ICES Journal of Marine Science: Journal du Conseil

ICES Journal of Marine Science: Journal du Conseil 2007 64(7):1333-1337; doi:10.1093/icesjms/fsm134

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© 2007 International Council for the Exploration of the Sea. Published by Oxford Journals. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

A note on non-random error structure in trawl survey abundance indices

Patrick L. Cordue

Innovative Solutions Ltd, PO Box 22235, Wellington 6441, New Zealand

Tel: +644 479 0151; e-mail: patrick.cordue{at}isl-solutions.co.nz

Cordue, P. L. 2007. A note on non-random error structure in trawl survey abundance indices. – ICES Journal of Marine Science, 64: 1333–1337.

Trawl survey time-series are routinely used in stock assessments to provide indices of relative abundance. There is the general assumption, for each year y, that the expected value of the trawl survey index (X_y) is related to the biomass (B_y) by a single proportionality constant q: E(X_y) = qB_y. In reality, the constant q varies due to many factors. An important factor, which is almost always ignored, is the role that non-trawlable ground can play in the variation of q. A general formula is derived for the q of a stratified random trawl survey. When the survey area contains non-trawlable ground, strata-specific data on the proportion of non-trawlable ground and on the preference of the species for trawlable/non-trawlable ground is required to weight stratum estimates correctly. In the absence of the correct stratum weights, a shift in the spatial distribution of the stock can combine with differing proportions of non-trawlable ground to introduce non-random error and possibly spurious trends into biomass indices. Each survey and species/stock should be considered on a case-by-case basis. A stratum-specific assessment of the proportion of non-trawlable ground is a pre-requisite for the production of trawl survey biomass indices.

Keywords: distribution shift, non-trawlable ground, spurious trend, trawl survey

Received 8 August 2006; accepted 26 July 2007.

	Introduction

Top Introduction Some simple examples A general formulation Special cases Discussion References

 
Trawl survey time-series are routinely used in stock assessments to provide relative abundance indices. In general, there is the assumption, for each year y, that the expected value of the trawl survey index (X_y) is related to the biomass (B_y) by a single proportionality constant q: E(X_y) = qB_y. In reality, q will vary somewhat from year to year because of a range of factors. In particular, the occurrence of non-trawlable ground within the survey area can cause variation in q and, if combined with a shift in distribution, can lead to spurious trends in biomass indices. This effect is illustrated with some simple examples, and a formula for q is developed given a general stratified random trawl survey design. Some special cases are considered and the implications for the calculation of trawl survey indices are discussed.

	Some simple examples

Top Introduction Some simple examples A general formulation Special cases Discussion References

 
The following examples are unrealistic (being both simple and extreme), but they are given to illustrate the basic problem of non-trawlable ground for trawl survey indices.

Consider a trawl survey with just two strata: stratum 1, which has 100% trawlable ground; and stratum 2, which has only 50% trawlable ground. Assume that in year 1, all the biomass (B) is in stratum 1, and that in year 2, all the biomass is in stratum 2. Also, for simplicity, assume that the average catch rate of each stratum's trawl stations is exactly equal to the average density of the biomass on the trawlable ground within each stratum.

Consider two methods of calculating the biomass indices:

Method 1: exclude non-trawlable ground from the scaling-up of catch rates;

Method 2: include non-trawlable ground in the scaling-up of catch rates.

For each method, consider two different "preferences" for the biomass about the nature of the ground:

Preference 1: biomass is only present on trawlable ground;

Preference 2: biomass is present on trawlable and non-trawlable ground, with equal average density.

For both methods, under either preference, the biomass estimate in year 1 is B. However, the different methods and preferences give different biomass estimates in year 2:

Preference 1 Preference 2 Method 1 B 0.5B Method 2 2B B

These simple examples show that non-trawlable ground can produce spurious trends in biomass indices whether or not it is included in the scaling-up of catch rates. The "correct" calculation depends on the relative preference that the particular species has for trawlable/non-trawlable ground. It should be noted that, in these examples, the behaviour of the species does not need to change from year 1 to year 2. There is an extreme shift in the distribution of the biomass (which could be caused, for example, by a shift in water temperature); it need not be due to a change in fish behaviour.

The magnitude of potential spurious trends and/or annual variation in q will be case-specific. Clearly, it will depend on the proportions of non-trawlable ground (within strata), the ground preference of the species, and the nature and extent of the distributional shift.

