Homogeneity of rates does not mean that the actual number of fatalities must be identical at all turbines but that the numbers do not vary significantly. In this context, it is difficult to precisely define “significantly” because the effect that variation in rates has on the model depends on the sizes, shapes, and characteristics of the search plots, and tolerance for potential bias depends on the objectives of the research. In theory, variation in rates can adversely affect accuracy. In practice, however, even substantial variation in mortality rates among turbines will rarely cause significant inaccuracies in estimating dwp.

Variation in wind patterns, turbine types, turbine operations (for example, curtailed versus uncurtailed), carcass type, and other factors may result in variation in carcass dispersion patterns. We refer to factors that vary among turbines or carcasses and that influence the actual carcass distributions as interacting covariates. These factors can be explicitly accounted for in the carcass distance models when the search plot layouts are imported from shapefiles or as x-y coordinates on a grid. With the simpler data types (list of distances, simple geometry, or R polygons), interacting covariates can be accounted for by fitting separate models for different levels of the covariate. For example, bats and small, medium, and large birds may each require their own models; likewise for curtailed and uncurtailed turbines, large turbines and small turbines, site A and site B, etc.

Variation in carcass distributions by direction from a turbine is referred to as anisotropy. If not accounted for, anisotropy can be a significant source of bias when the shapes of the search plots also vary by direction. Accounting for anisotropy introduces a considerable degree of statistical complexity along with more extensive data requirements. Perhaps the most straightforward and flexible way to account for anisotropy would be via Poisson regression of carcass counts on a coordinate grid, as discussed in Maurer and others (2020). The dwp package (version 1.0.1) does not include special functions for handling anisotropy.

Problems with extrapolating fitted models outside the search radius are illustrated clearly in an example dataset in which 100 carcass locations were simulated. Distances were drawn from a gamma distribution in which one-half the carcasses are expected to fall within 41 m of the turbine and 82 percent within 80 m. Searches were conducted in a circular plot centered at the turbine and with radius 80 m. All 81 carcasses inside the search radius were found and none of the 19 carcasses outside the search radius were found (fig. 3). The dwp package was used to fit 17 models from the exponential family of distributions.

All but a few (constant density, chi squared, Maxwell-Boltzmann, and inverse Gaussian) of the 17 models pictured in figure 4, appear to provide fairly close fits to the carcass data inside the 80-m search radius (fig. 4, gray bars), but just outside the search radius there is wide divergence among the models (fig. 4, lines). Although the vast majority (81 percent) of the carcasses lie inside the search radius, there is no guarantee that the models can accurately predict the final 19 percent of carcasses that lie outside the search radius. Indeed, the actual fraction of carcasses lying inside the search radius is 81 percent, but the models predict anywhere from 0 to 100 percent (ψ^ in table 1), depending on the model.

A traditional staple in model selection is the Akaike Information Criterion or AIC (Burnham and Anderson, 2002), which is a measure of how well a model fits the data given the number of parameters in the model. We use AICc, which is a slightly modified version of AIC that may be more accurate than AIC when sample sizes are small (Cavanaugh, 1997). To facilitate easy comparison of models, we define ΔAICc as the difference between a model’s AICc score and the minimum of the AICc scores of all models to be compared. The model with the most parsimonious fit has ΔAICc=0 and serves as the standard for comparison.

Thus, ΔAICc confirms the visual impression that the constant, chi-squared, and Maxwell-Boltzmann models give relatively poor fits to the data (fig. 4). Among the remaining models, it is not clear which provide the best fits to the data inside the search radius. AICc does capture some of the subtle distinctions among the model fits inside the range of data. However, outside the range of the data, the models diverge radically from each other, with the estimated fraction of carcasses lying inside the search radius ranging from 0 to 93 percent for the models with ΔAICc < 3. AICc provides no guidance for assessing model performance beyond the search radius.

As a rule of thumb, for two models that differ by more than about 7 in AICc, the one with the smaller AICc is preferable, and models with scores differing by less than about 2 are often considered statistically indistinguishable by this measure (Richards, 2005). If we only consider models with ΔAICc < 2, the range of estimated proportions of carcasses inside the search area narrows only slightly, to 0–84.3 percent. It is evident that AICc has extremely limited utility³ 3

That utility may even be exaggerated in this example because the ΔAICc<2 is far too restrictive, eliminating from consideration some models that appear to provide only marginally poorer quality fits within the range of data than the “best” models but without regard to the shapes of the curves outside the range of data.

If we assume that the number of carcasses is finite, we can eliminate from consideration all models that do not converge to zero or do not converge fast enough to guarantee that the estimated number of carcasses outside the search area is finite. We refer to models that predict a finite number of carcasses outside the searched area as extensible because they can be extended outside the search radius to give properly defined probability distributions⁴ 4

This is discussed further in sections, “Extrapolation Outside the Searched Area” and “Examples,” and in appendix 1.

The estimated fraction of carcasses inside the search radius for non-extensible models is an impossible 0 percent. Among the extensible models—which, for this data, include the truncated normal, gamma, Rayleigh, log- normal, Maxwell-Boltzmann, normal-gamma, chi-squared, xep1, xep02, xep012, and xep123—the estimated fraction ranges from a merely implausible 3.3 to 100 percent (table 1).

