Study Design Considerations for Aerial Surveys in the Spring and Fall to Estimate the Abundance of Pacific Walrus

Transcription

1 Study Design Considerations for Aerial Surveys in the Spring and Fall to Estimate the Abundance of Pacific Walrus Prepared by Bryan F.J. Manly Western EcoSystems Technology Inc Central Avenue, Cheyenne, Wyoming For Joel Garlich-Miller US Fish and Wildlife Service 1011 East Tudor Road, Anchorage, Alaska Pacific W alrus Sampling Page 1 of February 2002

2 Table of Contents Summary Introduction Aerial Line Transect Data Collection and Standardization United States Spring 1968 Survey... 6 United States Fall 1975 Survey... 6 Soviet Fall 1975 Survey... 6 United States National Marine Fisheries Service Spring 1976 Survey United States National Marine Fisheries Service Spring 1977 Survey United States Fall 1980 Survey... 7 Soviet Fall 1980 Survey... 7 United States Fall 1985 Survey... 7 Soviet Fall 1985 Survey... 7 Soviet Spring 1987 Survey... 7 United States Fall 1990 Survey... 8 Soviet Fall 1990 Survey Survey Design and Analysis for a Spring Survey The Generation of Simulated Data for Spring Surveys Sampling Design and Analysis Bootstrap Bias Correction Simulation Experiment 1: Spring Survey Design and Analysis for a Fall Survey The Generation of Simulated Transect Data for Fall Surveys Simulation Experiment 2: Fall Transects Over Water and Ice The Generation of Simulated Data From Land Haul-Out Sites Simulation Experiment 3: Fall Hall-Out Sites Combined Estimates From Line Transect and Haul-Out Site Surveys Discussion References Appendix A Counts from Haul-Out Sites in the Period Pacific W alrus Sampling Page 2 of February 2002

3 Summary! A database of the results from surveys of the Pacific walrus (Odobenus rosmarus divergens) has been constructed based on reports and papers and data files for the period 1968 to This database contains as a minimum for each survey the sampling dates, the locations of walrus groups, and the size of those groups, plus the locations of the start and end points of transect lines (in some cases derived from the walrus sighting information). Results are available for 11 surveys were survey results were available in data files, and for simulated samples from one survey where the results were presented in a figure and table.! Based on the results from Spring surveys, a model population was set up from which samples can be drawn to simulate a future Spring survey. A simulation study was carried out using this model population to assess the accuracy obtained with stratified random sampling, two-stage adaptive stratified sampling, and two-stage adaptive stratified sampling with bootstrap bias correction. This leads to the conclusion that adaptive sampling with bias correction gives the most satisfactory results, but that to achieve a CV of 30% or less may require as much as 80,000 km of transect sampling from the model population.! Based on the results from Fall surveys, a model population was set up for future line transect sampling over water and ice at this time. A simulation study then indicated that 20,000 km of transect sampling should give a CV of 20% or less for this model population.! The past counts for walruses at land haul-out sites were also used to simulate future counts at these sites, and it appears that two counts at each site should be sufficient to produce an estimated total count with a CV of 30% or less from this model population.! From both line transect sampling and counts at haul-out site it appears that two independent counts at each haul-out site and 20,000 km of line transect sampling should ensure a CV of 20% or less for estimating the total number of walrus in the real population.! It is noted that the study only considers the population of walruses that are available for sampling. Corrections for visibility and availability biases are needed in order to estimate the total number of walruses in the whole population. Pacific W alrus Sampling Page 3 of February 2002

4 1. Introduction This report concerns the design and analysis of aerial surveys to assess the abundance of Pacific walrus (Odobenus rosmarus divergens) in the Bering and Chukchi seas. In particular, the requirements of the project were: (1) to compile and standardize previous aerial survey data for walrus over sea ice and at haul-out sites on land; and (2) to use the past data to investigate the survey effort needed in future to estimate walrus abundance with a coefficient of variation of between 20% and 30%, for surveys in both Spring and Fall. The report is in four parts. First, the process used to collect and standardize the data on aerial line transect surveys is described. Second, a simulation study of sampling in the Spring is described, leading to information about the survey effort needed in order to obtain various levels of precision of estimation at that time of the year. Third, a simulation study of sampling in the Fall is described, covering sampling over ice and water, and land haul-out sites. This again leads to information about the survey effort required in order to obtain different levels of precision of estimation at that time of the year. Finally, the results of the Spring and Fall simulations are discussed in terms of when is the best time of the year to estimate Pacific walrus abundance. Throughout this report it is assumed that the population of walruses that is being considered consists of those that are visible. For sampling over water and ice, this means the individuals that are at the surface of the water or on the ice. For haul-out sites it means the individuals that are visible on the land. To obtain an estimate of the total population of all walruses corrections for visibility and availability bias are needed, presumably based on radio-tagging data. These corrections are not considered here. 2. Aerial Line Transect Data Collection and Standardization Past aerial survey data provided by the U.S. Fish and Wildlife Service (USFWS) were examined. Where possible the intention was to produce for each survey a data table containing one row for each observation on either a walrus group or an indication of the start or end of an aerial transect line, and one column for each of the variables that are described in Table 1. In some cases only part of the desired data were available, in which case missing values were coded as 0 for "unknown" because the correct values must be positive. There were 11 surveys for which data could be recorded at least to the extent of providing the date, location, and the group size for walrus observations. These surveys also provided enough information to define the start and end of transect lines, which are Pacific W alrus Sampling Page 4 of February 2002

