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Return to Reading Otoliths Return to Main PageIn the ideal haul, the trawl net drags along the bottom for 30 minutes at a speed of 3 knots, with a net opening width consistent with the width in other hauls in the survey. To account for the fact that actual hauls seldom match the ideal, the standard unit for comparing catch data is the weight of the catch taken per hectare, or catch-per-unit-effort (CPUE). CPUE is calculated as follows:
CPUE= 0.01Wijk Dij*Fgk*pijwhere Wij is the weight of the catch of species k caught at the jth station in the ith strata, Dij is the distance trawled, and pij is the width of the net opening. 0.01 is a constant to convert area trawled in square kilometers to hectares. Fgk is the relative fishing power correction for vessel g and species k. Two fishing vessels 20 miles apart conduct the trawls, moving parallel to each other. The correction factor is used to bring the catch rate of the vessel less efficient at capturing species k in line with the catch rate of the more efficient vessel. The correction factor varies by species, and is determined by NOAA statisticians at the end of the survey.
To determine the size composition of a species, the quantity of fish at each length must be determined. This is calculated by multiplying the CPUE for a given haul by the proportion of the fish at a size l for that haul, yielding population-at-length data. Proportion is used because the quantity caught and quantity counted varies from haul to haul. It is calculated by dividing the number of species k measured at size l by the total number measured for that haul. The quantity of fish k and length l from all the hauls in an area can then be added. This is repeated for all lengths obtained. Composition can be determined for any of the subareas, or for the entire survey area.
The equations for length vs. age and weight vs. length are the same whether the regressions are being computed in Sigma Plot, Excel, or by hand. I used Excel with Solver. For Weight vs. Length, the equation is W=aLb, where a describes condition and b describes growth. If b=3, growth is isometric: the fish has a consistent body form and specific gravity. If b is greater or less than 3, growth is allometric, with significant differences between different populations of the same species, or between the same population in different years (Ricker, 207-209). a is Fulton's condition factor for comparing fish of the same species, and may vary based on sex and season. When these elements are the same, greater weight at a given length implies better condition, giving a higher factor. (Ricker, 207-209). To determine the values for a and b, the natural logs of the lengths and weights are taken and the equation is converted from W=aLb to lnW=lna + blnL. The variables a and b are each linked to the contents of an adjoining cell and a probable value is entered into that cell. 3 is a good starting value for b, as many fish have isometric growth. The resulting model lnW is compared to the recorded lnW using sum of squares. The cells containing the sums of squares are entered into Solver as the target cells, to be minimized by changing the values in the cells linked to a and b. From this, Solver produces the values for the variables that yield the equation that most accurately models the data.
The equation for Length vs. Age is L= L∞(1-e-K(t-to)), where L∞ is the hypothetical length that a fish would approach if it grew indefinitely, K is the growth rate, t is the age, and to is the age the fish would have been at length=0 if it followed the model precisely (i.e., the t intercept) (Ricker, 221). This is known as Von Bertalanffy’s equation.
Although the Length vs. Age equation appears more imposing, it is easier to process in Solver than Weight vs. length. The variables are listed in a cell each. L∞ has to be entered as “Linf” and to as “to”. Each variable name is then linked with the contents of an adjoining cell (Insert:Name:Define), and a probable value is entered into that cell. ThE linking allows the variable names to be used in the formula based on the equation. This yields a model length based on the age. The difference between the model length and actual length is squared then a sum of squares is computed. The cells containing the sums are entered into Solver as the target cells, to be minimized by changing the values in the cells linked to the variables. From this, Solver produces the values for L∞, K, and to that make the equation that most accurately models the data. Once the equations have been obtained, graphs for length vs. age and weight vs. length can be easily plotted.
Population-at-Age data can be derived from these tables using Excel, starting with the length-at-age data.In a table of lengths vs. ages, the fish at each length are broken out by percent at age. For example, in the 140 cm category, 25% were aged at 4 years, 25% at 5 years, and 50% at 6 years. The number of fish at each length (from the population-at-length data) is multiplied by the percent of that length at each age: since 25% of the 15,305,132 fish at 140 cm were age 4, there were 3,826,283 fish at 140 cm and 4 years-of-age. When this has been done for all lengths, the number of fish in each age column can be added, yielding the total number of fish at that age.
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© 2004 Teresa Jewell
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