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Reducing Bias and Filling in Spatial Gaps in Fishery-Dependent Catch-per-Unit-Effort Data by GeostatisticalPrediction, II. Application to a Scallop FisheryJohn F. Waltera, John M. Hoenigb & Mary C. Christmanc
a National Oceanic and Atmospheric Administration–Fisheries, Southeast Fisheries ScienceCenter, 75 Virginia Beach Drive, Miami, Florida 33149, USAb Virginia Institute of Marine Science, College of William and Mary, Post Office Box 1364,Gloucester Point, Virginia 23062, USAc MCC Statistical Consulting LLC, 2219 Northwest 23rd Terrace, Gainesville, Florida 32605,USAPublished online: 30 Oct 2014.
To cite this article: John F. Walter, John M. Hoenig & Mary C. Christman (2014) Reducing Bias and Filling in Spatial Gaps inFishery-Dependent Catch-per-Unit-Effort Data by Geostatistical Prediction, II. Application to a Scallop Fishery, North AmericanJournal of Fisheries Management, 34:6, 1108-1118, DOI: 10.1080/02755947.2014.932866
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North American Journal of Fisheries Management 34:1108–1118, 2014C© American Fisheries Society 2014ISSN: 0275-5947 print / 1548-8675 onlineDOI: 10.1080/02755947.2014.932866
ARTICLE
Reducing Bias and Filling in Spatial Gaps in Fishery-Dependent Catch-per-Unit-Effort Data by GeostatisticalPrediction, II. Application to a Scallop Fishery
John F. Walter*National Oceanic and Atmospheric Administration–Fisheries, Southeast Fisheries Science Center,75 Virginia Beach Drive, Miami, Florida 33149, USA
John M. HoenigVirginia Institute of Marine Science, College of William and Mary, Post Office Box 1364,Gloucester Point, Virginia 23062, USA
Mary C. ChristmanMCC Statistical Consulting LLC, 2219 Northwest 23rd Terrace, Gainesville, Florida 32605, USA
AbstractFishery-dependent catch per unit effort (CPUE) comprises critical input for many stock assessments. Construction
of CPUE indices usually employs some method of data standardization. However, conventional methods based onlinear models do not effectively deal with the fact that samples are collected with a selection bias or with the problemof filling spatial gaps. Geostatistical interpolation methods can ameliorate some of the biases caused by both of theseproblems while remaining complementary to traditional linear model-based CPUE standardization. In this paperwe present geostatistical estimates of sea scallop Placopecten magellanicus CPUE from tows recorded by onboardobservers during an opening of Georges Bank Closed Area II in 1999. By selecting tows for which there was little prioreffort (on the basis of accumulated effort measured by vessel monitoring systems), we obtained tows that reflectedinitial abundance as closely as possible. These tows were used to obtain a variogram which was used in geostatisticalprediction of sea scallop CPUE. The kriged mean was substantially lower than the arithmetic sample mean, indicatingthat a geostatistical approach reduced the influence of repeated sampling in locations of extremely high CPUE andincreased the weight of isolated observations in areas of low CPUE. The results produced a map that was qualitativelysimilar to that obtained from a preseason fishery-independent survey. Overall differences between the two approacheswere driven by the extension of predictions into areas at the edges of spatial autocorrelation where kriging predictionsapproached the grand mean of the data set.
In a companion paper, Walter et al. (2014, this issue) usedsimulation to explore how geostatistical methods address theproblems inherent in interpreting catch-per-unit-effort (CPUE)data consisting of size-biased and clustered observations andthe need to interpolate between or extrapolate into areas lackingsamples (Walters 2003). The simulation results demonstrate thatthe spatial weighting properties of the geostatistical mean reducethe influence of sample clustering and the targeting of higher-abundance areas on estimated CPUE. Geostatistics provides an
*Corresponding author: [email protected] July 26, 2013; accepted June 2, 2014
autocorrelation-driven method for spatial interpolation, and theassociated geostatistical prediction errors provide a means of de-termining the prediction uncertainty according to the amount ofadjacent spatial information. In conjunction with conventionallinear model–based standardization methods, which account forfactors that affect catchability such as gear and season (Maunderand Punt 2004), geostatistical methods harness spatial autocor-relation for prediction and have the ability to ameliorate thebiases created by unequal sample selection probabilities.