Extensibility is a relatively permissive criterion that excludes only the most grossly inadequate models. Several dwp package functions automatically test for extensibility. Most notably, ddFit tags all fitted models as either extensible ($extensible = 1) or not extensible ($extensible = 0), and modelFilter returns a table of test results (including extensibility) for arrays of models to be compared.

Because gravity acts so inexorably on carcasses after turbine strikes, few carcasses are expected beyond 150 or 200 m (although this number will likely increase as turbines increase in height and rotor diameter.) However, because the search plots are inevitably finite and the models are fit with spatially limited data, sometimes a fitted model will perform well inside the search radius but have implausible tails, predicting far too many carcasses at great or short distances. A rough filter for tail plausibility identifies obviously inappropriate models. By default, models that predict that more than 1percent of carcasses lie beyond 200 m or more than 5 percent beyond 150 m fail the modelFilter for a plausible right tail.⁶ 6

If desired, users may override these default settings.

Because of the great size and height of turbines, we would expect a substantial fraction of carcasses to lie beyond 20 m and a significant fraction beyond 50 m. By default, modelFilter identifies models that predict that more than 50 percent of the carcasses lie within 20 m or more than 90 percent within 50 m as having implausibly heavy left tails. By this criterion, the chi-squared model is eliminated from further consideration for this dataset (table 1).

The tests of tail plausibility further narrow the range of the estimated proportion of carcasses inside the search radius from 3.3–100 percent to 69.5–93.4 percent (table 1), which is still a wide range but a vast improvement over using the extensibility criterion by itself (fig. 5).

In some cases, the apparently good fit of a model may strongly depend on one or a small set of data points, so that removing one or a small number of data points has a strong impact on the shape of the fitted model. The presence of these high influence points (Davison and Hinkley, 1997; Canty and Ripley, 2021) casts considerable doubt on the reliability of the model for the given dataset. Extrapolation would be especially worrisome for such models because their shapes are strongly dependent on one or a small number of fickle data points, and the effects would be felt well beyond the range of the data.

Several of the extensible models with plausible tails have points with high influence—the xep02, xep012, xep123, and normal-gamma models (table 1). Removing these from consideration further narrows the range of estimated proportions of carcasses inside the searched area to ψ^∈ (0.836, 0.931) (table 1) and eliminates much of the remaining clutter from the graphs of models still under consideration (fig. 5B–5C).

After models with implausible extrapolations or worrisome instabilities have been filtered out, criteria like AICc that measure the quality of a model’s fit strictly within the range of the data are useful for distinguishing among plausible models. AICc and other internal criteria are especially valuable when there are substantial unsearched areas inside the search radius, as would be the case with road and pad searches (Maurer and others, 2020) or with sites with forest or other thick vegetation, water, cliffs, poisonous snakes, or other features that render much of the area around turbines practically unsearchable. Predicting the proportion of carcasses in unsearched areas inside the search radius is an interpolation problem rather than extrapolation, and the quality of the fit to the data inside the search radius is directly relevant in a way that it is not when extrapolating outside the search radius.

In an extensive simulation study of the accuracy of carcass distribution models (app. 2 and 3), AICc was generally a poor guide for gaging the relative adequacy of models for extrapolating outside the search radius. Under some of the conditions tested, models with low AICc correlated with more accurate extrapolations; under other conditions, low AICc was correlated with less accurate extrapolations. However, under most conditions, there was no apparent correlation between AICc and suitability for extrapolation. One exception was that models with ΔAICc greater than (>) 10 frequently performed worse than models with ΔAICc < 10. For this reason, modelFilter compares model AICc scores and, by default⁷ 7

As with all modelFilter parameters, user has the option of overriding the default values.

After filtering out the lone extensible model with plausible tails, no high influence points, and ΔAICc > 10 (namely, the Maxwell-Boltzmann model), the two remaining models—truncated normal and Rayleigh—predict that, respectively, 84.3 and 83.6 percent of the carcasses lie inside the searched area. Both are in good agreement with the generating model in which 82 percent of the (simulated) carcasses are expected to lie inside the searched area. The fitted models closely match each other and appear to be good fits to the data (fig. 5). Also, the difference in their AICc scores is only 0.95, which is too small to make a statistically meaningful distinction between the models.

Several of the distributions are not well suited for modeling carcass distances. These include the inverse gamma (xepi0), inverse Gaussian, chi-squared, Pareto (xep0), and the exponential distributions. Because these distributions would rarely (if ever) be among the most appropriate models for carcasses, they are excluded from the dwp package’s list of 12 “standard” distributions but can still be used if specifically invoked. The standard models, which include xep1, xep01 (gamma), xep2 (Rayleigh), xep02, xep12, xep02, xep123, xep0123 (normal-gamma), truncated normal, Maxwell-Boltzmann, lognormal and constant, are the primary focus of this report. All the models are defined and briefly discussed in appendix 1.

