Abstract

Bacon, P. J., Gurney, W. S. C., McKenzie, E., Whyte, B., Campbell, R., Laughton, R., Smith, G., and MacLean, J. 2011. Objective determination of the sea age of Atlantic salmon from the sizes and dates of capture of individual fish. – ICES Journal of Marine Science, 68: 130–143.

The sea ages of Atlantic salmon indicate crucial differences between oceanic feeding zones that have important implications for conservation and management. Historical fishery-catch records go back more than 100 years, but the reliability with which they discriminate between sea-age classes is uncertain. Research data from some 188 000 scale-aged Scottish salmon that included size (length, weight) and seasonal date of capture on return to the coast were investigated to devise a means of assigning sea age to individual fish objectively. Two simple bivariate probability distributions are described that discriminate between 1SW and 2SW fish with 97% reliability, and between 2SW and 3SW fish with 70% confidence. The same two probability distributions achieve this accuracy across five major east coast Scottish rivers and five decades. They also achieve the same exactitude for a smaller recent dataset from the Scottish west coast, from the River Tweed a century ago (1894/1895), and for salmon caught by rod near the estuary. More surprisingly, they also achieve the same success for rod-caught salmon taken at beats remote from the estuary and including capture dates when some fish could have been in the river for a few months. The implications of these findings for fishery management and conservation are discussed.

Introduction

The Atlantic salmon (Salmo salar) has long been associated with human prosperity. Commercial fishing records going back centuries (Waite, 1831; Shearer, 1992; Summers, 1995) record catches of two morphs, the smaller grilse and the larger salmon. Around the early 20th century, when it became known that fish could be aged by growth rings on their scales (Calderwood, 1907), it was realized that the grilse were not simply stunted salmon, but rather salmon that had spent just one winter at sea (1SW) instead of a longer growing period involving two or more winters at sea (multi-sea-winter salmon, MSW; 2SW, 3SW, etc). Then, once the west Greenland feeding grounds of marine salmon had been discovered (Menzies and Shearer, 1957; Hansen, 1965; Netboy, 1968), it became apparent that, along with 1SW and 2SW salmon from North America, only 2SW European salmon fed there, with Europe's 1SW grilse going to feeding areas near the Norwegian sea (Jacobsen et al., 2001). About the same time, it was discovered that Scottish 1SW and MSW fish dominated in different freshwater habitats (lowlands and uplands, respectively; see Laughton and Smith, 1992; Walker and Walker, 1992; Webb, 1992; Smith and Johnstone, 1996; Smith et al., 1998). The sea ages of salmon therefore contain many important details of the lifestyles and requirements of the fish and their populations. The older MSW salmon are generally larger than 1SW fish, so are preferred by fishers. The economic value of salmon fisheries in Scotland is widely recognized and considerable (Radford et al., 2004). Whereas the Scottish 1SW fish return (i.e. “run” the rivers) mainly from May to about December, MSW salmon in Scotland run for most of the year, at least from January to about November, and their presence in late winter and spring extends the fishing season and brings important revenue to fishery owners and to the wider rural community outside the normal tourist season. Since 1952, catches of spring MSW salmon have declined, whereas those of summer salmon have remained fairly constant and those of 1SW grilse have increased (Anon., 2009).

Many Scottish fishery managers want to manage these sea-age components of their local populations differently. To do this effectively, they need to distinguish between management actions that can achieve this and longer-term natural variations in the ratio of 1SW to MSW fish that they probably cannot alter. Unfortunately, just a small minority of captured fish have their ages determined by scale reading. Assigning sea ages from angler opinion based on simple visual inspection or “size” rules is biased, with the resulting “grilse error” for rod-caught fish amounting to >20%, with seasonal biases (Dunkley et al., 1993; MacLean et al., 1996; A. F. Youngson, Environmental Research Institute, Thurso, pers. comm.).

Recent analysis of the fluctuations in numbers of a local upland population of predominantly 2SW salmon in the Girnock burn, Scotland (Gurney et al., 2010), suggests that despite characteristically wide variations in numbers between years, such data probably hold important signals about population status. Unfortunately, this analysis shows that because of the low signal-to-noise ratio in such data, 45 years is about the minimum period that will detect these signals. Greater precision could be obtained from longer series of data, but that goes back into the period when reliable scale-read ages are scarce or absent.

Here, we investigate the possibility that a pair of simple and objective criteria, date of return to the coast and fish size on that date, might provide a means by which the sea ages of salmon could be determined objectively with known precision. The findings suggest that good success can be achieved by a pair of canonical probability distributions. These distinguish 1SW from 2SW and 2SW from 3SW and, for each fish, produce a triad of probabilities [p1, p2, p3] that the fish is, respectively, [1SW, 2SW, 3SW], where p1 + p2 + p3 = 1.0000. Moreover, these relationships appear stable over five decades and across different river systems, and they do not vary between capture methods, even for capture by rods at sites removed from the estuary and during periods when some MSW salmon could have been in the river for months.

Methods

Some 80% of the data come from records of individual salmon caught by net-and-coble fisheries (see Shearer, 1992, for a description of this fishing method) in the estuaries of five east coast Scottish rivers (Spey, Dee, North Esk, Tay, and Tweed) on their return from the sea. A sample of the captured fish was measured by Fisheries Laboratory staff following standard protocols, and their ages were determined later by experienced personnel from scales taken at the time of capture, together with an assessment whether the fish had spawned previously. This net-caught portion of the dataset was analysed extensively by Bacon et al. (2009) to describe trends in the average length, weight, and condition of returning salmon between 1963 and 2006, and that manuscript should be consulted for further detail about methods and quality controlling of the data.

In the current analysis, this core dataset was supplemented with similar details from three other sources. First, some 8% of the records represent individual salmon caught by rod and line on fishing beats near the estuary (Spey, North Esk, and Tweed). Second, some 4% refer to salmon caught by rod and line away from the estuary. A small third set (1%) refers to coastal netted fish from sites on the Scottish west coast. Finally, a supplementary set, lacking the scale ages but including an evaluation of grilse or salmon by an experienced observer, consists of sizes and dates for individual salmon caught on the Tweed in 1894 and 1895 (Anon., 1896).

As the Scottish netting season, typically from February to August, is shorter than the rod seasons, typically from February to November, inclusion of the rod-caught fish extended the span of months from 7 to 10.

Outlying values

Bacon et al. (2009) removed from their analysis records having studentized residuals (SR) from a log(length) to log(weight) regression that were outside the range −6 to +6, because a few such large deviations could bias the averages calculated. As the present analysis focuses on discriminating between unusually small and large fish of different sea age, we here removed just seven records whose enormous values of SR clearly revealed mistyped decimal points in the underlying measurements. It is axiomatic that removing more outliers would have further improved the classification success reported below, especially because the residuals were kurtotic. However, this work aims to provide a method usable by anglers in the field so we conservatively retained all data which were not obviously corrupted. This may have led to ∼1–2% reduction in power.

Reliability of scale reading

Scale reading is not an entirely repeatable process. As it was possible that some disagreements between scale-read ages and our size-and-date assigned ages arose from errors in scale readings, we assembled a set of 200 North Esk scales (consisting of 119 outliers and 80 strong concurrences) that were re-read to assess repeatability. This procedure was carried out by Bryce Whyte (Marine Scotland Science), a person with 40 years experience of reading salmon scales. Many of the scales had originally been read by a different person. The sample of 200 scales was sufficient to show that a fair proportion of the outliers arose from scales that were particularly difficult to interpret. A larger sample, and a full blind trial, would be needed to investigate the causes of such scale-reading difficulties, but this was beyond the scope of the study.

Statistical methods

Our data consisted of individual records that included scale-determined sea age (a), at least one size measure (s, as length or weight), and day of the year captured (d). Almost no fish had scale-read sea ages of 4 years or more (89 of 186 594, or 0.05%), so the values predicted for a were restricted to 1, 2, or 3, where the last corresponds to three or more sea-age years. We modelled the distribution of sea age a as a function of size s and day of capture d. Therefore, using the records available, we estimate {p1(s, d), p2(s, d), p3(s, d)} where pa(s, d) is the probability that a fish of size s caught on day d has sea age a (=1, 2, 3). This allows us to predict the maximum probability sea age as the value of a corresponding to the highest of the three probabilities, but it also gives us a triad of probabilities for each fish [p1, p2, p3] that the fish is, respectively, [1SW, 2SW, 3SW], given its combination of (s, d) values.