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The widespread utilization of fishery observers and vesselmonitoring systems has created interest in using such data forstock assessment purposes (Deng et al. 2005; Mills et al. 2007;Palmer and Wigley 2009). Data from onboard observers con-stitute critical information for estimating bycatch and discardsin many fisheries, yet substantial logistical and theoretical is-sues exist with respect to the use of these data. First, one haslittle ability to control the spatial and temporal allocation ofobservations. Observed samples tend to be clustered, as theyare repeated observations from the same vessels in similar lo-cations, and they may not cover the entire spatial and tempo-ral extent of a fishery. Vessel monitoring systems can addressthe problem of repeated observations in the same location byplacing an observed sample in the context of the amount ofprior effort that has occurred at that location (Gedamke et al.2004).
Incomplete observer coverage and limited sampling re-sources invariably lead to gaps in sample coverage. These aremost problematic when the spatial gaps are likely to havenonignorable sampling bias (Little and Rubin 2002). Onemethod of overcoming such selection bias is to use spatialautocorrelation—the tendency of objects to be more alike atsmall distances than at large ones—as a spatial bridge to fill ingaps and, to the extent that there is spatial autocorrelation, toextrapolate to unobserved locations.
In this paper we address the question whether geostatisticalmethods allow us to use observer data alone to obtain the sameinsights into the spatial distribution and relative abundance ofsea scallops Placopecten magellanicus as were derived from anintensive fishery-independent survey. We use vessel monitoringsystem (VMS) effort data to address the issue of prior depletionby identifying a set of initial tows that occurred with very littleprior fishing effort from a larger set of 2,755 scallop dredgetows observed in a 1999 fishery on Georges Bank Closed AreaII (CAII) off Massachusetts into. From these tows we obtained avariogram and used it to predict relative abundance at the start ofthe fishery. We then evaluated the performance of geostatisticalpredictions by comparing variograms and predicted CPUE withestimates obtained from an intensive survey conducted 1 yearprior to the opening of the fishery.
METHODSGeneral geostatistical methods.—Geostatistics is a model-
based method of predicting the values of a spatial process froma set of samples (Webster and Oliver 2001). As with any model-based estimation method, the first step is to obtain a model ofthe process which, in this case, is a measure of spatial autocor-relation, the variogram. The second step is prediction, where themodel is used as a means of weighting samples to predict val-ues in unsampled locations. Geostatistics has found widespreadapplication in fisheries, particularly as an alternative to design-based estimators and for interpolating and mapping (Warren1998; Rivoirard et al. 2000; Petitgas 2001). Details regarding
the geostatistical modeling process used here can be found in acompanion paper (Walter et al. 2014).
Fishery and data description.—The North Atlantic sea scal-lop fishery occurs on the continental shelf from Cape Hatterasnorth to Canada. A major fishery occurs on Georges Bank, whereseveral large areas were closed in 1994 to protect groundfish(Figure 1). During these closures, sea scallop biomass accumu-lated to high levels (Murawski et al. 2000), permitting a limitedopening of Closed Area II to scalloping from June throughNovember 1999.
During the opening, observers hired by the National MarineFisheries Service’s Northeast Fisheries Science Center recordedthe catch per tow of sea scallops and bycatch species as wellas tow location and duration, sea condition, and vessel and gearcharacteristics for 2,755 tows from 35 vessels on a total of 40trips. On all vessels, New Bedford style scallop dredges wereused, with most vessels towing similar configurations consistingof paired 15-foot dredges with frame heights varying from 12 to21 in, chain bags with inside ring diameters of 3.5 in, and twinetops with 10-in meshes.
To obtain usable catch rates, we eliminated tows when itcould not be determined whether the catch came from the portdredge, the starboard dredge, or both and when the tows wereof poor quality (such as those involving flipped, tangled, or lostdredges). Tows with durations greater than 2 h or tow distancesgreater than 9 nautical miles (based on the distance betweenthe start and stop positions) were also eliminated, as these towscould not be rectified to spatial locations and may have entailedthe dredge being used as an anchor while the vessel processed thecatch. Catch was recorded in either whole weight or meat weightof kept and discarded scallops or as total bushel baskets of keptor discarded scallops. Tows in which these two measures couldnot be reconciled (the weight landed should be a multiple of thenumber of bushels) were removed. Two additional tows wereremoved as the scallop weights were misreported as bushels,resulting in an impossibly high bushel count.
