Strindberg And Buckland 2004

Zigzag Survey Designs in Line Transect Sampling Samantha STRINDBERG and Stephen T. BUCKLAND Zigzag survey lines are frequently used in shipboard and aerial line transect surveys of animal populations. Analysis proceeds assuming that coverage probability through the survey region is uniform. We show that the two types of zigzag design that are in wide use do not generally have this property, and explore the degree of bias in abundance estimates that can be anticipated. We construct a zigzag design for convex survey regions that has even coverage probability with respect to distance along a design axis. We also provide Horvitz-Thompson estimators that allow coverage probability to vary by location through the survey region. Key Words: Automated design algorithms; Coverage probability; Sighting surveys; Survey design; Systematic designs; Zigzag designs.

1. INTRODUCTION Shipboard and aerial line transect surveys are widely used for estimating the abundance of marine mammals (e.g., Borchers et al. 1998; Buckland et al. 2001, pp. 288–290) and seabirds (e.g., Clarke et al. 2003; Buckland et al. 2001, p. 290), and to a lesser extent, fish (e.g., Foote and Stefansson 1993; O’Connell and Carlile 1993; Chen and Cowling 2001). Aircraft are also widely used for surveys of terrestrial mammals (White, Bartmann, Carpenter, and Garrott 1989; Pojar, Dowden, and Gill 1995; van Hensbergen, Berry, and Juritz 1996; Trenkel, Buckland, McLean, and Elston 1997; Pople et al. 1998; Wiig and Derocher 1999). A design consists of a series of survey lines, along which the ship or aircraft travels, and observers record animals sighted, together with the distances of those observations from the survey line. A full description of the approach, including detailed discussion of survey design and analysis, was given by Buckland et al. (2001). Frequently in such surveys, the survey design comprises a series of parallel lines, either Samantha Strindberg is Associate Conservation Scientist and Biometrician, Living Landscapes Program, International Conservation Programs, Wildlife Conservation Society, 2300 Southern Boulevard, Bronx, NY 10460 (E-mail: [email protected]). Stephen T. Buckland is Professor of Statistics and Director, Centre for Research into Ecological and Environmental Modelling, The Observatory, Buchanan Gardens, St. Andrews KY16 9LZ, Scotland (E-mail: [email protected]). c
2004 American Statistical Association and the International Biometric Society Journal of Agricultural, Biological, and Environmental Statistics, Volume 9, Number 4, Pages 443–461 DOI: 10.1198/108571104X15601

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Figure 1. A line transect survey comprises a series of lines. Illustrated here within a simple rectangular survey region are examples of three commonly used designs: randomly spaced parallel lines (top); systematically spaced parallel lines (middle); and a continuous zigzag sampler.

randomly spaced or systematically spaced with a random start (Figure 1). The disadvantage of such a design is that the ship or aircraft must travel from one line to the next “offeffort,” that is, without searching for the animals of interest. If searching is carried out on these legs, the random design is compromised, as there tends to be greater effort along the boundaries of the survey region where animal density may be atypical, so that designbased estimation of abundance is biased. If searching is not carried out on these legs, resources are not used efficiently, especially in the case of shipboard surveys, for which ship time is expensive. Successive survey lines may be tens or even hundreds of kilometers apart, so that the loss of search effort is unacceptable on cost grounds. As a consequence, continuous zigzag designs are often preferred. These have the property that survey effort is continuous. Typically, a design comprises a single zigzag survey line, although for analysis purposes, successive legs of the zigzag are often assumed to be independent, just as the

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separate lines of a systematic parallel-line design are usually assumed to be independent (Figure 1). Thus, variance estimation proceeds as if the legs of the zigzag are random lines, independently placed within the survey region (Buckland et al. 2001, p. 238). Although variance estimation may be biased, systematic designs are efficient because systematic zigzags and systematic parallel lines are more evenly distributed through the survey region than are randomly spaced parallel lines (Figure 1). In a shipboard survey of a large region, lines tend to be long (perhaps several hundred kilometers) and resources typically allow few lines per stratum. Thus, design efficiency is an important advantage of systematic sampling. (If lines are shorter, then daylight hours are wasted in traveling from one line to the next for parallel-line designs.) We refer to the surveyed strip comprising the survey line together with its associated width 2w (w either side of the line) as a sampler. We define coverage probability π(u) at a point (x, y), denoted by u ≡ u(x, y), to be the probability that the sampler covers the point u, given a rule for randomizing the location of the sampler. Generally, zigzag survey designs do not provide uniform coverage probability throughout the survey region, yet the standard analysis methods assume that they do. This can lead to substantial bias in abundance estimates if animal density varies appreciably through the study area (Strindberg 2001). We address how abundance estimation should be modified to take account of uneven coverage probability, using a Horvitz-Thompson estimator and consider the properties of commonly used zigzag designs, demonstrating their potential for bias in estimates of animal abundance. We also construct a zigzag design within a convex-shaped survey region (termed “convex survey region” for brevity) that has even coverage probability with respect to distance along a design axis, and show how to modify the method for nonconvex survey regions. Annual surveys of Antarctic minke whales are conducted under the auspices of the International Whaling Commission. Due to the cost of operating ships in the Antarctic, and because of the size of the area, practical considerations dictate that zigzag designs be used. We illustrate the design algorithms of this article, using these surveys. The example also shows the importance of using an automated design algorithm that requires no subjective decisions; the ice-edge, which delineates the southern boundary of the survey region, can move by as much as 100 kilometers per day, so that the design must be developed as the survey progresses.

