LINE TRANSECT SAMPLING FROM A CURVING PATH

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1 LINE TRANSECT SAMPLING FROM A CURVING PATH LEX HIBY Conservation Research Ltd., 110 Hinton Way, Great Shelford, Cambridge, England hiby@ntlworld.com and M. B. KRISHNA Wildlife Conservation Society (India Program) Current address: The Heritage Ranga Rao Road, Shankarapura, Bangalore , India. SUMMARY. Cutting straight line transects through dense forest is time consuming and expensive when large areas need to be surveyed for rare or highly clustered species. We argue that existing paths or game trails may be suitable as transects for line transect sampling even though they will not, in general, run straight. Formulae and software currently used to estimate local density using perpendicular distance data can be used with closest approach distances measured from curving transects. Suitable paths or trails are those for which the minimum radius of curvature is rarely less than the width of the shoulder in the detection probability function. The use of existing paths carries the risk of bias resulting from unrepresentative sampling of available habitats, and this must be weighed against the increase in coverage available. KEY WORDS: Curved transect; Distance sampling; Line transect; Terrestrial survey; Trails. Biometrics 57, September Introduction Most of the literature on line transect sampling assumes, either explicitly or implicitly, that targets will be detected from straight line transects. For example, the assumption is implicit in the recommendation that the perpendicular distance to each detected target from the transect line should be estimated by recording radial distance and angle (as in, for example, Buckland et al., 1993). Skellam (1958), in generalising an earlier analysis by Yapp (1955), allowed both the observer and targets to move along arbitrary paths. However, curving transects do not appear to have received further consideration, probably because in aerial and shipboard surveys, or terrestial surveys in open areas, there is no difficulty in following straight lines. One exception is the contour survey technique introduced by Quang and Becker (1998) where a survey aircraft follows a height contour because of the difficulty of flying straight transects over very uneven terrain. Another situation in which it is difficult to use straight transects is terrestrial survey in dense forest. Cutting straight paths through dense forest can be time consuming and expensive (Walsh 1

2 and White, 1999, Karanth and Sunquist, 1992). It thus seems worth considering whether existing paths or game trails could be used as transect lines, allowing more extensive surveys to be completed for the same available effort. We suggest that, in certain situations, they can and that, furthermore, the data can be analysed using standard methods, e.g., by using the Distance software familiar to many researchers (Thomas et al., 1998). The only modification required is to record, as the detection distance y, the minimum distance from the path or trail to the target instead of recording radial distance and angle. This article describes a survey similar to the stake data experiment of Laake (1978), that was conducted in the Bannerghatta forest reserve near Bangalore, India, to check on the feasibility of using minimum distances. An obvious difference between contours and paths is that every point in a survey region lies on a height contour, allowing random selection of the contours used to ensure that the expected mean target density along the transect equals that in the survey area. Paths and trails (which, following Walsh and White, we will refer to generically as a path of least resistance, or PLR) may be numerous, allowing some degree of random selection. However, the expected mean density along the entire PLR network may differ from that in the survey region either because the network is used or avoided by the target species or because it simply fails to sample the available habitat representatively. This is a potentially serious problem, to which we will return in the Discussion; in the meantime, we restrict our attention to estimation of mean density on the chosen PLR section rather than over the survey region as a whole. If standard methods of line transect data analysis are to be used, the assumptions required for asymptotically unbiased estimation of mean density along a curving transect are identical to those required for a straight one. They are (slightly modified from section 1.5 of Buckland et al., (1993)): 1. Objects directly on the line are always detected. 2. Objects are detected at their initial location, prior to any movement in response to the observer. 3. Distances are measured accurately. For the estimated mean density along the transect to provide a valid estimate of animal abundance in the study area we also require the following: 4. Either animals are randomly distributed with constant density through the study area or the transects are located independently of animal density. Quang and Becker (1998) avoid the need for the first assumption by using independent observers; however they then need to assume that a target within the survey region has the same chance of being located at any right angle distance from 0 to w (the data being truncated at w ). The use of independent observers is not a practical option for terrestrial surveys in forests. Furthermore, the assumption of a uniform distribution for right angle distance will not hold for curving transects in general because it requires that w is less than the minimum radius of curvature. Because, in general, a 2

