Variation in the spatial distribution of fish length: a multi-annual geostatistics approach on anchovy in Biscay

Transcription

1 Internationa Counci for the Exporation of the Sea ICES / CIEM ICES CM 2003/Q:15 Variation in the spatia distribution of fish ength: a muti-annua geostatistics approach on anchoy in Biscay Pierre Petitgas, Jacques Massé, Patrick Greier and Pierre Beiois. Fish ength is an important ecoogica parameter as its muti-annua spatia pattern is reated to habitat occupation. Fish ength is aso an important eauation parameter in acoustic fisheries sureys as the acoustic response is a function of it. The paper describes the inter-annua ariation in the maps of fish ength and a way to perform mapping coherenty between years using geostatistics. The data are made of the mid-water peagic traw haus performed during IFREMER's acoustic sureys in Biscay, The data show a consistent spatia distribution of anchoy ength across the years with bottom depth being a correated ariabe. The regression of fish ength on bottom depth is incorporated into the kriging system so as to constrain the estimator to reproduce each year this consistent reation. Residuas from the regression contain the annua ariations. Their spatia and muti-year ariogram is modeed and annua kriging is performed eading to coherent inter-annua maps each haing its specificity. Inter-annua ariation in fish ength distribution was more important in the nursery grounds in front of Gironde estuary than in other areas. P. Petitgas, J. Massé, P. Greier and P. Beiois : IFREMER, BP 21105, F cedex 03, Nantes, France [te : , fax: , e-mai : Pierre.Petitgas@ifremer.fr, Jacques.Masse@ifremer.fr, Patrick.Greier@ifremer.fr, Pierre.Beiois@ifremer.fr]

2 Introduction Mapping fish ength is an important step in ecoogica studies for defining fish habitats and their roe in popuation dynamics. It is aso essentia in direct biomass eauation using acoustic sureys as the acoustic response is a function of fish ength. During acoustic sureys, a imited number of traw haus are undertaken conditionay to the obsered echo-traces, resuting in scattering the haus uneeny in the area. As a consequence, there are few haus (Tabe 1) to make a map of ength with. To oercome this difficuty, we used a coariate (e.g., bottom depth) to guide the mapping. This foowed the genera idea (Guisan and Zimmerman, 2000) that habitat can be predicted using a regression mode of abundance on enironmenta coariates. Mapping was performed by using the method of Externa Drift Kriging (e.g., Gai and Meunier, 1987; Chies and Definer, 1999; Rioirard and Wieand, 2001). Inter-year ariation was eidenced by comparing differences between maps with estimation error. Materia and Methods Data. The data come from the mid-water traw haus of acoustic sureys. Acoustic sureys are performed aong equay spaced transects traersing the entire shef aong the French coast of Biscay. For the purpose of identification of the echo-traces and estimating fish ength and age, mid-water traw haus are undertaken conditionay to the obsered echo-traces, in an opportunistic manner. For each hau, the catch is sorted and weighted by species. A random sampe is taken from the sorted catch of each species. Length of indiidua anchoy is measured with haf cm precision. Frequency of fish ength in the haf cm interas is then estimated for the entire anchoy catch in the hau. Mean anchoy ength of the catch is then deried. The data consist of the mean anchoy ength in the traw haus of the IFREMER acoustic sureys, and comprise 295 traw haus oer 16 sureys (Fig. 1). Basic statistics are in Tabe 1. The aerage number of haus per surey is had the argest anchoy obsered and 1990 the smaest. The statistica distribution of anchoy ength is cose to a gaussian (Fig. 2). Reation between ength and bottom depth. A inear reation between anchoy ength and bottom depth was eidenced and a inear regression was fitted by east squares (Fig. 3). Variography. The ariogram was estimated on the residuas of the regression. A spatia ariogram was estimated. It was the ariogram computed for pairs of points seperated by ector distance h and beonging to same year. A space-time ariogram was aso estimated. It was the ariogram computed for pairs of points seperated by ector distance h but beonging to different years. This proided a 3D ariogram mode to perform kriging in each year using data points of a years. Externa drift kriging. The inear reation between anchoy ength and bottom depth was used to impement externa drift kriging. The method amounts to adding constraints on the kriging weights aowing for the kriging estimator to conform to the shape of the spatia distribution proided by the "externa" coariate (i.e., the inear increase of the ength with increasing bottom depth). The ariographic structure used in the kriging is that of the residuas of the inear regression. Interpoation grid. The kriging interpoation grid was a 7 n.m. x 7 n.m. square grid defined within the poygon (Fig. 1) containing a surey data: from coast to shef break in ongitude, and from Gouf de Cap Breton to the ise of Yeu in atitude. The Externa drift method required that bottom depth be known at a sampe points as we as at a points of the interpoation grid. Bottom depth on the interpoation grid, was deried from a numerica mode of sea bottom depth (scae 5km x 5km), used in the hydrodynamic mode of IFREMER for Biscay (Lazure and Jegou, 1998). In each ce of the interpoation grid, we computed the aerage of a bottom depth aues of the numerica mode.