2. HORVITZ-THOMPSON ESTIMATOR If a survey design that gives uneven coverage probability is used, then a HorvitzThompson estimator (Horvitz and Thompson 1952) can be applied to obtain an unbiased estimate of abundance. Cooke (1985) was the first to suggest this approach for line transect sampling. When coverage probability is uneven, it is usually unknown. However, given automated algorithms for generating zigzag and other samplers, we can readily estimate coverage probability at any chosen point by simulation. To do this, we repeatedly generate

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realizations of a given design, and observe the proportion of realizations for which the generated sampler covers the chosen point. For estimating animal abundance, we must obtain this proportion for all points at which animals were detected. These coverage probabilities are estimated, whereas the Horvitz-Thompson estimator assumes that they are known. This presents no practical difficulty, however, as we can estimate them to any desired precision, simply by increasing the number of realizations of the sampler that we generate. In practice, except for strip transect surveys during which all objects within w of the line are counted, the effective half-width of search µ must also be estimated, and its precision is determined by the survey data, so that variance estimation should take this into account (this is discussed in the following). In conventional line transect sampling of a population when animals do not occur in groups or “clusters,” animal abundance N is estimated by An Nˆ = , 2µL ˆ

(2.1)

where A is the surface area of the survey region, n is the number of animals detected, µˆ is the estimated effective strip half-width, and L is the total length of line (Buckland et al. 2001). This may be obtained as a Horvitz-Thompson-like estimator as follows. If an animal i is located outside the sampled strips, it cannot be sampled. If it is inside the strip, and we do not condition on its distance from the line, the probability that it is detected is estimated by µ/w, ˆ where w is the half-width of the strip (Buckland et al. 2001, pp. 37–38). Let πc denote the mean coverage probability, where πc ≈ 2w×L/A for small w. (The exact value includes an additional term in w2 that depends on the angle of the sampler to the design axis and on the shape of the survey region. Modified designs for which the sampler does not overlap the region boundary that satisfy this result exactly are possible, but abundance estimates are then biased if animal density is not typical near the boundary.) Assuming even coverage probability, the probability that a sampled strip includes a given animal is πc . Hence the estimated inclusion probability pˆi of animal i is the product of these two quantities, and Nˆ =

n n n


An 1 1 1 = . = = pˆi (µ/w)π ˆ ( µ/w)(2wL/A) ˆ 2 µL ˆ c i=1

i=1

(2.2)

i=1

Note that this is not a strict Horvitz-Thompson estimator since the inclusion probability must be estimated when µ is unknown (Borchers et al. 1998). We now allow the probability that the sampled strip includes a given animal to vary by location. Denote the coverage probability at the location of detected animal i by πi . Then Nˆ =

n n n

1 1 w
1 = = . pˆi (µ/w)π ˆ µˆ πi i i=1

i=1

(2.3)

i=1

If πi = πc for all i (even coverage probability), this gives the same result as before. This equation may be further generalized by allowing animals within the strip to have their own