In general, when a significant fraction (for example, >10 percent) of carcasses are expected to lie outside the search radius and extrapolation is necessary to estimate the fraction of carcasses lying in the searched areas, it is essential to consider the suitability of models for extrapolation, using broad criteria like extensibility, tail plausibility, and stability (for example, the presence of high-influence points) before considering strictly internal criteria like AICc. If the search radius is long enough so that few carcasses are expected outside the search radius and a substantial fraction of the area inside the search radius is unsearched, then AICc and high-influence points become the critical criteria.

To determine which models are likely to perform well or poorly in predicting carcass numbers outside the search radius, it is necessary to step beyond traditional model selection criteria and make use of more general information about the distribution of carcasses. Most notably, the number of carcasses generated by a turbine will always be finite, and models that predict infinite carcasses outside the search radius should be eliminated from consideration. We refer to models that can be extended beyond the search radius to infinity and still predict finite numbers of carcasses as “extensible.” Among the original 17 models, 11 are extensible for the example data. Although many of the remaining, non-extensible models (xep0 (Pareto), xepi0 (inverse gamma), xep12, exponential, inverse Gaussian, constant) perform well according to AICc (table 1) and provide good fits to the data, they are not suitable for extrapolation outside the search radius and should not be used in estimating the fraction of carcasses inside the searched area.

The example illustrates some of the power and utility of the diagnostics. However, it remains a single example, and results specific to this particular dataset may not be generalizable. For example, the lognormal model fits relatively well within the range of this data but was implausible for extrapolation because it estimated that 97 percent of the carcasses lay beyond the 80 m search radius. For this dataset, lognormal is implausible, but for other datasets it may be the best choice.

With the exception of the xy-grid data, all the data types are converted to a common ring structure for analysis. A rings data structure is a list with six elements:

The analysis of rings data begins with the fitting of distance models. By default, the ddFit function fits 12 different generalized linear models (GLMs) to the data. Specifically, the models are Poisson regressions of carcass counts in 1-m rings, using the searched area (or the searched area in a given carcass class) as the exposure and the natural logarithm of the exposure as an offset. The fitted GLMs are then converted to parametric distributions. The models are discussed in greater detail in appendix 1.

The data structure for layouts defined on coordinate grids rather than shapes are somewhat different, but the analyses are similar.

A layout of the site on a grid of (x,y) coordinates marking the centers of 1 m² or 2 m ×2 m quadrats are the most flexible for modeling, but they are difficult to work with because of the computer memory requirements and slow processing speeds. A properly formatted, small (x,y) gridded layout dataset (layout_xy) is available in dwp. It is a standard data frame with x and y coordinates on 1- or 2-meter grids overlaying each turbine (unitCol) at the site. The coordinates may either be relative to the turbine location (that is, each turbine is assumed to be located at (0, 0) according to the grid coordinates listed for the turbine), or the coordinates are UTMs with the turbine locations given in a separate file (file_turbine). The number of carcasses found in each grid cell also must be provided (ncarcCol).

The importing and formatting is accomplished via initLayout, and there is no need for further processing before proceeding to the analysis, as discussed in section, “The (x, y) Grid Data.”

In this example, the site layout and carcass location data are imported from shapefiles. Detection probability varies with search class on the ground, as specified in the “Class” column in the shapefile. It is not necessary to know or estimate the detection probabilities in each search class. Rather, Class can be specified as a covariate in the model, and the relative detection probabilities will be implicitly accounted for in the model.

The first steps are to import the data from shapefiles (initLayout), format the data into rings (prepRing), and tally the carcasses discovered in each ring (addCarcass).

Once the site layout and carcass location data have been imported and formatted, use ddFit to fit a wide array of possible carcass distribution models to the data and use modelFilter as an aid in selecting an appropriate model for estimating dwp. Detailed help on any of the functions can be found by entering, for example, ?ddFit or ?modelFilter in R.

Now that a model has been selected, use estpsi to estimate ѱ, the probability that a carcass lies inside the searched area. The Greek letter ѱ is “psi” in English and short for “probability that search area includes carcass.” estpsi estimates ѱ at every turbine and accounts for the uncertainty in estimating the model parameters.

The “density-weighted proportion” (dwp) is the fraction of carcasses falling inside the searched area. The expected dwp is ѱ, but the uncertainty in estimating dwp is greater than the uncertainty in ѱ due to variation in the actual fraction of carcasses falling in the searched area given the probability of a carcass falling in the searched area. The problem is similar to flipping a coin 10 times. We know the probability of heads is 0.5, but the actual number of heads in 10 flips varies. estdwp accounts for that uncertainty.

When the dwp is estimated, it can be formatted⁸ 8

For dwp, GenEst requires a .csv file with a column for turbine name and a column for dwp estimates for each turbine. The dwp estimates for each turbine may be either (1) point estimates (single number), or (2) simulated values that account for the uncertainty in the estimates. In the latter case, the turbine and dwp columns will be of length nturb×nsim.

A series of more detailed examples that demonstrate how to run analyses on different data types and to highlight some of the difficulties that may be encountered in a dwp analysis are given in sections, “Examples 2–6.”

Bundled with the dwp package is an example dataset compiled from eagle carcass searches over several years at a large site. The searches were conducted with dogs out to a radius of approximately 100 m. We defined the search radius to be 100 m and assumed the plots were circular. The example dataset is formatted as a data frame, layout_eagle, which consists of three columns: DateFound, turbine, and r (which gives the distance from the carcass to the nearest turbine). The data frame structure is optional.