It may seem that it would be more natural to predict sea age directly from (s, d), and this is certainly possible, but our opinion is that it is important to have as much information as the data allow about both prediction and reliability. For example, if the estimated age distribution for fish X was the probability triad TX = (0.95, 0.05, 0.00), then we predict a sea age of 1 and are confident in so doing. However, if another estimated age distribution for fish Y was actually the triad TY = (0.50, 0.49, 0.01), then we may still predict that the sea age of fish Y = 1, but it is clear that this prediction has nothing like the reliability of that for fish X. In fact, for the TY example above, the data barely distinguish between 1SW and 2SW; we argue that this is the information that scientists and managers should have and interpret for themselves. We note, too, that although the triads TX and TY above are extremes in an obvious sense, the vast majority of (s, d) pairs in our records yielded estimates for the probabilities of sea age A similar in information content to the TX example above. However, there was also a small but important minority much closer in information content to example TY.

The probabilities were modelled and estimated using generalized linear modelling, and in particular, the continuation ratio logit model (Dobson, 2002). Details of this fitting procedure, as here applied, are outlined in Appendix 1.

Subsets of data

The above analyses were conducted on different spatial, temporal, and capture-method subsets of the data, and also with size represented by length, weight, and length derived from weight. As not all records had both length and weight measurements (although 95.2% did), the actual numbers of records included varied slightly between these size-metric comparisons.

The huge number of records available to us (186 594) meant that, as discussed by Bacon et al. (2009), tiny differences between subsamples (sites, capture methods) that had no useful biological meaning were statistically highly significantly different. However, as the aim of this work was to find a common way of determining the sea age of Atlantic salmon reliably, results are here reported as two types of classification success. First, we report the proportion of errors that resulted if the subset was classified according to the size-and-date information from the entire dataset. This showed the generality of the single canonical pair of probability distributions. However, if the discriminant functions varied somewhat between subsets, the classification success of each set predicted from its own size-and-date data is also provided.

When trying to establish reliable predictive equations, it is good practice to subdivide the data into one or more calibration and validation subsets and to ascertain whether the equation parameters derived from the calibration set(s) also predict values in the independent validation set(s) reliably. We initially followed that procedure. All the many subsets investigated reliably predicted each other to within a few per cent of r2, but, owing to the massive sample sizes, they were also usually very significantly different from each other. As this more complex procedure produced no extra biological insight, we report just the overall, canonical, findings.

Although visual assessments of sex supported the view that slightly more 1SW fish were male and slightly more 2SW fish were female, use of separate predictive equations for the sexes yielded no useful improvement in accuracy. Those endeavours are hence not reported here, and their proper investigation would require data not based on visual determinations of sex (especially for early-running male grilse).

Results

The full dataset (Table 1) contained 186 594 records of length, day of capture, and scale-determined sea age. Most records (95.2%) also had weight values. Our full analysis included many detailed local comparisons, such as North Esk estuary nets vs. North Esk near-estuary rod and line and North Esk near-estuary rod and line vs. North Esk non-near-estuary rod and line. These many local contrasts all showed that the single pair of canonical probability distributions gave robust discrimination which was of consistently high success. Therefore, for the sake of brevity, we here focus on the smaller set of comparisons given in Table 2.

Table 1.

Source and number of records analysed, by river system, location on the river, fishing gear, spawning status, and the abbreviations used in text.

River or site Location Gear Spawning Abbreviation Records 
Scale-read sea ages 
 Dee Estuary Net and coble Maiden DE1 5 731 
 North Esk Estuary Net and coble Maiden NE1 75 317 
 Spey Estuary Net and coble Maiden SP1 17 389 
 Strathy Coastal Fixed engine Maiden ST1 5 401 
 Tay Estuary Net and coble Maiden TY1 24 730 
 Tweed Estuary Net and coble Maiden TW1 32 590 
 West Coast Estuary Net and coble Maiden WC1 1 592 
 Dee Estuary Net and coble Repeats DERS 66 
 North Esk Estuary Net and coble Repeats NERS 668 
 Spey Estuary Net and coble Repeats SPRS 167 
 Strathy Coastal Fixed engine Repeats STRS 67 
 Tay Estuary Net and coble Repeats TYRS 189 
 Tweed Estuary Net and coble Repeats TWRS 199 
 West Coast Estuary Net and coble Repeats WCRS 13 
 North Esk Estuary Rod and line All NEMDR 5 697 
 Spey Estuary Rod and line All SPER 9 905 
 Tweed Estuary Rod and line All TWER 379 
 Dee River Rod and line All DNoER 130 
 Spey River Rod and line All SNoER 5 576 
 Tweed River Rod and line All TNoER 788 
Visual grilse/MSW determination 
 Tweed (1895) Berwick Nets All TWH.BN 300 
 Tweed (1895) Kelso Rod and line All TWH.RK 167 
 Tweed (1895) Unknown Unknown All TWH.US 729 
River or site Location Gear Spawning Abbreviation Records 
Scale-read sea ages 
 Dee Estuary Net and coble Maiden DE1 5 731 
 North Esk Estuary Net and coble Maiden NE1 75 317 
 Spey Estuary Net and coble Maiden SP1 17 389 
 Strathy Coastal Fixed engine Maiden ST1 5 401 
 Tay Estuary Net and coble Maiden TY1 24 730 
 Tweed Estuary Net and coble Maiden TW1 32 590 
 West Coast Estuary Net and coble Maiden WC1 1 592 
 Dee Estuary Net and coble Repeats DERS 66 
 North Esk Estuary Net and coble Repeats NERS 668 
 Spey Estuary Net and coble Repeats SPRS 167 
 Strathy Coastal Fixed engine Repeats STRS 67 
 Tay Estuary Net and coble Repeats TYRS 189 
 Tweed Estuary Net and coble Repeats TWRS 199 
 West Coast Estuary Net and coble Repeats WCRS 13 
 North Esk Estuary Rod and line All NEMDR 5 697 
 Spey Estuary Rod and line All SPER 9 905 
 Tweed Estuary Rod and line All TWER 379 
 Dee River Rod and line All DNoER 130 
 Spey River Rod and line All SNoER 5 576 
 Tweed River Rod and line All TNoER 788 
Visual grilse/MSW determination 
 Tweed (1895) Berwick Nets All TWH.BN 300 
 Tweed (1895) Kelso Rod and line All TWH.RK 167 
 Tweed (1895) Unknown Unknown All TWH.US 729 

The first two letters of the abbreviations refer to the river (DE, Dee; SP, Spey; etc); 1 as a suffix depicts first-time spawners, an RS suffix repeat spawners, etc.

Table 2.

Subsets of the data analysed, their abbreviations as used in the text, their composition relative to Table 1 names, and the numbers of fish records contained.

Abbreviation Composition (see Table 1Fish records 
allSA All scale-aged data 186 594 
allSA_63-73  All scale-aged data 1963–1973 47 150 
allSA_74-84  All scale-aged data 1974–1984 58 562 
allSA_85-95  All scale-aged data 1985–1995 51 513 
allSA_96-06  All scale-aged data 1996–2006 29 142 
allSA_W All scale-aged data with length and weight values 177 636 
allSA_Maidens DE1, NE1, SP1, ST1, TY1, TW1, and WC1 162 750 
allSA_Repeat DERS, NERS, SPRS, STRS, TYRS, TWRS, and WCRS 1 369 
allSA_EstRod NEMDR, SPER, andTWER 15 981 
allSA_NonEstRod DNoER, SNoER, and TNoER 6 494 
Tweed_1894-5 TWH.BN, TWH.RK, and TWH.US 1 196 
Abbreviation Composition (see Table 1Fish records 
allSA All scale-aged data 186 594 
allSA_63-73  All scale-aged data 1963–1973 47 150 
allSA_74-84  All scale-aged data 1974–1984 58 562 
allSA_85-95  All scale-aged data 1985–1995 51 513 
allSA_96-06  All scale-aged data 1996–2006 29 142 
allSA_W All scale-aged data with length and weight values 177 636 
allSA_Maidens DE1, NE1, SP1, ST1, TY1, TW1, and WC1 162 750 
allSA_Repeat DERS, NERS, SPRS, STRS, TYRS, TWRS, and WCRS 1 369 
allSA_EstRod NEMDR, SPER, andTWER 15 981 
allSA_NonEstRod DNoER, SNoER, and TNoER 6 494 
Tweed_1894-5 TWH.BN, TWH.RK, and TWH.US 1 196 

Overall predictions based on body length

Figure 1 shows the probability model fitted to the full-scale-aged dataset {allSA}, using length as the size measure. Note that the {allSA} dataset includes repeat-spawning fish. Figure 1a shows the 50 and 95% contours for the three probabilities p1, p2, and p3. Figure 1b shows the expected age of fish with particular (s, d) combinations, as a level plot with levels 0, 1.05, 1.5, 1.95, 2.05, 2.5, 2.95, and 3. As at any given (s, d), just two of the probabilities have values noticeably different from 0, the regions so produced bear simple interpretations related to our knowledge of a fish's sea age (see legend).

