the curtailed turbines were operating only under relatively high wind speeds.
We can analyze the data from the two turbine sets separately to see if there
are differences in the distance distributions and consequent $\hat{\psi}$.

The subset function from the dwp package can be used to split the layout_shape
data set into two parts: freely operating turbines (layout_free) and turbines
curtailed at low wind speeds (layout_curtailed). The two data sets can then be
analyzed separately and the results compared:

######## basic setup for freely operating turbines:

# data formatting

layout_free <- subset(layout_shape, select = "Turbine",

subset = paste0("t", c(1, 3, 4, 8, 11, 13, 14, 16, 20, 23)))

cod_free <- subset(cod, select = "Turbine",

subset = paste0("t", c(1, 3, 4, 8, 11, 13, 14, 16, 20, 23)))

rings_free <- prepRing(layout_free, scVar = "Class", notSearched = "Out",
silent = T)

rings_free <- addCarcass(cod_free, data_ring = rings_free, plotLayout =
layout_free)

# fitting the models

mod_free <- ddFit(rings_free, scVar = "Class")

#> Extensible models:

#> xep1

#> xep01

#> xep2

#> xep02

#> xep12

#> tnormal

#> MaxwellBoltzmann

#> lognormal

#>

#> Non-extensible models:

#> xep012

#> xep123

#> xep0123

#> constant

######## basic setup for curtailed turbines:

layout_curtailed <- subset(layout_shape, select = "Turbine",

subset = paste0("t", c(2, 5, 6, 7, 9, 10, 12, 15, 17, 18, 19, 21)))

cod_curtailed <- subset(cod, select = "Turbine",

subset = paste0("t", c(2, 5, 6, 7, 9, 10, 12, 15, 17, 18, 19, 21)))

rings_curtailed <- prepRing(layout_curtailed, scVar = "Class", notSearched =
"Out",

silent = TRUE) # "silent" cancels updates on the status of the calculations

rings_curtailed <- addCarcass(cod_curtailed, data_ring = rings_curtailed,

plotLayout = layout_curtailed)

mod_curtailed <- ddFit(rings_curtailed, scVar = "Class")

#> Extensible models:

#> xep1

#> xep01

#> xep2

#> xep02

#> xep12

#> xep012

#> tnormal

#> MaxwellBoltzmann

#> lognormal

#>

#> Non-extensible models:

#> xep123

#> xep0123

#> constant

The descriptive statistics for the fitted models can be summarized using the
function stats, which calculates estimated medians, modes, and the 75th, 90th,
and 95th percentiles according to models, and sorts the results by
$\Delta\textrm{AICc}$.

stats(mod_free)

#> median 75% 90% 95% mode p_win deltaAICc

#> xep01 30.6 44.3 59.5 70.1 23.5 0.985 0.00

#> xep2 30.6 43.3 55.9 63.7 26.0 0.996 0.64

#> xep02 30.4 41.9 53.2 60.2 26.8 0.999 1.51

#> xep12 30.5 42.3 53.6 60.7 27.0 0.998 2.35

#> lognormal 35.6 59.1 93.2 122.5 20.3 0.884 2.51

#> MaxwellBoltzmann 30.1 39.7 49.0 54.8 27.7 1.000 5.10

#> xep1 36.8 59.1 85.4 104.1 21.9 0.907 6.41

#> tnormal 30.8 41.3 50.8 56.5 30.3 1.000 9.61

stats(mod_curtailed)

#> median 75% 90% 95% mode p_win deltaAICc

#> MaxwellBoltzmann 35.0 46.1 56.9 63.6 32.2 0.998 0.00

#> xep12 35.7 47.2 58.1 64.7 33.6 0.997 1.02

#> xep02 36.0 48.6 60.9 68.6 32.3 0.993 1.06

#> xep2 38.6 54.5 70.3 80.2 32.8 0.969 1.09

#> xep01 38.8 55.9 74.9 87.9 30.1 0.946 1.95

#> tnormal 35.5 45.7 55.0 60.5 35.3 1.000 2.66

#> xep012 35.7 47.6 59.0 66.0 33.1 0.996 3.06

#> xep1 60.1 96.5 139.4 170.0 35.8 0.695 4.47

#> lognormal 64.2 116.5 199.2 274.6 29.4 0.632 4.93

In general, the carcasses at the curtailed turbines did tend to be somewhat
more distant from turbines than the carcasses at the freely operating turbines
as expected (fig. 24).. The magnitude of the estimated differences strongly
depended on the selected model (fig. 23A).The differences among the models
largely vanish when the models are restricted solely to the range of the data
and not extrapolated beyond the search radius (fig. 23B). This indicates that
the discrepancies among models are primarily due to the extrapolation beyond
the search radius and the models’ respective assumptions about the
distribution beyond the search radius, that is, in the tails of the
distributions. The heavy-tailed distributions (lognormal and xep1), predict a
relatively large fraction of carcasses beyond the search radius, while the
light-tailed distributions (truncated normal and Maxwell-Boltzmann) predict
relatively few carcasses beyond the search radius.

Figure 23: Comparison of descriptive statistics for the distance distributions
for the curtailed turbines (top row of dots in each quantile) and the
curtailed turbines (bottom row of dots in each quantile). The lines show the
difference between curtailed and free turbines. A. Estimated quantiles of
carcass distances (including extrapolations to account for carcasses beyond
the search radius); B. estimated quantiles of carcass distances solely within
the search radius.

Because the search radius is not long (square with half-width of 60 m or
diagonal radius of 82 m), one of the heavier-tailed distributions with degree
less than 2 is preferred (xep1, xep01, or lognormal). The lighter-tailed
distributions (Maxwell-Boltzmann and the xep models with a 2 or 3) tend to
converge rapidly to zero shortly beyond the most distant carcass and
underestimate the total outside the searched area. In both the curtailed and
free data sets, the xep01 (gamma distribution) was the highest-ranking,
relatively heavy-tailed distribution and had $\Delta AICc<2$. Thus, the xep01
distribution is an appropriate choice for both types of turbine.

Figure 24: Cumulative xep01 and xep2 distribution functions of carcass
distances at curtailed and free turbines at the Casselman site. Curves show
the maximum likelihood esimator. Boxplots show the interquartile ranges,
medians, and 90% CIs for estimated distances required to cover 10%, 20%, …,
90% of the carcasses.

The estimated probabilities of carcasses lying in the searched areas can be
calculated separately for the two sets of turbines (freely operating or
curtailed):

# estimated spatial coverage for curtailed turbines...

# ...using distance model fit to data from free turbines

psi_free <- estpsi(rings_free, model = mod_free["xep01"])

# ...using distance models fit to data from curtailed turbines

psi_curtailed <- estpsi(rings_curtailed, model = mod_curtailed["xep01"])

The estimated $\psi$ and $dwp$ for the free and curtailed turbines depends on
the configuration of the search plots, which vary by turbine, so the values
are not strictly comparable. However, a few meaningful patterns can be
discerned (fig. 25). First, the lower tails of the distributions for
$\hat{\psi}$ extend much lower for the curtailed turbines than for the free
turbines. This is due to the presence of more carcasses at greater distances
at the curtailed turbines and a smaller probability that the tail of the
distribution is resolved enough to avoid the possibility that many carcasses
lie beyond the search radius. Next, the uncertainty introduced in estimating
$dwp$ erases the clarity of the distinction between curtailed and free
turbines.

Figure 25: Boxplots of $\hat{\psi}$ and $\widehat{dwp}$ according to the xep01
distribution for curtailed and free turbines at the Casselman site.

After calculating the $\widehat{dwp}$’s for curtailed and free turbines
separately, the estimates combined into one data frame for export for GenEst.

ge_combined <- rbind(formatGenEst(dwp_curtailed), formatGenEst(dwp_free))

exportGenEst(ge_combined, file = "dwp_all.csv")

Although we choose the gamma distribution (xep01) for estimating $dwp$ for
export to GenEst, the Rayleigh distribution (xep2) provides an equally high-
quality fit and has the interesting and convenient property that the ratio of
the quantiles of two Rayleighs is equal to the ratio of their parameters
($\sigma_{1}/\sigma_{2}$), regardless of the quantile. Thus, measuring by the
fitted Rayleigh distribution, carcasses at the curtailed turbines were
$\sigma_{\textrm{curtailed}}/\sigma_{\textrm{free}}=1.258$ times more distant
from the turbines than were the carcasses at the free turbines. In other
words, the median, 90th percentile, 99th percentile, and so on were all 25.8%
greater at the curtailed turbines than at the free turbines.

# estimated spatial coverage for curtailed turbines...

# ...using distance model fit to data from free turbines

psi_free <- estpsi(rings_curtailed, model = mod_free["xep01"])

# ...using distance models fit to data from curtailed turbines

psi_curtailed <- estpsi(rings_curtailed, model = mod_curtailed["xep01"])

If not accounted for, the tendency for the carcasses to be more distant at the
curtailed turbines can result in substantial bias when estimating total
mortality. For example, using the distribution that was fit for the freely
operating turbines to estimate mortality at the curtailed turbines, we would
underestimate the mortality by some 55% on average at the curtailed turbines
due to systematic bias in the estimates of $\psi$ (fig. 26).

Figure 26: Comparison of estimates of the area correction factor
($1/\hat{\psi}$) at curtailed turbines using the correct distance model
(black) based on data from curtailed turbines versus using an incorrect
distance model (blue) derived from data from freely operating turbines. Boxes
show median and IQR. Whiskers show 95% and 99% CIs.

####  4.6.6 Carcass Size

Carcass size may also influence the distances that carcasses may fly, perhaps
even size differences among bats can influence the carcass dispersion patterns
around turbines. The Casselman data include a range of bat species of
different sizes, which we can class as either large (L) or small (S). The dwp
package provides tools for processing data sets with size class distinctions
and exporting the resulting dwp estimates to GenEst.

The following sections of code show how to split a data set on a carcass class
variable, fit separate distance models for each size class using a single
command, and create GenEst-ready dwp data with carcass class distinctions.

