# ESTIMATION OF PARAMETERS FOR SEVEN INDEX STOCKS OF SNAKE RIVER SPRING/SUMMER CHINOOK SALMON

In the associated example spreadsheet (Matrix example), we demonstrate
how we obtained the parameter estimates used in our demographic matrices
for Snake River spring/summer Chinook Salmon. These particular salmon
stocks have been relatively well studied; for most other stocks,
parameterization methods will have to be modified to accommodate far
less complete data.

To derive parameter estimates, we used current demographic data (1980 --
present) including redd counts and pit-tag studies. For the seven stocks
of Snake River Chinook examined here (Marsh Creek, Sulphur, Bear/Elk,
Johnson, Poverty Flat, Imnaha, and Minam), adults return as 3, 4, or 5
year olds to spawn and die \-- none are known to return at age 6 or
older. In addition, these stocks are all stream type salmon that spend
two winters in freshwater before migrating to the ocean. The matrices
only keep track of females \-- we therefore implicitly assume a 1:1 sex
ratio of eggs and equal survival probabilities of males and females. The
Basic structure of demographic matrices for spring/summer Chinook salmon
in the Snake River is as follows:

  -----------------------------------------------------------------------------------------
       1            2            3                    4                    5
  ---- ------------ ------------ -------------------- -------------------- ----------------
  1                              (1-)s~1~b~3~m~3~/2   (1-)s~1~b~4~m~4~/2   (1-)s~1~m~5~/2

  2    s~2~                                                                

  3                 s~3~                                                   

  4                              (1-b~3~)s~4~                              

  5                                                   (1-b~4~)s~5~         
  -----------------------------------------------------------------------------------------

s~x~ = probability of survival of females from age (x-1) to age x

m~x~ = number of eggs/female spawner of age x

b~x~ = propensity of females of age x to breed

= mortality of adult females during their migration upstream

**m~x~ (fecundities):** Estimates of average fecundity for most of the
stocks are referenced in Myers *et al*. (1998, Appendix C). Because
these data do not allow estimation of [age-specific]{.underline}
fecundities, we assumed constant fecundity for all three age classes.

+-----------------------------------+-----------------------------------+
| **Stock**                         | **Average Fecundity**             |
+-----------------------------------+-----------------------------------+
| Marsh, Bear, Sulphur              | 5607                              |
+-----------------------------------+-----------------------------------+
| #### Johnson & Poverty            | 4100                              |
+-----------------------------------+-----------------------------------+
| Imnaha                            | 4027                              |
+-----------------------------------+-----------------------------------+
| Minam                             | 4086                              |
+-----------------------------------+-----------------------------------+

**Step 1 of example spreadsheet:**

**(mortality of adults swimming upstream to spawn):** We used run
reconstructions by Beamesderfer *et al*. (1997) to calculate for each
stock and for each year (1980-1995) as:

= 1-(ps\* (1-harvest~sb~) \* Bon to Basin conversion \* (1-
harvest~ms~)),

where ps is prespawn survival = 0.9 in all years, Bon to Basin
conversion is survival of unharvested spawners from the Bonneville dam
to their spawning basin, harvest~sb~ is the rate of harvest in the
subbasin, and harvest~ms~ is harvest in the mainstem of the Columbia
River. We then found the average value of across years for each stock.

+--------------------------------------+-------------------------------+
| ## Stock                             | **Average**                   |
+--------------------------------------+-------------------------------+
| Marsh, Bear, Sulphur, Minam          | 0.496                         |
+--------------------------------------+-------------------------------+
| Johnson & Poverty                    | 0.420                         |
+--------------------------------------+-------------------------------+
| Imnaha                               | 0.458                         |
+--------------------------------------+-------------------------------+

**f**~x~ **(age distribution of female spawners):** Although age
frequencies do not enter the matrix directly, they are needed for
calculating the propensity to breed at age x (b~x~). We used age
frequency data for spawners from 1980-1995 (Beamesderfer *et al*. 1997).
Because these data are for both males and females, we corrected the age
frequencies of spawners for differences in the sex ratio of 3, 4, & 5
year olds. Specifically, we used data from Hall-Griswold and Cochnauer
(1988), White and Cochnue (1989), and Elms-Cockrom (1998) to find the
proportion of the total returning adults of age x that are female. To
find the f~x~\'s for females only, we multiplied the frequencies for
each year of the run reconstructions (which are total returning adults
of age x) by these proportions (and then re-scaled so the frequencies
summed to one). We then took the average across the time series.

  -------------------------- ---------------- ---------------- ----------------
  **Stock**                  **%female(3)**   **%female(4)**   **%female(5)**

  Marsh, Bear, Sulphur       0                49               67

  Johnson & Poverty          11               28               58

  Imnaha & Minam\*           11               28               58
  -------------------------- ---------------- ---------------- ----------------

\*No data were available for Imnaha and Minam. We used sex ratios from
Johnson & Poverty as a guess because Imnaha & Minam have hatchery fish,
which are known to spawn at younger age than wild fish.

