of the sampling occasions:

display math(eqn 1)
Table 1. Multistate model: model parameters and data to be collected (statistics)
Model parameters
inline image The number of carcasses of age 1 already on the ground immediately before the first sampling occasion
inline image The number of carcasses of age 2 already on the ground immediately before the first sampling occasion
inline image The number of new carcasses that appear between sampling occasions i and + 1 and that are still available for detection at time + 1. = 1,…,T−1.
inline image Carcass persistence probability in age-class 1: the probability that a carcass of age 1 on the ground at time i will remain available for detection until time + 1. = 1,…,T−1.
inline image The probability that a carcass of age 2 on the ground at time i will stay until time + 1
inline image The probability that a carcass of age 1 on the ground at time i and that persists until time + 1 enters age-class 2 before time + 1. = 1,…,T−1. Reverse transition from 2 to 1 is impossible.
inline image The probability that a carcass of age 1 on the ground at time i is detected then. = 1,…,T
inline image The probability that a carcass of age 2 on the ground at time i is detected then.
Statistics
inline image the number of unmarked carcasses first detected when of age 1 at time i. = 1,…,T
inline image the number of unmarked carcasses first detected when of age 2 at time i
inline image the number of marked carcasses detected when of age 1 at time i that were last detected as carcasses of age 1 at time j
inline image the number of marked carcasses detected when of age 2 at time i that were last detected as carcasses of age 1 at time j
inline image the number of marked carcasses detected when of age 2 at time i that were last detected as carcasses of age 2 at time j

Likelihood of the numbers of unmarked carcasses

The probability for a carcass to enter in age-class 1 vs. in age-class 2 is akin to the initial proportion parameter of multievent models (Pradel 2009), but the only possibility to enter the system in age-class 2 is at the first sampling occasion. In other words, we consider the possibility that a carcass becomes dry before being detected, but we exclude the possibility that this event occurs within a single time interval. The probability to enter in age-class 1, denoted π, is thus defined by inline image . Similarly, and following the standard formulation of superpopulation models, we introduce entry probabilities inline image defined by inline image. inline image is, among carcasses that enter the superpopulation in age-class 1, the fraction that enters the system (individual dies) between i and + 1 (= 0,…,T−1). Following Schwarz & Arnason (1996), the inline image are actual model parameters to be estimated, and the N's are estimated by backward transformation. Last, we define the intermediate quantities inline image corresponding to the probability that a carcass enters the system in age-class s before time i, is still present and in age-class s at time i and is not seen before time i (= 1,…,T), by the following recurrence relations:

display math(eqn 2)

Then, a multinomial distribution describes the u-statistics:

display math(eqn 3)

Likelihood of the recapture data

Following Brownie et al. (1993), we define the ‘survival’ transition matrix:

display math(eqn 4)

And the detection and non-detection matrices:

display math(eqn 5)

Then, Bij(rs), the probability of being detected in age-class s at occasion j given that previous detection was in age-class r at occasion i, can be found for any combination (i,j,r,s) in the (r,s)th cell of matrix Bij:

display math(eqn 6)

[if j=+ 1, the product is replaced by the identity matrix, and if ji, the matrix is filled with 0s]

The m-statistics follow multinomial distributions, for example for the ith line:

display math(eqn 7)

Identifiability

In the full time-varying model, which counts 6T-2 parameters in our multistate application, some parameters occur in the likelihood only as combinations and are thus not separately estimable, that is, the model is not identifiable (Schwarz et al. 1993). This is usually dealt with by putting some constraints on parameters describing the first and last time steps (Schwarz et al. 1993). In other words, a few assumptions about time variation are needed for the model to be identifiable. Alternatively, trial experiments can be designed to estimate the first and last detection probabilities directly with no identifiability problem. In the simulations presented below, we bypassed these issues by making the model time-constant. We also improved identifiability by replacing inline image in the likelihood by its estimator inline image .

Post hoc correction for early removals

NTot is the number of carcasses that are available for detection during at least one sampling occasion. However, scavengers can remove carcasses before the first sampling occasion following death; depending on the study design and persistence rate, these early removals can account for a large part of the total fatality number. Following Crosbie & Manly (1985) and Schwarz et al. (1993), it is here assumed that recruitment is uniformly distributed within each time interval. Then, the corrected estimate for the total number of fatalities occurring between the first and last searches is computed as:

display math(eqn 8)

where fU = 1 represents the density of the uniform distribution over one time interval. This correction is made post hoc using the maximum-likelihood estimates of model parameters, and the delta-method is used to compute sampling (co)variance. If recruitment is not uniformly distributed within time intervals (e.g. if most deaths occur on a particular night with exceptional weather or migration intensity), the corrected estimate can be seriously biased.