The first task with this data set is to add a “Size” variable to the carcass
data from the freely operating turbines, with LACI, LASE, and EPFU classed as
“L” and the other species as small.

cod_free$carcasses$Size <- ifelse(

cod_free$carcasses$Species %in% c("LACI", "LASE", "EPFU"), "L", "S")

It is always useful to check whether your command has the desired effect.
Tabulate the carcasses by size and species to verify that the split was done
correctly and that there are sufficient carcasses in each subset to do the
estimation.

table(cod_free$carcasses[, c("Species", "Size"), drop = TRUE])

#> Size

#> Species L S

#> EPFU 3 0

#> LABO 0 15

#> LACI 35 0

#> LANO 0 28

#> LASE 2 0

#> MYLU 0 18

#> PESU 0 12

NOTE: The drop = TRUE is necessary here because cod_free is an sf object,
which is a specially formatted data frame with GIS information attached. When
that extra information is attached, R does not recognize the sf object as a
data frame, so functions like table will not work. The GIS information is
dropped by using drop = TRUE in the subsetting. The result is a standard R
data frame that can be manipulated like any other data frame in R.

To estimate $\psi$ and calculate $\widehat{dwp}$ for use in GenEst when there
are distinctions among carcass classes, the $dwp$’s must be calculated for
each size class separately, which the dwp functions, estpsi and estdwp, can do
automatically if the data are properly formatted. Formatting involves adding
carcasses (feeding a carcass data set into addCarcass) to the description of
the site by rings (data_ring) with a directive to split the data sets by
carcass class (ccCol).

When a ccCol (carcass class column) is included in the argument list,
addCarcass creates separate ring structures for each carcass class, which in
turn are interpreted by dwp functions to estimate $\psi$, $dwp$, and format
the results for exporting to GenEst.

# split the ring data into separate structures for each size class

cod_free_size <- addCarcass(cod_free, data_ring = rings_free,

plotLayout = layout_free, ccCol = "Size")

# fit separate distance models to the different size classes:

mod_size <- ddFit(cod_free_size, scCol = "Class")

#> Extensible models:

#> xep1

#> xep01

#> xep2

#> xep02

#> xep12

#> xep012

#> xep123

#> xep0123

#> tnormal

#> MaxwellBoltzmann

#> lognormal

#>

#> Non-extensible models:

#> constant

#> Extensible models:

#> xep1

#> xep01

#> xep2

#> xep02

#> xep12

#> tnormal

#> MaxwellBoltzmann

#> lognormal

#>

#> Non-extensible models:

#> xep012

#> xep123

#> xep0123

#> constant

# view summary statistics for the resulting models:

stats(mod_size)

#> $L

#> median 75% 90% 95% mode p_win deltaAICc

#> xep2 28.4 40.1 51.7 59.0 24.1 0.999 0.00

#> xep1 30.8 49.4 71.4 87.1 18.4 0.951 0.22

#> xep12 28.4 41.8 55.5 64.4 21.8 0.994 1.66

#> xep01 29.0 44.4 62.1 74.6 19.7 0.976 1.72

#> xep02 28.4 41.0 53.5 61.4 23.3 0.997 1.78

#> lognormal 34.7 62.8 107.0 147.3 16.0 0.854 1.98

#> tnormal 28.7 40.4 51.2 57.7 27.1 1.000 3.26

#> xep123 28.3 40.7 51.6 57.9 23.9 1.000 3.44

#> xep012 28.4 42.2 56.7 66.2 21.1 0.992 3.69

#> xep0123 28.1 41.0 50.7 55.6 13.5 1.000 4.51

#> MaxwellBoltzmann 28.7 37.8 46.7 52.2 26.4 1.000 6.82

#>

#> $S

#> median 75% 90% 95% mode p_win deltaAICc

#> xep2 28.9 40.8 52.6 60.0 24.5 0.998 0.00

#> xep01 28.8 42.1 57.1 67.5 21.6 0.988 0.52

#> xep02 28.9 40.5 51.9 59.1 24.8 0.999 1.97

#> xep12 28.8 41.1 53.3 60.9 24.1 0.998 2.01

#> lognormal 32.1 53.2 83.9 110.1 18.3 0.910 2.04

#> xep1 32.3 51.8 74.9 91.3 19.2 0.941 2.57

#> MaxwellBoltzmann 29.1 38.3 47.3 52.9 26.8 1.000 6.15

#> tnormal 29.2 40.2 50.3 56.3 28.2 1.000 6.63

Selection of models to use in the calculations of $dwp$ must be done for each
size class separately. As with the earlier examples in the Casselman data
(sections 4.6.2 and 4.6.5), the xep01 model scores well in the modelFilter for
both size classes:

# Model filter scores for small carcasses

modelFilter(mod_size[["S"]])$scores

#> extensible rtail ltail aicc hin deltaAICc

#> xep2 1 1 1 1 0 0.0000000

#> xep01 1 1 1 1 0 0.5211667

#> xep02 1 1 1 1 0 1.9695832

#> xep12 1 1 1 1 0 2.0102368

#> lognormal 1 1 1 1 0 2.0444269

#> xep1 1 1 1 1 0 2.5680237

#> tnormal 1 1 1 1 0 6.6354677

#> MaxwellBoltzmann 1 1 0 1 0 6.1499831

#> xep123 0 0 1 1 0 0.1523155

#> xep0123 0 0 1 1 0 2.0836871

#> xep012 0 0 1 1 0 2.4650216

#> constant 0 0 1 0 0 54.2732536

# Model filter scores for large carcasses

modelFilter(mod_size[["L"]])$scores

#> extensible rtail ltail aicc hin deltaAICc

#> xep2 1 1 1 1 0 0.0000000

#> xep1 1 1 1 1 0 0.2207812

#> xep12 1 1 1 1 0 1.6626334

#> xep01 1 1 1 1 0 1.7142410

#> xep02 1 1 1 1 0 1.7826877

#> tnormal 1 1 1 1 0 3.2572103

#> xep123 1 1 1 1 0 3.4374762

#> xep012 1 1 1 1 0 3.6869378

#> xep0123 1 1 1 1 0 4.5038122

#> MaxwellBoltzmann 1 1 0 1 0 6.8218704

#> lognormal 1 0 1 1 0 1.9813879

#> constant 0 0 1 0 0 30.2262282

The same caveat about the tendency of light-tailed models to overestimate the
$\psi$—especially with bats—when the search radius is not large enough to
encompass all the carcasses (and with some cushion) that is mentioned in
sections 4.3, 4.5, and 4.6.5 and Appendix C applies here, so we again select
the xep01 model for both size classes.

There are two ways to call estpsi with data and models that include carcass
class distinctions. They both accomplish the same thing, namely, to estimate
$\psi$ for each carcass class.

  1. 1.

Pass the whole array of fitted models (mod_size, in this example) to the
estpsi function along with a vector of names of which models to select
(modnames):

psi_size <- estpsi(cod_free_size,

model = mod_size, # list of models for each carcass class

modnames = c(L = "xep01", S = "xep01")) # specify 1 model for each carcass
class

  1. 2.

Pass a list of models with exactly one model selected for each carcass class:

psi_size <- estpsi(cod_free_size,

model = list(L = mod_size[["L"]]["xep01"], S = mod_size[["S"]]["xep01"]))

Note that it is entirely possible (and easy) to select different model forms
for each class. For example, there would be some justification for choosing
xep01 for S but xep1 for L based on AICc scores. However, we opted to use the
same model form for each size class because the xep01 model, which was
selected as the “best” model for the pooled data, also worked well for each of
the size classes individually.

Once $\psi$ has been estimated, dwp can be calculated and formatted for
GenEst:

data_dwp <- estdwp(psi_size, ncarc = getncarc(cod_free_size), forGenEst = T)

dim(data_dwp) # nsim = number of simulated dwp's for each turbine and carcass
class

#> [1] 10000 3

head(data_dwp) # the first few rows of the data frame

#> turbine L S

#> 1 t1 0.714 0.538

#> 2 t3 1.000 0.400

#> 3 t4 0.833 0.800

#> 4 t8 0.750 1.000

#> 5 t11 0.667 0.500

#> 6 t13 1.000 0.833

# data_dwp can be used directly from the command line as the data_dwp argument

# in GenEst::estM, or it can be exported to a .csv file for importing into the

# GenEst GUI:

exportGenEst(data_dwp, file = "mydwp.csv")

##  Appendix A Brief Introduction to the Carcass Distributions

###  A.1 Model Anatomy

Estimating the expected proportion of carcasses lying in the searched area
involves three steps (fig. 2):

  1. 1.

modeling carcass density (carcasses per $\textrm{m}^{2}$) as a function of
location or distance from turbine;

  2. 2.

normalizing the fitted carcass densities to create a probability density
surface; and

  3. 3.

integrating the probability density over the searched area.

Each model is a Poisson regression, which is a generalized linear model (GLM)
with log link (McCullagh and Nelder 1983) such that

$y_{i}\sim\textrm{Poisson}(\lambda_{i})$ $\log(\lambda_{i})=XB$

so that

$\lambda_{i}=e^{XB}$

where where $y_{i}$ is the carcass count within measurement unit $i$ at an
average distance of $x_{i}$ m from the turbine151515The measurement unit for
xy-grid data is a 1 $\textrm{m}^{2}$ patch of ground. For the other data
types, it is a 1 m annulus with inner and outer radii at $x\pm 0.5$ m., and
$\lambda_{i}$ is the expected number of carcasses in that unit. In each model
(dropping the index, $i$), $XB$ is of the form

$p(x)+c+\log(\epsilon(x,c))$

where $c$ is an indicator variable for the search class, $p(x)$ is a
polynomial with coefficients to be estimated in the regression, and the offset
term, $\log(\epsilon(x,c))$, is the exposure or area ($\textrm{m}^{2}$) in
search class $c$ of the measurement unit.

