The Issue of Using Ordinal Quantities to Estimate the Vulnerability of Seabirds to Wind Farms

Abstract

The article is a follow-up study to research on analysing methodological approaches to estimate anthropogenic impacts on marine biota. The work examines relevant publications about the vulnerability of seabirds to the effects of offshore wind farms, relying upon the provisions of the measurement theory and the median theorem for ordinal quantities. It has been shown that the final ranking of sensitivity of certain bird species to wind farms, or indices of a risk of collisions with turbine blades and indices of a risk of birds’ displacement by wind farms, can vary at permissible monotonic transformations of the values of initial factors since all of these are estimated on ordinal scales. We conclude such estimates are incorrect. The summation of the two indices (exposure to collisions with turbine blades and to habitat change) of the birds’ species vulnerability in the context of the proposed models is incorrect. It has been demonstrated that the model based on dividing factors into primary and aggravation, when the latter are incorporated into the formula of computation through the exponent of primary factors, is incorrect: when primary factors have maximum values (equal to 1), the effect of aggravation factors is no longer taken into account; with some values of factors in the model, infinite vulnerability values can be obtained. The models are to operate within the entire presented range of initial data, but that is not fulfilled. Thus, none of the analysed models are correct, and an approach is needed, based on the use of metric values. Considering that the bird flight altitude primarily determines the impact of wind farms on birds, the proposals have been formulated to take into account the impact of various factors on the flight altitude.

1. Introduction

The present article is an extension of our first research on analysing basic methodological approaches in the estimation of anthropogenic impacts on marine biota [1], where relevant publications about the exposure of birds to oil spills were analysed. Offshore wind farms can also greatly affect populations of seabirds through bird mortality due to collisions with turbine blades and/or disturbance of avifauna habitats caused by vast fields of such installations. Conceivably, the construction of offshore wind facilities will become the largest technical project in Europe for marine habitats. The first wind farms were built in the early 1990s in Danish and Swedish coastal waters [2]. By the end of 2020, 5402 wind turbines had been built (116 wind farms), grid-connected in 12 European countries. Eight new marine wind power projects reached a final investment decision in four countries in 2020, and construction will commence in the years ahead [3].

Methodological approaches to the estimation of the effect of oil spills and marine wind farms on avifauna are very much the same. Additionally, mathematical models of computing vulnerability to one type of impact can be used in such computations for the other type of impact. Hence, of importance is the experience of estimating the vulnerability of seabirds to wind farms, and we will consider in detail the publications [2,4,5,6,7,8], related to this issue. The model described in [7] and presented as the “universal approach” (“transparent and tractable method to transpose the factor-mediated vulnerability assessment methodology into many different case studies”) is employed in publications on an estimation of the effect of oil spills on seabirds [9,10]. As in our first work [1], we do not provide a detailed review of publications on the impact, in this case, of wind farms on seabirds. References to studies on the impact of such anthropogenic objects on birds and wildlife, in general, can be found in [8]. This publication, in terms of the approach to assessing the sensitivity of wildlife to wind farms, is considered by us very briefly since the methodology recommended there is based on a simple summation of the sensitivity values as ordinal quantities. Such approach to calculations is analysed by us in [1] and it is shown that results based on arithmetic operations with such values are incorrect.

In analysing publications about the vulnerability of seabirds to wind farms, as in our first article [1], we proceed from the main provisions of the measurement theory (for more details, see [11,12,13,14,15,16,17,18,19,20,21,22]). One of the major postulates of this theory—a requirement for the algorithms of processing data—is stated as follows: conclusions are drawn based on the data, measured on the scale of a certain type, must remain unchanged at data transformations, permissible on this scale [15,16]; this is the first provision. The second dwells on arithmetic operations with quantities on an ordinal scale—addition and multiplication of ordinal quantities are indeterminate and impermissible. The third—the results, obtained by mathematicians as related to computing an important statistical characteristic of data arrays—their mean value on various measurement scales. In particular, only the terms of variation series (order statistics) constitute permissible means on an ordinal scale. We will further utilise these three provisions, having preliminarily justified the third provision. Ignoring them leads to partially wrong results and uncertain final conclusions (see [1]). The International Metrological Dictionary states as follows “…ordinal quantity: quantity, defined by a conventional measurement procedure, for which a total ordering relation can be established, according to magnitude, with other quantities of the same kind, but for which no algebraic operations among those quantities exist. NOTE 1 Ordinal quantities can enter into empirical relations only and have neither measurement units nor quantity dimensions. Differences and ratios of ordinal quantities have no physical meaning” (as quoted from [21]).

Our publication [1] examined, based on the measurement theory, the materials and the conclusions of the articles on the vulnerability of birds to oil [23,24,25,26,27]. It had been shown that ranking of the vulnerability of seabird species to oil spills, as a sum of factors [23], changes, if permissible transformation—a monotonic decrease in values, namely, replacement of scores 1, 3, 5 with 1, 2, 3, is applied to the initial data, obtained on an ordinal scale. The order of final vulnerability of bird species (what species is more vulnerable, what is less vulnerable) also changes, if both addition and multiplication of ordinal quantities are further performed when calculating (it has been demonstrated by the example of the model used in [25]). It is confirmed by the specified transformation of the initial data. Moreover, a conventional example has shown that a simple comparison between the products of two metric quantities on a ratio scale and the products of their corresponding ordinal values indicates incorrect results in the products [1].

In our first publication [1], we did not examine in detail the constraints on ordinal quantities as related to the calculation of the arithmetic mean value. The present article emphasizes the calculation of the arithmetic mean value on an ordinal scale since the publications discussed below [2,4,5,6,7] rely upon the use of these means. With allowance for this, the principal requirements for computing arithmetic mean on various scales are first considered. Furthermore, there has once more been given an illustrative example of comparing the products of ordinal and products of metric values of an array of two quantities. However, if in the work [1], each array of such quantities was comprised of three values, now the arrays of initial data, containing five values each, are examined, which conforms with the values of factors in the publications we have analysed. Then, based on the presented theoretical provisions of the measurement theory, general considerations, major methodological approaches, and mathematical models for computations, used in the mentioned publications, are examined.

The aim of the study: to analyse methodological approaches in articles on the vulnerability of seabirds to the impact of offshore wind farms, based on the provisions of mathematics to ordinal quantities.

2. Computation of Arithmetic Mean Values of Ordinal Quantities and Products of These Quantities

2.1. Computation of the Arithmetic Mean Value

Among all the methods for analysing data, the averaging algorithm has a significant place. For management decision-making, mean values are used, to replace an array of several numbers with one and compare arrays using means. To assess generalised indices in many mathematical models and algorithms, mean values, including arithmetic means, have also been employed.

There are numerous different mean values—the arithmetic mean, mode, median, geometric mean, harmonic mean, quadratic mean, … [28]. In the 1970s, it was found, which types of mean values can be used when analysing the data, measured on various scales. The arithmetic mean $X_m=\frac{X_1+X_2\dots +X_n}{n}$ is frequently used. Its use is so customary that the second word in the term is often omitted, and the mean is spoken of, implying the arithmetic mean. Such a tradition can lead to erroneous conclusions [16]. In all the works on the vulnerability of birds to wind farms, discussed below, initial data are evaluated on an ordinal scale, and arithmetic mean values of these data are used. However, there is no way to use such the mean on an ordinal scale. Let us demonstrate this, citing in detail the excerpts from the works of the Soviet and Russian mathematician A.I. Orlov [15,16].

According to A.L. Cauchy: any function $f\left(X_1,X_2,\dots X_n\right)$ is the mean value of the array $X_1,X_2,\dots X_n$, such that, at all possible values of arguments, the value of this function is not lower than the minimum of numbers $X_1,X_2,\dots X_n$, and not higher than the maximum of these numbers. All the types of mean values, enumerated in the above paragraph, are the Cauchy means.

It is apparent that permissible transformations of the scale (see [1]) change the value of the mean quantity. Though the conclusions on which array has the higher, or the lower mean, must not change (according to the condition of invariance of conclusions, assumed as the key requirement in the measurement theory).

