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E9-053.9 8-12 MLPP 316(b) Resource Assessment 

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Appendix A — Estimating Total Annual Entrainment 

Appendix A 

Estimating Total Annual Entrainment 

An estimate of total annual larval entrainment at an intake source (i.e., either the new combinedcycle [CC] units or Units 6 and 7) can be expressed as 

*=I 

;=1 

D; di 

v.. 

*=1

 L /=! 

•ijkl (Al) 

where 

x

iiki = measured density of larvae in the /th tow (/ = 1,2) within the k\h cycle (k - 1, ...,4) on the 

y'th day (_/ = 1,..., D() in the rth stratum (i =l,...,L); 

Vjjk = total water intake during the kth cycle {k = ],.. .,4) from theyth day [j = 1,..., £),-) in the 

rth stratum (z = 1,..., L); 

Dt = number of sampling days in the rth stratum of which dt are sampled (nominally dt = 2). 

Here, a temporal stratum will be defined as a 2- or 4-week period (i.e., depending on time of year) where 

in 2 days are selected for sampling. Equation (Al) can also be expressed in terms ofa volume-adjusted 

estimate where 

L v A 

v, *- i y=lL* 

4 1/ 2 

t=i

 L 

ijkl 

/=1 

(A2) 

and where 

d, 4 

j=\ k=\ 

Di 4 

j=\ *=i 

Nominally, d( will be 2 days for all temporal stratum. The variance of E [i.e., Equation (A2)] can be 

expressed as 

Var{E\E) = £ JL 

yVu 

<*,2i-4 

D. ij 

d, 

£

ij A A 4 S 2 -4zz ^ A i y=l *=1 

(A3) 

where 

P.6 

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P.7 

Appendix A — Estimating Total Annual Entrainment 

— \2 

_ _ \Xijkl ~ ^ijk ) 

st = '=' *l>*/ K*-1

) 

Xyk ~ 

ijkl 

N, ijk 

Njjk = total number of tows possible during the kth cycle (k = 1, ...,4) of they'th day 

(y = 1,...,dj) in the rth (i = \,...,L) stratum; 

and where 

— \2 

"(A-i)

 ; 

Ejj = total entrainment during they'th day (y = 1,..., Z),) in the rth stratum (/' = 1,..., L); 

A 

E,= 7 = 1 

D. 

Variance (A3) is based on the assumption that d( are a random sample from Z), days in the rth stratum 

(/ = 1,..., L). The variance also assumes the 2 tow volumes are a random sample ofthe intake water 

during the hh cycle (k = 1, ...,4) of they'th day (y = 1,..., dj). An unbiased variance estimator can be 

expressed (Appendix A) as 

Var(E\E) = % 

and where 

/=i 

- ,2 

_ 7 _ 

ViJ 

j </, 4 V2 ? 2 

n £,, n *->£-> 

ijk^xijkl 

D-. "• QJXM 2 

(A4) 

Z(V^) 

Z4 

1=^-

and further 

*2 = — 

x

ijU 

2__{XiJ^ Xijk) 

(2-1) 

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'1 " 

Appendix A — Estimating Total Annual Entrainment 

X

ijk ~ 

2 

2_,

 Xijkl 

/=] 

2 

The estimator for total annual entrainment for the Moss Landing Power Plant new combined-cycje units 

(ET) can then be written as 

hjT = t c c 

where Ecc is the estimate of total annual entrainment at the combined-cycle units, based on repeated use 

of Equation (2). The variance for the estimator of total annual power plant entrainment can then be 

written as 

Var^ET\ET^ = Var{^Ecc\Eccy (A5 ) 

Estimates of ET will be used in FH and AEL calculations to estimate annual effects of entrainment on 

fish stocks. 

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Appendix B — Derivation of the Variance and Estimated Variance of E 

Appendix B 

Derivation of the Variance and Estimated Variance of E 

Variance of E 

The variance of E can be derived by taking the variance in stages by first conditioning on the 

choice of d{ days, then taking expectation over all selections of dt of £> days within the temporal 

stratum. 

