Stock Abundance Indices Research Articles

Lessons from long-term monitoring of tropical rock lobsters to support fisheries management

Fishery-independent scientific surveys are highly valuable for monitoring lobsters and other fisheries. Yet, there are relatively few long-term monitoring surveys to support the provision of management advice. A number of challenges can prevent the continuity of long-term monitoring series. This paper uses the 35-year continuous annual dive surveys for Australia’s Torres Strait tropical rock lobster (TRL) to overview strengths and limitations of a long-term scientific survey, as well as lessons learnt from having overcome a range of challenges. Fishery-independent surveys are more reliable than fishery-dependent data and are more robust to external shocks such as market drivers and fuel prices. The primary purpose of the surveys is to provide a reliable index of stock abundance together with data to inform on size and age structure, plus model parameters and reference points. Here, we also highlight additional survey benefits including: yielding positive gains relative to cost, facilitating an adaptive management approach that serves as an early warning system, reducing and resolving inter-sector conflicts, reduced frequency of stock assessments, providing data from associated habitat monitoring, improved stakeholder buy-in, utility for development of Operating Models for Management Strategy Evaluation, contribution to achieving certification, contributing to biodiversity agreements and consistent cross-jurisdictional monitoring. We analyse lessons learnt from challenges such as changes in resource management, budget cuts, increased health and safety requirements, maintenance of capability and staff capacity, stakeholder buy-in, need to validate survey results, a pandemic and changing climate. Our experiences highlight why long-term scientific surveys are considered the gold standard for fisheries monitoring and we provide insights for building and maintaining key marine monitoring series.

Stakeholder-government collaboration in developing cost-effective fishery-independent surveys in rights-based and co-managed fisheries

Fishing rights and co-management are considered important aspects of fisheries management. But with rights comes a responsibility to contribute data that improves stock assessments and harvest strategies. This responsibility should include contributing data based on fishery-independent surveys (FIS) that provide robust measures of stock abundance and greater confidence in stock assessments, reducing risks to stock sustainability and resulting in more reliable management. FIS conducted with stakeholders provides a direct link between stock abundance indices and their own experience giving them ownership and a better appreciation of stock status. We examine case studies from Western Australian fisheries in four categories: FIS conducted using research vessels; FIS conducted on recreational fisheries using citizen-science; FIS conducted using volunteer industry vessels, and FIS using chartered industry vessels. These FIS cover fisheries with different gear, management, fishing sectors, value and number of participants. The case studies demonstrate that there are a number of ways that FIS designs may be implemented, with each fishery having a unique approach with a common element that they represent standardised surveys. We consider that all fisheries require some type of FIS and this paper discusses options of how this might be achieved. However, FIS can be expensive, so surveys must be cost effective to ensure that they endure. The approach to developing FIS should include collaboration with stakeholders in a co-management agreement. The FIS could be a requirement of having fishing rights and taken into account when developing management arrangements, e.g. allocation of part of the entitlements for a FIS.

Economic development in times of population decline—a century of European eel fishing on the Swedish west coast

Abstract The European eel stock is in a multi-decadal decline. When fishing yield diminished throughout Europe, the small-scaled fyke-net fishery for eel on the west coast of Sweden gradually increased. This contrary trend lasted from the early-1900s, until the 1990s when fishing restrictions eventually limited the catch. We identified the processes driving this aberrant historical development. Using data on the fisheries from 1914 to 2006, we analysed the relation of total landed quantities to stock abundance indices, weather conditions, technical development, and market indicators. No relation between landed volumes and abundance indices was found, but market price (inflation-adjusted) was clearly correlated. Weather and technical development had a minor influence on landed volumes. This indicates that the commercial eel fishery on the west coast developed due to increasing demand and increasing eel prices. We found no evidence that the local stock has been fully exploited, though the increasing catch must have gradually reduced the contribution to the international spawning stock. These results stress the importance of considering economic processes when interpreting historical catch data as a source of information on population size in stock assessments, and ultimately, understanding the collapse of the eel stock.

Diet Composition and Seasonal Fluctuations in the Feeding Habit of Snow Barbel (Schizothorax plagiostomus) in River Indus, Pakistan

Background:Schizothorax plagiostomus is widely distributed in river Indus and is most important food fish in Pakistan. The feeding habit of fish is directly related to the size of fish, its metabolic rate and environmental temperature. The accurate description of fish diet and feeding habit is a very important aspect in fisheries management for the purpose of species conservation, breeding and culture. The present work was aimed to investigate the specie abundance, the diet composition and seasonal variations in the feeding habit of Snow barbell Schizothorax plagiostomus.Materials, Methods & Results: A total of 1799 fish specimens were caught at the confluence of six tributaries along river Indus at Indus Kohistan, northeastern Pakistan. The fish were collected by 5-panels of gill net during first week of each month. The site specific Catch Per Unit Effort (CPUE) and season specific CPUE of fish fauna were assessed. For the gut content analysis 240 samples (99 male and 141 females) of S. plagiostomus were selected on monthly basis. Frequency of occurrence method and volumetric method were applied to record the different food items in the gut of S. plagiostomus. The physico-chemical parameters, NO3 concentration and dissolved Co2 of water from different localities of river Indus were recorded month wise by Hach sensION 156 meter, Horiba LAQUA Nitrate Meter and EA80 meter respectively. Significant difference was observed in water temperature during the four seasons. Except alkalinity no other water parameter showed significant variation across different localities. The results showed that highest Mean CPUE was observed for Darel Stream (0.55) and lowest for Jalkot stream (0.26). Peak abundance of fish was recorded in the month of November with a mean catch of 44.50, mean CPUE of 0.74 and mean Kruskal-Wallis rank value of 63.25. Spirogyra and Ulothrix occurred as maximum food items in the gut of S. plagiostomus during summer while their minimum amount occurred during autumn. According to the ranking index spirogyra and ulothrix ranked higher with significant difference in comparison to other food items. The results showed that S. plagiostomus is phytophagous in its feeding habit, which consumed mainly algae attached to stones and pebbles during the whole year. However, the presence of some secondary items such as animal matter, detritus, sand and mud might be due to the distinct availabilities of food along the seasons. The highest feeding activity of S. plagiostomus was recorded during summer while the lowest one occurred during autumn, spring and winter. Discussion: Catch per unit effort (CPUE) is an indirect measure of the abundance of a target species. It is used as an index of stock abundance in fisheries and conservation biology. During the study low fish fauna was found in River Indus as reported previously. Majority of the fish occurred in snow fed river tributaries in the study area as these tributaries are comparatively less turbulent. Previous studies have also recorded that Schizothoracine generally prefer clean waters. The present findings of gut contents analysis showed clearly that S. plagiostomus is a phytophagous fish which scrap and consumed spirogyra and ulothrix attached to stones and pebbles. Earlier it was reported that mouth of S. plagiostomusis is inferior, wide, with deep lower jaw having keratinized cutting edge and the lower lip is folded and expanded with numerous papillae making it best suited for scrapping algae attached to stones and pebbles. The highest feeding activity was observed during warmer months as compared to cold months. S. plagiostomus spawn twice in a year in autumn and in spring. The highest feeding activity of S. plagiostomus seems to be link with a reflex of recovery strategy due to physiological process of gonadal development.

Forecasting dynamics of red snapper (Lutjanus campechanus) in the U.S. Gulf of Mexico

Understanding dynamics and stock structures of fish is particularly relevant to assessments and management of marine living resources. Using the nonparametric, nonlinear time series (NLTS) approach, we modeled dynamics of red snapper (Lutjanus campechanus) represented by time series of two fisheries-independent abundance indices and two fisheries-dependent abundance indices in the eastern and western portions of the U.S. Gulf of Mexico (Gulf). Further, we examined regional dynamics of red snapper in the two areas and explored the utility of NLTS models in generating short term forecasts. Overall, red snapper in the eastern Gulf and western Gulf displayed distinct patterns in terms of the Gulf-wide ecosystem indicators, which likely implies different underlying regional dynamics of the species. Moreover, dynamic features of red snapper differed between the two regions. Specifically, the system dimension (mean±SE), i.e., the number of potential processes affecting the underlying dynamics of the species, was higher in the western Gulf (5.5±1.32) than in the eastern Gulf (3.75±1.44). The NLTS models exhibited significant skill (i.e., a measure of the goodness of fit (ρ) between observations and predictions) in forecasting red snapper abundance indices. The forecast skill (one year ahead) was 0.48±0.01 for indices representing the eastern Gulf and 0.33±0.05 for indices representing the western Gulf. The average dimension and forecast skill was 3±1.43 and 0.37±0.01 for fisheries-dependent indices, and 6.25±1.32 and 0.45±0.05 for fisheries-independent indices. These findings have implications for the Gulf red snapper fisheries in that the NLTS approach shows potential for forecasting stock abundance indices, and provides new information regarding the appropriate spatial scale for management of the species. Moreover, the ecosystem considerations in this study can be further explored to forecast the dynamics of red snapper for better assessments and management of the species.

The relationship among catch, fishing effort, and measures of fish stock abundance: implications in the Adriatic Sea

Relationships among catch, fishing effort, and measures of fish stock abundance have several implications for fisheries research. In this context, spatial and seasonal aspects are of significant importance for management decisions, especially when effort regulation schemes are used. In this paper, the multispecies trawl fishery in the Northern and Central Adriatic Sea was investigated, taking into account the heterogeneous distribution of fish stocks. Two approaches are presented depending on the availability (or not) of fishery-independent indices of stock abundance. The empirical results indicate that (i) aggregation and targeting behaviours affect catches by modifying the relationship between abundance and catch per unit effort and (ii) these relationships are not homogenous across space. Data from the Adriatic Sea is still insufficient to guarantee reliable estimations. However, these preliminary results call into question management decisions being made on the basis of catch per unit effort. Furthermore, the high heterogeneity between the northern and central areas of the sea basin calls for the adoption of spatially explicit management systems.

Using inshore fishery acoustic data on Atlantic herring (Clupea harengus) spawning aggregations to derive annual stock abundance indices

Acoustic data collected during fishing activities has been used to obtain nightly biomass and exploitation rate estimates for Atlantic herring (Clupea harengus) spawning components. Beginning in 2002, fishery based acoustic data were collected on the five major fall spawning aggregations in the sGSL and nightly biomass was estimated. Derivation of an annual seasonal index of herring biomass from fishery nightly biomass estimates have been problematic because of uncertainties associated with the behaviour of herring which accumulate prior to spawning, as well as a significant proportion of missing observations. We have developed a model which addresses these issues using nightly biomass estimates derived from acoustic data collected between 2002 and 2012 from the Fisherman's Bank fall spawning component gillnet fishery in the Southern Gulf of St. Lawrence. We conclude that these estimates identify annual changes in biomass and exploitation rate in this fishery. The method is generally applicable where the fishermen's search pattern covers the range of available fish schools during a fishing trip. Using inshore fishery biomass estimated from acoustic data to derive annual stock assessment indices could lead to the introduction of the proven reserve concept and empirical reference points to be incorporated into fishery management decisions, thus making the advisory process more transparent and responsive to local herring abundance.

Own-price elasticity of open access supply as a long-run measure of fish stock abundance

Extensive criticisms of the management of marine fisheries periodically arise for a variety of reasons. While the complexity of the US fisheries management process is partially to blame, these reasons manifest themselves in statements that include a perceived decline in living marine resource abundance levels, a proposed change in the regulatory infrastructure, or improvements in stock abundance that some feel should allow for new entrants. Following these criticisms are calls to end the federal management process or at the very least the replacement of the fishery management agency leadership. These demands driven by rent seeking behavior use short-term declines in abundance as justification when the long-run abundance measures would have indicated improvements in stock abundance and achievement of management objectives. Empirical, long-run, stock abundance indices are estimated using synthetic demand methodology, based on bioeconomic fishery theory, applied in a general and partial equilibrium theory framework to determine if maximum sustainable yield (MSY) management targets have been achieved and maintained for four fisheries considered to be successfully managed and one characterized as a socio-economic disaster. The abundance of a fish stock is more influence by extra-market and extra-fishery forces than by fisheries managers who base allocation decisions solely on single species stock assessments.

A new spatial framework incorporating uncertain stock and fleet dynamics for estimating fish abundance

AbstractRelative abundance indices based on catch and effort data can become biased unless consideration is given to the spatial dynamics of the fishery such as changes in either the spatial distribution of fishing effort or the range of the stock over time. The construction of such indices therefore needs to take into account features of the fishery itself. In this paper, a general framework is presented for developing more appropriate abundance indices based on fishery catch and effort data. In developing this framework, it adopts the approach of (i) developing a range of hypotheses which encompass the uncertainties in the spatial–temporal dynamics of the stock and the fishing effort, (ii) identifying the hypotheses underlying the different CPUE series, and (iii) evaluating the available information relative to these hypotheses as the basis for evaluating CPUE indices. Observations from the fishery for southern bluefin tuna (Thunnus maccoyii) are used to illustrate various hypotheses about the nature of the fishery which can be used to construct indices of stock abundance while a simple simulation framework is used to explore the implications of some of these hypotheses on the accuracy of such indices.

Constructing stock abundance indices from catch and effort data: Some nuts and bolts

Construction of annual indices of stock abundance based on the standardisation of catch and effort data remains central to many fisheries assessments. However, while the use of advanced statistical methods has helped catch rates to be standardised accounting for many explanatory variables, some of the more routine aspects of constructing abundance indices, such as how the index is constructed from the model parameters, receives little explanation in many analyses. This has lead to a lack of understanding as to how the indices are constructed and in some instances incorrect techniques have been applied. This lack of understanding can be a particular issue when interactions are used in the standardising model, especially those which contain interactions with the temporal effects over which the time-series of the abundance index is required. Other issues include the use of weighted model fits, the influence of anomalous data values, how best to impute missing values when required, the consequences of model mis-specification, and the use of random-effects. In this study, the basic approach for constructing abundance indices is outlined and a worked example using simulated data is presented to explore the nature of these issues and several techniques are suggested for dealing with them and to overcome potential biases. Finally, the methods presently used by ICCAT scientists for constructing abundance indices are reviewed.

Investigating the influence of length–frequency data on the stock assessment of Indian Ocean bigeye tuna

Most integrated stock assessment models are fitted to alternative sources of data like indices of abundance and length/age composition of catches in specific fisheries. While indices of abundance are often standardized over time, not much attention is paid to the temporal stability of the length/age data. A sequential approach to fitting model outputs to all sources of data, varying the weight given to the length composition data, for Indian Ocean bigeye tuna (Thunnus obesus) was examined in this paper. Logistic, double normal, and cubic spline selectivity functions were used to model the size composition of catches in the main industrial fisheries (longline and purse seine). Overall, there was a poor fit of stock assessment models to the individual length frequency observations collected from these fisheries, although marginal improvements of fit was made when temporally variable selectivity was implemented in the Stock Synthesis framework using the above described functions. The most influential factor in the assessment was the weighting of the length composition data relative to the indices of stock abundance. Contradictory signals between these two data sources have a large effect on spawning biomass dynamics, and inference based on these weightings can produce different management conclusions. We emphasized that understanding the data was the key to performing a well-calibrated stock assessment, and further refinements to the approach pursued in the analysis presented are discussed.

Management Strategy Evaluation of Fisheries Resources in Data-poor Situations Using an Operating Model Based on a Production Model

The allowable biological catch (ABC) in Japanese fishery management is currently determined by applying harvest control rules, which are categorized into two types depending on whether or not stock-size information is available. We evaluate management procedures (MPs) in data-poor situations using an operating model (OM) based on a production model. The OM incorporates uncertainties regarding its assumptions and the process and observation errors of population dynamics and fishing processes, while the values of the MP parameters are determined to avoid stock collapses and low catches. We evaluated the MP formulation ABC = δ×Ct ×(1+k(b/I)), where δ is a coefficient dependent on the stock status level on the stock size, Ct is the catch in year t, k is the weight coefficient, b reflects the trend in stock abundance index over time, and I is the mean of the stock abundance index I. This study shows that smaller values of k reduce the frequency of substantially low catches, particularly when there is significant uncertainty surrounding the stock status.In addition, the value of δ affects both the frequency of fishery collapse as well as stock and catch sizes. We conclude that more reliable stock abundance indices are necessary if the stock size and catch are to stabilize and MPs become more robust to uncertainties.

Killer whale (Orcinus orca) depredation effects on catch rates of six groundfish species: implications for commercial longline fisheries in Alaska

AbstractPeterson, M. J., Mueter, F., Hanselman, D., Lunsford, C., Matkin, C., and Fearnbach, H. 2013. Killer whale (Orcinus orca) depredation effects on catch rates of six groundfish species: implications for commercial longline fisheries in Alaska. – ICES Journal of Marine Science, 70: 1220–1232. Killer whale (Orcinus orca) depredation occurs when whales damage or remove fish caught on longline gear. This study uses National Marine Fisheries Service longline survey data from 1998–2011 to explore spatial and temporal trends in killer whale depredation and to quantify the effect of killer whale depredation on catches of six groundfish species within three management areas in Alaska: the Bering Sea, Aleutian Islands and Western Gulf of Alaska. When killer whales were present during survey gear retrieval, whales removed an estimated 54–72% of sablefish (Anoplopoma fimbria), 41–84% of arrowtooth flounder (Atheresthes stomias) and 73% (Bering Sea only) of Greenland turbot (Reinhardtius hippoglossoides). Effects on Pacific halibut (Hippoglossus stenolepis) and Pacific cod (Gadus macrocephalus) were significant in the Western Gulf only with 51% and 46% reductions, respectively. Overall catches (depredated and non-depredated sets) for all groundfish species significantly impacted by killer whale depredation were lower by 9–28% (p < 0.05). Effects on shortspine thornyhead (Sebastolobus alascanus) catches were not significant in any management area (p > 0.05). These results provide insight into the potential impacts of killer whale depredation on fish stock abundance indices and commercially important fisheries in Alaska and will inform future research on apex predator–fisheries interactions.

Development and application of an agent-based model to evaluate methods for estimating relative abundance indices for shoaling fish such as Pacific rockfish (Sebastes spp.)

Abstract Thorson, J. T., Stewart, I. J., and Punt, A. E. 2012. Development and application of an agent-based model to evaluate methods for estimating relative abundance indices for shoaling fish such as Pacific rockfish (Sebastes spp.). – ICES Journal of Marine Science, 69: 635–647. Many marine fish, including Pacific rockfish (Sebastes spp.), exhibit habitat-selective and shoaling behaviours, which can lead to imprecision when using survey data to estimate an annual index of stock abundance. We develop a spatial agent-based model (ABM) for Pacific rockfish, which generates data similar to those observed in existing bottom-trawl surveys and can represent various spatial and shoaling behaviours. We use the ABM to evaluate the performance of a model that uses mixture distribution methods to account for fish shoals and delta-methods to account for range expansion or contraction. This delta-mixture model is compared with conventional delta-generalized linear models (delta-GLMs) and a quantile regression delta-model. The delta-mixture increases precision by 15% relative to delta-GLMs in estimated abundance indices when shoaling behaviours are present, whereas precision is similar between delta-GLM and delta-mixture models when shoals are absent. The delta-quantile method has similar improvements over conventional delta-GLM methods, and the improved precision from delta-mixture and delta-quantile methods is decreased but not eliminated by decreased sampling intensities. These simulations represent the first evaluation of delta-mixture models for index standardization and show a substantial improvement over conventional delta-GLMs for shoaling species such as Pacific rockfish.

Environmental influences on stock abundance and fishing power in the silver-lipped pearl oyster fishery

Abstract Hart, A. M., Thomson, A. W., and Murphy, D. 2011. Environmental influences on stock abundance and fishing power in the silver-lipped pearl oyster fishery. – ICES Journal of Marine Science, 68: . Stock variability, fishing power, and the contributing environmental factors were examined for the Australian silver-lipped pearl oyster, Pinctada maxima. The approach was to compare the longer-term time-series of stock abundance derived from catch per unit effort against the shorter-term 0+ spat-settlement index to seek agreement on causal factors. Indices of stock abundance were established with generalized linear models that simultaneously examined the influence of technological and environmental factors, resulting in a predictive model with a 3-year forecast. The advent of global positioning systems caused a 30% increase in fishing power. A negative relationship between abundance and rainfall, and a positive relationship between abundance and temperature, was detected for both spat settlement and fishery abundance. Northerly winds (negative northings) from December to February significantly enhanced settlement, but easterly winds (negative eastings) in the main fishing month of May influenced fishing power positively. After standardizing for the effects of fishing power, a 150% increase in stock abundance of the exploited component of P. maxima stocks was detected between 2004 and 2009. A major contributor to this increase was an exceptionally high settlement of spat in 2005, associated with a rare combination of environmental conditions. Once this year class grows beyond the target size classes, abundance is predicted to decrease to the normal levels.

Multispecies estimation of Bayesian priors for catchability trends and density dependence in the US Gulf of Mexico

Fishery-dependent catch-per-unit-effort (CPUE) derived indices of stock abundance are commonly used in fishery stock assessment models and may be significantly biased due to changes in catchability over time. Factors causing time-varying catchability include density-dependent habitat selection and technology improvements such as global positioning systems. In this study, we develop a novel multispecies method to estimate Bayesian priors for catchability functional parameters. This method uses the deviance information criterion to select a parsimonious functional model for catchability among 10 hierarchical and measurement error models. The parsimonious model is then applied to multispecies data, while excluding one species at a time, to develop Bayesian priors that can be used for each excluded species. We use this method to estimate catchability trends and density dependence for seven stocks and four gears in the Gulf of Mexico by comparing CPUE-derived index data with abundance estimates from virtual population analysis calibrated with fishery-independent indices. Catchability density dependence estimates mean that CPUE indices are hyperstable, implying that stock rebuilding in the Gulf may be progressing faster than previously estimated. This method for estimating Bayesian priors can provide a parsimonious method to compensate for time-varying catchability and uses multispecies fishery data in a novel manner.

태평양 새치류의 어장분포와 어획량 경년 변동에 영향을 미치는 요인

This study includes spatial variations in the catch of billfishes in the Pacific Ocean and examines factors affecting interannual changes in the catch. Main billfish species caught by Korean tuna longline fishery were blue marlin and swordfish. A main fishing ground of the species was the tropical Pacific Ocean, while additional fishing ground of billfishes tended to be formed in the Pacific coast of Mexico in the El Nino periods. Further, the catch of billfishes was significantly related to CPUE (tons/average of the used hooks/vessel) in the entire Pacific Ocean as an index of stock abundance and equatorial SOI (EQSOI) as an index of El Nino event. Annual changes in the catch of billfishes in the Pacific Ocean could be regulated mainly by variations of stock abundance. In addition, increase of the density of billfishes in the tropical Pacific and additional formation of fishing ground by El Nino event possibly contribute to increase of the catch of billfishes in the Pacific Ocean. On the other hand, linear regression model may be more adequate in the analysis of relationships between fisheries data and indices made from using some environmental factors.

How reliable are the abundance indices derived from commercial catch–effort standardization?

Commercial catch-per-unit-effort (CPUE) data are often standardized to construct indices of stock abundance. The value of such standardization lies in the improvement in the proportionality between the derived index and true abundance. Using the Torres Strait rock lobster ( Panulirus ornatus ) fishery in Australia as an example, we first standardized the commercial CPUE data using a generalized linear model (GLM) and then fitted observation error models to the resulting abundance indices and independent abundance data (as estimated by research diver surveys) to examine the proportionality. While the GLM standardization greatly improved proportionality in comparison with the nonstandardized commercial catch rates, it could produce biased results if the model did not explicitly incorporate variables that had caused changes in fishing efficiency. As most catch–effort standardizations do not model the fishing power component simultaneously, this result may serve as a warning to the potential bias in stock abundance indices extracted from GLMs that are underfitted.

CPUE standardisation and the construction of indices of stock abundance in a spatially varying fishery using general linear models

Construction of annual indices of stock abundance based on catch and effort data remains central to many fisheries’ assessments. While the use of more advanced statistical methods has helped catch rates to be standardised against many explanatory variables, the changing spatial characteristics of most fisheries data sets provide additional challenges for constructing reliable indices of stock abundance. After reviewing the use of general linear models to construct indices of annual stock abundance, potential biases which can arise due to the unequal and changing nature of the spatial distribution of fishing effort are examined and illustrated through the analysis of simulated data. Finally, some options are suggested for modelling catch rates in unfished strata and for accounting for the uncertainties in the stock and fishery dynamics which arise in the interpretation of spatially varying catch rate data.

Using the EM algorithm to weight data sets of unknown precision when modelling fish stocks

Stocks of commercial fish are often modelled using sampling data of various types, of unknown precision, and from various sources assumed independent. We want each set to contribute to estimates of the parameters in relation to its precision and goodness of fit with the model. Iterative re-weighting of the sets is proposed for linear models until the weight of each set is found to be proportional to (relative weighting) or equal to (absolute weighting) the set-specific residual invariances resulting from a generalised least squares fit. Formulae for the residual variances are put forward involving fractional allocation of degrees of freedom depending on the numbers of independent observations in each set, the numbers of sets contributing to the estimate of each parameter, and the number of weights estimated. To illustrate the procedure, numbers of the 1984 year-class of North Sea cod (a) landed commercially each year, and (b) caught per unit of trawling time by an annual groundfish survey are modelled as a function of age to estimate total mortality, Z, relative catching power of the two fishing methods, and relative precision of the two sets of observations as indices of stock abundance. It was found that the survey abundance indices displayed residual variance about 29 times higher than that of the annual landings.