Let function $f\left(X_1,X_2,\dots X_n\right)$ signify the Cauchy mean for the quantities in parentheses, and the mean for the first array is lower than the mean for the second array:

$$f\left(Y_1,Y_2,\dots ,Y_n\right)<f\left(Z_1,Z_2,\dots ,Z_n\right)$$

(1)

where $Y_1,Y_2,\dots Y_n$ and $Z_1,Z_2,\dots Z_n$ are two arrays of ranking (score and, actually, ordinal quantities) estimates of one and the same object. Then, pursuant to the measurement theory to make the result of comparing means reliable, it is imperative that, for any permissible transformation $g$ from the group of permissible transformations on a respective scale, the following inequality be true:

$$f\left(g(Y_1),g\left(Y_2\right),\dots g(Y_n)\right)<f\left(g\left(Z_1\right),g\left(Z_2\right),\dots ,g(Z_n)\right)$$

(2)

i.e., the mean transformation of the values from the first array must also be less than the mean transformation of values for the second array. The formulated condition therewith must be met for any two arrays $Y_1,Y_2,\dots Y_n$ and $Z_1,Z_2,\dots Z_n$ and any permissible transformation $g$. Mean values, satisfying the formulated condition, are permissible on an appropriate scale. As per the measurement theory, only permissible mean values can be used when analysing the opinions of experts and other data, measured on a particular scale. The median theorem provides the answer as to which mean quantities are permissible for an ordinal scale. According to the publication [16], a somewhat simplified version of the median theorem as well as its complete and exact text [15] is given in the Appendix A to this article.

The median theorem. Among all the Cauchy means, only the terms of variation series (order statistics) constitute the permissible means on an ordinal scale.

According to this theorem, the median (with an odd sample size) can be employed in particular as the mean for the data, measured on an ordinal scale. When the sample size is even, it is worthwhile using one of two central terms of the variation series: left or right median. Sample quartiles, minimum and maximum, deciles, etc., can be used. However, the theorem prohibits the use of the arithmetic mean, geometric mean, etc., for ordinal quantities (i.e., measured on ordinal scales).

Let a numerical example be given, illustrating inappropriate use of the arithmetic mean $f\left(X_1,X_2,X_2\right)=\left(X_1+X_2+X_2\right)/3$ on an ordinal scale. E.g., there is an array of data 1, 2, 3. Let $Y_1=Y_2=Y_3=2$, $Z_1=1$, $Z_2$=1, $Z_3$= 3. Then, $f\left(Y_1,Y_2.Y_3\right)=\frac{2+2+2}{3}=2$, which is greater than $f\left(Z_1,Z_2,Z_3\right)=\frac{1+1+3}{3}=1.66$. Let strictly increasing transformation $g$ be such that $g\left(1\right)=1$, $g\left(2\right)=2$, $g\left(3\right)=10$. There can be plenty of such transformations. Hence, $f\left(g\left(Y_1\right),g\left(Y_2\right),g\left(Y_3\right)\right)=\frac{2+2+2}{3}=2$, which is less than $f\left(g\left(Z_1\right),g\left(Z_2\right),g\left(Z_3\right)\right)=\frac{1+1+10}{3}=4$. It is clear that due to permissible, i.e., strictly increasing transformation of values, the ratio of the arithmetic mean values changed.

Hence, according to the measurement theory, no arithmetic mean can be used on an ordinal scale.

Let us give one more example of the incorrectness of arithmetic mean calculation of ordinal quantities (Figure 1). Generally, the values of range limits of the quantities presented by ordinal values are completely unknown.

For scores (ordinal quantities), we have the arithmetic mean X_{mean(ordinal)} = (1 + 2 + 3 + 4 + 5)/5 = 3, not depending on which values the metric quantity takes in a particular range.

Let us show that for metric values on the ratio scale (unknown to those who asses them as one or other scores), the arithmetic mean can be any value corresponding to the limits of ranges with ranks 2, 3, or 4.

For the minimum values of the quantities in presented ranges: X_{mean(metric min)} = (0 + 10 + 200 + 210 + 360)/5 = 780/5 = 156. This corresponds to the second range (as rank 2). For the maximum values in the ranges: X_{mean(metric max)} = (10 + 200 + 210 + 360 + 500)/5 = 1280/5 = 256 (fourth range, as rank 4). Intermediate values in these ranges can give arithmetic mean corresponding to range 3: X_{mean(metric int)} = (5 + 100 + 205 + 340 + 390)/5 = 208 (as rank 3).

Only one conclusion from this example and others: the calculation of arithmetic mean for ordinal quantities can lead to incorrect results. Therefore, for ordinal values, speaking of means, it is necessary to indicate that this is a median and calculate the mean as a median. However, even with such values (medians) obtained on an ordinal scale (since this is an ordinal quantity), arithmetic operations must not be carried out.

2.2. Multiplication of Ordinal Quantities

As in the first article [1], we will turn our attention to the multiplication of parameters, expressed by ordinal quantities. Such an operation on an ordinal scale is not defined (actually prohibited), as stated in the International Metrological Dictionary [21]. However, for clarity, an imagined example will be still considered once more, namely, the results of the multiplication of two arrays of the initial data on two different scales. Each of the initial arrays constitutes known metric values on a ratio scale and their corresponding ranks (values on an ordinal scale). Arrays of a greater number of values (five, in this case) provide a more complete and clearer picture of comparing products of metric quantities and their corresponding ranks, than smaller arrays (e.g., of three values as in [1]).

The initial values of arrays are presented in

Table 1. Initial data (columns X and Y—names of quantities, Xm and Ym—values of quantities on a metric scale; Xr and Yr—ranks or values of quantities on an ordinal scale).

X	Xm	Xr	Y	Ym	Yr
A	10	1	T	1000	5
B	60	2	U	800	4
C	70	3	V	100	3
D	150	4	W	30	2
E	300	5	Z	1	1

, and the results of their multiplication are in

Table A1. Results of multiplication of initial data (XY—the point name; Xm × Ym—a product of metric quantities; R(Xm × Ym)—rank of the product in the general array Xm × Ym; Xr × Yr—a product of rank values of initial data; R(Xr × Yr)—rank of the product in the general array Xr × Yr).

XY	Xm × Ym	Rm = R(Xm × Ym)	Xr × Yr	Rr = R(Xr × Yr)
AZ	10	1	1	1
AW	300	6.5	2	2.5
AV	1000	8	3	4.5
AU	8000	13	4	7
AT	10,000	15	5	9.5
BZ	60	2	2	2.5
BW	1800	9	4	7
BV	6000	11	6	12
BU	48,000	18	8	14
BT	60,000	20	10	16.5
CZ	70	3	3	4.5
CW	2100	10	6	12
CV	7000	12	9	15
CU	56,000	19	12	18.5
CT	70,000	21	15	20.5
DZ	150	4	4	7
DW	180	5	6	12
DV	15,000	16	12	18.5
DU	120,000	22	16	22
DT	150,000	23	20	23.5
EZ	300	6.5	5	9.5
EW	9000	14	10	16.5
EV	30,000	17	15	20.5
EU	240,000	24	20	23.5
ET	300,000	25	25	25

of Appendix B. The range of products on a metric scale is from 10 to 300,000, and on an ordinal scale is from 1 to 25. Comparing the results of products Xr × Yr and Xm × Ym in a graphic form will not be fully representative due to a very large range (more than four orders) of products Xm × Ym. We, therefore, present the comparison of ranks (R) of the products of these quantities: Rm = R(Xm × Ym) and Rr = R(Xr × Yr) (Figure 2 and Table A1 of Appendix B). The main result of the comparison is that instead of the monotonic increasing points series, which can be combined by the monotonic increasing line, we have the wide field of points.

It is seen that single points, corresponding to product Xm × Ym, are positioned below or coincide along the axis R(Xr × Yr), as compared to those with lower values R(Xm × Ym). In this example, there are 13 such “discrepant” points (more than half).

For example, let point AW (rank Rm = 6.5, highlighted in red and larger in size) be compared with points, of which position is in conflict with the position of point AW on the graph (also highlighted in red, though standard in size). Point AW is to the right of points BZ (Rm = 2), CZ (Rm = 3), DZ (Rm = 4), and DW (Rm = 5) along the axis R(Xm × Ym), and has the same Rm value as that of point EZ (Rm = 6.5), which corresponds to the ranked position of products of their metric quantities Rm. However, point AW along with the axis R(Xr × Yr) is located lower than four points CZ, DZ, DW, and EZ, and its Rr value is the same as that of point BZ, though for point AW product Xm × Ym = 300 is greater than for the mentioned four points (several-fold for some of them) and its value is the same as that of Xm × Ym for point EZ (Table 1). Taking this into account, the Xr × Yr value of point AW must also be higher (and Rr, eventually), than the Rr values of these 4 points. The position of point AW must also be the same as the position of point EZ on the Rr scale, and together they must be localised above these 4 points. Figure 2 also shows two additional points larger in size: BT (highlighted in green) and EU (highlighted in brown), different from the other points, highlighted in the appropriate colour.