Models of this form are distinguished by the form of the polynomial, whether,
for example, it is linear, quadratic, or some other form. It is not necessary
to know or estimate the detection probabilities in each search class. Rather,
specifying $c$ as a covariate in the model accounts for the relative detection
probabilities among search classes. Because the search class term serves only
to modify the intercept, we can ignore the search class term, $c$, in this
discussion and consider the simple case where all the area within the search
radius is searched. Then, the exposure would be proportional to $x$, and the
form simplifies, to $\lambda(x)\propto e^{p(x)+log(x)}=xe^{p(x)}$.

We abbreviate such models as xep followed by digits to indicate the form of
their polynomial. For example, xep12 would have a polynomial with linear and
quadratic terms and would expand to

$e^{a+\beta_{1}x+\beta_{2}x^{2}+\log(\varepsilon(x))}$

We represent a $\log(x)$ term by “0”, and a $1/x$ term by “i”. Thus xep01
would correspond to $p(x)=\beta_{0}\log(x)+\beta_{1}x$ and xepi0 to
$p(x)=\beta_{i}1/x+\beta_{0}\log(x)$. In practice, the constant term, $a$, is
a nuisance parameter that is lost when the model is normalized to integrate to
1.

Normally, if the leading coefficient161616The leading coefficient is the
$\beta$ associated with the highest power of $x$ in the polynomial. of $p(x)$
is negative, the model corresponds to a probability distribution which can be
extended beyond the search radius, and the model can be extrapolated to
estimate the fraction of carcasses falling outside the search radius. However,
if the model also has a $\log(x)$ term, that coefficient ($\beta_{0}$) must be
greater than -2 for the model to yield a proper probability distribution.
Similarly, a $1/x$ coefficient ($\beta_{i}$) must be negative. In all cases,
if the leading coefficient is positive, the model is not extensible beyond the
search radius and cannot be used to estimate the fraction falling outside the
search radius.

All the extensible fitted Poisson regression models can be converted into
probability distributions in the exponential family simply by normalizing the
fitted model by dividing by its integral from 0 to $\infty$. In this way, some
of the models give rise to familiar distributions, like the normal, lognormal,
gamma, and exponential. For example, a normalized xep01 model is a gamma
distribution, and a normalized xep model with polynomial
$p(x)=\beta_{0}\log(x)+\beta_{1}\log(x)^{2}$ is a lognormal distribution.
However, although all the models do give rise to distributions in the
exponential family, most do not have familiar names. In all cases, the ddFit
function automatically normalizes the fitted distance functions whenever
possible, thereby converting them to probability distributions. The package
also includes functions for calculating the probability density (PDF), the
cumulative distribution (CDF), quantiles, and random deviates for all the
distance distributions in much the same format as the d/p/q/r family of R
functions for other distributions. For the fitted distance distributions in
the dwp package, the functions ddd, pdd, qdd, and rdd serve as analogs to R’s
dnorm, pnorm, qnorm, and rnorm for the normal distribution.

In addition to the xep models which have offset = $\log(\textrm{exposure})$ to
account for the area searched in each ring, ddFit will fit a few models that
have a modified offset that serves dual purposes. In addition to accounting
for area sampled in each ring, the modified offset alters the shape of the
distribution in a standard way as the distance changes. These xep-like
distributions with modified offset that are fit by ddFit include the normal,
exponential, $\chi^{2}$, Maxwell-Boltzmann, and inverse Gaussian. Finally, a
model that assumes the carcass density is constant throughout the searched
area is fit (table 4).

Table 4: Poisson Regressions, Standard Distributions Distribution | GLM Form and Density | Extensibility
---|---|---
Constant | $y\sim\textrm{offset}(\log(\varepsilon(x)))$ | NA
| $f(x)\propto x$ |
xep1 | $y\sim x+\textrm{offset}(\log(\varepsilon(x)))$ | $\beta_{1}<0$
| $f(x)\propto xe^{\beta_{1}x}$ |
xep01 (gamma) | $y\sim\log x+x+\textrm{offset}(\log(\varepsilon(x)))$ | $\beta_{0}>-2,\beta_{1}<0$
| $f(x)\propto xe^{\beta_{0}\log x+\beta_{1}x}$ |
xep2 (Rayleigh) | $y\sim x^{2}+\textrm{offset}(\log(\varepsilon(x)))$ | $\beta_{2}<0$
| $f(x)\propto xe^{\beta_{2}x^{2}}$ |
xep02 | $y\sim\log x+x^{2}+\textrm{offset}(\log(\varepsilon(x)))$ | $\beta_{0}>-2,\beta_{2}<0$
| $f(x)\propto xe^{\beta_{0}\log x+\beta_{2}x^{2}}$ |
xep12 | $y\sim x+x^{2}+\textrm{offset}(\log(\varepsilon(x)))$ | $\beta_{2}<0$
| $f(x)\propto xe^{\beta_{1}x+\beta_{2}x^{2}}$ |
xep012 | $y\sim\log x+x+x^{2}+\textrm{offset}(\log(\varepsilon(x)))$ | $\beta_{0}>-2,\beta_{2}<0$
| $f(x)\propto xe^{\beta_{0}\log x+\beta_{1}x+\beta_{2}x^{2}}$ |
xep123 | $y\sim x+x^{2}+x^{3}+\textrm{offset}(\log(\varepsilon(x)))$ | $\beta_{3}<0$
| $f(x)\propto xe^{\beta_{1}x+\beta_{2}x^{2}+\beta_{3}x^{3}}$ |
xep0123 | $y\sim\log x+x+x^{2}+x^{3}+\textrm{offset}(\log(\varepsilon(x)))$ | $\beta_{0}>-2,\beta_{3}<0$
| $f(x)\propto xe^{\beta_{0}\log x+\beta_{1}x+\beta_{2}x^{2}+\beta_{3}x^{3}}$ |
Lognormal | $y\sim\log x+\log^{2}x+\textrm{offset}(\log(\varepsilon(x)))$ | $\beta_{1}<0$
| $f(x)\propto xe^{\beta_{0}\log(x)+\beta_{1}\log(x)^{2}}$ |
Truncated Normal | $y\sim x+x^{2}+\textrm{offset}(\log(\varepsilon(x))-\log x)$ | $\beta_{2}<0$
| $f(x)\propto xe^{\beta_{1}x+\beta_{2}x^{2}}$ |
Maxwell Boltzmann | $y\sim x^{2}+\textrm{offset}(\log(\varepsilon(x))+\log x)$ | $\beta_{2}<0$
| $f(x)\propto x^{2}e^{\beta_{2}x^{2}}$ |

By default, a standard set of 12 distributions is fit, although others can be
fit as well (table 5). Enter ?ddFit for details on the function, including how
to specify which distributions to fit.

Table 5: Poisson Regressions, Supplementary Distributions Distribution | GLM Form and Density | Extensibility
---|---|---
xep0 (Pareto) | $y\sim\log x+\textrm{offset}(\log(\varepsilon(x)))$ |  $\beta_{0}<-2$ and $x\in[1,\infty)$
| $f(x)\propto x^{\beta_{0}+1}$ |
xepi0 (inverse gamma) | $y\sim 1/x+\log x+\textrm{offset}(\log(\varepsilon(x)))$ |  $\beta_{0}<-2$ and $\beta_{i}<0$)
| $f(x)\propto x^{\beta_{0}+1}e^{\beta_{i}/x}$ |
Chisquared | $y\sim\log(x)+\textrm{offset}(\log(\varepsilon(x))-x/2)$ | $\beta_{0}>-2$
| $f(x)\propto x^{\beta_{0}+1}e^{-x/2}$ |
Exponential | $y\sim x+\textrm{offset}(\log(\varepsilon(x))-\log(x))$ | $\beta_{1}<0$
| $f(x)\propto e^{\beta_{1}x}$ |
Inverse Gaussian | $y\sim 1/x+x+\textrm{offset}(\left(\log(\varepsilon(x))-5/2\log(x)\right)$ | $\beta_{i},\beta_{1}<0$
| $f(x)\propto\sqrt{1/x^{3}}e^{\beta_{1}x+\beta_{i}\cdot 1/x}$ |

###  A.2 Model Characteristics

There are never universally accepted criteria for determining which model is
definitively the “best.” When the model must be extrapolated well beyond the
range of data, the problem becomes much more difficult. How well a model fits
within the range of the data may give very little indication of how well it
will fit outside the range of data. Conventional tools like Akaike Information
Criterion (AIC) can be very useful for comparing how well various models fit
the data, but two models that are virtually indistinguishable within the range
of data can diverge dramatically outside the range of data. However, there are
some useful guidelines for model selection that are discussed briefly here and
in greater detail in the discussions of examples (sections 4.2-4.6) and in
Appendix C.

  1. 1.

Models with lower AICc scores give better fits within the range of the data.
Models with AICc scores within about 4 are considered indistinguishable by
this measure. AICc scores are less informative about model reliability for
extrapolating beyond the search radius, but models with $\Delta\textrm{AICc}$
scores above 10 do not seem to fare as well as models with
$\Delta\textrm{AICc}<5$.

  2. 2.

Models with heavy tails tend to overestimate the fraction of carcasses falling
outside the search radius when the search radius is long. The weight of the
tail is primarily a function of the degree of the polynomial. For example, the
tail of an xep distribution with degree 2 will have a lighter tail than one
with degree 1. The weight of the tail also depends somewhat on the other terms
in the polynomial, with lower order terms contributing to lighter tails.

  3. 3.

Road & pad searches (RP) or other search patterns in which the proportion of
area searched per ring decreases with distance from turbine and the search
coverage is low but non-zero for a broad range of distances greater than about
40 meters can lead to reduced stability in the models. In this situation:

     * •

there will be a tendency to overestimate the fraction of carcasses within the
search radius if no carcasses are found at great distances, and we recommend
using a relatively heavy-tailed distribution these situations to account for
the possibility of significant numbers of carcasses beyond the search radius,
and

     * •

there will be a tendency to strongly underestimate the fraction of carcasses
within the search radius if carcasses are found at great distances, and we
recommend using a relatively light-tailed distribution in these cases.

  4. 4.

Distributions of smaller carcasses tend to have heavier tails than
distributions of larger carcasses because the smaller carcasses are more
strongly affected by wind and are blown farther under high wind conditions and
are not flung as far under light wind conditions, as demonstrated in ballistic
simulations in appendices B and C.