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individual effective strip half-widths µi : Nˆ =

n n n


1 1 1 = =w . pˆi (µˆ i /w)πi µˆ i πi i=1

i=1

(2.4)

i=1

The µi may be modeled as a function of their distance from the line and of other covariates (Marques and Buckland 2003). If animals occur in clusters, and the ith detected cluster is of size si , then the above formulas yield estimates of cluster abundance, and animal abundance may be obtained by n n replacing i=1 1/pˆi with i=1 si /pˆi in each of the above expressions. Variances for the various estimators may be estimated by bootstrapping transects (Borchers et al. 1998; Marques and Buckland 2003). For zigzag survey designs these are usually taken to be the separate legs of the zigzag sampler. This tends to be more robust than the analytic variance for the Horvitz-Thompson estimator, because it does not require the assumption that different detections on the same transect are independent, and uncertainty due to estimating µ is readily incorporated into the bootstrap variance. The simple percentile method is generally adequate for estimating confidence limits (Buckland et al. 2001, p. 82). If analytic variances are preferred, Huggins (1989) showed how to include a component into the variance of the Horvitz-Thompson estimator when the inclusion probabilities are estimated. However, in the context of zigzag samplers, as for standard systematic sampling methods, some joint inclusion probabilities are zero if a single sampler is used. It is generally impractical to conduct surveys using several zigzag samplers, randomly located over the survey region, so that the strategy of bootstrapping sampler legs has strong practical appeal.

3. PROPERTIES OF ZIGZAG SURVEY DESIGNS In this and the section that follows, we assume that the survey region is convex. Methods for nonconvex regions are considered in Section 5.

3.1

COVERAGE PROBABILITY

A zigzag survey line is defined with respect to a design axis G that, without loss of generality, starts at the origin and runs along the x-axis of a Cartesian coordinate system. Let the angle of the line to the design axis G be either θ(x) or 180◦ − θ(x) at distance x, where θ(x) may vary in the range [0, 90◦ ). Conventional zigzag designs keep θ(x) constant within a leg, but we do not impose that constraint here. To use standard distance sampling methodology, we would like a zigzag sampler to provide even coverage probability, so that coverage probability at point u is π(u) = πc , ∀u ∈ R. To date, survey designs based on zigzag samplers have been ad hoc, with no check that coverage probability is at least approximately uniform (e.g. Buckland, Cattanach, and Gunnlaugsson 1992; Barlow 1994; Branch and Butterworth 2001; Hammond et al. 2002; Hedley and Buckland 2004).

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Figure 2. The sampler s intersected by an infinitesimal strip perpendicular to the design axis G of width δx, height H(x) and area (to first order) H(x)δx. The height hs (x) of s is determined by its width 2w and angle θ(x), where hs (x) = 2w / cosθ(x). The filled parallelogram of area hs (x)δx indicates the covered proportion of the strip. The h (x) h (x)δx = s . marginal coverage probability πx (x) is therefore given by πx (x) = s H(x)δx H(x)

Let hs (x) denote the height of the zigzag sampler s at location x along the design axis G. Figure 2 shows that hs (x) is given by hs (x) =

2w . cosθ(x)

(3.1)

The coverage probability π(u) achieved by a zigzag sampler typically varies spatially. We consider how it varies along the design axis (the horizontal component πx (x)), integrated over y. Similarly, we use the term “vertical component” to denote coverage probability at point y, πy (y), after integration over x. Let H(x) denote the height of the survey region at x. For all types of zigzag design, the horizontal component is given by πx (x) =

hs (x) . H(x)

(3.2)

This is illustrated in Figure 2. The vertical component πy (y) does not have an equivalent analytic form. Its value depends on the type of zigzag sampler, the survey region shape and the total survey effort L. To ensure that coverage probability will be even throughout the region, a solution is to select a random orientation for G, provided we select a zigzag sampler such that πx (x) as given by Equation (3.2) is constant. The algorithms for generating each of the zigzag samplers described in this article are readily automated, so it is easy to estimate the combined vertical and horizontal coverage probability as a function of location by simulation. If this were considered to vary too much from a constant probability, the Horvitz-Thompson

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estimator, which does not require the assumption of an even coverage probability, can be used as described in Section 2.

