Abstract: The main goal in estimating population abundance is to maximize its accuracy and precision. This is difficult when the survey area is large and resources are limited. We implemented a feasible adaptive sampling survey applied to an aggregated population in a marine environment and compared its performance with five classical survey designs. Specifically, larval walleye pollock (Theragra chalcogramma) in the Gulf of Alaska was used as an example of a widespread aggregated population. The six sampling designs included (i) adaptive cluster, (ii) simple random, (iii) systematic, (iv) systematic cluster, (v) stratified systematic, and (vi) unequal probability. Of the five different adaptive estimators used for the adaptive cluster design, the modified Hansen-Hurwitz performed best overall. Of the six survey designs, the stratified systematic survey provided the best overall estimator, given there was accurate prior information on which to base the strata. If no prior information was available, a systematic survey was best. A systematic survey using a single random starting point with a simple random estimator performed as well as and sometimes better than a systematic cluster survey with two starting points (clusters). The adaptive cluster survey showed no advantages when compared with these two designs and furthermore presented substantial logistical challenges.
Resume : L'objectif principal poursuivi dans l'estimation de l'abondance d'une population est l'amelioration de l'exactitude et de la precision. Cela est difficile quand la surface inventoriee est grande et les ressources limitees. Nous avons mis au point un inventaire adaptatif d'echantillonnage pratique pour une population a distribution contagieuse dans un environnement marin et nous avons compare sa performance en fonction de cinq plans d'inventaire classiques. Nous utilisons, en fait, des larves de goberges (Theragra chalcogramma) de l'Alaska du golfe de l'Alaska comme exemple specifique d'une population a large repartition et a distribution contagieuse. Les six plans d'inventaire consistent en (i) un plan adaptatif avec regroupements, (ii) un plan aleatoire simple, (iii) un plan systematique, (iv) un plan systematique avec regroupements, (v) un plan systematique stratifie et (vi) un plan a probabilites inegales. Des cinq estimateurs adaptatifs utilises dans le plan adaptatif avec regroupements, l'estimateur modifie de Hansen-Hurwitz donne le meilleur resultat global. Des six plans d'inventaire, l'inventaire systemique stratifie fournit le meilleur estimateur global, etant donne qu'il existe de l'information prealable precise pour determiner les strates. Lorsqu'il n'y a pas de renseignements prealables, l'inventaire systematique fonctionne le mieux. Un inventaire systematique avec un seul point de depart aleatoire et avec un estimateur aleatoire simple fonctionne aussi bien, et souvent mieux, qu'un inventaire systematique avec regroupements avec deux points de depart (regroupements). L'inventaire adaptatif avec regroupements ne presente aucun avantage par rapport aux deux plans precedents et, de plus, il cree de serieux problemes de logistique.
[Traduit par la Redaction]
Introduction
The most common requirement in resource management, studies of population dynamics, and many other subjects is to estimate mean abundance of spatially aggregated populations. The usefulness of these estimates are dependent on their bias and precision. Yet, in most field studies, traditional survey procedures are used without much background information on how to improve the bias and precision of the mean and standard error estimators. This can lead to inaccurate fish stock assessments or hypothesis tests and general misrepresentation of the spatial distribution.
The goal of this project was to compare the effectiveness of six sampling methods on patchy distributions (i.e., adaptive cluster, random, systematic, systematic cluster, stratified systematic, and unequal probability). The criteria for comparing designs were the bias, precision, and mean squared error (MSE) of the estimates. Because MSE is a measure of both bias and precision, it was used as a final criterion for determining the best estimator. We computed these estimates of the mean density and their standard error (SE) by simulating many patchily distributed populations with different degrees of patchiness and then repeatedly sampling these populations using each sampling method.
A primary motivation for this study was to decide if using an adaptive cluster design (Thompson 1990) would yield a more precise estimate. This design has received much attention in recent years with studies of patchily distributed populations, primarily because it allows concentrated sampling around initial samples that meet a predetermined criterion (typically areas of high density) and because it is reported to yield a smaller variance than random sampling. It can be particularly useful when more individuals of interest need to be collected for other studies.
In our study, we examine the performance of adaptive cluster sampling by sampling from realistic, simulated populations. Other examples of adaptive sampling applied to simulated patchily distributed populations include Christman (2000), Brown (2003), and Su and Quinn (2003). Christman (2000) looked at adaptive cluster sampling of rare, spatially clustered populations. In contrast, our population of interest is patchy or clustered, but not rare, and covers a large geographic extent. Our simulated populations are typical of many ubiquitous but patchy marine species, such as adult Pacific hake (Merluccius productus) and shortspine thorny-head (Sebastolobus alascanus) along the west coast of North America, adult arrowtooth flounder (Atheresthes stomias), rex sole (Glyptocephalus zachirus), and Pacific sandlance (Ammodytes hexapterus) in the Gulf of Alaska, and many species of ichthyoplankton; hence, our survey design comparisons have broad applicability. To further broaden the scope of our study, we simulated populations with a wide range of patchiness.