Taking this into consideration, it can be stated that, actually, in this case, numerous values of products of ordinal quantities will not reflect the products of metric quantities in the right manner. Thus, the results of arithmetic operations with ordinal quantities (given that a researcher is unaware of real metric values on a ratio scale and uses ordinal quantities, as expert estimates, instead of them) provide a distorted picture of reality, since such operations on an ordinal scale are indefinite. Further, in this case, it is not known, which values are “out” of the overall picture. Thus, in the above example: if metric values of the initial data are unknown, it is unclear, which products are inappropriate, among other things, for example, it is unknown, whether the value, corresponding to point AW is wrong, or the values, corresponding to the remaining five considered points.

It must be stressed that, in consideration of the foregoing as related to the conventional example (Figure 2), even the general breakdown of all values of products of two arrays into the highest and the lowest product-wise values remains in question. It can be stated only for two extreme points—minimum and maximum values. For further examination in Section 3, we need the following conclusion. If ordinal quantities are multiplied, some of their resulting products do not correspond to the respective products of metric values, deviating from an overall series of the entire array of products, when the products of metric values increase, an increase in their corresponding ordinal values must be observed as well. Such products of ordinal quantities reflect the situation, if ever, very roughly. A researcher is unaware of which products of those obtained are wrong, where an approximate boundary at least between the groups of products is (what group is larger than the other), and whether it exists at all.

Thus, this more extended than in [1] example, also demonstrates that the product of ordinal quantities fails to correspond to those for metric quantities. In this case, 13 of 25 (52%) products of ordinal quantities do not correspond to the products of the respective metric quantities. It suggests that the results of computations based on the products of ordinal quantities can contain several dozens and more percent of wrong data, and thereby, if maps were drawn according to them, such maps would be wrong, specifically, if three and more, rather than two, parameters are multiplied. Although, as it follows from the graph in Figure 2, the general trend towards the growth in Xr × Yr, as the Xm × Ym values increase, partially persists. However, a researcher is unaware of the wrong values, inhibiting the growth in Xr × Yr, as values Xm × Ym increase. One can be certain about only two extreme values—minimum and maximum. Accordingly, the particular maps, drawn based thereupon, will be wrong.

3. Brief Description of the Approaches to the Estimation of the Impact of Offshore Wind Farms on Seabirds

Let us consider relevant approaches to the estimates of the impact of wind turbines on seabirds, described in works [2,5,6,7]. Additionally, the wildlife sensitivity mapping manual [8] is briefly considered. Much attention is given to publication [7], where a “universal approach” to estimate the anthropogenic impact on marine biota is proposed. The parameters and models, used in these publications [2,5,6,7], are given in the

Table 2. Mathematical models and used designations of the initial data, given in the discussed publications (designations of quantities, given in the original articles, remain unchanged; some designations are contracted, though the expansions of abbreviations are given; see detailed explanations in the original articles).

[2]	[5]	[6]	[7]
Wind farm Sensitivity Index: $$\mathit{W}\mathit{S}\mathit{I}={{\displaystyle \sum }}_{\mathit{s}\mathit{p}}(\mathbf{ln}\left(\mathit{d}\mathit{e}\mathit{n}{\mathit{s}}_{\mathit{s}\mathit{p}}+\mathbf{1}\right)\times \mathit{S}\mathit{S}{\mathit{I}}_{\mathit{s}\mathit{p}}),$$ (3) $SS{I}_{sp}$—Species-specific Sensitivity Index: $$\mathit{S}\mathit{S}{\mathit{I}}_{\mathit{s}\mathit{p}}=\frac{\left(\mathit{a}+\mathit{b}+\mathit{c}+\mathit{d}\right)}{\mathbf{4}}\times \frac{\left(\mathit{e}+\mathit{f}\right)}{\mathbf{2}}\times \frac{\left(\mathit{g}+\mathit{h}+\mathit{i}\right)}{\mathbf{3}},$$ (4) $den{s}_{sp}$—density of each species on the area,   (A) flight behaviour: $a$—flight manoeuvrability; $b$—flight altitude; $c$—percentage of time flying; $d$—nocturnal flight activity;   (B) general behaviour: $e$—disturbance by ship and helicopter traffic; $f$—flexibility in habitat use;   (C) status: $g$—biogeographical population size; $h$—adult survival rate; $i$—European threat and conservation status. Economic Zone of Germany in the North Sea.	Collision Risk Score: $$\mathit{C}\mathit{R}\mathit{S}=\mathit{e}\times \frac{\left(\mathit{f}+\mathit{g}+\mathit{h}\right)}{\mathbf{3}}\times \mathit{C}\mathit{I}\mathit{S},$$ (5)   Disturbance/Displacement Score: $$\mathit{D}\mathit{i}\mathit{s}\mathit{t}/\mathit{D}\mathit{i}\mathit{s}\mathit{p}\mathit{S}=\left(\left(\mathit{i}\times \mathit{j}\right)\times \mathit{C}\mathit{I}\mathit{S}\right)/\mathbf{10},$$ (6)   CIS—Conservation Importance Score [4]: $$\mathit{C}\mathit{I}\mathit{S}=\left(\mathit{a}+\mathit{b}+\mathit{c}+\mathit{d}\right),$$ (7) $a$—percentage of the biogeographic population in Scotland; $b$—adult survival rate; $c$—UK threat status; $d$—Birds Directive status; $e$—flight altitude; $f$—flight manoeuvrability; $g$—percentage of time flying; $h$—nocturnal flight activity; $i$—disturbance by wind farm structures, ship and helicopter traffic; $j$—habitat specialization. Scottish waters.	Wind farm Sensitivity Index: $$\mathit{W}\mathit{S}{\mathit{I}}_{\mathit{w}\mathit{i}\mathit{n}\mathit{d}\mathit{f}\mathit{a}\mathit{r}\mathit{m}}={{\displaystyle \sum }}_{\mathit{s}\mathit{p}}(\mathbf{ln}\left(\mathit{d}\mathit{e}\mathit{n}{\mathit{s}}_{\mathit{s}\mathit{p}}+\mathbf{1}\right)\times \mathit{S}\mathit{S}{\mathit{I}}_{\mathbf{max}\left(\mathit{c}\mathit{o}\mathit{l}\mathit{l}\mathbf{,}\mathit{d}\mathit{i}\mathit{s}\mathit{p}\right)}),$$ (8)   $SS{I}_{\max \left(coll/disp\right)}$—maximum of two rank values: $$\mathit{S}\mathit{S}{\mathit{I}}_{\mathit{c}\mathit{o}\mathit{l}\mathit{l}}=\mathit{e}\times \frac{\left(\mathit{f}+\mathit{g}+\mathit{h}\right)}{\mathbf{3}}\times \mathit{C}\mathit{I}\mathit{S},$$ (9) $$\mathit{S}\mathit{S}{\mathit{I}}_{\mathit{d}\mathit{i}\mathit{s}\mathit{p}}=\left(\left(\mathit{i}\times \mathit{j}\right)\times \mathit{C}\mathit{I}\mathit{S}\right)/\mathbf{10},$$ (10)   Wind farm Sensitivity Index for collision: $$\mathit{W}\mathit{S}{\mathit{I}}_{\mathit{c}\mathit{o}\mathit{l}\mathit{l}}={{\displaystyle \sum }}_{\mathit{s}\mathit{p}}(\mathbf{ln}\left(\mathit{d}\mathit{e}\mathit{n}{\mathit{s}}_{\mathit{s}\mathit{p}}+\mathbf{1}\right)\times \mathit{S}\mathit{S}{\mathit{I}}_{\mathit{c}\mathit{o}\mathit{l}\mathit{l}}),$$ (11)   Wind farm Sensitivity Index for displacement: $$\mathit{W}\mathit{S}{\mathit{I}}_{\mathit{d}\mathit{i}\mathit{s}\mathit{p}}={{\displaystyle \sum }}_{\mathit{s}\mathit{p}}(\mathbf{ln}\left(\mathit{d}\mathit{e}\mathit{n}{\mathit{s}}_{\mathit{s}\mathit{p}}+\mathbf{1}\right)\times \mathit{S}\mathit{S}{\mathit{I}}_{\mathit{d}\mathit{i}\mathit{s}\mathit{p}}),$$ (12) $SS{I}_{coll}$—Species Sensitivity Index to wind farm collision (scores 1–5); $SS{I}_{disp}$—Species Sensitivity Index to wind farm displacement (scores 1–4);   CIS—Conservation Importance Score: $$\mathit{C}\mathit{I}\mathit{S}=\left(\mathit{a}+\mathit{b}+\mathit{c}+\mathit{d}\right),$$ (13) $a$—score for highest percent of biogeographic population in England in any season; $b$—adult survival score; $c$—UK threat status score; $d$—Birds Directive score; $e$—estimated percentage at blade height; $f$—flight manoeuvrability; $g$—percentage of time spent flying; $h$—nocturnal activity; $i$—disturbance susceptibility; $j$—habitat specialization. English waters.	Any individual vulnerability or species sensitivity: $$\mathit{r}={\mathit{a}}^{\mathbf{1}-\frac{\mathit{g}}{\mathit{g}+\mathsf{\gamma }}},$$ (14) $a$—any group of primary factor; $g$—any group of aggravation factor; $\gamma$—parameter for the influence of aggravation over primary factors.   Community vulnerability to collision: $${\mathit{C}}_{\mathit{J}}={{\displaystyle \sum }}_{\mathit{i}}^{\mathit{s}}\frac{{\mathit{p}}_{\mathit{i}\mathit{j}}}{\mathbf{1}-{\mathit{c}}_{\mathit{i}}{\mathit{s}}_{\mathit{i}}},$$ (15)   Community vulnerability to disturbance: $${\mathit{D}}_{\mathit{J}}={{\displaystyle \sum }}_{\mathit{i}}^{\mathit{s}}\frac{{\mathit{p}}_{\mathit{i}\mathit{j}}}{\mathbf{1}-{\mathit{d}}_{\mathit{i}}{\mathit{s}}_{\mathit{i}}},$$ (16) pij—proportional abundance of the ith species at location j; ci—individual vulnerability to collision: $${\mathit{c}}_{\mathit{i}}={\left({\mathit{F}}_{\mathit{i}\mathbf{1}}\times {\mathit{F}}_{\mathit{i}\mathbf{2}}\right)}^{(\mathbf{1}-\frac{\frac{{\mathit{F}}_{\mathit{i}\mathbf{3}}+{\mathit{F}}_{\mathit{i}\mathbf{4}}}{\mathbf{2}}}{\frac{{\mathit{F}}_{\mathit{i}\mathbf{3}}+{\mathit{F}}_{\mathit{i}\mathbf{4}}}{\mathbf{2}}+\mathsf{\gamma }})},$$ (17) di—individual vulnerability to disturbance: $${\mathit{d}}_{\mathit{i}}={\left({\mathit{F}}_{\mathit{i}\mathbf{5}}\right)}^{(\mathbf{1}-\frac{{\mathit{F}}_{\mathit{i}\mathbf{6}}}{{\mathit{F}}_{\mathit{i}\mathbf{6}}+\mathsf{\gamma }})},$$ (18) si—species sensitivity: $${\mathit{s}}_{\mathit{i}}={(\frac{{\mathit{F}}_{\mathit{i}\mathbf{7}}+{\mathit{F}}_{\mathit{i}\mathbf{8}}+{\mathit{F}}_{\mathit{i}\mathbf{9}}}{\mathbf{3}})}^{(\mathbf{1}-\frac{{\mathit{F}}_{\mathit{i}\mathbf{10}}}{{\mathit{F}}_{\mathit{i}\mathbf{10}}+\mathsf{\gamma }})},$$ (19) $F_1$—proportion of time spent flying; $F_2$—proportion of time spent at blade height; $F_3$—flight manoeuvrability; $F_4$—nocturnal flight activity; $F_5$—disturbance by ship and helicopter traffic; $F_6$—habitat flexibility; $F_7$—species status in the international red list; $F_8$—species status in the bird directive; $F_9$—species status in the national red list; $F_{10}$—adult survival rate. Bay of Biscay.