##  Appendix B Mechanistic Model of Carcass Deposition

To explore properties of carcass density models and their performances on
known data, we generated carcass distributions according to a ballistics model
similar to those used by Hull & Muir (2010) and by Prakash and Markfort
(2020). Models are fit to simulated carcass dispersions under several
different scenarios with different carcass types (eagle and bat), wind
regimes, and carcass search protocols. Since the actual (simulated) carcass
distributions are known, models can be compared with each other based on how
well they are able to predict the true data under a variety of conditions.

###  B.1 The Model

Physics puts a fairly strict limit on how far carcasses may fall from the
turbine. Gravity inexorably pulls them to the ground after a brief time.
Excluding cases where 1) a bird or bat is not killed immediately but is able
to fly or walk a great distance after getting struck by a turbine blade, 2) a
carcass is carried a great distance by a scavenger, 3) a carcass is swept
upward or downward by a wind with a strong vertical component, or 4) a carcass
is blown or rolls a great distance after landing, we can use well-known
differential equations from ballistics modeling to get a qualitative
understanding of carcass spatial distributions around a turbine. Of particular
interest is the general shape of the tails of the carcass distribution, which,
in turn, informs the choice of parametric distribution for empirical modeling
of carcass distributions and estimation of the proportion of carcasses landing
in the searched areas.

Mechanistic differential equation models can be useful in addressing simple,
qualitative questions about carcass dispersion. Which general shapes of
distributions are best in which types of situations? How well does AICc
perform for identifying models that will most accurately predict the fraction
of carcasses landing in the searched area? How do carcass size and wind speed
broadly affect the distribution of carcasses on the ground?

However, a quantitative understanding that accurately predicts the actual
distributions of carcasses in the field as a function of environmental
covariates such as wind profile and turbine size is much more difficult and
may not be feasible. The precise distribution critically depends on a number
of parameters and assumptions that are difficult to validate in the field.
Factors such as wind speed and direction at the time of strike, strike
location on the blade, carcass velocity at the time of strike, change in
carcass velocity upon strike (angle of the blade, transfer of momentum from
blade to carcass), drag coefficient and its change during flight as carcass
changes shape and direction, rotor speed at time of collision, carcass
fluttering or spinning after collision, dependence of wind velocity on
position (including wind shear and wake effect), turbulence behind the blade,
displacement by scavengers after landing, and even the type of ground cover
(which affects the wind gradient with height) all have a significant impact on
the resulting carcass distribution on the ground, but none of them are known
with any practical, quantitative degree of reliability.

If we make a few simplifying assumptions, the ballistics equations are not
difficult. We start by assuming that the animal is killed upon impact with the
turbine blade and that the wind is blowing horizontally (with no vertical
component) and normal to the plane of the blades. Like Prakash and Markfort
(2021) and Hull and Muir (2011), we ignore the wake effect behind the turbine,
but we do include wind shear.

To calculate a carcass landing position after being struck by a turbine, we
first define a coordinate system centered at the base of the turbine to track
carcass position in 3 dimensions, $s=(x,y,w)$, where $x$ is the horizontal
position in the plane of the turbine blades, $y$ is height above the ground,
and $w$ is the horizontal distance normal to the plane of the turbine blades
in the downwind direction. We then use ballistics differential equations to
describe the trajectory of the carcass as a function of time.

Define $x$ and $y$ to be the horizontal and vertical displacements of a
carcass from the turbine base, and $w$, the distance downwind from the
turbine. If $s(t)$ is the position vector of the carcass as a function of time
and $v(t)=s^{\prime}(t)$ is the velocity, then the pair $(s(t),v(t))$ is a
6-dimensional vector which satisfies a set of first order differential
equations:

$s^{\prime}(t)=v(t)$
$v^{\prime}(t)=-\mathfrak{y}\left(v(t)-\mathpzc{w}(s(t))\right)\lVert
v(t)-\mathpzc{w}(s(t))\rVert-G$

where $\mathpzc{w}(s(t))$ is the wind velocity at $s(t)$,
$\mathfrak{y}=9.807/{y^{\prime}_{T}}^{2}$ (with $y^{\prime}_{T}$ = terminal
velocity of the carcass) is an aerodynamic constant associated with each
carcass type, and $G=[0,9.807,0]$ is the acceleration due to gravity. Given an
initial carcass strike point on the blade and an initial carcass velocity, we
track the trajectory of the carcass through space using the method of lines
with $\Delta t=0.01$ seconds. The final position of the carcass on the ground
is $(x,w)$ when $y=0$.

This set of governing equations is similar to those used by Prakash and
Markfort (2021) but allows wind speed to vary with carcass position, thereby
enabling the accommodation of wind shear and wake effect. Like Prakash and
Markfort (2021) and Hull and Muir (2011), we ignore the wake effect. We do
account for wind shear through the Wind Power Law, or
$\mathpzc{w}(y)=\mathpzc{w}(y_{n})\cdot{y/y_{n}}^{a}$, where $\mathpzc{w}(y)$
is the wind speed at $y$ meters above the ground, $y_{n}$ is the height of the
nacelle, and $a$ is the Hellman coefficient (Counihan 1975).

In addition, we assume that the cross-sectional area of the carcass presented
to the wind ($A$) and the drag coefficient ($C_{d}$) are constant, so the term
$0.5\,C_{d}A\rho/m$ (Prakash and Markfort (2021); Hull and Muir (2011))
simplifies to $9.807/{y^{\prime}_{T}}^{2}$ after noting that the terminal
velocity is the vertical velocity at which the wind resistance equals the
force of gravity so there is no longer any vertical acceleration. In other
words, terminal velocity, {y’_T}, is the velocity such that
$\rho/2C_{d}A/m\cdot{y^{\prime}_{T}}^{2}-g=0$ from Prakash and Markfort’s
(2021) equations 3 and 4 with no acceleration in the horizontal directions,
giving $\rho/2C_{d}A/m=G/{y^{\prime}_{T}}^{2}=\mathfrak{y}$. Thus, the
terminal velocity, which is relatively easy to measure and is relatively well-
known, is used in lieu of the product of parameters with largely unknown
values ($C_{d}$ and $A$).

###  B.2 Simulation Parameters

We used the ballistics model described in the previous section (appendix B) to
explore the effects of carcass type, wind regime, strike position, and flight
speed on carcass distributions on the ground (appendix B.3). Random carcass
locations were generated from the carcass distributions and then sampled to
create simulated data sets. Distributions were fit to the simulated data sets
using ddFit. Statistical properties of the models are explored and discussed
in Appendix C to provide guidance on dwp model selection.

Parameter values used in the carcass dispersion models and simulations are a
mix of fixed constants (such as turbine specifications) that are constant
across the simulation scenarios and variable parameters that may differ among
carcasses and simulation scenarios (table 6).

Table 6: Parameters used in the ballistics model and simulations Parameter | Value
---|---
Fixed across scenarios |
nacelle height | 80 m
blade length |  $r_{0}=45$ m
tip speed ratio | 6
acceleration due to gravity | $G=9.807\,\mathrm{m/s^{2}}$
Hellman coefficient | $a=0.22$
strike position on the blade | random uniform
Varying among scenarios |
terminal velocity |  $V_{t}=8.8$ m/s for bats, and $V_{t}=25$ m/s for eagles
wind speed | low, moderate, high
wind profile | constant, varying
carcass initial horizontal velocity | 0, varying (up to 8 m/s)
search plot type | cleared, roads and pads

The Hellman coefficient governs the wind shear or the tendency for wind speed
to increase with height. It is a measure of the coarseness of the landscape
texture (Kaltschmatt et al. 2007), with smaller values corresponding to
smoother surfaces and smaller changes in wind speed with height. Values
typically range from about 0.05 for very smooth ground to 0.5 or 0.6 for
cities with high-rise buildings.

The terminal velocity is a measure of an object’s air resistance and is an
amalgam of the drag coefficient ($C_{d}$), the cross-sectional area presented
to the wind, the air density—all of which are unknown—and the object’s mass.
The terminal velocity is used in lieu of the product of parameters with
largely unknown values. We assume a value of 8.8 m/s for the terminal velocity
for bats, which we calculated from data given by Grodsky et al.
(2011).171717Grodsky et al. (2011) reported that bats took an average of 10–12
seconds to fall 91.4 m in an experiment. Terminal velocity is the velocity,
$y^{\prime}$, at which the carcass is no longer accelerating as it falls but
is falling at a constant rate, that is
$y^{\prime\prime}=-G+A(y^{\prime}_{T})^{2}=0$, where $y^{\prime}_{T}$ is the
terminal velocity, $A$ is the projected area, and $G$ is the acceleration due
to gravity. Solving for $y$, we get
$y(t)=y_{0}-\log(\cosh(3.1305t\sqrt{A}))/A$. Then, taking the initial height
as $y_{0}=91.4$ and the final height after 11 seconds as $y(11)=0$, we get
$A=0.1263$ and a terminal velocity of $y^{\prime}_{T}=8.8$ m/s for bats, which
is in good agreement with the findings of Prakash and Markfort (2020). We used
a terminal velocity of 25 m/s for eagles, which is slightly less than it is
for live cats (Whitney and Mehlhaff, 1987).

Turbine nacelle height and blade length match those of the 1.8 MW Vestas V90
turbine. We assume the turbine is operating at a tip speed ratio of 6, that
is, that the tips of turbine blades are moving at 6 times the wind speed,
while the wind varies.

Simulation scenarios assume that strike position on the blades is uniformly
distributed and that the initial velocity of the carcass after being struck is
equal to the sum of the blade and carcass velocities at the point of impact.

