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/ ( -Length of track in patch i, /-Total number of patches 
encountered, L-Sum of length in patches, L r Length of 
patch i, fl-Mean CPUE within patch i, Z^-Mean patch 
CPUE in all patches, 
Bj-Patch biomass, 
1l,d, 
A— =AD.p, 
t 
B- B 0 + B 1 , B-Total biomass 
=$B=AD 0 (1 - p) + AD 1 p=A(D i) (1 - p) + D x p). 
Biomass variance 
After defining the variables, we derived the variance of 
the overall biomass estimate (V[B]): 
V[B]=A 2 V[D 0 {l-p) + D lP ]. 
We used the definition of the variance of a sum: 
V[B]=A 2 (v[D 0 {l~p)] + V[D l p] + 2Cov[D 0 (l-p),D 1 p]). 
We applied the definition of the variance of a sum again: 
Appendix 
This appendix outlines the method with which we 
derived TAPAS variance estimators. Capital letters 
denote random variables; lower case letters denote real- 
ized values of random variables. This formulation rede- 
fines l'/L as an estimate of p, which is the proportion of 
the survey area in the patches, so that the properties 
of the binomial distribution can be used to capture the 
variability of track lengths of patches. Overbar notation 
refers to the mean, and hat notation refers to a sample 
estimate. 
Definitions 
V[B}=A 2 (v[D 0 -D 0 p] + V[D 1 p\ + 2C°v[D 0 {l-p),D l p}), 
'V[D 0 ] + V[D 0 p] + 2Cov[D 0 ,-D 0 p] + V[D 1 p] 
+2Cov[D 0 (l- p),D x p] 
V[B]=A- 
We removed constants and re-arranged the equation so 
that covariance terms were at the end: 
V[B]=A' 2 
V[D 0 ] + V[D 0 p]+V[D lP ]-2Cov[D 0 ,D 0 p] 
+2 Cov\_D 0 (1 - p),D x p\ 
p-Proportion of survey area in patches, 
t_ 
t ’ 
a-Total area swept by bottom trawl, A-Total area of 
sampling area, Z) 0 -Mean background CPUE, B 0 -Back- 
ground biomass, AD 0 (l-p), f-Total track length, Z'-Total 
track length within patches, 
/ 
We defined parts to simplify the derivation with the 
delta method: 
P=V[D 0 p], 
Q=V[D lP }, 
R=Cov[D 0 ,D 0 p ], 
S=Coo[/) 0 (l-p),/) 1 p], 
V[B]=A 2 (V[D 0 ] + P + Q-2R + 2S). 