	A general formulation

Top Introduction Some simple examples A general formulation Special cases Discussion References

 
The ideal stratified random trawl survey design has all non-trawlable ground blocked-off from the survey strata. In practice, this is not usually possible (because not all non-trawlable ground is identified a priori), and a survey design will include contingencies if a random station is in a non-trawlable location. If most strata have little non-trawlable ground, this should not present a problem. However, when there are some strata with a large proportion of non-trawlable ground and/or many strata with some non-trawlable ground, then a distributional shift in biomass could distort the biomass indices.

Before moving to the fully general formulation, the equations for an "ideal" trawl survey design need to be developed to introduce the notation and the relevant concepts.

Consider a stratified random trawl survey with all non-trawlable ground blocked-off from the survey strata. Let C_ij be the catch rate of the jth tow in the ith stratum (kg km^–2), a_i the area of the ith stratum (km²), n_i the number of random trawls in the ith stratum, b_i the biomass in the ith stratum (kg), and d_i the average density in the ith stratum (kg km^–2).

The biomass index is

where

Let w be the areal availability (the proportion of the total biomass which is in the survey area), u the vertical availability (the average proportion of the biomass in the water column, which is in front of the net after vertical herding), and v the vulnerability (the average proportion of the biomass in front of the net, before horizontal herding, which is actually caught).

Assume, for the moment, that u and v are the same for all strata. Then, because the trawl stations are randomly allocated, it follows, for a given stratum, that the C_ij are independent and identically distributed:

and

Let B be the total biomass (kg) and q the trawl survey proportionality constant. Then

which is the usual expression assumed for q (at least in New Zealand; Francis, 1989; Cordue, 1996).

Now consider a more general trawl survey design and initially just consider a single stratum. It is divided into grids of equal size (following a Gulf of Alaska survey which has areas with much non-trawlable ground; see Martin and Clausen, 1995; Munro and Hoff, 1995). Some grids have no trawlable ground, and if they are initially selected, a replacement grid will be chosen. For selected grids, a single trawl station is allocated within the grid. Label the grids:

trawlable: 1, ..., g.

non-trawlable: g+1, ..., m.

Let a be the area of each grid (km²), b_j the biomass in grid j (kg), t_j the proportion of trawlable ground in grid j, d_j the average density on trawlable ground in grid j (kg km^–2), and e_j the average density on non-trawlable ground in grid j (kg km^–2). Then

Assume that n trawlable grids are chosen at random with replacement (a convenient and reasonable approximation—assuming there is a large number of grids in the stratum).

Let C_jk be the catch rate in grid j_k (kg km^–2) k = 1, ..., n, and let the biomass estimate for the stratum be

The inclusion of a weight is for generality (it will be useful to be able to apply specific weights in special cases).The expected value of each C_jk is obtained from conditional expectation on the random grid selection:

As there is an equal probability of selecting any of the trawlable grids, it follows that

and hence,

Now, the biomass in the stratum is

Adding and subtracting d_j inside the square brackets gives

Let and define p as the proportion of stratum biomass on trawlable grids, so that

Then,

and hence,

If the average densities on trawlable and non-trawlable ground are equal, then the second term in the above equation is zero. However, if they are not, then the biomass estimate is not proportional to the stratum biomass.

The equation above is fairly easily extended to the full survey area. Assume that there are n strata (with n_i trawls in each stratum) and let Y_i be the biomass estimate for the ith stratum. Let the index for a given year be X:

Then,

where the previous notation has been generalized for the ith stratum. For purposes of generality, the vertical availability and vulnerability are no longer assumed constant across strata.

Let

and

r_i is distinguished because it is the proportion of biomass on the trawlable grids divided by the proportion of trawlable grids. Also let a_i = am_i be the area of the ith stratum (km²), B the total biomass (kg), and h_i = B_i/B. Then

and

Finally, the above equation can be generalized to a multiyear time-series indexed by y:

The only parameters in the above equation that typically do not vary annually are the stratum areas. The second term in the equation is zero if all trawlable grids have 100% trawlable ground, and it will be small for species which have similar densities on trawlable and non-trawlable ground.

	Special cases

Top Introduction Some simple examples A general formulation Special cases Discussion References

 
Two simplifying assumptions are made to derive an initial special case from which all the other (specific) special cases are derived. Return to a particular year and assume i, j u_i = u, v_i = v, d_ij = e_ij for some constants u, v, .

The constant describes a species-specific ground "preference". If = 1, then there is no preference; if > 1, then trawlable ground is preferred; if < 1, then non-trawlable ground is preferred.

The general special case is:

where

and

The averages are for trawlable grids only.

The first approximation only requires

The second approximation requires

and, in addition, that for each stratum the average of the non-trawlable grid densities (e_ij), over all grids is equal to that over trawlable grids.

