ICES Journal of Marine Science 59: 000–000. 2002 doi:10.1006/jmsc.2001.1176, available online at http://www.idealibrary.com on
Høst, G., Berg, E., Schweder, T., and Tjelmeland, S. 2002. A Gamma/Dirichlet model for estimating uncertainty in age-specific abundance of Norwegian spring-spawning herring. – ICES Journal of Marine Science, 59: 000–000.
An approach to estimating the statistical uncertainty in age-specific abundance of
Norwegian spring-spawning herring is presented. The method is applied to data from a survey on the overwintering stock in the Vestfjord system in northern Norway in
December 1996. It is based on building separate statistical models for acoustic and biological data. The acoustic data are modelled as a gamma-transformed spatial random field and fitted to echo readings from the survey. The biological data are fitted to a two-stage model using multinomial sampling from Dirichlet-distributed age proportions. Uncertainty in the abundance by age distribution is obtained by bootstrapping. Previous assessments have a priori taken the abundance by age from a skewed distribution, such as gamma or log-normal families. In contrast, this analysis results in an estimated abundance by age that seems symmetric and does not display heavy tails.
2002 International Council for the Exploration of the Sea
Key words: acoustics, parametric bootstrap, random field, space-time analysis, statistical model, trawl data.
Received 21 December 2000; accepted 25 November 2001.
G. Høst and E. Berg: Norwegian Computing Centre, PO Box 114 Blindern, N-0314
Oslo, Norway; Tel: +47 22 852500; fax: +47 22 697600; e-mail: Gudmund.Host@nr.no.
T. Schweder: Department of Economics, PO Box 1095 Blindern, University of Oslo,
N-0317 Oslo, Norway. S. Tjelmeland: Institute of Marine Research, PO Box 1870
Nordnes, N-5817 Bergen, Norway. Correspondence to G. Høst.
The Norwegian spring-spawning herring is the largest fish stock in the Northeast Atlantic; its spawning stock is currently about 6 million tons but historically it reached >10 million tons. The stock spawns along the
Norwegian coast, feeds in the Norwegian Sea and overwinters in the Vestfjord system in northern Norway
(Røttingen, 1990).
Management of the stock is based on calibrating a simple population dynamics model to time trends in survey data, as is the case for most fish stocks in the
Northeast Atlantic. Uncertainty is assessed independently of the factors that a ff ect it: transect density, biological sampling, annual environmental e ff ects that may lead to varying target strength. The goal of the present paper is to contribute to establishing a method by which the uncertainty in a survey can be assessed directly from the observations. In future, surveys are likely to be even more data-rich than at present, allowing for elaborate statistical methodology. The proposed framework is a first attempt to meet these situations.
Adhering to the precautionary approach to fisheries management, uncertainty in abundance estimates should be directly reflected in catch quotas: greater uncertainty should result in smaller quotas. For Norwegian springspawning herring, an important source of uncertainty is that in the abundance estimates from research surveys.
Here we present a methodology for estimating the distribution of herring in the Vestfjord system, Norway, in December 1996. The method involves statistical space-time modelling of echo abundance, modelling the distribution of age/length samples from trawl samples, and finally combining these to obtain the probability distribution for abundance by age for the survey.
The assumption made in previous assessments of the stock (ICES, 1999) that the error distribution of surveys is constant across survey series and years is unfortunate, because factors a ff ecting the observation model (number of samples, environmental conditions, geographic
1054–3139/02/000000+00 $30.00/0 2002 International Council for the Exploration of the Sea

