The suggested citation for this analytic report is:

*Thorley, J.L. and Dalgarno, S. (2018) Lower Columbia River Rainbow Trout Spawning Analysis 2018. A Poisson Consulting Analysis Report. URL: http://www.poissonconsulting.ca/f/934446658.*

## Background

Each spring in the Lower Columbia River (LCR) below the Hugh L. Keenleyside Dam (HLK) and in the Lower Kootenay River (LKR) below the Brilliant Dam, thousands of Rainbow Trout spawn. Since 1992, BC Hydro has stabilized the spring discharge releases from HLK to protect Rainbow Trout redds from dewatering.

The primary goal of the current analysis to estimate the abundance of spawners and the stock-recruitment relationship.

### Data Preparation

The aerial spawner counts were conducted by Mountain Water Research. The age-1 recruitment estimates were provided by the Fish Population Indexing Program.

The study area was divided into three sections: the LCR above the LKR, the LKR and the LCR below the LKR. Redd and spawner counts upstream of Norns Creek Fan and downstream of Genelle were excluded from the section totals because they constitute less than 0.1% of the total count and were not surveyed in all years. The redd and spawner counts for the Right Upstream Bank above Robson Bridge were also excluded as they appear to be primarily driven by viewing conditions (and constitute less than 2.5% of the total). Viewing conditions were classified as Good or Poor. If information on the viewing conditions was not available a decline in the redd count of more than one third of the cumulative maximum count for a particular section was assumed to be caused by poor viewing conditions.

The data were prepared for analysis using R version 3.5.0 (R Core Team 2017).

### Statistical Analysis

Model parameters were estimated using Bayesian methods. The Bayesian estimates were produced using JAGS (Plummer 2015) and Stan (Carpenter et al. 2017). For additional information on Bayesian modelling in the BUGS language, of which JAGS uses a dialect, the reader is referred to Kery and Schaub (2011). For additional information on Bayesian modelling in the Stan language the reader is referred to Stan Development Team (2017).

Unless indicated otherwise, the Bayesian analyses used uninformative uniform or normal prior distributions (Kery and Schaub 2011, 36). The posterior distributions were estimated from 1,500 Markov Chain Monte Carlo (MCMC) samples thinned from the second halves of three chains (Kery and Schaub 2011, 38–40). Model convergence was confirmed by ensuring that \(\hat{R} < 1.1\) (Kery and Schaub 2011, 40) for each of the monitored parameters (Kery and Schaub 2011, 61).

The parameters are summarised in terms of the point *estimate*, standard
deviation (*sd*), the *z-score*, *lower* and *upper* 95% credible limits
(CLs) and the *p-value* (Kery and Schaub 2011, 37, 42). A p-value of
0.05 indicates that the lower or upper 95% CL is 0. The estimate is the
median (50th percentile) of the MCMC samples, the z-score is
\(\mathrm{mean}/\mathrm{sd}\) and the 95% CLs are the 2.5th and 97.5th
percentiles.

The results are displayed graphically by plotting the modeled
relationships between particular variables and the response(s) with the
remaining variables held constant. In general, continuous and discrete
fixed variables are held constant at their mean and first level values,
respectively, while random variables are held constant at their typical
values (expected values of the underlying hyperdistributions)
(Kery and Schaub 2011, 77–82). When informative the influence of
particular variables is expressed in terms of the *effect size* (i.e.,
percent change in the response variable) with 95% confidence/credible
intervals (CIs, Bradford, Korman, and Higgins 2005).

The analyses were implemented using R version 3.5.0
(R Core Team 2017) and the `jmbr`

and `smbr`

packages.

### Model Descriptions

#### Area-Under-The-Curve

The spawner abundance and spawn timings were estimated from the aerial fish and redd counts for the three sections using an Area-Under-The-Curve (AUC) model.

Key assumptions of the AUC model include:

- Spawner abundance varies by river section.
- Spawner abundance varies randomly by year and section within year.
- Spawner observer efficiency is between 0.8 and 1.0.
- Number of redds per spawner is between 1 and 2.
- Spawner residence time is between 14 and 21 days as determined in a previous year’s analysis.
- Redd residence time is between 30 and 40 days.
- Spawner arrival and departure times are normally distributed.
- Spawner arrival duration (SD of normal distribution) varies randomly by section within year.
- Peak spawner arrival timing varies randomly by year.
- The residual variations in the spawner and redd counts are described by separate Negative Binomial distributions.