Simulation study

The multistate model was fitted to simulated data using the maximum-likelihood approach, in R (R Development Core Team, 2010) and with the optimization function nlm. We considered a few representative scenarios (Appendix S1, Supporting information part 1) but do not claim that this simulation study allows drawing conclusion for the rest of the parameter space.

Small samples

Fatalities involving protected species are expected to be rare, and thus, NTot will usually be small. Two issues arise when sample size is small: imprecision of the estimates (small sample bias and instability of the numerical routine) and numerous zero detection events, that is, sampling occasions during which no new, unmarked carcass is detected. The first issue can be partially addressed by adding experimentally planted carcasses to the sample of naturally occurring ones. The second issue can lead to what is commonly referred to as the boundary parameter issue in capture–recapture models (Lebreton et al. 1992). To investigate whether this induced a bias on the estimation of NTot, we ran a second simulation analysis. We considered only one age-class, and a small sample of 12 carcasses, which entered in a staggered fashion over nine sampling occasions: inline image . Carcass persistence was 0·8, and carcass detection was 0·5. This scenario was expected to loosely mimic a context of raptor management, with crew members searching an extensive area every 2 weeks, for example. We simulated 50 data sets under this scenario.

Regression model

In this section, we assume a direct relationship between carcass age and model parameters, that is, deterministic instead of stochastic transitions between age-classes, with no need for onsite technicians to document the estimated age of the carcasses. Such regression models are a modification of the models developed by Kendall (2006) and Pledger et al. (2009). The modification consists in introducing a distribution for the age of carcasses present at the start of the study, whereas typical applications assume all individuals enter the system after or immediately before the start of the study (Matechou E., Pledger S., Efford M., Morgan B.J.T. and Thomson D.L., unpublished data). The likelihood is individually based. Model parameters are similar to the multistate model, except for the full age specificity (Table 2). In the following, the age of a carcass is the number of occasions it has been present on site (constant time interval between occasions).

Table 2. Regression model: notations for model parameters and data to be collected
Model parameters
inline image The total number of carcasses that are available to capture during at least one of the sampling occasions.
inline image Entry probability: the proportion from the N carcasses that enters the system (individuals die) between sample occasions i and + 1; = 0,…,T−1. = 0 represents carcasses already present at the first sampling occasion.
inline image The probability that a carcass had entered the system x time periods before the start of the study inline image In practice, we set a maximum age: x ≤ X.
inline image Carcass persistence probability: the probability that a carcass of age a on the ground at time i persists until time + 1. = 1,…,T−1. inline image In practice, we set a maximum age: inline image
inline image Carcass detection probability: the probability that a carcass of age a on the ground at time i is detected at that time. = 1,…,T.
Data collected from naturally occurring carcasses
inline image Detection history for kth individual carcass: ωk,i = 1 if the carcass is detected at sampling occasion i, and ωk,i = 0 if not detected.
inline image The first sampling occasion at which carcass k is detected.
inline image The last sampling occasion at which carcass k is detected.
C The total number of detected carcasses

Probability of detection histories

For each detection history ωk, the probability of occurrence is the sum of the probabilities conditional on two disjoint events: entry before or during the study. To compute the former, we define negative time occasions, so that time occasion –x is situated x + 1 time intervals before the start of the study.

display math(eqn 9)

where X represents the maximum age of the carcasses that are present at first sampling occasion, arrival (k) is the first occasion after the entry of the kth carcass, and β′x denotes the probability that a carcass enters the study area between occasions –x and –+ 1, and survives up to occasion 1. The conditional probabilities are then expressed as functions of persistence and detection probabilities:

display math(eqn 10)

and:

display math(eqn 11)

[In equations 10 and 11, the products are replaced by a one when the end index is lower than the start index].

To compute parameters β′x, we assumed that the entry rate was uniform before the first sampling occasion. We further assumed that age-specific persistence before the first sampling occasion could be approximated by the temporal average of age-specific persistence after the first occasion, which we denoted inline image. Then, we could write:

display math(eqn 12)

with inline image (0) = 1.

Probability of missing a carcass

Let P0 be the probability of missing a carcass given that it was available during at least one occasion. As in detected carcasses, P0 is the sum of the probabilities conditional on two disjoint events: entry before or during the study.

display math(eqn 13)

The conditional probabilities are expressed as functions of persistence and detection probabilities:

display math(eqn 14)

and:

display math(eqn 15)

[In equations 14 and 15, the products are replaced by a one when the end index is lower than the start index].