2 G. Høst et al.
coverage) may vary from year to year and between survey series. More important at present, the assessment is highly sensitive to assumptions made about the functional form of the survey error distribution, i.e. assuming a log-normal distribution or a gamma distribution with constant CV yields assessments that di ff er substantially (ICES, 1999). In this paper, we seek to establish the survey error distribution directly from the survey observations made each year, so making the assessment less dependent on subjective assumptions.
The approach is to build a statistical model that accounts for the variability observed in the various sources of data. The two main sources of data are acoustic data from a transect survey and age/length samples from research trawling. We model the acoustic data as gamma-distributed random variables and account for possible space-time autocorrelations. In addition, we take the observed age composition as multinomial samples from Dirichlet-distributed proportions. The statistical distribution for abundance by age is inferred by parametric bootstrapping from the model.
The data consist of echo integrator values and samples of age/length composition from a survey conducted in Vestfjord, Ofotfjord, and Tysfjord (the Vestfjord system) in the period 4–18 December 1996. The survey design and the method of abundance estimation based on survey data used so far are described by Røttingen et al.
(1994). Data were collected according to the standard protocol, with no view to the methodology presented here. However, we believe that our methodology can be used on all acoustic surveys carried out in the Vestfjord system for assessment of Norwegian spring-spawning herring.
There are 1837 depth-integrated echo readings from
747 di ff erent spatial locations. Each echo reading is integrated back-scattered energy from a strip of length one nautical mile. The echo recordings have not been contaminated by the presence of other species. The location of the survey area, the sampling intensity, and the time-averaged echo readings are shown in Figure 1.
The number of samples at each location varies between one and 16. The entrance of Tysfjord is the region most densely sampled in time, and the greatest abundance of herring is in the inner part of Vestfjord.
The marginal distribution of the echo readings is highly skewed. Skewed spatial data are often modelled on a logarithmic scale, but other marginal transformations may be more appropriate. Figure 2 is a quantilequantile plot of gamma- and log-transformed data against quantiles of the normal distribution. Except for the lower tail, the gamma-transformed data form a reasonably straight line, but the log-transformed data are more curved. This suggests that the gammatransformation is a better choice for the present data, an issue pursued in the next section.
The Vestfjord system has a complex geometry, as can be seen from Figure 1. To estimate the large-scale patterns in the herring distribution, such as the spatial trend, a geometric simplification is used. For this purpose, s is geographic coordinate and d( s ) a transformation of the two-dimensional geographic coordinate s into a one-dimensional ‘‘fjord coordinate’’. For each location s , d( s ) gives the corresponding distance between s and the entrance to Vestfjord, as depicted in Figure 3.
The choice of d( 0 ) is for convenience, and does not a ff ect the analysis.
In Figure 3, the fjord coordinate is the distance measured along the line drawn. The geographical coordinates are projected onto the line shown to give a fjord coordinate. Figure 4 shows the measured echo readings as a function of the fjord coordinate. The mean of the echo readings seems to increase to a maximum 20–40 nautical miles from the entrance to Vestfjord, then decays to almost zero before increasing farthest from the fjord entrance. Later, it will be shown that the fjord coordinate is a useful tool for capturing the main spatial features of an echo field.
As a rule, 100 fish are sampled at random from each trawl. The length of each fish in this sample is measured and its age determined. All samples consist of
Norwegian spring-spawning herring only. For the current analysis, there are 21 trawl samples of herring, all but two consisting of 100 fish. The sampling locations are shown in Figure 5.
About half the samples are taken from the area outside the Tysfjord–Ofotfjord entrance, roughly where the echo density is greatest. Figure 6 depicts the mean length at age for each sample. Although the curves seem similar, a tendency towards parallel lines suggests that investigating possible location e ff ects may be important in the future.
For a given location s and time t, the density (number of individuals per unit area) of herring by age and length is taken to be a realization of a multivariate random field in space-time. The joint distribution of this random field is generated by an abundance field and an age/length composition field. These fields are taken to be independent. The data source for abundance is the echo values, integrated over strips of one nautical mile along the survey transect. The age/length composition is also a random field, and the data source is a set of trawl samples giving joint age/length histograms for various locations and times. The aim here is to build the likelihood for each data source, then to multiply the two

Gamma/Dirichlet model for Norwegian spring-spawning herring 3
Figure 1. Survey area and echo readings data – (a) the number of readings at each location, (b) mean of the echo readings in each coordinate (the greater the number/value, the darker the shading).
to obtain the complete likelihood for herring density by age, length, space, and time. This complete likelihood will contain the information needed to infer summary statistics, such as area- and length-integrated numbers of herring of each age group.
The echo at location s and time t is denoted by Z( s ,t).
Exploratory data analysis suggests that the marginal distribution of the echo readings may be fitted to a gamma-distribution (see Figure 2). Therefore, Z( s ,t) is modelled as a gamma-transformed Gaussian random field. A gamma random field may be constructed in several ways. The approach taken here is to use a transformation of a suitable Gaussian random field Y( s ,t). Let
Statistical analysis of the acoustic data is based on geostatistical techniques, as recommended by ICES
(1993). However, an important extension is that the time dimension is specifically incorporated in the model.
Y( s , t)= f (d( s )) + ( s , t)
E{Y( s , t)}= f (d( s ))
(1)
(2)