Wind speeds behind the turbine and at the height of the nacelle were either
constant at 4, 8, or 12 m/s or varying with low, moderate, or high average.
Winds in the low wind speed regime were assumed to be Weibull distributed with
mean of 5.32 m/s and standard deviation of 2.78 m/s. The high wind speed
regime was also assumed to be Weibull distributed, with mean of 9.18 m/s and
standard deviation of 2.1 m/s. The moderate wind speed regime mimics the winds
measured at a site in the Netherlands, with a mean of 7.5 m/s, standard
deviation of 1.25 m/s, and having a probability density function (PDF) of wind
speeds of $f(w)=10^{-0.04(w-7)^{2}-0.8}$, which approximately fits the
moderate wind profile in fig. 2 (row $\mathrm{W_{80}}$, column JJA) of He et
al. (2013) . In all cases, no carcasses were produced when wind speeds were
<3.5 m/s. Wind direction was assumed to be uniformly distributed, with no
prevailing direction.

Animal flight velocity was assumed to be constant in half the scenarios and
variable in half. In the constant scenarios, carcasses attain the same
velocity as the turbine blade at the point of impact upon being struck,
implying a coefficient of restitution of $e=1$, following Hull and Muir (2010)
and Prakash and Markfort (2020). In the variable scenarios, flight speed
relative to the ground was assumed to be 8 m/s with 0 vertical component but
horizontal direction varied uniformly between 0 and 2$\pi$, with the component
of carcass initial velocity in the plane of the blades ($x$) erased at the
moment of impact and the component normal to the plane of the blades ($w$)
added to the wind velocity.

The zone where carcasses are likely to fall would be expected to extend to a
radius at least as great as the turbine blades ($r_{0}$). Carcasses may land
outside that range if the impact from the blade knocks them beyond $r_{0}$ in
the plane of the blades or if the wind carries them beyond $r_{0}$ in the
windward direction. Figure 27 compares the carcass fall zones for eagles and
bats at wind speeds of 4, 8, and 12 m/s, showing images of the disk spanned by
the turbine blades projected onto carcass dispersion patterns on the ground.
The contours are mappings of concentric rings around the turbine hub to the
position on the ground where an eagle or bat carcass would be expected to fall
under the given wind speed and carcass type under the conditions of the
simulation scenario (table 6).

Eagle carcasses are large, and, compared to small carcasses, their
trajectories are less affected by the wind. Like a baseball getting struck by
a bat, an eagle directly hit by a turbine blade is flung in the direction
dictated by the blade, with relatively little modification of its subsequent
trajectory by the wind. In the simulations, under low wind speed conditions (4
m/s) eagle carcasses fell near the line of the blades. With stronger winds,
carcasses landed significantly downwind (fig. 27).

By contrast bat carcasses are small and may be carried aloft by the wind for
great distances, like popcorn thrown into the air on a windy day. In the
simulations, even at the relatively low wind speed of 4 m/s, carcasses struck
near the top of the swept area were blown over 50 meters, significantly beyond
the $r_{0}=45\mathrm{m}$ blade length (fig. 27). Greater windspeeds had little
effect on the distribution of carcasses in the plane of the blades (left-right
in fig. 27) but a substantial effect on the distribution of carcasses in the
downwind direction (bottom to top in fig. 27). With a wind of 12 m/s,
carcasses near the top of the swept area were blown close to 200 m.

Figure 27: Projection of turbine-carcass strike points onto the ground.
Turbine is located at (0, 0), with wind blowing north (bottom to top) at a
constant velocity. Lines are projected images of turbine-carcass strike points
onto the ground. Contours correspond to strike points on the blades in
concentric rings with radii of 5, 10, …, 45 m. Shading marks the span of the
turbine blades in the x direction.

###  B.3 Distributions of Carcass Distances

Because bats are be more heavily influenced by the wind than eagles are and
are blown farther, a smaller proportion of bats are expected to fall very
close to the turbine, as reflected in the histograms of simulated carcass
distances shown in figs. 28 and 29.

The eagle histograms are qualitatively similar to those constructed for large
birds by Hull and Muir (2011, fig. 2). Under most of the simulated conditions,
the distributions have relatively flat tops over much of the range of
distances but then drop precipitously. The most consequential difference
between our model and that of Hull and Muir (2011) is that the latter neglects
the downwind direction. Because eagle carcasses are typically not blown far by
the wind after being struck, the downwind direction plays a relatively minor
role in determining eagle distance distributions, which explains the
similarity between the patterns in fig. 28 and those observed by Hull and Muir
(2011).

The bat histograms (fig. 29) show relatively few carcasses very near the
turbines and more of a classic bell shape (albeit, somewhat right-skewed) than
the eagle histograms. That these bat histograms bear little resemblance to
those of Hull and Muir (2011, fig. 2) and Prakash and Markfort (2020, fig. 9)
is due to the importance of the downwind direction, which is included in our
model but neglected in the other two.

Figure 28: Histograms of distances of eagle carcasses to turbine. Upon
collision with a turbine blade, carcass velocity takes on the velocity of the
blade plus the flight speed. Flight speed is either assumed to be 0, in which
case initial carcass velocity equals turbine blade velocity at point of
impact, or variable, in the range of wind speed ± 8 m/s. Figure 29: Histograms
of distances of bat carcasses to turbine. Upon collision with a turbine blade,
carcass velocity takes on the velocity of the blade plus the flight speed.
Flight speed is either assumed to be 0, in which case initial carcass velocity
equals turbine blade velocity at point of impact, or variable, in the range of
wind speed ± 8 m/s.

##  Appendix C Fitting Parametric Distributions to the Dispersion Patterns

The general shape of the actual distribution of carcasses varies with the
ballistics parameters and environmental conditions. The performance of a
fitted distribution depends on how well the distribution family’s shape
conforms to the actual distribution and, perhaps even more critically, how
accurately the fitted model reflects the distribution of carcasses lying
outside the searched area.

In practice, the search radius is finite, and a fitted model does not “know”
what fraction of carcasses lie within the search radius and has to “guess”
based on its general shape. If the general shape happens to align well with
the true distribution, the prediction of the fraction lying outside the search
radius will be accurate; if not, the error may be substantial.

The ballistics modeling suggests that eagle carcass distributions are often
likely to be relatively flat near the turbine but decline precipitously with
distance at some point (fig. 28), so that, under many conditions, eagle
distributions would be expected to be light-tailed. Thus, relatively heavy-
tailed distributions such as the lognormal and xep1 would usually not be a
good match for eagle distributions. By contrast, the simulations suggest that
bat carcass distributions will tend to have few carcasses very near the
turbines, and, compared with the eagle distributions, a more gradual climb to
a narrower peak (fig. 29) and have a different suite of distributions with the
best fits.

Although some broad descriptions of model characteristics and their general
applicability can be made, it may not be possible to determine _a priori_
which particular models will provide the best predictions in a given
situation. Factors like wind regime and species determine the actual
distribution of carcasses, but other factors—like search radius and plot shape
(for example, cleared plots or roads & pads)—can heavily influence the aptness
of a model for predicting the fraction of carcasses outside the searched area.
Selecting a specific model for a specific situation must be done on a case by
case basis.

###  C.1 Model Accuracy in Predicting $\psi$

The accuracy of a carcass distribution model for predicting the probability
that a carcass lies in the searched area depends both on how well the model
conforms to the shape of the actual distribution and how the fit within the
searched area can be extended outside the searched area. There are two types
of unsearched area: 1) area outside a search radius, and 2) unsearched area
within the search radius. Spatial prediction for carcasses outside the search
radius involves extrapolation, and accuracy requires making effective use of
additional information, apart from the data itself because a good or even
perfect fit within the range of data gives no guarantee of a good or even
plausible fit outside the range of the data. Prediction for unsearched areas
within the search radius involves interpolation, and the relative quality of a
model’s fit to the data within the searched area (for example, AICc) is a
reasonable guide for model selection for interpolation. Thus, wind conditions,
carcass type, search radius, and plot shape are all important factors in
determining which models will provide the most accurate predictions.

We tested the model predictions for the standard $dwp$ models (Appendix A) in
a variety of simulation scenarios (table 6), with $n=1000$ replicates for each
scenario. For each replicate, a total of 200 random carcass distances were
generated from the distance distributions arising from the ballistics models
for eagles and bats under the wind regimes described in table 6 and Appendix
B.3 and shown in figs. 28 and 29. Random directions were generated as
uniform(0, $2\pi$) deviates. Carcasses that lay within the searched area (fig.
30) were found and included in the data used for fitting the models; carcasses
falling outside the searched area were not included. For replicates in which
at least 5 carcasses were found, each of the standard $dwp$ models was fit and
$\hat{\psi}$ was calculated using the maximum likelihood estimator for the
model parameters whenever the model was extensible. For each simulation
scenario, box plots of $\hat{\psi}$ are presented for each of the standard
$dwp$ models (figs. 31 \- 42), ordered by degree of the polynomial in the
model (table 4).

Figure 30: Search areas used in the simulations. Searches were conducted out
to radii of 50, 100, or 150 m from the turbine (circles) and included either
all carcasses within the search radius (“cleared plot”) or only those
carcasses on the roads and turbine pad (“RP”). Figure shows carcasses found on
an RP search out to 150 m.

####  C.1.1 Accuracy of $\hat{\psi}$: Eagles under constant winds and cleared
search plots

For eagles under constant wind conditions in the simulations, the distribution
(PDF) of carcasses was fairly flat out to a certain point but then dropped
rapidly to zero (fig. 31). Under these conditions, a short search radius
misses the precipitous decline in the probability density, and the empirical
models have great difficulty estimating the fraction of carcasses within the
search radius.

Figure 31: Estimated $\psi$ for the standard models with simulated M = 200
eagles with winds constant and cleared plot searches. Boxes show sample IQR
with median; whiskers extend to the most extreme points within 1.5 IQR of the
box; points beyond 1.5 IQR of the box are shown as small circles. Red lines
show the true $\psi$.

At 4 m/s, the carcass PDF (fig. 31, histogram on bottom left) begins a long
decline as distance increases beyond approximately 10 m. The heavy-tailed
lognormal continues the gradual decrease past the search radius, unable to
accommodate the sharp drop beginning at about 75 m. As a result, the lognormal
regularly underestimates $\psi$ under these conditions as the model implicitly
assumes that there is a substantial number of carcasses past the search
radius. By contrast, the distributions with degree $\geq\\!2$ have an
inherently greater acceleration in the rate of decline in the PDF and tend to
predict fewer carcasses within the search radius than do the heavier-tailed
distributions. With a search radius of 100 m, all carcasses fell within the
search plot, and all the fitted distributions except the lognormal routinely
estimated $\hat{\psi}>90\%$.

