Background: Mathematical Methods

These series of equations are based off of Wakabayashi et al. (1985):

Wakabayashi, K., R. G. Bakkala, and M. S. Alton. 1985. Methods of the U.S.-Japan demersal trawl surveys, p. 7-29. In R. G. Bakkala and K. Wakabayashi (editors), Results of cooperative U.S.-Japan groundfish investigations in the Bering Sea during May-August 1979. Int. North Pac. Fish. Comm. Bull. 44.

Data Collection Process

For each station (indexed by \(j\)) contained in stratum (indexed by \(i\)), the total catch weight and numbers are recorded for all fish and most invertebrate taxa for the estimation of total biomass and abundance. For a subset of fish and invertebrate taxa, the lengths of either all individuals or a subsample of representative individuals are recorded for estimation of size composition. Individuals from a second level of subsampling below the length subsample are aged for the calculation of an age-length key, which is used to expand the length subsample to the complete length- and age-distribution of the sampled area.

Catch Rates

All trawl catch weights and numbers are standardized using the area swept by the trawl, \(E_{ij}\), of station \(j\) in stratum \(i\). \(W_{ijk}\) is the total catch weight of taxon \(k\) at station \(j\) in stratum \(i\). \(R_{ijk}\) is the total catch weight per area swept (also known as “weight-CPUE”, units \(kg/km^2\)) for taxon \(k\) at station \(j\) in stratum \(i\):

\[R_{ijk} = \frac{W_{ijk}}{E_{ij}}\]

\(N_{ijk}\) is the total numerical catch of taxon \(k\) at station \(j\) in stratum \(i\) enumerated either directly from every individual caught in the trawl sample or expanded from a representative subsample of individuals. In the case where a subsample is taken, the expanded total numerical catch \((\hat N_{ijk})\) is estimated by dividing the total catch weight by the mean individual weight \(h_{ijk}\):

\[\hat N_{ijk} = \frac{W_{ijk}}{\hat h_{ijk}}\]

\[h_{ijk} = \frac{w_{ijk}}{s_{ijk}}\]

where \(w_{ijk}\) and \(s_{ijk}\) are the total subsampled weight and numbers of taxon \(k\) at station \(j\) in stratum \(i\).

The estimate of numerical catch per area trawled (also known as “numerical CPUE”, units \(no/km^2\)) is calculated similar to weight-CPUE:

\[\hat S_{ijk} = \frac{\hat N_{ijk}}{E_{ij}}\]

Biomass and Abundance Calculations

Total estimated biomass or abundance (and the associated estimated variances) of taxon \(k\) in stratum \(i\) is the product of average CPUE and the area of stratum \(i\) (\(A_i\)):

\[\hat B_{ik} = A_i \bar R_{ik}\]

\[\bar R_{ik} = \frac{\sum_{j = 1}^{n_i} R_{ijk}}{n_i}\]

with estimated variance:

\[\hat{Var}(\hat B_{ik}) = A_i^2 \hat{Var}(\bar R_{ik})\]

\[\hat{Var} (\bar R_{ik}) = \frac{\sum\limits_{j=1}^{n_i} (R_{ijk} - \bar R_{ik})^2 }{n_i(n_i-1)}\]

Since strata are independent, the total biomass and variance estimates across a subarea or the entire survey region are calculated as the sum of the stratum-level estimates of total biomass and variance estimates, respectively, that are contained within the subarea or region. As an example, the total estimated biomass for the entire survey area is:

\[\hat B_{k} = \sum\limits_{i=1}^I \hat B_{ik}\]

where \(I\) is the total number of strata, with estimated variance:

\[Var(\hat B_{k}) = \sum\limits_{i=1}^I Var(\hat B_{ik})\]

Total abundance (\(\hat P_{k}\)) is calculated similar to total biomass, replacing weight-CPUE with numerical CPUE.

Size Composition

For a subset of taxa, all individuals (or subsampled individuals) of taxon \(k\) at station \(j\) in stratum \(i\) were lengthed to the nearest 1-cm bin (indexed by \(l\)) and classified by sex, indexed by \(m\) where 1 = Male, 2 = Female, and 3 = Unsexed (\(s_{ijklm}\)). The recording of unsexed individuals occurs either because the sex of individual could not be determined or the sex of individual was not collected.

