A generalised random encounter model for estimating animal density with remote sensor data

Summary Wildlife monitoring technology is advancing rapidly and the use of remote sensors such as camera traps and acoustic detectors is becoming common in both the terrestrial and marine environments. Current methods to estimate abundance or density require individual recognition of animals or knowing the distance of the animal from the sensor, which is often difficult. A method without these requirements, the random encounter model (REM), has been successfully applied to estimate animal densities from count data generated from camera traps. However, count data from acoustic detectors do not fit the assumptions of the REM due to the directionality of animal signals. We developed a generalised REM (gREM), to estimate absolute animal density from count data from both camera traps and acoustic detectors. We derived the gREM for different combinations of sensor detection widths and animal signal widths (a measure of directionality). We tested the accuracy and precision of this model using simulations of different combinations of sensor detection widths and animal signal widths, number of captures and models of animal movement. We find that the gREM produces accurate estimates of absolute animal density for all combinations of sensor detection widths and animal signal widths. However, larger sensor detection and animal signal widths were found to be more precise. While the model is accurate for all capture efforts tested, the precision of the estimate increases with the number of captures. We found no effect of different animal movement models on the accuracy and precision of the gREM. We conclude that the gREM provides an effective method to estimate absolute animal densities from remote sensor count data over a range of sensor and animal signal widths. The gREM is applicable for count data obtained in both marine and terrestrial environments, visually or acoustically (e.g. big cats, sharks, birds, echolocating bats and cetaceans). As sensors such as camera traps and acoustic detectors become more ubiquitous, the gREM will be increasingly useful for monitoring unmarked animal populations across broad spatial, temporal and taxonomic scales.

S1. Step length s N Number of steps per simulation d Distance moved in a time step m S Probability of remaining stationary -A Maximum turning angle rad Table S1. List of symbols used to describe the gREM and simulations. '-' means the quantity has no units.

S2. SUPPLEMENTARY METHODS
S2.1. Introduction. These supplementary methods derive all the models used. For continuity, the gas model derivation is included here as well as in the main text. The calculation of all integrals use in the gREM is included in the Python script S3.
S2.2. Gas model. Following ?, we derive the gas model where sensors can capture animals in any direction and animal signals are detectable from any direction (θ = 2π and α = 2π). We assume that animals are in a homogeneous environment, and move in straight lines of random direction with velocity v. We allow that our stationary sensor can capture animals at a detection distance r and that if an animal moves within this detection zone they are captured with a probability of one, while animals outside the zone are never captured. In order to derive animal density, we need to consider relative velocity from the reference frame of the animals. Conceptually, this requires us to imagine that all animals are stationary and randomly distributed in space, while the sensor moves with velocity v. If we calculate the area covered by the sensor during the survey period we can estimate the number of animals the sensor should capture. As a circle moving across a plane, the area covered by the sensor per unit time is 2rv. The number of expected captures, z, for a survey period of t, with an animal density of D is z = 2rvtD. To estimate the density, we rearrange to get D = z/2rvt. S2.2.1. gREM derivations for different detection and signal widths. Different combinations of θ and α would be expected to occur (e.g., sensors have different detection widths and animals have different signal widths). For different combinations θ and α, the area covered per unit time is no longer given by 2rv. Instead of the size of the sensor detection zone having a diameter of 2r, the size changes with the approach angle between the sensor and the animal. For any given signal width and detector width and depending on the angle that the animal approaches the sensor, the width of the area within which an animal can be detected is called the profile, p. The size of the profile (averaged across all approach angles) is defined as the average profilep. However, different combinations of θ and α need different equations to calculatep. Thisp is the only thing that changes We have identified the parameter space for the combinations of θ and α for which the derivation of the equations are the same (defined as sub-models in the gREM) ( Fig. S2.1). For example, the gas x 1 is used in SE and NE models (including the gas model). x 2x 4 are used in NW and SW models. The sector shaped detection region is shown in grey. Animals are filled black circles and the animal signal is an unfilled sector. The animals direction of movement is indicated with an arrow. The profile p is shown with a red line. (a) Animal is directly approaching the sensor at x 1 = π 2 . (b) Animal is directly approaching the sensor at x 2 = π 2 . x 2 then decreases until the profile is perpendicular to the edge of the detection region. (c) When the profile is perpendicular to the edge of the detection region, x 3 = θ. (d) x 4 measures the angle between the left side of the detection region and the profile. model becomes the simplest gREM sub-model (upper right in Fig. S2.1) and the REM from (?) is another gREM sub-model where θ < π/2 and α = 2π.