At 12 m/s (fig. 31, right column), the situation changes dramatically. The
density of carcasses (PDF) is largely flat but gradually increasing with
distance from the turbine until about 125 m, when the density drops rapidly to
zero. The short search radius ($r=50$ m) entirely misses the decrease, and
almost all the fitted distributions vastly overestimate the fraction of
carcasses within the search area as their implicit assumptions about how the
density would converge to zero are in error, resulting curves that
underestimate the extent of the flat part of the density. The relatively
heavy-tailed lognormal and xep1 distributions are forced to decrease gradually
from their peak, and their degree of overestimation of $\psi$ in this scenario
is modest compared to the lighter-tailed distributions.

The distributions of carcasses within the first 100 m from a turbine were
similar for winds of 8 and 12 m/s (fig. 31, histograms), so the $r=50$ and 100
m search radii yield similar fitted distributions (fig. 31, comparing the 8
m/s panels with the 12 m/s panels for $r=50$ and 100). However, the accuracy
of the estimates depends on the target (fig. 31, red horizontal lines).

####  C.1.2 Accuracy of $\hat{\psi}$: Eagles under constant winds and constant
flight speeds for road & pad searches

Under constant wind conditions, predictions across the fitted distributions
were more consistent in the road & pad searches than in the cleared plot
searches (fig. 32). In each panel, the distinctions among the fitted
distributions were slight. However, the lognormal almost always predicted a
smaller fraction of carcasses within the searched area than did all other
models. The only exceptions were for a short search radius was short ($r=50$
m) and the wind speed was 8 or 12 m/s, in which case xep1 had the smallest
(and most accurate) estimates for $\psi$. When the search radius was not short
($r=100$ or 150 m), there was little distinction among the model fits.

Figure 32: Estimated $\psi$ for the standard models with simulated M = 200
eagles with winds constant and roads & pads searches. Boxes show sample IQR
with median; whiskers extend to the most extreme points within 1.5 IQR of the
box; points beyond 1.5 IQR of the box are shown as small circles. Red lines
show the true $\psi$.

####  C.1.3 Accuracy of $\hat{\psi}$: Eagles under varying winds and constant
flight speed for cleared plot and road & pad searches

The eagle distributions and fitted models under variable winds (figs. 33, 34)
were similar to those under constant winds (figs. 31), 32). Refer to sections
C.1.1 and C.1.2 for discussion.

Figure 33: Estimated $\psi$ for the standard models with simulated M = 200
eagles with winds variable and cleared plot searches. Boxes show sample IQR
with median; whiskers extend to the most extreme points within 1.5 IQR of the
box; points beyond 1.5 IQR of the box are shown as small circles. Red lines
show the true $\psi$. Figure 34: Estimated $\psi$ for the standard models with
simulated M = 200 eagles with winds variable and roads & pads searches. Boxes
show sample IQR with median; whiskers extend to the most extreme points within
1.5 IQR of the box; points beyond 1.5 IQR of the box are shown as small
circles. Red lines show the true $\psi$.

####  C.1.4 Accuracy of $\hat{\psi}$: Eagles under varying winds and flight
speeds (cleared plot searches, road & pad searches)

The eagle distributions under variable winds and flight speeds (figs. 35, 36)
were notably less flat than the eagle scenarios in which the effect of initial
flight speed was nullified upon impact by the turbine blade. The predicted
proportion of carcasses within the searched area generally decreased with the
order of the fitted distribution, with the accuracy increasing with search
radius. With road & pad searches, there was a marked tendency for models to
overpredict $\psi$, especially when the search radius was short ($r=50$). That
tendency to overpredict was fully compensated for in the lognormal and xep1
models, which, as heavy-tailed distributions, naturally tend to underpredict.

Figure 35: Estimated $\psi$ for the standard models with simulated M = 200
eagles with winds variable, flight speed variable and cleared plot searches.
Boxes show sample IQR with median; whiskers extend to the most extreme points
within 1.5 IQR of the box; points beyond 1.5 IQR of the box are shown as small
circles. Red lines show the true $\psi$. Figure 36: Estimated $\psi$ for the
standard models with simulated M = 200 eagles with winds variable, flight
speed variable and roads & pads searches. Boxes show sample IQR with median;
whiskers extend to the most extreme points within 1.5 IQR of the box; points
beyond 1.5 IQR of the box are shown as small circles. Red lines show the true
$\psi$.

####  C.1.5 Accuracy of $\hat{\psi}$: Bats

Bat carcass distributions tended to have fewer carcasses near the turbine and
a somewhat more elongated right tail than did the eagle distributions (figs.
37-42). As with the eagles, the models tended to overpredict by a substantial
margin when the search radius was short ($r=50$ m), variation among model
predictions was much smaller for road & pad searches than for cleared plot
searches, and $\hat{\psi}$ tended to decrease with the degree of the model (as
reflected in the decreasing trend in the boxes in figs. 37-42).

Figure 37: Estimated $\psi$ for the standard models with simulated M = 200
bats with winds constant and cleared plot searches. Boxes show sample IQR with
median; whiskers extend to the most extreme points within 1.5 IQR of the box;
points beyond 1.5 IQR of the box are shown as small circles. Red lines show
the true $\psi$.

The same pattern naturally arises with the road & pad searches. The sample is
likely to not have any carcasses at great distances because the search
coverage is too low there. The light-tailed distributions drop off quickly, so
there’s not much probability in the tails and they overestimate the fraction
of carcasses in the search radius. As the search radius increases, the heavy-
tailed distributions tend to think there’s still a lot of carcasses farther
out, and consequently they tend to underestimate the fraction falling in the
searched area.

Figure 38: Estimated $\psi$ for the standard models with simulated M = 200
bats with winds constant and roads & pads searches. Boxes show sample IQR with
median; whiskers extend to the most extreme points within 1.5 IQR of the box;
points beyond 1.5 IQR of the box are shown as small circles. Red lines show
the true $\psi$. Figure 39: Estimated $\psi$ for the standard models with
simulated M = 200 bats with winds variable and cleared plot searches. Boxes
show sample IQR with median; whiskers extend to the most extreme points within
1.5 IQR of the box; points beyond 1.5 IQR of the box are shown as small
circles. Red lines show the true $\psi$. Figure 40: Estimated $\psi$ for the
standard models with simulated M = 200 bats with winds variable and roads &
pads searches. Boxes show sample IQR with median; whiskers extend to the most
extreme points within 1.5 IQR of the box; points beyond 1.5 IQR of the box are
shown as small circles. Red lines show the true $\psi$. Figure 41: Estimated
$\psi$ for the standard models with simulated M = 200 bats with winds
variable, flight speed variable and cleared plot searches. Boxes show sample
IQR with median; whiskers extend to the most extreme points within 1.5 IQR of
the box; points beyond 1.5 IQR of the box are shown as small circles. Red
lines show the true $\psi$. Figure 42: Estimated $\psi$ for the standard
models with simulated M = 200 bats with winds variable, flight speed variable
and roads & pads searches. Boxes show sample IQR with median; whiskers extend
to the most extreme points within 1.5 IQR of the box; points beyond 1.5 IQR of
the box are shown as small circles. Red lines show the true $\psi$.

###  C.2 Akaike Information Criterion, Model Accuracy, and $\psi$

The Akaike information criterion (AICc) is a well-tested and commonly used
tool for model selection and has proven to be effective for distinguishing
among models by the relative quality of the fit to the data. However, its
validity is limited to the scope of the data, and there is no guarantee that
models that perform well within the scope of the data will perform well—or
will even be remotely plausible—when extended outside the scope of the data,
as must be done when predicting the proportion of carcasses that lie outside
the searched area ($1-\psi$). To test the utility of AICc for prediction of
$\psi$, we summarized the results from the simulations in section C.1 by
$\Delta\mathrm{AICc}$, binning the $\hat{\psi}$ values by the
$\Delta\mathrm{AICc}$ scores of the fitted model. For example, for eagles
under constant wind speeds of 4 m/s and searched on cleared plots out to a
radius of $r=50$ m, we generated 1000 simulated data sets, fit the 11 standard
$dwp$ models, and binned the $\hat{\psi}$ values according to their
$\Delta\mathrm{AICc}$ scores (fig. 43, upper left panel). The boxplot of
$\hat{\psi}$ values for $\Delta\mathrm{AICc}=0$ summarizes the collection of
$\hat{\psi}$’s for the best-fitting model in each of the 1000 replicate data
sets. The boxplot for $\Delta\mathrm{AICc}$ = 0–1 is a summary of the
$\hat{\psi}$ values for models with $\Delta\mathrm{AICc}\in(0,1]$, and the
rightmost boxplot in each panel is for models with $\Delta\mathrm{AICc}\geq
10$.

####  C.2.1 Eagles

For eagles under constant wind conditions and the initial carcass position and
velocity are taken to be that of the rotor at the point of impact, the
distribution (PDF) of carcasses was fairly flat out to a certain point but
then dropped rapidly to zero (fig. 28, left panels). Under these conditions, a
short search radius misses the precipitous decline and empirical models have
great difficulty estimating the fraction of carcasses within the search
radius. The key feature of the distribution—namely, the point marking the
beginning of the steep drop—is absent from the data, so measures of how well
the models fit within the range of the data (AICc, for example) cannot make
meaningful distinctions among the models for extrapolating beyond the search
radius.