4
2
0
–2
G. Høst et al.
Cov{Y( s , t ), Y( s , t )}=c( 0 s s
0 , / t t / ; ) (3)
Here, f ( · ) is a vector of low-order polynomials and is a vector of coe ffi cients, as is common in (universal)
Kriging models (Cressie, 1993). Further, d( s ) is the fjord coordinate, as explained in the next section. The zeromean residual term ( s ,t) has covariance function c( · ; ), where =(
2
, s
, t
) are the covariance parameters to be estimated. Thus, we propose that the large-scale spatial features depend only on the distance from the entrance to Vestfjord, while other features will be reflected in the residual field ( s ,t).
The relation between the Y-field and the original scale
Z-field is defined by the one-to-one transformation
Y( s ,t)= {G[Z( s ,t);a,b]} (4)
–4
–2 0 2
Quantiles of standard normal distribution
Figure 2. Quantiles of gamma-transformed (dots) and logtransformed (triangles) echo readings plotted against quantiles of the standard normal distribution.
where is the standard normal cumulative distribution function and G the cumulative distribution function for the gamma distribution with shape parameter a and rate parameter b. With no trend in Y, Z would, following
Equation (4), be gamma-distributed at each location in the space-time domain. However, owing to the trend f ,
Z is only approximately gamma-distributed. For this
68.5
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53.4
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0
61.1
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16.0
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Longitude (degrees E)
17.0
17.5
Figure 3. The path used to calculate the fjord coordinate, with some distances marked (in nautical miles). The fjord coordinate of any given location s
0 in the fjord is set equal to the fjord coordinate of the closest point on the full line.

Gamma/Dirichlet model for Norwegian spring-spawning herring 5
600 000
400 000
200 000
0
20 40
Fjord coordinate (nautical miles)
Figure 4. Echo readings as a function of fjord coordinate.
60 80 reason the process defined by Equations (1)–(4) is referred to as a gamma-transformed random field, to distinguish it from the gamma-random field of Wolpert and Ickstadt (1998).
To model the age/length field in space and time is di ffi cult. Here, we introduce some general notation to serve as building blocks for future work, in which the present simplified approach will be generalized.
Normally, age and length compositions may be regarded as random fields in space-time, but the present model of age and length compositions does not take such spatial or temporal structure into account.
For a given location in space-time, probability distributions for age and length compositions need to be specified. Here, we use a semi-parametric model for the age composition and re-sample from the observed length data conditional on age. For sampling of independent individuals of categorical variables, the distribution will be multinomial. However, the ages of individual fish in each trawl sample are likely to be strongly dependent.
In fact, a
2
-test of fit to a common multinomial distribution rejects five of 21 trawl samples at the 5% level. This suggests that a model allowing for excess variability in age frequencies should be used instead of a multinomial distribution.
In the approach presented here, we distinguish between the true underlying age composition of herring in the fjord and the sampled age composition. We take the true age proportions at a given time and location to follow a multivariate Dirichlet distribution. Further, we take each observed age composition as a multinomial sample conditional on the Dirichlet-distributed proportions at that time and location.
Denoting a as the underlying proportions of individuals at age a, it is assumed that a are Dirichletdistributed with parameters and a
; a=1, . . ., k, where k is the number of age-classes. The Dirichlet distribution
(Aitchison, 1986) is a multivariate distribution with density
In the Dirichlet distribution, each a follows a beta( a
) distribution. The first and second moments of under this Dirichlet distribution are a
, a

6 G. Høst et al.
68.5
68.4
68.3
68.2
68.1
68.0
67.9
14.5
15.0
15.5
16.0
16.5
Longitude (degrees E)
17.0
17.5
Figure 5. Locations of the trawl samples taken. Several samples were sometimes taken from the same or almost the same spatial location, but this is not visible on the Figure.
Denote by x a the observed count of individuals at age a in a sample i of n individuals. Conditional on (n, a
), x a follows a binomial (n, a
) distribution and, therefore, it has moments
E{x a
}=n
Var{x a a
}=n a
(1 a
)
The parameter may be regarded as the ‘‘e ff ective sample size’’ in the Dirichlet distribution. We assume that is constant for the survey, whereas the age proportions a may vary with time and location. In this approach, is estimated by one parameter for the whole survey by maximum likelihood. Then, a is estimated for each space-time sample i by a,i
= (x a
/ n) i
, with the constraint k a = 1 a,i
= ; i=1, . . ., 21.
Now, a bootstrap sample of the observed age composition in trawl sample i may be generated by first drawing a Monte Carlo sample
* a,i
; a=1, . . ., k from a Dirichlet distribution with parameters , a,i
; a=1, . . ., k, then drawing a Monte Carlo sample x
* a,i
; a=1, . . ., k from the multinomial distribution conditional on
* a,i
; a=1, . . . k and n i
. The Dirichlet sampling is done by combining gamma-distributed samples.
For generating the distribution of herring abundance by age, an age/length distribution for each measured echo reading in space-time is needed. This is achieved by using the closest trawl sample for each echo reading, giving 21 sub-regions (some sub-regions are identical, owing to identical trawl locations). Each sub-region has a unique age/length composition. The allocation of echo measurement locations to trawl locations is shown in
Figure 7. The centre of the radiating lines in this map shows the location of the trawl sample; the radiating lines extend to the acoustic readings assumed to be represented by the trawl sample.