Identifiability

In this model, the age specificity in persistence and detection probabilities is used to make inference about the ages of the carcasses at first detection. This model is thus similar to a mixture model with as many heterogeneity classes as there are possible ages at first detection (Pradel 2009; Pledger, Pollock & Norris 2010), and is therefore identifiable. However, in practice, low sample size and lack of strong age specificity lead to ‘plateaus’ in the likelihood, that is, large areas of the parameter space for which the likelihood hardly varies. Such ill-conditioning can make it difficult for the optimization procedure to find the maximum likelihood when that maximum is not different enough from surrounding values. Furthermore, if the assumption of uniform entry rate before first occasion is not met, then additional parameters are needed to describe the βx [equation 12], in which case the model becomes truly non-identifiable. These identifiability issues can be addressed by using known-age data, that is, carcasses of which the age at first detection is perfectly known. Trial experiments (planting fresh carcasses at known times and locations) provide such a data set. For each trial carcass, the recorded data are then: the first and last occasion the carcass was present, and its detection history inline image (detections by a person different from the one that planted the carcass). Then, the probability of observing history inline image is as follows:

display math(eqn 16)

where inline image and inline image are the first and last sampling occasions a fake carcass is present on site.

If C naturally occurring carcasses have been found and inline image fake ones have been planted, and under the usual assumption of independence of fate, the joint likelihood is as follows:

display math(eqn 17)

where θ is a vector containing all model parameters, including NTot.

Simulation study

In our simulations (Appendix S1, Supporting information part 2), we used the three-parameter logit-quadratic function (equation 18): first persistence probability may increase with age (representing the activity of scavengers, heterogeneity and drying) and then it may decrease with age (decay).

display math(eqn 18)

where, A, B, Ci are parameters to estimate. We note that in actual estimation situations, this parametric form can yield flawed results by constraining symmetry on the logit scale (i.e. when the persistence probability increases at young ages but does not decrease afterwards, this model often nonetheless estimates a spurious decline at old, poorly documented ages). We did not consider age specificity in detection probability in our simulations.

Current study designs and New Jersey application case

Trial experiments are often organized to deal with the detection probability. Externally obtained carcasses are planted in the search area; the same technicians that would look for ‘naturally occurring’ carcasses report whether they found the trial carcasses. Trial carcasses are also used to document persistence rate (Smallwood et al. 2010; Huso 2011; Barrientos et al. 2012). The typical study design that combines carcass searches and trial experiments, however, differs from the situations covered in the previous two sections: (1) naturally occurring carcasses are not marked and left where found but are instead permanently removed upon detection, (2) trial experiments and carcass searches are not always simultaneous, and (3) carcasses are not assigned to an age-class. We describe below a modified version of the superpopulation model that combines both types of data, at the cost of a model simplification. This model is available in R-package carcassCMR (Appendix S3, Supporting information).

Carcass surveys

Removing the age specificity (superscripts) from the multistate model, but otherwise maintaining the same notation, we write for T:

display math(eqn 19)

Then, a multinomial distribution describes the u-statistics:

display math(eqn 20)

Detection trials

Detection trials consist of planting carcasses (or any item of which the detection probability is similar to that of a carcass) prior to a scheduled search. Noting Di the number of planted items at time i, and di the number of those items detected during the carcass search, detection probability pi is then estimated following:

display math(eqn 21)

Survival trials

Planted carcasses are then monitored by a crew member who knows where they are. In the following, we note inline image the number of carcasses still present at time + 1 (the number present at time i being inline image ). Survival probability inline image is then estimated following:

display math(eqn 22)

The joint likelihood is the product of the components described in equations (20-22).

New Jersey application case

The study was conducted by the New Jersey Audubon Society at the Jersey Atlantic Wind, LLC/Atlantic City Utilities Authority wind power facility, near Atlantic City, NJ (39°22′N, 74°27′W). There were five turbines. Below each turbine, we defined a search plot as a square with sides of 130 m. Only 24–85% of each search plot was searchable; however, the rest consisting of inaccessible rooftops, clarifying and aerating pools, and tidally influenced marsh areas. The superpopulation thereby corresponded to carcasses falling within the searchable area. The latter did not vary through time. For this study, we focused on the months of August and September 2007 (post-breeding migration period). This included a total of = 23 sampling occasions separated by intervals ranging from 2 to 5 days. We selected only the data from bats: Hoary bat Lasiurus cinereus and Eastern red bat Lasiurus borealis, which we pooled together.