.

3.1. Article of S. Garthe, O. Hüppop [2]

Main provisions of the article. To one extent or another, this publication marked the beginning of a number of works on estimating the vulnerability of seabirds to the impact of wind farms. As far as the vulnerability index WSI (wind farm sensitivity index) is concerned, it was based upon publications [26,29] (we have considered the first in [1]).

Article [2] presents the index of sensitivity $WSI$ to wind farms (see Table 2—Formula (3)) for 26 species of seabirds. The index is applied to compute seasonal (winter, spring, summer, autumn) and annual maps of the vulnerability of the exclusive economic zone of Germany in the North Sea with a grid of 6′ latitudes× 10′ longitudes each (total area~120 km²). The final maps provided three levels of concern about the localisation of wind farms depending on percentile WSI. The WSI value is defined by the sum of the products of the natural logarithm of density of each bird species and its species-specific sensitivity index SSI_sp (sensitivity index). Bird density data were taken from ship observations with some adjustments or correction factor 1.3, based on the Baltic Sea data. Bird databases were also involved. Density in a grid cell for each species was obtained by dividing the sum of bird individuals, registered on a transect, by the total transect area, covered by cruises.

To calculate the sensitivity index SSI_sp, the authors take nine factors into account. Each was rated on a 5–score scale of 1 (low vulnerability of seabirds) to 5 (high vulnerability). Five of these factors are based on real data about birds (factors b, c, f, g, h), though factor f is partially estimated based on observations. The remaining four (a, d, e, i) can be assessed only based on subjective judgements, also accounting for the experience of observation at sea. Experts corrected all primary estimates independently. The authors organized the nine vulnerability factors into three groups, comprising (A) flight behaviour (factors a–d), (B) general behaviour (factors e–f) and (C) status (factors g–i). For each group, an average score of the respective factors was calculated. These average scores were subsequently multiplied by each other to give the species-specific sensitivity index (SSI) for each species.

Let us note factors b, c, g, and h, with underlying quantitative data, which is necessary for further analysis.

Flight altitude (b): 5–score scale to account for the bird flight altitude is converted from six altitude classes (allowing for 50 and 90 percentiles) with ranges: 1 class—0–5 m; 2, 5–10 m; 3, 10–20 m; 4, 20–50 m; 5, 50–100 m; 6, >100 m. In all likelihood, this is a percentage of birds of a given species, flying within the specified altitude ranges. Flight time percent (c) or percentage of birds of a given species, flying over a given area, was assumed in the following ranges: 1 score—0–20%; 2, 21–40%; 3, 41–60%; 4, 61–80% and 5, 81–100%. Biogeographic size of populations (g): score 1 is assigned to the population containing more than 3 mln. bird individuals; 2, 1–3 mln.; 3, 500,000–1 mln.; 4, 100,000–500,000 and 5 for less than 100,000 bird individuals. Survival rate (h) for various species of birds: score 1, if h < 0.75; 2, h > 0.75–0.80; 3, h > 0.80–0.85; 4, h > 0.85–0.90; 5, h > 0.90.

It is also worth noting that as compared to the further described works, factor i incorporates both the population threat status, and the status of birds’ conservation in Europe.

Analysis of particular provisions of the article. Let us verify whether the conclusions obtained in the examined article are invariant. Conclusions based on the results of computations on any scale must not change at any permissible transformation on this scale (see Section 2). Thus, for example, all computations can be made in miles or feet, rather than kilometres, though these are permissible transformations on a ratio scale, and in this case, we always arrive at the same conclusions based on the results of computations, irrespective of utilised measurement units. Let us now proceed from values 1, 2, 3, 4, 5 to values, e.g., 1, 2, 3, 40, 500. Any monotonic transformation of the initial ordinal data can be used to check the obtained final results. After such transformation, the ranking of species SSI_sp will change radically. Thus, among the five most vulnerable species (Black-throated diver, Red-throated diver, Velvet scoter, Sandwich tern, and Great cormorant), one new—Red-necked grebe (third by the degree of vulnerability) instead of Velvet scoter (which will become the 16th) will appear. Species occupied 1 and 2 places (Black-throated diver and Red-throated diver), will switch places. Species at the end of the list will also change. If Northern fulmar previously closed the list, it is the 15th on the list now, and its place will be occupied by Black-legged kittiwake, who held second to last place together with Black-headed gull (24–25th place). Mew gull will become the 25th on the list, though previously it had the 20–21^st place, and Black-headed gull will not lose its 24th position. The list will change substantially! However, if the problem lies in evaluating the most vulnerable species, the list (variation series), in this case, of such five species depends on the monotonic transformation of initial data for computations and can both remain unchanged, and change by one or several species. Of importance is that the use of particular values of ordinal quantities for the initial data changes the position of certain species in the final ranking of vulnerability SSI_sp at monotonic transformations (since ordinal quantities—the marks of the values of variation series—are merely numbers, rather than values, with which arithmetic operations can be performed).