After the raw data have been defined in a vector or a column in a data frame or array, they need to be formatted as a rings object (section, “Ring Structure”).

After the data have been formatted as a rings object, use ddFit to fit parametric distributions to the data.

Of the 12 standard models that are fit by default (app. 1), all but 3 are extensible for this dataset. With the exception of xep1 and the lognormal, all the extensible models are in close agreement with each other and fall within a tight band (fig. 6). Because the search coverage within 100 m of a turbine is 100 percent and there is no unsearched area inside the search radius, the estimated dwp is the estimated fraction of carcasses falling within 100 m, which is the y-value of the cumulative distribution functions (CDFs) at x = 100 (fig. 6). Thus, it can be seen that all models except for xep1 predict that more than 95 percent of the carcasses lie within 100 m, and thus there is little distinction among the models.

There appears to be little distinction among the fitted models for the eagle data beyond xep1 and lognormal predicting somewhat lower ѱ than the other distributions (fig. 6). The question is whether it is the xep1, lognormal, or one of the other seven models that is giving the most accurate prediction of ѱ and dwp.

A primary aid in model selection is the function modelFilter, which gives a table of filtering test results, which are discussed in section, “Tools for Model Selection” and illustrated in section, “Examples.” The filter gives a 0 (fail) or 1 (pass) for each test for each model, along with ΔAICc.

All extensible models passed all the filtering tests (value = 1). The lognormal had the lowest AICc, but all the extensible models had ΔAICc < 10. When there is little to no unsearched area inside the search radius as with the eagle data, AICc is of little to no value for distinguishing among models with ΔAICc < 10 because the dwp prediction does not involve any interpolation but is strictly a matter of extrapolation to the area outside the search radius. In general, the heavy-tailed lognormal and xep1 distributions tend to be less well-suited to eagle carcass distributions and should be used with caution (app. 3). The other fitted distributions are remarkably similar for the eagle data, with all predicting at least 98.3 percent of the carcasses inside the search radius, which is given as p_win in the summary stats table for the fitted models:

Thus, regardless of which model we choose (excluding xep1 and lognormal), dwp^ will be practically the same, so we will select xep01 (gamma) because, among the non-heavy-tailed distributions, it is the top model in terms of AICc. The fitted parameters of this model are in the parms element of the corresponding model object. Obtaining confidence intervals for parameters is demonstrated in section, “Model Fitting and Analysis.”

The probability that a carcass will lie inside the searched area can be estimated using estpsi, which returns a vector of simulated ψ^ values that account for the uncertainty in estimation of ѱ, and a confidence interval can be calculated using the R base function, quantile.

Then, dwp^ is calculated from the estimated psihat and the number of carcasses using estdwp. A confidence interval can once again be calculated using quantile.

As expected (app. 4), the confidence interval for dwp is wider than the confidence interval for ѱ. The dwp^  can now be formatted for GenEst and exported.

Simple layouts can be read from .csv files using initLayout or can be entered by hand and then preformatted using initLayout. There is an example of a simple geometry site description (layout_simple) bundled into the package. The dataset has search plots with different shapes at each of four turbines (fig. 7).

The layout_simple raw dataset is formatted as a data frame and needs to be preformatted using initLayout to convert it into a simpleLayout object which can be recognized by other dwp functions.

Once the site layout data have been preformatted using initLayout, use the prepRing function to process the layout data into 1 m concentric rings with the area (in square meters) searched in each ring at each turbine.

After formatting the site layout data into rings, use addCarcass to incorporate carcass data into the rings data structure for analysis. The carcass data for simple geometry layouts should be organized in a data frame with each row representing a carcass and columns for the turbine ID and the distances from the turbine at which the carcasses were found.

After adding carcasses to the site layout formatted as a rings object using prepRing and add_carcass, the carcass distribution models can be fit using ddFit.

The lognormal distribution predicts only 30.6 percent of the carcasses to lie within the maximum search radius among the turbines (120 m), and the gamma predicts 82.8 percent. All other extensible models predict in excess of 95 percent (fig. 8) within 120 m. If there were very little unsearched area within 120 m, then the six extensible models other than lognormal and gamma would give the same >95 percent for  dwp^, and there would be no practical difference among them. However, there is a great deal of unsearched area within the maximum search radius at the site. The models are quite different from one another between about 40 and 100 m from the turbine, and the choice of model is likely to have a significant impact on the resulting dwp prediction.

The first step in model selection is to use the modelFilter function.

The fitted lognormal and gamma (xep01) distributions evidently have implausibly heavy right tails, as they both fail the rtail model filter test (table 2). There is at least one high-influence point for the xep02 and xep1 models, thus failing the hin test (table 2). Only three models—the truncated normal (tnormal), xep123, and Rayleigh (xep2)—pass all the diagnostic tests. The truncated normal and xep123 CDFs are virtually indistinguishable, whereas the Rayleigh differs markedly (fig. 8). Because there is a great deal of unsearched area at the site, prediction of ѱ and dwp relies strongly on interpolation, so a good fit within the range of the data is crucial, and models with large AICc scores—like the Rayleigh (xep2)—can be eliminated. The truncated normal has the lowest AICc score among the models that pass all the diagnostic tests, and that is the one we select for calculating dwp^ for these example data.