5 considered to be the basic sample unit for the surveys. In a 12th survey a report indicates where walrus groups were found, allowing some artificial data to be generated for the types of survey results that could have occurred. The sources of data and other information for these surveys are described below, in chronological order. The data for the 12 surveys are recorded in the spreadsheet WALR-SUR.WK1. Table 1 Variables recorded, where possible, from past aerial surveys of Pacific walrus. The definitions of specific codes are provided with each data set. Variable Description 1 ObsNum The observation number (row in the data), shown as 1, 2, 3,... 2 Date Date of survey, coded 1 for the first day of sampling, 2 for the second day, and so on. 3 Time The time of day in HH:MM format. 4 ArTyp The aircraft type, with different types coded as 1, 2, 3, etc. 5 Line The survey line, numbered consecutively 1,2, 3, etc. within each survey data, with each (approximately straight) survey line being considered as a basic sample unit. 6 ObsTyp The observation type, either 1 for the start of a new survey line, 2 for an observation on a walrus group, or 3 for the end of the survey line. 7 DistLst The distance (km) from the current observation to the previous observation, always set at 0 when ObsTyp = 1. 8 DistSt The distance of the observation from the start of the survey line, always set at 0 when ObsTyp = 1. 9 NumObs Number of observers recording walrus groups. 10 Lat The latitude of the observation in decimal format, e.g is 0.70 of the way between 64 N and 65 N. 11 Long The longitude west of the observation in decimal format, with values above 180 allowed. 12 SurTyp The survey type with 0 for unknown, 1 for open water, and 2 for ice-edge. 13 Oktas Octals of ice from 1 (none) to 8 (all ice). 14 LnW dth The width of the line surveyed from left to right (m). 15 Alt The altitude of the flight (m). 16 Speed The speed of the flight (km/hr). 17 DistLn The distance of a walrus group from the flight path (m). Always 0 for ObsTyp 1 and Habitat The walrus habitat, with 0 for unknown, 1 for water and 2 for ice. Always 0 for ObsTyp 1 and Count Number of walrus in the group, with 0 for unknown and for ObsTyp 1 and Stratum The location of the observation in the 5 geographical regions (strata) shown in Figure 1. Pacific W alrus Sampling Page 5 of February 2002

6 United States Spring 1968 Survey Field notes and maps were available for this survey of marine mammals in the Bering Sea which was conducted from April Times of observations were recorded, and positions calculated on the assumption of a constant speed. Flight lines were defined as straight segments on the flight maps. The altitude was stated (once) in the field notes to be 300 feet, and only records for walrus within 0.5 miles of the flight path were considered. United States Fall 1975 Survey Background information about this survey is provided by Estes and Gilbert (1978) and Estes, J.A. and Gol'tsev, V.N. (1984). It was carried out on five days in the period 1-12 September, The data were provided by the U.S. Fish and Wildlife Service in four files (TIME75US.TXT, OB75US.TXT, LINE75US.TXT and FLT75US.TXT), which between them provided most of the information required for the variables in Table 1. Soviet Fall 1975 Survey This survey was carried out in conjunction with the United States Fall Survey in the same year. Background information is provided by Gol'tsev. (1976) and Estes and Gol'tsev (1984) and Gol'tsev (1976). The survey was carried out on three days in the period from 20 September to 6 October Walrus observation were provided by the U.S. Fish and Wildlife Service in the file SOV75.TXT, but without information on the locations of the start and end of survey lines. The first observation each day has therefore been taken as the start of survey line 1, lines are arbitrarily ended at the first observation after 200 km, and a new line started at the next observation, and so on until the last line ends with the last observation. For example, if walrus groups are recorded at 180km, 203km and 220km, then the first line was ended at 203 km, and the second line started at 220 km. However, any walrus groups at the start and end of these artificial survey lines have been ignored to simulate survey lines of known lengths. United States National Marine Fisheries Service Spring 1976 Survey This survey was for all marine mammals, but only the Pacific walrus observations are considered here. Background information is provided by Braham et al. (1984), and the survey data were provided by the U.S. Fish and Wildlife Service in the file AERIAL.TXT. Surveys were carried out on 15 days in the period from 15 March to 22 April United States National Marine Fisheries Service Spring 1977 Survey This survey is assumed to be similar to the 1976 Spring survey described by Braham et al. (1984), although no paper or report is available to describe the survey methods. The Pacific W alrus Sampling Page 6 of February 2002