With the foregoing as the background, as to the computation of means in Section 2.1, it can be stated that only the values of one of the factors in each group can be taken as the arithmetic means from Formula (3). If, e.g., we have four factors of group A with values 1, 1, 1, 5, the “ordinary” arithmetic mean will be equal to (1 + 1+1 + 5)/4 = 2. However, with allowance for the median theorem requirements, the median is to be taken as the mean, i.e., 1. It two-fold differs from the first. This alone (the use of “ordinary” arithmetic means, rather than the median) in any case will change the final conclusions (vulnerability ratios between species).

Likewise, in view of the foregoing in Section 2.2, it can be stated that since SSI_sp is a product of ordinal quantities, these SSI_sp values fail to correspond to the real situation, since many SSI_sp values are incorrect when being compared to one another. Let us note for further use: all mean quantities from Formula (4), with allowance for what has just been said, can be considered ordinal quantities.

Once again, we turn our attention to the replacement of real metric values with their corresponding ordinal quantities, involving the replacement of corresponding ranges as well. Thus, for factor g—biogeographic population size—ratio max/min ≥30. Given that, for survival rate h, this ratio is merely about 0.95/0.73 = 1.30. However, real variation ranges of all other factors are also transformed into the range of ordinal quantities 1–5 (max/min = 5/1 = 5). Clearly, the ratio between individual values of ranges for each factor, including a particular max/min ratio, significantly affects the final value of SSI_sp. Additionally, replacement of real and, most likely, different variation ranges of the employed parameters with one and the same range (in this case, range 1–5) seriously distorts the final result of computations, even if the operation of multiplying ordinal quantities was justified. Let us also note that for five factors a, d, e, f, i, no ratio between real minimum and maximum values are known even approximately (in addition, on principle, we do not measure factor i in its usual sense), however, for ordinal values, the ratio is taken as equal to 5 ( =5/1) (also see our first article [1]).

Summing up parameters b and c (percent scores (percentages) of birds at a particular altitude + percent scores of flying birds in relation to the total number of birds in the area) can hardly be correct as well. It is clear by default that species vulnerability SSI_sp is proportional to the percentage of birds, flying at the height of turbine blades’ operation in relation to the total quantity of birds in a cell. Further, this percentage, in its turn, constitutes the product of the percentage of birds, flying at the height of blades’ operation (b as decimal quantities) and the percentage of flying birds (c as decimal quantities as well) relative to the total number of birds at the specific section (vulnerability~b × c). Additionally, all the remaining factors should be taken into account for this specific percentage. This remark is also relevant to the following works [5,6,7].

Thus, all the results in work [2] are obtained based on arithmetic operations with ordinal quantities, which is unacceptable. With allowance for this, the conclusions are non-invariant in relation to the permissible monotonically increasing transformation of data—ranking of species by SSI_sp changes significantly. When computing SSI_sp, the arithmetic mean values are used, rather than the variation series values. Products of ordinal quantities provide certain incorrect (and unknown to a researcher) values. For all the factors, whole-number values within the 1–5 range are taken, which fails to correspond to the real ratio between the values of ranges for various factors. Accordingly, the maps of vulnerability, drawn based on such calculations, are also incorrect.

Of certain importance is the provision, referred to by S. Garthe and O. Hüppop [2]: “Debate on the effects of human activities on wildlife necessitates risk and impact assessments [30] even where the database might be poor [31]”. Indeed, the database can be low-quality, incomplete, deficient, and erroneous. Nevertheless, with the methodology of dealing with any data, any database must be mathematically justified, appropriate, and consistent with generally recognised methodological requirements, and the measurement theory requirements as well.

3.2. Article of R. Furness et al. [5]

Main provisions of the article. The authors mostly follow the work [2], also using the same data, considering revisions and alterations due to new research data. Vulnerability indices are computed for 38 species of birds, dwelling in Scottish waters. No mapping of vulnerability/sensitivity is performed. Calculations use ten factors (Table 2), actually, these are the same factors as in the work [2]. Four factors (e, f, g, h), incorporated in the risk of collision with blades of wind turbines Collision risk score (CRS) (Formula (5)), are linked to the bird flight agility and flight behaviour, and two factors (i, j) (Formula (6)), defining the Disturbance/displacement score (Dist/DispS) index of birds,—to using specialisation of habitat by birds and their susceptibility to disturbances. Each of these formulas contains the Conservation importance score (CIS) index, as a multiplier. It is defined, using four factors (Table 2, Formula (7)), rather than three as in [2]. Two factors (c and d) instead of one (i), proposed in [2], were applied as the conservation status. All the factors, except for e (percentage of birds, flying at the height of turbine blades’ operation), are estimated as ordinal quantities in the 1–5 whole-number value range as well. Moreover, as opposed to factor g, where scores are attributed according to the population size of each bird species (as in [2]), factor a is used in the work, where scores are attributed to the percentage of the biogeographic population in the Scottish waters. The scores were estimated as follows: 1, <1%; 2, 1–4%; 3, 5–9%; 4, 10–19%; 5, ≥20% [4]. Since this index can vary depending on the season, in computations, the authors used the maximum season-wise score for each species.

What is principally new and important in [5] is the breakdown of a single index of vulnerability into two. The first—Collision risk score (Formula (5))—for the risk of collisions with the blades of wind turbines. The second—Disturbance/displacement score (Formula (6))—for the risk associated with avoidance/displacement of seabirds depending on the degree of disturbance on the part of wind turbines, ships, and helicopters, and habitat change.

For CRS in (Formula (5)) as compared to (Formula (4)), factor e is given a greater significance (a higher weight). This factor is taken out as an individual multiplier, assuming that this is the main risk factor of collision with wind turbines. Such an approach to employing the factor of “flight altitude” is more logical and justified, since the death of birds directly from wind turbines, first and foremost, depends on what percentage of the total number of birds fly at the height of operation of the turbines’ blades. Of note is that this factor is a metric quantity, varying from 1 to 35% [5], though it can also take the zero value, which other factors lack. It can be further used in making vulnerability maps. Of importance in computing anthropogenic impact is also the use of the percentage of the biogeographic population in the mapped area, here, in Scottish waters, rather than the total size of the population of birds of each species, because the larger this percentage, the stronger the corresponding negative effect on the overall population.

Analysis of particular provisions of the article. Actually, this article contains all the flaws of the work [2]. The principal ones include unacceptable arithmetic operations with ordinal quantities, and, as a result, the uncertainty of some results due to using the operation of multiplying such quantities, and the arithmetic means for them in computations.

As in the case with the analysis of the work [2] results, constancy of conclusions at a permissible transformation of the initial data as to CRS (see Formula (5) in Table 2) can be estimated. As should be expected, the final ranking of species will depend on the type of transformation, i.e., it can both change, and remain unchanged.

Of additional note is that the article contains no rationale for the multiplication of parameters i (disturbance/habitat displacement) and j (habitat specialisation).

Overall, to date, it is unclear, how two indices CRS and Dist/DispS can be jointly used. Everything suggests that each species of bird can be exposed to such different influences simultaneously, though CRS and Dist/DispS indices are still calculated separately, due to insufficient knowledge and information about this problem, and they are in no way compared and combined (not summed up).