Var{E) = Edi [ Var^) + Vardj [E^ ) 

= E, 

i ( 

/=i 

V,, ^ HlfEfS l

ijkl 

K

 ri y> 2 *• i=\ J 

+ Var, 

( L(V ^ v„. Si 1

! 

* = i

 z /=i 

2 "ft 

c

(/*/ 

- £ , 

f T. \ Z

( = 1 

^ zz 

d, 4 y* 1 -

^j* / 

v w y y=i *=i 

+ ^ 0 

L 

I 

i=l 

V

 yi )j=\ 

(Bl) 

^/v V r f .Ai ^ 

/•=i KVU 

\ 

V

 Ni*) 

*ijkl i 

*>iPk-l 

• + 

i'=l 

f „ \2 

yVu d, 

=z 

/=i 

vTi 

I J 

rf.21-

D{ i J 

% J A 4 -+fzz^ 

^ i y=i *=i 

'i--*-\. 

AT *vH 

where 

P.9 

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P.10 

Appendix B — Derivation of the Variance and Estimated Variance of E 

A — \2 Zte-5> 

c2 __= | 

E>~ (A- 0 ' 

A 

Z*> 

ZW=I 

A 

and where 

A',-; 

— \2 

Z J (*//*/ ^y* j 

x

ijU 

l=\ 

K-0 

z** {/« 

y -

 /=1 

Ny*/ 

and furthermore 

4

 Niju 

E(j = Z Z xijki = tota ' entrainment for they'th day (y = 1,..., D) in the rth stratum (/' = 1,..., L). 

k=\ 1=1 

The finite population correction [i.e., 

V

 Nm) 

] will be nearly one in all cases and can be ignored. 

Estimated Variance of E 

The variance for E is 

^(%) = Z 'SLY 

yVi) 

( d-^ 

v A y 

d-. A 4 1 — 

Nu \ Uk

 J 

x

ijU 

i y=i *=i 

(B2) 

The term 

D, 4 

1 -

N., V

 ly'Jk A y y j/2 J 

n £->*—i 'Jk o 

M j=\ k=\ z 

in Equation (B2) can be unbiasedly estimated by the quantity 

( ~ ^ 

x

ijkl 

di 4 

ZZ^ 

y=i k=\ 

1-

Nijk) 

x

iju 

(B3) 

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Appendix B — Derivation ofthe Variance and Estimated Variance of E 

p.11 

when 

s] =M 

x

ijkl 

^\xijkl xijk) 

(2-1 ) 

However, 

£|4„ K 

Instead, 

\k=\ l l=\ J 

taking the variance in stages 

Varfa) = Edi[var[E^,)] + ^Far^,)] 

= £, 

4 y 

y

ijk Var\ ^f ,

X

'jkl 

/=1 

+ Van Var\ 2 "Uk 

.k=\ 

2 YI 

_ l r

 X'jkl 

1=1 

dt 

)_ 

V

1 

4

 Vijk K~N

»J 

x

ijU 

Vard,[Ey] 

Df 4 izz 

u

i j=\ k=\ 

>jk 1 — 

N„ \ ijk J 

"ijkl 

+ S: E„ 

Hence, 

(

 d-^ 

v Ay 

E

>J 

= d; 

(

 d-^ 

v Ay 

E

" Di ^i-^lzzV

2 

v

ijk 

x

ijk 

(B4) 

which has a positive bias of 

D; 

1 -

d{ 

v

l 

\ D 4 '>'* 

V Ay 

zzj=\ k=\ 

1 — 

% ; 

*y* 

(B5) 

In rum, the bias (B5) can be estimated by the quantity 

E9-053.9 B-3 MLPP 316(b) Resource Assessment 

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Appendix B — Derivation of the Variance and Estimated Variance of E 

P.12 

)-it$-

ijk 1 x

ijk (B6) 

An estimator of VaryE £jcan then be expressed by taking into account Equations' (B3-B6) as 

/=! 

y-n 

\ Ay 

4 -

( d V'

 4 

'-fZZV

 ui )j=\ k=\ 

vi ijk 1 - — x

ijk 

( 

d{ 4 <Jk 

+zzy=i *=i 

».

