were tabulated above Bonneville Dam as were 

 counted in that year at Bonneville. 



In developing the model of chinook salmon 

 mortality at Bonneville Dam expressed in equa- 

 tion (10), we assumed that the carcasses of all 

 fish dying below the dam were recoverable. We 

 also assumed that all tagged carcasses introduced 

 into the river below the dam were recoverable. 

 In the event that these assumptions are unjusti- 

 fied, a somewhat more general mortality model 

 can be used. This more general model provides for 

 the possibility that a certain percentage of car- 

 casses released or dying below the dam become 

 unrecoverable. 



To derive this model, we define the following 

 quantities: 



rb = fraction of carcasses of fish dying be- 

 fore counting or released below the 

 dam that are recoverable. 



T = number of tagged carcasses introduced 

 into the river below the dam. 



D'o = number of carcasses of fish dying be- 

 fore counting that are unrecoverable. 



Note that Rx = rbT so that a more general 

 equation for estimating Y, than equation (3) is 



Y, = rbTi 



a - 



(12) 



An equation for D'o that is analogous to equation 

 (6) for Do can be written 



(20 20 \ 



ZCiqi + EDiqi) 

 i=l i=l / 



(13) 



Three relationships analogous to those given in 

 equations (7), (8), and (9), respectively, can then 

 be written as follows: 



/ 20 20 \ 



Yi = m(^I: Cq^ + E Diqij - Do - D'„ 



/ 20 20 \ 



= (far. - farb + rb)M E C^qi + Z ^idi) 

 \ 1=1 i=i / 



20 



rbZDiQi = Y, - Mar, £ Ciq; 



(14) 



(15) 



20 



= fa[Y, 



+ (1 - fara + farb " rb) 



/ 20 20 \ -] 



M(^Z Ciqi + E Diq^jJ (16) 



Equations (14), (15), and (16) may be used to 

 derive a general expression for M in the same way 

 that equations (7), (8), and (9) were used to derive 

 the expression for M given in equation (10). 

 The result, which allows for the possibility of 

 carcasses of fish dying or being introduced into 

 the river below the dam becoming unrecoverable, 

 is as follows: 



M = 



Y, 



(rbZCiqi + Y.Vl -Q + f, 



20 



a' a 2^ Ciqi 

 i=l 



M. Z Ciqi = fa(Y. + Do + D'o) 



i=l 



(17) 



where the value of ^i from (12) is used. Note that 

 equations (12) and (17) reduce to equations (3) 

 and (10), respectively, when rb = 1. 



In estimating that 16.78 percent of the chinook 

 salmon run was destroyed near Bonneville Dam, 

 we assumed that all mortalities occurred below 

 the dam (i.e., that fa = 0). As has already been 

 pointed out, it is likely that some mortality oc- 

 curs as a result of fish being swept back over the 

 spillway after they have been counted. There- 

 fore, it is of considerable interest to explore the 

 effect that this mortality of counted fish would 

 have on our estimate of M. Because some of the 

 carcasses of fish that die by being swept back 

 over the spillway could be so severely mutilated 

 as to be rendered unrecoverable (i.e., have no 

 chance of floating and being recovered), this 

 factor must also be considered. 



To explore the various possibilities, fa (fraction 

 of deaths occurring after counting) and ra (frac- 

 tion of carcasses of fish dying after counting that 

 are recoverable) were assumed to take on various 

 pairs of values, and M was calculated for each 

 assumed pair of values by using equation (10). 



For fa the following values were assumed: 0.0, 

 0.125, 0.250, 0.375, 0.500, 0.625, 0.750, 0.875, and 

 1.0. The same values were assumed for ra, and M 

 was calculated for all possible pairwise combina- 

 tions of these values. From these results, we con- 

 structed an isopleth diagram giving values of M 

 corresponding to values of fa and ra (fig. 8). 



Figure 8 shows that our estimate, M = 0.1678 

 (based on the assumption that fa = 0) is a mini- 



480 



U.S. FISH AND WILDLIFE SERVICE 