3 target may lie at right angles to more than one point in a curving transect, the right angle distance must be specified as the distance to the nearest of those points if it is to be defined for all targets. But the locus of points at a given minimum distance from the transect is reduced in length once that distance exceeds the minimum radius of curvature. Fortunately, in order to estimate mean target density on the transect, it is not necessary to assume a uniform distribution for right-angle distance, provided the assumption of certain detection on the transect holds. In the next section we consider how suitable data might be collected and provide simulated and practical examples. 2. Methods For targets detected from a curving transect we need to measure the minimum distance from the transect to the location occupied by the target when it was detected. Clearly, for targets that are sessile or do not move before detection, the probability of detection will tend to one as that distance tends to zero. The simplest option is for the observer to estimate the point of closest approach of the PLR to the location and measure the distance from there using a tape measure or optical range finder. Thus, for mobile targets, a reference point (such as a tree of a particular species) close to the location of the target when it was detected would have to be noted and kept in view till the closest approach point was reached. This implies some reduction in effort but, given that detection rates are usually low, the reduction is likely to be small. If it is not practical to walk to the detection location or use a rangefinder, the minimum distance could also be calculated by taking compass bearings from the point of closest approach and from another point on the transect a measured distance away. Let f (d) represent the frequency distribution for this closest approach distance to detected targets d. Then the mean target density on the transect, D (0), is related to the expected number of detections, E (n), and the length of the transect (around the curves), L, by the same general formula as derived by Burnham, Anderson and Laake (1980) for distance data collected from straight line transects, i.e., D ( 0) = E ( n) f (0) (2L). To see that this formula still applies, consider the region within distance w of the PLR (Figure 1). Detection is impossible for targets at locations more than w from the path, or the data are truncated at w. Similarly any detections before or after boundaries perpendicular to the start and end of the PLR are censored. The area of the region is less than 2 Lw because of the folding that occurs to the inside of each curve at distances greater than the radius of curvature. As mentioned in the Introduction, this shortens the locus of points lying at any such distance from the PLR. Let h ( be the mean length of the loci at minimum distance y to either side of the PLR, D ( the mean target 3

4 density at this distance and g( the probability of detection for a target at this distance. Then the expected number of detections equals D( 2h( g ( dy. w 0 But w f ( = D( 2h( g( D( 2h( g( dy = D( 2h( g( E( n). 0 Therefore, f ( 0) = D(0)2L E( n) and D( 0) = E ( n) f (0) 2L. Thus, just as for the straight line transect case, estimation of D (0) depends on estimation of f (0) from the recorded detection distances. There has been extensive research on the problem of estimating f (0) where f ( relates to perpendicular distances, which is summarised in Buckland et al. (1993) and the Distance software. The results are also applicable when f ( relates to closest approach distances, provided f ( retains a shoulder near the origin (i.e., f '(0) = 0 ) and otherwise reduces with y smoothly enough to permit close approximation using one of the robust models available in the Distance software. A number of simulations were carried out using sine-wave transects of different wavelengths and a hazard rate detection process (with independent detection probability over successive time increments dependent on radial distance and angle from the observer). Because targets to the inside of the curve are passed more slowly by the observer they are more likely to be detected than targets at the same distance to the outside of the curve (Figure 2). This asymmetry averaged out to have a negligible effect on the shape of g (. However, with very short wavelengths the shoulder in f ( was reduced because the reduction in h (, referred to earlier, then occurred within the distance ( y *, sa over which almost all targets were detected. With a sawtooth form of transect this reduction occurs over all y because the minimum radius of curvature is then zero, so that f ( shows no shoulder at all. A loss or reduction of the shoulder in the observed frequency distribution of detection distances will jeopardise reliable estimation of f (0). Therefore, in a real survey the PLR used should have few curves with radius of curvature much less than * y. Equivalently, the area swept out by a line extending out to * y on the left and right of the transect should not be much less than 4 * 2Ly. To investigate the likely reduction in shoulder width for a survey conducted from a real path, we carried out an experiment similar to the stake data survey of Laake (1978). We chose a section of a path running through the Bannerghatta reserve near Bangalore in India (Figure 3) and placed 117 small squares of newspaper at random locations on either side of the path. The bearing along the section varied from 3 to 105. So we selected a principal axis at 50 and placed targets at distances from 0 to 50 feet from the path at right angles to this axis. The length of the axis was 504 feet so