Trial experiments were conducted between July 2007 and August 2010 by placing a total of 57 small bird or bat carcasses (<35 g) in the searchable area within 24 h of a scheduled search (with a maximum of one trial per month and two trial carcasses per turbine and trial). Forty seven of these carcasses persisted until the next scheduled search. The detection trial consisted of recording whether carcasses were detected during that scheduled search. The persistence trial consisted of visiting carcass locations every day for 7 days, to monitor their continued presence or their removal.

Detection and daily persistence probabilities were assumed constant through time, similar between bats and birds of similar size and independent of carcass age. The first two assumptions were required in order to use trial data in lieu of capture–recapture data. We considered three models with different distributions of carcass entries through time: (1) full time dependence, (2) inline image depends logit-linearly on the length of the interval between occasions i and + 1 and (3) linear increase with time on the logit scale. The varying length of the intervals between sampling occasions was accommodated by introducing a daily survival probability and raising it to the appropriate power.

Sampling protocols with marking of carcasses

In the first two sections, we described models for which carcasses are marked upon detection and left where found to be subjected to the detection process again. Currently, those models are deemed inapplicable because of the stationary nature of carcasses (‘Current study design and New Jersey application case’). Yet, the loss of information about carcass location between successive sampling occasions can be achieved by dividing a large survey area into multiple subunits and assigning crew members to different units at each occasion, so that no crew member surveys the same sample unit twice in a row. Alternatively, one may use a double observer protocol, where the efficiency of the second observer is estimated using the carcasses found by the first observer, and reciprocally (e.g. Nichols et al. 2000).

Various methods can be used to mark naturally occurring carcasses, from sticker notes to passive integrated transponders (PIT tags), individual alphanumeric tags and geo-location. We also note that because the likelihood of the multistate model is based on summary statisticsinline image , it is not strictly necessary that carcasses are individually marked. ‘Batch marks’ may be used, that is, time-specific colour marks (spray paint, wool threads, etc.) can be applied to carcasses upon detection. One consideration for any form of marking is that the mark should not draw the attention of the crew members to the marked carcasses.

Four variables then describe the sampling effort: (1) the proportion of turbines that are monitored, denoted K; (2) the number of sampling occasions, denoted T, that is, the number of times field crews visit the search area; (3) the interval between successive sampling occasions, denoted I; and 4) optionally, the number of trial carcasses used to boost sample size and document persistence and detection probabilities, denoted F. For a fixed daily removal rate by scavengers, persistence probability decreases exponentially with I.

We were then interested in the optimal design of a capture–recapture carcass survey, that is, the optimal combination of inline image values. For this purpose, we devised equation 23 to describe the utility of a carcass survey: utility increases with K because small K's induce a risk of sampling a non-representative set of turbines; utility increases with study duration I × T because short durations induce a risk of sampling a non-representative period (e.g. miss migration peak); utility increases with diminishing bias (in our case bias is linked to the cross multiplication needed to extrapolate from the sampled turbines to the whole farm); and utility diminishes with increasing standard error of the estimate for superpopulation size (Mackenzie & Royle 2005).

display math(eqn 23)

where inline image was the expected bias (in absolute proportional value), inline image was the expected coefficient of variation of the estimate of fatality number, and u1, u2 and u3 were parameters of which the values depend on the objectives of the stakeholders.

We then defined the components of the cost function:

display math(eqn 24)

where c0 was the fixed overhead component, c1 the price of visiting the wind farm once, c2 the price of searching under one additional turbine once technicians were on site, and c3 the price of acquiring and deploying one trial carcass. We assumed that there was no additional cost when increasing the interval between sampling occasions.

We focused on the optimal study design for the following set of three constraints: (1) Fixed cost: we wanted to find the optimal allocation of resources that maximized utility for a given budget. (2) Fixed study duration: we assumed that among the requirements and specifications was the fact that the study had to document fatality number over a predetermined period of time, for example one migratory season. (3) Fixed number of trial carcasses: we assumed that the number of trial carcasses was constrained by factors other than cost. Under these three constraints, there was a single set of K, T, I and F values that maximized utility. For illustration purposes, we computed this set of optimal values for c= 10, c= 10, c= 1, c= 0 (since F was fixed), u= 0 (since duration was fixed), u= 1 and u= 10. Study duration was fixed to × = 90 days and number of trial carcasses to = 100 carcasses.

