Variance estimation for systematic designs in spatial surveys

Summary In spatial surveys for estimating the density of objects in a survey region, systematic designs will generally yield lower variance than random designs. However, estimating the systematic variance is well known to be a difficult problem. Existing methods tend to overestimate the variance, so although the variance is genuinely reduced, it is over‐reported, and the gain from the more efficient design is lost. The current approaches to estimating a systematic variance for spatial surveys are to approximate the systematic design by a random design, or approximate it by a stratified design. Previous work has shown that approximation by a random design can perform very poorly, while approximation by a stratified design is an improvement but can still be severely biased in some situations. We develop a new estimator based on modeling the encounter process over space. The new “striplet” estimator has negligible bias and excellent precision in a wide range of simulation scenarios, including strip‐sampling, distance‐sampling, and quadrat‐sampling surveys, and including populations that are highly trended or have strong aggregation of objects. We apply the new estimator to survey data for the spotted hyena (Crocuta crocuta) in the Serengeti National Park, Tanzania, and find that the reported coefficient of variation for estimated density is 20% using approximation by a random design, 17% using approximation by a stratified design, and 11% using the new striplet estimator. This large reduction in reported variance is verified by simulation.     

用距离取样技术估算森林生境中的梅花鹿密度

作者于 1999年 9、 10月在日本国日光国立公园的一个梅花鹿 (Cervusnippon)高密度区域实施了 16次夜间探照样带距离法取样调查。采用车载探照记数和激光测距仪测距 ,记录了梅花鹿的结群大小和距样带中心样线的垂直距离。应用距离取样技术 (Distancesamplingtechnique ,DST)计算的结果表明 ,梅花鹿的密度为12 5 8头 /km2 (95%CI 10 0 7-157 3 ) ,在两侧 80m宽度的调查样带内梅花鹿的总发现率为 52 5% ;对线样带法的传统动物密度计算方法 (King s ,Hayne s ,Leopold s ,Gates sⅠ ,Gates sⅡ ,Gates sⅢ和Frye s方法 )的结果验证表明 ,这些传统密度计算方法过高估算梅花鹿的密度 ,其结果为距离取样技术 (DST)计算结果的 1 76(Frye′s法 ) -6 10 (Hayne′s法 )倍。因此 ,作者建议在使用线样带法测定动物密度时应尽量避免使用这些传统计算方法 ;不同地区间及同一地区不同季节间应用线样带法调查得到的动物个体记数值之间的比较存在偏差 ,应该使用距离取样技术计算动物的密度     

Line transect sampling from a curving path

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. Formulas 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.     

Accommodating unmodeled heterogeneity in double-observer distance sampling surveys

Mark-recapture models applied to double-observer distance sampling data neglect the information on relative detectability of objects contained in the distribution of observed distances. A difference between the observed distribution and that predicted by the mark-recapture model is symptomatic of a failure of the assumption of zero correlation between detection probabilities implicit in the mark-recapture model. We develop a mark-recapture-based model that uses the observed distribution to relax this assumption to zero correlation at only one distance. We demonstrate its usefulness in coping with unmodeled heterogeneity using data from an aerial survey of crabeater seals in the Antarctic.     

Predicting and correcting bias caused by measurement error in line transect sampling using multiplicative error models

Line transect sampling is one of the most widely used methods for animal abundance assessment. Standard estimation methods assume certain detection on the transect, no animal movement, and no measurement errors. Failure of the assumptions can cause substantial bias. In this work, the effect of error measurement on line transect estimators is investigated. Based on considerations of the process generating the errors, a multiplicative error model is presented and a simple way of correcting estimates based on knowledge of the error distribution is proposed. Using beta models for the error distribution, the effect of errors and of the proposed correction is assessed by simulation. Adequate confidence intervals for the corrected estimates are obtained using a bootstrap variance estimate for the correction and the delta method. As noted by Chen (1998, Biometrics 54, 899-908), even unbiased estimators of the distances might lead to biased density estimators, depending on the actual error distribution. In contrast with the findings of Chen, who used an additive model, unbiased estimation of distances, given a multiplicative model, lead to overestimation of density. Some error distributions result in observed distance distributions that make efficient estimation impossible, by removing the shoulder present in the original detection function. This indicates the need to improve field methods to reduce measurement error. An application of the new methods to a real data set is presented.     

