Sampling design, mapping and assessing by ...

GIS/Spatial Analyses in Fishery and Aquatic Sciences (Vol.6) Conan - Sampling design, mapping and assessing by geostatistics made clear (089-116)

ⒸInternational Fishery GIS Society, 2016

Sampling design, mapping and assessing by geostatistics made clear Part II: Results, Discussion and Conclusions Gerard Y. Conan Marine Geomatics, 3307 Rte 910 Baltimore NB, E4H 1V2, Canada Phone: +1 (506) 734 3215 E-mail: [email protected]

Abstract Geostatistics were initially developed for the mining industry, but have more recently been applied to mapping and assessing fisheries resources. Bridging with standard statistical methods used for processing fisheries data is required as concepts and terminology used for mining and fishing data were developed independently. We set emphasis on the assumptions and the ensuing practical use for mapping and assessing marine fisheries resources. We provide basic information for developing computer approximations for geostatistical equations that cannot readily be fully solved analytically. We document: 1) Mapping of a resource by geostatistics on Mercator marine charts. 2) Assessing a resource by geostatistics, setting confidence limits and correcting bias. 3) Assessing a resource regionally, in different fishing subareas or within an area devoted to a special use such as public works. 4) Optimization of sampling design by minimizing estimation variance and by minimizing route and costs. 5) Exporting optimized routes, data point information and resource maps designed by geostatistics into marine navigation system plotters commonly used by the Fishing Industry.

Keywords Geostatistics, Kriging, Semivariance, Semivariogram, Cross Validation, Spatial Correlation, Spatial Covariance, Fisheries Resources, Assessing, Mapping, Sampling Design, Snow Crab, Chionoecetes opilio, Simulated Annealing, Trawling.

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3. Results 3.1 Annual concentration of patches The data set is taken from the 2010 commercial snow crab biomass survey, Southern Gulf of Saint Lawrence. In these data the concentrations of commercial snow crab patches considerably vary from year to year (Figure 2). It does not appear possible to design sampling strata prior to an assessment.

Figure 2. Location of patches vary from year to year. Over the past 20 years the position and size of the patches of commercial snow crab concentrations have varied year after year. This precludes the design of a stratified sampling procedure.

3.2 An example of geostatistic assessment compared with the standard arithmetic mean approach 3.2.1 Semivariogram The first step consists in calculating an experimental semivariogram and trying a fit with several models. We display here a fractal model fitted to a relative semivariogram (Figure 3).

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Figure 3. Example of semivariogram. The spatial variable is the abundance in tonnes of commercial snow crab in the Southern Gulf of Saint Lawrence in 2010. The numbers of paired observations for each lag or distance in Km are presented along the curve. A Log/Log transformation is used for adjusting a fractal model.

3.2.2 Chart The corresponding Mercator marine chart is generated by geostatistics. Abundance estimates in Tonnes per Km2 are calculated as kriging averages over cells of 1 square nautical mile. This produces a color coded map of abundance (Map 1). 3.2.3 Calculation of overall biomass The overall biomass is calculated here within the polygon formed by the 20 and 200 fathoms isobaths inside fisheries regulation limits. The overall biomass is calculated here by one step kriging average over the whole polygon. The confidence limits are calculated from the estimation variance of the kriging average. The area was calculated from the portion of spherical area delimited by the 20-200 fathom isobaths polygon. The overall biomass was also calculated by summing up the estimates in each one square mile cells of the map (Map 1). The results are similar but accurate confidence limits cannot be simply calculated by this method, as spatial covariance exist between the cells. Finally a simple estimate of biomass by arithmetic mean of all sampling unit stations located within the polygonal contour was calculated for comparison. The confidence limits were calculated from the variance 91

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of the arithmetic mean. The area was calculated from the portion of spherical area delimited by the 20-200 fathom isobaths polygon (Map 1).

Map 1.

Mercator nautical chart for Commercial snow crab biomass in 2010 in the Southern Gulf of Saint Lawrence. The calculations were made by kriging within a grid of 1 by 1 nautical mile cells, using a spherical semivariogram. The small numbers on the chart identify location of sampling stations. The vector graphics representation allows magnification to any size from an HPGL-2 file, without loss of definition.