Models with θ = 2π are described first (the gas model described above and SE1). Then models with θ > π are described (NE then SE). Finally models with θ < π (NW then SW) are described.
S2.3. Model SE1. SE1 is very similar to the gas model except that because α ≤ π the profile width is no longer 2r but is instead limited by the width of the animal signal. We therefore get a profile width of 2r sin(α/2) instead.p SE1 = 1 π 3π 2 π 2 2r sin α 2 dx 1 eqn S1 This profile is integrated over the interval [ π 2 , 3π 2 ] which is π radians of rotation starting with the animal moving directly towards the sensor (Fig. S2.2a).
S2.4. Models NE1-3. When the detection zone is not a circle, we have more complex profiles and need to explicitly write functions for the width of the profile for every approach angle. We then use these functions to find the average profile widthp for all approach angles by integrating across all 2π angles of approach and dividing by 2π.
There are three submodels within quadrant NE ( Fig. S2.1). Note that NE1 covers the area α = 2π as well as the triangle below it as these two models are specified exactly the same, rather than happening to have equal results.
These models have up to five profiles.
(2) At x 1 = θ/2, the right hand side of the profile cannot be r wide as the corner of the 'blind spot' limits its size to being r cos(x 1 − θ/2) wide ( Fig. S2.3a).
(3) The third profile is only found in NE3. If α < 4π − 2θ, then at x 1 = θ/2 + π/2, when the profile is perpendicular to the edge of the blind spot, the whole right side of the profile is invisible to the sensor (Fig. S2.3b). This gives a profile size of just r. (4) At some point, the sensor can detect animals once they have passed the blind spot giving a profile width of r +r cos(x 1 +θ/2) ( Fig. S2.3c). From x 1 = π, if the animal signal is wide enough to be detected in this area, this is the wider profile. This then defines the split between NE1 x -θ/2 Figure S2.3. Three of the integrals in NE models. The sector shaped detection region is shown in grey. Animals are filled black circles and the animal signal is an unfilled sector. The animals direction of movement is indicated with an arrow. The profile p is shown with a red line.
Dashed red lines indicate areas where animals cannot be detected. (a) The second integral in NE with width r + r cos(x 1 − θ/2) (b) The third integral in NE3. α/2 is labelled. As it is small, animals to the right of the detector cannot be detected. (c) After further rotation, α/2 is now bigger than the angle shown and animals to the right of the detector can again be detected. and NE2. In NE1, with α > 3π − θ, the animal signal is wide enough that at x 1 = π the animal can immediately be detected past the blind spot and so this profile is used. In NE2, with α < 3π − θ, the latter profile is reached at 5π/2 − θ/2 − α/2. (5) Finally, common to all three models, at x 1 = 2π − θ/2 the profile becomes a full 2r once again.
(1) As α is less than π the profile is smaller than 2r, even when the sensor width is a full diameter. The profile width starts as 2r sin(α/2). (2) Similar to NE, at a certain point the blind spot of the sensor area limits the profile width on one side. This gives a profile width of r sin(α/2) + r cos(x 1 − θ/2) ( Fig. S2.4).
x 4 π + x -θ 4 (a) The right side is r cos(π − x 4 ) = −r cos x 4 respectively. In both images the sector shaped detection region is shown in grey. Animals are filled black circles and the animal signal is an unfilled sector. The animals direction of movement is indicated with an arrow. The profile p is shown with a red line.