In general, for cleared plots and fixed wind speeds, $\Delta\mathrm{AICc}$ was
largely unrelated to accuracy in predicting $\psi$. For search radii of
$r=50,\,100$ m, $\hat{\psi}$ tended to increase with $\Delta\mathrm{AICc}$
(fig. 43), presumably because the heavier-tailed models tended to fit better
within the range of data than the lighter-tailed models. However, the actual
simulated carcass distributions are light-tailed, so $\Delta\mathrm{AICc}$
gives an unreliable measure of the aptness of the model fits. For example, at
8 m/s, and a short search radius (fig. 43, top middle panel), the higher the
$\Delta\mathrm{AICc}$, the poorer the prediction. However, with a longer
search radius and a corresponding greater fraction of carcasses within the
search radius, the greater the AICc, the better the prediction, the best-
fitting models had by far the least accurate predictions (fig. 43, center
panel).

When the search radius was close to the maximum carcass distance, as in the
high winds, long search radius example, the AICc tended to successfully select
the model with the best estimate of $\psi$ (bottom right panel). When
windspeeds dropped (4m/s or 8m/s) but search radius remained long (bottom row
left and center, or middle row at left), few if any carcasses fell beyond the
search radius and all models did well at determinig that $\psi=1$. These
scenarios are effectiely interpolation rather than extrapolation given the
length of the search radius and the shape of the generating distributions.

Figure 43: AICc and estimated $\psi$ for the standard models with simulated M
= 200 eagles with winds constant and cleared plot searches. Boxes show sample
IQR with median; whiskers extend to the most extreme points within 1.5 IQR of
the box; points beyond 1.5 IQR of the box are shown as small circles. Red
lines show the true $\psi$.

Because $\Delta\mathrm{AICc}$ applies strictly to the fit of models within the
range of data, we would expect $\Delta\mathrm{AICc}$ to perform better when
the fraction of carcasses within the search radius is higher, as illustrated
in fig. 43 for $r=150$, where close to 100% of the carcasses lie within the
search radius, and, in the case where there are distinctions among models
(lower right panel), greater $\Delta\mathrm{AICc}$ values are associated with
poorer fits. However, in the center panel (8 m/s and $r=100$), 90% of the
carcasses lie within the search plot but the best predictors tended to be the
models with the poorest fits within the range of the data. In this simulation
scenario, the distribution of carcasses was largely flat within the 100 m
search radius but dropped rapidly to zero just outside the search radius,
requiring a light-tailed distribution in order to capture the abrupt end of
the distribution. However, it is the heavy-tailed distributions that fit best
within that first 100 m from the turbine, hence the inverse relationship
between the quality of the fit within the range of data and the adequacy for
prediction outside the range of the data.

With road & pad sampling, much of the spatial prediction is in extending the
models to the unsearched area within the search radius. Because interpolation
to unsearched areas within the range of the data plays a large role in
prediction of $\psi$ from road & pad data, we would expect
$\Delta\mathrm{AICc}$ to be a better indicator of model accuracy than it was
for the cleared plot searches. Indeed, lower $\Delta\mathrm{AICc}$ values do
seem to be somewhat associated with more accurate prediction in the road & pad
scenarios (fig. 44, fig. 46 and fig. 48) than in the corresponding cleared
plot scenarios (fig. 43, 45, and 47, respectively).

Figure 44: AICc and estimated $\psi$ for the standard models with simulated M
= 200 eagles with winds constant and roads & pads searches. Boxes show sample
IQR with median; whiskers extend to the most extreme points within 1.5 IQR of
the box; points beyond 1.5 IQR of the box are shown as small circles. Red
lines show the true $\psi$. Figure 45: AICc and estimated $\psi$ for the
standard models with simulated M = 200 eagles with winds variable and cleared
plot searches. Boxes show sample IQR with median; whiskers extend to the most
extreme points within 1.5 IQR of the box; points beyond 1.5 IQR of the box are
shown as small circles. Red lines show the true $\psi$. Figure 46: AICc and
estimated $\psi$ for the standard models with simulated M = 200 eagles with
winds variable and roads & pads searches. Boxes show sample IQR with median;
whiskers extend to the most extreme points within 1.5 IQR of the box; points
beyond 1.5 IQR of the box are shown as small circles. Red lines show the true
$\psi$. Figure 47: AICc and estimated $\psi$ for the standard models with
simulated M = 200 eagles with winds variable, flight speed variable and
cleared plot searches. Boxes show sample IQR with median; whiskers extend to
the most extreme points within 1.5 IQR of the box; points beyond 1.5 IQR of
the box are shown as small circles. Red lines show the true $\psi$. Figure 48:
AICc and estimated $\psi$ for the standard models with simulated M = 200
eagles with winds variable, flight speed variable and roads & pads searches.
Boxes show sample IQR with median; whiskers extend to the most extreme points
within 1.5 IQR of the box; points beyond 1.5 IQR of the box are shown as small
circles. Red lines show the true $\psi$.

####  C.2.2 Bats

Fig. 49 shows bat carcass distributions and estimated probabilities that
carcasses lie in the searched area for constant wind speeds of 4, 8, and 12
m/s and cleared plot search radii of 50, 100, and 150 m. As with eagles, AICc
does not appear to be a useful predictor of model performance in estimating
$\psi$ except in cases where the search radius is large enough to encompass
nearly the whole of the distribution.

For bat carcasses, $\Delta\mathrm{AICc}$ appears to be even less reliable for
selecting the most accurate models for $\psi$ than it was for eagle carcasses.
In some cases, the worse-fitting models (high $\Delta\mathrm{AICc}$) gave more
accurate predictions (for example, fig. 49, upper right panel, with 12 m/s
winds and search radius of $r=50$ m); in some cases, the best-fitting models
gave the most accurate predictions (for example, fig. 49, center panel, with 8
m/s winds and search radius of $r=100$ m); in most cases, though, there was
little discernible relationship between $\Delta\mathrm{AICc}$ and accuracy
(figs. 49-54) .

Figure 49: AICc and estimated $\psi$ for the standard models with simulated M
= 200 bats with winds constant and cleared plot searches. Boxes show sample
IQR with median; whiskers extend to the most extreme points within 1.5 IQR of
the box; points beyond 1.5 IQR of the box are shown as small circles. Red
lines show the true $\psi$. Figure 50: AICc and estimated $\psi$ for the
standard models with simulated M = 200 bats with winds constant and roads &
pads searches. Boxes show sample IQR with median; whiskers extend to the most
extreme points within 1.5 IQR of the box; points beyond 1.5 IQR of the box are
shown as small circles. Red lines show the true $\psi$.

The road & pad searches yielded accurate predictions for $\psi$ for bats under
constant or variable windspeeds (figs. 50181818The missing boxplots in the
upper right panel of fig. 50 are due to so few models being able to fit viable
distributions to the data because the carcass PDF did not start its decline to
zero until well beyond the search radius of 50 m., 52, and 54), except when
the search radius was too short to capture the distribution’s dropoff in the
right tail, in which case models tended to vastly overestimate $\psi$. The
$\Delta\textrm{AICc}$ had very little predictive value because
$\Delta\textrm{AICc}$ measures relative fit within the range of the data but
gives no information on how well the model extrapolates beyond the data or
whether a given model happens to have a tail that matches that of the actual
carcass distribution.

Figure 51: AICc and estimated $\psi$ for the standard models with simulated M
= 200 bats with winds variable and cleared plot searches. Boxes show sample
IQR with median; whiskers extend to the most extreme points within 1.5 IQR of
the box; points beyond 1.5 IQR of the box are shown as small circles. Red
lines show the true $\psi$. Figure 52: AICc and estimated $\psi$ for the
standard models with simulated M = 200 bats with winds variable and roads &
pads searches. Boxes show sample IQR with median; whiskers extend to the most
extreme points within 1.5 IQR of the box; points beyond 1.5 IQR of the box are
shown as small circles. Red lines show the true $\psi$.

Under the conditions of fig. 52, it is remarkable how performance appears to
be almost entirely independent of AICc.

Figure 53: AICc and estimated $\psi$ for the standard models with simulated M
= 200 bats with winds variable, flight speed variable and cleared plot
searches. Boxes show sample IQR with median; whiskers extend to the most
extreme points within 1.5 IQR of the box; points beyond 1.5 IQR of the box are
shown as small circles. Red lines show the true $\psi$. Figure 54: AICc and
estimated $\psi$ for the standard models with simulated M = 200 bats with
winds variable, flight speed variable and roads & pads searches. Boxes show
sample IQR with median; whiskers extend to the most extreme points within 1.5
IQR of the box; points beyond 1.5 IQR of the box are shown as small circles.
Red lines show the true $\psi$.

##  Appendix D Mortality Estimation and the Distinction between $\psi$ and
$dwp$

Accurate estimation of the fraction of carcasses lying in the searched area
depends on both the probability that a carcass lies in the searched area
($\psi$) and the number of carcasses that were found in the searched area
($X$). For example, if $\psi=0.2$ and $X=2$ carcasses were found at turbine A,
the most likely total number of carcasses would be $\hat{M_{A}}=2/0.2=10$, but
$M_{A}$ could readily be anywhere in its 90% credible interval191919Credible
interval is a Bayesian analog to the confidence interval of classical
statistics. Credible intervals for $M_{A}$ given $X_{A}$ and $\psi$ can be
calculated in the dwp package using, for example, MCI(postM(x = 2, g = 0.2)).
of [4, 26], and the fraction of carcasses lying in the searched area at the
turbine ($dwp_{A}$) would likely be in the interval [2/26, 2/4] or 7.7–50%. By
contrast, if $\psi=0.2$ at turbine B too, but $X_{B}=20$ rather than 2, then
the credible interval for $M_{B}$ would be [73, 109], and the fraction of
carcasses lying in the searched area at turbine B ($dwp_{A}$) would likely be
in a much narrower interval—[20/139, 20/73] or 14.4–27.4%. $dwp$ properly
reflects the greater relative information content of $X=20$ compared to $X=2$,
but $\psi$ does not.