In 1999, vessel speeds were not available in the observerdatabase, so we assumed an average vessel speed of 5 knots(Gedamke et al. 2004). To georeference each tow we used themidpoint of the start and stop positions, and to obtain a towdistance we used the product of the tow duration and averagevessel speed. We converted latitude and longitude to an equidis-tant scale in nautical miles with a Universal Transverse Mercatorprojection in North American Datum 83, Zone 19.
We used bushels (1 bushel = 35.239 L) of kept and discardedwhole sea scallops per nautical mile to measure abundance. Inthe few (<5%) cases in which only one dredge was observed, wedoubled the catch to approximate the catch had two tows beenobserved based upon the significant correlation between catch inport and starboard dredges observed in the fishery-independentsurvey described below (r2 = 0.9734, P ≤ 0.001, df = 528). Wedid not correct for selectivity because the mean shell heightswere greater than 4.33 in in 95% of the tows, well above theshell height (3.3 in) of 100% selectivity for commercial dredge
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FIGURE 1. Sea scallop area closures off the East Coast of North America. The study area is the cross-hatched section of Closed Area II, which was opened forfishing in 1999. The contour line is the 50-fathom curve.
gear with 3.5-in rings used in the 1999 fishery (Serchuk andSmolowitz 1980).
Processing VMS data.—The U.S. North Atlantic scallop fish-ery has a universal VMS that records and transmits vessel posi-tions to satellite receivers. We obtained 90,944 individual VMSrecords representing the total fishing effort from June 16 throughNovember 13, 1999, in the exemption area of CAII (Figure 1).Each record consisted of the latitude and longitude, the time anddate, and the time differential between the record and the previ-ously recorded position for the vessel. The average straight-linevelocity of the vessel (representing a minimum velocity becausethe path may not have been straight) can be inferred from thedistance between the current and previous positions divided bythe elapsed time. We assumed that vessels with a velocity lessthan 5.5 knots were fishing and used this time at speed to re-flect fishing effort (Gedamke et al. 2004). For most records theelapsed time was close to 1 h, as the VMS systems were pro-grammed to report vessel position every hour. In a small fraction
(0.8%) of records the elapsed times were greater than 3 h, andwe removed these anomalous records because it was impossibleto spatially assign effort to a particular location.
Geostatistical prediction of fishing effort.—It is critical thatCPUE indices reflect abundance at a recognizable time withrespect to the processes of biomass addition and depletion. Withfishery-dependent data there is little ability to control the timingof data collection with respect to removals. VMS systems canbe used to assign a prior level of fishing effort to each observedcatch. If there are enough catches at a common level of effort,they can be used to generate a CPUE index with a similar—andknown—level of prior effort.
For each observed catch rate we obtained two distinct geo-statistical predictions of fishing effort from the VMS data.The first represents prior fishing effort at the location of thetow. The second represents the total fishing effort that occurredover the course of the fishery at the location of the tow. Becausethe trajectories of the tows were not known exactly, it was not
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GEOSTATISTICAL PREDICTION II 1111
possible to determine precisely how much effort had occurredpreviously over the exact path of a tow. Note that the goal ofthis method was to provide a measure of relative effort at the lo-cation of an observed tow rather than to determine the absolutearea swept and bottom contact time, as was the goal of Millset al. (2007).
To obtain the predicted prior and total fishing effort, wesummed VMS fishing effort by assigning each record to thenearest node on a 1-nautical mile × 1-nautical mile grid cov-ering CAII. Grid nodes with no effort received a value of zero.This provided summed fishing effort on a spatial grid. Thesepoint estimates of fishing effort were then used to obtain a var-iogram so that effort could be predicted anywhere within thefished area.
For each observed tow all prior effort over the entire fisherywas accumulated over the spatial grid. A kriged prediction offishing effort was then obtained at the midpoint of the tow toprovide a smoothed estimate of the prior effort that had occurredin its general proximity. To determine how much effort wouldhave subsequently occurred around the location of the tow, alleffort was summed over the season and the process repeated.Hence, for each observed tow, it was possible to determine howmuch effort had occurred prior to that tow and how much effortwas going to occur after it. For example, if a tow had a priorfishing time of 5 h and a total fishing time of 100 h, this towoccurred within the first 5% of the total effort that would occurat that location.