Stratified random survey (no grids)
The approximations are exact if there is only a single grid in each stratum. Of course, then there is only one trawl in each stratum. However, as only expectation is being considered, the equations are correct for any number of "independent identically distributed" trawls in each stratum. Therefore, when each g_i = m_i = 1 and each _i = 1, the equations apply to a stratified random trawl survey without any grids (i.e. just randomly allocated stations) with the usual estimator.

The specific special case is

where z is the proportion of biomass on the non-trawlable ground:

When a species has no ground preference, the equation reduces to the familiar q = uvw. When the preference is for trawlable ground, then q > uvw, and similarly, when the preference is for non-trawlable ground, q < uvw. Annual variation in z will cause annual variation in q. This could occur by a simple shift in biomass between strata. The assumption of a constant proportion of biomass on non-trawlable ground would need to be justified by analysis (unless the spatial distribution of the species covers little non-trawlable ground).

If all non-trawlable ground is blocked off from the survey area, the equation reduces to q = uvw (because z = 0). However, the areal availability w must now account for biomass on the non-trawlable ground excluded from the survey area.

Finally, for this special case, if and the t_i are known, then there is no need to set each _i = 1. Indeed, with such knowledge, the expectation of each stratum estimator can be made proportional to stratum biomass using

Given d_i = e_i, this follows almost immediately from

and

The proportionality constant is q = uvw, and because all non-trawlable ground is retained in the survey area, there is no concern about the annual variability of w.

All trawlable grids 100% trawlable
This case appears similar to a stratified random trawl survey with all non-trawlable ground blocked off. However, in this case, the grid design is retained and the non-trawlable grids are still within the survey area.

As each t_ij = 1, each f_ij = 0. Also, the approximate formula for r_i is simplified:

This formula reveals the problem with retaining the non-trawlable ground within the strata for any species which has a ground preference (when = 1, _i can be set to 1, and q = uvw as usual). Any unfortunate annual variation in h_i will cause variation in q which will create non-random error in the abundance indices (when fitted with a single q). The problem can be solved by choosing the stratum weights appropriately:

This is analogous to the previous special case and yields the familiar q = uvw. However, this can only be done if the trawlable grid-proportion is known for each stratum and the ground preference is known (for species of interest).

Of course, if all non-trawlable grids are removed from the strata (g_i = m_i), then the _i can be set to 1, and q = uvw (but this simply shifts the problem to w, which will have larger annual variation because of the inclusion of the non-trawlable ground).

	Discussion

Top Introduction Some simple examples A general formulation Special cases Discussion References

 
There is little discussion in the primary literature on the potential deleterious effects of non-trawlable ground on trawl survey indices. Potential between strata variation of other factors, such as vertical availability, has also received little or no attention in terms of the effects on trawl survey indices. There is no mention of such factors in the 2004 and 2005 ICES workshop reports on survey design and data analysis (ICES, 2004, 2005).

In the grey literature, there are no doubt extensive comments on the potential problems for individual surveys for species that favour non-trawlable ground. There is perhaps less comment on the problems associated with species that favour trawlable ground. The only relevant general reference I have been able to find is Francis (1989). He was concerned with defining a standard approach and terminology for the analysis of "biomass bottom trawl surveys" (in New Zealand). His definitions of "areal availability" and "vertical availability" are somewhat different from those used here. In his formulation, they were specified weights in the calculation of biomass indices. "Areal availability" was allowed to be stratum-specific (to deal with fish on non-trawlable ground) and "vertical availability" was allowed to be station-specific (e.g. if acoustic data were available for each station). Of course, the distinction between assumed values (in the calculation of indices) and "true" values was implicit. An explicit distinction was made by Cordue (1996) when deriving equations for trawl survey qs.

In the general statistical literature concerned with population surveys, there is much discussion of an analogous problem, that of "non-response" from a portion of the population owing to the inability or unwillingness to respond. A typical example is a "telephone survey", where some portion of the population does not have telephones, and of those that do, some portion will refuse to participate. There is, of course, some potential for bias, because the non-responders' characteristics may be different from those of the responders. Solutions to the problem involve obtaining data from the non-responders by some other means, then either demonstrating that the characteristics of interest are not too different, or post-stratifying on response/non-response.

The analogy with fish densities on trawlable/non-trawlable ground is clear—the densities on non-trawlable ground are "non-responsive". Therefore, the "solution" is to sample the non-responders by some other means. The uses of acoustics and/or longline/pot surveys are possible options for obtaining information on fish densities on non-trawlable ground. Of course, every sampling method has its own associated bias, perhaps dependent on ground type, and these would need to be accounted for when estimating densities and combining density estimates across methods. One can envisage methods for integrating supplementary observations into a trawl survey design or for using supplementary data to estimate stratum weights (but the development of such methods is beyond the scope of this paper).