Adaptive line transect sampling

Adaptive line transect sampling offers the potential of improved population density estimation efficiency over conventional line transect sampling when populations are spatially clustered. In adaptive sampling, survey effort is increased when areas of high animal density are located, thereby increasing the number of observations. Its disadvantage is that the survey effort required is not known in advance. We develop an adaptive line transect methodology that, by varying the degree of adaptation, allows total effort to be fixed at the design stage. Relative to conventional line transect surveys, it also provides better survey coverage in the event of disruption in survey effort, e.g., due to poor weather. In analysis, sightings from the adaptive sections are downweighted in proportion to the increase in effort. We evaluate the methodology by simulation and report on surveys of harbor porpoise in the Gulf of Maine, in which the approach was compared with conventional line transect sampling.     

Estimating seasonal density of blue sheep (<Emphasis Type="Italic">Pseudois nayaur</Emphasis>) in the Helan Mountain region using distance sampling methods

The monitoring of animal populations is necessary to conserve and manage the rare or harvest species and to understand the population change over several years. We used distance sampling methods to estimate seasonal density of blue sheep in a 2,740 km² area of Helan Mountain region by walking along 32 transect lines from winter 2003 to autumn 2005. In all, 367–780 blue sheep were observed in 91–143 groups in the surveys during the seasons. Observed mean group size ranged from 3.42 to 8.35 individuals; encounter rate, the number of groups detected per kilometre, varied from 0.19 to 0.30 during the seasons. A hazard rate key function with cosine series expansion and a half-normal key function with either cosine or simple polynomial series expansion were the best fitting models based on the lowest value of Akaike’s information criterion (AIC). Density estimates varied between 3.627 sheep per square kilometre in spring 2004 and 4.635 per square kilometre in summer 2005. There were no detectable differences in estimated density among seasons (P = 0.887). The estimated density of blue sheep was negatively correlated with the total number of deaths (P < 0.05), number of sub-adult males’ deaths (P < 0.05), number of sub-adult females’ deaths (P < 0.05), and number of male lambs’ deaths (P < 0.05). We concluded that distance sampling surveys should be used to monitor long-term population trends to provide the best quantitative estimates of blue sheep populations in the Helan Mountains region.     

Estimating the population size of the Sanje mangabey (Cercocebus sanjei) using acoustic distance sampling

The Sanje mangabey (Cercocebus sanjei) is endemic to the Udzungwa Mountains, Tanzania, and is classified as Endangered due to its putatively declining population size, habitat degradation and fragmentation. Previous population size estimates have ranged from 1,350 to 3,500 individuals, with the last direct survey being conducted 15 years before the present study. Previous estimates are now thought to have underestimated the population due to a limited knowledge of group and habitat size, nonsystematic approaches and the use of visual methods that are not suitable for surveying the Sanje mangabey with its semi-terrestrial and elusive behaviors. We used an acoustic survey method with observers recording the distinctive “whoop-gobble” vocalization produced by mangabeys and point transect distance sampling to model a detection function and estimate abundance. Twenty-eight surveys were conducted throughout the two forests where Sanje mangabeys are found: Mwanihana forest in the Udzungwa Mountains National Park (n = 13), and the Uzungwa Scarp Nature Reserve (n = 15). Group density was found to be significantly lower in the relatively unprotected Uzungwa Scarp forest (0.15 groups/km²; 95% CI: 0.08–0.27) compared to the well-protected Mwanihana forest (0.29 groups/km²; 95% CI: 0.19–0.43; p = .03). We estimate that there are 1,712 (95% CI: 1,141–2,567) individuals in Mwanihana and 1,455 (95% CI: 783–2,702) in the Uzungwa Scarp, resulting in a total population size of 3,167 (95% CI: 2,181–4,596) individuals. The difference in group density between sites is likely a result of the differing protection status and levels of enforcement between the forests, suggesting that protection of the Uzungwa Scarp should be increased to encourage recovery of the population, and reduce the threat of degradation and hunting. Our results contribute to the reassessment of the species' IUCN Red List status and informing management and conservation action planning.     