Obtaining initial tows.—In this relatively short fishery open-ing, during which depletion was expected to be substantial, wewanted to obtain catch rates that reflected abundance prior to thefishery opening as closely as possible. Given the extremely highlevel of observer coverage (30% of vessels) in the opening ofCAII, we felt that this could be accomplished by selecting towsthat had occurred when there was very little prior accumulatedeffort. We used both a relative-effort criterion (which selectedtows with less than 5% of the accumulated effort at the locationof the tow) and an absolute-effort criterion (which selected towswith less than 12.5 h of accumulated effort). The absolute-effortcriterion was necessary because the second tow in a particularlocation with little effort would have been excluded accordingto the 5% criterion, even though there had been very little totalprior effort. The absolute criterion of 12.5 h represented the 5thpercentile of effort accumulated over the entire fishing area aswell as a low level of total effort covering an area ≤9.8% of thearea over which VMS data were accumulated (a circle with aradius of 1 nautical mile).
Meeting assumptions and estimation of sea scallop CPUEvariograms.—Geostatistical analysis assumes stationarity, i.e.,that the spatial process has a constant expected mean and vari-ance over the entire region (Cressie 1993). An additional as-sumption is that of isotropy, or that the correlation betweenobservations at locations is the same in all directions (Websterand Oliver 2001). We examined the data sets for evidence ofnonstationarity (trend) by plotting the data values against the
north–south and east–west directions and found little evidenceof nonstationarity. We evaluated geometric anisotropy throughplots of the autocorrelation in various angles of rotation and var-ious axis scaling ratios and concluded that corrective measureswere unwarranted.
After selection of the fishery tows we obtained an empiri-cal variogram of sea scallop catch per nautical mile using therobust method of Cressie and Hawkins (1980). To fit theoreti-cal variograms we tested spherical, exponential, and Gaussianmodel forms, but the spherical model fit best and is the only onepresented:
γ(h) = C0 + C
2
[3h
a−
(h
a
)3]
for h < a,
γ(h) = C0 + C for h ≥ a, (1)
where a is the range of autocorrelation in the data, C0 is thenugget, C is the partial sill (C0 + C = the total sill), h isthe distance between the points, and γ(h) is the value of thesemivariogram at a given distance.
To fit each of these models to the empirical semivariogram,we employed a weighted least-squares function (Cressie 1993):
K∑J=1
|N (h( j))|{
γ(h( j))
γ(h( j); )− 1
}2
, (4)
where K is the number of variogram distance bins or lags,|N (h( j))| is the number of pairs of points in each lag, θ isthe set of variogram parameters (nugget, partial sill, and range),γ(h( j)) is the modeled semivariance value at distance bin h(j),and γ(h( j)) is the observed semivariance value at distance binh(j)). Fits were obtained in SPLUS 2000 (Mathsoft, Inc.) usingnonlinear bounded minimization. Spherical theoretical modelswere selected on the basis of the smallest objective function forboth the fishery and survey data sets (described below; Table 1;Figure 2).
Comparison with fishery-independent survey.—Kriged pre-dictions of sea scallop CPUE were obtained with the best-fittingvariogram model and predicted on a 30-nautical mile × 30-nautical mile grid over the exemption area of CAII. We comparethese estimates with kriged estimates from a survey conductedover 6 weeks in August–October 1998, prior to the fishery open-ing. This survey was a collaborative effort between the scallopindustry and the scientific community and consisted of 547 ob-servations from random and systematic grid locations. Theseobservations were collected on six commercial scallop vesselsusing gear similar to that used in the 1999 fishery except thatthe tows were 10-min straight-line tows at 5 knots.
For comparison with the fishery data in 1999, we convertedthe numbers of sea scallops in 1998 to the numbers in 1999by incrementing them for growth and decrementing themfor mortality. Scallops were assigned to 0.196-in shell-height
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TABLE 1. Objective function value, estimated range, sill and nugget, kriged mean CPUE, and pointwise correlations for three variogram models for the fisherydata set.