NW1 is the first model with θ < π. Whereas previously the focal angle has always been x 1 , we now use different focal angles. x 2 and x 3 correspond to γ 1 and γ 2 in (?) while x 4 is new. They are described in Fig. S2.2b-d.
There are five different profiles in NW1.
(1) x 2 has an interval of [π/2, θ/2] which is from the angle of approach being directly towards the sensor until the profile is parallel to the left hand radius of the sensor sector ( Fig. S2.2b). During this interval the profile width is 2r sin (θ/2) sin(x 2 ) which is calculated using the equation for the length of a chord . Note that while rotating anti-clockwise (as usual) x 2 decreases in size.
(2) From here, we examine focal angle x 4 (note that x 3 is used in later models, but is not relevant here.) The left side of the profile is a full radius while the right side is limited to −r cos(x 4 − θ) ( Fig. S2.5a). (3) At x 4 = θ − π/2, the profile is perpendicular to the edge of the sensor area. Here, the right side of the profile is 0r giving a profile size of r. Figure S2.6. Profile sizes when an animal approaches from behind in models NW2-4. If α is relatively large, animals can be detected when approaching from behind. Otherwise animals cannot be detected. The sector shaped detection region is shown in grey. Animals are filled black circles and the animal signal is an unfilled sector. The animals direction of movement is indicated with an arrow. (a) If α/2 is less than π − θ/2, as is the case here, then the width of the profile when an animal approaches directly from behind is zero. (b) If α/2 > π − θ/2 the profile width from behind is 2r sin (θ/2) sin(x 2 ).
(4) When x 4 = π/2 the angle of approach is from behind the sensor, but we can once again be detected on the right side of the sensor ( Fig. S2.5b). Therefore the width of the profile is r − r cos(x 4 ). (5) Finally, we have the x 2 profile, but from behind.
S2.7. Models NW2-4. The models NW2-4 have the five potential profiles in NW1 but not all profiles occur in each model, and the angle at which transitions occur are different. Furthermore, there is one extra profile possible.
(1) When approaching the sensor from behind, there is a period where the profile is r wide as in NW1 profile (3).
(2) At some point after profile (1) animals to the right of the sensor can be detected again. If this occurs in the x 4 region, the profile width becomes r − r cos(x 4 ) as in NW1.
(3) However, as α is now less than 2π, animals to the right of the sensor may be undetectable until we are in the second x 2 region. In this case, when we first enter the second x 2 region, the profile has a width of r cos(x 2 − θ/2). This occurs only if α ≤ 3π − 2θ. This inequality is found by noting that animals to the right of the sensor can be detected again at x 4 = 3π/2 − α but the x 2 region starts at x 4 = θ. The new profile in x 2 will only occur if θ < 3π/2 − α/2 which is rearranged to find the inequality above. This defines the boundary between NW2 and NW3. (4) As α ≤ 2π it is possible that when the angle of approach is from directly behind the sensor the animal will not be detected at all. This is the case if α/2 ≤ π − θ/2 ( Fig. S2.6). This inequality (simplified as α ≤ 2π − θ) defines the boundary between NW3 and NW4. S2.7.1. Model NW2. NW2 is bounded by α ≥ 3π − 2θ, α ≤ 2π and θ ≤ π (Fig. S2.1). NW2 has all five profiles as found in NW1. However, the change from the r profile (third integral) to the r − r cos(x 4 ) profile (fourth integral) occurs at x 4 = 3π/2 − α/2 instead of at x 4 = θ.
NW3 does not have the fourth integral from NW2 as animals are not detectable to the right of the sensor until after the x 4 region has ended and the x 2 region has begun. Therefore the second x 4 integral has an upper limit of θ and the profile after has a width of r cos(x 2 −θ/2) and is integrated with respect to x 2 . The final integral starts at x 4 = 3π/2−α/2−θ/2 and has the full width of 2r sin(x 2 ) sin(θ/2).