3.3. Article of G. Bradbury et al. [6]

Main provisions of the article. This work follows publications [2,5] to a great extent. The formulas from [5] are used. The work objectives are to present the sensitivity index SSI for 54 species of seabirds, dwelling in the English waters; map densities of seabirds for these waters; combine these two results for seabird sensitivity mapping (maps of general sensitivity index WSI_windfarm distribution—Table 2, Formulas (8)–(13)). SeaMaST GIS–package (Seabird Mapping and Sensitivity Tool) is used for mapping. WSI_windfarm index is calculated for separate segments of 9 km² (3 km × 3 km) as a sum WSI for all the accounted bird species (Table 2, Formula (8)). Indices SSI_coll and SSI_disp are computed according to Formulas (9) and (10). Prior to being used in WSI_windfarm, they are ranked: values SSI_coll for 5 ranges—score 1 for values 0–42; 2, 67–187; 3, 200–400; 4, 420–817; 5, 960–1470; SSI_disp values for 4 ranges: score 1 for values 1–5; 2, 6–8; 3, 10–18; 4, 22–32; maximum score—either SSI_coll or SSI_disp is selected for each species. To forecast the density of the species of seabirds (denc_sp), the authors used estimates of parameters, obtained in the DSM–model [32]. Due to the inhomogeneity of data, bird density maps were made only for 32 species. Seasonal (summer and winter): (1) maps of sensitivity to the risk of collision with turbine blades, employing only SSI_coll values (Formula (9)); (2) maps of sensitivity to bird habitat flexibility, employing SSI_disp values (Formula (10)); and (3) general maps of sensitivity WSI_windfarm (Formula (8)) were drawn. To compute SSI_coll and SSI_disp, ten factors are used (Table 2), actually, these are the same factors as in the publication [5], though accounting for coverage and including updated data.

Analysis of particular provisions of the article. Overall, all the remarks, given as to the two previously discussed publications, also relate to this work. However, there are several considerations on the presented algorithm in terms of comparing and summing up SSI_coll and SSI_disp.

The authors of the examined work draw attention to the fact that “the top rank ‘Very High Risk’ was not assigned for displacement concern, acknowledging the lower risk to populations compared to collision risks”. SSI_coll and SSI_disp values after computation by Formulas (9) and (10) are ranked (for SSI_coll—5 scores, for SSI_disp—4 scores), since, according to the authors, “the two resulting scales should not be compared in a quantitative way but only in terms of the species ranking within one scale”. Actually, the work contains a comparison between calculated score values (sub-ranges) SSI_coll and SSI_disp of two scales, which are then taken to one scale after repeated ranking. When two quantities are compared on any scale (the greater of two is found), it is actually a comparison of quantitative values: one value is higher than the other. If the values are compared on an ordinal scale, they are estimated in an expert way: one is strictly higher than the other, though on a metric scale (on one and the same ratio scale or an interval scale) the values of these two quantities can be unknown. Further, it is not excluded that comparing values are on different metric scales. Then, such comparison is not possible.

Indeed, SSI_coll and SSI_disp must be summed up, though not in this way: it is impossible in the context of the proposed model. These values should be estimated in terms of quantity and compared on one and the same scale. If this is possible, basically, such values for the impact of collisions and habitat change can be identical or very close: on one part, for the birds, flying at the height of blades’ operation, on another part, for the birds, flying beyond the area of their impact (if the impact estimates are made on various ratio scales, in any case, some kind of comparison between these impacts can be performed). Then, the computation, if made with such values, by Formulas (8)–(10), would provide an almost two-fold lowered value of WSI_windfarm, since one of the impacts would not be taken into account. This problem, as we see, must be solved in terms of a strictly metric approach, and, as it has been above-mentioned, at the moment, it is not solved, since there is not enough knowledge and information about mutual consideration of these impacts. Analysis of this problem is beyond the scope of the present work. If it is impossible to compare SSI_coll and SSI_disp on a single metric scale, two different maps of vulnerability need to be drawn: based on SSI_coll and SSI_disp.

3.4. Article of G. Certain et al. [7]

Main provisions of the article. The article, based on the development and enhancement of the approaches from works [2,5], suggests a more common, and, as the authors write, “universal approach” to vulnerability—to not only the impact of wind turbines at sea on birds but also, on the whole, to any anthropogenic impact on any biological components of the ecosystem. Bird vulnerability to wind turbines at sea in the Bay of Biscay and vulnerability of benthos in the Barents Sea to the impact of bottom trawling are evaluated. We shall further consider only the impact on seabirds since, for various biological objects, basic methodological approaches are almost the same. The algorithm, proposed in [7], as has been said at the beginning of our article, is applied in the work [10] to estimate the vulnerability of marine avifauna to oil.

Indices of sensitivity to the risk of collision and the risk of avoidance of wind turbines for 30 species of seabirds are computed in the model of G. Certain et al. [7]. Final calculations in the form of maps were made for the 5 km × 5 km areas for winter and spring periods. Abundant species based on the results of winter aero- and ship observations in spring were taken into account with allowance for quantity p_ij, proportional to the number ith of species at the jth segment. As a result, a diagnostic panel with four maps for the mapped region is presented: (1) vulnerability of seabird communities to collisions with wind turbines (Formula (15)), (2) vulnerability of seabird communities to a disturbance on the part of wind turbines (Formula (16)), (3) map, reflecting a certain extent, the relative number of birds in the mapped region based on the A_ij value, and (4) an integral (synthetic) map, reflecting this information in a combined manner.

Three groups of factors are considered for each bird species, when calculating their vulnerability, as in the work [2]. The analysis of the Formula (4) for SSI_sp by G. Certain et al. [7] enables substantiation of the equations for these groups of factors: c_i—the individual vulnerability of seabirds to collisions with wind turbines (Formula (17)), d_i—the individual vulnerability to concern and disturbance of normal vital activity of birds due to wind turbines and habitat change (Formula (18)), s_i—species sensitivity determined by relative indices of elements, that describe the species conservation status and their recoverability after the exposure (Formula (19)).

G. Certain et al. [7] account for ten various factors (Table 2). Six factors (F₁–F₆), relating to c_i and d_i, are almost identical to those in [2]. Four factors (F₇–F₁₀) are taken into consideration to estimate species sensitivity: three factors of conservation status (F₇–F₉) instead of one i (European threat and conservation status) as in [2] and F₁₀—a survival rate of adult bird species, the same as in [2], though there is no factor of the biogeographic population size. As compared to [5,6], factor a (percentage of biogeographic population) is not used, and the third factor F₇ is added to the two conservation ones. All the factors are estimated within the range of whole-value numbers 1–5, as in the three previous works, but taken to the scale 0.2, 0.4, …, 1.0. The values of factors are actually the same as in works [2,5], though accounting for mapping area and with some alterations and updated data.

An essential provision of the considered article includes the breakdown of anthropogenic factors into primary (a), which directly define vulnerability or sensitivity, and aggravation (g), which can be inessential as autonomous, though increasing the already existing vulnerability or sensitivity. Formula (14), based on a hierarchy of factors, is proposed, which, according to the authors, can be clearly formulated.

Analysis of particular provisions of the article. The authors write that “As these factors are all expressed on the same scale, they can be mathematically combined together”. However, all of them are specified on an ordinal scale, and failure to take this into account leads to a number of problems and inconsistencies.

Compensation between factors and their hierarchy. G. Certain et al. [7] combine factors somewhat differently than in the three previously discussed works, accounting for, as they write, their different interaction with one another. The authors consider two ways of their combining: averaging and multiplication, as was suggested in [2] and further used to a certain extent in the works [5,6]. G. Certain et al. [7] note that averaging allows for compensation between factors, namely, a high value of one factor can be compensated with a low value of another (what is suitable for the factors of various natures), and multiplication can be convenient (appropriate), when factors are interacting, or when they are conditional to each other.

Overall, all these statements as to the two specified operations and their relationship with compensation and hierarchy are not quite correct. One cannot fully agree with the statement of G. Certain et al. [7], that averaging, or multiplication of factors does not admit (does not account for) any hierarchy between factors. One of the factors from the product can have the exponent, which is greater or less than one, hence, the contribution of this factor will differ upward or downward from that of others, depending on whether the factor value is greater or less than one. In summing up or averaging several factors, the most important factor can have a multiplier other than 1. Then, when summing the factors up (fundamentally different, though on one and the same scale), if one of them has multiplier 2, and others 1, it is clear that as the value of this factor increases, its contribution to the final result will grow two times faster than the contribution of the others (although, it is not necessarily that its contribution will be two times greater than of the others). However, in this case, such a multiplier has no relation to the remaining factors, i.e., when this factor changes, the change in the final quantity, which it affects, depends only on the change of this factor with multiplier 2.