Line Transect Sampling in Small Regions

This article develops an approach to estimating population abundance from line transect surveys that uses a calibration survey to estimate the detection function, which is then employed as a weight function in constructing the abundance estimate. Nonparametric methods of estimating the detection function via local regression and via a kernel density estimator are considered. The proposed methods are evaluated using a set of Western Australian plant data and weed enumeration data.     

Double-Observer Line Transect Methods: Levels of Independence

Summary .  Double-observer line transect methods are becoming increasingly widespread, especially for the estimation of marine mammal abundance from aerial and shipboard surveys when detection of animals on the line is uncertain. The resulting data supplement conventional distance sampling data with two-sample mark–recapture data. Like conventional mark–recapture data, these have inherent problems for estimating abundance in the presence of heterogeneity. Unlike conventional mark–recapture methods, line transect methods use knowledge of the distribution of a covariate, which affects detection probability (namely, distance from the transect line) in inference. This knowledge can be used to diagnose unmodeled heterogeneity in the mark–recapture component of the data. By modeling the covariance in detection probabilities with distance, we show how the estimation problem can be formulated in terms of different levels of independence. At one extreme, full independence is assumed, as in the Petersen estimator (which does not use distance data); at the other extreme, independence only occurs in the limit as detection probability tends to one. Between the two extremes, there is a range of models, including those currently in common use, which have intermediate levels of independence. We show how this framework can be used to provide more reliable analysis of double-observer line transect data. We test the methods by simulation, and by analysis of a dataset for which true abundance is known. We illustrate the approach through analysis of minke whale sightings data from the North Sea and adjacent waters.     

Line Transect Sampling in Small and Large Regions

Melville and Welsh (2001, Biometrics 57, 1130-1137) consider an approach to line transect sampling using a separate calibration study to estimate the detection function g. They present a simulation study contrasting their results with poor results from a traditional estimator, labeled the "Buckland" estimator and referenced to Buckland et al. (1993, Distance Sampling: Estimating Abundance of Biological populations). The poor results from the "Buckland" estimator can be explained by the following observations: (i) the estimator is designated for untruncated distance data, but was applied by Melville and Welsh to truncated distance data; (ii) distance data were not pooled across transects, contrary to standard practice; and (iii) bias of the estimator was evaluated with respect to a fixed rather than a randomized grid of transect lines. We elaborate on the points above and show that the traditional methods perform to expectation when applied correctly. We also emphasize that the estimator labeled the "Buckland" estimator by Melville and Welsh is not an estimator recommended by Buckland et al. for practical survey applications.     

First record of Alicia mirabilis (Anthozoa: Actiniaria) from the Aegean Sea and density assessment with distance sampling in a site of high abundance

The occurrence of the actiniarian Alicia mirabilis in the Aegean Sea is documented for the first time from five sites in the Saronikos Gulf. The density of A. mirabilis was assessed in one site (Lychnari bay) with line transects (underwater distance sampling with SCUBA). Its density ranged between 0 and 4.3 individuals are^-1 (1 are=100 m²) with a mean of 1.1±0.5 individuals are^-1. The species was found in a variety of substrates (sandy bottoms, weeds, seagrass beds, rocks, litter) and its distribution was aggregated. Distance sampling was an efficient way to estimate the density of A. mirabillis and is proposed as a good choice for density estimations of actiniarians and other benthic fauna.     