3.2

GENERAL CONSTRAINTS FOR THE ZIGZAG SAMPLER

Consider a survey region R and a zigzag sampler s that forms an angle of either θ(x) or 180◦ − θ(x) with design axis G, where sampler angle θ(x) ∈ [0, 90◦ ). In general, θ(x) varies with distance x along G. Suppose the boundary of the survey region is defined by a set of straight edges E = {e1 , e2 , . . . , eK }, and let φ = {φ1 , φ2 , . . . , φK } denote the set of angles formed by each of the ek with G, where k = 1, . . . , K. Let EU denote the set of “upper” edges that can be intersected by s as it moves in the direction of increasing y. Let EL denote the set of “lower” edges that can be intersected by s as it moves in the direction of decreasing y. Depending on whether an edge ek belongs to the set EU or EL and the angle φk of ek , there is a corresponding allowable range for the zigzag angle θ(x) that prevents the sampler s from “bouncing outside” the survey region R. Figure 3 shows four categories of survey region edge along with the corresponding allowable sampler angle range. Thus, if we consider a survey region R with edge angles φ for a selected design axis orientation, the smallest possible sampler angle (corresponding to the least effort L), which ensures that s remains within R, can be calculated by considering the edge that has the greatest absolute slope with respect to G (any edge ek perpendicular to G, i.e., φk = 90◦ , is not considered as the sampler is not required to “bounce off” such edges). Let φ denote the angle that such an edge forms with G. Then the smallest possible allowable sampler angle is θ = φ if 0 ≤ φ < 90◦ or θ = 180◦ − φ if 90◦ < φ ≤ 180◦ . (Note that the boundary is taken to be part of the survey region and the sampler is allowed to coincide with the boundary). The minimum amount of effort, that is, sampler length L, required to avoid bouncing outside R can be estimated by applying the formula: L=

(xmax − xmin ) , cosθ

(3.3)

where xmin and xmax define the minimum and maximum values of x within the survey region R. When the ends of R are not lines perpendicular to G the formula shown in (3.3) is an approximation and will overestimate the amount of effort required.

4. ZIGZAG SAMPLER ALGORITHMS 4.1

THE EQUAL-ANGLE ZIGZAG SAMPLER

For the equal-angle zigzag sampler, θ(x) = θ for all x. The design may be implemented by selecting a random (uniformly distributed) point along the design axis, and a sampler orientation with respect to the design axis of θ or 180◦ − θ, with equal probability. The sampler is extended from the selected point in both directions until it meets the upper and lower

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Figure 3. A survey region edge ek belongs to either the set of “upper” or the set of “lower” edges (EU or EL ) that can be intersected by the zigzag sampler s as it moves in the direction of increasing or decreasing y, respectively. Consider a coordinate system whose origin O can be located at any point along ek . The sampler angle (either θ(x) or 180◦ − θ(x)) may vary with distance x along the design axis G. The restrictions on θ(x), which ensure that s remains within the boundary of R as it intercepts ek , are determined by its edge set and the angle φk (indicated by the dotted line in the diagrams) of ek . As s approaches O from the left (as shown in diagrams 1 and 4) or from the right (diagrams 2 and 3) θ(x) must be such that s passes through the portion of the quadrant indicated by the vertical lines fill and the quadrant indicated by the diamond fill, which is completely accessible (i.e., s can take on any orientation in that quadrant and still remain within the boundary of R).

region boundary. At each point of intersection of the sampler with the boundary (termed a “waypoint”), the sampler “bounces off” the boundary. This is achieved by reflecting the sampler in a line perpendicular to G passing through the intersection of the sampler with the boundary. This procedure is continued until the sampler covers the entire x-value range of G. Designs based on an equal-angle zigzag sampler are denoted by SEAZ . An equal-angle zigzag sampler is generated using a specified value for either θ or L. We may obtain θ from L or vice versa from the relationship shown in Equation (3.3). As noted earlier the value obtained will either be exact or a conservative approximation depending on whether or not the ends of R are lines perpendicular to G. As θ is constant, Equation (3.1) yields hs (x) =

2w =h cosθ

say, from which Equation (3.2) gives πx (x) =

h , H(x)

∀x ∈ R.

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Figure 4. The construction of an equal-spaced zigzag sampler (solid line) using the intersection points of a systematically spaced set of parallel lines (dashed lines) with the boundary of the survey region R. The sample end segments are constructed using the intersection of the dotted and dot-dashed line.

Thus, the coverage probability is a function of distance along the design axis and is inversely proportional to the height of the survey region.

4.2

THE EQUAL-SPACED ZIGZAG SAMPLER

To construct the equal-spaced zigzag sampler, a set of equally spaced parallel lines is constructed perpendicular to, and randomly positioned along, the design axis G and spanning its full length. Alternate intersections of these lines with the region boundary form the waypoints of the zigzag sampler, as illustrated in Figure 4. Thus, if the parallel lines pass north to south, the zigzag is formed by linking the north intersection of one line with the boundary to the south intersection of the next line, and so on. This fails to define the sections of sampler between xmin and the first parallel line, and between the last line and xmax . We chose to define these sections as shown in Figure 4. We denote designs based on an equal-spaced zigzag sampler by SESZ . An equal-spaced zigzag sampler thus has waypoints equally spaced along the design axis. For a specified spacing c, it is possible to estimate the expected total sampler length for a realization of the design. The actual sampler length varies about this value, depending on the shape of the survey region and the particular realization of the design. Similarly, for a fixed total zigzag length L, an estimate of the spacing required to generate that total line length can be calculated. To estimate L given c or vice versa the following formula is applied:   2 (xmax − xmin ) A L= + c2 . xmax − xmin c The formula relies on the knowledge that the average height of the survey region R is given by the surface area A of R divided by its width (xmax − xmin ) with respect to G. Thus L can be approximated by multiplying the length of the hypotenuse of the right-angled triangle, whose other edges include this average height and the spacing c, by the number of spacings