5 targets were placed at a mean density of 117/50,400 or targets per square foot. As they were placed at pre-determined locations, irrespective of the vegetation occurring there, some were in the open and others partially obscured. Observers were then asked to walk along the path and record the minimum distance (not the distance at right angles to the principal axis) to each detected target. 3. Results Figure 4 shows the frequency distribution of detection distances as measured from the point of closest approach of the path. It also shows the fitted f ( as given by program Distance using ungrouped detection distances and the Akaike Information Criterion for selection of the detection function. The chosen function was the half-normal, which gave an estimate of for f (0) and for D (0) (95% confidence limits, ). The labelling of results by Distance is slightly misleading here because, due to the effect of h (, the fitted f ( function may no longer be proportional to the detection function. The fact that the shape of the detection function is no longer available is not important because, given the f (0) estimate, it is not required for estimation of target density. The frequency distribution in Figure 4 shows a distinct shoulder. To investigate the effect of path curvature on that distribution, the path was accurately mapped using a tape measure and compass (Figure 5a) and the h ( function calculated. Figure 5b shows that, for this path, there is no significant reduction in h ( out to beyond the shoulder width. To have generated a noticeable erosion of the shoulder in f ( the path would have needed to be more jagged. The simulated paths shown in Figure 5a to the right of the real path, were generated by adding perturbations of up to 10 and 15 feet to waypoints measured along the the real path. The corresponding h ( functions now show a decline well within the detection function shoulder data collected from such paths would not have allowed reliable estimation of target density. 4. Discussion The use of paths or trails as transects for distance sampling is not generally accepted and represents poor survey practice according to Buckland et al. (1993, p. 18). However it is the lack of random design, not the lack of straightness, that is the real problem. Curvature of the track poses no serious theoretical or practical problems provided a substantial proportion of detections occur within the radius of curvature. The fact that detection distances are generally short in scrub and forested habitats and that there is a natural tendency for paths and trails to avoid sharp turns suggests that, in terms of curvature, most would be suitable as transects. We had no difficulty in determining the closest 5

6 approach of the path to each target detected during the experimental survey, though for mobile targets a second observer might be required to mark the detection location while the first moved to the nearest point on the PLR. The lack of random transect placement does pose two potentially serious problems. If the target species either uses or avoids the PLR network it will be very difficult to extrapolate the estimated density on the transect to the whole survey region. Short transects might be cut at right angles to the PLR to try to measure the relative change in D ( with y, but a calibration based on those results would require a map of the entire PLR network and its minimum distance from every point in the survey region. Furthermore, any direct effect of the PLR on target distribution is likely to vary with habitat type. So in practice it will be necessary to assume that no such effect exists (the same assumption is, of course, required in relation to cut transects). Whether that is a reasonable assumption will depend on the species and the type of PLR used. The chosen PLR can be any route allowing access to the survey region, so it may be possible to avoid using existing paths of a width likely to be used by the target species. The second problem is that the PLR network may fail to sample the available habitats representatively. Existing paths may avoid more difficult terrain or access along paths may have led to changes in habitat nearby. The mean target density measured on the transects may therefore be biased as an estimate of mean density for the survey region as a whole. A partial solution, given a breakdown of the region in to different habitat areas, is to analyse the data stratified by habitat type. A stratified analysis is likely to be preferred in any case, even given random transect placement. But correlation between habitat and the PLR network may also exist at spatial scales that are too small to permit a stratified analysis, e.g., existing paths may tend to follow certain levels of vegetation density or elevation. It may be possible to select sections that counteract such effects and avoid those with an obvious risk of correlation with target distribution, for example, like those that follow watercourses. However, without a genuine random element of transect placement it will be impossible to avoid all risk of bias, and this must be weighed against the increased coverage available. It should also be weighed against the other sources of bias inherent in forest surveys. One potentially serious bias is actually reduced by the use of a curving transect. On a straight transect there is a risk that some animals will be seen as they cross the transect well ahead of the observer and will therefore be assigned a very small or even zero perpendicular distance. This amplifies the effect of random target movement, which is normally considered to be negligible, and may cause serious upward bias. 6