3.3 Effects of the semivariogram model on charting Minute differences in semivariogram modeling can generate differences in maps. In Figure 4 we display bottom temperature charts generated by using spherical and exponential models. The data are identical. The adjustment of the models are quite comparable. However they do generate differences in the map at locations where sampling units are scarce.

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Figure 4. Effects of the semivariance model on charting. A spherical model and an exponential model are fitted to the same data set. The fits are quite similar but produce a modification of patterns in the south east where the sampled stations are distant from one another. Minute modifications in model used can have important effects.

3.4 Semivariance steadily increasing as distance increases A Fractal semivariogram model as presented in Figure 3 can model a semivariance increasing steadily as a function of lag (distance). The effect is encountered for semivariance in acoustic data for schools of fish such as herring. The effect is also frequently encountered for relative semivariances of biomass of benthic crustaceans such as snow crab and Alaskan red king crab (Figure 5). A lack of homoscedasticity is corrected by a fractal model.

3.5 Drop of semivariance as distance increases A drop of semivariance as distance (lag) increases (Figure 5) is a very common effect. Semivariance first increases up to a sill, remains stable for a while and then drops again. The effect is often masked when not considering the full extent of the data. The drop can often be corrected by a using a relative semivariogram (Figure 6) and a fractal model. 93

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Figure 5. A drop of semivariance when points in a pair are spread over large distances (Lags). This may reveal a semivariance to mean relationship.

Figure 6. A relative semivariance may correct a semivariance to mean relationship. A subsequent fit to a fractal model corrects for homoscedasticity.

3.6

Logarithmic transformation and other Gauss normalizing transformations of the spatial variable abundance data

Such transformations have been tried on present snow crab data. No benefits have been found. Such transformations appear to mask most important features in the semivariograms.

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3.7 Detecting a hot spot A hot spot is a location where abnormally high values of the spatial variable are encountered. Such hot spots occur when snow crab congregate at certain stages of their life cycle for feeding, molting, mating or incubating egg batches. These hot spots can be detected on a semivariogram as a sharp peak for small distances (lags). An example is given in Figure 7.

Figure 7. Detecting a hot spot. At hot spots the value of the geographic variable is exceptionally high. The usual models will not adequately fit the resulting semivariance peak. The models will not adequately describe the geographic extent of the effect and may bias the kriging average.

3.8 Selecting an optimal number of iteration points for averaging semivariance. A good numerical approximation is required for averaging semivariance in the linear set of kriging equations (Equation 3). A small number of iteration points may generate erratic values for estimated kriging average and its estimation variance. A very large number of iterations may be quite processing time consuming.

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Preliminary trials are recommended for insuring an adequate precision but avoiding excessive processing time. The following trials were made for processing the Southern Gulf of Saint Lawrence commercial snow crab biomass survey data charted in Map 1. The effect of the number of iterative numerical integration points on the estimate of Kriging average is presented in Figure 8. A minimum of 800 points is required for stabilizing the estimate.

Figure 8. Effect of the number of numerical integration points on the kriging average. For this set of data the kriging average estimate stabilizes only once the semivariance has been averaged over 800 to 1000 swept points inside the area being assessed.

The effect of the number of iterative numerical integration points on the estimation standard deviation of the kriging average is presented in Figure 9. A minimum of 800 points is required for reaching a stabilized minimized estimation standard deviation.

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Figure 9. Effect of the number of numerical integration points on the estimation standard deviation of the kriging average. This confirm the results displayed in Figure 7. Eight hundred to 1000 points must be swept over the area assessed.

Alternatively, a precision index can be used. A 95% precision index equal to the percentage of 2 standard deviations divided by the kriging average can be plotted against the number of numerical integration points ( Figure 10).

Figure 10. Effect of the number of numerical integration points on the precision of the kriging average. The precision index is defined here as the percentage of +/- 2 standard deviations over the kriging average estimate. When 800 to 1000 points are swept over the area assessed the precision reached is 12 %.