The ideal trawl time-series consists of a series of unbiased estimates of qB for a constant q. Any structure in the errors introduced by variation in q is undesirable. However, annual variation in q is only of concern if it introduces a spurious trend in the biomass indices. In a properly designed trawl time-series, covering an appropriate area, with appropriate standardization of gear and gear operation, there would appear few circumstances that could lead to such a trend. Certainly, a radical change in fish behaviour could cause such trends. However, for some surveys, a simple redistribution of fish across strata (which does not require a change in fish behaviour) can lead to unwanted variation in q, which could introduce non-random errors and/or a spurious trend in the biomass indices.

This brief communication is aimed at alerting researchers to the possibility that indices from their trawl survey may need re-evaluation. The main focus of the paper is non-trawlable ground, but potential variation across strata in vertical availability should also be considered (e.g. for semi-demersal species, where u could depend on depth).

Researchers who produce biomass indices from trawl survey data should reconsider the trawl survey areas and the stocks of interest in relationship to potential interactions between stock distribution and non-trawlable ground. If the trawl survey area is believed to cover most of the spatial distribution of the stock each year, and there is little non-trawlable ground within the boundaries of the survey area, then non-trawlable ground is not an issue. If the proportion of trawlable ground varies substantially across strata, then they need to consider carefully how they are calculating the indices. The correct formulation requires information on the ground preference of the species. If detailed quantitative data are available, then appropriate stratum weights can be applied when calculating the indices. Of course, each time-series, and each stock of interest, needs to be considered on a case-by-case basis.

In the absence of quantitative data on the ground preference of a stock/species of interest, I suggest the following sequential procedure for the re-evaluation of trawl survey data.

1. If the trawl survey area covers at least 90% of the spatial distribution of the stock and the proportion of non-trawlable ground enclosed within the boundaries of the trawl survey area is < 10%, ignore any potential effects from non-trawlable ground.
   
2. Otherwise, consider the preference of the species:
   1. if it predominantly inhabits non-trawlable ground, use a different survey method (e.g. acoustics, longline);
      
   2. if it has an equal preference for trawlable and non-trawlable ground, include non-trawlable ground in the scaling-up of average catch rates;
      
   3. if it predominantly inhabits trawlable ground, exclude non-trawlable ground in the scaling-up of average catch rates.
      
   
   
3. If the preference for a species of interest is not known (and cannot be deduced), then analyse the existing time-series about spatial distribution:
   1. if there has been no obvious shift in distribution (relative to the occurrence of non-trawlable ground), the existing indices are probably usable;
      
   2. if there has been a shift, then the existing indices are unreliable—try to obtain some data on the relative preference of the species.
      
   
   

Of course, the above approach requires that the researcher first quantify, to some extent, the proportion of trawlable ground in each stratum. Although here I have considered only random designs, the same general conclusions apply to any stratified design (e.g. fixed stations).

	Acknowledgements

 
The topic of this paper came to my attention during a review meeting on Alaskan rockfish in June 2006 at the Alaskan Fisheries Science Center. My attendance at the meeting, as a member of the review panel, was funded by the Center for Independent Experts (CIE). An initial draft of the equations in this paper was part of my CIE report. I am grateful to the CIE and to participants in the review meeting for discussions on this topic. Also, an earlier draft of this paper was improved thanks to the comments of two anonymous reviewers.

	References

Top Introduction Some simple examples A general formulation Special cases Discussion References

 

Cordue P. L. A model-based method for bounding virgin biomass using a catch history, relative biomass indices, and ancillary information. New Zealand Fisheries Assessment Research Document 96/8. (1996) 48.

Francis R. I. C. C. A standard approach to biomass estimation from bottom trawl surveys. New Zealand Fisheries Assessment Research Document 89/3. (1989) 4.

ICES. Report of the Workshop on Survey Design and Data Analysis (WKSAD) 21–25 June 2004, Aberdeen, UK. ICES Document CM 2004/B: 07. (2004) 65.

ICES. Report of the Workshop on Survey Design and Data Analysis (WKSAD) 9–13 May 2005, Sête, France. ICES Document CM 2005/B: 07. (2005) 67.

Martin M. H., Clausen D. M. Data report: 1993 Gulf of Alaska bottom trawl survey. (1995) US Department of Commerce, NOAA Technical Memorandum NMFS-AFSC-59. 217.

Munro P. T., Hoff R. Z. Two demersal trawl surveys in the Gulf of Alaska: implications of survey design and methods. (1995) US Department of Commerce, NOAA Technical Memorandum NMFS-AFSC-50. 139.

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