We needed an explicit function describing inline image and inline image as a function of K, T, I F. Our model was, however, deemed too complex for an analytic approach, in particular because the analytic computation of inline image would be based on asymptotic variances, whereas in the situations of interest here, sample sizes were expected to be low (Carother 1973; Crosbie & Manly 1985). Therefore, we conducted a series of simulations in which we fixed K, T, I, F, and directly estimated inline image and inline image from these data sets, and then extrapolated to obtain a continuous function. Simulations were conducted under the following scenario: there were 100 turbines; one animal died each day; the turbine causing death was selected at random (uniform distribution of carcasses); the carcasses had a constant 5% daily probability of being removed by scavengers (inline image = 0·95, resulting in expected persistence time of 20 days); at each of T equally spaced sampling occasions, the carcasses had a 70% probability of being detected (= 0·70). K was allowed to vary from 5 to 95%, T from 3 to 15 occasions, I from 1 to 90 days and F from 10 to 1000 carcasses. We simulated and analysed 20,736 data sets. We then investigated the dependence of inline image and inline image on K, T, I, F using linear models (generalized additive models indicated that nonlinearities were weak once explanatory variables K, T, I, F were log-transformed).

Results

Multistate model

Simulation study

The results are presented in full in Appendix S1, part 1 (Supporting information). Even in a data-rich system root-mean-square error reached 12% if detection probability was low. If neglecting the age-structure, there was a systematic bias on the estimate for NTot, which depended mostly on the sign of the difference p(2)p(1). The difference φ(2)−φ(1) and the value of µ(12) had no apparent influence on the bias, but they affected the root-mean-square error: positive φ(2)−φ(1) and high µ(12) increased the risk of error (Appendix S1, Supporting information: Table S1).

Small samples

(1) A zero detection event always led to the estimation of a zero entry probability. Some of these zero estimates occurred during intervals with non-zero mortality. (2) Sometimes, even when no mortality occurred during a given interval, carcasses were detected at the end of it which had been missed during previous occasions. Entry probabilities could then be estimated at non-zero values when the actual value was zero (44% of cases for interval #7 in our simulations). (3) However, there was no systematic bias on inline image (average bias below 0·001%). Thus, even if the phenology was not well estimated, the estimate for the total number of carcasses over the study period was approximately unbiased.

Regression model

The results of the simulations are in Appendix S1 (Supporting information), part 2. Adding more trial carcasses did not markedly reduce root-mean square error or bias (probably because we simulated a situation with already a lot of naturally occurring carcasses). Root-mean square error could reach 40% when detection probability was low.

New Jersey application case

The AIC of the model with entry rate depending on interval length was lowest: AIC = 141·2, vs. 165·5 for the model with full time dependence and 143·3 for the model with logit-linear variation; the latter estimated zero entries before the first occasion and constant entry rate afterwards. In the full time-varying model, all occasions during which no new carcass was detected yielded an estimated zero entry probability (as expected; see ‘Small samples’ above). Daily persistence probability was around 80%. The probability for a carcass to persist over one interval ranged from 33% to 65% depending on interval length. The estimates of N* were >2 times larger than the number of carcasses actually detected during surveys (Table 3), despite the fact that surveys were conducted every 2–5 days.

Table 3. Parameter estimates for the New Jersey application case (2007 fall migration). Total number of observed carcasses in this study was 27. NTot is the number of bat carcasses present during at least one occasion. N*, the total number of fatalities, is estimated using equation 5. φ is the daily carcass persistence probability. P is the detection probability. Standard errors (SE) are computed using the delta-method
  inline image N* inline image P
Est.SEEst.SEEst.SEEst.SE
Model with entry probability depending on interval duration55·510·476·412·30·750·020·310·07
Model with fully time-dependent entry probability46·49·262·615·40·810·030·350·07

Optimal sampling design with marking of carcasses

The average coefficient of variation for the estimate of fatality number was 10·6% with a standard deviation of 6·9 (over all simulated data sets). The effects of sampling design variables K, T, I and F on the coefficient of variation were in the directions expected (higher CV for low K, low T, high I and low F), but interactions between variables had to be taken into account (Appendix S2, Supporting information: Table S3).

The average proportional bias (in true not absolute value) over all simulated data sets was −0·8% (as expected since the method was unbiased), but with a standard deviation on the estimate for carcass number of 17·8%. There was structure in the estimated bias, indicating that the probability of a large bias (in absolute value) increased for low K, low T, high I and low F. As for the coefficient of variation, interactions between variables had to be taken into account (Appendix S2, Supporting information: Table S4).