Variable Spherical variogram Exponential variogram Gaussian variogram
Objective function value 2,317.13 2,888.44 2,615.4Estimated range (nautical mile) 9.176 3.331 4.465Sill 6.93 8.30 5.93Nugget 3.28 2.14 4.29Kriged mean (bushels per nautical mile) 4.78 4.786 4.812Pointwise correlation with spherical model predictions 1 0.980 0.982
bins, and the numbers were adjusted for size-selectivity usingthe selectivity curve of DuPaul et al. (1989). The size at themidpoint of each bin was converted to age using a von Berta-lanffy growth model obtained from National Marine FisheriesService surveys of Georges Bank in 1998, namely, shell height(in) = 6.378 · {1 − exp(−0.3374 · [age − 1)]}. The age wasadvanced 1 year and a new shell height after 1 year of growthwas obtained. The numbers of scallops were decremented for1 year of natural mortality ( = 0.1/year; Merrill and Posgay1964) and then expanded using the selectivity curve to reflectthe expected catch of scallops at length in 1999. We convertedthese numbers into bushels per tow using a regression obtainedfrom the 1998 survey of the number of scallops of a given
shell height per bushel (number of scallops per bushel =9214.3·(shell height)−2.803; r2 = 0.77, df = 1,034). Hereafterthese converted tow observations will be referred to as thesurvey tows. Identical conversions have been used with similardata (Gedamke et al. 2004) and to project future abundance forsetting quotas (NEFSC 2007).
Prediction area versus the degree of extrapolation—As de-scribed in the companion simulation paper (Walter et al. 2014),the kriging variance provides a means of determining the de-gree to which a prediction is a product of adjacent spatial in-formation or extrapolation of distant data points. This degree ofextrapolation (equation 9 in Walter et al. 2014) is a function ofthe strength of autocorrelation and the number and proximity
FIGURE 2. Empirical and fitted variograms for the 1998 survey data and the fishery catch rates; nm = nautical miles.
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FIGURE 3. (a) Kriged sea scallop abundance (bushels per nautical mile) for the fishery data set (contours of relative abundance are shown); (b) kriged surveyabundance; (c) fishery minus survey predictions at all prediction points; and (d) total fishing effort as obtained from VMS records (shading) and 2,755 fishery tows(points). The symbols in panels (a), (b) and (d) represent individual dredge tows.
of adjacent samples and varies from 0% to 100%. We plottedcontours of the degree of extrapolation for the fishery CPUE.Extrapolation is important because as the degree of extrapola-tion approaches 100% the kriged prediction converges on themean of the observed data. Thus, in the absence of data withinthe range of autocorrelation, kriging predicts the sample meanabundance in areas beyond the range of autocorrelation.
Some areas of high effort were not observed until substantialeffort had occurred. Due to our desire to assign observed towsto a common level of prior effort, some important fishing areaswere left with no observations. Unlike gaps in coverage createdby the absence of catch and effort, these are problematic becauseeffort did occur but the observed CPUE may only reflect abun-dance after substantial depletion. The kriging variance coupledwith VMS data provides a means of identifying these areas byplotting contours of the sum of the relative effort (on a scale of0% to100%) and the degree of extrapolation. The values rangebetween 0 and 200 but are useful in a relative sense to identifyareas that stand out as locations of critical uncertainty. Theyare uncertain because of the absence of adjacent information(an uncertainty shared with unfished areas) but are critical tointerpreting overall catch rates because they are heavily fished.
RESULTS
Obtaining Initial TowsOf the 2,755 total tows, 289 occurred with ≤5% of the total
fishing time at that location. The union of the two criteria (ei-ther less than 5% locally or less than 12.5 h total) gave 1,076observations, which we will refer to as the fishery tows in theremainder of this paper. It is important to note that these towsdid not necessarily correlate with the timing of the fishery, i.e.,an initial tow at a location could occur at the end of the seasonif it took that long until the area was visited by a vessel. Towlocations were recorded only to the nearest minute of latitudeand longitude, so some tows shared a midpoint position. In thesecases we used only the first (in time) tow at a location. Becauseour criteria for determining initial tows required approximationand were prone to bias from some level of prior removal, weexplored the sensitivity of the results to various cutoff levels ofprior effort. These tows covered much of the survey area, thoughselection according to prior effort left some gaps in observedsamples in areas where there was high fishing effort (Figure 3a).Also, no tows were observed in the far western region, wherelittle to no fishing effort occurred.
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TABLE 2. Kriged means, prediction errors, and arithmetic means and standard errors for the survey and fishery data.
Data Kriged mean Kriging prediction error Arithmetic mean Arithmetic standard error
Fishery 4.78 2.40 5.95 4.07Survey 5.11 2.28 5.15 5.53
Variogram FitsThe variogram of the total effort captured the spatial distribu-
tion of fishing cumulated over the entire season. An exponentialvariogram model was chosen over a spherical model on the ba-sis of objective fitting criteria described in greater detail below.The parameter estimates for the fitted model comprised a sillof 14,150,000, a nugget of 470,000, and a range of 4 nauticalmiles.
