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1 Appendices 1 A1. Calculation of the Height of a Spherical Cap 2 A2. Surface Area of a Spherical Rectangle 3 A3. Calculation Of 1 Appendices 2 A1. Calculation of the height of a spherical cap 3 A2. Surface area of a spherical rectangle 4 A3. Calculation of fields of vision and lateral surface angles of cropped detectors (cameras) 5 A4. Non-uniformly distributed approach directions: derivation of equations 10 and 11 6 A5. Step-by-step guide to using the density estimation formula 7 1 8 A1. Calculation of the height of a spherical cap 9 The top section of an acoustic – i.e. conical – detection zone is a spherical cap. The surface area 10 of this cap is determined by the radius of the corresponding sphere (here, given by the cone’s slant 11 height s) and the cap’s height h. The following trigonometric calculations yield the formula for h that is 12 used in equation 5 of the main text (cf. Figure S1 for notation): 휙 14 퐴퐶 = 푠 cos and 퐴퐵 = 퐴퐷 = 푠 2 13 As such, 휙 15 ℎ = 퐷퐶 = 퐴퐷 − 퐴퐶 = 푠 (1 − cos ) (푆1.1) 2 16 17 Figure S1. Cross section of an acoustic detection zone, showing the measurements needed to calculate 18 the height h of the spherical cap (shaded region). The detector is located at the vertex A, which is also the center 19 of the sphere of radius s that the spherical cap is a part of. The angle 흓 is the opening angle of the detector. See 20 text for calculations. 21 2 22 A2. Surface area of a spherical rectangle 23 The top section of a camera-trap’s detection zone is a spherical rectangle whose surface area 24 can be calculated by considering the proportion of the surface area of the sphere of radius s that it 25 covers. To determine this proportion, we use the concept of solid angles. Intuitively, a solid angle can be 26 understood as a measure for the amount of the field of view that is covered by a given object from some 27 particular observation point. Formally, an object’s solid angle is defined as the area of the unit sphere 28 that is blocked by the object from the view of an observer located at the center of the sphere. In our 29 case, the object is the rectangular frame that is determined by the horizontal and vertical fields of vision 30 of the camera, κ and λ (Fig. S2), which has solid angle Ω, calculated as follows (see proof below): 휅 휆 tan tan 2 2 32 Ω = 4 arcsin (푆2.1) 휅 휆 √tan2 + 1√tan2 + 1 2 2 31 The area of the spherical rectangle atop a camera’s detection zone is then: Ω 33 푆 = 4휋푠2 = 푠2Ω (푆2.2) 푅 4휋 34 because solid angles are measured in steradians (sr), a unitless quantity, and the maximum solid angle 35 for the complete unit sphere is 4π sr. The proportion of the unit sphere occupied by the solid angle of Ω 36 the camera frame is then . This proportion is multiplied by the surface area of the sphere of radius s, 4휋 37 which equals 4휋푠2, to obtain the surface area of the spherical rectangle as in equation S2.2. 38 To see the validity of equation S2.2, we note that the solid angle of a rectangle with side lengths 39 a and b that is located at a distance c in front of an observer (Fig. S2) is given by 푎푏 40 Ω = 4 arcsin (푆2.3) √푎2 + 푐2 √푏2 + 푐2 41 Khadjavi (1968), and that: 3 휅 휆 42 푎 = 푐 tan ; and 푏 = 푐 tan (S2.4) 2 2 43 where κ and λ are the horizontal and vertical fields of vision (Fig. S2). Substituting a and b from equation 44 (S2.4) into (S2.3) we obtain equation (S2.1). 45 46 47 Figure S2: Section of the cropped detection zone of camera traps. The dark sections are used in the 48 calculation of the vertical and diagonal fields of vision (FOVs) λ and γ, and the lateral angles μ and ν. The 49 rectangle ABCD, with side lengths a and b, is one quarter of the image frame delimited by the vertical and 50 horizontal FOVs, and is located a distance c from the detector at E. 4 51 A3. Calculation of fields of vision and lateral surface angles of cropped detectors (cameras) 52 Application of our methods requires knowing the horizontal, vertical, and diagonal fields of 53 vision of the camera (κ, λ and γ), as these are all required to calculate the opening angles (μ and ν) of the 54 disk sectors that make up the sides of the camera detection zone. However, camera manufacturers 55 often only provide the horizontal field of vision (FOV). Here we show how to calculate all the unknown 56 angles