To illustrate in another, more realistic way and validate the accuracy of the
$dwp$ estimator, we ran sets of 1000 simulated collisions between bats and a
turbine and the resulting dispersion of carcasses on the ground near the
turbine. Carcass distances were generated according to a gamma distribution
with 60% of the carcasses lying within 50 m of the turbine and 90% within 100
m (that is, a gamma distribution with shape= 1.7744 and rate = 0.0355).
Carcass directions were uniformly distributed between $0^{\circ}$ and
$360^{\circ}$. The search area was confined to a fairly large turbine pad and
a moderately wide access road, where the visibility was good and search
conditions were easy. Carcasses were scavenged and removed from the field with
persistence times randomly distributed as Weibull(shape = 0.64, scale =
1.705), with median persistence time of 1 day and 90% of the carcasses removed
within ~6 days. Searches were conducted on the road & pad area out to 150 m
from the turbine every 5th day over a period spanning 150 days. Searcher
efficiency for a carcass in the first search after arrival was 80% and
$0.8\cdot 0.75^{k-1}$ on the $k$th search. Results for one set of 1000
carcasses are displayed in fig. 55.

Figure 55: Simulated carcass dispersion and discovery (n = 1000). The
horizontal tan strip in the center represents the searched area.

For each simulated data set with 1000 carcasses, ddFit from the dwp package
was used to fit a gamma distribution to the dispersion of the recovered
carcasses. Then, a 90% confidence interval for the probability that a carcass
would lie in the searched area was calculated using estpsi, a 90% confidence
interval for the fraction of carcasses lying in the searched area was
calculated using estdwp, and the actual fraction of carcasses lying in the
searched area was recorded. The process was repeated 1000 times to create 1000
confidence intervals for $dwp$ calculated using estdwp and 1000 $\psi$ using
estpsi. Following this process, each simulated 90% confidence interval should
have a 90% chance of including the actual fraction of carcasses in the
searched area.

Figure 56: Confidence intervals for the fraction of carcasses lying in the
searched area ($dwp$) using a. estpsi or b. estdwp from the dwp package as the
estimator.250 confidence interval are plotted for each, with black to indicate
CIs that cover the true $dwp$ and red to indicate missing the true $dwp$. The
blue lines in the center show the target, actual $dwp$ for each simulated CI.
The CIs are ordered by increasing target, actual $dwp$. Coverages are the
proportion of 1000 simulated CIs that cover the target $dwp$.

The proportion of intervals that do include (or “cover”) the actual parameter
(namely, the fraction of carcasses in the searched area) is referred to as the
_coverage_ and is an essential test of estimator performance. Fig. 56 displays
250 simulated CIs for the fraction of carcasses lying in the searched area
according to $\hat{\psi}$ and estdwp as estimators. The CIs based on
$\hat{\psi}$ appear to be too small and routinely miss the true dwp. This is
wholly expected because $\psi$ is the probability of a carcass lying in the
searched area rather than the actual, realized fraction of carcasses lying in
the searched area and is inherently much less variable than dwp. Coverage for
the nominal 90% CIs was only 26.8%. This is not a problem with $\psi$ or the
calculation of $\hat{\psi}$; rather, it is a confusion of $\psi$ as dwp and an
attempt to use $\psi$ to do dwp’s job. By contrast, coverage for the nominal
90% CIs based on dwp was 90.5%, which is as close to 90% as can be expected
for even a perfect estimator in such a small simulation.

##  Appendix E Accounting for the Uncertainty in dwp: Technical Details

To be able to use the software effectively, it is not necessary to understand
the details of how the dwp package accounts for the uncertainty in dwp. The
details are offered here for completeness.

The uncertainty in $\widehat{dwp}$ given the number of carcasses in the
searched area ($m_{\textrm{in}}$) and the probability of a carcass lying in
the searched area ($\psi$) is accounted for by sampling from the posterior
distribution of $M$ given $m_{\textrm{in}}$ and $\psi$ or, more precisely,

$\textrm{Pr}(M=m\mid
m_{\textrm{in}},\psi)=\frac{\textrm{Pr}(m_{\textrm{in}}\mid
M=m,\psi)\cdot\textrm{Pr}(M=m)}{\sum_{m}\textrm{Pr}(m_{\textrm{in}}\mid
M=m,\psi)\cdot\textrm{Pr}(M=m)}$

where $m_{\textrm{in}}\sim\textrm{binomial}(M,\psi)$ and $\textrm{Pr}(M)$ is
the integrated reference prior, $M\propto\sqrt{(}M+1)-\sqrt{(}M)$ (Berger et
al. 2012; Dalthorp et al. 2017). In practice, $\psi$ is not known but
estimated using ddFit from the dwp package. The uncertainties in $\hat{\psi}$
and $M\mid(\psi,m_{\textrm{in}})$ are assessed via a three-step process:

  1. 1.

simulate the regression parameter estimates from the fitted GLM as
multivariate normals (Nelder and Wedderburn 1972) to account for uncertainty
in the fitted model (using ‘ddSim‘, which is called by estpsi but can be
called directly if desired),

  2. 2.

integrate the simulated models over the searched area (estpsi) to account for
uncertainty in $\hat{\psi}$; and

  3. 3.

calculate a simulated
$\widehat{dwp}_{i}=\frac{m_{\textrm{in}}}{M\mid(m_{\textrm{in}},\hat{\psi})}$
for each simulated value of $\hat{\psi}$, using a random draw from the
posterior distribution of $M\mid(m_{\textrm{in}},\hat{\psi})$.

The vector of estimated dwp values generated in this way incorporates the
uncertainty in $M\mid(m_{\textrm{in}},\hat{\psi})$, yielding accurate CIs for
dwp.

###  E.1 $dwp$ and the Estimation of $M$

After being properly formatted (using the function exportGenEst), the
estimated $dwp$ can be used with GenEst to estimate total mortality. At a site
with a single search class with a single turbine, a simplified version of the
model202020The estimator for $M$ that is used in GenEst is difficult to write
down succinctly. Readers interested in finer detail are directed to the GenEst
technical manual (Dalthorp et al. 2018). for estimating $M$ is
$\hat{M}=X/(\hat{g}\cdot\widehat{dwp})$, where $X$ is the number of carcasses
found, $\hat{g}$ is the estimated detection probability for carcasses lying in
the searched area during the span of the monitoring field season, and
$\widehat{dwp}$ is the estimated fraction of carcasses lying in the searched
area.

Note that
$\hat{M}=X/(\hat{g}\cdot\widehat{dwp})=\frac{X/\hat{g}}{\widehat{dwp}}=\hat{m}_{\textrm{in}}/\widehat{dwp}$,
which incorporates both the uncertainty in estimating the number of carcasses
within the searched area ($m_{\textrm{in}}$) and the uncertainty in the
fraction of carcasses lying in the searched area ($\widehat{dwp}$). Thus,
$\hat{M}$ accounts for the two major sources of uncertainty discussed
previously, and in theory it should give accurate confidence intervals, so
that a 90% confidence interval should have a 90% chance of including $M$.

To verify the accuracy, we ran a simulation of the processes of carcass
deposition and discovery, field trials for estimating $g$, and distance
modeling to estimate $dwp$. Then, we used GenEst to estimate mortality for
each turbine and the total for the site. We tallied the number of 90%
confidence intervals that included $M$, calculated the _coverage_ as the
fraction of the CIs that included $M$, and compared the coverages with the
target coverage of 90%. The parameters for the simulation scenarios:
Simulation:

  1. 1.

generate carcass distances for $M=300$ or 1000 carcasses as
$~{}\textrm{gamma}(\alpha=1.774,\beta=28.17)$

  2. 2.

arrival at site with three turbines, with layouts as in fig. 57

  3. 3.

searches in within squares with radius 120 meters, days = 0, 5, …, 150

  4. 4.

Weibull persistence with parameters:

     * •

Easy visibility: $\alpha=0.64$, $\beta=1.705$

     * •

Moderate visibility: $\alpha=0.64$, $\beta=7.37$

     * •

Difficult visibility: $\alpha=0.64$, $\beta=9.47$

  5. 5.

Searcher efficency with parameters:

     * •

Easy visibility: $p=0.8$, $k=0.75$

     * •

Moderate visibility: $p=0.3$, $k=0.75$

     * •

Difficult visibility: $p=0.15$, $k=0.75$

  6. 6.

summary parameters

     * •

$\psi$ = 0.898, 0.254, 0.0967 for turbines t1, t2, and t3, resp.

     * •

$g$ = 0.283, 0.285, 0.179 for search classes E, M, and D, resp.

The estimated coverage probabilities range from 85% and 95% for all
combinations of mortality ($M=300,\,1000$) and field trial carcasses
($n=10,\,20,\,100$) at all turbines and the site as a whole (fig. 58), with a
single exception. Coverage for total mortality for the site as a whole when
$n=10$ and $M=1000$ was 84.8%, with 14.8% of the confidence intervals missing
high (that is, $\hat{M}_{lwr}>M$) and 1.8% missing low. This indicates a
tendency to slightly overestimate the total under these conditions. With
discrete random variables ($M$ and $n$), coverage probabilities for 90% CIs
are guaranteed to differ from the nominal 90% except for in rare
circumstances, so the small deviations from the target coverage is neither a
surprise nor cause for alarm, especially when $n=10$ (Madsen et al. 2019).
When $M=1000$, the CIs appear to be slightly too wide at turbine 3, as
coverages approach 95%, with approximately equal probability of the CI missing
high or low (fig 58).

Figure 57: Maps of search areas for three turbines. Dashed lines indicate
search boundaries. Figure 58: Coverages for nominal 90% confidence intervals
for the three turbines at a site and for the total (solid circles) and the
proportion of confidence intervals that fail to cover M, missing high
($\hat{M}_{lwr}>M$, indicated by $\times$) or missing low ($\hat{M}_{upr}<M$,
indicated by $\bigcirc$).

## Acknowledgements

We thank Bat Conservation International, and Avangrid Renewables for providing
the carcass location data used in our bat examples and Norwegian Institute of
Nature Research for carcass location data used in our eagle examples. We
appreciate the statistical advice provided by L. Madsen, although if there are
any errors, they are entirely our own. Funding for this research was provided
by the U. S. Geological Survey’s Ecosystems Mission Area Wildlife Program, the
U. S. Department of Energy, Bat Conservation International and the American
Wind Wildlife Institute. Any use of trade, firm, or product names is for
descriptive purposes only and does not imply endorsement by the U.S.
Government.

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