### Douglas J. McCauley, Kevin A. McLean, John Bauer, Hillary S. Young, and Fiorenza Micheli. 2012. Evaluating the performance of methods for estimating the abundance of rapidly declining coastal shark populations. Ecological Applications 22:385–392.

Appendix H. Algorithms that model the operation of simulation-implemented survey methods.

To examine how the geometry and mechanics of a survey affect the density estimates they yield we mathematically modeled the operation of point, belt and video surveys as implemented in our simulation environment. Algorithms are based on the assumption that all sharks in the simulation are uniformly distributed and travel in random directions with a uniform distribution on the direction of travel (Appendix D). These assumptions generally hold true except in the case of severely concave survey geometries. Each equation predicts the number of unique sharks observed in each of these three survey methods given the specified survey conditions.

 Point: πR² + 2DRST (H.1)

R = radius (m); D = shark density (sharkes/m²); S = shark speed (m/s); T = survey time (s).

To model the operation of the point count surveys Eq. H.1 we consider an infinitesimally small area A near the edge of what the observer sees, of size δV. There are DδV sharks in A. We assume that each shark travels for an infinitesimal amount of time, δt, moving at speed S, and we assume that they will not change direction in time δt, so they move y = Sδt in time δt.

Because we assume the sharks travel in a random direction, the number of sharks that reach the observation area in time δt is the number of sharks that pick a direction for which the observation area is less than y away from the shark. Because y is infinitesimally small, we can approximate the edge of observation area near A as a straight line.

In this case, if the distance from A to the observation area is x, then the chance of picking a direction from a uniform distribution such that the distance is less than y from the observation area is (1/π)acos(x/y), as long as 0 ≤ xy, and is otherwise zero. Thus, the number of sharks from area A that enter the observation area is ƒ(A) = (1/π)acos(x/y)DδV.

With this value, we can then calculate the total number of sharks anywhere in the simulation that travel into the observation area in time δt. Sharks must be somewhere in the region from radius R and radius R + y away from the observer. The formula ƒ(A) does not depend on direction, so let ƒ(rV = ƒ(A), let δV = Rδθδr and integrate by polar coordinates: This provides a value for the number of sharks that cross into the observation area in time δt. Therefore, we have 2DRS sharks per second entering the observation area.

A point observer who observes for time T can therefore expect to see πR²D sharks in the initial review of the observation area (first term in the equation) and 2DRST additional sharks over the course of the observation (second term).

 Belt: WLD + ( 1 - K ) 2 DSKT + K 1 DSWT + K² 1 DST (H.2) L π L π L π

W = width (m); L = length (m); D = shark density (sharks/m²); K = max shark detection distance (m); S = shark speed (m/s); T = survey time (s).

The equation for density of sharks seen using the belt transect method Eq. H.2 is derived following logic similar to that described for the point count.

We assume that over the course of the belt observer’s travels, they see sharks equal to the expected number of sharks in the observation area. For this assumption to hold, we assume that sharks leave unobserved parts of the belt survey area at the same rate other sharks enter. This yields the first term of the equation, WLD.

The time until the belt observer is K away from the end of the transect is ((L-K)/L)T. During this period, the observer may see additional sharks that were not originally in the survey area as they swim in from the lateral sides of the transect. We assume any sharks entering from the frontal edge of the transect were already counted in the WLD term. As the sides of the belt observation area are each K long, that gives 2K of edges along which sharks could enter. Using the same formula for ƒ(A) computed for the point count, we have (1 - K/L)(2/π)DSKT sharks which swim in from the sides of the observation area during the first (1 - K/L)T time. This gives the second model term.

This leaves (K/L)T time in which the observer is observing a decreasing area, from distance K from the end to distance 0. There are two lateral edges and the frontal edge in which sharks can enter the observation area. Note that we now count sharks entering along the frontal edge of the survey. When there is time t left in the observer’s advance, the length of each lateral edge is t(L / KT)K, and the length of the frontal edge is W, for a total of 2t(L/T) + W edge from which new sharks can enter. Integrating from t = 0 to t = K/L, and once again using ƒ(A) to determine the number of sharks that enter the observation area, we get a total of (K/L)(1/π)DSWT + (K²/L)(1/π)DST sharks (the third and fourth terms in the equation).

 Video: R²θD T + T 1 DS(2R + 2Rθ) (H.3) 30 15 π

R = radius (m); θ = observation angle (both left and right); D = shark density (sharks/m²); S = shark speed (m/s); T = survey time (s).

Note that θ is half of the total angle seen by the observer, so θ = π is a whole circle. The video records data 2 s every 30 s. Because these intervals are invariant they are treated in the equation as constants.

With this information we can also calculate the number of sharks expected for the video observer Eq. H.3.

We assume that every time the camera turns off and then turns back (28 s later), the sharks have completely reset to a uniform distribution. This assumption may cause overestimation of shark density estimates, but is sufficiently valid to be operationally useful.

Using these assumptions, each time the camera turns on and records data, it captures R²θD sharks. It turns on T/30 times, for a total of R²θD(T/30) sharks (the first term in the equation).

Each time the camera turns on, it is then active for 2 s, during which time additional sharks can enter the observation area. There are 2R length of lateral edges and 2Rθ length of frontal edges (e.g. outside the observed radius but within the observed angle) in which sharks may enter. The total time the camera is in this state during the simulation is T/15.

Again using ƒ(A) from the point observer, we get (T/15)(1/π)DS(2R + 2Rθ) sharks entering the video observation area during periods when the camera is on (second term in the equation).

The largest contribution of these algorithms is their capacity to help identify the mechanisms that influence how each method estimates shark abundance. Survey geometry is clearly important. We see that the area of a given survey determines the initial number of sharks recorded during a survey. Additionally, the shape and circumference of the survey influence the likelihood that sharks will enter a survey area and be counted as the survey advances. Mechanical details of the surveys are also important. The delay programmed into the intermittently recording video survey permits sharks the opportunity to mix in the region of the camera and thus increases the likelihood that new individuals will be recorded by this method. While these algorithms could be turned around and used to translate shark density estimates from field surveys into estimates of “true” shark density, the error associated with their predictive capacity and the almost certain deviations between simulation assumptions (e.g. shark distribution and movement) and the realities of coastal environments suggest that such translations be made with great caution.

TABLE H1. Shark density values reported from unfished and fished simulation runs and predicted shark density estimated produced using algorithms derived above. Percent error represents deviation between simulation and algorithm density estimates.

 Unfished Fished Simulation Algorithm % Error Simulation Algorithm % Error Belt (B) 0.20 0.20 0.0 0.03 0.02 -50.0 Belt large (BL) 2.11 2.20 4.1 0.22 0.22 0.0 Point (P) 0.21 0.27 22.2 0.03 0.03 0.0 Point large (PL) 0.50 0.60 16.7 0.05 0.06 16.7 Video (V) 5.41 6.66 18.8 0.39 0.66 40.9

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