A field test of the distance sampling method using Golden‐cheeked Warblers

ABSTRACT Criteria for delisting Golden‐cheeked Warblers (Dendroica chrysoparia) include protection of sufficient breeding habitat to ensure the continued existence of 1000 to 3000 singing males in each of eight recovery regions for ≥10 consecutive years. Hence, accurate abundance estimation is an integral component in the recovery of this species. I conducted a field test to determine if the distance sampling method provided unbiased abundance estimates for Golden‐cheeked Warblers and develop recommendations to improve the accuracy of estimates by minimizing the effects of violating this method's assumptions. To determine if observers could satisfy the assumptions that birds are detected at the point with certainty and at their initial locations, I compared point‐transect sampling estimates from 2‐, 3‐, 4‐, and 5‐min time intervals to actual abundance determined by intensive territory monitoring. Point‐transect abundance estimates were 15%, 29%, 43%, and 59% greater than actual abundance (N= 156) for the 2‐, 3‐, 4‐, and 5‐min intervals, respectively. Point‐transect sampling produced unbiased estimates of Golden‐cheeked Warbler abundance when counts were limited to 2 min (N= 154–207). Abundance estimates derived from point‐transect sampling were likely greater than actual abundance because observers did not satisfy the assumption that birds were detected at their initial locations due to the frequent movements and large territory sizes of male Golden‐cheeked Warblers. To minimize the effect of movement on abundance estimates, I recommend limiting counts of singing males to 2‐min per point. Counts for other species in similar habitats with similar behavior and movement patterns also should be limited to 2 min when unbiased estimates are important and conducting field tests of the point‐transect distance sampling method is not possible.     

Density and population size of mammals introduced on a land-bridge island in southeastern Brazil

The introduction of alien species is one of the main threats to the conservation of native species, especially in island ecosystems. Here, we report on the population growth of 15 species of mammals introduced in 1983 on the island of Anchieta, an 828 ha land-bridge island in southeastern Brazil. We estimated the density of mammals through 296 km of line transect census. Five species introduced became extinct (coypu, brocket deer, six-banded armadillo, nine-banded armadillo, maned three-toed sloth); six became over-abundant (marmoset, coati, agouti, seven-banded armadillo, and capybara); one has a stable population (capuchin monkey). Anchieta Island has the highest density of mammals in the entire Atlantic forest (486.77 ind/km²), especially nest predators (232.83 ind/km²) and herbivores (253.58 ind/km²). Agoutis (Dasyprocta spp.) and marmosets (Callithrix penicillata) were, by far, the species with the highest population growth. The high density of mammals in this island may have strong consequences for plant recruitment and bird diversity.     

Some comments on anderson and pospahala's correction of bias in line transect sampling

ANDERSON and POSPAHALA (1970) investigated the estimation of wildlife population size using the belt or line transect sampling method and devised a correction for bias, thus leading to an estimator with interesting characteristics. This work was given a uniform mathematical framework in BURNHAM and ANDERSON (1976). In this paper we show that the ANDERSON-POSPAHALA estimator is optimal in the sense of being the (unique) best linear unbiased estimator within the class of estimators which are linear combinations of cell frequencies, provided certain assumptions are met.     

Stipa tenacissima aerial biomass estimated at regional scale in an Algerian steppe, using geostatistical tools

Data from systematic sampling may be independent or autocorrelated. In the latter case geostatistical tools are used to identify the spatial patterns within the universe sampled. Special formulas have been derived by Russo & Bresler (1982) to estimate the variance of a value averaged over several transect samples. We applied these formulas to the green biomass of the dominant perennial steppe grass, Stipa tenacissima or alfa, in a 400 km² region in North-West Algeria; thirty years ago, this region was considered one of the best sites for alfa. A two-level sampling design was implemented with stratification of the region and systematic sampling within each stratum; globally the study included fifteen transects, representing 713 1 m² quadrats. Autocorrelation up to 500 meters was detected in five semi-variograms or correlograms, which were fitted to linear models with a sill. Biomass averaged only 165 (±55) kg ha^-1. We discuss the processes that have lead to the rapid degradation of alfa steppes in northern Algeria and the variation in spatial patterns of alfa stands. Ignoring autocorrelation in systematic sampling leads to biased estimates of variances and standard errors.