### Stock-Recruitment

The relationship between the number of spawners and the resultant number of age-1 fish was estimated using a Beverton-Holt stock-recruitment model (Walters and Martell 2004):

\[ R = \frac{\alpha \cdot S}{1 + \beta \cdot S} \quad,\]

where \(S\) is the spawners (stock), \(R\) is the recruits, \(\alpha\) is the recruits per spawner at low density and \(\beta\) determines the density-dependence.

Key assumptions of the stock-recruitment model include:

- The recruits per spawner at low density (\(\alpha\)) is normally distributed with a mean of 90 and a SD of 50.
- The recruits per spawner varies with the percent of redds dewatered.
- The residual variation in the number of recruits is log-normally distributed.

The mean of 90 for \(\alpha\) was based on an average of 2,900 eggs per female spawner, a 50:50 sex ratio, 50% egg survival, 50% post-emergence fall survival, 50% overwintering survival and 50% summer survival.

The carrying capacity is \(\alpha / \beta\).

### Model Templates

### Area-Under-The-Curve

```
data {
int<lower=0> nObs;
int<lower=0> nSection;
int<lower=0> nYear;
int Year[nObs];
int Section[nObs];
real Doy[nObs];
int Spawners[nObs];
int Redds[nObs];
parameters {
vector<lower=5,upper=10>[nSection] bSpawnerAbundanceSection;
real<lower=0,upper=2> sSpawnerAbundanceYear;
vector[nYear] bSpawnerAbundanceYear;
real<lower=0,upper=2> sSpawnerAbundanceSectionYear;
vector[nSection * nYear] bSpawnerAbundanceSectionYear;
real<lower=0.8,upper=1.0> bSpawnerObserverEfficiency;
real<lower=0.0,upper=2.0> bReddObserverEfficiency;
real<lower=1, upper=2> bReddPerSpawner;
real<lower=14, upper=21> bSpawnerResidenceTime;
real<lower=30, upper=40> bReddResidenceTime;
real<lower=100, upper=150> bPeakSpawnerArrivalTiming;
real<lower=0,upper=21> sPeakSpawnerArrivalTimingYear;
vector[nYear] bPeakSpawnerArrivalTimingYear;
real<lower=log(10), upper=log(50)> bSpawnerArrivalDuration;
real<lower=0,upper=2> sSpawnerArrivalDurationSectionYear;
vector[nSection * nYear] bSpawnerArrivalDurationSectionYear;
real<lower=0, upper=2> bDispersionRedds;
real<lower=0, upper=2> bDispersionSpawners;
model {
vector[nObs] eSpawnerAbundance;
vector[nObs] ePeakSpawnerArrivalTiming;
vector[nObs] eSpawnerArrivalDuration;
vector[nObs] eRedds;
vector[nObs] eSpawners;
bSpawnerAbundanceYear ~ normal(0, sSpawnerAbundanceYear);
bSpawnerAbundanceSectionYear ~ normal(0, sSpawnerAbundanceSectionYear);
bPeakSpawnerArrivalTimingYear ~ normal(0, sPeakSpawnerArrivalTimingYear);
bSpawnerArrivalDurationSectionYear ~ normal(0, sSpawnerArrivalDurationSectionYear);
for (i in 1:nObs) {
eSpawnerAbundance[i] = exp(bSpawnerAbundanceSection[Section[i]] + bSpawnerAbundanceYear[Year[i]] + bSpawnerAbundanceSectionYear[((Section[i] - 1) * Year[i]) + Year[i]]);
ePeakSpawnerArrivalTiming[i] = bPeakSpawnerArrivalTiming + bPeakSpawnerArrivalTimingYear[Year[i]];
eSpawnerArrivalDuration[i] = exp(bSpawnerArrivalDuration + bSpawnerArrivalDurationSectionYear[((Section[i] - 1) * Year[i]) + Year[i]]);
eRedds[i] = eSpawnerAbundance[i] * bReddPerSpawner * bReddObserverEfficiency * (normal_cdf(Doy[i], ePeakSpawnerArrivalTiming[i], eSpawnerArrivalDuration[i]) - normal_cdf(Doy[i], ePeakSpawnerArrivalTiming[i] + bReddResidenceTime, eSpawnerArrivalDuration[i]));
eSpawners[i] = eSpawnerAbundance[i] * bSpawnerObserverEfficiency * (normal_cdf(Doy[i], ePeakSpawnerArrivalTiming[i], eSpawnerArrivalDuration[i]) - normal_cdf(Doy[i], ePeakSpawnerArrivalTiming[i] + bSpawnerResidenceTime, eSpawnerArrivalDuration[i]));
}
Redds ~ neg_binomial_2(eRedds, 1/bDispersionRedds);
Spawners ~ neg_binomial_2(eSpawners, 1/bDispersionSpawners);
..
```

Template 1. AUC model.