 N«*J 

=z 

1=1 

V > 

• y 

/ 

4 ( d.^ 

v Ay 

F. 2 

rf,- 4 V* 

5

£. + ^zzM y=l A=l 

1 -

K, V 'y

ykJ 

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P.13 

1 

I 

Appendix C — Estimating Proportional Entrainment and the ETM Calculations 

Appendix C 

Estimating Proportional Entrainment and the ETM 

Calculations 

The empirical transport model {ETM) attempts to estimate the total annual mortality probability 

for larvae from power plant entrainment. The annual estimate is based on periodic daily 

probabilities of entrainment mortality. The calculations will assume all larvae entrained die. 

The daily probability of entrainment can be defined as 

abundance of entrained larvae;: 

u P = — 

'' abundance of larvae in source population^ 

= probability of entrainment on they'th day (y - 1,..., dt) 

ofthe rth temporal stratum (/' = 1,.... L). 

In turn, the daily probability can be estimated and expressed as 

pT 

P =-£- M« - S. (Cl) 

where 

Ej: = estimated abundance of larvae entrained on they'th day (y = 1,.. .,df) ofthe rth 

stratum (/ = \,...,L); 

IL = estimated abundance of larvae at risk of entrainment from the source populations on 

they'th day (y = 1,..., dt) ofthe /th stratum (/ = 1,..., L). 

Estimating Daily Entrainment 

The estimate of total daily entrainment (E{j ) at the combined-cycle units can be written as 

E' = ESC (C2) 

wher -cc e Er is the estimate of daily entrainment for the combined-cycle units intake. 

E9-053.9 C-1 MLPP 316(b) Resource Assessment 

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Appendix C — Estimating Proportional Entrainment and the ETM Calculations 

P.14 

At any one intake site, the estimate of entrainment can be expressed as 

4 V 2 VZfZ 

*• 1=1 

l

ijkl 

k=\ 

with associated variance 

Vt. 

which can be estimated by 

jk 

K

 N'jk ) 

*=I 

f

 2 ^ 

H^K)=z^ 

k=\ 

^

 NUk J 

"ijkl 

Typically, the finite population correction [i.e., 

large. 

V

 NUkJ 

(C3) 

(C4) 

(C5) 

] can be ignored for NiJk is exceedingly 

Estimating Daily Numbers of Larvae at Risk 

With the well-defined and agreed-upon sources of Monterey Bay (MB), Moss Landing Harbor 

(MLH), and Elkhorn Slough (ES), the daily abundance of larvae at risk can be estimated by 

hj = VMB • DMB,. + VMLH • DMLH.. + VES • DESj, (C6) 

where V denotes volume at Monterey Bay (MB), Moss Landing Harbor (MLH), or Elkhorn 

Slough (ES), and D denotes an estimate of average density. The variance of Expression (C6) 

can be written as 

Var(Rv\Rv)=V2

B.Var(bMBi. E^MBa + VMLH V«r{bMW>. DMLHij J 

+ V2

ES-Var(bESi. DESi). 

(C7) 

The individual variances within Formula (C7) describe temporal-spatial variance in density 

within a source population during the day of sampling. Within Moss Landing Harbor and 

Monterey Bay, four sampling stations have now been proposed (TWG meeting September 15, 

1999). With Elkhorn Slough, 2 fixed-location sites were selected for sampling (i.e., The Dairies 

and Kirby Park). Ideally, tow samples would be collected probablistically through time and 

space during a sampling day at a potential source population. 

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P.15 

Appendix C — Estimating Proportional Entrainment and the ETM Calculations 

Variance for Daily Estimate of PMt. 

The variance for the daily estimate of PM, can be expressed as 

Var{PM\PM^±Var 

which by the Delta method can be approximated by 

{** 

E

^ 

Var^PM^PM^-jL 

\2 

4

 + « (C8) 

and can be estimated by 

ra,(PMv\PMv) = (PMijflcvfijlEjj)2 + CV^R^ 

where 

CVl WlVaiffi) 

v 

ETM Calculations 

By combining Equations (Cl), (C2), and (C6), the estimate of daily entrainment mortality 

can be written as 

p -

M,ij 

E? 