S2.8. Model REM. REM is the model from (?). It has α = 2π and θ ≤ π/2 ( Fig. S2.1). It has three profile widths, two of which are repeated, once as the animal approaches from in front of the sensor and once as the animal approaches from behind the sensor.
(1) Starting with an approach direction of directly towards the sensor, and examining focal angle x 2 , the profile width is 2r sin(x 2 ) sin(θ/2).
(2) When the profile is perpendicular to the radius on the right hand of the sector sensor region, we instead examine x 3 where the profile width is r sin(x 3 ).
(3) At x 3 = π/2 the profile becomes simply r and this continues for θ radians of x 4 . (4) The x 3 profile is then repeated with an approach direction from behind the sensor. (5) Finally the x 2 profile is repeated, again with an approach direction from behind the sensor.
S2.9. Models NW5-7. In the models NW5-7, the sensor has θ ≤ π/2 as in the REM. As α ≥ π a lot of the profiles are similar to the REM. Specifically, the first three profiles are always the same as the first three profiles of the REM. This is because when an animal is moving towards the sensor, the α ≥ π signal is no different to a 2π signal. However, when approaching the sensor from behind, things are slightly different. The animal can only be detected by the sensor if the signal width is large enough that it can be detected once it has passed the sensor.
(1) Starting with an approach direction of directly towards the sensor, and examining focal angle x 2 , the profile width is 2r sin(x 2 ) sin(θ/2).
(2) When the profile is perpendicular to the radius edge of the sector sensor region, we instead examine x 3 where the profile width is r sin(x 3 ).
(3) At x 3 = π/2 the profile becomes simply r and this continues for θ radians of x 4 . (4) If α ≤ 2π + 2θ, the animal becomes undetectable during this profile when x 3 has decreased in size to π − α/2. This inequality marks the boundary between NW7 and NW6. (5) If instead α ≥ 2π + 2θ then the animal does not become undetectable during the x 3 focal angle. Instead the profile has width greater than zero for the whole of the x 3 angle. The x 2 profile starts with width r cos(x 2 − θ/2) as only animals approaching to the left of the sensor are detectable. (6) During this second x 2 profile the signal width needed for animals to be detected to the left of the detector is increasing while the angle needed for animals to be detected to the right of the detector is decreasing. Therefore, either the left side becomes undetectable, making both sides undetectable (this occurs if α ≤ 2π − θ as in NW6) (7) or the right becomes detectable (if α ≥ 2π − θ as in NW5), making both sides detectable and giving a profile width of 2r sin(x 2 ) sin(θ/2).
S2.9.1. Model NW5. NW5 is bounded by α ≥ 2π − θ, α ≤ 2π and θ ≤ π/2 ( Fig. S2.1). It is the same as REM except that it includes the extra profile in x 2 (the fifth integral) where only animals approaching to the left of the profile are detected.
S2.9.2. Model NW6. NW6 is bounded by α ≤ 2π − θ, α ≥ 2π + 2θ and θ ≤ π/2 (Fig. S2.1). NW6 is the same NW5 except that as α ≤ 2π − θ, animals that approach from directly behind the detector are not detected. Therefore at x 2 = α/2 + θ/2 − π/2 the profile width goes to zero and therefore the last integral in NW5 is not included. Figure S2.7. The first profile in SW models is limited by either α or β depending on whether α < β. The sector shaped detection region is shown in grey. Animals are filled black circles and the animal signal is an unfilled sector. The animals direction of movement is indicated with an arrow. (a) As α/2 < θ/2 the profile width is limited by the signal width rather than the sensor region. The profile width is 2r sin (α/2) (b) As α/2 > θ/2 the profile width is limited by the sensor region, not the signal width. The profile width is 2r sin (θ/2) sin(x 2 ).