### Stock-Recruitment

```
model {
bAlpha ~ dnorm(90, 50^-2) T(1,)
bAlphaDewatered ~ dnorm(0, 2^-2)
bBeta ~ dnorm(-5, 5^-2)
sRecruits ~ dunif(0, 2)
for(i in 1:length(Stock)) {
log(eAlpha[i]) <- log(bAlpha) + bAlphaDewatered * Dewatered[i]
log(eBeta[i]) <- bBeta
eRecruits[i] <- eAlpha[i] * Stock[i] / (1 + eBeta[i] * Stock[i])
Recruits[i] ~ dlnorm(log(eRecruits[i]), sRecruits^-2)
}
..
```

Template 2.

## Results

### Tables

### Area-Under-The-Curve

Table 1. Parameter descriptions.

Parameter | Description |
---|---|

`bDispersionRedds` |
Clustering parameter for `Redds` |

`bDispersionSpawners` |
Clustering parameter for `Spawner` |

`bPeakSpawnerArrivalTiming` |
Intercept for `ePeakSpawnerArrivalTiming` |

`bPeakSpawnerArrivalTimingYear[i]` |
Effect of `i` ^{th} `Year` on
`bPeakSpawnerArrivalTiming` |

`bReddObserverEfficiency` |
Redd observer efficiency |

`bReddPerSpawner` |
Number of redds per spawner |

`bReddResidenceTime` |
Redd residence time (days) |

`bSpawnerAbundanceSection[i]` |
Intercept for `log(eSpawnerAbundance)` by
`Section` |

`bSpawnerAbundanceSectionYear[i]` |
Effect of `Section` by `Year` on
`bSpawnerAbundanceSection` |

`bSpawnerAbundanceYear[i]` |
Effect of `i` ^{th} `Year` on
`bSpawnerAbundanceSection` |

`bSpawnerArrivalDuration[i]` |
Intercept for
`log(eSpawnerArrivalDuration)` |

`bSpawnerArrivalDurationSectionYear[i]` |
Effect of `Section` by `Year` on
`bSpawnerArrivalDuration` |

`bSpawnerObsEfficiency` |
Spawner observer efficiency |

`bSpawnerResidenceTime` |
Spawner residence time (days) |

`Doy[i]` |
Day of the year of `i` ^{th} count |

`ePeakSpawnerArrivalTiming[i]` |
Expected `Doy` of peak spawner arrival for
`i` ^{th} count |

`eSpawnerAbundance[i]` |
Expected spawner abundance for `i` ^{th}
count |

`eSpawnerArrivalDuration[i]` |
Expected SD of spawner arrival timing for
`i` ^{th} count |

`Redds[i]` |
Observed number of redds on `i` ^{th} count |

`Section[i]` |
Section of `i` ^{th} count |

`Spawners[i]` |
Observed number of fish on `i` ^{th} count |

`sPeakSpawnerArrivalTimingYear` |
SD of `bPeakSpawnerArrivalTimingYear` |

`sSpawnerAbundanceSectionYear` |
SD of `bSpawnerAbundanceSectionYear` |

`sSpawnerAbundanceYear` |
SD of `bSpawnerAbundanceYear` |

`sSpawnerArrivalDurationSectionYear` |
SD of `bSpawnerArrivalDurationSectionYear` |

`Year[i]` |
Year of `i` ^{th} count |

Table 2. Model coefficients.

term | estimate | sd | zscore | lower | upper | pvalue |
---|---|---|---|---|---|---|

bDispersionRedds | 0.0637051 | 0.0061613 | 10.374498 | 0.0524675 | 0.0767115 | 7e-04 |

bDispersionSpawners | 0.3539547 | 0.0288701 | 12.316814 | 0.3026709 | 0.4170390 | 7e-04 |

bPeakSpawnerArrivalTiming | 117.6892961 | 2.3854974 | 49.321980 | 113.0233654 | 122.3957975 | 7e-04 |

bReddObserverEfficiency | 0.5645274 | 0.1217781 | 4.722354 | 0.3765360 | 0.8171811 | 7e-04 |