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that fit between xmin and xmax . Again, this formula will approximate values for L or c unless the ends of R are lines perpendicular to G. The equal-spaced zigzag sampler can be partitioned into K segments. For each sampler segment sk the sampler angle θ(x) remains constant. That is, θ(x) = θk for every x ∈ sk , where k = 1, . . . , K. Thus, for each sampler segment sk , from Equation (3.1), the height hk of sk is given by hs (x) = hk =

2w cos(θk )

∀x ∈ sk ,

k = 1, . . . , K.

Equation (3.2) yields πx (x) = πk =

hk H(x)

∀x ∈ sk ,

k = 1, . . . , K.

As for the equal-angle zigzag sampler, the coverage probability varies with distance along the design axis. It is inversely proportional to the height of the survey region within segment k. However, the angle of the sampler, and hence the constant of proportionality hk , differs between sampler segments. This allows a closer approximation to even coverage probability, with a smaller value for hk in shorter segments.

4.3

THE ADJUSTED-ANGLE ZIGZAG SAMPLER

The equal-angle zigzag and, to a lesser degree, the equal-spaced zigzag do not achieve an even coverage probability because the surface area of a sampler subsection falling within any strip of R is not proportional to the surface area of that strip. We can construct a zigzag sampler that does have this property. From Figure 2, we simply need to ensure that sampler height hs (x) is proportional to survey region height H(x). Then πx (x) = πc and from Equations (3.1) and (3.2) together with the approximation for mean coverage probability this gives 2w 2w × L = . cosθ(x) × H(x) A Thus, the angle θ(x) must be continually adjusted in proportion to the survey region height H(x). At any position along G, the angle of the sampler is given by   A A −1 , ∀x ∈ Rs ⊂ R, θ(x) = cos ≤ 1, ∀x, (4.1) LH(x) LH(x) where Rs denotes the sampled subregion of R. Note that even coverage probability for the above continuous sampler is achievable only if H(x) ≥ (A/L) for all x ∈ R. As for other zigzag samplers, the procedure relies on being able to “bounce off” a survey region boundary at an angle θ(x) to G. This is possible only when the absolute value of the slope of the boundary of R is smaller than the absolute value of the slope of the sampler. (Note that boundary sections at an angle of 90◦ , that is, with an infinite slope, need not be considered as the sampler terminates when it reaches such

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sections.) This condition always holds as θ(x) → 90◦ , which occurs as L → ∞. For smaller values of θ(x), subregions may occur in which a continuous sampler with constant coverage probability along the design axis is not achievable. When H(x) = (A/L), θ(x) = 0◦ , and at smaller values of H(x), even a sampler that is parallel to the design axis over-samples an infinitesimal strip at x. For most surveys, bias from this source is likely to be minimal, but if it is considered to be a significant problem, a solution is to reduce search effort along this section of transect, for example, by traveling “off-effort” for short segments. In the special case of a rectangular survey region of height H and width W , whose top and bottom edge runs parallel to the design axis, the sampler angle is given by     H ×W W −1 −1 θ(x) = cos = cos (L ≥ W ), L×H L independent of x. In this case, all three zigzag samplers considered here are equivalent. To implement the adjusted-angle zigzag sampler, first select a starting point at random within the survey region, and determine the corresponding angle θ(x) from Equation (4.1). Select from θ(x) and 180◦ −θ(x) with equal probability to determine the angle of the sampler to the design axis G. Extend the sampler from the selected point in both directions along G, while continually adjusting the angle according to Equation (4.1). When it intersects the upper or lower boundary of R, it is reflected in a similar manner to the equal-angle zigzag is, except that the sampler angle is determined by the height of R at that x-coordinate. This procedure is continued until the sampler covers the entire x-value range of G. We denote designs based on an adjusted-angle zigzag sampler by SAAZ .

