SCI. MAR., 62 (Supl. 1): 117-133 SCIENTIA MARINA 1998

NEPHROPS NORVEGICUS: COMPARATIVE BIOLOGY AND FISHERY IN THE MEDITERRANEAN SEA. F. SARDÀ (ed.)

The application of geostatistics in mapping and assess- ment of demersal resources. Nephrops norvegicus (L.) in the northwestern Mediterranean: a case study*

FRANCESC MAYNOU

Institut de Ciències del Mar (CSIC), P. Joan de Borbó, s/n, 08039 Barcelona, Spain.

SUMMARY: The application of geostatistics in mapping and assessing demersal resources is reviewed. The basic sum- mary of the methodology is illustrated with a case study on for Nephrops norvegicus off Barcelona (NE Spain) over commercial fishing grounds. The geostatistical analysis revealed that N. norvegicus is distributed over the continental slope in high-density patches of ~ 7 km, alternating with areas of low density. Mapping by disjunctive helped visu- alize the spatial distribution of the resource and identified areas where the resource abundance was of commercial level.

Key words: Norway lobster, Nephrops norvegicus, geostatistics, , kriging, resource assessment.

RESUMEN: APLICACIÓN DE MÉTODOS GEOESTADÍSTICOS EN LA CARTOGRAFÍA Y EVALUACIÓN DE RECURSOS DEMERSALES. ESTUDIO DE NEPHROPS NORVEGICUS EN EL MEDITERRÁNEO NOROCCIDENTAL. – Se presenta una revisión sobre la aplicación de métodos geoestadísticos para el cartografiado y evaluación de recursos demersales. El resumen básico de la metodología se ilustra con el estudio del recurso de cigala (Nephrops norvegicus) en caladores comerciales de la flota de Barcelona (NE España). El análisis geoestadístico permitió poner en evidencia que la cigala se distribuye en manchas de alta densidad de ~ 7 km, alternando con áreas de baja densidad. La cartografía por kriging disyuntivo permite visualizar la distribución espacial del recurso e identificar áreas en las que la abundancia del recurso alcanza niveles comerciales.

Palabras clave: Cigala, Nephrops norvegicus, geostadística, variograma, kriging, evaluación de recursos.

INTRODUCTION its corresponding precision index (Cochran, 1977). When the random (or variations thereof) is The classical methods used to obtain abundance carried at the appropriate spatial scale, it effectively estimates in demersal fish populations include bot- obliterates any underlying spatial structure. However, tom trawl surveys using standard trawling gears and the scale of spatial distribution of the target species is specific sampling strategies, e.g. they are methods usually unknown and this factor may introduce bias based on sampling theory. The sampling points (trawl in the computation of abundance estimates. In exper- hauls) can be randomly positioned over the sampling imental trawl surveys, when the average distance area, yielding an unbiased estimate of abundance and between hauls is smaller than the zone of spatial influence of the underlying spatial phenomenon, the abundance of organisms at one point can be partially *Received 12 October 1997. Accepted 20 February 1998. predicted from neighbouring points.

APPLICATION OF GEOSTATISTICS IN N. NORVEGICUS IN THE MEDITERRANEAN 117 The existence of spatial structure in the distrib- Poulard (1989); Simard et al. (1992); Sullivan ution of organisms (or any other ecological vari- (1991). They have also been applied to pelagic fish able) originates spatial between and acoustic survey data (Anonymous, 1990, 1991; pairs of samples. Spatial autocorrelation can be Petitgas, 1993) and ecology (Rossi et al., 1992). defined as the departure from in the In the geostatistical literature, an autocorrelated spatial distribution of organisms or ecological vari- variable is termed a regionalized variable. The ables, and it is analogous to a coeffi- example of regionalized variable which will be cient. The concept of spatial autocorrelation can used in this article is the density of N. norvegicus also be seen as the non-null probability of predict- over the fishing grounds off Barcelona (NE Spain). ing the value of a sample from neighbouring sam- The describing the spatial ples. The spatial autocorrelation present in a given autocorrelation is called structure function. An can be analyzed, modeled mathematically example of structure function used in this article is and incorporated into spatially-explicit models the variogram. Other examples of structure func- which can be used to produce unbiased abundance tions are the autocorrelation functions used in spa- estimates of demersal resources. tial ecology, such as Moran’s or Spatial statistical methods were developed inde- Geary’s correlogram. pendently by two major lines of research: Human In this review, the use of geostatistical tools for (and ecology) and mining geology. In assessing the spatial structure, mapping and estima- the 1950’s and 1960’s autocorrelation models were tion of demersal resources is highlighted. These first produced in (Moran, 1950; methods are illustrated with the case study of N. Geary, 1954; Berry and Marble, 1968) and later norvegicus and the underlying spatial phenomena adopted in ecology by the influential work of Cliff that originate the spatial structure in this species are and Ord (1973). During the 1970’s and 1980’s, discussed. these models were widely applied in ecology and in the 1990’s they have become mainstream statistical methods, although their use in marine ecology is METHODS still limited (Jumars et al., 1977; Jumars, 1978; Mackas, 1984; Legendre and Troussellier, 1988). Sampling design Excellent reviews of this methodology in ecology can be found in Sokal and Oden (1978), Legendre Two surveys (GEOESC-I and GEOESC-II) and Fortin (1989) and Legendre (1993). were specifically designed for the mapping and In mining geology, spatial statistical methods assessment of the harvestable resource of Norway were first developed in the 1960’s in the context of lobster on the fishing grounds off Barcelona. For ore-reserve estimation. These methods were termed the purposes of this review, only the results for geostatistics. The theoretical basis of geostatistics GEOESC-I will be presented. The survey site was was laid out by Matheron (1962, 1963, 1969 and chosen over muddy bottoms with gentle slope and 1971). Later influential reviews and developments depth contours parallel to the coast, limited by two include the work of Matheron (1973, 1976), submarine canyons (Fig. 1). A regular grid 1 by 2 Journel and Huijbregts (1978), Isaaks and nautical miles was set parallel to the coast and a Srivastava (1989), and more recently, Cressie start location for each tow was randomly selected (1991). Geostatistics expanded from mining geolo- within each cell. The total area covered by the gy to other domains of the earth sciences in the sampling design was 790 km2, comprising 115 1980’s (e.g. Burrough, 1983), and particularly to cells, although due to logistical and practical con- fisheries science (Conan, 1985; Conan et al., 1988). straints only 59 locations could be sampled during Since the late 1980’s and in the 1990’s, geostatistics the first cruise. The depth of the sampling loca- has been widely used for the estimation of abun- tions varied from 141 to 730 m, encompassing the dance of demersal resources. To the pioneering whole depth of Norway lobster in the area applications of G.Y. Conan cited earlier, we can of study. add: Conan et al. (1992); Conan et al. (1994); The experimental sampling gear was a specially Freire et al. (1992); González Gurriarán et al. designed otter trawl drawn by a single warp (1993), Maynou et al. (1996); Maynou et al. (‘Maireta System Trawl’, Sardà et al., 1994). The (1998); Pelletier and Parma (1994); Petitgas and codend stretched mesh was 12 mm in order to