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3.9 Standing Biomass vs. Fishable Biomass Standing Biomass is the overall abundance of the resource, it is useful for defining quotas. It does not depend on the concentration of the resource in patches. It is useful for calculating potential income. Fishable biomass depends on the concentration of the resource into patches that can be profitably harvested by a commercial fleet. In Table 1 we have contrasted estimates of standing biomass and fishable biomass for commercial snow crab harvestable off North East Nova Scotia in 2013 (Map 2). It is assumed that the sill for profitability is 2 Tonnes/Km2 and that a fishing gear unit covers a 0.3x0.3 cell. These figures have been arbitrarily set for illustrating the technique and do not imply any management issues. The biomass outside the fishable concentrations in low density areas far exceeds the biomass inside fishable high density concentrations. Table 1.

Example of Standing Biomass vs Fishable Biomass. The area assessed is the polygonal contour defined in Map 2. The standing biomass is the total biomass of commercial snow crab as calculated by kriging in one step over the whole area. The fishable biomass is the amount of biomass within 0.3x0.3 nautical mile cells exceeding the profitability sill of 2 tonnes per square Kilometer. This value is arbitrarily chosen to illustrate the technique. Only 2757 tonnes out of 7440 would be worth fishing if the profitability sill conformed to actual economic constraints.

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3.10 Sampling design We compare the precision of biomass estimates achievable by different sampling designs. We locate stations inside square cells organized as a grid covering the area to be assessed. Inside each cells a station is either located randomly or evenly (at the center). The route between stations is optimized by simulated annealing. The data are from the North East Nova Scotia commercial snow crab 2013 survey (Map 2). The semivariogram displayed in Figure 5 is used for the calculations.

Map 2.

Surveyed stations and chart of commercial snow crab abundance in 2013 off North East Nova Scotia. A high power electric cable is to be laid across the commercial snow crab concentrations . The cable corridor is drawn at scale.

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The cells of equal area on the sphere are designed without using plane conformal (orthomorphic) projection coordinates. The longitude intervals in degrees used for the cells is adjusted from North to South in order to correct for an equal size in Nautical Miles (Figure 11).

Figure 11. Design of cells of equal area on the sphere without using plane conformal (orthomorphic) projection coordinates. The longitude cell intervals in degrees are adjusted from North to South in order to correct for an equal size in Nautical Miles. This results in an East West shift of the cells from one latitudinal row to the next.

The resulting grid maps, stations and optimized routes are presented in Figures 12 and 13. It is assumed that the ship is steaming at 8.5 knots between stations, spending 45 minutes on station, working 12 hours a day and steaming at night between stations. These values are reasonable but 100

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arbitrary, no specific recommendations are made. The purpose is to illustrate a technique for logically improving survey design. Precision and cost of precision vs. days at sea and precision vs. number of stations are useful for estimating what is the best affordable sampling design within budget.

Figure 12.Stations located at the center of the cells on the grid. The shortest route is searched by simulated annealing.

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Figure 13. Stations located at random within each cells on the grid. The shortest route is searched by simulated annealing.

Precision as a function of number of stations and number of days at sea is presented in Figures 14 and 15. It is noticeable that precision can be improved much beyond the 30 station limit predicted by the arithmetic mean calculations which do not take into account spatial covariance. Even spacing of stations provides a better precision than randomly setting the stations; however the discrepancy tends to disappear when the number of stations increases.

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Figure 14. Precision of kriging average estimate as a function of number of stations sampled. The precision index is the percentage of two estimation standard deviation divided by the kriging average. A precision of 10% can be achieved for 200 stations, it is presently of 25% for some 70 stations. If the stations were set at even intervals, the precision for 70 stations would be enhanced to 18%.

Figure 15. Precision of kriging average estimate as a function of number of days at sea. It is assumed that the ship is steaming at 8.5 knots between stations, spending 45 minutes on station, working 12 hours a day and steaming at night between stations.