Having selected the truncated normal distribution (tnormal) for each turbine, we can predict the probability that a carcass lies inside the searched area using estpsi. Because estpsi integrates the selected model over the searched area (as with the “volcano” in fig. 2), both the searched area (formatted as rings_simple) and the selected model are required as arguments to estpsi.

The prediction of dwp depends not just on the probability of a carcass lying inside the searched area (ѱ) but also on the number of carcasses discovered inside the searched area, so the estdwp function requires both ψ^ and the carcass counts.

The estimated dwp’s for each turbine can now be plotted (fig. 9) and exported to a .csv file for use in GenEst.

If the shape of the search plots is somewhat complex, but there is only one search class, you may import your site layout data in R polygon format. An R polygon is simply two columns of data giving the x and y coordinates of the vertices of a polygon. For dwp analysis, the polygons must be associated with turbines in a standard data frame with three columns:

There must be one polygon for each turbine searched, and polygons must not be self-intersecting. A polygon site layout dataset with R polygons is included as part of the dwp package and can readily be loaded and viewed:

To begin the analysis, the polygons must be converted to a polygonLayout object which has been error-checked and converted to a properly formatted list. This can be done using the initLayout function with argument dataType = "polygon". Once the polygon layout has been formatted, it can be plotted (fig. 10) and prepared for GLM analysis by tallying the searched area in concentric 1m rings around each turbine using the function prepRing.

A simulated carcass dataset for this polygon site (carcass_polygon) is included in the package. It is a simple, two-column data frame with each row representing a carcass and columns for turbine ID (turbine) and distances from the turbine (r, in meters) at which carcasses were discovered in the carcass surveys. The carcass data are added to the ring structure via addCarcass to complete the data preparations for analysis.

After adding carcasses to the formatted rings_polygon, carcass distance models can be fit using ddFit and assessed using these functions:

For details on the use of these and other useful functions in the package, enter ?dwp.

The extensible models give a broad range of predictions of the probability of a carcass lying inside the search radius, from a low of 2.3 percent for the lognormal distribution to a high of 100 percent for the Maxwell-Boltzmann. The modelFilter provides useful guidance on model selection (table 4).

The lognormal model failed the right tail test because it predicts implausibly low probabilities of carcasses lying within 80, 120, 150, and 200 m, whereas the Maxwell-Boltzmann model failed the left tail test because it predicts implausibly high probability of carcasses lying within 50 m.

The xep02 and xep012 models fail the hin test (table 4) because they have high influence points, that is, points that are out of line with the others according to the model but that have a significant impact on the shape of the fitted curve. This is a red flag that signals an instability in the model for the given data.

The remaining fitted distributions (tnormal, xep1, xep123, xep2) are broadly comparable according to the diagnostic criteria tested. However, the distributions differ markedly between 30 m and the maximum search radius of 85 m (fig. 11), so other considerations come into play. In particular, there is a tendency for the light-tailed distributions to overestimate ѱ when the search radius is short or moderate and there is substantial unsearched area inside the search radius as with this example (app. 3), so the Rayleigh would not be recommended here. Also, when there is substantial unsearched area, it is important to have a low AICc score, so tnormal and xep1 are to be commended over xep123.

The two remaining models (tnormal and xep1) have AICc scores differing by only 0.65, well below a minimal threshold to be statistically meaningful. The xep1 distribution has a heavier tail (fig. 11) but the truncated normal has much more frequency of very small ψ^ values than does xep1 (fig. 12) and therefore a less stable fit.

After selecting the xep1 model, we can calculate dwp^ and export for use in GenEst or other programs using this parameter estimate.

The initLayout function reads GIS shapefiles using the st_read function from the sf (simple features) R package for importing and managing GIS data. The importing of the shapefiles and subsequent formatting of the data for dwp is done behind the scenes in the package and does not assume that the user has any familiarity with the sf package. However, users with a good working knowledge of sf may be able to use that to their advantage in managing the data and displaying results.

The importing and formatting of shapefile data involves four steps in R:

To read shapefiles, the initLayout function requires “simple features” files, including a polygon or multi- polygon shapefile that delineates the area searched at each turbine and a points file that gives the locations of the turbines on the same coordinate system as the polygons. The shapefiles must have their three standard component files (.shp, .shx, and .dbf) stored in the same folder.

The polygons and turbine point files must have a column with turbine IDs, which must be syntactically valid R names, that is, contain combinations of letters, numbers, dots (.), and underscores (_) only and not begin with a number or a dot followed by a number. Other characters are not allowed: no spaces, hyphens, parentheses, etc. are allowed in turbine IDs.

Search areas may be defined for any number of search classes that affect detection probabilities on the ground but are not expected to affect the distances that carcasses fly after being struck, for example, vegetation type or ground texture. Interacting covariates that affect the actual distances that carcasses fly—like wind speed, turbine height, carcass size—present special difficulties that are difficult to model using the kinds of automated procedures used in most of the convenient dwp functions and are not implemented in the basic package functions. Savvy R users may be able to use the dwp functions as a basis for more elaborate analyses, including accounting for anisotropy (or carcass dispersion patterns that vary with direction from turbines) (Maurer and others, 2020) or analyzing dependence of dispersion on other interacting variables.