based on a single known angle and the aspect ratio of the image frame. First, in Appendix A3a, 57 we show how to determine the vertical FOV from the horizontal FOV if it is not already provided by the 58 manufacturer; in Appendix A3b, we show how to determine the diagonal FOV from the horizontal and 59 vertical FOVs; and in Appendix A3c, we show how to determine the lateral opening angles using all the 60 FOVs. 61 A3a. Calculation of the vertical FOV from the horizontal FOV 62 Some manufacturers provide both the horizontal and vertical FOVs, κ and λ. If only the 63 horizontal FOV is provided, the vertical FOV can be obtained from the horizontal FOV and the aspect 64 ratio, q, as: 휆 휅 65 tan = 푞 tan (푆3.1) 2 2 66 To see this, consider the pyramid ABCDE in figure S2, which represents one quarter of the base 휆 67 of the cropped detection zone. Defining 퐴퐵 = 푏, 퐴퐷 = 푎, and 퐴퐸 = 푐, we obtain 푎 = 푐 tan and 푏 = 2 휅 휅 휆 68 푐 tan , given that 퐴퐸퐷̂ = and 퐴퐸퐵̂ = . Because we also have 푏 = 푞푎, where q is the user-selected 2 2 2 69 aspect ratio (most commonly 3/4 or 9/16), we obtain (S3.1). 70 A3b. Calculation of the diagonal FOV from the horizontal and vertical FOVs 5 71 If a diagonal field of vision of a camera detector is not provided by the manufacturer, it can be 72 determined from the detector’s horizontal and vertical fields of vision, κ and λ. Even if the diagonal FOV 73 is provided, we suggest using the calculations here for consistency with the calculations of other angles. 74 For a rectilinear, non-distorted lens, γ is given by: 휆 휅 76 γ = 2 arctan √tan2 + tan2 (푆3.2) 2 2 75 77 To see this, consider the triangle ABC in Fig. S2, and note that 퐴퐵2 + 퐵퐶2 = 퐴퐶2. Furthermore, we have 휆 휅 훾 78 퐴퐵 = 푐 tan , 퐵퐶 = 푐 tan , and 퐴퐶 = 푐 tan , from which it follows that 2 2 2 휆 2 휅 2 훾 2 79 (푐 tan ) + (푐 tan ) = (푐 tan ) 2 2 2 휆 휅 훾 80 ⇒ √푐2 (tan2 + tan2 ) = 푐 tan 2 2 2 83 푐tan2휆2+tan2휅2=푐tan훾2#(푆and 81 and thus, equation (S3.2). 82 A3c. Calculation of the lateral opening angles μ and ν from the fields of vision 84 The sides of the base of the cropped detection zone are disk sectors, whose surface areas are 85 calculated by multiplying the total surface of a disk of radius s by the proportion of the disk that is 휇 휈 86 occupied by the disk sectors. These proportions are given by and , respectively, where μ and ν are 2휋 2휋 87 the frontal and lateral opening angles of the detection zone (Fig. S2), and which can be calculated from 88 the diagonal, vertical, and horizontal fields of vision, γ, λ, and κ as: 훾 훾 cos cos 휈 = 2 arccos 2 and 휇 = 2 arccos 2 (푆3.4) 89 휅 휆 cos cos 2 2 6 휇 휈 휈 90 To see this, note that = 퐷퐸퐶̂ and = 퐵퐸퐶̂ in Figure S3. DEC is a right triangle so 퐷퐸 = 퐶퐸 cos . 2 2 2 휅 91 Moreover, 퐴퐸 = 퐷퐸 cos from which it follows that 2 휈 휅 92 퐴퐸 = 퐶퐸 cos cos (푆3.5) 2 2 93 We also have EAC is a right triangle, so: 훾 94 퐴퐸 = 퐶퐸 cos (푆3.6) 2 95 Equating (S3.5) and (S3.6) we obtain: 휈 휅 훾 96 퐶퐸 cos cos = 퐶퐸 cos 2 2 2 훾 cos 휈 2 97 cos = 휅 2 cos 2 98 and thus, the first equation S3.4. The second equation is obtained in the same manner, only using λ 99 instead of κ. 7 100 A4. Non-uniformly distributed approach directions: derivation of equation 12. 101 In natural systems it is likely there will be some bias in movement, which invalidates the 102 assumption of equally likely directions of approach to the detection zone. Our simulations show that any 103 error in the estimation of density caused by such biases can be prevented by setting up multiple 104 detectors facing in different directions. However, if only a few detectors are available or if the sampling 105 region is not large enough to set multiple detectors independently, strong direction bias should be 106 addressed in the calculation of the mean profile area. 107 A4a. Derivation of formula in equation 12 108 To calculate the mean projected area of a detection zone, 푝̂, for the case of non-uniformly 109 distributed approach directions, we derive formulae for the detection zone’s projected area, p(ω, θ), 110 for all possible directions of approach (ω, θ), and then weight these according to the probability 111 distribution of approach directions P(ω, θ), which equals P(ω)P(θ) in case of independence, as follows: 116 푝̂ = ∬ 푃(휔) 푃(휃) 푝(휔, 휃) 푑휃 푑휔 (푆4.1) 112 As seen in Fig.
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