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3.11 Regional estimates In the Southern Gulf of Saint Lawrence (Map 1) snow crab fishing quotas are allotted each year to specific fishing fleets. The fishing grounds have been arbitrarily partitioned in sub fishing areas opened to each of these fleets. It is useful to calculate the commercial crab biomass present in each sub area before the opening of the fishing season. Overall polygon kriging provides such estimates with confidence limits. Off the North East coast of Nova Scotia (Map 2) a project is presently considered for laying a high power electric cable across the commercial snow crab concentrations. We have used overall polygon kriging inside the cable corridor to estimate the commercial snow crab biomass and recruit numbers potentially affected by the project. Confidence limits are calculated for each of the estimations. These two examples illustrate how preexisting data can be used to provide regional estimates on biomass and numbers within any sub polygon within or even outside at the margins of a surveyed area. It is preferable but not mandatory to have stations located inside the sub polygon. Actual values are not provided here as they may imply management recommendations.

3.12 Sub categories within a species may have different geographic distributions When designing a survey it is frequently assumed that different biological categories within a species will spatially distribute according to the same spatial patterns. This is not the case for snow crab. We have mapped the spatial distributions of immature males, females and mature females as well as commercial snow crab, which are mostly morphometrically mature males as defined by Conan and Comeau (1992) and Comeau and Conan (1986). The spatial distribution of immature males and females are quite similar, but strikingly differ after maturity (Map 2 and Figures 16, 17 and 18). The semivariograms also differ.

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Figure 16. Concentrations of immature female snow crab.

Figure 17. Concentrations of immature male snow crab.

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Figure 18. Concentrations of mature female snow crab. A hot spot is noticeable in the north west.

3.13 Effects of bottom temperature and depth on the geographic distribution of biomass and number of individuals The spatial distribution of snow crab at depth and bottom temperature have been monitored over the past 25 years. It appears that benthic stages of snow crab are found within a layer of water called the Cold Intermediate Layer (CIL). This layer is separated in summer from a warmer less saline surface layer and year round from a warmer more saline deep layer. The concentrations of snow crab at different stages move as a function of the vertical interfaces (mixing layers) of these layers. We present maps of commercial snow crab concentrations, bottom temperature and depth off the North East coast of Nova Scotia. The effect is similar, snow crabs are encountered in a gulley which is likely seasonally filled with CIL water flowing from the Gulf of Saint Lawrence. The presence and possibly the abundance of snow crab depends on the presence of this water mass (Map 3). It is not depth per se but the presence of the water mass on bottom that defines the habitat. Subsequently we do not use depth as an auxiliary variable for mapping and estimating abundance of snow crab by kriging. 106

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Map 3.

Effects of bottom temperature and depth on location of concentrations of commercial snow crab. The bottom temperature is an indicator of a water mass favorable as snow crab habitat. This water mass varies in depth from year to year and seasonally so does the snow crab geographic distribution.

4. Discussion and conclusions 4.1 Some comments on the present data analysis and possible generalizations I shall review and comment the present results identifying possible potential generalizations for similar data, especially for surveys of benthic resources. Following Jumars (1978) it is important to distinguish the sampling distribution skewness effect from the spatial autocorrelation or covariance effect. Correcting skewness does not necessarily affect spatial covariance effects. Kriging in its most accepted versions does not make inferences about the skewness of the sampling distribution, it takes advantage of spatial covariance to enhance estimates and at least partially correct bias generated by preferential sampling. 107

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For the present data normalizing skewed sampling distributions by log transforms and other enhanced methods did not improve calculations. The transformations affect the quality of the semivariogram diagrams and generate bias in the estimates that are not easy to correct. Year to year changes in geographic distributions are quite common among mobile species, but frequently overlooked. Cold blooded species naturally search for sites and water masses suiting varying physiological states in their life cycle. Water masses vary seasonally and from year to year, their interfaces are not plane and horizontal, there are upwellings and downwellings and mixing layers forming after storms. Subsequently grounds may or may not offer stable favorable habitats. Using information from a previous survey to design sampling strata for the next is not advisable for snow crab and may not be advisable for most mobile benthic species. Stratification is not logically possible as no information is available for designing strata before the survey. Discrete partition of survey area into strata is not required as kriging is designed to work in a continuum and use each station as a stratum with an optimized weight. Depth is not a determinant factor for the presence of snow crab for the same reasons as above. Snow crab normally inhabit the Cold Intermediate Layer (CIL) of water at temperatures usually ranging from -1 to 2 degrees C. but with a wide upper mixing layer reaching 4 to 5 degrees C. The CIL deepens from Spring when it is close to surface to Fall when it reaches 80 m in certain areas. However snow crab can be found year round in fairly shallow water, isolated in basins retaining cold intermediate CIL water formed over the previous winter. This is quite common in the Gulf of Saint Lawrence where such geomorphological depressions were formed during previous glaciations and subsist as fjords or isolated “holes”. It is therefore not advisable to use depth as a third dimension for kriging or as an external function modeling a trend. Depth is frequently used for defining strata in fisheries resource surveys, its effect should be verified each time before proceeding. The movements of the CIL can be monitored independently using water density and sigma-t. Salinity measures are not available for the present data collected aboard a chartered fishing vessel. But temperature per se is usually a sufficiently good reference for monitoring snow crab movements. However, short term history must be replayed as some snow crab can be left behind when the CIL limits move vertically. Further, some snow crab may enter the mixing layers at least for brief time lapses.