bReddPerSpawner | 1.4073038 | 0.2835794 | 5.071441 | 1.0219257 | 1.9621557 | 7e-04 |

bReddResidenceTime | 32.0573562 | 2.1215308 | 15.373843 | 30.1023635 | 37.8457172 | 7e-04 |

bSpawnerAbundanceSection\[1\] | 7.3285175 | 0.2006368 | 36.512773 | 6.9273329 | 7.6751803 | 7e-04 |

bSpawnerAbundanceSection\[2\] | 6.6294694 | 0.2014922 | 32.867142 | 6.2187510 | 6.9717098 | 7e-04 |

bSpawnerAbundanceSection\[3\] | 7.8697039 | 0.2047534 | 38.406725 | 7.4624092 | 8.2333417 | 7e-04 |

bSpawnerArrivalDuration | 3.1721320 | 0.0358610 | 88.478686 | 3.1037130 | 3.2424805 | 7e-04 |

bSpawnerObserverEfficiency | 0.9059636 | 0.0563495 | 16.026934 | 0.8040606 | 0.9938792 | 7e-04 |

bSpawnerResidenceTime | 19.6810469 | 1.3042694 | 14.856393 | 16.1809839 | 20.9313131 | 7e-04 |

sPeakSpawnerArrivalTimingYear | 8.3850071 | 1.8649550 | 4.641040 | 5.8098781 | 12.9778684 | 7e-04 |

sSpawnerAbundanceSectionYear | 0.1478012 | 0.0301194 | 4.990114 | 0.0983256 | 0.2168681 | 7e-04 |

sSpawnerAbundanceYear | 0.6497327 | 0.1333599 | 5.019832 | 0.4592398 | 0.9897478 | 7e-04 |

sSpawnerArrivalDurationSectionYear | 0.1806274 | 0.0287631 | 6.339208 | 0.1339776 | 0.2447777 | 7e-04 |

Table 3. Model summary.

n | K | nchains | niters | nthin | ess | rhat | converged |
---|---|---|---|---|---|---|---|

481 | 16 | 3 | 500 | 1 | 154 | 1.024 | TRUE |

### Stock-Recruitment

Table 4. Parameter descriptions.

Parameter | Description |
---|---|

`bAlpha` |
Intercept for `eAlpha` |

`bAlphaDewatered` |
Effect of `Dewatered` on `log(bAlpha)` |

`bBeta` |
Intercept for `log(eBeta)` |

`Dewatered[i]` |
Proportional redd dewatering in `i` ^{th} spawn year |

`eAlpha` |
Expected number of recruits at low density |

`eBeta` |
Expected density-dependence |

`eRecruits[i]` |
Expected `Recruits` |

`Recruits[i]` |
Number of age-1 recruits from `i` ^{th} spawn year |

`sRecruits` |
SD of residual variation in `Recruits` |

`Stock[i]` |
Number of spawners in `i` ^{th} spawn year |

Table 5. Model coefficients.

term | estimate | sd | zscore | lower | upper | pvalue |
---|---|---|---|---|---|---|

bAlpha | 107.8514990 | 41.1785294 | 2.693460 | 39.5864437 | 199.1224960 | 0.0007 |

bAlphaDewatered | 0.1813919 | 0.0612213 | 2.967033 | 0.0604210 | 0.3047707 | 0.0053 |

bBeta | -5.3907348 | 0.4371864 | -12.456668 | -6.4832232 | -4.7450059 | 0.0007 |

sRecruits | 0.2232654 | 0.0469900 | 4.879267 | 0.1581004 | 0.3388865 | 0.0007 |

Table 6. Model summary.

n | K | nchains | niters | nthin | ess | rhat | converged |
---|---|---|---|---|---|---|---|

17 | 4 | 3 | 500 | 1000 | 1413 | 1 | TRUE |

### Figures

### Stock-Recruitment

## Acknowledgements

The organisations and individuals whose contributions have made this analysis report possible include:

- BC Hydro
- Darin Nishi
- Margo Dennis
- Guy Martel
- Philip Bradshaw
- James Baxter

- Mountain Water Research
- Jeremy Baxter

- Poisson Consulting
- Robyn Irvine

- Dam Helicopters
- Duncan Wassick

- Highland Helicopters
- Phil Hocking
- Mark Homis

- Golder Associates
- Dustin Ford
- Demitria Burgoon
- David Roscoe

- Additional Support
- Clint Tarala
- Crystal Lawrence
- Gary Pavan
- Gerry Nellestijn
- Jay Bowers

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