Gamma/Dirichlet model for Norwegian spring-spawning herring 7
35
30
25
20
2 4 6 8
Age (years)
10 12
Figure 6. Mean length plotted against age, one curve for each of the 21 trawl samples. For each sample, a line is drawn between the mean lengths at the di ff erent ages.
The method for generating a bootstrap sample of herring abundance by age is as follows. First, draw a sample of the echo field from the fitted space-time field.
For each sub-region and trawl sample j, draw an age/ length composition as follows: draw the ages from the hierarchical Dirichlet-multinomial model described above, then resample the lengths from observed lengthsat-age. By averaging the squared lengths we obtain a mean-squared length l
2 j to use for each sub-region j.
Now, the conversion from echo backscatter Z( s i
,t i
) to density of herring per square nautical mile, ( s i
,t i
), may be done according to the formula where the conversion factor is taken from Foote et al.
(1997). By summing over all echo readings within the sub-region, herring density and age composition for that sub-region is obtained. A bootstrap estimate of herring density for the whole survey is obtained by repeating the process for other sub-regions, and weighting the sub-regions according to their spatial contribution to the total survey area. The whole procedure is repeated 1000 times to generate 1000 bootstrap estimates of abundance by age for the survey. These bootstrap estimates may be used as input to stock assessment models.
The echo readings were fitted to the model, Equations
(1)–(4), as follows. First, the transformation parameters a and b were estimated. It is necessary to account for the roughly 5% zeros in the echo readings, because the gamma distribution does not have support at zero. The zero recordings are considered as gamma-distributed random variables from the interval (0,1), also with parameters a and b. We iterate to find stable estimates of a and b. In other words, we impute random values on
(0,1) from the current a and b estimates, then re-estimate a and b using observations and imputed values, then reimpute, etc. The procedure converges after only a few iterations. Alternatively, a model that allows for zero values could have been fitted. However, it is unlikely that the results here are sensitive to how zeros are

8 G. Høst et al.
68.5
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67.9
14.5
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Longitude (degrees E)
17.0
17.5
Figure 7. Assignment of echo readings to the nearest trawl sample. The rectangle in the lower right corner shows the distortion of the map plot; undistorted, it would be a square.
( handled, because there are very few zeros in the present data.
A third order polynomial in the fjord coordinate gave a reasonable fit to the spatial trend on the Y-scale. The estimated coe ffi cients in this polynomial were
0
,
8.08
10
1
,
5
)
2
,
3
)=( 3.70,0.403, 0.0109, and the fitted gamma parameters were (aˆ, b )=(0.293,
5.73
10
6
). The fitted trend and transformed data on the Gaussian scale are shown in Figure 8.
Subtracting the fitted trend from the transformed data results in fitted residuals. A smoothed empirical spacetime variogram of the residuals is shown in Figure 9. For a given lag in space and time, the empirical space-time variogram is the average of squared di ff erences of residuals within the lag. For example, if there was strong positive space-time correlation in the residual field, the empirical variogram would be anticipated to be small for small lags and bigger for larger lags in space and/or time.
There are no indications of residual spatial correlation, but some tendency for temporal correlations at lags of half a day and one day. These temporal e ff ects may be due to diurnal vertical movements of herring, but because they are quite small, they are not considered in the analysis.
Figure 10 shows quantiles of the fitted residuals on a transformed scale against quantiles of the standard normal distribution. The fit seems reasonable, but the upper tail is too light. The maximum likelihood estimate of the parameter in the Dirichlet distribution was 8.43, the value used when sampling from the multinomial.
Figure 11 shows 1000 bootstrap samples of the total number of herring in the survey. The total survey area used in converting from herring density to absolute number of fish is 326.5 nautical miles
2
. The 95% confidence interval is (21 187 10
6
, 24 771 10
6
), with a mean of 22 895 10
6 fish.
The fitted density curve seems quite symmetric and does not have heavy tails. Using the data directly gives a point estimate of 22 708 10
6 fish, as indicated by the vertical line in Figure 11. The point estimate is located almost in the centre of the bootstrap distribution of simulated estimates, indicating that the proposed estimator has negligible bias. Figure 12 shows the same bootstrap samples disaggregated into the various ageclasses. The picture is slightly more complex, with some

Gamma/Dirichlet model for Norwegian spring-spawning herring 9
2
0
–2
–4
20 40
Fjord coordinate (nautical miles)
60 80
Figure 8. Transformed echo readings as a function of fjord distance. The fitted third-degree polynomial trend is shown.
skewness at high ages. The number of individuals varies between age-classes as a result of variability in recruitment. In particular, the 1983 year-class was very strong, giving a relatively high abundance of 13-year olds.
Table 1 shows the correlations between the abundance in the various age-classes.
The approach taken here is introductory and exploratory, and it may be enhanced in various ways. However, the main framework is believed to be a promising tool for further research. Some limitations and areas for improvements are discussed below.
The introduction of a fjord coordinate seems helpful in explaining the large-scale spatial features of the echo field. In future, a more elaborate definition may be possible, perhaps allowing for topographic e ff ects in this field. Moreover, the approach to modelling the random field of echo readings is quite crude, but it was chosen to allow for spatial dependence. Diggle et al.
(1998) give a generalized linear mixed-model approach to spatial modelling of non-Gaussian data, using Markov Chain
Monte Carlo methodology for inference. Wolpert and
Ickstadt (1998) construct a spatial gamma random field by an inverse Le´vy measure algorithm. Their approach gives independent and gamma-distributed volume over disjoint areas, with appropriate scaling under spatial aggregation and refinement. In contrast, the simpler approach taken here is a convenient tool to explore the main e ff ects and may serve as a starting point for more elaborate modelling.
The question of whether the acoustic data are spatially correlated or not is di ffi cult and important. Intuitively, positive correlation would be anticipated up to some spatial and temporal lag. However, the analysis does not reveal such correlations in the spatial domain.
The reason may be that the spatial averaging of the back-scattered energy masks such spatial correlation, i.e.
that the correlation range is <1 nautical mile. If this is the case, there is a good rationale for sampling with the given resolution, because under this spatial sampling plan the echo residuals may be treated as virtually uncorrelated. Alternatively, for uncorrelated residuals one could use Generalized Additive Models (Borchers et al.
, 1997), although some marginal transformation of the raw data may still be necessary.
In the temporal domain, there is a problem that the
‘‘time series’’ are not systematically sampled to uncover such temporal structure. An alternative design may be

10 G. Høst et al.
5
4
3
2
1
0
6
5
4
Time (da
3 20 ys) 2
15
10
1
5
Fjord coordinate (nautical miles)
0 0
Figure 9. Smoothed version of the empirical space-time variogram of the residuals.
25
30
2
1
0.0004
Mean = 22 895 SD = 901 CV = 0.039
0.0003
0
–1
0.0002
–2
0.0001
–3
–2 0 2
Quantiles of standard normal distribution
Figure 10. Quantile-quantile plot of the residuals and the standard normal distribution, plotted with the 1:1 line.
0.0
20 000 21 000 22 000 23 000 24 000 25 000 26 000
Total population (millions of fish)
Figure 11. Empirical probability density of the total population of Norwegian spring-spawning herring, using 1000 simulations.
chosen if the temporal structure had been the main target of the study. Diurnal variability in the echo readings attributable to vertical migration has been reported by Vabø (1999) and Huse and Korneliussen
(2000). By analysing acoustic data on a finer spatial scale and disaggregating in the vertical, it may be possible to incorporate these e ff ects into the statistical model in future.
Depth e ff ects may also be important in trawl sampling. In particular, age compositions may depend on depth. Here, the sampled age compositions were used directly to reflect spatial dependence. In this case, spatial variability of the age-composition field would depend on the sampling configuration. Alternatively, one could introduce spatial smoothing of the age compositions in terms of a hierarchical model (Wikle et al.
, 1998), a

