Hanselman et al.: Application of an acoustic-trawl survey design to improve estimates of rockfish biomass 
395 
Part P: 
We rearranged the terms (covariance between D 0 and 
p was assumed to be zero): 
P=E 
dDoP 
dP n 
2 E 
dP„P 
dP n 
E 
dD 0 P 
dp 
{ dP 0 p 
dp 
2 
V[p] + 
(l-B[p])V[B 0 ]+E[pfF(D,] + 
Cov[P 0 ,p ] 
V[B]=A 2 
(E[0„]-E[D,])V[p] 
+2E[p](E[D 1 ]-E[D 0 ])Cov[D 1 ,p] 
P=E[pfv[P 0 ] + E[Djv[p] 
+2 E[P 0 ]E[p]Cou[P 0 ,p]. 
We estimated the biomass variance by replacing expected 
values with sample statistics: 
Part Q: 
Q=E 
dP { p 
dP l 
2 
V[D X ] + D 
dP { P T 
dp 
+2 E 
dP.P 
dP x 
Cov \. D v p - 
(i-p) 2 y[4]+p 2 v[A] 
+(d 0 -4) 2 v[p] 
+2p(P 1 -B 0 }C6v[P 1 ,p^ 
\ 
whereas 1 
was the finite population correction. 
q=e[ p ] 2 v[p 1 ] + e[p 1 ] 2 v[p] 
+2E[D x ]E[p]Cov[D l ,p). 
Part R: 
R=E 
+E 
d(P 0 p) 
L9D 0 J 
dP 0 \ 
+E 
+E 
pD 0 ] 
E 
d{P 0 p) 
dp 
l dp 
pD 0 l 
E 
~d(P 0 p)' 
JPo. 
dp 
pD 0 ' 
E 
d(P 0 p) 
. rj P _ 
[ 3D 0 . 
V[D 0 ] 
y[p] 
Cov[P 0 ,p] 
Coo [a -D 0 ] 
R=E[p]V[D 0 ] + E[D 0 ]Cov[D 0 ,p]. 
We aggregated the parts: 
V[B]=A 2 (V[P 0 ] + P + Q-2R + 2S), 
v[p 0 ] + e[p] 2 v[p 0 ] + e[p 0 Jv[p] 
+2 E[P 0 ]E[p]Cov[P 0 ,p] + 
EipfvlP^ + ElP.fvip] 
+2E[D 1 \E[p]Cov[D l ,p\ + 
-2(E[p]V[D 0 '] + E[D 0 ]Cov[D 0 ,p]) + 
f E[l -p]E[A]Cou[D 0 ,pp 
+2 -ElDMD^Vip] 
E[D 0 ]E[p]Cov[D 1)P ] , 
V[B]=A 2 
Derivation of the variance of the estimates of p, D 0 , and D, 
Variance of the estimate of p: 
Each patch accounted for some proportion of the total 
length of the trackline so that Pi -LJt. We were inter- 
ested in the overall proportion of the trackline that was 
in the patches, or p. The parameter p was considered to 
be a parameter of a binomial distribution. In a binomial 
distribution, an estimate of p is X/n, where X was the 
number of successes in n discrete observations. In our 
TAPAS application, the total of the discrete observations 
was h L (the number of 100-m segments along the survey 
trackline) and X was the number of these observations 
that were in a patch. Our sample estimate of X/n was p 
with the binomial estimated variance: 
^[.] = p<l-p) 
n L 
These n L observations could have been assumed to be 
independent, but there was likely some spatial correla- 
tion. For our application, variogram analysis of acoustic 
backscatter data indicated that the range parameter 
was ~12 km. This range resulted in an effective sample 
size that was much smaller than the total number of 
discrete sampling units, and variance was underesti- 
mated. The value of n L used in the variance equation 
should reflect this autocorrelation. In our application, 
we divided our total trackline length (-1200 km) by the 
variogram range parameter (-12 km), a calculation that 
yielded an n L -100. 
Variance of the estimate of D 0 : 
The variance in D 0 was the straightforward random 
sampling estimator shown as the variance of P 0 in 
Table 1. 