[y»a'k MBU MLH ^MLHu T v E S •*«. ) 

El 

IJ 

R, 

(C9) 

If the species has a single spawning period per year, then the estimate of total annual mortality 

can be expressed by 

D-a PM=\-flfl{\-PMiJY

iJ 

1=1 7=1 

(CIO) 

where 

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Appendix C — Estimating Proportional Entrainment and the ETM Calculations 

D\. = number of days represented by the jth sample (j =\,...,dj) in the ith temporal stratum 

{i = l.-,L). 

Alternatively, if the species has multiple overlapping spawnings, then an estimate of total annual 

entrainment can be based on the formula 

PM=\-tif^-^ijf (Cll) 

i=l y'=l 

where 

fy = estimated annual fraction of total larvae hatched during the survey period 

represented by they'th sample in the /th temporal stratum. 

Formula (C11) is based on the total probability law where 

P{A) = fjP(A\Bi).P{Bi). 

1=1 

In the above example, the event A is larval survival and event B is hatching with P[B) 

estimated by fy. 

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Appendix D — Delta Method for Calculating Variance 

Appendix D 

Delta Method for Calculating Variance 

Variance for PE, 

Using the delta method (Seber, 1984), variance of PF, can be effectively approximated by 

Var{PEi)=Var 

I N > 

= Var{Nt) 

W + f.MJ 

+ Var{ft) < -N. ^ 

Ua + /,)v 

+ Var{Ai)\ -N, 

\ 2 

0 + /H V 

___(__i+______!+___(__) 

Nf d+y;-)2 42 

= PE?[CV(N()

2

 + CT(1 + /) 2 + CV(A,)2] . 

P.17 

can be estimated from 

Variance for SA 

S_=-

F-R-SESL 

where: 

F = average egg mass per female per year; 

R = reproduction longevity, average number of years of reproduction for a female; 

SE = egg survival rate; 

SL = survival of larvae from hatching to time of entrainment. 

The variance of SA based on the delta method is then estimated by the approximate formula 

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Appendix D — Delta Method for Calculating Variance 

Var{SA) = S] Var{F) , VarjR) , Var{SE) Var{SL) 

/?' Si si 

For the example of monkeyface eel, the variance of SA is estimated as 

Var{SA) = (0.0001388): (4,667)2

 (2.08)2 (0.0373)2 (0.0314) 

• + • + + • 

(32,000)' (11.75)2 (0.4240)2 (0.0904)2 

= 0.0000000035 

or 

SE{SA) = 0.00005905 

Variance for AEL 

The estimator of adult equivalent loss is 

A A 

ACtlu — 'Hi-p ' O j 

with exact variance 

Var{ AEL) = Var{ET )-S2

A+ Var{SA f • E2

 + Var{ET )2 • Var{SA) . 

Using the variance formula in conjunction with the monkeyface eel data results in an estimated 

variance of 

Var{AEL) = (197,677,101)2(0.0001388)2 +(0.00005905)2(160,544,555)2 +(192,677,1 Ol)2 (0.00005905) 

= 934,541,905.8 

or 

SE{AEL) = 30,5703 . 

Variance for FH 

The estimator of hindcast fecundity lost is 

.__ 7* 

FH = 

SE'SL- FT 

where 

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Appendix D — Delta Method for Calculating Variance 

ET = estimated total entrainment of larvae; 

SE - survival probability for eggs; 

SL = survival of larvae from hatching to time of entrainment; 

FT = estimated average total lifetime fecundity = F • R. 

Using the Delta method, an approximate variance estimator is 

Var{FH) = FH' Var{ET) , Var{SE) , Var{SL) , VarjF) , VarjR) 

A ^ r ^—r 1 TT r j— ~r ~z~z 

st R' 

For the example of monkeyface eel, the variance of FH is calculated to be 

Var{FH) = (11,140) (192,677,101)2

 (0.0373)2 (0.0314)2 (4,667)2 (2.08) 

• + + • 

or 

(160,544,555)2 (0.4240)2 (0.0904)2 (32,000)2 (11.75)2 

= 201,208,630 

SE{FH) = 14,185 

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