3 3D bock kriging with externa drift. Z : fish ength at sampe point. Z : aerage fish ength in bock. λ : kriging weight appied to sampe point. : matrix of coariance between sampe points and β in the kriging neighbourhood (points C 3D β and β may beong to the same year or to different years, therefore the use of the 3D coariance structure) D C 3 : ector of aerage coariance between sampe points and a points of the bock to be estimated (points and bock may beong to the same year or different years) C : aue of the aerage spatia coariance in bock to be estimated (this aue does not depend on the year) 0 f : ector of unit aue (to be used in the first constraint imposed on the kriging weights) 1 f : ector of the bottom depths at sampe aues 0 f : aerage aue 1 on the bock to be estimated 1 f : aerage bottom depth on the bock to be estimated The kriged bock estimate is : Z k = λ Z with weights respecting the non bias conditions: λ f = f, = 0, 1 The kriging system is : 3D λ C µ f β λ β f β = f = C 3D whateer whateer ( µ are Lagrange parameters to be estimated with the kriging weights) 2 3D The estimation ariance (kriging ariance) is: σ = C λ C µ f k + The method does not require the aues of the coefficients of the regression, ony the functions f. In each year, the eft handside of the kriging system is unchanged, as the neighbourhood is unchanged for D each bock to be estimated. The right handside ( C 3 ) changes with the year. As a resut, the estimate and its kriging ariance differ in each year because of the 3D coariance structure but the neighbourhood configuration remains fixed for a years. Choice of neighbourhood. The neighbourhood was a circe of fixed dimension with an at most gien number of sampe points retained for kriging. Different trias were performed. Vaues retained where those that enabed to respect non bias of fish ength in each year (mean kriged aues as cose as possibe to the mean sampe aues). The circe radius retained was 36 n.m. with at most 10 points (Tabe 2). Resuts Regression. The inear regression of ength on bottom depth (Fig. 3) expained 45% of tota ength ariance. The sope was significant (Tabe 3). Variography. The residuas of the regression showed a good spatia structure but no structure between years (Fig. 4). If two points beonged to the same year, the spatia structure inking their two aues

4 was gien by the spherica ariogram. If two points beonged to different years, the spatia structure inking their two aues was a pure nugget effect. Map of fish ength. Maps for each year are presented on Figures 5-8. The aerage map is presented on Figure 9 with mean standard estimation error. The map reproduces we the increase of ength with increasing bottom. The area where the smaest fish is distributed is that in front of the Gironde. The map of the estimation error shows that the inner part of the shef is estimated with a good precision. Most of the traw haus occur each year in that part (Fig. 1). Inter-year ariation in the maps Tota ariance in a bock is the sum of two ariance terms: σ tot = σ E + σ I. σ E was the aerage 2 across a years of the estimation ariance (aerage kriging ariance) and σ I was the ariance 2 between bock means across a years (inter-year ariance of bock estimates). The terms σ E and 2 σ I were computed in each bock. The inter-year ariance is ower in aue than the aerage estimation ariance. The inter-year ariance is aso highest where the estimation ariance is owest, i.e., in the inner part of the shef (Fig. 10). This coud coud resut from the number of different years present in the neighbourhoods. In an area where the number of sampe points is high, with many different years contributing sampes, the kriging estimates wi be precise and wi aso ary between years. A exampe of such area is the inner part of the shef. In contrast, in an area samped more scarcey spatiay and across the years (e.g., the north part of the map), the kriging estimates wi be ess precise and wi ary ony for those years contributing sampes. The impact of the number of different years in the neighbourhoods was not seere (Fig. 11), meaning that the inter-year ariation obsered was rea but possibe to detect in those areas that were intensiey enough samped. The 2 inter-year ariance was expressed as the ratio σ 2 / I σ and mapped (Fig. 12, eft). The areas most tot ariabe were in the inner part of the shef which concerned the habitats of the smaer fish. Another possibiity to quantify inter-year ariation on the maps is to test bock by bock for the a difference in means between the aerage of the kriged aues across years ( M, Fig. 9) and the k 2 yeary kriged aue ( Z, Figs. 5-8). The ariance of the aerage was σ tot and the ariance of the 2 kriged aue was the yeary kriging ariance σ k. The statistica distribution of the estimates was considered gaussian (gaussian distribution of fish ength). The foowing test was performed in each a k 2 2 bock, in each year: ε = M Z / σ tot + σ k ; if ε was greater than 1.96, then the means were considered significanty different. In each bock, we counted the number of times the test concuded to a significant difference in means and expressed the count as a frequency across a years (probabiity of difference, Fig. 12, right). The resut is sighty different concuding that the area of the Gironde is the most prone to ariation between years. The inter-year ariance proides information on how different are the aues where as the test concudes quaitatiey on whether it is different. It can be concuded that the area most prone to ariation is the Gironde area but that the differences are obsered a aong the inner shef. Discussion and concusion The present approach is a soution to the mapping of particuar ariabes for which ony a sma number of obserations are aaiabe each year, typicay for fish ength in acoustic sureys. By buiding a time series oer the years and anaysing the drift (regression on enironmenta coariates) and the residua correation structure (3D-ariogram), the approach aows for mapping each year using the muti-year data set.