Figure 1.

The canonical probability model. (a) 95% (heavy) and 50% (light) contours for the probabilities p1 (green), p2 (blue), and p3 (red) of a fish of length L (cm) caught on a given day of the year have sea age 1, 2, or 3, respectively, calculated from the probability model described in the section “Statistical methods” fitted to the canonical {allSA} dataset. (b) Level plot of the expected age E{a} = p1 + 2p2 + 3p3 calculated from the canonical probability model with levels (0, 0.95, 1.05, 1.5, 1.95, 2.05, 3). As for any given length–day combination one of the three probabilities is essentially zero, the coloured regions bear the following interpretation: 95% certain 1SW (dark green), 1SW or 2SW but more likely 1SW (light green), 1SW or 2SW but more likely 2SW (light blue), 95% certain 2SW (dark blue), 2SW or 3SW but more likely 2SW (purple), 2SW or 3SW but more likely 3SW (light red), and 95% certain 3SW (red).

Figure 1.

The canonical probability model. (a) 95% (heavy) and 50% (light) contours for the probabilities p1 (green), p2 (blue), and p3 (red) of a fish of length L (cm) caught on a given day of the year have sea age 1, 2, or 3, respectively, calculated from the probability model described in the section “Statistical methods” fitted to the canonical {allSA} dataset. (b) Level plot of the expected age E{a} = p1 + 2p2 + 3p3 calculated from the canonical probability model with levels (0, 0.95, 1.05, 1.5, 1.95, 2.05, 3). As for any given length–day combination one of the three probabilities is essentially zero, the coloured regions bear the following interpretation: 95% certain 1SW (dark green), 1SW or 2SW but more likely 1SW (light green), 1SW or 2SW but more likely 2SW (light blue), 95% certain 2SW (dark blue), 2SW or 3SW but more likely 2SW (purple), 2SW or 3SW but more likely 3SW (light red), and 95% certain 3SW (red).

Figures 2a–c show scatters of datapoints for individuals of given sea ages superimposed over the probability contours from the canonical model. To assist in interpretation, where the data are numerous enough to permit kernel-density estimation, we also superimposed density contours for the appropriate data subset. For each sea-age class, it is clear that the majority of datapoints for each sea age (within the outermost elliptical contour) are the correct side of (i.e. between) the appropriate probability contours, with just a tiny proportion of scale-read points in regions of the (s, d) plane where their (s, d)-derived probability values assign them as more likely to be a different sea age from the scale reading.

Figure 2.

Comparing the probability model and data. (a–c) Data from the {allSA} dataset compared with the probability model fitted to the same data. (d–f) Repeat spawners {allSA_Repeat} plotted against the canonical probability model. (g–i) The canonical data degraded to monthly dates and 1/2 lb weight categories plotted against the canonical probability model. All panels show the 50% (light) and 95% (heavy) contours for p1 (green), p2 (blue), and p3 (red), as in Figure 1a, on which, where there are sufficient data to permit kernel-density estimation, are superimposed scatterplots of the data for a single sea age with density contours at 5, 20, 35, 50, 65, and 80% of the peak density. (a), (d), and (g) show 1SW data, (b), (e), and (h) show 2SW data, and (c), (f), and (i) show 3SW data.

Figure 2.

Comparing the probability model and data. (a–c) Data from the {allSA} dataset compared with the probability model fitted to the same data. (d–f) Repeat spawners {allSA_Repeat} plotted against the canonical probability model. (g–i) The canonical data degraded to monthly dates and 1/2 lb weight categories plotted against the canonical probability model. All panels show the 50% (light) and 95% (heavy) contours for p1 (green), p2 (blue), and p3 (red), as in Figure 1a, on which, where there are sufficient data to permit kernel-density estimation, are superimposed scatterplots of the data for a single sea age with density contours at 5, 20, 35, 50, 65, and 80% of the peak density. (a), (d), and (g) show 1SW data, (b), (e), and (h) show 2SW data, and (c), (f), and (i) show 3SW data.

Figures 1 and 2 reveal why the combination of length and date was needed to separate the sea-age groups reliably. Most of the 2SW fish and nearly all the 3SW fish had arrived before day 160, by which date the run of 1SW fish was only just starting, so date was highly informative in separating 1SW from MSW fish. However, before day 150, length was clearly the better discriminator between 2SW and 3SW fish, and after day 160, length was also needed to discriminate between 1SW and 2SW fish. It is not particularly helpful to try and quantify the relative degrees of importance of the two factors, because a precise answer depends on the relative numbers of sea-age fish and their arrival periods, factors that vary between rivers and years, and which appear much less stable than the age-discriminations described here.

Table 3 gives the classification success statistics for the entire dataset {allSA}, as derived from the probability distribution illustrated in Figure 1. This shows that scale-read sea ages were correctly predicted by the algorithm for 98.7% of individual fish known to be 1SW, 97.0% of those known to be 2SW, and 70.8% of those known to be 3SW. Note, however, that for algorithm-predicted 3SW fish, a greater proportion of 85% (instead of 70.8%) was confirmed as correct by scale reading. For 1SW fish, the main misdiagnosis (1.3%) was, unsurprisingly, as 2SW. Scale-read 2SW fish were mainly misdiagnosed as 1SW (2.1%), whereas a smaller proportion, just 0.9%, was wrongly identified as 3SW. For scale-read 3SW fish, essentially all misclassifications (29.1%) were stated as 2SW.

Table 3.

Prediction efficiency table of sea age observed from scale reading for the {allSA} dataset, derived from sizes and dates from the same dataset, using length as the size measure, with the predicted sea age for each record defined as the sea age having the maximum probability for its combination of size and date of capture.

 Sea age predicted from size and date
 
Scale-read sea age p1 p2 p3 
Observed sea age 1 (%) 98.7 1.3 
 (Count) (98 460) (1 301) (2) 
Observed sea age 2 (%) 2.1 97 0.9 
 (Count) (1 696) (78 955) (723) 
Observed sea age 3 (%) 29.1 70.8 
 (Count) (1) (1 590) (3 865) 
 Sea age predicted from size and date
 
Scale-read sea age p1 p2 p3 
Observed sea age 1 (%) 98.7 1.3 
 (Count) (98 460) (1 301) (2) 
Observed sea age 2 (%) 2.1 97 0.9 
 (Count) (1 696) (78 955) (723) 
Observed sea age 3 (%) 29.1 70.8 
 (Count) (1) (1 590) (3 865) 

Weight or length as alternative size measures

Length is arguably a more appropriate size measurement than weight in this context, because it does not decrease during periods of fasting, which include much of a salmon's journey back to the coast and its time in freshwater (see Bacon et al., 2009, for a brief discussion). However, many historical commercial records report fish weights rather than lengths, so it is important to establish the predictive ability of body weight too. We constructed a data subset ({allSA_W}, see Table 2) of 177 636 records having both length and weight values. Unsurprisingly, these pairs of values are highly correlated. A regression of length against the cube root of weight yields an r2 value of 0.9546. Subsequent calculations are therefore made in terms of the metric length derived from weight, LW, defined for an individual of weight W as LW = 0.2133 + 43.61W1/3.

We acknowledge that an even higher value of r2 could be obtained for the entire {allSA_W} data by fitting a set of equations, one for each sea age, in which the slopes and intercepts vary between the sea ages. However, because our aim was to provide a technique to assess the sea age of salmon for which sea age was not recorded, this option was not open to us. An effect of assuming a single regression slope, when in fact the slopes increase marginally with sea age, is to produce a distribution of residuals that is slightly kurtotic (see the QQ plot, Appendix 2, Figure A1). Only 92.16% of the residuals from the regression were within the expected 95% central region, with some 4% of residuals in each of the tails, where just 2.5% are expected for a perfectly normal curve.