For further analysis, let the approach [7] be assumed and considered in terms of the permissibility of actions with the used quantities, i.e., conventionally taking them as a metric, rather than ordinal. It is essential to analyse general formulas of computing c_i, d_i, s_i (Formulas (14) and (17)–(19)).

Individual vulnerability (c_i) to collisions with turbine blades. For this quantity (Formula (17)), the authors distinguish the factors in the context of their hierarchy. Factors F_i1 (percentage of time in flight) and F_i2 (percentage of time at the height of blades’ operation in relation to the total flight time) are deemed the key factors of collision with turbines’ blades and directly define vulnerability. Manoeuvrability (F_i3) and nocturnal activity (F_i4) are aggravating factors that can worsen the existing vulnerability.

Time spent by a bird at the height of the blades’ operation F_i2, is considered depending (determined by) on F_i1, hence, the authors use a multiplication ratio between them. Multiplication of F_i1 and F_i2 produces the percentage of all birds, flying at the height of turbines’ blades operation relative to all flying birds of the species in this section. Conversely, the authors assume that it is beyond reason to assume any relationship between manoeuvrability and nocturnal activity, at least, as related to their contribution to vulnerability to collision with wind farms. Additive ratios between F_i3 and F_i4 are therefore used [7]. These two different quantities, of which the meaning and impact on the final result are not clearly described, are defined in an expert manner and summed up.

Moreover, at the value $F_{i1}\times F_{i2}=1$, i.e., when all the birds from a segment fly and all of them fly at the height of the blades’ operation, there is merely not any influence (aggravation) neither on the part of F_i3 nor F_i4 (if Formula (17) is followed): 1, in any case, remains unchanged, i.e., at any values F_i3 and/or F_i4, c_i will remain to be 1. However, it is very weird! The same relates to Formulas (17) and (18). As to Formula (18): if F_i7, F_i8, F_i9 = 1 (out of values 0.2–1), which corresponds to a very high degree of conservation at all levels, then always s_i = 1 irrespective of F_i10. However, it appears that the proposed model is not quite functional on the whole (also see uncertainty in values C_j and D_j below).

Individual vulnerability (d_i) to concern and disturbance of normal vital activity of birds due to habitat change and disturbance of birds in relation to a wind farm, ship traffic, and helicopters, is computed by Formula (18). It is assumed that this is a combination of a primary factor—the intensity of behavioural response to anthropogenic activity (F_i5) and a factor of aggravation—flexibility in using habitat (F_i6). No influence of F_i6 on F_i5 at F_i5 = 1 has been reported in the foregoing. It implies that at the such maximum value of F_i5, various values of habitat flexibility (F_i6) affect in no way an individual vulnerability of species to disturbance and habitat displacement d_i. However, it is hardly correct as well.

Species sensitivity s_i depends on the combination of four factors, three of which are associated with regulatory conservation statuses of species at various levels, namely, international, European, and national—F_i7-F_i9, as primary factors, and the parameter of adult survival rate of species (F_i10), as aggravation factor. The value of s_i is computed by the Formula (19).

Uncertainties in some final values C_j and D_j. As has been shown above, all three parameters (c_i, d_i, s_i), included in the denominator of Formulas (15) and (16) can take values equal to 1. Then, we will obtain C_j and D_j with an infinite value, which is unacceptable! Formulas (models) must be applied within the entire presented (permissible) range of variation in the initial data, which is not observed in the assumed model. Hence, this model cannot be regarded as correct and adequate.

Total bird vulnerability to the impact of wind farms. As in the previous works, the total vulnerability is not calculated. It does not allow for ultimately assessing the final vulnerability of such projects relative to seabirds.

General considerations on accounting for diurnal and nocturnal activity. This article, as the three above-mentioned, combines four parameters in one formula: flight time percentage, flight height, flight manoeuvrability, and nocturnal flight activity (Formulas (4), (5), (9) and (17)). Nevertheless, here, no assumptions are made, no rationale or proof is provided as related to the fact that the first three factors change somehow or remain unchanged during nocturnal flight. E.g., all three such factors change equally proportional to nocturnal activity. Or they, and vice versa, change variously: for some species, the values of some factors decrease, and for others—increase. However, any model is built based on proofs and rationale, or, at least, assumptions. Unfortunately, neither this article nor the other discussed above contain anything of these. The parameter of nocturnal activity and other factors (factor) are just summed up. However, it is incorrect, even if the fact is omitted that all these factors are ordinal quantities. In any case, the approach, proposed in the analysed works, to nocturnal activity, likewise to the other factors, as to ordinal quantities, is incorrect and leads, when used in calculations, to wrong final decisions.

In view of the conducted analysis of the article [7], it can be stated that the conclusions and recommendations obtained therein for estimating the anthropogenic impact of offshore power plants on seabirds cannot be considered correct and justified. Hence, the proposed approach cannot be applied to estimate other anthropogenic impacts on the particular components of marine biota as well.

3.5. The Wildlife Sensitivity Mapping Manual [8]

Main provisions of the Manual. This Manual [8] is a comprehensive compendium of the information necessary to develop wildlife sensitivity mapping approaches to inform renewable energy deployment. Such maps should be a standard precursor to all renewable energy plans and development. We focus only on the part of “Step-by-step approach to wildlife sensitivity mapping”, namely, on the assessments of the biological objects sensitivity for the developed maps. The Manual states that data relating to the Natura 2000 network, collected on the base of a 10 × 10 km grid, can provide a good basis for data generation.

We will give a brief summary of the basic, somewhat simplified, approach to assessing sensitivity for mapping a selected area (without classifying certain core features, such as protected areas, as no-go sites and less sensitive, secondary locations, as sites where development could prove problematic and where caution is advised). More complex mapping exercises assign sensitivity by weighing features in relation to known parameters that increase sensitivity [8]. Taking into account the recommendations given in the Guidance, it is proposed to identify the types of renewable energy infrastructure and which biota species and their habitats are likely to be affected. Next, to compile distributional datasets on sensitive species, habitats, and other relevant factors. Then, to develop a sensitivity scoring system for the species presented in the area. On the basis of GIS, to generate corresponding maps and develop a system for interpreting data on these maps.

In terms of developing a sensitivity scoring system for wildlife, it is necessary to select the main factors that determine the sensitivity of the species in the mapped area. These factors are species characteristics (species behaviour, species morphology and migratory behaviour); habitat characteristics (habitat fragility, and habitat dependence); population dynamics (proportion of global/regional/national population); conservation status (global, EU, regional or national conservation status) [8].

The Guidance notes that once a list of at-risk species and habitats has been created, these can be scored in terms of the level of their sensitivity. Such lists should be based on a thorough investigation of the scientific literature and through consultation with key experts [8].

On a theoretical example, for four species of biota, scores are given for morphological, behavioural, and population dynamic characteristics, as well as for conservation status. The following scores are accepted for the first three characteristics: 1 (low), 2 (medium), 3 (high), 4 (very high sensitivity); and for conservation status: 0 (low), 2 (medium), 4 (high), 6 (very high). The following ranges of overall sensitively scores are accepted for the summed scores: Medium (3–8), High (9–14), and Very High (15–20). Besides, any species scoring 3 or 4 for morphology/behaviour/population dynamics is automatically in the High category [8]. In other words, for the first species, the sum of the scores is 5, since according to the morphology parameter, it has a score of 3 (high sensitivity), then the result (sensitivity score) is assigned to the High range (9–14).

Next, in the general case, for each grid cell, the sensitivity scores of all presented species are summed up, if necessary, with weight coefficients, such as the number (density) of species or proportion of the global or regional population of each species present.

Analysis of particular provisions of the Manual. Analysis of calculations based on the use of ordinal quantities in arithmetic operations (which is used in this methodology), were held in the first work [1]. It is shown that such calculations give uncertain and incorrect results: conclusions based on the results of simple summing of ordinal quantities are changing with permissible monotonous transformations on such scale. It is obvious that this conclusion does not depend on whether the weight coefficients are used (even specified on the ratio scale) or not. Summing and multiplication of ordinal quantities are unacceptable according to ([21], and other references in [1]). In addition, the range of changes of all values of sensitivity characteristics is identical (1–4), except for conservation status. It also aggravates the negative situation, since the ranges, in reality, are most likely different.