Spherical variogram models provided the best fit to the bothfishery and survey data sets. However, the parameters differedfor the two data sets, notably in the magnitude of the sill and therange of autocorrelation. The fishery CPUE had an estimatedrange of 9.147 nautical miles, shorter than the estimated surveyrange (11.453 nautical miles) and a total sill (partial sill +nugget) approximately 55% of the survey variogram total sill(Figure 2).
Geostatistical PredictionWe obtain kriged abundances from the survey data on the
same 30 × 30 prediction grid as for fishery tows (Figure 3b).Geostatistical predictions of sea scallop CPUE measured inbushels per nautical mile for the entire study area for the fisheryand survey data differed by only 6.5% (4.78 and 5.11, respec-tively; Table 2). However, the arithmetic mean values for thetwo data sets showed a divergence of 15.5%, indicating that thespatial weighting of kriging for the fishery data gave an estimatemore similar to that of the survey.
To compare the two spatial maps, we subtracted kriged frompredicted values on each grid node and plotted the differences(Figure 3c). The survey data provided more extensive cover-age of the entire region and showed well-defined areas of highCPUE in the central area and on the eastern edge. The areasof highest discordance were at the far eastern edge, where thefishery data had gaps in observations. Though observer cover-age eventually extended to almost the entirety of the fished area,there was substantial prior effort for the tows in some locationseven within the 5% cutoff. In particular, for the tows in two areasof extremely high effort (dark shaded areas on the far right andat the top center of Figure 3d) between 156 and 164 h of effortoccurred before a tow had been observed. Thus, some observa-tions would have been influenced by prior depletion and maynot have reflected initial abundance. The effects of this can beseen in the lower predicted abundances from the fishery data inthe far right and the upper area (Figure 3c). In contrast, for thefar western unfished region, the fishery predictions were higherthan the survey values, as prediction extended beyond the range
of autocorrelation where the geostatistical predictions revertedto the arithmetic mean.
Sensitivity of the Kriged Means to Effort CutoffBecause the effort level cutoff was used to approximate the
unfished conditions, we explored the sensitivity of the krigedand arithmetic means to varying the percentage of prior effortallowed on the location of a tow (Figure 4). Increasing the levelof prior effort reduced both the kriged and arithmetic means, asgreater effort brought in observations with greater levels of de-pletion. Nevertheless, the kriged means were consistently lowerthan the arithmetic means, reflecting the differential weightingof sample observations.
Sensitivity of the Kriged Means to VariogramFunctional Form
Since the empirical variogram was estimated from data witherror, there is no certainty that the best-fitting theoretical modeltruly represents the spatial autocorrelation of the process. Forthis reason we obtained geostatistical abundance estimates witheach of three fitted theoretical models and found minor differ-ences in the overall predicted mean for fishery CPUE (Table 1),indicating that the functional form of the variogram had littlesubstantive impact upon predicted CPUE.
Prediction Area versus the Degree of ExtrapolationAs observed in the plot of VMS coverage, the fishery did
not fish in the far western area of CAII and consequently no
FIGURE 4. Sensitivity analysis of kriged (K) and arithmetic mean (A) CPUEas a function of the percentage of total prior VMS fishing effort allowed.
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GEOSTATISTICAL PREDICTION II 1115
FIGURE 5. Contours of percent extrapolation, defined as 100 times the kriging variance divided by the sum of the variogram nugget and sill. Locations are the1,076 initial fishery tows.
tows were observed from this area (Figure 3d). The contoursof the degree of extrapolation for the fishery data (Figure 5)indicate that areas in the far upper left were beyond the rangeof autocorrelation. Predictions at these locations reverted to thearithmetic mean and represented 100% extrapolation of catchrates in the observed locations. Where the degree of extrap-olation was relatively low (<50%), the mean kriged fisheryand mean kriged survey values were correlated (Figure 6). Be-yond this level of extrapolation, however, the kriged surveyvalues declined as predictions largely extended into the un-fished areas, whereas the kriged fishery predictions increasinglybecame a product of extrapolation and converged on the arith-metic sample mean. Given that the biomass of sea scallops wasnot evenly distributed, 86% and 96%, respectively, of the abun-dance estimated from the survey occurred within areas boundedby the 35% and 50% contours in Figure 5, indicating that al-most all of the available biomass could be contained withinan area where little extrapolation of the fishery data would benecessary.