Table 4 compares the classification success achieved for the {allSA_W} dataset between scale-read ages and predictions based on different size-metric combinations. Using either weight or its transform LW to predict sea age is detectably less effective than using observed lengths, especially for distinguishing between 2SW and 3SW salmon. However, for 1SW and 2SW fish, the success achieved with the W or LW combinations remains high, at >97.9 or >95.8%, respectively (Table 4).

Table 4.

Prediction efficiency, presented as the percentage of observed individual fish of a given scale-read age predicted to be ages 1, 2, or 3, based on the size and date probability distribution derived from the {allSA_W} dataset using length (L), weight (W), or weight at equivalent length (LW) as the basis for fitting or prediction.

  Scale-read observed 1
 
Scale-read observed 2
 
Scale-read observed 3
 
Fitted to Predicted from p1 p2 p3 p1 p2 p3 p1 p2 p3 
L L 98.7 1.3 0.0 2.2 96.9 0.9 0.0 30.4 69.6 
W W 98.2 1.8 0.8 3.4 95.8 0.8 0.1 37.7 62.2 
LW LW 98.3 1.7 0.0 3.4 95.8 0.8 0.1 37.4 62.6 
LW L 98.9 1.1 0.0 2.4 96.7 0.9 0.0 29.2 70.8 
L LW 97.9 2.1 0.0 3.1 96.1 0.8 0.0 38.8 61.1 
  Scale-read observed 1
 
Scale-read observed 2
 
Scale-read observed 3
 
Fitted to Predicted from p1 p2 p3 p1 p2 p3 p1 p2 p3 
L L 98.7 1.3 0.0 2.2 96.9 0.9 0.0 30.4 69.6 
W W 98.2 1.8 0.8 3.4 95.8 0.8 0.1 37.7 62.2 
LW LW 98.3 1.7 0.0 3.4 95.8 0.8 0.1 37.4 62.6 
LW L 98.9 1.1 0.0 2.4 96.7 0.9 0.0 29.2 70.8 
L LW 97.9 2.1 0.0 3.1 96.1 0.8 0.0 38.8 61.1 

Table 4 also includes the findings of two further assessments, comparing observed lengths L with a probability distribution derived from LW, and vice versa. Not unexpectedly, using length derived from weight as the predictor produced the least satisfactory result of the group, confirming our belief that weight is a less satisfactory predictor than length. Slightly bizarrely, the best result of the group was achieved using length as a predictor but calculating the probability distribution for sea ages from the lengths derived from weights. We see no way that this can be systematic, and conclude that it represents an accidental side effect of the fact that the GLM fitting process maximizes the overall likelihood, rather than minimizing the error rate.

Prediction efficiency for major subsets of data

Table 5 compares the efficiency with which the single overall pair of equations, fitted to the entire {allSA} dataset (Prob|{allSA}), compares with fitting separate probability models (of the same form) to those same major subsets of the data (Prob|{self}), for four subsets which might be expected to show different relationships from each other. The outcomes for three groups of first-time spawners (estuary nets, near-estuary rods, non-estuary rods) and an overall group of repeat spawners are displayed in Table 5.

Table 5.

Predictive efficiencies from the overall {allSA} probability distribution compared with those from the subset of a specific probability distribution for various data subsets.

 Prob|{allSA}
 
Prob|{self}
 
Data subset p11 p22 p33 p11 p22 p33 
allSA_Maidens 99.0 97.7 72.4 99.2 97.5 73.7 
allSA_EstRod 99.0 94.0 67.3 98.5 95.2 77.1 
allSA_NonEstRod 98.7 92.5 42.0 97.3 95.4 51.0 
allSA_Repeat 60.0 52.3 47.7 89.4 82.4 24.6 
 Prob|{allSA}
 
Prob|{self}
 
Data subset p11 p22 p33 p11 p22 p33 
allSA_Maidens 99.0 97.7 72.4 99.2 97.5 73.7 
allSA_EstRod 99.0 94.0 67.3 98.5 95.2 77.1 
allSA_NonEstRod 98.7 92.5 42.0 97.3 95.4 51.0 
allSA_Repeat 60.0 52.3 47.7 89.4 82.4 24.6 

Columns 2–4 show the percentage of correct identifications of 1SW, 2SW, and 3SW fish using the {allSA} probabilities, and columns 5–7 show the same quantities using the subset of specific probabilities. pnn (p11, p22, p33) signify the probabilities that the sea age predicted from (size, date) was n given that the scale-read sea age was also n.

The results in Table 5 also emphasize the fact that the degrees of improvement in classification effectiveness when fitting to the same data subset were generally minor, and sometimes negative (given the massive sample sizes, some of the changes might be statistically significant if subjected to rigorous testing, but we here continue to focus on the tiny magnitude of any improvements and hence the ubiquity of the overall pair of predictive equations). Indeed, the sea-age predictions from size and date for all three groups of first-time spawners were functionally equally effective, no matter whether the prediction was based on their own subset or on the entire {allSA} dataset. Only for the small group (2% of fish) of repeat spawners were predictions based on the {allSA} data less effective (50 vs. 60%) than those based on their own dataset (80%, Table 5, excluding 3SW fish). Figures 2d–f illustrate the difficulties posed by repeat-spawner data. Repeat spawners that first spawned as grilse appear to form two clouds of points. One cloud is broadly coincident with first-time grilse spawners, although slightly larger in size on the same date, but which occur over a similar range of dates. The second cloud represents fish which apparently next returned to spawn at slightly larger size but over a much wider span of dates. These topics are discussed further in the section “Biological implications” below.

Interstock and river-catchment comparisons

Similar comparisons to those reported above can be made between different stock (or river-catchment) subsamples. Table 6 lists the results for a set of first-time spawners from five Scottish east coast catchments, plus a collection of sites on the west coast of Scotland, and Figures 3a–c and d–f illustrate the findings for the Spey and Tweed, respectively.

Table 6.

Predictive efficiencies compared from size-at-return-date probability distributions derived from the global dataset ({allSA}, columns 3–5) as opposed to the same data subset (columns 6–8), with columns 3–8 showing the percentage of correct identifications for each sea age, the latter with emboldening denoting trivial improvement of {self} over {allSA}.

  Prob|{allSA}
 
Prob|{self}
 
Prob|{self} − Prob|{allSA}
 
Data subset (all first-time spawners) Sample size p1 p2 p3 p1 p2 p3 p1 p2 p3 
Dee maidens (DE1) 5 731 98.2 97.5 82.6 99.5 98.0 74.3 1.3 0.5 −8.3 
North Esk maidens (NE1) 75 317 99.5 97.7 71.0 98.4 97.7 77.7 −1.1 0.0 6.7 
Spey maidens (SP1) 17 389 98.8 97.9 75.3 99.4 97.4 76.4 0.6 −0.5 1.1 
Tay maidens (TY1) 24 730 97.3 98.2 81.6 99.3 98.2 75.5 2.0 0.0 −6.1 
Tweed maidens (TW1) 32 590 99.1 97.3 45.0 99.0 97.7 45.4 −0.1 0.4 0.4 
West coast maidens (WC1) 1 592 98.5 96.3 0.0 99.8 25.0 75.0 1.3 −71.3 75.0 
Total across catchments 157 349          
Minimum (%)  97.3 97.3 45.0 98.4 97.4 45.4 −1.1 −0.5 −8.3 
Mean (%)  98.6 97.7 71.1 99.1 97.8 69.9 0.5 0.1 −1.2 
s.d. (%)  0.9 0.3 15.3 0.4 0.3 13.7 1.2 0.4 6.0 
Maximum (%)  99.5 98.2 82.6 99.5 98.2 77.7 2.0 0.5 6.7 
  Prob|{allSA}
 