7 ACKNOWLEDGEMENTS We thank Krishna Narain for providing facilities and help for conducting the field experiment; Basavaraj for field assistance; and Ullas Karanth, Russell Leaper, V. Srinivas, L. Shyamal, and N. Samba Kumar for their encouragement and comments on the draft. REFERENCES Buckland, S. T., Anderson D. R., Burnham K. P., and Laake J. L. (1993). Distance Sampling. London: Chapman and Hall Burnham, K. P., Anderson, D. R., and Laake, J. L. (1980) Estimation of density from line transect sampling of biological populations. Wildlife Monographs, No.72, supplement to: The Journal of Wildlife Management 44. Karanth, K. U. and Sunquist, M. E. (1992). Population-structure, density and biomass of large herbivores in the tropical forests of Nagarhole, India. Journal of Tropical Ecology 8, Laake, J. L. (1978) Line transect estimators robust to animal movement. MS Thesis, Utah State University, Logan, UT, USA. 55pp. Quang, P. X., and Becker, E. F. (1998) Aerial survey sampling of contour transects using doublecount and covariate data. In: Marine Mammal Survey and Assessment Methods, L. McDonald (ed), Rotterdam: Balkema Skellam, J. G. (1958). The mathematical foundations underlying the use of line transects in animal ecology. Biometrics 14, Thomas, L., Laake, J. L., Derry, J. F., Buckland, S. T., Borchers, D. L., Anderson, D. R., Burnham, K. P., Strindberg, S., Hedley, S. L., Burt, M. L., Marques, F. F. C., Pollard, J. H., and Fewster, R. M Distance 3.5. Research Unit for Wildlife Population Assessment, University of St. Andrews, UK. Available: Walsh, P., and White, L. (1999). What it will take to monitor forest elephant populations. Conservation Biology 13, Yapp, W. (1955). The theory of line transects. Bird Study 3,

8 Figure 1. A target at minimum distance y from a curving transect. The locus of points at minimum distance w from the transect lies beyond the minimum radius of curvature, and its average length h (w) is therefore less than the transect length, L. Figure 2. The observer moves from left to right along a curving transect and targets are detected using a hazard rate detection process. Bands of shading illustrate detection probability from white (near one) to black (near zero). Targets to the inside of a curve have a higher detection probability than targets at the same distance to the outside. 8

9 Figure 3. Section of a path in Bannerghatta reserve near Bangalore, India, used for a field trial of survey for sessile targets from a curving transect. Small squares of newspaper were placed at distances of up to 50 feet at right angles to the average direction of the path 9

10 Figure 4. The frequency distribution of the minimum distance from the experimental transect to detected targets and the fitted half-normal f ( function from program Distance. (a) (b) Figure 5. a. Plot O illustrates the shape of the experimental transect used and plots P10 and P15 are transect shapes obtained by randomly shifting the waypoints measured on the real transect by up to 10 and 15 feet. b. The corresponding h( functions show that, for the jagged transects, the locus of points at minimum distance y from the transects declines in length as y increases. HIBY, LEX AND KRISHNA, M.B. (SEPTEMBER 2001) BIOMETRICS 57, LINE TRANSECT SAMPLING FROM A CURVING PATH. Received February Revised January Accepted January