Gamma/Dirichlet model for Norwegian spring-spawning herring 11
0.008
0.006
0.004
0.002
0.0
0.010
0.006
0.002
0.0
Mean = 1010
SD = 45 CV = 0.044
900 1000 1100
Millions of
2-year-old fish
Mean = 804
SD = 40 CV = 0.05
700 750 800 850 900 950
Millions of
6-year-old fish
0.004
0.003
0.002
0.001
Mean = 1900
SD = 98 CV = 0.052
0.0
1600
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0.004
0.0
1800
Millions of
3-year-old fish
Mean = 557
SD = 33 CV = 0.059
500 550
2000
600
2200
650
Millions of
7-year-old fish
0.0008
Mean = 12 202
SD = 520 CV = 0.043
0.0015
Mean = 5514
SD = 234 CV = 0.042
0.0006
0.0010
0.0004
0.0002
0.0
11 000 12 000 13 000 14 000
0.04
0.03
0.02
0.01
0.0
Millions of
4-year-old fish
Mean = 104
SD = 10 CV = 0.093
80 90 100 110 120 130 140
Millions of
8-year-old fish
0.0005
0.0
5000 5500 6000
Millions of
5-year-old fish
0.30
0.25
0.20
0.15
0.10
0.05
0.0
Mean = 6
SD = 2 CV = 0.242
4 6 8 10
Millions of
10-year-old fish
12
Mean = 53
SD = 5 CV = 0.096
Mean = 53
SD = 4 CV = 0.084
Mean = 691
SD = 37 CV = 0.054
0.008
0.006
0.004
0.002
0.08
0.06
0.04
0.02
0.010
0.008
0.006
0.004
0.002
0.0
0.0
40 50 60
Millions of
11-year-old fish
70
0.0
40 50 60 70
Millions of
12-year-old fish
600 650 700 750 800 850
Millions of
13-year-old fish
Figure 12. Empirical probability densities of di ff erent age-classes of Norwegian spring-spawning herring, using 1000 simulations.
Table 1. Correlation matrix of Norwegian spring-spawning herring age populations, from 1000 simulations.
Age 0 2 3 4 5 6 7 8 10 11 12 13
7
8
5
6
2
3
4
10
11
12
13
1.0
0.9
0.8
0.6
0.6
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0.1
0.6
0.9
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0.7
0.4
0.4
0.3
0.3
0.1
0.3
0.1
0.3
0.8
0.7
1.0
0.9
0.4
0.4
0.2
0.1
0.2
0.3
0.4
0.6
0.4
0.9
1.0
0.5
0.6
0.3
0.2
0.3
0.7
0.6
0.6
0.4
0.4
0.5
1.0
0.9
0.8
0.4
0.8
0.4
0.9
0.5
0.3
0.4
0.6
0.9
1.0
0.8
0.7
0.8
0.6
0.9
0.5
0.3
0.2
0.3
0.8
0.8
1.0
0.7
1.0
0.1
0.7
0.2
0.1
0.1
0.2
0.4
0.7
0.7
1.0
0.6
0
0.3
0.5
0.3
0.2
0.3
0.8
0.8
1.0
0.6
1.0
0.1
0.7
0.1
0.1
0.3
0.7
0.4
0.6
0.1
0
0.1
1.0
0.6
0.6
0.3
0.4
0.6
0.9
0.9
0.7
0.3
0.7
0.6
1.0
mixture model (Ferna´ndez and Green, submitted), or a semi-parametric smoothing of the average age composition field. This may be an important topic for future research.
Other topics for further study may involve analysis of alternative spatial trend models, and checking for spatially heterogeneous variance of the acoustic data. To be able to investigate these problems in more detail,

12 G. Høst et al.
several data sets need to be analysed. The problem of building a spatial model for age/length composition is regarded as most pressing, and for such work, intensive biological sampling in space-time would be useful.
Our uncertainty estimate of herring abundance may be an underestimate, because some sources of uncertainty may not have been taken into account. Some such sources may be uncertainty in the backscatter-toabundance conversion formula, and uncertainties in age readings and survey area. Uncertainty in the abundance estimate attributable to these and other additional sources has been di ffi cult to assess in this study.
The research done at the Norwegian Computing Center was funded by the Institute of Marine Research in
Bergen.
Aitchison, J. 1986. The statistical analysis of compositional data. Monographs on Statistics and Applied Probability.
Chapman & Hall, London. 416 pp.
Borchers, D. L., Buckland, S. T., Priede, I. G., and Ahmadi, S.
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