4.4

COMPARISON OF ZIGZAG SAMPLERS

As noted previously, general practice is to estimate animal abundance assuming that the zigzag sampler design attains even coverage probability πc . Under that assumption, and given knowledge of animal density D(x, y) within the area covered by the sampler, we could obtain a Horvitz-Thompson estimator (Horvitz and Thompson 1952) of abundance as  xmax  ys (x)+hs (x) 1 ˆ NHT = D(x, y)dydx, πc xmin ys (x) where xmin and xmax define the full range of x within the survey region, and ys (x) is the lower y-extreme of the sampler at distance x along the design axis. This equation can equivalently be written as  1 Nˆ HT = D(u)z(u)du, (4.2) πc R where z(u) is an indicator function, equal to one if u ∈ Rs , and zero otherwise. These results are not useful for analysis of real survey data, because D(x, y) (equivalently D(u)) cannot be observed. However, they allow us to evaluate bias arising from assuming that the equal-angle and equal-spaced zigzag samplers provide even coverage

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Figure 5. A trapezoidal survey region R, whose design axis G and edge coincide with the x-axis. Within R three types of zigzag sampler are illustrated: equal-angle (dotted line), equal-spaced (dashed line), and adjusted-angle (solid line).

probabilities, given an assumed form for D(u), without the need to simulate. Strindberg (2001) evaluated the bias in Nˆ HT for three cases: D(u) = αx + β, ∀u ∈ R where α, β are positive constants; D(u) = αx + βy + γxy + δ, ∀u ∈ R, where α, β, γ, δ are positive constants; and D(u) = α(x)y 2 + β(x)y + γ(x), ∀u ∈ R where α(x) =

4(Dmax (x) − Dmin (x)) , (H(x))2

β(x) = −α(x)H(x), γ(x) = Dmax (x), and Dmin (x), Dmax (x), are the minimum and maximum density with respect to y for a given x. The functional form of Dmin (x) and Dmax (x) is such that Dmax (x) > Dmin (x) > 0, ∀x ∈ R. The survey region was taken to be a trapezium with vertices at (0, 0), (0, 100), (120, 20), and (120, 0), respectively (Figure 5). Each of the three zigzag samplers was constructed with the same total length L. To compare them visually, we started all three from the origin in Figure 5, although in practice,

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Figure 6. Coverage probability πx (x) plotted against x for the three zigzag samplers in Figure 5. Also shown is the height of the trapezium (not to scale) as a function of distance along the design axis, which indicates that the equal-angle zigzag (dotted line) has too low coverage probability where the survey region is wide, and too high where it is narrow. For the equal-spaced zigzag (dashed line), the coverage probability changes for each change in sampler angle. The coverage probability for the adjusted-angle zigzag (solid line) remains constant, because the angle varies as a smooth function of the trapezium height.

their location within the survey region would be random. The curve of the adjusted-angle zigzag becomes especially apparent in Figure 5 as the height of the survey region becomes small. Also apparent is the much better approximation of the equal-spaced zigzag sampler to the adjusted-angle sampler than that of the equal-angle sampler. This is reflected in Figure 6, which shows that the equal-spaced sampler provides less variable coverage probability than does the equal-angle sampler. Strindberg’s (2001) evaluations of bias conform to expectation, with the equal-spaced sampler (SESZ ) yielding much lower bias under the assumption of even coverage probability than the equal-angle sampler (SEAZ ). Using Equation (4.2), the model D(u) = αx + β with α equal to .02, .2, or 2 and β equal to .05, .5, or 5 gave percentage biases in Nˆ HT ranging from 4.5% to 28.6% (median = 27.1%) for SEAZ and from 1.0% to 6.1% (median = 5.8%) for SESZ . The model D(u) = αx + βy + γxy + δ with α equal to .002, .02, or .2, β equal to .003 or .03, γ equal to .001 or .01, and δ equal to .005 gave percentage biases in Nˆ HT ranging from 1.1% to 26.0% (median = 10.5%) for SEAZ and from .2% to 5.6% (median = 2.2%) for SESZ . The model D(u) = α(x)y 2 + β(x)y + γ(x) with Dmin (x) = αd x + βd , Dmax (x) = γd Dmin (x), ∀x ∈ R and αd equal to .0001, .001, .01, .1, 1, or 10, βd equal to .3 or 3 and γd equal to 2.5, gave percentage biases in Nˆ HT ranging from .05% to 28.6% (median = 17.4%) for SEAZ and from .01% to 5.8% (median = 3.6%) for SESZ .

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These simulations give a clear message. If conventional analyses that assume even coverage probability are used, then equal-spaced zigzag samplers give substantially lower bias than equal-angled samplers. Bias from this source can be eliminated entirely by using the adjusted-angle zigzag sampler, at the cost of increased complexity of design.