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Within the snow crab species different benthic stages have different spatial distributions. Such categories may exclude each other for biological reasons. For instance, cannibalism occurs between morphometrically mature males (commercial category) and immature or juvenile stages. Certain categories may congregate such as molting individuals or egg carrying females, creating hot spots. It is therefore advisable not to process data from all biological categories at a time. The semivariograms will differ and standard geostatistics may not even be advisable in the case of strong clumping into hot spots. Enhancing the precision of the estimates and the quality of maps is possible by using spatial covariance, as shown by kriging. The possibilities depend on the shape of the semivariogram. Frequently semivariance/inverse covariance increase with distance between stations up to a distance or range at which a sill in semivariance is reached. In such a case, setting stations at a distance apart smaller than the range will enhance precision of resource estimations. This is verified even beyond the upper limit of 30 stations defined by standard statistics. If the data is poorly sampled with no precise location of the stations, varying swept area or poorly measured swept area at the stations, a large nugget effect will eventually mask semivariance/spatial covariance effects. The benefits of geostatistics will be lost and an arithmetic mean estimator may as well be used for the sake of simplifying calculations. On Map 1, three different values are provided for overall commercial snow crab biomass surveyed in 2010 in the Southern Gulf of Saint Lawrence. Two values are from geostatistics calculations involving all stations within and outside of the assessed polygon. The third estimate was obtained from the arithmetic mean for stations located inside the polygon. The geostatistics estimates are almost identical. The first estimate results from overall 1 step kriging, with calculation of estimation variance and precision of the estimate. The second estimate results from kriging one by one the 1 nautical square mile cells used for mapping and summing, there is no possibility of variance calculations due to covariance between cells. The arithmetic mean estimate is the largest with the widest confidence limits. This may result from a slightly biased sampling. The position of the stations being chosen slightly preferentially at the time of setting the trawl on station. This is a subjective effect that cannot be fully avoided as the captain decides for the exact position and timing depending on sea conditions. A few meters apart can make a difference between the 109

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quality of two hauls. Kriging has allowed here to correct some bias in the estimated biomass and to correct the variance by using covariance. Such effects are quite frequent. However it may also occur that the calculated variance of the arithmetic mean be smaller than the kriging estimation variance due to highly preferential locations of the stations. This is frequently the case when commercial fishing locations are used, fishermen will naturally set their gear on the most similarly productive spots. Kriging will detect this behavior and slightly correct if some stations were set on less productive spots. However the correction of bias cannot be comprehensive in the absence of full geographic information. Kriging Average and its estimation variance vs. Arithmetic Mean and variance of the mean are not equivalent estimators of the same mathematical beings. Within the geostatistics approach the variance of the arithmetic mean, for sampling units taken within a defined assessed area, is an estimator of the variance of this arithmetic mean if the contours of the area were preserved but shifted all over an infinite field having the same mean and spatial variance/covariance properties as the assessed area. The Kriging average is simply an estimator of the resource inside the assessed area, given the location of the stations and its estimation variance measures possible estimation errors resulting from the spatial variance/covariance structures detected inside the area. The calculations made on standing biomass versus fishable biomass simply show how misleading may be estimates obtained from preferential commercial or sport fishing on resource concentrations. The resource can be far more abundant outside of the concentrations than inside the concentrations. Without an independent non preferential sampling it is quite impossible to relate standing biomass to fishing biomass. Kriging can correct some bias due to samples preferentially located such as in the case of cpue's or fishing concentrations. Nevertheless inferences made from cpue's only can be quite misleading. In the present case of benthic snow crab numbers or biomass, there is no spatial periodicity in the data as shown by Conan and Maynard (1987) and no risk of bias by setting stations a period apart. In such cases, the precision of estimates can be improved by planning even sampling on a grid instead of random sampling. This contradicts general sampling design habits but is quite evident with the present data. When the number of stations increases the effect tends to disappear. This can most likely be generalized to other resources. Spatial periodicity in benthic marine living 110