The shapefiles for site layout should include at a minimum a turbine ID and a “geometry” that gives the coordinates of the vertices for polygons that delineate the search areas. Other covariates (for example, search class, species, size, turbine operational constraints) are optional. When a shapefile is read into R using sf::st_read (as with dwp’s initLayout), it should look something like the following, showing necessary geographic information in a header and a summary of the first 10 polygons or “features”:

Shapefiles are read into R using the initLayout function with the names of the search files (including the file path if they are not stored in your working directory in R) along with the name of the column that contains turbine IDs. Other arguments for initLayout are for reading other file and data formats and are ignored when reading shapefiles.

Notice that the turbine IDs (1, 2, …) in the raw data file (shown above) are not syntactically valid R names. initLayout will convert these to simple, syntactically valid names, and alert the user that conversion has taken place. In general, users should use the initLayout function rather than sf::st_read directly for importing the shapefiles because initLayout employs some additional error-checks and formats the data for further processing and analysis:

The polygons in the searchpoly.shp shapefile are expressed as collections of (x, y) coordinates in meters relative to a reference point. In this example, there are a total of 178 polygons (or “features”) that delineate distinct, non-intersecting search areas at 23 turbines. Search classes include Easy, Moderate, Difficult, Very Difficult, and Out. Maps of the searched areas at three of the turbines are shown in figure 15.

After the data are imported into R, they must still be formatted for analysis:

For a large site with multiple search classes, data formatting may take several minutes. The result is an S3 object of class rings, which features a characterization of the searched areas at each turbine, including the total area and number of carcasses in each search class in each concentric, 1-m ring from the turbine to the search radius (stored in rdat, which is used in fitting the distance models) along with some other data. The ring structure is efficient and convenient for analysis. A full description of the data structure is provided in the section, “Rings Structure.”

After formatting the search plot configurations into rings for analysis, carcasses must be added to the rings. If the search plots are delineated into separate search classes, carcasses must be added by a two-step process of (1) importing the shape data, and (2) adding the data to the rings. If there are no search class distinctions, then carcasses may be added via a data frame with columns for turbine ID (unitCol) and for the carcass distances from turbine (r).

When reading carcass observation data from a shapefile into a rings structure with search class distinctions, dwp assigns each carcass to the proper ring and search class. The rings data are required, but there are two ways to insert the carcass data into the rings. The recommended approach is to provide the plotLayout data. Then, addCarcass determines the proper ring and search class to assign each carcass to by calculating the distances from the carcass (x, y) coordinates in the carcass observation data to the turbine locations stored in the plotLayout data, which has been previously read and formatted via initLayout and prepRing. Search class assignment is determined by the search class in plotLayout that is associated with the carcass (x, y) coordinates. If a carcass location falls in an unsearched area, the addCarcass function aborts with an error. The calculations are all done safely “under the hood” and directly from the imported shapefile data without needing further user input.

An alternative approach would be leave out the plotLayout data (plotLayout = NULL) to add carcasses directly from the carcass observation data into the rings data. Carcasses are assigned to rings based on distances calculated from the carcass locations (x, y) coordinates in the carcass observation data to the turbines (x, y) coordinates in the $tcenter array in the data_ring. Carcasses are assigned to search classes as indicated by the scVar in the carcass observation file. This can introduce errors in interpretation, because the model assumes that the “search classes” apply to the characteristics of the general area where a carcass is found (fig. 15), but often the search classes that are tagged to individual carcasses in carcass observation data apply to the search conditions in the immediate vicinity of carcasses. For example, if a carcass is found in a small Difficult patch in a large Moderate area, and the site map (plotLayout) includes only that Moderate area but not the Difficult patch, the carcass should be tallied as Moderate. This potential problem in data interpretation is not an issue if the plotLayout is included in the addCarcass function call.

If there are no class distinctions among the search areas, carcasses may be added to rings from a data frame with columns for turbine ID and carcass distance. For example, the cod argument in the addCarcass function call may be a data frame rather than an imported carcass location shapefile:

Now that the site layout data have been formatted into rings and the carcass data have been inserted into the ring structure, the model fitting and analysis can proceed. For most of the input formats (excepting the grid data), the analyses follow the same path regardless of the initial data format. For the Casselman data, we do a basic analysis first but then indulge some more sophisticated analyses as an introduction to some of the more advanced questions that can be addressed using the software.

The basic modeling and analysis is similar, regardless of the initial data format. Several tools for fitting, interpreting, and evaluating models are available. However, the shapefiles allow covariates to be included in the models, enabling greater power to address more difficult and varied questions and to account for important factors that the other data formats cannot easily address.

The analysis begins with the fitting of distance models using ddFit with ring data or grid data. The basic function call requires only a formatted dataset as returned by the prepRing function (or initLayout for grid data). With the shapes data (rings_shape, as imported and formatted earlier) in the example discussed in this section, the search plots are divided into areas with different detection probabilities or search classes, which are stored in the Class column in the data and should be included in the model as scVar = "Class".