5 The method appied foowed that used by Rioirard et a. (2001) on herring in the Northern North Sea. It is worthy to note that the spatia and inter-year correation structure for anchoy in Biscay is simiar to that for herring around Shetand. Reation of fish ength with bottom depth woud be indicatie of a genera ecoogica gradient that organises fish popuation geographicay, in a seas. The differences between years and the sma number of sampe points per year are such that the interyear spatia structure is obsered as a nugget effect (no correation structure between years on the residuas). The 3D correation structure obsered coud aso be obsered esewhere, making the entire approach genera for mapping fish ength for acoustic sureys. In contrast to the anaysis on herring in the Northern North Sea (2001), the si of the ariogram across years was here in Biscay simiar to that in a gien year, meaning herring around Shetand aried in mean ength across years perhaps more than did anchoy in Biscay. In the present method used, the map is drien by the regression (constraints on the weights) as we as by the correation structure in the residuas (kriging). In comparison, the use of a GLM or GAM woud resut in driing the map by the drift ony (the regression). Significatie differences between years in the maps were eidenced by comparing the inter-annua difference in the estimates with that in the estimation ariance. The inter-annua ariation in the maps was ocaised near the Gironde and in the inner part of the shef. This coud resut from the bioogy, in particuar from inter-year difference in fish growth at age. References Chiès, J.-P. and Definer, P Geostatistics: modeing spatia uncertainty. Wiey, New York. Gai, A. and Meunier, G Study of a gaz reseroir using the externa drift method. In: Geostatistica case studies, pp Ed. by G. Matheron and M. Armstrong. Hoand: D. Reide Pub. Co. Guisan, A. and Zimmermann, N Predictie habitat distribution modes in ecoogy. Ecoogica Modeing, 135: Lazure, P. and Jégou, A.-M D modeing of seasona eoution of Loire and Gironde pumes on Biscay Bay continente shef. Oceanoogica Acta, 21(2): Rioirard, J., Simmonds, E., Fernandes, P. and Bez, N Geostatistics for estimating fish abundance. Backwe Science, Oxford, 206p. Rioirard, J. and Wieand, K Correcting for the effect of dayight in abundance estimation of juenie haddock (Meanogrammus aegefinus) in the North sea: an appication of kriging with externa drift. ICES Journa of Marine Science, 58:

6 Tabe 1: Basic statistics of anchoy ength in each year and oera: nb = nb of sampe points, mean = aerage of sampe points, std = standard deiation, min = minimum, max = maximum Year nb mean std min max A Tabe 2: Basic statistics of sampe points in the kriging neighbourhoods Number of different years Number of points aerage min 3 5 max Tabe 3: Statistics of the inear regression of fish ength on bottom depth. Nb of data points is 295. Residua standard error: on 293 degrees of freedom. Mutipe R-Squared: F-statistic: on 1 and 293 degrees of freedom, the p-aue is 0. Coefficient aue Std. Error t aue Pr(> t ) Intercept sope

7 Fig. 1: Location of the traw haus containing anchoy (295 haus oer 16 sureys). The poygon for defining the contour of the mapping is aso shown. Fish ength (cm) Quanties of Standard Norma Fig. 2: QQ-pot of anchoy ength (aerage ength per traw hau).

8 mean Length (cm) Bottom Depth (m) Fig. 3: Linear regression of anchoy ength (mean ength in traw catch) with bottom depth. Variogram Distance (Nm) Fig. 4: Variographic structure for the residuas of the regression, considering pairs of points beonging to the same year (diamonds) and to different years (crosses).

9 mean L(cm) mean L(cm) 1984 mean L(cm) 1985 mean L(cm) 1986 Fig. 5: Maps of anchoy ength obtained by Muti-year Externa drift Bock Kriging

10 mean L(cm) 1987 mean L(cm) mean L(cm) 1990 mean L(cm) 1991 Fig. 6: Maps of anchoy ength obtained by Muti-year Externa drift Bock Kriging

11 mean L(cm) 1992 mean L(cm) 1993 mean L(cm) 1994 mean L(cm) 1997 Fig. 7: Maps of anchoy ength obtained by Muti-year Externa drift Bock Kriging

12 mean L(cm) mean L(cm) 2000 mean L(cm) mean L(cm) 2002 Fig. 8: Maps of anchoy ength obtained by Muti-year Externa drift Bock Kriging

13 mean L(cm) - a years Std Estimation Error - a years Fig. 9: Aerage map of anchoy ength (aerage of the yeary maps), , with mean standard estimation error (square root of the aerage of the yeary kriging ariances). inter-year ariance aerage kriging ariance Fig. 10: Scatter pot between inter-year ariance of bock estimates and kriging ariance of bock estimates.

14 inter-year ariance Nb of years Fig. 11: scatter pot between the inter-year ariance and the number of different years in the kriging neighbourhood. Each point on the graph corresponds to one neighbourhood and one bock Inter-year ariance (percent) a years - prob. difference Fig. 12: Inter-year ariation between maps. Left: map of the inter-year ariance, expressed in percent of tota ariance. Right: number of times the yeary map differed significanty from the aerage map, expressed in frequency across years.