5. SURVEYS OF MINKE WHALES IN THE ANTARCTIC Annual surveys of minke whales are conducted in the Antarctic under the auspices of the International Whaling Commission. The Antarctic is divided into six different management areas, and in most years, a single management area is surveyed. These surveys provide a good illustration of the value of automated design algorithms. Whale density is highest at the ice-edge, so sampling effort is concentrated within the ice-edge stratum. The ice-edge is very irregular, and can move by up to 100 kilometers in a single day, so it is not possible to lay down the design in advance. Instead, an algorithm must be defined, to allow the transects to be determined as the survey progresses. The zigzag samplers were defined for convex survey regions. For nonconvex regions, depending on the characteristics of the region, the amount of available effort and the orientation of the design axis, it may be possible to generate the zigzag samplers. If, however, some parts of the region are inaccessible to the sampler, the properties explored here do not hold. Minke whale surveys within the ice-edge strata can provide an extreme departure from convexity. To ensure that the entire survey region is accessible and that coverage probabilities are nonzero throughout the region, one of three approaches can be employed. These are illustrated in Figure 7 and are as follows: 1. When departure is mild, the problem may be overcome by designing the survey within a convex hull around the survey region. Sections of sampler that fall outside the survey region are then not sampled. It is advantageous to use a convex hull rather than a minimum bounding rectangle of the survey region for survey design purposes, as the relative sampler discontinuity is less and a more even coverage probability πx (x) can be achieved by using the former. However, Figure 7(a) shows that this solution is far from satisfactory for this particular ice-edge stratum. 2. Consider dividing the stratum into subregions, each of which is convex. This strategy can be successful, but again, for the ice-edge stratum of Figure 7, far too many subregions would be required. Instead, define convex hulls around almost convex subregions of the original survey region, giving a reasonable design as shown in Figure 7(b). In this case, three subregions were defined, and a convex hull was established around each one. The solution is not ideal as inefficiency arises from sampler discontinuity and the need for relocation between subregions, but an advantage is that the orientation of the design axis can be separately defined for each subregion. 3. The last approach attempts to generate the zigzag samplers within the nonconvex survey region itself. In the example shown in Figure 7(c), this approach minimizes the sam-

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Figure 7. The survey stratum WN and a more complex ice-edge stratum WS in the Antarctic. (a) A single realization of a zigzag sampler is shown in the convex hull of WS. The sampler is discontinuous (because sections outside the survey region are not covered) and thus inefficient. (b) WS has been divided into three almost convex sub-regions, and a zigzag sampler placed within the convex hull of each, reducing the sampler discontinuity. (c) A zigzag sampler is generated in the nonconvex WS giving the least sampler discontinuity in this instance. Within WN three types of zigzag sampler are illustrated: equal-angle (dotted line), equal-spaced (dashed line) and adjusted-angle (solid line).

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Table 1. Estimates of Pod Abundance for the WN Stratum of the 1987–1988 International Decade of Cetacean Research survey of Antarctic Minke Whales Assumed zigzag sampler

Estimate

95% confidence interval

Constant coverage probability Equal-angle

4260 3740

(510, 17040) (430, 14790)

pler discontinuity. It may, however, not be a viable solution for certain types of nonconvexity, especially when attempting to generate the adjusted-angle zigzag sampler. Illustration of the Horvitz-Thompson estimators of Section 2 is not straightforward, because automated algorithms for zigzag samplers have not been available until now. However, we use the “west north” (WN) stratum of the 1987–1988 surveys for illustration, as the realized design is a close approximation to an equal-angle zigzag design. Within this stratum of area 148,820 nautical miles squared, there were seven legs of effort, totaling 1,500 nautical miles. Half of each leg was covered in “passing mode” and half in “closing mode.” In “closing mode,” the survey vessel ceases searching when a pod (school) of whales is detected, and closes with it, to confirm pod size. Because this procedure can create some bias in the estimated encounter rate, “passing mode” is also used, in which pod sizes are not confirmed, but encounter rate is likely to be less biased. Full details of the analysis of the 1987–1988 survey were given by Borchers and Haw (unpublished). Within the WN stratum, 44 minke whale pods were detected within 1.5 nautical miles of the trackline, of which 33 were detected while in passing mode. We do not attempt a full reanalysis of the data here. We use the estimated effective strip half-widths from Borchers and Haw (unpublished): .555 nautical miles with coefficient of variation 40.3% for passing mode, and .418 nautical miles with coefficient of variation 25.2% for closing mode. These were found using a truncation distance of 1.5 nautical miles, so that the survey strips are three nautical miles wide. For each pod detected, the probability that it would have fallen within a survey strip for a single random design of the equal-angle zigzag sampler was estimated from 10,000 realizations of the design. Conventional analysis assumes that coverage probability is the same everywhere, in which case each pod has the same probability of 1,500 × 3/148, 820 = .0302. We show the resulting estimates of numbers of minke whale pods in Table 1. Assuming constant coverage probability, the estimated pod abundance Nˆ CCP was obtained using Equation (2.3) for each of closing mode and passing mode, and summing the two estimates (with µˆ i equal to .555 for pods detected while in passing mode, and .418 for closing mode). The estimated pod abundance Nˆ EAZ was obtained similarly, using the coverage probabilities corresponding to each sighting achieved by an equal-angle zigzag design. Bootstrapped confidence intervals were obtained using the percentile method. Parametric bootstrap replicates of the effective strip half-width were generated, assuming a log-normal distribution, and nonparametric resampling of effort legs was conducted to quantify variability in encounter rate. The equal-angle sampler’s coverage probabilities varied between .0207 and .0557,