What also sets our paper apart from other simulations is the focus on a realistic implementation of an adaptive sampling design where the sampling unit is extremely small when compared with the entire area to be surveyed, a scenario that is common in large marine surveys. Our simulated population contained almost 9000 sampling units, whereas, for example, Thompson (1991) used a population with 400 sampling units, and Christman (2000) used a population with 75 sampling units. Our implementation includes setting the criterion for initiating adaptive sampling, controlling sample size, and avoiding negative variance estimates (which occurs in Horvitz-Thompson estimates). These issues have not been adequately addressed in real applications or in simulated samples. Much of the literature a priori uses species presence to initiate adaptive sampling, makes no attempt to control sample size, and seldom yields negative variances due to artificially simple populations. Those studies that have introduced methods to control sample size include Lo et al. (1997), Christman (2003), and Su and Quinn (2003). Each of these methods introduces bias into the estimator, as does our method. We measured the bias, or systematic error, by simulating realistic populations and then repeatedly sampling from these populations and comparing the true values to the repeated estimates.
Common estimators used for adaptive cluster sampling include the modified Hansen-Hurwitz (HH) and Horvitz-Thompson (HT) estimators of the mean and variance (Thompson and Seber 1996). In addition to these two, we also looked at two alternate estimators of the variance of the HT mean, the Yates-Grundy-Sen (Sen 1953; Yates and Grundy 1953) and the Brewer-Hanif (Brewer and Hanif 1983), as well as one other estimator of both the mean and variance, the Hajek-Sarndal (Hajek 1971; Sarndal et al. 1992). Patchy distributions usually yield large variances, regardless of survey design (Andrew and Mapstone 1987), but several studies suggest that using the modified HT estimator in an adaptive cluster survey yields the most precise estimate (Hanselman et al. 2003; Salehi 2003; Su and Quinn 2003). These conclusions, however, were dependent on the degree of aggregation, "neighborhood" definition, size of "networks", within-network variance, size of "initial sample", size of "sampling unit", and the "criterion" used to begin adaptive sampling (see Appendix A for definitions of terms that are in quotes).
We apply these methods to estimating mean density (number per 10 [m.sup.2]) of larval fish in the ocean, and specifically to the patchily distributed walleye pollock (Theragra chalcogramma) (Stabeno et al. 1996). We patterned our simulated populations on the distribution of walleye pollock larvae as recorded from ichthyoplankton surveys in Shelikof Strait, Alaska, conducted by the National Oceanic and Atmospheric Administration (NOAA) Alaska Fisheries Science Center (AFSC). We built the framework used in this study by modeling the spatial distribution in terms of patches and using parameters that describe these patches based on actual historical data.
Materials and methods
Population simulations
Artificial populations were generated that have the same spatial characteristics as walleye pollock larvae in early May in Shelikof Strait, Alaska. The primary characteristic is the spatial pattern of its patches, that is, areas of high density (Stabeno et al. 1996). We examined the observed larval walleye pollock spatial distributions from nine spring ichthyoplankton surveys conducted by AFSC between 1986 and 1998 (Table 1). We chose surveys from the time interval shortly after the eggs hatched into larvae and before the larvae became widely dispersed, so that the patches were still coherent.
To examine the effect of patchiness on the performance of the various survey designs, we simulated two sets of populations: 50 populations that we considered to be very patchy and 50 populations that were much less patchy. We defined high-patchy populations to have patches that were small in geographic extent, but had large maximum densities within the patch. Conversely, we defined low-patchy populations to have patches that were large in geographic extent, but with maximum densities that were less extreme than the high-patchy population (Fig. 1). We adopted these definitions both because they are intuitive and because Brown (1996) found that these factors (along with the number of patches) were the most important in determining the benefits of adaptive sampling over random sampling. By simulating populations with a wide range of patchiness, the conclusions drawn from these populations may be extrapolated to many other natural populations that have various degrees of patchiness.
The area of the simulated populations was defined by plotting the stations from cruises in Table 1 and delineating the area that was usually surveyed and contained larvae (Fig. 2). A grid of 8857 "cells", where each cell is 1 nautical mile x 1 nautical mile (1 n.mi. = 1.852 km) sea-surface area, was created inside this survey area. A density was simulated for each cell based on patch parameters estimated from historical data. This density was defined as the number of fish in the water column below a 10 [m.sup.2] surface area within the cell. We used the index i to refer to the ith cell; [x.sub.i] and [y.sub.i] to refer to the east-west and north-south location of the centroid within cell i, respectively, in units of meters; and Z([x.sub.i], [y.sub.i]) to refer to the simulated density, in units of number per 10 [m.sup.2], within cell i. See Appendix B for details of the simulation of populations.
Survey simulations
The simulated surveys were based on quadrat sampling, which is common practice in ecological surveys. The geographic range of the simulated population was divided into a grid of 1 n.mi. x 1 n.mi. contiguous cells; this was our spatial sampling frame and each cell was a quadrat, the sampling unit. A sample of cells was selected and the population density within a cell was observed and multiplied by a sampling error term, [[epsilon].sub.i]. The density of fish "caught" in the ith sampled cell (number per 10 [m.sup.2]) was designated as
z([x.sub.i], [y.sub.i]) = [z.sub.i] = Z([x.sub.i], [y.sub.i]) x [[epsilon].sub.i]
where Z([x.sub.i], [y.sub.i]) is the simulated density of the population at that cell (see eq. B1 in Appendix B), and [[epsilon].sub.i] is a random lognormal variate, representing small-scale variability about the densities within a cell. The …
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