Search class variables (like vegetative cover, for example) affect the detection probability of carcasses but are not expected to affect how far carcasses will lie from the turbines. There is no need to estimate the detection probabilities for the various search classes; the model automatically estimates relative detection probabilities among search classes and accounts for the differences. Although the relative detection probabilities are estimated (for example, the odds of detection in class A may be 2.52 times as great as in class B), the absolute detection probabilities (g in GenEst and eoa) are not. A simple example of fitting distance models using ddFit is given below. A number of options to ddFit are available; enter ?ddFit in R for a more detailed information on arguments and return values.

As the models are fit, their names are listed on the console as either “extensible” or “non-extensible,” depending on whether or not the model can be extended outside the search radius and converted to a probability distribution. Non-extensible models generally are not appropriate for dwp analysis but may be interesting in their own right. The plot function can be used to graph the fitted models (fig. 16).

The default for plot is to graph the cumulative distribution function (CDF) for each extensible, fitted model, although any subset of models (for example, distr = c("xep01", "xep1")) also may be plotted (fig. 17). By default, the best model according to a default model filter (?modelFilter) is highlighted, but any model can be highlighted at the user’s discretion by setting the mod_highlight argument equal to the name of the model. Alternatively , the probability density functions (type = "PDF") rather than the CDF can be plotted, but the probability density is limited strictly to inside the search radius (extent = "win"). For more details on plotting options, enter ?plot in R.

To estimate ѱ, the probability of a carcass lying inside the searched area and dwp, the fraction of carcasses lying inside the searched area, a model must be selected. If it is desired to account for all carcasses lying outside the areas searched—including areas outside the search radius—an extensible model is required.

The graph of the CDF shows the estimated probability that a carcass lies within a given distance from the turbine. In this example, the two heaviest-tailed models (lognormal and xep1) predict about 20 percent of the carcasses lie outside the search radius of around 88 m, whereas the other extensible models predict virtually zero carcasses outside the search radius. In some datasets, choice of model can have a profound impact on the estimated fraction of carcasses falling inside the search radius, dwp^. Some of the important issues involved in model selection are discussed in the appendixes. The modelFilter function is designed to address many of the issues.

The probability that a carcass lies inside the search radius (srad)—or any other specified distance—can be estimated for any of the extensible, fitted distributions using the pdd function, which is an analog to pnorm for distance distributions.

The lines shown in figures 16 and 17 are for the maximum likelihood estimators of the models, which represent the most likely or “best” estimate for the given distribution. However, due to random variation, the distributions are estimated with uncertainty. That uncertainty can be accounted for by simulating from the asymptotic distribution of the parameter estimators using the ddSim (distance distribution simulation) function, and confidence intervals can be constructed (fig. 18).

It is also straightforward to calculate confidence intervals for quantiles of distance distributions using the qdd function, which is the analog of qnorm but for distance distributions. For example, to calculate 90 percent confidence intervals for the 90th percentile of distances at which carcasses lie, we can use the same sets of simulated parameters used in estimating the probabilities that carcasses lie inside the search radius, and calculate the 90th percentiles as qdd(0.9, ...):

Thus, according to the xep02 model, carcasses had an estimated 90-percent probability of falling within 56.1 m of the turbine with a 90-percent confidence interval of [51.4, 63.5] m (qdd(0.9, dmod_shape["xep02"])), whereas the lognormal model estimates a substantially greater distance of 118.5 m with a confidence interval of [82.1, 221.1] m (qdd(0.9, dmod_shape["lognormal"])). The xep01 model was similar to the xep02, and the xep1 more like the lognormal (fig. 19).

The pdd function can be used to estimate the CDF of carcass distribution as a function of distance from a turbine, which is related to but distinct from the probability, ѱ, that a carcass lies inside the searched area. ѱ is estimated by integrating the PDF of the fitted carcass distribution over the areas searched at each turbine. This can be accomplished for any fitted model and site layout using the estpsi function, and a summary figure can be viewed using plot (fig. 20).

The probabilities of lying in the searched area ranged from around 55 percent for turbines t11 and t15 to 85–90 percent for turbines t17–t23. The combined total was around 75 percent. More precise statistics can be extracted from the psiHat object returned from the estpsi function. In this case, that is psi01, which is a matrix with a column of nsim simulated values,ψ^, for each turbine, with the uncertainty in the estimation of ѱ at each turbine captured by the variation in values in the turbine’s column.

The ψ^values for xep01 and xep02 varied by a few percent (fig. 21), although the 0.005 quantiles sometimes differed from one another by more than 10 percent.

The expected fraction of carcasses lying inside the searched area (that is, the mean dwp) is equal to the probability of a carcass lying inside the searched area (ѱ), but the uncertainty in the estimate of the actual fraction of carcasses lying inside the searched area ( dwp^) is greater than the uncertainty in the estimated underlying probability (ψ^). Recall, the situation is analogous to a coin toss. We know the probability of a head (equivalent to ѱ) is 50 percent, and the expected fraction of times a coin comes up heads (equivalent to dwp) in n coin tosses is n/2. However, the actual fraction of heads in a sequence of n flips will usually not be exactly n/2 but a little more or a little less. That additional, binomial uncertainty, in addition to the uncertainty in estimating ѱ, needs to be captured in our estimate of dwp (Maurer and others, 2020). The distinction between ѱ and dwp is discussed further in appendix 4.