Prob|{self}
 
Prob|{self} − Prob|{allSA}
 
Data subset (all first-time spawners) Sample size p1 p2 p3 p1 p2 p3 p1 p2 p3 
Dee maidens (DE1) 5 731 98.2 97.5 82.6 99.5 98.0 74.3 1.3 0.5 −8.3 
North Esk maidens (NE1) 75 317 99.5 97.7 71.0 98.4 97.7 77.7 −1.1 0.0 6.7 
Spey maidens (SP1) 17 389 98.8 97.9 75.3 99.4 97.4 76.4 0.6 −0.5 1.1 
Tay maidens (TY1) 24 730 97.3 98.2 81.6 99.3 98.2 75.5 2.0 0.0 −6.1 
Tweed maidens (TW1) 32 590 99.1 97.3 45.0 99.0 97.7 45.4 −0.1 0.4 0.4 
West coast maidens (WC1) 1 592 98.5 96.3 0.0 99.8 25.0 75.0 1.3 −71.3 75.0 
Total across catchments 157 349          
Minimum (%)  97.3 97.3 45.0 98.4 97.4 45.4 −1.1 −0.5 −8.3 
Mean (%)  98.6 97.7 71.1 99.1 97.8 69.9 0.5 0.1 −1.2 
s.d. (%)  0.9 0.3 15.3 0.4 0.3 13.7 1.2 0.4 6.0 
Maximum (%)  99.5 98.2 82.6 99.5 98.2 77.7 2.0 0.5 6.7 

The bottom four rows indicate the small variation in the estimated catchment probability values when summarized across catchments (not adjusting for sample sizes) for each column, but excluding data from the Scottish west coast (WC1), which has too few records of 2SW and 3SW fish to be reliable on its own. The west coast sample had just 108 fish and just 4 3SW fish, so is excluded from the summary statistics subtable.

Figure 3.

Comparing the probability model and catchment data. (a–c) Data from the Spey maiden spawner dataset compared with the canonical probability model. (d–f) Tweed maiden spawner dataset compared with the canonical probability model. (g and h) Tweed 1894/1895 fishery-officer-aged dataset compared with the canonical probability model. All panels show the 50% (light) and 95% (heavy) contours for p1 (green), p2 (blue), and p3 (red) as in Figure 1a, on which, where data allow kernel-density estimation, are superimposed a scatterplot of the data for a single sea age with density contours at 5, 20, 35, 50, 65, and 80% of the peak density. (a), (d), and (g) show 1SW data, (b) and (e) show 2SW data, (h) shows MSW data, and (c) and (f) show 3SW data.

Figure 3.

Comparing the probability model and catchment data. (a–c) Data from the Spey maiden spawner dataset compared with the canonical probability model. (d–f) Tweed maiden spawner dataset compared with the canonical probability model. (g and h) Tweed 1894/1895 fishery-officer-aged dataset compared with the canonical probability model. All panels show the 50% (light) and 95% (heavy) contours for p1 (green), p2 (blue), and p3 (red) as in Figure 1a, on which, where data allow kernel-density estimation, are superimposed a scatterplot of the data for a single sea age with density contours at 5, 20, 35, 50, 65, and 80% of the peak density. (a), (d), and (g) show 1SW data, (b) and (e) show 2SW data, (h) shows MSW data, and (c) and (f) show 3SW data.

Usually, predictions based on the size-at-date distributions for the same catchment predicted scale-read sea ages marginally better than predictions based on the overall Scottish dataset {allSA}. Predictions for 1SW by either method exceed 97% for all sites, and predictions for 2SW fish by either method exceeded 97% if the small west coast 2SW sample is excluded (at 96.3 and 25%, respectively; Table 6). Again, 3SW fish were predicted much less well, at just 45% for each method. Excluding the west coast sample, which had just 108 fish of 2SW and only 4 of 3SW, a detailed comparison shows that five groups (Dee 3SW, North Esk 1SW, Spey 2SW, Tay 3SW, and Tweed 1SW) were actually better predicted by the overall data than by their own catchment-specific data. Although the tabulated differences between prediction methods are almost certainly statistically significant, owing to the massive sample sizes, it is of note that 5 of 15 were worse when predicted by catchment-specific data, 6 of 15 improved prediction efficiency by ≤0.6%, and just four groups showed improvements of <0.6% (1.1–2.0%).

Excluding the small west coast sample, the local datasets produced, on average, just 0.5% and 0.1% improvements for 1SW and 2SW fish, respectively, whereas the local-data-based predictions actually averaged 1.2% worse for 3SW fish. These differences in classification success were small compared with both the 97% baseline success and the likely stochastic variations in the sea-age composition of the small data subsets and were not pursued further. The lesser success achieved by local-data-based predictions for 3SW fish presumably arises from the scarcity of 3SW fish in all datasets, such that that the overall {allSA} set is less likely to be stochastically biased.

The effectiveness of the probability distribution determined from the {allSA} data in determining the sea ages at a catchment level is illustrated in Figure 3 for first-time spawners from the Spey and Tweed, respectively. As in Figure 2, for both catchments, the great bulk of the points for each scale-aged group (as shown by their ellipsoidal contours) fall between the appropriate probability contours, with just a handful of extreme deviants. Possible explanations for these extreme deviations are explored below.

Potential causes of extreme outliers

The very high sea-age prediction success achieved by the size-at-date algorithms, combined with the unusual distributions of points for repeat spawners and the knowledge that, although highly reliable, scale reading is not immune from errors, led us to investigate possible reasons for the more extreme outliers. Therefore, a sample of scales was selected to have their sea ages re-read. Unknown to the reader, 80 of these were deemed by the algorithm to have reliable initial scale ages and 119 were deemed to be dubious. The experienced scale-reader was asked to rank the ease of interpretation of all scales re-read (easy, difficult, or very hard). Of the 80 scales, the algorithm predicted had originally been reliably aged, consisting of 47 1SW, 30 2SW, and 3 3SW fish, 78 scales (96%) were classed as easy to read, just 2 (4%) as difficult to read, and none as very hard to read. All 80 were reclassified to the same sea age as their original classification.

The difference in reclassification probability between the algorithm-designated reliable and dubious classes was very highly significant (forumla, p ⋘ 0.001), but there was no evidence that the chances of re-ageing the dubious scales varied between the combinations of original and algorithm-suggested sea-age-change classes (forumla, 0.5 > p > 0.3).

Of the 119 dubious scales, 4 were discovered to have had their dates incorrectly transcribed from the scale-packets (February rather than July, i.e. “2” misread as “7”). Of the 115 remaining dubious ones, just 67 (58%) were classed as easy to read, 40 (35%) as difficult, and 8 (7%) as very hard to read. An average of just 71% of the dubious scales was reclassified to the same sea age, and the remaining 29% to different sea ages. Interestingly, only 1 of these 33 reclassifications was to a new sea age that was not the same as the revision suggested by the size-at-date algorithm (initially read as 3SW; algorithm suggested 1SW; re-read as 2SW). One scale (out of 115) was reclassified as 2?SW, and a second was noted as of dubious reliability by the original scale-reader.

Of the 52 reasons given for scales being hard to interpret, 26 (50%) referred to probable checkmarks, 14 (27%) to probable so-called “droppers”, i.e. salmon that enter one estuary, spend a period of some weeks in freshwater, then return to the sea (and presumably subsequently re-enter another river (Smith et al., 1998), and 1 (2%) to a probable kelt (late-returning kelts that netsmen happen to catch along with newly entered fish).

To summarize, a small proportion of the scale records, probably ∼1%, proved problematic to read. Of the records identified as outliers by the size-at-date algorithm, some 30% were subsequently ascribed a different age on second reading. This second reading was not (could not be) fully independent of the first (because fish lengths are written on the scale-packets and scale-readers traditionally use that information to guide their designations). Around 33% of the algorithm outliers seem to be ascribable to potential scale-reading problems, such as checkmarks, but most (67%) were consistently designated to a common sea age by both readers. The data in this preliminary validation exercise are insufficient to allow us to calculate the likelihoods that they would agree by chance about problematic scales, because the two assessments were not fully independent and the sample size too small.

Temporal stability of the probability distributions

The temporal stability of the age–length–date relationships was tested at two levels, first within decades of the {allSA} dataset and second for the Tweed 1894/1895 data. We remind the reader that those Tweed data only give the recorder's opinion whether the fish were 1SW or MSW (and do not distinguish at all between 2SW and 3SW fish), but that they are entirely independent of the data used to derive the probability equations.