118 F. MAYNOU 100m

Barcelona 200m

400m

600m 800m 1000m Latitude (N)

Serola

Merenguera

Can Pere Negre

Longitude (E)

FIG. 1. – Sampling area of the experimental survey GEOESC-I. Crosses indicate mid position of trawl hauls. Main commercial fishing grounds for Norway lobster are shown. retain small individuals not normally available to We consider the density of Norway lobster as -2 the commercial fishing gear. The actual opening of the variable Z(xi): Number of individuals · km at the trawl was measured using an acoustic system location xi, whose properties are relatively constant (SCANMAR) and stabilized at 14.0 m width by 2.0 over the area at the spatial and temporal scales of m height. During the survey, tows were made par- study. For instance, it can be reasonably accepted allel to the depth contours. The duration of each that the abundance of Norway lobster does neither tow was set to exactly 15 min (time of effective vary abruptly from one location to a neighbouring trawling). The towing speed varied between 2.3 and location nor shows a clear spatial trend, based on 2.6 knots ( 2.5 knots). Start and end locations what is known from the biology of the species. for each tow were taken by GPS. The actual surface These assumptions can be verified in practice by an covered by each tow was computed from the GPS exploratory data analysis previous to geostatistical and the SCANMAR readings. Effective time of the analysis. survey was from 7:30 to 20:00 h each day. The sur- Under these conditions, the structure of spatial vey was completed between 27 April and 5 May variability of Norway lobster abundance can be 1991. studied by computing an appropriate structure The total catch of N. norvegicus was counted, function, for instance a variogram. The computa- weighed and measured. The density of Norway lob- tion of the variogram is summarized in Fig. 2, see ster was computed (number of individuals · km -2) also eq. (1) of the Appendix. A vector of distance from the total surface covered by each tow. classes or lags (h) is first specified, together with an angular tolerance. Then, from each sampling point

xi, the distance to each other location is computed Statistical methods and arranged according to the specified distance classes. The squared difference of density values The fundamentals of geostatistics, with empha- for each pair of samples pertaining to a given lag is sis on the methods employed here, are laid out in then computed. Finally the estimated variogram the Appendix in a rather mathematical way. What value for a given lag is obtained, dividing the sum follows is an elementary summary of linear geosta- of squared differences by the number of pairs of tistics and disjunctive kriging that should suffice in sampling points pertaining to this lag. This process order to understand the methodological basis of our is illustrated in Fig. 2 considering the first 6 sam- analysis and results from a fisheries perspective. pling points in our data set.

