Stochastic Age-frequency Estimation Using the Von Bertalanffy Growth Equation

The method of estimating age frequency from length frequency via the von Bertalanffy growth equation is deterministic and yields biased results. Most of the biascan be removed by incorporating a stochastic element in the von Bertalanffy relationship. The stochastic element is based on estimated probabilities of lengths by intervals at age. the probabilities being estimated from variances in lengths-at-age. Based on age-length samples from the Pacific bonito fishery the stochastic method gives improved age-frequency estimates over those obtained by the deterministic method. The stochastic application may be generalized to all growth models includingdiscontinuous growth such as in crustaceans. Complex population dynamics techniques rely heavily on age-structure information. Frequently, appropriate assessment techniques for a stock require an estimate of the age frequency of that stock. For example yield-per-recruit analysis (Ricker 1958) is computed on the dynamic relationship between growth and mortality: Mortality rates when computed via cohort analysis (Murphy 1965) are based on estimated age frequency. For some species accurate aging methods are not available. When feasible, determining the age of fish and consequently computing an age frequency are most accurately accomplished by visual inspection of scales, otoliths, or other structures (Ricker 1958). Such visual inspection is time consuming and often expensive. To reduce the cost and time of estimating the age structure of a fisheries catch, age frequency is usually estimated from sampled length frequency, the agelength relationship being described by either an age-length key or a growth curve such as the von Bertalanffy growth curve (Ricker 1958). The growth curve method is used when there are insufficient data to construct an age-length key. Age-length keys work on the principle that age can be estimated from length using information contained in a previously or concurrently aged sample from the population. As long as the proportion of length-at-age remains the same for all ages. then the age-length key will yield unbiased ISouthwest Fisheries Center La Jolla Laboratory. National Marine Fisheries Service. NOAA. 8604 La Jolla Shores Drive. La Jolla. CA 92038. ‘1837 Puterbaugh Street, San Diego. CA 92103. Manuscript accepted June 1982. FISHERY B U L L E T I N : VOL. 81. NO. I . 1983. estimates of age for any sampled lengths from that population. However, since the estimated parameters of an age-length key-proportions of age-at-length-are dependent on the sampled population used to construct the key, the application of the key to the population with altered age structures can yield inaccurate results. Kimura (1977) and later Westrheim and Ricker (1978) demonstrated that under conditions of varying year-class strength and substantial overlap of lengths between ages, age-length keys can yield nearly useless estimates of numbers-at-age. Clark (1981) effectively removes age-length key bias by first proportioning numbers in length intervals a t age over time and then using the matrix of these proportions standardized over time to compute least-squares estimates of age frequency from the vector of length frequency. Effective applications of many stock assessment growth and mortality based methods require that ages are expressed in fractions of years (Ricker 1958; Lenarz et al. 1974). The large number of aged fish required to construct asufficient key for a large number of ages isdifficult and expensive to attain. Even with Clark’s bias correction procedure, the construction of a sufficient key can present difficulties due to data needs. In this paper we deal specifically with the von Bertalanffy growth equation and the application of stochastic methods to reduce or eliminate biases. However, it should be noted that the method presented here may be applied to any growth equation as well as to cases where no growth equation has been fitted or where growth is discontinuous as in crustaceans. The von Bertalanffy growth equation mathe-