S2.9.3. Model NW7. NW7 is bounded by α ≥ 2π + 2θ, α ≥ π and θ ≥ 0 (Fig. S2.1). It is similar to NW6 but does not include the last integral as during the x 3 profile, at x 3 = π − α/2 the signal width is too small for any animals to be detected, so the profile width goes to zero.
S2.10. Model SW1-3. The models in SW1-3 are described with the two focal angles used in models NW2-4, x 2 and x 4 . As α ≤ π an animal can never be detected if it is approaching the detector from behind. This makes these models simpler in that they go through the x 2 and x 4 profiles only once each.
There are five potential profile sizes.
(1) At the beginning of x 2 , with an approach direction directly towards the sensor, the parameter that limits the width of the profile can either be the sensor width, in which case the profile width is 2r sin (θ/2) sin(x 2 ).
(2) Or the signal width can be the limiting parameter, in which case the profile width is instead 2r sin(α/2) (Fig. S2.7) (3) The next potential profile in x 2 has a width of r sin(α/2) − r cos(x 2 + θ/2) as the right side of the profile is limited by the width of the sensor region while the left side is limited by the signal width. However, the angle at which the profile starts depends on whether the first profile was 1) or 2) above. If the first profile is profile 1) then the profile is limited on both sides by the sensor region and then the left side of the profile becomes limited by the signal width. This happens at x 2 = π/2 − α/2 + θ/2. If however the first profile was 2) then the first profile is limited by the signal width. We move into the new profile when the right side of the profile becomes limited by the sensor region. This occurs at x 2 = π/2 + α/2 − θ/2. (4) In the x 4 region the left side of the profile is always r sin(α/2) while the right side is either 0, giving a profile of r sin(α/2). (5) Or limited by the sensor giving a profile of size r sin(α/2) − r cos(x 4 − θ).
As α is large the first profile is limited by the size of the sensor region giving it a width of 2r sin (θ/2) sin(x 2 ). It is the only one of the three SW models to start in this way. Later on, still with x 2 as the focal angle the left side of the profile does become limited by the signal width. So at x 2 = π/2 − α/2 + θ/2 the profile width becomes r sin(α/2) − r cos(x 2 + θ/2).
As we enter the x 4 region, the profile remains limited by the signal on the left and by the sensor on the right, giving a profile width of r sin(α/2) − r cos(x 4 − θ). Finally, at x 4 = θ − π/2 the right side of the profile becomes zero and the profile is width is r sin(α/2).
S2.10.3. Model SW3. SW3 is bounded by α ≤ 2θ − π and θ ≤ π (Fig. S2.1). SW3 is similar to SW2 except that the profile does not become limited by sensor at all during the the x 4 regions. Therefore, at x 4 = 0 the profile is still of width 2r sin(α/2). Only at x 4 = θ − π/2 − α/2 does the profile become limited on the right by the sensor region.
S2.11. Model SW4-9. As α < π, animals approaching the sensor from behind can never be detected, so unlike REM, the second x 2 and x 3 profiles are always zero. The six models are split by three inequalities that relate to the models as follows. (1) Models with α ≤ π−2θ have no x 4 profile. This is because at x 4 = 0, the signal width is already too small to be detected as can be seen in Fig. S2.8a where α/2 < π/2 − θ which simplifies to give the previous inequality.
(3) Finally, models with α ≤ 2θ have a second profile in x 2 where to one side of the sensor α is the limiting factor of profile width, while on the other side θ is (Fig. S2.8b). This gives a width of r sin(α/2) − r cos(x 2 + θ/2). This profile does not occur in models with α ≥ 2θ.

S3. SUPPLEMENTARY SCRIPT: SYMBOLIC ALGEBRA PYTHON SCRIPT
This script uses the SymPy package (?), a computer algebra system to calculate the equations for p in the various models and to perform unit checks on the results.