APPLICATION OF GEOSTATISTICS IN N. NORVEGICUS IN THE MEDITERRANEAN 119 obvious pattern in our variogram, it is always pos- sible to contemplate other geostatistical models or α + tol. assume that there is no spatial structure in our vari- able at the scale of study. α The behaviour of the variogram near the origin Z3 Z5 requires a detailed study. A variogram may show a Z6 discontinuity near the origin, called the nugget effect in geostatistics. The nugget effect can be

northing attributed to measurement error, micro-scale vari- Z4 α - tol. ability or small-scale spatial structure. For instance, below ~ 1 km little can be said from our data set, as Z2 Lag + tol. Lag this is the average tow length in our study. A nugget Lag - tol. component is usually included in most theoretical Z1 variogram models. easting The variogram fitted can be used to predict the abundance of our variable at locations not sampled

FIG. 2. – Computation of the experimental variogram for distance ( by kriging) or over a user-defined class Lag and direction α. Starting from the sampling point Z1 and region (block estimation by kriging). The estimate specifying α ±tol, sampling points Z3, Z4 and Z5 would be inclu- ded in the computation of the omnidirectional variogram. For a at a point or block is obtained from linear or non- directional variogram specifying a direction α and an angle tole- linear weighting of the observed values. A structure rance ±tol, only the sampling points Z4 and Z5 would be taken into account. The computation of the experimental variogram proceeds function (for instance the variogram fitted) is used in this manner using eq. (1) of the Appendix, starting from each to optimally compute the kriging weights (linear sampling point at a time. estimation) or to estimate the kriging function (non-linear estimation). Kriging has the important property that for each estimate, an associated esti- The shape of the variogram gives clues to the mation can be computed. spatial structure of the variable under study. In The point estimation process is useful to pro- ecology, the structure functions known as autocor- duce an accurate map of the resource. For instance, relograms (analogous to the variogram) are often by setting a grid of cells over the study area we can the object of the study. They can also be tested sta- estimate the density of Norway lobster at each node tistically, and for this reason are often preferred, in (or cell) and colour-code the abundance values to ecological work, in order to make inferences on the obtain a map of the resource. In this sense, point spatial structure of ecological variables (Legendre kriging is an interpolation method with the advan- and Fortin, 1989). In this review, we will assume tage that it yields a precision index for each esti- that the variogram suffices to reveal the spatial mate on the map. The non-linear method of dis- structure of N. norvegicus. Then, by analogy with junctive kriging (Yates et al., 1986; Appendix sec- Sokal and Oden’s (1978) simulations employing tion 3.2) is employed here as a to illustrate , we will attempt to establish the spa- the use of kriging in mapping and in estimating the tial characteristics of Norway lobster in the study probability that Norway lobster density is above a area with the aid of the variogram. certain profit threshold. Another aspect of geostatistics is the possibility of estimating the density of our variable at points not sampled or over areas where a global estimate RESULTS is needed. This aspect is known as kriging and can be regarded as spatial prediction or estimation. In Exploratory Data Analysis order to proceed with kriging, the experimental var- iogram has to be fitted to a theoretical variogram Before building the geostatistical model, it is model (some models are given in the Appendix, advised to start with an exploratory data analysis. section 2.2). When the experimental variogram has This preliminary analysis should help check for been computed correctly and there exists a clear inconsistencies of the assumptions of the geostatis- underlying spatial structure, it is often easy to tical model with the data or to suggest a specific choose and fit a theoretical model. When there is no geostatistical model to apply, i.e. linear or non-lin-

120 F. MAYNOU ear geostatistics, models with drift. A simple way to reveal a meaningful variogram. In the GEOESC-I perform the exploratory data analysis is illustrated survey the maximum distance in an E-W direction in Fig. 3, with the aid of scatterplots of Norway lob- is ~ 43 km and ~ 25 km in a N-S direction. The ster density vs. some ancillary variables. More maximum distance in the field of study is ~ 50 km, refined methods can be found in Cressie (1991, pp. so our first choice of h is to set a distance class at 30-51). each 2 km, from 0 to 48 km, yielding a total of 25 The dispersion plot of Norway lobster density distance classes. The tolerance specified is ± 1 km. vs. depth did not evidence any depth-related trend In Fig. 2 the variogram computation is illustrated, (Fig. 3a). Nephrops density was related neither to see also Appendix, eq. (1). time of the day (Fig. 3b) nor to a given geographi- The variogram computed in this manner is cal direction, such as easting or northing (Fig. 3c shown in Fig. 4, bottom. The figure shows the esti- and 3d). Hence, a geostatistical model with no spa- mate of γ(h) at each distance class hj, together with tial drift can be applied to the variable density of the number of pairs used to compute this estimate. Norway lobster. The spatial predictor to choose can The number of pairs used to compute each γ^ is use- be based on a linear predictor (Ordinary Kriging, ful in order to know the quality of the estimate. The Table A.1 of the Appendix) or a non-linear predic- weighted method of fitting the vari- tor (Disjunctive Kriging, Table A.1 of the ogram is an automated procedure for giving more Appendix). weight to those distance classes with a higher num- ber of pairs. Variogram computation A way to obtain a well-defined experimental variogram, easier to fit, is to create a vector of dis- The variogram computation starts by choosing a tance classes with unequal spacing. For instance, vector h of distance classes, including a tolerance h= {0.5 ± 0.5, 1.5 ± 0.5, 2.5 ± 0.5, 3.5 ± 0.5, 5.0 ± range for unequally spaced samples, as is our case. 1.0, 7.0 ± 1.0, 9.0 ± 1.0, 12.5 ± 2.5, ..., 42.5 ± 2.5} The choice of h is based on the maximum distance yields a variogram (Fig. 4, top) which is easier to fit in the field of study divided by 10 or 20, in order to to the spherical model (Appendix, section 2.2.1.a, ensure a sufficient number of distance classes and Fig. A.1) and with higher resolution at the small

FIG. 3. – Preliminary data analysis of Norway lobster total density vs. some factors that could invalidate the stationary assumptions.

APPLICATION OF GEOSTATISTICS IN N. NORVEGICUS IN THE MEDITERRANEAN 121 distance classes, thus making it easier to estimate Experimental in an east (0º), north-east the nugget parameter (c0 in Appendix, section (45º), north (90º) and north-west (135º) directions 2.2.1.a). The manual and weighted least squares were produced and they did not show any evidence (w.l.s.) fits to the variogram in Fig. 4 (top) are given of anisotropy (Maynou, 1995). The east and north- in Fig. 5. Note that in variogram fitting, only those east variograms (along the field of study) showed distance classes |h|

FIG. 4. – Experimental variogram of Norway lobster density using two different lag vectors. Bottom: an equally-spaced vector does not resolve accurately the small distance classes. Top: an unequally-spaced vector allows to better resolve the small distance classes and is easier to fit to a theoretical model. See text for vector specification.

122 F. MAYNOU FIG. 5. – Comparison of a manual fit and an automated fit (weighted linear squares) to the experimental variogram in Fig. 4 (top). The variogram model is a spherical variogram (Appendix, section 2.2.1.a). The parameters of the manual fit are: Nugget= 4.50 x 105, sill= 1.75 x 106, range= 7.0 km. The parameters of the automated fit are: Nugget= 4.23 x 105, sill= 1.70 x 106, range= 6.46 km.

Kriging variable Nephrops density. The first cut-off corre- sponded to the usual Norway lobster catch of To produce a precise map of the density of the Barcelona fishermen (for the purpose of this exer- resource in the study area, we set first a regular grid cise, 15 kg/day). Using the appropriate conversion of 90 x 60 cells and estimated the density of factors from our experimental trawl and fishing Nephrops at each cell by disjunctive kriging strategy and simulating the commercial fisherman’s (Appendix, section 3.2). Each cell had a surface of strategy we arrive at ‘usual’ catch of 1528 ind · km-2 0.25 km2. The cells falling outside an irregular or 31 kg · km-2 (Maynou, 1995). In the same way, polygon bounding the data set were not computed. we specified a ‘high’ catch cut-off value (25 Disjunctive kriging requires approximating the kg/day), corresponding to 2,546 ind/km2 or 51 non-linear predictor by a Hermite polynomial. We kg/km2 (Maynou, 1995). The first cut-off level (Fig. chose K=10 to truncate the Hermite polynomial. 6, bottom, left) revealed areas with probability The map in Figure 6 (top left) was produced using p>0.6 where fishing was economically profitable. the algorithm of Yates et al. (1986). The map pro- These areas were the Serola, Can Pere Negre and duced by the disjunctive kriging technique does not Merenguera fishing grounds. The second cut-off differ from a map produced by ordinary kriging, level (Fig. 6, bottom, right) indicated that only in which was presented in Maynou et al. (1998). Serola could a ‘high’ catch be expected, with prob- Patches of high density of Nephrops can be ability p > 0.6. This agrees with the fact that most observed over the fishing grounds known as Can (over 90%) Norway lobster landed at the Barcelona Pere Negre and Serola, with densities over 3600 ind fish market is caught in the Serola fishing grounds · km-2 at some cells. The kriging variance obtained (Sardà and Lleonart 1993). when solving the kriging system can be used as a precision index to evaluate the quality of the esti- mate at each cell. The kriging DISCUSSION for each cell is shown in Fig. 6 (top, right). To illustrate the use of disjunctive kriging in In our geostatistical study of N. norvegicus, we computing densities above a user-specified thresh- uncovered a spatial pattern of high-density areas old, we tried two levels of economic profit in the (patches), alternating with low-density areas. This

APPLICATION OF GEOSTATISTICS IN N. NORVEGICUS IN THE MEDITERRANEAN 123 FIG. 6. – Results of disjunctive kriging. Top left: Map of N. norvegicus density in the study area, scale units are ind · km-2. Top right: Precision index (standard estimation of the point estimate) of the previous map. Bottom left: Probability levels for the first cut-off level: 1,528 ind · km-2 or 15 kg · day-1. Bottom right: Probability levels for the first cut-off level: 2,546 ind · km-2 or 25 kg · day-1.

patchy distribution is relatively stable throughout tion exists in snow crab due to its complex spatial the year and across biological categories, as shown behaviour. by Maynou et al. (1998) studying the variograms of Our results are not directly comparable to the juveniles, adults, males and females in two differ- results of Fariña et al. (1994) who studied the dis- ent seasons of the year (spring and autumn). The tribution of Norway lobster in Galician waters (NW range of the variograms fitted varied from 6 to 10 Spain, Atlantic Ocean). They found a range of ~100 km, with a mean of ~ 7 km. Our results can be con- km in the spatial distribution of this species, but trasted with the results of Conan et al. (1988, 1996) their minimum distance among stations was of the in studies of the spatial distribution of the snow order of the range obtained in our study. Thus, their crab (Chionoecetes opilio). These authors showed results showed rather a large-scale areal distribu- that considerable sexual and size spatial segrega- tion of Nephrops.

124 F. MAYNOU High-resolution mapping by kriging has been ACKNOWLEDGEMENTS used by fisheries biologists as a tool to forecast accu- rately the location and the spatial characteristics of The research project NEMED, funded by the an exploited resource. In this sense, several applica- EC, DG. XIV(MED92/008) provided the opportu- tions illustrate the potential of ordinary kriging to nity to carry out the sampling cruises GEOESC-I mapping and estimation of resource abundance, such and GEOESC-II. I thank all participants in the as the contributions of Conan et al. (1988, 1992, cruises for their assistance during the fieldwork, 1994, 1996), Sullivan (1991), Simard et al. (1992) especially Dr. Joan Cartes, Dr. Joan B. Company and others cited in the reference list. Disjunctive and Dr. F. Sardà. My sincere appreciation to Dr. G. kriging has not previously been used in a fisheries Conan for introducing me to geostatistics. context to our knowledge. 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126 F. MAYNOU APPENDIX – SPATIAL STATISTICS 1.2. Stationarity

1. DEFINITIONS The stationarity assumptions are taken on the first and second order moments of the distribution 1.1. Regionalized variables and random function (Journal and Huijbregts, 1978; Myers, functions 1989). The first order does not depend on The object of study of geostatistics (or surface location: pattern analysis) are continuously varying phe- nomena, i.e. phenomena which take a value at E [Z(x)] = m, ∀ x. each point of . These phenomena are termed regionalized variables in geostatistics. The second order moments (centered) are: A regionalized variable can be defined as a function taking a value z(x) at each point of the E [(Z(x) - m(x))2] = Var[Z(x)], variance; n space R , where z(x) only depends on x (the geo- E [(Z(x1) - m(x1)) (Z(x2) - m(x2))] = 2 graphical location). In this article, n=2; i.e., we E [Z(x1) Z(x2)] - m , ∀ x; 2 2 consider only two dimensions of space. The den- E [(Z(x1) -Z(x2)) ]-E [Z(x1)-Z(x2)] = 2 sity of N. norvegicus (Number of individuals/unit E [(Z(x1)-Z(x2)) ], ∀ x. area) is then a regionalized variable z(x). The bot- tom trawl sampling used to obtain the density of Letting x1=x and x2=x+h (where h is a vector of N. norvegicus over the study area (A) can be distance) the last two expressions become: regarded as a realization of the regionalized vari- able, and the observed densities at the sampling E [Z(x) Z(x+h)] - m2 = C(h), points xi are noted Z(xi). covariogram function (E1); The set of realizations of z(x) is called a ran- E [(Z(x)-Z(x+h))2]/2 = γ (h), dom function, {Z(x), x ∈ A} (Journel and variogram function (E2). Huijgbregts, 1978), which is defined by its distri- bution function: Under the second-order stationarity hypothesis, the following identities are useful:

Fx ...x (z1 ... zn) = P {Z(x1) ≤z1 ... Z(xn) ≤zn}. 1 n Var [Z(x)] = C(0), It is not possible to make γ (h) = C(0) - C(h). with a single realization of the random function, because its distribution function cannot, in gener- Both, γ(h) and C(h) can be used to characterize al, be known. Thus some assumptions on the sta- the structure of spatial variability of Z(x) by means tistical behaviour of the random function have to of their estimators (see below). Variations of these be made in order to proceed with the statistical functions are also employed in the geostatistical lit- analysis. Geostatistical models can become erature, such as correlogram, relative variogram, increasingly elaborate by increasing the number madogram, etc. (Cressie, 1991; Maynou et al., of assumptions, but it is preferable to keep the 1996). assumptions to a minimum in order to describe in γ (h) is more general than C(h) because it a realistic manner the phenomenon being ana- assumes only that the variance of the increments is lyzed. finite. The concept of finite variance of the incre- Most assumptions consider that the regional- ments in the variogram is called the intrinsic ized variable repeats itself in space and the sam- hypothesis (Matheron, 1971). This hypothesis is pling is representative of the regionalized vari- weaker than the second-order stationarity hypothe- able, i.e. if we were to repeat the sampling over sis and includes it. the same study area A we would obtain the same Second-order stationarity hypothesis and the results (statistically). In linear geostatistics (the intrinsic hypothesis require an unlimited domain, simpler and more widely used geostatistical meth- but in practice the analysis is limited to a finite ods in fisheries science), the assumptions taken field. Within the specific spatio-temporal scale of are of stationarity and isotropy. the problem to be addressed it is recommended to

APPLICATION OF GEOSTATISTICS IN N. NORVEGICUS IN THE MEDITERRANEAN 127 check that our data do not depart to a great extend illustrated by considering a water column (R3): the from these hypotheses. Preliminary exploratory spatial continuity over the plane x1, x2 is of the same data analysis can be used to verify the adequacy of nature, while in x3 (vertical axis) it is fundamental- the data to the assumptions being taken (Cressie, ly different and depends on the gravity (Margalef, 1991; Maynou et al., 1998). 1991, pp. 136-139). In order to incorporate zonal anisotropy in the spatial modeling of Z(x) it is nec- 1.3. Anisotropy essary to obtain independent or external informa- tion. If the regionalized variable under study shows the same spatial structure on every direction (α) then the phenomenon is isotropic. When there 2. STRUCTURE FUNCTIONS exists anisotropy, then γ (h) and C(h) are functions of a also: γ (h,α) and C(h,α). There exist two types 2.1. Experimental variogram and covariogram: of anisotropy (David, 1977; Isaaks and Srivastava, 1989). In the case of geometric anisotropy the spa- The underlying autocorrelation function of a tial continuity in one of the directions is different random function Z (x) is often unknown but under but of the same nature than in other directions. The the stationarity conditions detailed above, it can be problem can be reduced to isotropy by correcting estimated from the observations Z (xi). Expe- the units of the anisotropic axis (affine transforma- rimental variograms γ^ (h), or covariograms tion of the Euclidean space by an anisotropy coef- C^ (h), are computed first and later fitted to a theoret- ficient, Cressie, 1991). ical variogram function. The spatial autocorrelation In the case of zonal anisotropy the nature of the descriptors are computed as follows (estimators spatial variability is different in one of the direc- obtained by the method of moments, Matheron, tions (Isaaks and Srivastava, 1989). This can be 1962):

FIG. A – Some common variogram models.

128 F. MAYNOU 1 among samples. γ(h) reaches the sill only asymp- ^ (h) = ––––––– (Z(x )-Z(x ))2, (1) totically in the exponential model and oscillates γ ∑ i j 2N(h) N(H) around the sill in the wave (or hole-effect) model. where h is a vector of distance, including a user- 3 h 1 h3 c0 + c’ – – - – – h < a specified tolerance in irregular samplings, and a. Spherical model: γ (h) = ( 2 a 2 a3) ,

N(h)={(xi, xj): xi-xj=h; i,j=1...n,} is the number of {c0 + c’ h ≥ a pairs used in computing the experimental vari- ogram for each distance class. The experimental when h=a, γ(h) = C(0). covariogram is (Matheron, 1962): h - – b. Exponential model: γ(h) = c + c’ 1 - e a , 0 { } ^ 1 – – C(h) = ––––––– (Z(xi)-Z) (Z(xj)-Z), (2) when h=3a, γ (h) = 0.95 C(0). N(h) ∑N(H) h a sin – – 1 n ( a ) where: Z = –– Z(x ). c. Wave model: γ(h) = c + c’ 1 - ––––––––– , n ∑ i 0 i=1 { h }

The vector h can be computed for specified directions α yielding directional variograms or reaches the sill periodically, for h=π a and multi- covariograms. ples.

2.2. Variogram models 2 h - ––2 d. Gaussian model: γ(h) = c0 + c’ 1 - e a , Given that γ (h) is of more general use than C(h) { } or other related functions, the remaining of this review is illustrated with the variogram function for h=,3a γ(h) = 0.95 C(0). and the experimental variogram. An advantage of using γ (h) is that the variogram is defined for 2.2.2. Non-transition models: Their general b intrinsically stationary processes, which is a class equation is γ (h)=c0 + p h . When b=1, the equation including second order stationary processes. Also, γ^ yields the ; when 0

The models depicted include a constant c0, which is experimental variogram to a theoretical variogram known as the nugget effect (see below). The for- model. The most common method consists in mulae used to compute the variogram function are graphically fitting the two or three parameters that as follows: define the theoretical model. This simple method is 2.2.1. Transition models: γ(h) grows continu- very useful for well-defined experimental vari- ously up to a certain range α where it stabilizes ograms, which easily suggest the variogram func- around the sill C(0)= c0 + c’. The range α is the dis- tion to choose. Some mathematical methods for tance beyond which there is no spatial correlation variogram fitting are reviewed by Cressie (1991,

APPLICATION OF GEOSTATISTICS IN N. NORVEGICUS IN THE MEDITERRANEAN 129 pp. 90-104) and include maximum likelihood 2.3. Nugget effect methods, least squares methods or cross-valida- tion. However, a very accurate fitting of the vari- Although strictly γ(0)=0, the function γ(h) often ogram model is not essential when using the fitted shows a discontinuity at the origin due to small- model in spatial interpolation, as our own scale spatial phenomena. It is termed the nugget

(Maynou, 1995) or Sheshinski’s (1979) simulation effect and is simbolized as c0 in the equations. It results show. originates from the fact that at a scale smaller than the minimum distance between samples, γ(h) can- 2.2.3 Some problems encountered when using not be studied from the experimental data set. It can Matheron’s estimator: 1) This estimator is not be interpreted in a number of ways, depending on robust to outliers in the data set. A robust estimator the phenomenon under study: as white noise has been proposed by Cressie and Hawkins (1980). (Matheron, 1962), measurement error, or existence 2) The fitting of the experimental variogram of spatial structure at distances smaller than min reduces the spatial variability of the original data ||xi-xj|| (Cressie, 1991). When the nugget effect is set. This implies that the estimation variance com- observed over the entire range of h (flat variogram), puted by kriging (see below) underestimates the the absence of spatial structure at the scale of study real variance. An approach to this problem was can be assumed. investigated by Bárdossy et al. (1988, 1990) using To ascertain the contribution of a measure- fuzzy set theory. ment-error nugget effect to the overall variance of

TABLE A.1. Types of kriging classified according to assumptions on the data and on the predictor. m(x)=E[Z(x)] is the mean of the regionalized variable, which can be constant (m) or not (m(x)) over the field of study. Z(x) and Y(x) are random functions. See review in Cressie (1991).