  --------------------- ---------------- ---------------- ----------------
  **Stock**             **avg. f~3~**    **avg. f~4~**    **avg. f~5~**

  Marsh                 0.000            0.221            0.779

  Bear                  0.000            0.212            0.788

  Sulphur               0.000            0.216            0.784

  Johnson               0.013            0.222            0.765

  Poverty               0.018            0.212            0.770

  Imnaha                0.014            0.331            0.655

  Minam                 0.008            0.444            0.548
  --------------------- ---------------- ---------------- ----------------

**Step 2 of example spreadsheet:**

**b~x~ (age-specific propensity to breed):** Snake River spring/summer
Chinook salmon are not known to breed beyond age 5, so we set b~5~ = 1.
Knowing the age distribution of spawners (f~x~) and the age-specific
survival probabilities (s~x~), it is easy to estimate b~3~ and b~4~ by
solving a set of simultaneous equations (Ratner *et al*. 1997)

f~x~ = b~x~l~x~/b~i~l~i~ for x = 1 to 5,

where l~x~ = p~x~

where p~1~ = s~1~ and p~x~ = (1-b~x-1~)s~x~ for x \> 1.

In words, p~x~ is the probability of survival from age x-1 to x
discounted by the proportion of individuals of age x-1 that migrated (or
at least tried to migrate) back to the spawning grounds, and l~x~ is the
probability of surviving from age 0 to age x (product of the p~x~'s from
age 0 to x).

  ------------------- ------------------------- -------------------------
  **Stock**           **b~3~**                  **b~4~**

  Marsh               0                         0.185

  Bear                0                         0.177

  Sulphur             0                         0.181

  Johnson             0.009                     0.189

  Poverty             0.012                     0.181

  Imnaha              0.010                     0.288

  Minam               0.005                     0.393
  ------------------- ------------------------- -------------------------

To calculate the b~x~'s, use Excel Solver to change b~3~ (cell E32) and
b~4~ (cell E33) so that B32 = H32 and B33 = H33. Specify B33 as the
target and set to its minimum. B33 is not actually minimized, but you
have to set something as the target to get the solver to run.

**Step 3 of example spreadsheet:**

To estimate productivity for each stock, we assumed that productivity
does not depend on the density of spawners (see section VII). Our
estimate of productivity was simply the median ln(R/S) from 1980-1990
(Table VII-3b).

  ---------------------------- ------------------------------------------
  **Stock**                    **median ln(R/S)**

  Marsh                        -0.515

  Bear                         -0.300

  Sulphur                      -0.084

  Johnson                      0\. 080

  Poverty                      0.156

  Imnaha                       -0.358

  Minam                        -0.687
  ---------------------------- ------------------------------------------

**Step 4 of example spreadsheet:**

**s~2~ (survival during downstream migration and first year in ocean):**
To estimate survival during downstream migration, we used current (A1)
hydro rates, generated by PATH (spreadsheet provided by Chris Toole).
Current estimates of survival during in-river migration average 0.3224.
However, under current operations, an average of 77.45% of fish are
transported by barge. As in the PATH models, we assume 2% barge
mortality (s~b~ = 0.98).

Few data are available with which to estimate estuary and early ocean
survival. However, survival during the first year in ocean is thought to
generally be between 2-10% (Nickelson 1986, Pearcy 1992). For now, we
assume s~e~ = 7%.

Combining these, s~2~ = ((1-pt) \* s~d~ + pt\*s~b~)\* s~e~

((1-0.7745)\*0.3225 + 0.7745\*0.98)\*0.07 = 0.0582

**s~3~, s~4~, & s~5~ (survival of adults in the ocean):** There are no
direct estimates of adult survival in the ocean, but there is very
little harvest of spring Chinook salmon in the ocean. As in Ratner *et
al*. (1997), we set s~3~ = s~4~ = s~5~ = 0.8. This is based on Ricker\'s
(1976) estimate of 20% adult mortality per year in the ocean.

**s**~1~ **(survival from egg to LGR):** s~1~ was found by
simultaneously solving the Euler equation,

(1-)l~x~(m~x~/2)b~x~^-x^ = 1

and ^T^ = e^medianln(R/S)^,

where T (generation time) = (1-)xl~x~(m~x~/2)b~x~^-x^

(as in Ratner *et al*. 1997).

Using Excel Solver, the target cell is H55, set to Value of 1, by
changing cells C30 and G46, subject to the constraint that G47 = G48.

  ----------------------------------- -----------------------------------
  **Stock**                           **s1**

  Marsh                               0.039

  Bear                                0.044

  Sulphur                             0.049

  Johnson                             0\. 064

  Poverty                             0.068

  Imnaha                              0.043

  Minam                               0.047
  ----------------------------------- -----------------------------------

**Step 5 of example spreadsheet:**

The final step is to summarize the various demographic rates in a matrix
format. Population growth rate can be found either by projecting a
starting population vector forward until the growth rate stabilizes or
by solving for the dominant right eigenvalue of the matrix. We used
Matlab 5 to find analytical solutions.