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with an abundance estimate that is 520 pods lower than for the conventional analysis. This suggests substantial bias occurs when the properties of the zigzag samplers are ignored, although the high variability in these estimates makes it difficult to draw strong conclusions.

6. DISCUSSION For some survey regions, when the available survey effort is limited, it will be impossible to avoid a boundary and sampler orientation combination that forces the zigzag sampler out of R. This problem could be resolved by letting the sampler run along the boundary to minimize the discontinuity, while maintaining the same total line length, although this leads to over-sampling along the edge. The effect on coverage probability could be assessed by simulation. Depending on the shape of the survey region, it may be possible to avoid the problem by suitable choice of orientation of the design axis. Failing this, a pragmatic option is to approximate the shape of the survey region in a way that avoids the problem. For populations known to exhibit a density gradient, the zigzag sampler orientation should usually be chosen so that each leg as far as possible cuts across all density levels. This minimizes variation in encounter rate among lines (Buckland et al. 2001, pp. 238–239), so that better precision in the estimated abundance is achieved. For zigzag designs, this is accomplished by placing the design axis approximately parallel to the density contours. In our example, minke whales have highest density along the ice edge, so that the design axis is chosen to be roughly parallel to the ice edge. If nothing is known about density gradients within the population, and assuming that the sampling unit is taken to be a single leg of the zigzag sampler, then a sampler orientation that maximizes the number of legs is advantageous. This increases the sample size for variance estimation and yields a more precise estimate of variance. Zigzag samplers have an associated width, so overlap at the boundaries of the survey region occurs between successive legs of the sampler. This overlap is not a problem provided detected animals that fall within the area of overlap are recorded for one of the legs and that the effort in the area of overlap contributes to total effort only once. Another problem is that part of the sampler may fall outside the survey region at the boundary. This can lead to a coverage probability that is smaller than expected. Usually, the sampler width is very small compared to the sampler length, so that overlap and other edge effects are negligible. Many ship and aerial surveys span large geographic regions, so that the loss of survey effort in transiting from one line to another of a parallel-line design can be substantial. In these circumstances, zigzag surveys should be considered. In the case of shipboard surveys, if lines of a parallel-line design are say 20 kilometers apart, it may take an hour to traverse from one line to the next. Given total line length, line separation and number of lines, a straightforward calculation reveals the percentage of potential search effort lost during such traverses. A zigzag design eliminates almost all of this loss. Aircraft travel faster, and a break between lines can be helpful, giving observers the chance of a short rest, which improves their concentration while on-effort. The operation costs per hour are also typically much less than for a ship. Hence a greater loss in potential survey time might be deemed

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acceptable. However, if loss of search effort is of the order of 10% or more relative to a zigzag design, we suspect that most survey planners would opt for a zigzag design. We recommend that equal-angle zigzag samplers not be used. In practice, equal-spaced zigzag samplers are likely to prove satisfactory in most surveys, typically with low bias, even if constant coverage probability is assumed. Increasingly, surveys are conducted with sophisticated navigation systems, and in these circumstances, the adjusted-angle zigzag sampler may be practical, and offer a worthwhile improvement over equal-spaced zigzag samplers when the shape of the survey region is irregular.

ACKNOWLEDGMENTS We thank the two referees for their thorough and constructive reviews. The research for this article was completed within the School of Mathematics and Statistics at the University of St Andrews, Scotland.

[Received September 2002. Revised February 2004.]

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