The contours of the sum of the degree of extrapolation andrelative fishing effort (Figure 7) identify two locations of highestcritical uncertainty, one in the top center and the other at the farright. These areas had extremely high effort (Figure 3d) but few
fishery observations without substantial prior effort, such thatthey could not have reasonably been attributed to initial unfishedconditions.
FIGURE 6. Comparison of the kriged means obtained from the fishery andsurvey data sets as they are increasingly a product of extrapolation. Predictionsare averaged for bins partitioned according to the percent extrapolation for thefishery predictions.
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1116 WALTER ET AL.
FIGURE 7. Contours of the degree of extrapolation plus the percent of total VMS effort, which identify areas of critical uncertainty due to high effort and a highlevel of extrapolation.
DISCUSSIONWe obtained several important results regarding the perfor-
mance of geostatistical methods for obtaining an index of abun-dance using fishery data. First, we found that the sea scallopobserver data provided a variogram that was different from thesurvey data due to the lower sill but similar in the range of auto-correlation, thus providing a variogram that generally reflectedthe most critical component of spatial autocorrelation neces-sary for geostatistical modeling. Next, the geostatistical esti-mator produced a much lower mean abundance than the arith-metic sample mean, indicating that geostatistical methodologyreduced the influence of the clustered and repeated observationscommon in fishery-dependent data. Furthermore, while the geo-statistical mean abundance from the fishery data was lower thanthat from the survey data, it closely tracked abundance in areaswith low levels of data extrapolation. It is rare to use fishery-dependent CPUE to estimate absolute abundance; it is muchmore common to use it as a relative abundance measure andthen to estimate fishery catchability within a stock assessmentmodel. In these situations a geostatistical treatment of fishery-dependent catch rates is likely to better track abundance thanCPUE derived from a generalized linear model (in essence anarithmetic mean).
These results mirror simulation results (Walter et al. 2014)that suggest that the geostatistical mean will be less biased thana sample mean whereby all data points receive equal weightunder conditions of sample selection similar to those of fishery-dependent catch rate data. Furthermore, when changes in thearea fished over time represent a substantial problem (Walters2003; Carruthers et al. 2011), the ability to fill in spatial gaps willprovide a more robust treatment of CPUE data. Geostatisticaltreatment of fishery data also readily facilitated mapping, which,in this case, was quite similar to the survey map of sea scallopabundance. Lastly, the methodology of partitioning observedcatch rates using VMS data elucidated some of the possibili-ties and limitations of obtaining relative abundance informationfrom fishery-dependent data that are valuable considerations forfuture applications.
As with any model-based predictive method, geostatisticsrequires obtaining a model for prediction, in this case the vari-ogram. While geostatistical prediction does not require design-based or random sampling, obtaining the variogram requires thatsamples be at least representative of the spatial process of in-terest (Webster and Oliver 2001), which fishery-dependent datamay not ensure (Paloheimo and Dickie 1964). In this applica-tion, the observer data provided a variogram with a similar range
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GEOSTATISTICAL PREDICTION II 1117
but a reduced sill. The range is the most critical variogram pa-rameter for kriging because it affects the absolute distance overwhich prediction can be made and it has the greatest potentialto alter the predicted abundance.
In contrast, the sill and nugget parameters do not influencethe geostatistical mean other than through their effect upon therange when all three parameters are jointly estimated. The silland nugget scale the absolute value of the kriging predictionvariance. As the kriging variance is only used in a relative man-ner to identify areas of uncertainty, this relative scaling has neg-ligible impact upon the kriged means. The reduced sill for thefishery data matched the simulation results that predict lowersills in the biased data due to underestimation of variability(Silliman and Berkowitz 2000). Most tows occurred in areas ofhigher abundance, reducing the frequency of low tows. Also,commercial tow lengths (8 km, but rarely straight-line) werelonger than the survey (1.8 km straight-line), so they integratethe patchy sea scallop distribution over a greater area.
The variogram functional form with the best fit was a spher-ical model for both the survey and the fishery. Kriged meanswere relatively insensitive to variogram functional form, largelybecause the different models fitted to the same data providedsimilar spatial patterns with only slight differences at short spa-tial scales. The potential for substantial bias occurs at longerspatial scales approaching or exceeding the true range of auto-correlation. For each of the three variogram models, it appearsthe sea scallop observer data provided ranges of autocorrelationsimilar to the survey, even if the sill was underestimated.
