Calculation of receiver area
To determine the possible area where each fish was detected, initially the percentage of
detections for each receiver was calculated. Receivers with percentage detections fewer than
5% were considered negligible and were excluded from area calculations. For receiver
locations where the number of detections was greater than 5% of the total detections, the
overall area can be calculated.
The positioning of the receivers allows perfect coverage of the outer reef flat, reef crest and
base in Pioneer Bay, Orpheus Island. This coverage causes the range of each receiver to
intersect with the receivers adjacent to its position. Therefore, for the case where two
receivers, i and j for example, have their detection range areas intersect, if detections were
shown for receiver i and not receiver j, the detection area for receiver i is not simply the area
of a circle of equal receiver radius, r (m). Instead, the middle area between the two
intersections, Ei,j (m2), should be determined and subtracted from this area to calculate the
detection area based solely around receiver i, Ai (m2), receiver j, Aj (m2) and the combined
area of receivers i and j, Ai,j (m2) (Fig S1).
Fig. S1 Intersection of two receiver detection ranges for receivers i and j: a) schematic
showcasing intersection area, b) constructed geometry to determine the intersection area.
To determine the respective areas of receivers i and j, first the centre-centre distance, di,j (m),
between the receivers is approximated using global positioning system (GPS) coordinates. As
the range radius is identical for all receivers, the angle, θi,j, created by a radial line generating
from the centre of the receiver to the intersection with the adjacent receiver is calculated
using the principles of an isosceles triangle such that.
𝜃𝑖,𝑗 = cos−1
𝑑𝑖,𝑗
2𝑟
(1)
The resulting area of the intersection between the two receivers can then be calculated as,
𝐸𝑖,𝑗 = 𝑟 2 (2𝜃𝑖,𝑗 − sin 2𝜃𝑖,𝑗 )
With the areas of receivers i and j separately being,
(2)
𝐴𝑖 = 𝜋𝑟 2 − 𝐸𝑖,𝑗
(3)
𝐴𝑗 = 𝜋𝑟 2 − 𝐸𝑖,𝑗
(4)
and the combined area if a detection was detected at receivers i and j is,
𝐴𝑖,𝑗 = 𝐴𝑖 + 𝐴𝑗 + 𝐸𝑖,𝑗 = 2𝜋𝑟 2 − 𝐸𝑖,𝑗
(5)
Although these equations hold true for intersections of two receivers, due to their positioning,
some cases create intersections between three receivers for example, i, j and k (Fig S2). In
this instance, a number of possible outcomes are present. These include, detections at only
receiver i, receiver j or receiver k, detections at either receivers i and j, receivers i and k or
receivers j and k, or detections at all receivers i, j and k. In either of these circumstances, the
area for all segments must be calculated. This is achieved by first generating a polygon
consisting of a number of triangles that form when drawing lines between each intersection
and centre points of the receivers (Fig S2b).
Fig. S2 Intersection of three receiver detection ranges for receivers i, j and k: a) schematic
showing intersection areas, b) constructed geometry to determine the inner polygon area, c)
showcased angles to determine the outer circular areas not included in the inner polygon’s
geometry.
The distances between each centre point, di,j (m), di,k (m) and dj,k (m), are again determined
through GPS coordinates. The polygon area, Pi,j,k (m2), can then be calculated by summating
the areas of the four triangles drawn with Heron’s formula. For the three outer triangles, their
respective areas, pi,j (m2), pi,k (m2) and pj,k (m2), can be calculated by,
𝑠𝑥,𝑦 =
2𝑟 + 𝑑𝑥,𝑦
2
𝑝𝑥,𝑦 = 𝑠𝑥,𝑦 (𝑠𝑥,𝑦 − 𝑑𝑥,𝑦 )(𝑠𝑥,𝑦 − 𝑟)
(6)
2
(7)
where x and y are the respective possible combinations of receivers i, j and k. The inner
triangle area, pi,j,k (m2), is determined by,
𝑠𝑖,𝑗,𝑘 =
𝑑𝑖,𝑗 + 𝑑𝑖,𝑘 + 𝑑𝑗,𝑘
2
𝑝𝑖,𝑗,𝑘
(8)
(9)
= 𝑠𝑖,𝑗,𝑘 (𝑠𝑖,𝑗,𝑘 − 𝑑𝑖,𝑗 )(𝑠𝑖,𝑗,𝑘 − 𝑑𝑖,𝑘 )(𝑠𝑖,𝑗,𝑘 − 𝑑𝑗,𝑘 )
Thus, the overall inner polygon area is,
𝑃𝑖,𝑗,𝑘 = 𝑝𝑖,𝑗 + 𝑝𝑖,𝑘 + 𝑝𝑗,𝑘 + 𝑝𝑖,𝑗,𝑘
(10)
The remaining area of the circle can be determined by finding the outer angle, φ, for the
various receivers i, j and k as shown in Fig S2c. For all angles shown by α and γ, these can be
calculated using a modified eqn (1) with α and γ replacing θ, and the centre-centre distance
di,j replaced with the corresponding centre-centre distances (Fig S2). The inner angle β is
determined using the cosine rule for all receivers i, j and k. Therefore, the outer angle is
calculated for each receiver as,
𝜑𝑥 = 2𝜋 − 𝛼𝑥 − 𝛽𝑥 − 𝛾𝑥
(11)
where subscript x is replaced with i, j and k for each corresponding receiver. The outer area
for each receiver, Oi (m2), Oj (m2) and Ok (m2), not considered by the inner polygon is then,
𝑂𝑥 =
1 2
𝑟 𝜑𝑥
2
(12)
Again with subscript x being replaced with i, j and k for each corresponding receiver. The
total combined area, Ai,j,k (m2), of receivers i, j and k can now be found using eqn (13).
𝐴𝑖,𝑗,𝑘 = 𝑂𝑖 + 𝑂𝑗 + 𝑂𝑘 + 𝑃𝑖,𝑗,𝑘
(13)
The individual areas of each receiver, Ai (m2), Aj (m2) and Ak (m2), are determined by
considering the intersection of only each pair of receivers within the group, and subtracting
this combined area of the two intersecting receivers from the total combined area. This
summarizes to,
𝐴𝑖 = 𝐴𝑖,𝑗,𝑘 − 𝐴𝑗,𝑘
(14)
𝐴𝑗 = 𝐴𝑖,𝑗,𝑘 − 𝐴𝑖,𝑘
(15)
𝐴𝑘 = 𝐴𝑖,𝑗,𝑘 − 𝐴𝑖,𝑗
(16)
Throughout the process of calculating the areas for each pair of receivers, the lens areas Ei,j
(m2), Ei,k (m2) and Ej,k (m2) are determined. Therefore, the middle area, mi,j,k (m2), of the
intersection can be determined, as this area, dependent on the positioning of the receivers,
must be subtracted or added to each receiver area based on a number of combinations. This
area can be found using simple geometry such that,
𝑚𝑖,𝑗,𝑘
=
𝐴𝑖,𝑗,𝑘 − 𝐴𝑖 − 𝐴𝑗 − 𝐴𝑘 − 𝐸𝑖,𝑗 − 𝐸𝑖,𝑘 − 𝐸𝑗,𝑘
(−2)
(17)
Conducting these calculations for each receiver allows for the calculations of a number of
areas. These calculations form a tedious process, and a MATLAB code was generated to
conduct these calculations, and given appropriate percentage detections at each receiver, this
code displays the total detection area of the reef fish where total detections exceeded 5% of
all detections.