For the decadal subsets of the {allSA} data, the results tabulated in Table 7 show both that the classification accuracy remains similarly high and also that it is very stable over time. Although the efficiency of age-assignment generally increases slightly when predicted from that same decade's data as opposed to the entire {allSA} set, the improvements are tiny. For 1SW fish, the changes vary between −0.2 and +0.4%, and for 2SW fish, the changes vary between −0.1 and +0.8%. The changes for 3SW fish, which have much smaller sample sizes, are somewhat larger; over the five decadal periods the changes ranged between −5.7 and +6.3%. In the first three decades, when the numbers of recorded 3SW fish were non-trivial, the absolute accuracies for the 3SW fish were generally >70%, but in the later two decades, when 3SW fish numbers were extremely small, the accuracies decline to around 60 and 40%, respectively (Table 7).

Table 7.

Predictive efficiencies of the probability distribution derived from the canonical dataset {allSA}, compared with those derived from various historical datasets, generically denoted by {self}, with columns 2–4 showing the percentage correct identifications of 1SW, 2SW, and 3SW fish using the {allSA} probabilities, and columns 5–7 showing the same quantities when predicted from size and date from the same subset.

 Prob|{allSA}
 
Prob|{self}
 
Data subset p11 p22 p33 p11 p22 p33 
allSA 98.7 97.0 70.8 98.7 97.0 70.8 
allSA_63-73 97.5 97.4 74.0 97.9 97.9 80.3 
allSA_74-84 98.5 97.4 74.6 98.6 97.8 68.9 
allSA_85-95 99.0 97.0 57.5 99.0 96.9 58.6 
allSA_96-06 99.6 95.4 40.3 99.4 96.2 45.4 
Tweed_1894-5 97.7 95.7 n.a. 95.0 98.3 n.a. 
 Prob|{allSA}
 
Prob|{self}
 
Data subset p11 p22 p33 p11 p22 p33 
allSA 98.7 97.0 70.8 98.7 97.0 70.8 
allSA_63-73 97.5 97.4 74.0 97.9 97.9 80.3 
allSA_74-84 98.5 97.4 74.6 98.6 97.8 68.9 
allSA_85-95 99.0 97.0 57.5 99.0 96.9 58.6 
allSA_96-06 99.6 95.4 40.3 99.4 96.2 45.4 
Tweed_1894-5 97.7 95.7 n.a. 95.0 98.3 n.a. 

n.a., not available.

The Tweed data from 1894 and 1895 are illustrated in Figures 3g and h, with Figure 3h showing MSW (2SW + 3SW) fish. The classification accuracies for that dataset (bottom row, Table 7) show 1SW successes (relative to the observer's opinion) of 97.7% compared with 98.7% for the {allSA} baseline; we note that this disagreement could be attributed to the observer's opinion rather than the algorithm being incorrect. For MSW fish, the accuracies change from [97.0%, 70.8%] to an amalgam of 95.7%. The amalgamated value is as high as it is partly because the discrimination is simpler if the 2SW to 3SW distinction is omitted and also because the Tweed has never had many 3SW fish.

The comparisons summarized in Table 7 show that the relationships between size and dates of capture that separate the sea-age classes have varied little over time.

Predictions from less precise data

The data analysed thus far are high resolution, yielding fork lengths to 1 cm, weights to 0.01 kg, and dates to 1 d. However, many long-term historical records that fisheries managers want interpreted are much less precise, e.g. monthly lists of fish weighed to 0.5 or 1 lb (Imperial; 0.22 or 0.45 kg). The consequences of such reduced precision were investigated using our {allSA_W} dataset. Weights were degraded to 0.5 lb intervals and the dates to monthly midpoints (full technical detail is given in Appendix 3). The sea ages of the degraded values were then predicted using the equations derived from the undegraded LW values. The resulting prediction efficiencies (Table 8) remained high (98.0, 95.4, and 62.7% for p11, p22, and p33, respectively), decreasing by just a fraction of 1% from those obtained with the undegraded data (down by 0.3, 0.4, and 0.1%, respectively), as illustrated in Figures 2g–i. It is therefore clear that historical data at these much lower levels of precision still contained extremely valuable information.

Table 8.

Percentage predictive efficiency for less precise (degraded) versions of the {allSA_W} dataset, using LW as the size measure, giving the classification success probabilities derived from the scale-aged information from the same dataset.

 Scale-read 1SW
 
Scale-read 2SW
 
Scale-read 3SW
 
Data accuracy p1 p2 p3 p1 p2 p3 p1 p2 p3 
Original accurate data 98.3 1.7 0.0 3.4 95.8 0.8 0.1 37.4 62.6 
Month/0.5 lb 98.0 2.0 0.0 3.6 95.4 0.9 0.1 37.2 62.7 
Month/1 lb 97.2 2.8 0.0 3.2 96.0 0.8 0.1 38.8 61.2 
 Scale-read 1SW
 
Scale-read 2SW
 
Scale-read 3SW
 
Data accuracy p1 p2 p3 p1 p2 p3 p1 p2 p3 
Original accurate data 98.3 1.7 0.0 3.4 95.8 0.8 0.1 37.4 62.6 
Month/0.5 lb 98.0 2.0 0.0 3.6 95.4 0.9 0.1 37.2 62.7 
Month/1 lb 97.2 2.8 0.0 3.2 96.0 0.8 0.1 38.8 61.2 

The predicted sea age for each record is the most probable sea age for its size and date of capture.

Sea age as traditionally determined from fixed-size boundaries

In the past, a number of rules-of-thumb, involving weight limits that did not vary with season, was used to indicate whether salmon were more or less likely to be grilse or MSW fish. We compare a range of these in Table 9.

Table 9.

Predictive success table, comparing the efficiency of the {allSA} length-based probability distributions (top row), which vary with date of capture, with a series of time-invariant delimiters, based on LW converted from historical classes of (Imperial) pound-classes (see text for detail).

 Scale-read 1SW
 
Scale-read 2SW
 
Scale-read 3SW
 
Boundary p1 p2 p3 p1 p2 p3 p1 p2 p3 
{allSA} length 98.7 1.3 0.0 2.2 96.9 0.9 0.0 30.4 69.6 
6 lb 59.0 41.0 – 2.1 98.0 – – – – 
8 lb 87.8 12.2 – 18.3 81.7 – – – – 
10 lb 97.5 2.5 – 47.7 52.3 – – – – 
16 lb – – – – 93.2 6.8 – 45.6 54.4 
18 lb – – – – 97.4 2.6 – 68.0 32.0 
 Scale-read 1SW
 
Scale-read 2SW
 
Scale-read 3SW
 
Boundary p1 p2 p3 p1 p2 p3 p1 p2 p3 
{allSA} length 98.7 1.3 0.0 2.2 96.9 0.9 0.0 30.4 69.6 
6 lb 59.0 41.0 – 2.1 98.0 – – – – 
8 lb 87.8 12.2 – 18.3 81.7 – – – – 
10 lb 97.5 2.5 – 47.7 52.3 – – – – 
16 lb – – – – 93.2 6.8 – 45.6 54.4 
18 lb – – – – 97.4 2.6 – 68.0 32.0 

The date-invariant rules performed appreciably worse than the {allSA} probability functions that vary with date. A rule of 8 lb was widely used by netsmen to distinguish grilse from the more-valuable salmon (around the 1980s, for example, the value per lb of commercially caught grilse was <£1.50 if under 7 lb in weight, but £2 per lb for salmon, classified as >7 lb). Table 9 shows that, over an entire season, such a rule probably resulted in them wrongly classifying 12.2% of true grilse as salmon, but that extra profit was apparently more than what was lost (assuming the rule was applied consistently) by them misclassifying 18.3% of true salmon as grilse. An alternative 6-lb rule would have been more profitable: over the season such a rule would elevate 41% of true grilse to the more-valuable salmon, and only penalize profits marginally by misclassifying 2.1% of true salmon as grilse. In contrast, a 10-lb rule, if applied over the entire season, would classify 97% of true grilse correctly, but exempt 47.7% of true salmon.

More complex rectangular rules (below/above weight W after date D) could clearly be investigated, but the curved nature of the constant probability contours means that they would always be inferior.

Discussion

A primary purpose of the work reported here is to devise an objective criterion by which time-stamped, length-classified counts of adult salmon returning to spawn can be separated into sea-age classes without the necessity to read sea ages from scale annuli. A key finding is that a combination of fish length and return date predicts sea age with great reliability over the whole year and irrespective of the sea-age composition of the sample. Equations based on either one alone would give differing rates of success if the return period, size of fish, or sea-age composition changes.