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resources data is unlikely to exist and should at least be checked by using periodograms, before being referred to. In the present calculations ordinary kriging has been used without incorporating any detrending or anisotropy. The concepts of detrending and anisotropy are sometimes confused, particularly by some users of the thin plate splines developed by Duchon (1976). Thin plate splines use an unvarying local function h2Log|h| instead of a flexible semivariogram for modeling spatial variability. In kriging anisotropy means a deformation of the semivariogram when it is calculated in different directions. The patches would be elongated preferentially in a given direction. This has been tested for these data and the results are negative. The effect can be be easily corrected by a change of units and a rotation of axes. However the correction is not required as the effect does not exist for these data. As a matter of fact we never encountered anisotropy in any set of marine species benthic data. The effect is probably limited to some geologic formations observed in mining. A trend would occur when the variable mean would vary in a direction along the fisheries grounds, either in a simple linear way or as a function of order 2 or more. Geostatisticians would prefer to process a trend at the same time as the system of linear kriging equations (3), advocating that it is not acceptable to detrend prior to processing the data when in the presence of spatially correlated data. Elegant analytical developments have been made. However, practically 1) we have never encountered such trends for benthic living resource data, 2) using the analytical development would imply an independent knowledge of the semivariance in the absence of the trend, 3) analytic developments are limited to punctual estimates; estimates over an irregular polygonal surveyed area assorted with an estimation variance do not appear to be readily available. Presence/absence of an apparent trend may be simply a question of scale; for instance, a trend in altitude on the side of a hill disappears when the whole mountain range is surveyed. If the sampled area is adequately covered by a grid of sampling stations, there should be no need to worry about a trend. There is a variation of the kriging set of equations (3) allowing to incorporate a trend generated independently from the spatial coordinates as a function of an external variable. Depth is sometimes used as an external variable. We have discussed the effect of depth on the abundance of snow crab, and eliminated this option for ecological and physiological reasons. Further a plot of density in biomass or numbers at station versus depth does not show any correlation. 111

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Optimizing the route by searching the shortest way is readily achieved by simulated annealing. The algorithm solves a mathematical dilemma which could not be resolved by standard procedures. Simulated annealing is actually said to provide an approximate, but efficient solution. We have successfully tested this method for a number of stations exceeding 350 in the Southern Gulf of Saint Lawrence. We have simplified data import and export to/from the kriging mapping and assessing software by using simple CSV ASCII format that can be generated by any text editor or spreadsheet. The coast line, sampled variables, contour of the area to be estimated can be entered this way. The results can be exported to Nobeltec charts as part of integrated navigation software.

4.2 Limitations of the kriging method used The quality and adjustment techniques used for choosing and fitting a model to the semivariogram are sometimes criticized. However, in the process of interpolating and extrapolating, kriging attempts to fit a function to observed variations of the data. Other methods using inverse of the distance, cubic splines, thin plate splines (Duchon, 1976) use a fixed local function, in the latter case h2Log|h|. Using the experimental semivariogram for calculating average semivariograms in the set of linear equation 3 is discouraged by many geostatisticians. This would not lead to single solutions for the set of linear equations. We have not experienced this problem while processing our data. Theoretically, there are good reasons for using semivariogram models based on experimental semivariogram. As a condition, the model used should be of positive definiteness. Further, the standard models should of course conforms to the data of the experimental semivariogram. Unfortunately this is frequently not the case. Rather than trusting theory before nature, set of equations 3 may at times have to be adapted to fit some actual conditions of the data. The quality of the data should often be checked for sampling artifacts when the so called nugget effect is not negligible. We believe that a nugget effect is usually a sampling artifact rather than a structural feature. If it is assumed that the sampling is punctual, the quality of the data should also be checked for uniformity of the size of the sampled swept areas covered by each sampling unit. Large swept area will have regularized semivariances resulting in better confidence limits but poorer chart definitions. Regularization over very different swept areas will 112