7 data come from the same file (AERIAL.TXT) that provided the Spring 1976 data. Surveys were carried out on ten days in the period from 2 April to 15 May United States Fall 1980 Survey This survey is described by Johnson et al. (1982), and is referred to as the second coordinated U.S. - U.S.S.R. survey of walruses, the first coordinated survey being carried out in The survey data were provided by the U.S. Fish and Wildlife Service in four files (TIME80US.TXT, OB80US.TXT, LINE80US.TXT and FLT80US.TXT) which between them provided the required information. The survey was carried out on five days in the period from 15 to 20 September Soviet Fall 1980 Survey This survey was the Soviet part of the 1980 coordinated survey. It is described by Fedoseev (1981). The data were provided by the U.S. Fish and Wildlife Service in three files (OBS80SOV.TXT, LIN80SOV.TXT and FLT80SOV.TXT), which between them provided the required information. Surveys were carried out on 25 and 26 September United States Fall 1985 Survey This survey was the United States part of the third coordinated U.S. - U.S.S.R. survey of walruses. It is described by Gilbert (1986, 1989). The data were provided by the U.S. Fish and Wildlife Service in four files (TIME85US.TXT, OBS85US.TXT, LINE85US.TXT and FLT85US.TXT), which between them provide the required information. Surveys were carried out on eight days during the perion from 20 September to 1 October Soviet Fall 1985 Survey This survey was the Soviet part of the 1985 coordinated survey. It is described by Fedoseev and Razivalov (1986). The data were provided by the U.S. Fish and Wildlife Service in four files (SOV85WL.TXT, SOV85FL.TXT SEG85S.TXT and LINE85S.TXT), which between them provided the required information. Surveys were carried out on five days in the period from 26 September to 7 October Soviet Spring 1987 Survey This survey is described by Fedoseev et al. (1988). No data are available, but flight lines, the positions of walrus groups, and the numbers in groups are provided by Fedoseev et al. in tables and maps. Because of the need for some data from the eastern part of the Bering Sea in Spring, the maps and walrus numbers by Fedoseev et al. were used to simulate data by randomly placing transect lines on the map and recording the positions and sizes of the walrus groups intercepted by these imaginary flight lines. Pacific W alrus Sampling Page 7 of February 2002

8 United States Fall 1990 Survey This survey was the United States part of the fourth coordinated U.S. - Soviet walrus survey. It is described by Gilbert et al. (1990, 1992). The data were provided by the U.S. Fish and Wildlife Service in two files (US90WL.TXT and US90FL.TXT). Surveys were carried out on seven days during the period from 28 September to 5 October Soviet Fall 1990 Survey This survey was the Soviet part of the fourth coordinated survey. It is described by Gilbert et al. (1990, 1992) along with the description of the United States part of the survey. The data were provided by the U.S. Fish and Wildlife Service in two files (SOV90WL.TXT and SOV90FL.TXT). Surveys were carried out on 12 days during the period from 16 September to 1 October Survey Design and Analysis for a Spring Survey One of the uses for the data described in the last section is to represent the type of data that might be obtained from future surveys, and to aid in planning those surveys (Manly, 1992). In this section the past Spring data is used to examine the type of results that might be expected from surveys at this time of the year, with different amounts of sampling effort. The Generation of Simulated Data for Spring Surveys Data are available from four Spring surveys. The results from these surveys have been used to produce a model population that can be sampled to simulate the outcome from sampling the real population at some future time. The survey lines for the four surveys are shown in Figure 1. The method used to simulate data from a future Spring survey is as follows:! The observations from each of the past surveys were allocated to the five strata shown in Figure 2. The transects in each strata were then joined together to form one long transect line for the model population.! A sample unit is defined to be a length m of transect (in km), and the sample size n is the number of such units that are selected for sampling. Pacific W alrus Sampling Page 8 of February 2002

9 ! Each of the n sample units is obtained independently by selecting a random place to start along the population transect line, and using the results from the next m km as the sample data. For example, if the sample unit size is m = 200 km, and n = 5 of these are to be selected, then the sample will consist of 1000 km of transect, with each 200 km section chosen randomly and independently along the population transect. This sample selection process is a type of bootstrapping that is design to maintain the type of clustering of walrus groups that is observed with real data. Table 2 shows a summary of the model population after it was set up as just described. Stratum E is only represented by the artificial data that were generated based on the results of the Soviet Spring 1987 survey. The means and standard deviations of walrus counts in 200 km segments of transect are shown in Table 3. The standard deviations vary from being about twice the mean to four times the mean in the different strata, indicating the considerable degree of clustering in walrus distributions along past survey transect lines. There is one further step in the process of simulating artificial data that will usually be necessary. The densities shown in Table 2 will give rise to varying total number of walruses, depending on the areas of walrus habitat in the different strata, which vary from year to year. In order to adjust the densities to give a fixed total population size of walrus for standardization of the simulation results, each walrus group size in the model population can be multiplied by the scaling factor that produces this fixed total size. Pacific W alrus Sampling Page 9 of February 2002

10 Table 2 Characteristics of the model population constructed from the data from four Spring surveys of Pacific walrus. The assumption is made that the transect width is constant at km (1 mile). Stratum Survey A B C D E Total Number of W alrus Observations Total Distance Sampled (km) Total Total W alrus Sampled Total Density of W alrus Per Square km Total Pacific W alrus Sampling Page 10 of February 2002

11 Figure 1 Sampled transect lines for the four Spring samples that have been used as a basis for simulating data for a new Spring survey. Pacific W alrus Sampling Page 11 of February 2002

12 Figure 2 The sampling region for a Spring survey of Pacific walrus, with five strata indicated. Pacific W alrus Sampling Page 12 of February 2002

13 Table 3 Means and standard deviations of walrus counts in 200 km segments of transect in the model population. Stratum Mean Count SD of Counts A B C D E Sampling Design and Analysis It is assumed that stratified sampling will be used, based on the strata used to set up the model population (Figure 2), with the sample units consisting of transect lines with a length of m km. However, as well as the usual type of stratified sampling with proportional allocation of sampling effort, a two-stage adaptive procedure developed by Francis (1984) for fisheries stock assessment has also been investigated. This adaptive procedure involves allocating some part of the sampling effort to strata based on what is known about the population, and then allocating out the second stage of sampling effort using the information obtained from the first stage. Francis' (1984) approach is often more practical than the stratified adaptive cluster sampling of Thompson (1991), and Brown (1999) found that it also gave a lower mean square error for most of a number of simulated populations. Its use therefore seems worth investigating with walrus sampling. Suppose that stratified random sampling is carried out with a total sample size of n 2 units, and a sample size of n i units in the ith stratum, which has a total area of A i km. In the ith stratum the n i sample units will each provide an estimate of the density of walrus 2 per km. Let the mean of these estimates be denoted by D i, and let the sample standard deviation calculated in the usual way be denoted by S i. Then an unbiased estimator of the mean density of walrus over the entire area sampled is with estimated variance S D = (A i / A) D i, (1) i = 1 Pacific W alrus Sampling Page 13 of February 2002

14 S 2 2 Var(D) = (A i / A) (S i /n i), (2) i = 1 where A = A 1 + A A S is the total area. These equations are standard for stratified sampling. The estimated total number of walrus in the area A is with estimated variance W = A.D, (3) 2 Var(W) = A Var(D). (4) Two-stage adaptive stratified sampling method works as follows. A first sample of size n f is allocated to the S strata with the sample size n 1i in stratum i, and the data collected. Equations (1) and (2) then give the estimated overall density with its variance at that point. Consideration is then given to the question of what would be the best stratum in which to take one more sample unit, in the sense of getting the maximum reduction in Var(D). This is easily found by increasing n i by one for each of the strata, and recalculating Var(D) keeping everything else constant. The extra sample unit is allocated to the stratum in which it gives the maximum reduction in Var(D). Consideration is then given to which would be the best stratum for adding another sample unit. Again this is easily found by recalculating Var(D) with the extra sample unit added to the first stratum, then the second stratum, and so on. The allocation of the second stage unit proceeds like this with sample units added one at a time until n s second stage units have been allocated. The second stage sample is then collected and analysed using equations (1) and (2) as if the full sample was just an ordinary stratified random sample. Francis (1984) recommended on the basis of simulation results that the first stage sample should be about 75% of the total number of sample units. He noted that although equation (1) is biased with adaptive sampling this is usually more than offset by the reduction in variance and the more normal distribution obtained for D. In fact, a type of bootstrapping can be used to remove the bias if necessary (see below), but this increases the variance and is therefore not necessarily desirable. Bootstrap Bias Correction A bootstrap bias correction can be applied to the estimate from two-stage sampling after the second stage sample has been taken. Here the following procedure was used: (a) The data from all strata were combined and standardized to have a mean of zero and a standard deviation of one. Pacific W alrus Sampling Page 14 of February 2002

15 (b) To produce a bootstrap set of first stage data from stratum i, the required number of values were obtained by randomly sampling with replacement from the pool of values set up in (a) and then adjusted using the equation X' = M i+ SX, i where M iis the mean and S i is the standard deviation in stratum i for the real data. The adjustment step ensures that the samples from strata have the same means and standard deviations as the original data. Data are generated in this way for all strata. (c) The sample from step (b) is used to decide on the optimum sample sizes in the strata for the second stage sample, and the second stage sample is selected as described in (b). (d) The full bootstrap sample is analysed using equations (1) to (4), and the relative error in estimating the total number of walrus for the bootstrap sample is determined as (e) r = (W B - W) / W, (5) Where W is the estimate from the original sample and W B is the estimate from the bootstrap sample. Steps (b) to (d) are repeated many times to estimate the mean value of r accurately. The bias-corrected estimate of walrus numbers for the original sample is then W C = W / (1 + r - ), (6) where r - is the mean of r. Equation (6) is based on the assumption that the expected value of W is approximately the total number of walrus multiplied by 1 + r -. Simulation Experiment 1: Spring In order to simulate sampling in the strata shown in Figure 2 it is necessary to determine the area in each of the strata that can be considered to be walrus habitat. To this end, information was collected on the area with between 15% and 80% ice for each of the years from 1979 to The results are shown in Table 4, with the years sorted according to the total amount of ice. Based on the results in this table, three years were chosen to represent the range of ice cover that might be obtained in the future. The years selected were 1987 (low), 1984 (medium) and 1988 (high). Although there were three years with lower ice cover than 1987 they were not selected for use because of the complete lack of walrus habitat in one or two of the strata. In practice, in years with no habitat in some strata all of the survey effort might be shifted to the strata with walrus habitat. Pacific W alrus Sampling Page 15 of February 2002

16 2 Table 4 Areas (km ) in strata with ice cover between 15% and 80% in the years 1979 to 1988, on either 15 or 16 April, with years in order of the total area of ice. Stratum Year A B C D E Total A factorial design was used for the simulation experiment, with the factors being: A: the strata areas were either from the low, medium or high ice year; B: the total number of walrus in the population available for sampling was either 150,000, 200,000 or 250,000; and C: the number of sample units (transects of length 200 km) was 50, 100, 200 or 400, for the whole survey. For each of these 3x3x4 = 36 scenarios, results were obtained with (i) stratified sampling with allocation proportional to area, (ii) two-stage stratified sampling with the first stage proportional to area, and (iii) two-stage stratified sampling with the first stage proportional to area plus a bootstrap correction for bias in estimation. The comparison in all cases was Pacific W alrus Sampling Page 16 of February 2002

17 made as a result of generating and analysing 1000 sets of data from the model population. Bootstrap bias corrections were made using 500 bootstrap samples. The results of this experiment are shown in Table 5, in terms of three summary statistics related to the estimate of the total number of walrus. The first is the root mean square error (RMSE), which reflects the contribution of both the bias and the variance of an estimator. The second is the mean relative error (MRE) expressed as a percentage, which is the average value of RE = 100x(Estimate - True Value)/(True Value). The third is the coefficient of variation (CV) expressed as a percentage, which is one hundred times the standard deviation of RE. Each summary statistic is provided for sampling with proportional allocation, two-stage adaptive sampling, and two-stage adaptive sampling with bootstrap bias correction. Values shown in bold are where the MRE is 5% or less, and the CV is 30% or less. Such situations might be considered acceptable in terms of the accuracy of estimation. The primary results of this simulation study are:! Adaptive sampling without bootstrap bias correction always gives the smallest RMSE, and for this reason might be considered to always be "best", except that it exhibits some bias. However, the bootstrap bias correction removes about half of the bias, with little increase in the RMSE and CV, so that this might be considered to be on balance even better.! There are serious problems with the amount of sampling effort required to obtain estimators with reasonable properties. With low ice cover (where there was assumed to be less spread in the location of walrus groups in the population) it was just possible to get acceptable estimates with 200 sample units (a total flight path of 200x200 = 40,000 km), but as many as 400 units might be needed with medium or high ice cover.! The total number of walrus does not seem to be a crucial factor in terms of MRE and the CV, although the RMSE obviously increases with the total number. Pacific W alrus Sampling Page 17 of February 2002

18 Table 5 Results from the simulation experiment on Spring sampling. Results shown in bold are possibly acceptable results, where the coefficient of variation (CV) is 30% or less, and the mean relative error (MRE) is 5% or less. The root mean square error (RMSE) is in thousands. Ice Level W alrus Numbers ('000) Sample Units Proportional Allocation Adaptive Sampling Adaptive & Bias Correction RMSE MRE (%) CV (%) RMSE MRE (%) CV (%) RMSE MRE (%) CV (%) Low Medium High A final comment concerns the accuracy with which the CV can be estimated from the survey data. With all scenarios the CV was under-estimated on average, with a high bias when the true CV was large. For example, for adaptive sampling with bootstrap bias correction the worse case true CV shown in Table 5 is 79%. However, the mean of the estimated CVs based on the generated data was 40%, with a standard deviation of 19%. This is clearly unsatisfactory. The situation does improve as the true CV goes down. Thus the minimum CV shown in Table 5 with adaptive sampling with bias correction is 22%. For Pacific W alrus Sampling Page 18 of February 2002

19 this the mean of the sample estimates is 18, an under-estimate with a standard deviation of 4%. This is better, that the result for large CVs, but is still not totally satisfactory. The bias is not a problem just with adaptive sampling as it is also present with the stratified sampling with proportional allocation. 4. Survey Design and Analysis for a Fall Survey A Fall survey of walrus is more complicated than a Spring survey because the walrus can be in several different areas. They may be around the ice edge in the north of the Chukchi Sea, off the coast of Russia in the Chukchi Sea, at a number of land haul-out sites along the Russian coast in the Chukchi and Bering Seas, or at a number of land haul-out sites on the Alaskan coast. For the surveys over sea and ice the approach used for studying the design of a Spring survey can be applied, more or less unchanged. Therefore, this arm of a Fall survey will be considered first. The Generation of Simulated Transect Data for Fall Surveys There are four United States and Russian Fall surveys in 1975, 1980, 1985 and 1990 for which aerial line survey data are available. These eight surveys did not have a common protocol, but nevertheless the results will be treated as typical of what might be obtained in future, at least as far as the counts and their locations along the transect lines. Flight lines are shown in Figure 3. The mean counts per km vary considerably for the eight surveys, and it was decided that this allowed for the consideration of four strata in a simulated survey, varying from one with quite a high density, to one with a rather low density. Stratum A (high density) was constructed from the U.S. surveys in 1980 (5.32 walrus/km) and 1975 (1.46 walrus/km). Stratum B (medium density) was constructed from the U.S. survey in 1985 (1.22 walrus/km) and the Russian 1980 survey (1.17 walrus/km). Stratum C (low density) was constructed from the Russian surveys in 1985 (0.92 walrus/km) and 1975 (0.66 walrus/km). Finally, Stratum D (very low density) was constructed from the U.S survey (0.65 walrus/km) and the Russian 1990 survey (0.19 walrus/km). A model population was constructed in the same way that the model population was constructed for Spring sampling, as described in Section 3. That is, the real survey data for the two surveys within a stratum were joined together to form one long line, which could then be resampled to produce simulated data. Table 6 shows a summary of the model population. The sample units used for simulated data were 100 km long and the table therefore shows the mean and standard deviation in each of the strata for the number of walrus per unit. Pacific W alrus Sampling Page 19 of February 2002

20 Figure 3 Sample transects for joint United States - Russian surveys carried out in the Fall of 1975, 1980, 1985 and Pacific W alrus Sampling Page 20 of February 2002

21 For the simulation of Spring sampling the sample unit used was 200 km long. For Fall a shorter length is appropriate because of the smaller sample areas involved, and the need for flexibility in sampling along the ice edge and off the coast of Russia. For the simulation of the aerial line transect part of the Fall survey it was assumed that the area suitable for this type of sampling can be divided into four strata, corresponding to those for the model population. These might for example be the east and west sections of the ice edge, and the area off the Russian coast divided into north-west and south-east 2 sections. These strata are also assumed to be of equal size (25,000 km each) for simplicity, although in reality the sampled area varies from year to year, and even within years. Simulation Experiment 2: Fall Transects Over Water and Ice Data were generated for 12 different scenarios, corresponding to all combinations of four levels of the walrus population size on ice and water (25,000, 50,000, 100,000 and 200,000), and three sampling intensities (100, 200 or 400 sample units of 100 km each). The results are summarized in Table 7, in the same format as was used for Table 5, except that results are not given for different levels of ice cover. Table 6 Characteristics of the model population constructed from the data from eight Fall surveys of Pacific walrus. The assumption is made that the transect width is constant at km (1 mile). Stratum A B C D Total Observations Length (km) Area (km ) W alrus on Line Density (W alrus/km ) Mean W alrus per 100 km SD of W alrus per 100 km As was the case for the simulation of Spring sampling, two-stage adaptive sampling without bias correction always give the smallest RMSE, but the bootstrap bias correction process might be favored because it largely removes the bias with a relatively small increase in the RMSE. The results indicate that about 150 sampling units (15,000 km) should give a CV of less than 30% for surveys of the type being considered, and that 200 sample units (20,000 km) will reduce the CV down to close to 20%. Pacific W alrus Sampling Page 21 of February 2002

22 Table 7 Results from the simulation experiment on Fall sampling. Results shown in bold are possibly acceptable results, where the coefficient of variation (CV) is 30% or less, and the mean relative error (MRE) is 5% or less. The root mean square error (RMSE) is in thousands. W alrus Numbers ('000) Sample Units (100 m) Proportional Allocation Adaptive Sampling Adaptive & Bias Correction RMSE MRE (%) CV (%) RMSE MRE (%) CV (%) RMSE MRE (%) CV (%) The Generation of Simulated Data From Land Haul-Out Sites Information from surveys in the period from 1975 to 1990 provided at least one count at each of 38 haul-out sites, as listed in Appendix A. The average counts at the sites and their percentages of the total of these averages are shown in Table 8. To better understand the nature of the variation in the counts from one haul-out site, a log-linear model was fitted to all of the data provided in Appendix A, using Genstat (Lawes Agricultural Trust, 2001). The year was introduced as a factor with four levels, and the site as a factor at 38 levels. Models were considered with site effects only, with site effects and year effects, and with site effects, year effects and interaction effects. The interaction effects were not significant (p = 0.89) but the site effects were highly significant (p < 0.001), and the year effects were significant at the 5% level (p = 0.025). The interaction effect was therefore removed from the model, but the other terms left in. The model was fitted using an estimated heterogeneity factor. This accounts for the extent to which the variation in the observed counts exceeds what is expected on the assumption that these counts follow Poisson distributions (which is the basis for the loglinear model). In fact, the estimation of this factor was the main reason for fitting the model in the first place. With this factor estimated, the standardized deviance residuals are approximately normally distributed, indicating that the model is a reasonable representation of the data. The estimated heterogeneity factor is , indicating that the variance of repeated counts at one site in one year is approximately times the expected count at that site, because the variance of the Poisson distribution is equal to the expected value. Pacific W alrus Sampling Page 22 of February 2002

23 Table 8 Average counts at haul-out sites for surveys during the period 1975 to 1990, with the percentages that these are of the total of the averages. Average % of Total Russian 1 Herald Island Sites 2 Cape Blossom Somnitelnaya Spit Davidova Spit Gavai Kolyuchin Island Idlidlya Island Cape Serdtse-Kamen Cape Inkigur Inchoun Cape Dezhnev Big Diomede Island Cape Nunyagmo Cape Kriguigun Arakamchechen Island Nunyangan Island Rudder Spit New Haulout Meechkin Spit Russkaya Koshka Cape Navarin Burunneyeh Island Dezhnev Bay Anastasia Bay Bogoslava Island Cape Tiomney Capes Sery and Anana Cape Skladchatyi Verkhoturova Island Cape Golenishcheva Cape Semionova Cape Govena U.S. 33 Hall Island (St Matthew) Sites 34 Cape Newenham Cape Peirce Twin Islands Round Island Cape Seniavin Total The very large heterogeneity factor demonstrates clearly that the variation between counts at a site is very extreme. Furthermore, as some of the counts provided in Appendix A are close together in time, there is the possibility that the variation may be slightly underestimated because of positive serial correlation in time, leading to similar counts from one day to the next. Pacific W alrus Sampling Page 23 of February 2002

24 To generate simulated counts at haul-out sites, the following procedure was adopted:! The 38 sites shown in Table 8, together with the percentages of the average counts at these sites were assumed to represent the sites to be sampled in any year, with the assumed total number of walrus available for sampling at haul-out sites being allocated to the 38 sites according to the stated percentages, to give an expected count at each site.! The simulated count at a site is a random value from a log-normal distribution with mean m and variance m, truncated to the nearest integer. Simulation Experiment 3: Fall Hall-Out Sites For the third simulation experiment, 20 scenarios were simulated using the Resampling Stats (2000) add-on for Excel, with a two factor factorial design. Past surveys show counts ranging from about 40,000 to 160,000 walruses at haul out sites (Estes and Golt'sev, 1984; Fedoseev and Razlivalov, 1986; Gilbert, 1989; Gilbert et al., 1992). The first factor was therefore the expected total count at four levels, which were 25,000, 50,000, 100,000 and 200,000 walruses. The second factor was the level of sampling effort, at five levels. This was the number of independent counts made for each haul-out site, which varied from one to five. For each of the 20 scenarios, 5000 sets of data were generated, and summarized in terms of the percentage CV. Bias was negligible. The results are summarized in Table 9. When there are two or more counts at each haul-out site it is possible to estimate the variance of the estimated site mean from the replicate results, and hence estimate the variance of the total count by the sum of the estimated variances for all the sites. The estimated CV is then the estimated standard error divided by the estimated total number of walruses, and multiplied by 100. Table 9 also shows the mean values obtained for these estimated CVs. It is seen from Table 9 that the CV decreases with the expected total walrus count, which is an outcome of the assumption that the variance of counts is proportional to the mean count. Nevertheless, it appears that two counts at each site should be sufficient to obtain a CV of less than 30%, with the estimated CV being biased slightly low. It is of course important for these counts to be independent, and estimating the same quantity. The data in Appendix A indicate that independence should be achieved if the counts are at least 5 days apart, and the expected counts should also be similar as long as they are both made at about the same time in the Fall. The CV will also be reduced further if more than two counts are made at the colonies where most walruses are likely to be found. Pacific W alrus Sampling Page 24 of February 2002