The probabilistic technique described, when applied to a large multiyear dataset, achieves >97% accuracy in discriminating 1SW and 2SW fish, and generally discriminates 2SW and 3SW fish with ∼95 and ∼70% accuracy (Table 3). We are unaware of a large, rigorous, blind trial of scale-reading reliability that shows whether this method achieves >97% success.

Figure 4 illustrates the practical applicability of these methods by separating the North Esk first-time spawner {NE1} dataset of fish caught (note that these numbers depend on both fishing and sampling effort and are not reliable surrogates for either population numbers or relative survival) and sampled into its sea-age components using probabilistic methods. Two results are shown, based on distributions fitted to both the data themselves and to the full dataset {allSA}. As we would expect from the whole dataset test results reported above, reconstruction of the 1SW and 2SW time-series is essentially perfect. Although not unexpected, the 3SW time-series shows detectable deviations between observation and reconstruction, the reconstructed series shows all the qualitative features of the observations. We also observe that using distributions fitted to the data themselves yields very little improvement over those fitted to the global dataset. Finally, we see from the lower row of panels in Figure 4 that these probabilistic methods achieve comparable performance using categorized data with quite wide category limits. Figure 4 and Table 2 also show that the method is robust to considerable variation in the relative proportions of each sea-age group in the samples analysed.

Figure 4.

Recovering the sea-age-class composition from the North Esk first-time spawner data using length and date of capture. Fish numbers are of salmon caught and sampled. The upper row shows results accurate to 1 cm and 1 d, with panels left to right for 1SW, 2SW, and 3SW fish, respectively. The solid lines show the observed numbers determined from scale ages. Open squares joined by dotted lines show counts of individuals with a given maximum probable sea age determined from the probability distribution fitted to the appropriate dataset. Filled squares show counts from ages determined on a probability distribution fitted to the {allSA} dataset. The lower row shows comparable results where the predictions were made with degraded data having only 30-d temporal resolution and 2-cm length resolution.

Figure 4.

Recovering the sea-age-class composition from the North Esk first-time spawner data using length and date of capture. Fish numbers are of salmon caught and sampled. The upper row shows results accurate to 1 cm and 1 d, with panels left to right for 1SW, 2SW, and 3SW fish, respectively. The solid lines show the observed numbers determined from scale ages. Open squares joined by dotted lines show counts of individuals with a given maximum probable sea age determined from the probability distribution fitted to the appropriate dataset. Filled squares show counts from ages determined on a probability distribution fitted to the {allSA} dataset. The lower row shows comparable results where the predictions were made with degraded data having only 30-d temporal resolution and 2-cm length resolution.

Although here applied only to the marine-return life stage of Atlantic salmon, the method is clearly general and could in principle be applied to other life stages and other fish species. However, there is no guarantee that such endeavour would result in such clear and convenient outcomes as reported above.

Biological implications

Although it has long been known that grilse are smaller on average than salmon and that average 2SW salmon are smaller than the average 3SW fish, it is also widely believed that it is hard to separate grilse from salmon by size. There are three surprising aspects of our present findings. First is that paired values (date of coastal return and size) allow discrimination between grilse and MSW fish with ∼97% confidence, identify 2SW with 95% confidence, and 3SW with ∼70% reliability. Second is that the pair of probability distributions that achieve this vary only trivially by site within Scotland, between recent decades, for Scottish data from a century ago and by the method of capture. Third is that the date of capture for rod-caught fish at beats well removed from the estuary and in seasons (October and November) when MSW fish could have been in the river for months seems a good surrogate for the date of coastal return. On reflection, however, these surprises become less startling.

The general von Bertalanffy growth equation (von Bertalanffy, 1938) widely applied to many fish species shows that fish growth rates decline proportionately to the difference between current size and an asymptotic maximum. Lester et al. (2004) recently showed that the von Bertalanffy model is, for iteroparous fish, a special case of a mechanistic model that posits that fish divert increasing proportions of assimilated food to reproductive stores as they grow (and age). If one allows that Scottish Atlantic salmon follow a similar assimilate-allocation rule, although 98% of them are effectively semelparous (Bacon et al., 2009), one would expect larger size differences between younger age classes. Bacon et al. (2009) showed that the average sizes of 1SW and 2SW salmon increase with month during their year of coastal return, implying that the growth rule may also apply at finer time-scales than 1 year. 1SW and MSW fish interbreed, have similar shapes and conditions (Bacon et al., 2009), and have broadly similar rates of ocean survival (rates per month or year, not over the total periods). Assessments carried out by the ICES Working Group on North Atlantic salmon (ICES, 2009) use a common mortality of 0.03 month−1 for both 1SW and MSW fish. Therefore, if one assumes that sea age is partly heritable (for which there is growing evidence; Gardner, 1976; Gjerde, 1984; Gjerde and Gjedrem, 1984; Hankin et al., 1993; De Leaniz et al., 2007), then evolutionary arguments mitigate against the likelihood of (mature) fish of different sea ages having greatly overlapping sizes. If oceanic mortality, m, was broadly similar across sea ages, the shorter sea-age life history would outperform the longer unless the size (and hence breeding resource) difference was of the order of the inverse of the survivals. This is (1 − m)1/2 for 1SW to 2SW, but just (1 − m)2/3 for 2SW to 3SW. This expectation is commensurate with our finding that 2SW and 3SW fish have a larger proportion of fish in the overlap zone than do 1SW and 2SW.

Our analysis confirms the fact that single size thresholds that do not vary with day of the year are relatively poor indicators of sea age (Table 9). This is unsurprising, given the sloping nature of the probability contours in Figure 1. MacLean et al. (1996) showed that rod fishers' subjective categorization of grilse as MSW fish (the grilse error) varies with season. More recently, it has been shown that the extent of grilse error (as judged for fish with subsequent scale-read ages) varied not only with season, but was much larger for rod-caught reports (error proportions of 0.50–0.20, depending on month, consistent with the earlier findings of Dunkley et al., 1993) than for net reports (0.15–0.05; A. F. Youngson, pers. comm.). Our results show that, on any given date, the size distribution is clearly bimodal. Netsmen will often handle many fish simultaneously in a day and are therefore much more able to observe and judge how this demarcation changes over a season. In contrast, most anglers will see fewer fish in a month than many netsmen in a day (week) and are therefore less able to observe the changing size boundaries. The probability equations here defined are appreciably more accurate than both, being based on many more fish than even netsmen handle.

Figure 2 shows that repeat-spawning grilse, though somewhat larger than first-time spawning grilse, are smaller at a total sea age of 2 (their additional year at sea plus their first-time spawner-age of 1SW) than are first-time 2SW spawners of the same total age. Combined with the fact that repeat spawning is rare in Scotland (<2%; Bacon et al., 2009), these small growth increments mean that our overall sea-age allocation success remains high for Scottish salmon, even when repeat spawners are represented in the total sample {allSA}. In circumstances where repeat-spawning frequency is much higher (e.g. Finland, up to 21%, Niemela et al., 2006; Canada, up to 30% on Miramichi, Moore et al., 1995), the overall success rate might be proportionally lower. It would be instructive to repeat these analyses with such data.

Re-reading a subset of scales where the original scale reading disagreed with the algorithmic ageing from size and date indicated that a fair proportion of such discrepancies result from scales that are hard to interpret. The method and sample size of scales re-read does not allow us to quantify the likely magnitude of the difficulty, but it does indicate that a proportion of scales yield uncertain values.

Bacon et al. (2009) reported differences in mean salmon length (for 1SW and 2SW fish) between rivers that change coherently across years. At first sight, this sits ill with the current findings that single separating functions perform almost equally well across both rivers and decades. However, the findings are not incompatible. The explanation is simply that the degree of site and annual change are such that even their combined effects only move a tiny percentage of individual salmon sufficiently far out of the wide regions of individual variation (ellipsodal contours, Figure 1) that they cross the probability boundaries.

It would be interesting to know whether Atlantic salmon populations elsewhere in Europe or North America showed similar degrees of separation, and particularly if their probability functions were similar to the Scottish ones (a web-based data-comparison system to facilitate such evaluations can be found at http://www.mathstat.strath.ac.uk/outreach/salwrd).