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obscure semivariance patterns in a semivariogram. A difficult case results from a resource concentrated exclusively into hot spots of unknown geographic location. Standard kriging techniques will not allow to process efficiently such data. An adaptive sampling strategy may be required in first identifying location of the hot spots and then designing a sampling grid strategy for each hot spot. We encountered such spatial distributions while sampling benthic Chilean squat lobsters (Langostino), Pleuroncodes monodon and Cervimunida johni.

4.3 Other methods Discernment is required. The geostatistical methods we adapted, developed and used are by no way the unique tools for processing spatially distributed resource data. If the assumptions used are adequately tested and fit well the data to be processed, similar results can be obtained using methods based on very different assumptions but modeling well the data within existing limits. Such were the conclusions of an International Commission for the Exploration of the Sea the author co-chaired by the author (ICES. 1989).

4.4 What analytic developments would be useful Separating nugget effect and instrumental error from structural error in the kriging equations could be useful. Using a model allowing for instrumental error may enhance the quality of the confidence limits set on kriging average. Relative semivariograms often fit a fractal model. This is the only model offering a functional and possibly realistic, interpretation. However most geostatisticians do not accept relative semivariograms use in calculation of kriging average and its estimation variance. Cross validation should be adapted to objectively take into account the number of parameters being used when comparing fits by different kriging methods. However cross validation will never be a fully efficient technique; the functionality of kriging for extrapolating and interpolating beyond existing stations needs to be tested, rather than how well existing stations can inter predict each others' value.

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Acknowledgments / Disclaimer This research document was sponsored by the Fishing Industry, associations of fishermen harvesting snow crab in the Southern Gulf of Saint Lawrence, Canada. Most of the data is presented by permission. It was collected through collaborative agreements between Fishermen Associations and the Department of Fisheries and Oceans Canada (DFO) with DFO maintaining ownership of the data. No official endorsements are made about quality of the data, analyses or any direct implications concerning management of the resource. I wish to thank two anonymous referees. Their constructive input was addressed and additional information was incorporated in the text.

References Černý, V. 1985.Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm. Journal of Optimization Theory and Applications 45: 41–51. Comeau, M. and Conan, G.Y. 1992. Morphometry and gonad maturity of male snow crab, Chionoecetes opilio. Canadian Journal of Fisheries and Aquatic Sciences, 49: 2460–2468. Conan, G. Y. 1985. Assessment of shellfish stocks by geostatistical techniques. ICES CM 1985/K:30, 24 pp. Conan, G.Y. and Comeau, M. 1986. Functional maturity and terminal molt of male snow crab, Chionoecetes opilio. Canadian Journal of Fisheries and Aquatic Sciences, 43: 1710–1719. Conan, G.Y., M. Moriyasu, E. Wade and M. Comeau, 1988. Assessment and spatial distribution surveys of snow crab stocks by geostatistics. ICES C.M. 1988/K:10, 23 pp. Conan, G.Y., Comeau, M., Gosset, C., Robichaud, G. and Garaïcoechea, C. 1994. The Bigouden Nephrops trawl and the Devismes trawl, two otter trawls efficiently catching benthic stages of snow crab (Chionoecetes opilio) and American lobster (Homarus americanus). Canadian Technical Report of Fisheries and Aquatic Sciences, 1992 : 28 pp. Conan, G.Y. and Maynard, D.R. 1987. Estimates of snow crab (Chionoecetes opilio) abundance by underwater television a method for population studies on benthic fisheries resources. Journal of Applied. Ichthyology. 3: 158–165. David, M. 1977. Geostatistical ore reserve estimation. Elsevier. Amsterdam. 364 pp.

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