The calculation of the size composition comes from two expansions. First, the proportion of observed individuals of sex \(m\) in length bin \(l\) of taxon \(k\) at station \(j\) in stratum \(i\) (\(z_{ijklm}\)) was used to expand the length-frequency subsample to the total numbers of individuals per area swept (\(\hat S_{ijk}\)) to calculate the estimated number per area swept of individuals of sex \(m\) and length bin \(l\) of taxon \(k\) at station \(j\) in stratum \(i\) (\(\hat S_{ijklm}\)).

\[z_{ijklm} = \frac{s_{ijklm}} {\sum\limits_{m=1}^3\sum\limits_{l=1}^{L_{ijkm}} s_{ijklm} }\]

\[\hat S_{ijklm} = \hat S_{ijk} z_{ijklm}\]

where \(L_{ijkm}\) is the maximum length bin observed for taxon \(k\), sex \(m\) at station \(j\) in stratum \(i\). Second, \(\hat S_{ijklm}\) was expanded to the total estimated stratum-level numerical abundance to calculate the estimated number of individuals of sex \(m\) in length bin \(l\) of taxon \(k\) in stratum \(i\) (\(\hat P_{iklm}\))

\[\hat P_{iklm} = \hat P_{ik} \frac{\sum\limits_{j=1}^{n_i} \hat S_{ijklm}} {\sum\limits_{j=1}^{n_i}\sum\limits_{m=1}^3\sum\limits_{l=1}^{L} \hat S_{ijklm} }\]

In the Bering Sea survey regions, only hauls with associated length-frequency data are included in the size composition calculations. In the GOA and AI an “average size composition” \(\bar z_{iklm}\) is computed using the hauls in the same stratum and year and then used to impute the size composition for that haul:

\[ \bar z_{iklm} = \frac{\sum\limits_{j=1}^{n_i} s_{ijklm}}{\sum\limits_{m=1}^{3}\sum\limits_{l=1}^{L_{ikm}}\sum\limits_{j=1}^{n_i} s_{ijklm}}\]

Age Composition

Within each haul, a second subsample is drawn from the length subsample to collect otoliths for determining individual age, indexed by \(x\). The number of observed individuals of age \(x\), sex \(m\), length bin \(l\), taxon \(k\), station \(j\) in stratum \(i\) is \(v_{ijklmx}\). An age-length key characterizing the proportion of ages by taxon \(k\), length bin \(l\), and sex \(m\) is created by pooling age data across stations and strata within a given survey region (\(y_{klmx}\)).

\[y_{klmx} = \frac{v_{klmx}}{\sum\limits_{x=1}^{X_{klm}} v_{klmx}}\]

where \(X_{klm}\) is the oldest age of taxon \(k\) in length bin \(l\) and sex \(m\). The age-length key is used to transform the size compositions to age compositions, total number of individuals of age \(x\), sex \(m\), taxon \(k\) and stratum \(i\) \((\hat Y_{ikmx})\).

\[\hat Y_{iklmx} = \hat P_{iklm}y_{klmx}\]

\[\hat Y_{ikmx} = \sum\limits_{l = 1}^{L_{ikm}}\hat Y_{iklmx}\]

The estimated mean length (mm) at age \(x\) and sex \(m\) for taxon \(k\) in stratum \(i\) \((\bar d_{ikmx})\) is a weighted average of length using distribution of \(\hat Y_{iklmx}\) across length bins as the weights.

\[\bar d_{ikmx} = \frac{\hat Y_{iklmx}d_l}{\sum\limits_{l=1}^{L_{ijkm}} \hat Y_{iklmx}}\]

where \(d_l\) is the length value associated with length bin \(l\), with estimated weighted variance:

\[\hat {Var}(\bar d_{ikmx}) = \sum\limits_{l=1}^{L_{ijkm}} (\frac{\hat Y_{iklmx}}{\sum\limits_{l=1}^L \hat Y_{iklmx}} (d_l - \bar d_{ikmx})^2)\]