The uncertainties in estimating ѱ are captured in estpsi, and the uncertainties in estimating dwp given ѱ are captured using the estdwp function. The function requires the estimated ѱ and the number of carcasses observed at each turbine. The number of carcasses is required because the fewer the carcasses, the greater the relative uncertainty. Again, the coin toss analogy is illustrative. It would not be surprising to get anything from 0 to 100 percent heads in 4 coin tosses, but it would be highly unusual to get less than 35 percent or more than 65 percent heads in 40 coin tosses.

The fraction of carcasses lying inside the searched area at each turbine is calculated using estdwp:

The structure of the dwp estimate is the same as that of ψ^, namely, a matrix with a column for each turbine, incorporating the uncertainties in the estimated dwp’s for each turbine and for the total at the site.

Paired boxplots of ψ^and estimated dwp clearly reveal the greater uncertainty in estimating dwp compared to ψ (fig. 22).

The estimated dwp can be formatted for use in GenEst either by setting forGenEst = TRUE in the argument list for estdwp, or by using the formatGenEst function on the estimated dwp.

The formatted dwp01 can now be used as data_dwp = dwp01 in the argument list in a command line call to the estM function in GenEst (assuming that the other required arguments can be supplied as well) or can be exported to a .csv file for use in the GenEst GUI using the exportGenEst function.

In the previous example, we included a covariate to account for variability in carcass detection probabilities on the ground. Interacting covariates—that is, variables that affect the actual distributions of carcasses rather than simply affecting detection probabilities on the ground—present challenges that are difficult to overcome in a generic way that would be appropriate for automation in dwp.

The Casselman data provide an example of an interacting covariate, which can be analyzed by fitting separate sets of models for different covariate levels. In particular, the site represented in the “shapes” files has turbines that were under two different operating protocols, freely operating or curtailed at low windspeeds. We would expect that the carcasses at the curtailed turbines would be distributed farther from the turbines than carcasses at the non-curtailed turbines because 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 ψ^.

The subset function from the dwp package can be used to split the layout_shape dataset into two parts—freely operating turbines (layout_free) and turbines curtailed at low wind speeds (layout_curtailed).

The two datasets can then be analyzed separately and the results compared:

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 ΔAICc.

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. 23). The magnitude of the estimated differences strongly depended on the selected model (fig. 24A). The differences among the models largely vanish when the models are restricted solely to the range of the data and not extrapolated outside the search radius (fig. 24B). This indicates that the discrepancies among models primarily are due to the extrapolation outside the search radius and the models’ respective assumptions about the distribution outside 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 outside the search radius, whereas the light-tailed distributions (truncated normal and Maxwell-Boltzmann) predict relatively few carcasses outside the search radius.

Because the search radius is not long (square with half-width of 60 m and diagonal radius of 82 m), one of the heavier-tailed distributions with degree less than 2 is preferred (xep1, xep01, or lognormal). The light-tailed distributions (Maxwell-Boltzmann and the xep models with degree 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 datasets, the xep01 (gamma distribution) was the highest-ranking, relatively heavy-tailed distribution and had ΔAICc < 2. Thus, the xep01 distribution is an appropriate choice for both types of turbine.

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

The estimated ѱ and dwp for the free and curtailed turbines depend 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 ψ^ 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 outside the search radius. Next, the uncertainty introduced in estimating dwp erases the clarity of the distinction between curtailed and free turbines.

After calculating the dwp^’s for curtailed and free turbines separately, the estimates combined into one data frame for export for GenEst.

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 (σ1/σ2), regardless of the quantile. Thus, measuring by the fitted Rayleigh distribution, carcasses at the curtailed turbines were σcurtailed/ σfree=1.258 times more distant from the turbines than were the carcasses at the free turbines. That is, the median, 90th percentile, 99th percentile, and so on were all 25.8 percent greater at the curtailed turbines than at the free turbines.

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 percent on average at the curtailed turbines due to systematic bias in the estimates of ѱ (fig. 26).

Carcass size also may 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 datasets with size class distinctions and exporting the resulting dwp estimates to GenEst.

The following sections of code show how to split a dataset 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 dataset 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.

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.

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 ѱ and calculate dwp^ for use in GenEst when there are distinctions among carcass classes, the dwp’s must be estimated 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 dataset into addCarcass) to the description of the site by rings (data_ring) with a directive to split the datasets 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 ѱ, dwp, and format the results for exporting to GenEst.

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, “Modeling Fitting and Analyses” and “Additional Analyses for an Interacting Covariate” ), the xep01 model scores well in the modelFilter for both size classes:

The same caveat about the tendency of light-tailed models to overestimate the ѱ—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, "Example 3—Simple Geometry,” “Example 5—The (x, y) Grid Data,” “Additional Analysis for an Interacting Covariate,” and appendix 3 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 ѱ for each carcass class.

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 ѱ has been estimated, dwp^ can be calculated and formatted for GenEst: