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Fishery Bulletin 99(4) 
1996) and EVA software (Petitgas and Prampant 2 ) to ana- 
lyze and visualize spatial distributions of sardine eggs/ 
min and to compute a variogram for each of the three 
age groups of sardine eggs. Two basic assumptions of the 
variogram (Eq. 1) are intrinsic stationarity and isotropy. 
Intrinsic stationarity means that a constant mean exists 
and the variance of egg density is defined by the mag- 
nitude of h. Isotropy means that spatial correlation and 
the range of correlation do not change with direction. The 
variogram is normally expressed as a function of three 
parameters: range, sill, and nugget effect. The range is 
the distance beyond which the observations are not corre- 
lated. The sill is the variance of the random field and is the 
asymptotic value of y(h) . The nugget effect measures the 
micromeasurement error and the white noise for h close to 
0 (Cressie, 1991). 
Ideally, for h close to 0, the variogram ( y(h )) will be close 
to zero because the observations tend to be similar. As h 
increases, the observations become h units apart and tend 
to be different, or co v(h) decreases, and the variogram in- 
creases. At a certain distance, h*, cov(h) approaches zero, 
and for h >h*, the variogram approaches its asymptote 
(sill). The distance, h*, is the range. The range (h*) was 
estimated from a model that best fits the data and was 
used to estimate the diameter of the patch of sardine eggs 
because eggs whose distances are less than h* nmi are 
correlated and thus are likely to be in the same patch. 
Conversely, eggs that are more than h* nmi apart are no 
longer correlated and thus are assumed to be in different 
patches. We chose the robust (or stable) estimator of the 
variogram (Cressie and Howkins, 1980; Cressie, 1991): 
gram: range, sill and nugget effect. The range was then 
used as the estimate of the diameter of the patch for each 
of the three age groups and total number of eggs. 
Results of the 1996 pilot CUFES survey 
Spatial correlation and patch size of sardine eggs 
For each of the three age groups and the total number 
of eggs, the four-directional variograms of the residuals 
of ln(eggs/min+l) from LOESS (Fig. 3) indicated that the 
variograms for transects at 0°, 45°, and 135° were clearer 
than the cross-transect (90°), particularly for the 1-day-old 
eggs, 2-day-old eggs, and total egg category. The variogram 
in the direction of within-transect (0 degree) had the clear- 
est signal because intervals between adjacent collections 
were the shortest. Because the variograms for each direction 
looked similar, we used the variogram in the within-transect 
direction to assess the spatial correlation of sardine eggs. 
The spherical model was chosen to fit the variogram 
(Cressie, 1991): 
y(h;6) -c 0 +c s (3 / 2)(||/r||/a s ) - (1/ 2)(||/z||/a s )' 
c 0 + c s , 
h = 0 
0 < INI < a s (3) 
No- 
where c 0 = the nugget effect; 
c s = the practical sill (variance - nugget); 
a s = the range; and 
II h || = Euclidean distance. 
T l 4 
| u(x) - u(x + h)\ V ~ 
2 ~(/l) — _N(h) 
* ~ [0.457 + 0.494/|Afi/i)|]Afi/i) 4 ’ 
where |M/j)| = the number of distinctive pairs; and 
h = the distance (nmi) between any two 
locations. 
To avoid possible trends, we first ran a local regression 
model (LOESS) of ln(eggs/min+l) against line distance (the 
distance computed from the survey lines, y-axis) and sta- 
tion distance fic-axis) (Chamber and Hastie, 1992) where 
the reference point (0,0) is a pseudo station in Mexico (sta- 
tion 260 on CalCOFI line 980; Fig. IB). We then constructed 
a variogram for the residuals from the local regression 
model (LOESS) in four directions clockwise from the tran- 
sect (0° ,45°, 90°, and 135°) to examine possible aniso- 
trophy. We chose natural logarithm (In) transformed data 
because the distribution of eggs was skewed. Finally, we 
used an interactive S+ function (model. variogram function) 
to determine the estimates of parameters for each vario- 
2 Petitgas, P., and A. Prampant. 1993. EVA (estimation vari- 
ance), a geostatistical software on IBM-PC for structure charac- 
terization and variance computation. CM1993/D:65, 81 st meeting 
of ICES. IFREMER laboratoire d’Ecologie Halieutique, BP 
21105,44311 Nantes (France), Cedex 03, 33 p. 
For total number of eggs (all eggs combined), the range 
(2) of the residuals of ln(eggs/min+l) was 22.2 km (12 nmi), 
sill (variance in the random field) was 0.4, and nugget was 
0.05. For 1-day-old eggs, the range was 14.8 km (8 nmi) 
and the sill was 0.3. For 2-day-old eggs, the range was 18.5 
km (10 nmi), the sill was 0.15, and nugget was 0.005, and 
for 3-day-old eggs, range was 22.2 km ( 12 nmi), the sill was 
0.065, and nugget was 0.005 (Fig. 4). Because the maxi- 
mum range was 22.2 km (12 nmi), eggs collected more 
than 22.2 km (12 nmi) apart were considered uncorrelat- 
ed; therefore, transect lines spaced intervals of 12 nmi or 
greater were considered independent. The data also indi- 
cated the gradual dispersion of egg sardine patches with 
time as described by Smith (1973) because the diameter of 
sardine egg patches increased from 14.8 km for 4-27 h old 
eggs to 22.2 km for eggs 52-75 h old. 
Conversion of CUFES egg density to full-water- 
column abundance and distribution of egg stages 
Egg counts from 91 paired samples collected with the 
CUFES and CalVETs during leg 1 of the 1996 survey 
(Table 1) were used to derive a conversion factor from 
eggs/minute of CUFES sample to CalVET catch (R). We 
used a regression estimator to compute the ratio of eggs/ 
minute from the CUFES to eggs/tows from CalVETs, R = 
p v /p x , where y = eggs/minute; x = eggs/tow; and R = the 
catch ratio. The estimator of R is R = Z(x x y )/Z(x 2 ). 
