Somerton and Kobayashi: Inverse method for mortality and growth estimation 



369 



Although the various approaches may differ in the 

 specifics of their application, all are based on the 

 premise that length-frequency data must first be cor- 

 rected for selection bias before they can be utilized to 

 estimate growth and mortality. Here we introduce a 

 new approach which reorders and joins the processes 

 of data correction and parameter estimation. This 

 approach, which is based on a stock assessment tech- 

 nique known as synthesis modeling (Methot 1989, 

 1990), will herein be referred to as the inverse method 

 for mortality and growth estimation (IMMAGE). The 

 use of IMMAGE is examined to estimate growth and 

 mortality rates from biased length-frequency data. 

 Additionally, the performance of IMMAGE, using 

 Monte Carlo simulation, is compared with approaches 

 used to correct selection bias in length-frequency data 

 prior to parameter estimation. 



Materials and methods 



IMMAGE vs. bias correction 



To understand how IMMAGE works and why it is an 

 inverse method for obtaining estimates of growth and 

 mortality parameters, the bias-correction approach 

 should first be examined (Fig. la-d). One variant of 

 the bias-correction approach might include (1) esti- 

 mating the unbiased length-frequency distribution 

 (Fig. lb) by dividing the observed length-frequency 

 distribution (Fig. la) by length-specific estimates of 

 capture probability; (2) converting the unbiased length- 

 frequency distribution to an age-frequency distribution 

 (Fig. Ic) using age and length information; and (3) 

 estimating the instantaneous mortality rate (M) as the 

 slope of a straight line fit to the logarithms of numbers 

 at age (Fig. Id). 



The IMMAGE approach, if applied to the same data, 

 would include (1) choosing initial values for M and the 

 number of day-0 larvae (Ng); (2) estimating an un- 

 biased age-frequency distribution (Fig. le) based on the 

 values of M and Nq; (3) estimating the unbiased 

 length-frequency distribution (Fig. If) from the age- 

 frequency distribution using age and length informa- 

 tion; (4) estimating the observed (i.e., biased) length- 

 frequency distribution (Fig. Ig) by multiplying the 

 unbiased length-frequency distribution by estimates of 

 capture probability; and (5) iteratively varying M and 

 Nq, and repeating steps 2-4, until the best fit is 

 achieved between the estimated and observed length- 

 frequency distributions. 



Thus both approaches estimate M by fitting a mor- 

 tality model. However, in the bias correction approach, 

 the model is fit to numbers-at-age derived from the 

 observed length-frequencies; while in the IMMAGE 

 approach, the model is fit to the observed length- 



frequencies themselves. Growth parameters are esti- 

 mated by IMMAGE in a similar manner, except a 

 growth model rather than a mortality model is fit to 

 the length-frequencies. 



To estimate the observed length distribution, 

 IMMAGE requires specification of a process model and 

 ancillary data. The process model contains parameters 

 that are iteratively varied to achieve the best fit to the 

 observed length-frequency distribution; the ancillary 

 data are parameters assumed to be known. For growth 

 estimation, the process model consists of a growth func- 

 tion describing the mean length-at-age and a variance 

 function describing the variance in length-at-age. An- 

 cillary data include estimates of the capture probabil- 

 ity at each length. For mortality estimation, the pro- 

 cess model consists of a mortality function describing 

 the instantaneous mortality rate at age or length. An- 

 cillary data include the mean and variance in length- 

 at-age, and the capture probability at each length. 

 Growth and mortality process models are not restricted 

 to any particular form and may include linear or 

 nonlinear functions. 



The performance of IMMAGE and several of the bias 

 correction approaches to parameter estimation was 

 examined by using a Monte Carlo simulation model. 

 For growth parameter estimation, bias correction ap- 

 proaches were not examined because no application to 

 larval fishes could be found in the literature. For mor- 

 tality parameter estimation, three bias correction ap- 

 proaches were examined: (1) elimination of the biased 

 portions of the observed length-frequency distribution, 

 (2) division of the observed length-frequency distribu- 

 tion by estimates of capture probability (correction), 

 and (3) elimination of the biased ages from a corrected 

 age distribution. 



Monte Carlo model 



The Monte Carlo simulation model is designed to mimic 

 the sequence of steps typically used in growth and mor- 

 tality studies. A central feature of this model is the 

 simulated collection of three types of data: length- 

 frequency samples, selection samples, and ageing 

 samples. 



Length-frequency samples are either unbiased, rep- 

 resenting random samples drawn from a larval fish 

 population, or biased, representing samples collected 

 with a plankton net. Selection samples are two indepen- 

 dent length-frequency samples, one biased and the 

 other unbiased, used to estimate length-specific cap- 

 ture probabilities. Such samples represent those that 

 might be produced by an experiment to estimate the 

 length-selection characteristics of a plankton net (i.e., 

 day to night catch comparisons). Ageing samples are 

 length-frequency samples in which each length mea- 