If we assume that the 1998 survey extrapolated to 1999 abun-dances represented our best estimate of true relative abundance(or at least the relative abundance upon which the 1999 quotawas set), then the geostatistical estimator provided an overallCPUE only 6% lower than the survey. In contrast, the arith-metic sample mean of the fishery data was 15.5% higher thanthe arithmetic mean of the survey data, indicating that the com-bination of kriging and using only the first observation at alocation reduced the influence of clustered observations in lo-cations of high abundance. Furthermore, the map of relativeabundance qualitatively resembled the survey map. In the hypo-thetical situation in which there was no survey, the geostatisticalprediction obtained with observations partitioned by prior effortrepresents a substantially improved treatment of the data versusalternatives which might assume independent observations.
Nevertheless, some substantial differences remain betweenthe two geostatistical predictions involving extrapolation of thearithmetic mean into unfished areas beyond the range of auto-correlation and lower predicted fishery abundances in a heavilyfished central region. At the far western edges of the open areaof CAII there was very little effort, presumably because therewere few sea scallops there, an observation corroborated by thesurvey (Figure 3b). In these areas predictions were increasinglybased on extrapolation—to the point that at the range of autocor-relation they reverted to the arithmetic mean, which was muchhigher than the abundance in this area based on the 1998 survey.
This reversion to the mean is generally a convenient property ofkriging in most situations; in this case, however, the mean of thefished areas is not representative of much of the unfished areas.
Conversely, in the central areas of extremely high fishing ef-fort, the fishery data predicted lower abundance in the centralridge than the survey. This result was incongruous with the ex-pected results but was due to two factors: allowing some priordepletion to occur and the gaps in sample coverage in areas ofhigh sea scallop densities, fishing effort, and exploitation rates.Particularly in the high-abundance area on the far eastern edge,few observed samples were included in the set of initial catchesand low abundance was predicted on the basis of adjacent sam-ples. Though observer coverage eventually extended to some ofthese areas of high effort, there were few observed tows before60–80% of the effort had been expended, so these observationslikely suffered from substantial prior depletion and would nothave reflected initial abundance.
In practice, it may be unlikely that geostatistics (or any othermodel-based methodology) will fill in very large gaps in samplecoverage. It is in this situation that the geostatistical predictionvariance becomes valuable in identifying the areas of greatestuncertainty. Though less critical for assessing fishery removalsbecause of low effort, areas where the extrapolation level washigh (the far western edges of Figure 7) are important to includefor developing time series of CPUE (i.e., the spatial gap-fillingrequirement of Walters [2003]). For bycatch monitoring, catchrates obtained under the assumption that the observed areasrepresent unobserved locations could be misleading (Lewisonand Crowder 2003) when substantial fishing effort occurs inunobserved locations. In a bycatch monitoring situation, identi-fication of the areas of “critical uncertainty” (Figure 7), whereboth the degree of extrapolation and the level of fishing effortare high, may be critical for evaluating the adequacy of bycatchestimates and observer sample coverage.
While there are numerous methods of standardizing fishery-dependent catch rate data to account for lack of control of thesampling process (Maunder and Punt 2004), most rely uponthe assumptions of linear modeling theory, i.e., that samplesare independent, identically distributed, and collected withouta selection bias. Geostatistics provides a means of spatiallyweighting observations, thus partially reducing the influenceof selection bias and providing a means of extrapolation intounsampled areas, at least to the extent of the range of autocorre-lation. Furthermore, geostatistical prediction can be combinedwith traditional CPUE treatments based upon linear models bystandardizing to account for factors such as vessel type, gearconfiguration, and seasonal effects prior to modeling the var-iogram and making geostatistical predictions (Petitgas 2001).Furthermore, geostatistics provides a map of relative abundancethat is useful in understanding the dynamics of fishery catchand bycatch (Kulka et al. 1996) and allows extrapolation of in-formation into unsampled locations, a desirable property whenfishery-dependent data leave gaps in sample coverage. Even forfishery-independent survey data, a similar type of data-informed
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extrapolation may be useful when there are gaps in sample cov-erage (NEFSC 2007).
ACKNOWLEDGMENTSWe thank William DuPaul, Courtney Harris, Roger Mann,
Paul Rago, and Doug Vaughan for reviewing this manuscript.This research was conducted by J.F.W. while he held a NOAA–Sea Grant Joint Graduate Fellowship in population dynamics.This is VIMS contribution 3399.
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