Kope and Botsford: Recruitment of Oncorhynchus tshawytscha in central California 



261 



Figure 2 



Chinook salmon stock-recruitment curves for the main stem of the 

 Sacramento River. Upper cui-ve was fitted to the data for brood years 

 1962-66 and the lower curve was fitted to 1967-83. 



degrees of freedom is less than the number of years 

 in the series. Linear trends in the time-series reflect 

 variability on time scales that: (a) can easily be removed 

 by detrending, and (b) are not detectable with existing 

 data. We have removed linear trends from our time- 

 series because they may obscure a relationship on 

 faster time scales. 



Intraseries correlation due to variability on inter- 

 mediate time scales was accounted for in evaluating 

 the significance of computed correlation coefficients. 

 The variance of computed crosscorrelation coefficients 

 can be approximated by 



Var [r,„(0] 



ri 



h1 



P,,{i) P,nM) 



(1) 



ulation series. Stock-recruitment relationships for the 

 Sacramento, Feather, and American Rivers were ex- 

 amined by plotting recruitment against spawning 

 stock, and fitting Ricker stock-recruitment curves to 

 the data by the standard linear-regression method. This 

 approach can result in substantial biases in the esti- 

 mates of stock-recruitment parameters that result from 

 the correlation between recruitment and subsequent 

 spawning stock (Walters 1985, Kope 1988). However, 

 because we use stock-recruitment curves only as a 

 visual aid in interpreting the stock-recruitment data, 

 potential bias was not evaluated. 



To examine possible relationships between recruit- 

 ment and the environment, correlations were computed 

 between time-series of recruitment and quarterly aver- 

 ages of the environmental variables at all lags that had 

 potential biological ineaning. These quarterly averages 

 are referred to as: winter, January-March; spring, 

 April-June; summer, July- September; fall, October- 

 December. Correlations with individual streatn flows 

 were computed at lags that corresponded to the winter, 

 and to spring while the fish were resident in the 

 streams as fry or migrating downstream. Correlations 

 with delta flows and diversions were computed for the 

 spring while the fish were migrating to sea, and cor- 

 relations with oceanographic variables were computed 

 from the spring in the year that fish migrated to sea 

 through the summer 2 years later when most fish 

 mature and leave the ocean to spawn. Correlations in- 

 volving abundance indices (i.e., catch and spawning 

 escapement series) were computed at lags that as- 

 sumed the indices consisted primarily of 3-year-old fish. 



Variability on all time scales can contribute to the 

 variance of time-series and to the magnitude of calcu- 

 lated correlation coefficients. However, variability on 

 greater than annual time scales involves intraseries 

 correlation, which implies the effective number of 



where n is the number of data pairs, P,rj{i) and Pyy{i) 

 are the autocorrelation functions of the two time-series 

 variables at lag i, and r\y{t ) is the computed cross- 

 correlation coefficient between the two time-series at 

 lag t (Botsford and Wainwright unpubl., Kope and 

 Botsford 1988). When no intraseries correlation is pres- 

 ent, this expression simplifies to 



Var[r,,(0] = 



1 



n 



(2) 



Because the real values of the autocorrelation functions 

 of time-series variables are unknown, and computed 

 autocorrelation coefficients must be used instead, ex- 

 pression (1) can sometimes produce variance estimates 

 smaller than expression (2). However, expression (2) 

 places a lower bound on the possible variance of com- 

 puted crosscorrelation coefficients because it approx- 

 imates the variance in the best possible case, when no 

 intraseries correlation is present. We estimated the 

 significance of computed crosscorrelation coefficients 

 by assuming a normal distribution with variance given 

 by the greater of expressions (1) and (2). This strategy 

 has performed the best in giving appropriate rejection 

 rates in Monte Carlo simulations for independent ran- 

 dom series with varying degrees of intraseries correla- 

 tion (Botsford and Wainwright, unpubl. data). 



Results 



Exploration of possible density-dependence revealed 

 no clear relationships that could be removed prior to 

 examining the influence of environmental variables. 

 The stock-recruitment relationship for the upper Sacra- 

 mento River appears to show a decrease in the equi- 

 librium stock size that coincides with the closure of Red 

 Bluff Diversion Dam in 1966 (Fig. 2). This supports the 