Assumptions Linear predictor Non-linear predictor

m and γ (h) known Simple Kriging (Matheron, 1962)

m stationary, γ (h) estimated Ordinary Kriging Disjunctive Kriging

from Z(xi) (Matheron, 1962) (Matheron, 1976) Indicator Kriging (Journel, 1983) Isofactorial Models (Armstrong and Matheron, 1986a,b)

Z(x)=m(x)+Y(x), m(x) Universal Kriging deterministic, Y(x) (Matheron, 1969) stationary and γ(h) unknown

Z(x)=m(x)+Y(x), filtering of Intrinsic Random m(x), general covariance of Functions of order k (IRF-

Y(x) computed from Y(xi). k) (Matheron, 1973)

Z(x)=m(x)+Y(x), m(x) Markov Random Fields known, distribution of Y(x) (MRF) (Besag, 1974) multinormal, covariance function known.

Normalization of data: Trans-Gaussian Kriging Z(x)=φ(G(x)), where G(x) is (Cressie, 1991) multinormal

Logarithmic transformation Lognormal Kriging of data (Rendu, 1979; Journel, 1980)

130 F. MAYNOU the phenomenon it is necessary to sample with 3.1. Ordinary kriging replicates. When the nugget effect is thought to originate from a small-scale spatial phenomenon, 3.1.1. Point kriging: If Z(x) is a stationary ran- it is advised to sample with different spatial inten- dom function, the difference R(x)=Z(x0) - Z*(x0) is sities, with areas of increased sampling effort at also a random function, where Z(x0) is the real value small distances. The contribution of c0 to the of Z(x) at x0 and Z*(x0) is its estimator. As Z(x0) is overall estimation variance is important and spe- unknown, R(x) is also unknown but under the sta- cial attention has to be given to correctly model tionary hypotheses specified in section 1.2 of the this parameter (Brooker, 1986). Appendix, the computation of its two first moments becomes possible:

2 3. SPATIAL PREDICTION mE = E [R(x)] and σE = Var [R(x)], called the estimation variance One of the most important aspects of spatial modeling regards the possibility of making statis- With these two moments, we can specify the tical inference of a spatially autocorrelated vari- quality of the estimation: able. Here we will focus on the aspects of spatial 2 prediction (estimation) of the value of Z(x) at mE= 0, unbiased and σE small. points not sampled (interpolation) or over areas where a mean or global estimate has to be Matheron (1963) proposed a linear unbiased obtained. The process of estimating the value of estimator Z*(x0) of Z(x0) from the observed values

Z(x) at unknown locations is termed kriging Z(xi). This estimator and its variance can be com- (Matheron, 1963). Kriging over a grid of points is puted analytically and minimized under certain very useful to obtain an accurate of a conditions (Matheron, 1971; Journel and resource and in this regard can be seen as an Huijbregts, 1978). The linear estimator of Z(x0) is: interpolation or mapping method (point kriging).

Estimating the value of Z(x) over a bounded area Z*(x0) = Σi wiZ (xi), (3) is known as block kriging and can be used as a direct method of biomass assessment in fisheries. where wi are the kriging weights attributed to each The predictor (estimator) of Z(x) at x is noted Z(x ), subject to = 1 in order to guarantee unbi- 0 i Σi Z*(x0) and its estimation error Z(x0) - Z*(x0). The asedness. The vector of weights to be given to each theoretical estimation variance is: E(Z(x0) - observed value is obtained by solving a linear krig- 2 Z*(x0)) . Theoretical considerations (Cressie, ing system of equations. Kriging is then an opti- 1991, p. 109) indicate that it is possible to obtain mized weighting of the samples using their auto- two general classes of unbiased predictors: linear correlation structure (Matheron, 1971). The system predictors and non-linear predictors. Combining of equations is (Matheron, 1971): the several assumptions and the two classes of n predictors, different types of kriging are obtained. -Σwj γ (xi, xj) - µ + γ (x0, xi) = 0, i = 1...n j=1 They are summarized in Table A.1, with the most (4) n significant reference citations added. Here we { Σwi = 1 j=1 will illustrate in detail the linear geostatistical method of ordinary kriging and the non-linear where µ is a Lagrange parameter and γ(a,b) is the method of disjunctive kriging. The former is the value of the fitted variogram function from point a general method employed in most geostatistical to b. The kriging (or estimation) variance is fisheries papers (Conan et al., 1988; Maynou et (Matheron, 1971): al., 1996, 1998; Simard et al., 1992). The latter has a specific interest in fisheries as it allows esti- n mating the probability that Z*(x ) be above a cut- 2 w γ (x ,x ) + µ (5) 0 σ (x0) = Σ i 0 i off value. It can be seen that the simpler kriging k j=1 methods require less assumptions and less para- meters to be estimated and they are probably more realistic.