25 Table 9 Coefficients of variation (CV) obtained for different expected counts of walruses at haul-out sites, with from one to five independent counts made at each site. The mean of the estimated CVs (Mean Est) from the generated samples are also shown. 1 Count per Site 2 Counts per Site 3 Counts per Site 4 Counts per Site 5 Counts per Site Expected Count CV (%) Mean CV (%) Mean Est 1 Est CV (%) Mean Est CV (%) Mean Est CV (%) Mean Est One count per site does not allow the CV to be estimated. Combined Estimates From Line Transect and Haul-Out Site Surveys The total estimated number of walruses from a Fall survey will be the sum of the counts from the line transect sampling over ice and water and the counts from haul-out sites. The joint United States - Russian surveys in 1975, 1980, 1985 and 1990 produced the estimates shown in Table 10, suggesting a total population size of between 200,000 and 250,000, with the number at haul-out sites varying from about one-quarter to threequarters of this total. Table 10 Estimates from joint United States - Russian surveys of the number of walruses at land haul-out site, off the Russian coast, and near the ice edge. Year Haul-Out Sites Off the Coast Ice Edge Total ,852 33, , , ,000 68, , , , ,531 62, , ,047 22,503 16, ,039 Based on the results in Table 10, three reasonable scenarios are that there are (a) 50,000 walruses at haul-out sites and 150,000 on water and ice, (b) 100,000 walruses at haul-out sites and the same number on water and ice, and (c) 150,000 walruses at haulout sites, and 50,000 on water and ice. The estimated total number of walruses is in each case the sum of the separate estimates for the haul-out sites and water and ice, with a variance that is the total of the individual variances. On this basis, Table 11 shows the Pacific W alrus Sampling Page 25 of February 2002

26 CVs that would be obtained with from one to three surveys of haul-out sites, and 100, 200 or 400 sample units (100 km) of line transects over water and ice. For all three scenarios, two counts at each haul-out site and 200 sample units of transect (20,000 km) are sufficient to ensure a CV of less that 20% for the simulated populations. Table 11 Standard errors (SE) and percentage coefficients of variation (CV) for total walrus estimates based on adding counts at haul-out sites and estimates from line-transect samples over water and ice. Walrus Numbers 100 km Transect Haul-Out Ice/Sea Haul-out Counts Units Sampled SE CV Discussion The first simulation experiment (Section 3) suggests that for a Spring survey of walrus in the Bering Sea it might be possible to obtain a CV of 30% or less by flying 40,000 km of transects if there is low ice cover, but that as much as 80,000 km may be needed with medium or high levels of ice. In Fall, the sampling problem is more complicated because of the possibly large numbers of walruses at land haul-out sites. However, the results from the second and third simulation experiments suggest that at this time of the year a CV as low as 20% may Pacific W alrus Sampling Page 26 of February 2002

27 be obtainable with two counts of each potential haul-out site (the minimum needed to get an estimate of variance) and approximately 20,000 km of transect sampling. This would seem to make a Fall survey more attractive than a Spring survey. However, there is a reservation that must be taken into account with respect to the sampling of land haul-out sites. This is that the estimation of the amount of variation from repeated counts in one year is based on little data from sites with some very large counts (Appendix A). If this variance is under-estimated or has been incorrectly modeled then the simulation experiment 3 may be too optimistic in terms of the CV obtainable from only two counts at each potential haul-out site. For this reason, it would be desirable to plan for more than two counts at the land haul-out sites with potentially very large numbers of walruses present. As noted in Section 1, this report is only concerned with the counting of walruses that are available for observation. Corrections for the bias due to walrus being invisible because they are under the surface of the sea or in some other inaccessible area have not been taken into account, although they are clearly essential for obtaining a valid estimate of the actual population size. Finally, it has been assumed that line transect sampling will be carried out with lines of a fixed length (200 km in the Spring and 100 km in the Fall). In practice it may be desirable to use variable length lines. This would require a modification of the estimation equations, for example using ratio estimation. However, the accuracy obtained should be similar to what is obtained from fixed length lines so that the results reported here are still a good guide to the total sampling effort required. References Braham, H.W. et al. (1984). Habitat partitioning by ice-associated pinnipeds: distribution and density of seals and walruses in the Bering Sea, April In Soviet-American Cooperative Research on Marine Mammals, Vol. 1 - Pinnipeds (Eds. F.H. Fay and G.A. Fedoseev), pp NOAA Technical Report NMFS 12. Brown, J.A. (1999). A comparison of two adaptive sampling designs. Australian and New Zealand Journal of Statistics 41: Estes, J.A. and Gilbert, J.R. (1978). Evaluation of an aerial survey of Pacific walrus (Odibenus rosmarus divergens). Journal of the Fisheries Research Board of Canada 35: Estes, J.A. and Gol'tsev, V.N. (1984). Abundance and distribution of the Pacific walrus, Odobenus rosmarus divergens: results of the first Soviet-American joint aerial survey, Autumn In Soviet-American Cooperative Research on Marine Mammals, Vol. 1 Pacific W alrus Sampling Page 27 of February 2002