The most unexpected aspect of our results was to find that using the date of rod capture as a surrogate for the date of coastal return at beats distant from the estuary and even in autumn did not greatly reduce the success of the size-and-date assignment of sea age. However, Thorley et al. (2007) showed that rod-recapture rates of salmon decline rapidly with days after river entry, and Smith and Johnstone (1996) and Smith et al. (1998) showed from radio-tracking studies that salmon movements become less frequent with time after river entry. The declining rate of movement and the rapidity of the rod-recapture decline suggest that most salmon that have been in the river for a month or so may be extremely hard to catch by rod. The overall rod-capture dataset would therefore consist mainly of newly arrived, fresh-run fish, resulting in the high classification success for such rod captures reported here. This argument becomes stronger if one takes the vertical extents of the size variation on a given date (across the ellipsoidal contours of Figure 2) and rotates them into the horizontal. This indicates that the typical size uncertainty variation equates to some 25 d, a period similar to that in which large declines in movement and recapture chances are reported and to the monthly time degradation to which the ageing algorithm is robust (Figure 2, Table 8).

Management implications

Short-term benefits

A key challenge facing Scottish fisheries managers is to balance the conservation of stocks of MSW salmon while still maintaining a fishery. Sportsmen not content to catch and release (C&R) are therefore obliged to focus on the season when grilse can be taken and kept, but important components of the MSW stocks overlap this period. At present, even owners wishing to conserve these MSW fish and anglers content to return them are largely unable to make reliable distinction (20–50% grilse-error rate for rods) between 1SW and MSW fish at the time of capture on the riverbank, before fish have been killed or scales could be read. Our method (Figure 2) provides 98% confidence using a single Scotland-wide relationship, the seasonal values of which could be read off a graph easily carried in the angler's pocket (example graphs and tables, with explanatory information, can be downloaded as a single-page A4 leaflet from the weblink given above). Anglers in Scotland using such an approach would clearly then end up releasing a small proportion of 1SW repeat spawners in mistake for 2SW fish, but a proportion as small as 2% should not really be of great concern.

We have shown that such size-and-date distinctions are better than size alone. Length, which can be measured with simple equipment, at least on dead fish, gives slightly more accurate prediction of sea age, but weight, which is also easily determined for rod-caught salmon destined for release, discriminates virtually as effectively as length. Our sea-age discrimination method could, in principle and in future, be used by both rod and net fishers.

As an added incentive to anglers, the same method would give confidence (70%) whether their MSW fish were 2SW or the scarcer 3SW. More proactively, as net-derived data decline, anglers could be encouraged to report their records of size and date which, combined with scale samples, would allow fishery scientists to check whether the relationship evident over the past 50 years does indeed hold good in the future. Such scales when read to give freshwater age and sea age would also allow adult salmon to be ascribed their year of birth, another useful metric for understanding population dynamics.

In principle, salmon managers outside Scotland could use the same C&R approach to conserve MSW stocks. Once suitable algorithms were known (the web application cited above allows both comparison of other data to the Scottish canonical set and the calculation of local probability distributions, given local data provided by the user), the same advantages as for Scotland would apply if repeat-spawner frequency was low. Indeed, there is still potential for useful improvement even if repeat spawning was more common, although the extent of improvement would need to be assessed.

Longer-term understanding

Gurney et al. (2010) recently demonstrated that long (>40 year) time-series of numbers of adult salmon returning to local spawning burns potentially yield important insights about population processes. Perhaps even longer runs of data would permit more detailed insight and help distinguish between fluctuations in salmon abundance attributable to long-term environmental processes, as opposed to those that can be influenced by management interventions. Knowing the sea ages of adult salmon is important to that analysis. The approach described here provides an objective means, at least for Scottish salmon, of determining sea ages, and the ratio of such fish in populations from samples of fish recorded in angling logbooks. We propose to address this issue in future. However, developing a robust method along these lines would require further datasets, of which we are currently unaware, representing many other Scottish rivers (the cited weblink includes data entry and validation routines, plus a contact for discussion of the potential value of any such historical records).

Future prospects

Most of the records on which the insights of this paper are founded come from studies instigated by fisheries scientists and carried out with the cooperation of salmon netsmen. During the past decade or so, most nets have ceased to operate, removing this vital source of important, representative data. Although one might have expected that similar data from angling, which depends on fish responding to lures, would yield biased results compared with net capture, it appears, fortunately and somewhat surprisingly, that in this instance salmon angling data are quite adequate. If salmon managers are to continue to be informed reliably in future, then this replacement source of information from anglers will need to be utilized. At present, there is no widespread recording of such data, nor any mechanism to collect and interpret such information. Even less is there any mechanism to collect representative sets of scales from such rod-caught fish, which would be crucial to establishing whether the underlying relationships reported here reliably held good into the future.

We therefore end by stressing the value of good, comprehensive record-keeping at salmon fishing beats (including, where possible, the collection of scales and simple biometric details), and emphasizing the importance of preserving old record books for beats and fisheries (especially when beats change ownership) and of properly collating and interpreting the information contained in them. Example pro formae and brief explanations of the use of such records can be found at http://www.mathstat.strath.ac.uk/outreach/salwrd. The weblink provided is presented as a first stage in providing a mechanism to facilitate such communal efforts and a discussion forum for their interpretation and use.

Acknowledgements

We thank the many Fisheries Laboratory staff who contributed to scale collection and reading and to data curation over the past five decades. We are also grateful to the netsmen, fishery owners, gillies, and anglers who provided access to collect the samples used in this analysis. We particularly acknowledge the meticulous approach of James Tosh, Fisheries (Woodall) Prizeman, St Andrews University, to the study he undertook in 1894 and 1895 on the Tweed and the detailed records he left. The far-sightedness of the then Fishery Board for Scotland in commissioning such work is also acknowledged. We thank Philip McGinnity for constructive discussions, and Jason Godfrey, the editor, and two anonymous referees for constructive comments that considerably improved earlier drafts. The work was supported by the Scottish Government (SF0280). Funding to pay the Open Access publication charges was provided by Marine Scotland.

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Appendix 1: Statistical methods

Suppose that we have N records of {sea age, size, day} in the form {(ai, si, di): i = 1, 2, …, N}. The likelihood of our data is given by  

formula
where zi = (si, di) and nij = 1 if ai = j and 0 otherwise (j = 1, 2, 3).

We now define the three probabilities as suitable functions of z = (s, d), then estimate the parameters of these functions by maximizing the likelihood function L. This can be done in a variety of ways, but one of the simplest and most flexible is the continuation ratio logit model in which L is reformulated as the product of two binomial likelihoods, so L = L1L2, where  

formula
 
formula
and q1(zi) = p1(zi), q2(zi) = p2(zi)/(1 − p1(zi)). Note that q1 = p1 is the probability that the sea age is 1, and q2 = p2/(1 − p1) is the probability that the sea age is 2 conditional on it being >1.

These two likelihoods are estimated separately using logistic regression, with the logit of both of the binomial probabilities q1 and q2 modelled as quadratics in the two covariates (s, d), i.e. Logit(q) = β0 + β1s + β2d + β3s2 + β4sd + β5d2. All terms were significant. The probabilities q1 and q2 were estimated, then transformed back to the original probabilities p1 and p2 for each pair (s, d).

Appendix 2: Normality of the log(length) and log(weight) relationship

Constructing a QQ plot of the residuals from the log–log length-to-weight relationship for the {allSA} dataset (Figure A1) shows that the relationship is extremely close to a normal distribution within ±2 s.d., but beyond that the tails are strongly extended from a normal expectation, indicating strong kurtosis.

Figure A1.

QQ plot for normality of residuals of log(weight) against log(length).

Figure A1.

QQ plot for normality of residuals of log(weight) against log(length).

Appendix 3: Less precise data

Cases where less precise data are available were investigated as follows. The example assumes that dates of capture are only recorded in monthly intervals and that weights are recorded in intervals of 0.5 lb Imperial (0.2268 kg). The effects of this on prediction ability were investigated with the {allSA_W} dataset by fitting a probability distribution to LW. Weights W and days of year (DoY) were then degraded according to  

(A1)
formula
 
(A2)
formula

Finally, degraded weights were converted to degraded LW values and sea ages predicted from the probability distribution fitted to the undegraded LW values. The resulting graphs initially look somewhat bizarre, but the classification efficiency table (Table 8, which also gives results for a weight accuracy of 1.0 lb) shows that the success rates for [1/2 lb and monthly dates] is only trivially worse than those achieved by the original data.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/2.5), which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.