APPLICATION OF GEOSTATISTICS IN N. NORVEGICUS IN THE MEDITERRANEAN 131 The kriging variance can be used to give a pre- These variance terms can only be computed ana- cision index to the point estimates Z*(x0) or to con- lytically when working over simple geometric struct confidence intervals for the estimate, e.g. a shapes, and they have to be solved by numerical for Z*(x0) at the 0.05 p-level is approximations in cases where global estimates

Z*(x0)± 1.96 σk(x0), under the assumption of nor- have to be produced over irregular contours or in mality of the data Z(xi) (Cressie, 1991). places of complex topography (Conan et al., The kriging variance is affected by several 1994). factors, such as the type of kriging or the sampling The problem of change of support is encoun- design. In order to minimize the variance and to tered when trying to estimate the global fish abun- obtain precise estimates it is useful to bear in mind dance over a large area (in our case study ~ 100 that: km2) from trawl hauls which can be considered - The difference between the real underlying point samples (~ 0.01 km2). variogram function and the fitted model from the Other types of kriging, e.g. universal kriging or experimental variogram may be important. When kriging with intrinsic random functions of order k solving the kriging system of equations (4), the fit- can be more suitable to specific applications, espe- ted model is considered correct and the kriging cially when the data have a clear spatial trend, variance does not include the difference between which can be modeled and included in these more the underlying variogram function and the fitted advanced types of kriging (Matheron, 1969; 1973; model. An approach to this problem has been inves- Besag, 1974; Rendu, 1979; Journel, 1980; Cressie, tigated by Bárdossy et al. (1988, 1990) using fuzzy 1991). For a fisheries application, see Sullivan set theory. (1991), who modeled the spatial trend stemming - The actual sampling design (equally or from the dependence of fish abundance with depth. unequally spaced samples, preferential sampling) influences the kriging variance. 3.2. Non-linear geostatistics and disjunctive

- The data itself Z(xi) can have high or low vari- kriging ance. Specially, the ratio c0 to c’ is very important.

High c0/c’ ratios contribute to a high estimation Non-linear geostatistics are of interest in fish- variance. It is essential to give special attention to eries science as it allows computing the probabili- the modeling of the first distance classes of the ty that an estimated level of abundance is higher experimental variogram to accurately assess the than a user-defined cut-off value. This cut-off nugget value (Brooker, 1986; Cressie, 1991). value can be chosen, for instance, as a value beyond which the resource becomes economically 3.1.2. Global kriging and change of support: exploitable. In mathematical terms, the problem is The mean density of the resource, Z(B), over the to find an approximation to the conditional expec- estimation area B is (Matheron, 1971): tation P{Z(x0)≥z0 | Z(x)}, from the data values only. One such approximation is illustrated by disjunc- 1 tive kriging (Matheron, 1976). Other non-linear Z(B) = –––– Z(x)dx , (6) methods are indicator kriging (Journel, 1983) or ∫B |B| isofactorial kriging models (Armstrong and Matheron, 1986a; 1986b). where |B| is the area (or volume): |B| = ∫B dx. The An essential step in disjunctive kriging is to . global abundance of the resource is ZG(B)=Z(B) find a transformation φ of the original data, |B|. Z(x)=φ(Y(x)) such as to guarantee marginal nor-

The estimation of Z(B) from the data Z(xi) is mality in the bivariate distribution F1,2(z1, z2). Then complicated by the problem of change of support F1,2 is expressed in terms of F1 and F2 (the margin- (Matheron, 1971; Journel and Huijbregts, 1978). al distribution functions). The non-linear estima- The samples are defined over a support b, while tor is: Z(B) is defined over a block |B|. Additional vari- ance terms (Journel and Huijbregts, 1978, pp. 77- n 123, including computer programs) are added to f (Z(x )), Z*(x0) = Σ i i the kriging equations given above to include the i=1 error made when estimating Z(B) from the Z(xi). which cannot generally be solved analytically.

132 F. MAYNOU The f are approximated by expansions of i n K -1 Hermite polynomials of order K, as follows: Z*(x0) ≅Σ Σaikηk (φ (Z(xi))). i=1 k=0 Its estimation variance is given by: K

hi (y) = fi(φ (y)) = aikηk (y), i = 1...n, kΣ=0 K K n 2 2 k σ dk (x0) ≅ bk - bk aik ρ 0,i. K kΣ=0 kΣ=0 iΣ=1 φ(y) = bkηk (y), Σk=0 The value of K chosen to truncate the Hermite where ηk are Hermite polynomials of order K and polynomial is an important and subjective factor the coefficients aik are n x K unknowns. The k=1...K that may influence the quality of the estimation. system of equations is: Values of K=10 or higher can be recommended. An additional practical problem is the need to invert K n k k Σaik ρi,j = bk ρ0,j , j = 1...n, matrices of n x n, which can slow down the com- i=1 puting process considerably. The estimation of Z(B)

is analogous to Z(x0), taking into consideration the where ρi,j=1-γY (h)/C(0), Rendu (1980). These aik are point-to-block correction as discussed in 3.1.2 of employed in the computation of approximate esti- the Appendix. Disjunctive kriging algorithms can mator: be found in Yates et al. (1986).

APPLICATION OF GEOSTATISTICS IN N. NORVEGICUS IN THE MEDITERRANEAN 133