When the budget was low, the optimal study design was to sample about one-third of the turbines, with the minimum number of occasions and intervals between occasions that exceeded expected carcass persistence time (Table 4: line ‘Cost = 175’). As the budget increased, the optimal number of sampling occasions increased from about 3–5, and the interval between occasions decreased from about 30–20 days (20 days being the expected persistence time), while the proportion of turbines to sample increased more moderately from about 38 to 45% (Table 4). Then, there was a budget threshold after which the optimal percentage of turbines to sample became 100%; this was initially traded against fewer occasions and longer intervals (Table 4: line ‘Cost = 350’). Then, as budget increased above the threshold, the optimal number of occasions increased again and optimal interval decreased.

Table 4. Optimal study design for various fictional budget allowances, when study duration is fixed to 90 days and number of trial carcasses is fixed to 100 carcasses. The cost and utility functions used for this example are described in equations 23 and 24. K is the proportion of monitored turbines (in %), T is the number of sampling occasions, and I is the interval in days between sampling occasions. Daily persistence probability was 0·95 (expected persistence time: 20 days) and detection probability was 0·75 at each occasion
BudgetOptimal KOptimal TOptimal I
17538·03·227·9
20041·33·525·7
25044·74·221·4
30045·85·017·9
3501003·030·0
4001003·526·1
4501003·923·0
5001004·420·7
10001008·910·1

Discussion

Our method was motivated by (1) the fact that the problems encountered by carcass searchers are exactly those that led biostatisticians to develop capture–recapture models to estimate animal population size (Williams, Nichols & Conroy 2002) and (2) the fact that many, if not most, carcass surveys comprise several sampling occasions. In addition, compared to other approaches (Huso 2011; Korner-Nievergelt et al. 2011), our approach has the potential to provide estimates that are less biased because time variation and age variation can be included in all parameters, including entry probabilities, and because we accommodate the fact that carcasses may persist undetected for several intervals, and thus, detected numbers at any given occasion may be smaller or larger than the numbers that died during the interval immediately preceding the occasion (which is sometimes referred to as ‘bleed-through’). Indeed, in the New Jersey application case, the probability to persist over one time interval was always more than 30% and often around 65%. This constituted a strong violation of the assumptions of Huso (2011) and would have induced a strong overestimation of fatality numbers by the latter model. In Table 4, we also show that the optimal sampling design is generally one in which carcasses have a non-negligible probability to persist more than one time interval (20–60% depending on budget).

Whether or not ancillary trial experiments need to be conducted depends on the specifics of the study at stake. They are required, for example, if, for legal or other reasons, ‘naturally occurring’ carcasses need to be removed and discarded upon detection. If one person conducts all field operations for the duration of the survey, then they are also needed to estimate detection probability. In any other situation, they are not strictly necessary, but they constitute a potentially necessary sample size booster. Importantly, trial carcasses can be used to estimate P0, the probability of missing naturally occurring carcasses, even if no such naturally occurring carcass is recorded during the surveys (M. Huso et al., unpublished data). Whenever possible, we suggest that trial experiments be conducted at the times and locations of carcass searches, so that trial data can be used to estimate time specificity of persistence and detection parameters. In conclusion, our basic superpopulation capture–recapture approach can be tailored to the specifics of most kinds of survey protocols.

We believe that the method meets various existing needs for estimates of bird and bat mortality in wind farms. First, the permitting process for new wind farm sites in the United States and elsewhere requires projections of the expected number of deaths of threatened and endangered birds and bats. The projection models are informed by surveys of bird and bat activity at the proposed sites, and by fatality rate estimated at wind farms with similar characteristics. Our method will allow those fatality rate estimates to be produced. Once ‘take’ of endangered species in the United States is permitted, compensatory mitigation activities (e.g. habitat protection or management, power line retrofitting) may be mandated. Estimates of take are clearly required in order to establish the appropriate level of mitigation. Finally, the actual operation of turbines at existing wind farms may be modified in order to minimize bird and bat mortality. For example, turbine speeds and even times of operation may be adapted. Estimation of the functional relationships between these operational control variables and focal species mortalities will also be made possible by the method proposed herein. In summary, we believe that the models developed here will be useful for informing a wide variety of questions and management decisions about wind farm mortality of birds and bats.

Acknowledgements

We thank R. Barker and S. Pledger for reviewing a previous draft of this manuscript and M. Huso for discussing carcass surveys with us.

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