3.6. Some Considerations on Taking into Account the Factor of the Seabirds Flight Altitude and the Impact of Other Factors on It

As appears from the considered above works, many factors impact the bird vulnerability to offshore wind farms. At the same time, the bird flight altitude h_sp is a key influencing factor. In turn, other factors impact it as wind speed and direction, rain and precipitation, visibility and cloudiness, time of the day, season, foraging, migration, distance to coast, habitat type and spatial arrangement, offshore wind farms, fishing boats and ships, and the others [33]. Therefore, initially, it is important to know the dependence of this key parameter on these factors. However, such consideration is absent in the analysed works. We very briefly dwell on the impact of different factors on h_sp since the main aim of our research is to show, based on provisions of mathematics with ordinal data, the incorrectness of conclusions in the analysed articles.

There are developed models (methods) for taking into account the dependence of h_sp on the series of influencing factors (for example, see [34,35]). They explain from about 10–67% till 95% of the h_sp variability. In an elementary case, the method of the research and calculation h_sp = f(factors) is brought to the following. For single species of birds, the measurements of their flight altitude (altitude range) and factors affecting h_sp are carried out for areas with wind farms and for areas where wind farms are absent. Different methods can be used for the detection the flight altitude, including visual observations, a laser rangefinder, a radar of sea observation [34]. As we understand, data on h_sp as results of visual observations are hardly suitable for further analysis since they have a large inaccuracy. The value h_sp should be measured with sufficient accuracy for solving such issue. Detailed and accurate observations of the bird location relative to the position zone of wind farms area and blades of single installations are also important. The most suitable instrument, in this case, is radar. The 5 cm radar is ideal for bird detection [36]. Military radars, due to use of several rays combined into one [37], make it possible to obtain three target coordinates that, in this case, gives the opportunity to solve practically all issues of ornithological observations. Based on the result of the field research, an array of measurement results is formed. Then, using one or another mathematical tool, for example, the factor analysis, the most significant factors (x₁, x₂, x₃, …), which impact the birds’ flight altitude (h_sp), are detected. At the final stage, the corresponding regression equation h_sp = f(x₁, x₂, x₃, …) of the flight altitude dependence on most significant factors is calculated. Another method is presented in [35].

4. Conclusions and Recommendations

The authors of the considered publications contributed greatly to the development of the methodology for computing the vulnerability of seabirds to offshore wind farms. They obtained and revised quite correct, as of the time of studies, metric values of a number of parameters, necessary to assess bird vulnerability to offshore wind farms—the percentage of birds flying at various altitudes, as well as at the height of wind turbine blades’ operation; densities of distribution (or quantities proportional to them) of birds in the mapped regions. Of importance are also the proposals to divide bird vulnerability into two components: vulnerability to collision with blades of wind turbines and vulnerability to habitat displacement due to the impact of wind farms and infrastructure to build and maintain them. It is likely that these two components must be summed up somehow to find the total vulnerability and draw the respective maps. The existing assumptions, based on using ordinal quantities, give no way of solving this problem.

In all the examined works, arithmetic calculations (addition and multiplication) use ordinal quantities, and the arithmetic means for them. Such an approach cannot be deemed correct, taking into account the unacceptability of arithmetic operations with ordinal quantities [21] and the median theorem [15]. It has been additionally shown by a conventional example that the multiplication of even two arrays of original ordinal quantities leads to uncertain final results. Hence, the derived final maps of vulnerability cannot be deemed correct as well. All this finally leads to wrong conclusions and can lead to wrong management decisions.

Conclusions obtained in work [2] as to the values of the species-specific index of bird sensitivity to wind farms (SSI_sp) are incorrect. Based on the example, it has been demonstrated that the order of values of the species-specific index of species sensitivity (SSI_sp values) changes considerably at a monotonic transformation of ordinal values of initial data (factors). This conclusion is also valid for all the remaining considered works, where the multiplication of the arithmetic “mean” values and certain initial data as ordinal quantities are performed.

The separation of general sensitivity into the species index of bird sensitivity to collisions with wind farms and the species index of bird sensitivity to the habitat change in [5,6] is quite logical. Taking into account metric quantity (percentage of birds, flying at the height of blades’ operation), it is more correct than its use as an ordinal quantity. However, in this case, the use of ordinal quantities in computations as well does not allow for arriving at the correct definite conclusions.

The work [7] provides a description of the “universal approach” based on formulated definitions, assumptions, and mathematical statements, which, according to the authors, could have been used in the estimates of anthropogenic impacts on biota. It is likely that the suggested approach, namely, dividing all factors into primary and aggravation, when the latest factors are incorporated in the formula of computation through the exponent of primary factors, can be very helpful in the future. However, such formulas (models) must be applied within the whole presented (permissible) range of variation in the initial data, which is not observed for the assumed model. In this regard, and in view of the fact that arithmetic operations with ordinal quantities are used in computations (with their arithmetic mean values as well), the proposed model cannot be considered correct.

Concerning the work [7], it should be noted that a fundamental difference between operations (multiplication and addition) is as follows. Any initial parameters, as metric quantities, can be generally combined multiplicatively by multiplication/division, if they are specified on ratio scales, albeit being in different measurement units, and additively by addition/subtraction only in cases when they have either one and the same scale of ratios or differences, or (what is the same) identical measurement units (see [1,17,21]). All such operations are permissible only on metric scales, with that, to the fullest extent only on a ratio scale. On an ordinal scale, such operations are not defined [1,21]. It should also be noted that additions of quantities, different in nature, are possible if they are previously taken to the same scale. These statements are valid for the other analysed works [2,5,6].

In the Manual [8], ordinal quantities are used in a simple algorithm of calculations (arithmetic calculations of the sensitivity of certain species and overall sensitivity for each grid cell of the map area). However, it is unacceptable, because it leads to uncertain results—the conclusions are changed with permissible monotonic transformations on a scale. Such methodology is contrary to the requirements of measurement theory. Calculations based on ordinal quantities give very rough and often incorrect results that can mislead specialists.

Thus, the models of calculating the seabird’s vulnerability to offshore wind farms presented in the considered articles do not allow correct calculations. In such models, which provide arithmetic operations with input quantities, it is necessary to use metric values on the ratio scale as the latter. However, a simple replacement of ordinal values by metric ones, in the analysed models, will lead to the summation of quantities with different measurement units, which is unacceptable. The development of fundamentally different models is required.

There are metric models [38,39] that are used in practice. However, only the risk of bird collisions with turbines or blades is assessed in them, without taking into account disturbance of habitats, feeding, and barrier effects. At the same time, although in [39] it is proposed to take into account the ordinal data for the nocturnal activity of birds [2], it is only as a guiding line for an expert assessment of the quantitative value in the range of 0–100%. Everything is quite strict in these two works [38,39] from the common methodology point of view, though it is possible to argue about the accuracy (uncertainty) of the calculation on such models. Using estimates of the uncertainty of the initial quantitative data, it is possible to assess the final uncertainty of the obtained results and to compare calculations for such models. However, for the four analysed works, it is impossible to assess such uncertainty of either the initial data or the final results, since they are based on ordinal values. Although there are attempts to make such assessments in [40], they are also made on the basis of arithmetic operations, including those with ordinal values, which are initially incorrect.

Methods of expert quantitative assessments can also be used to calculate the bird’s vulnerability to wind farms. These are methods based on multi-criteria approaches, in particular, on the Analytic Hierarchy Process (AHP) method proposed by T.L. Saaty [17]. Assessments obtained by this method are also considered as values on a ratio scale. The AHP method, based on the construction of a hierarchical structure, apparently makes it possible to assess seabirds’ individual characteristics or parameters associated with various types of impact. An example of such an approach is given in [41]. The results of such assessments of individual characteristics and parameters are necessary for the assessment of bird vulnerability, including for the assessment of their conservation value, and/or their final vulnerability to one or other impacts of wind farms, as it seems, can be used in different quantitative models (but not in models of Table 2). However, all these approaches require further analysis.

It should be emphasized once more that the data can be low-quality, incomplete, deficient, and erroneous. However, the methodology of dealing with any data or any database must be mathematically justified, correct, and consistent with the generally accepted methodological requirements, including those of the measurement theory.

In our opinion, in the future, in calculations of the effect of wind farms on seabirds, and any anthropogenic impacts on marine biota, and other components of the marine ecosystem, one should rely on both the use of only metric values on a ratio scale, correct mathematical models, that operate within the entire range of the employed original values.