Lower Duncan River Kokanee AUC Analysis 2017

The suggested citation for this analytic report is:

Thorley, J.L. (2018) Lower Duncan River Kokanee AUC Analysis 2017. A Poisson Consulting Analysis Report. URL: http://www.poissonconsulting.ca/f/1498798685.

Background

Every fall, Kokanee from Kootenay Lake spawn in the Duncan-Lardeau system. Since 2008 aerial surveys have been conducted in the Lower Duncan River (LDR) to enumerate the number of kokanee spawners. The management questions are:

  1. What is the spawn run timing, fry emergence timing, and relative intensity of kokanee spawning in the Lower Duncan River? What potential operational/environmental cues affect this variable?
  2. What are the timing/cues of kokanee spawners in Meadow Creek and Lardeau River systems?
  3. What are the relative distribution of kokanee spawners in the Lower Duncan River, Meadow Creek and Lardeau River? What potential operation/environmental/physical cues (e.g., temperature, velocity, depth, cover, substrate) affect this variable?; and
  4. What physical works or operational constraints could be implemented to minimize operational conflicts associated with recommended kokanee spawning operations?

Data Preparation

The 2017 survey data were provided by LGL Limited in the form of Excel tables. They were imported into the existing SQLite database. The data were clean and tidied (Wickham 2014) using R version 3.4.3 (R Core Team 2016). When there were repeated counts for a channel the higher of the two values was choosen. The mean daily discharge and water temperature values at Duncan River below Lardeau (DRL) were extracted from BC Hydro’s environmental database for the Kootenays, which is maintained by Poisson Consulting. The annual abundances of Kokanee spawning in Meadow Creek and the Lardeau River were provided by the Ministry of Forests, Lands and Natural Resource Operations.

Statistical Analysis

Model parameters were estimated using Bayesian methods. The estimates were produced using JAGS (Plummer 2015). For additional information on Bayesian estimation the reader is referred to McElreath (2016).

Unless indicated otherwise, the Bayesian analyses used uninformative normal prior distributions (Kery and Schaub 2011, 36). The posterior distributions were estimated from 1500 Markov Chain Monte Carlo (MCMC) samples thinned from the second halves of 3 chains (Kery and Schaub 2011, 38–40). Model convergence was confirmed by ensuring that \(\hat{R} \leq 1.1\) (Kery and Schaub 2011, 40) and \(\textrm{ESS} \geq 150\) for each of the monitored parameters (Kery and Schaub 2011, 61). Where \(\hat{R}\) is the potential scale reduction factor and \(\textrm{ESS}\) is the effective sample size.

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

Where relevant, model adequacy was confirmed by examination of residual plots for the full model(s).

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 3.4.3 (R Core Team 2015) and the mbr family of packages.

Model Descriptions

Observer Efficiency

The aerial observer efficiency was estimated from ground counts using an overdispersed Poisson regression. Key assumptions of the observer efficiency model include:

Area-Under-the-Curve

The aerial spawner counts for the Lower Duncan River and its sidechannels were analysed using a hierarchical Bayesian Area-Under-the-Curve (AUC) model (Hilborn, Bue, and Sharr 1999). Key assumptions of the AUC model include:

The emergence timing was calculated from the daily water temperature at DRL assuming 950 Accumulated Thermal Units (ATUs). The relationship between peak spawn timing and the mean september water temperature at DRL was tested using linear regression. Neither relationship was significant (p > 0.3).

Model Templates

Observer Efficiency

model{

  bEfficiency ~ dunif(0, 3)
  sDispersion ~ dunif(0, 5)

  for (i in 1:nObs){
    eAerial[i] <- Ground[i] * bEfficiency
    eDispersion[i] ~ dgamma(sDispersion^-2, sDispersion^-2)
    Aerial[i] ~ dpois(eAerial[i] * eDispersion[i])
  }
..

Template 1. Model definition.

Area-Under-The-Curve

model{
  bAbundance ~ dnorm(10, 5^-2)
  bPeakTiming ~ dunif(-14, 14)
  bDuration ~ dnorm(2, 1^-2)

  bEfficiency ~ dnorm(1.40, 0.38^-2) T(0.89, 2.40)
  bResidenceTime ~ dunif(7, 14)

  sAbundanceYear ~ dunif(0, 5)
  sPeakTiming ~ dunif(0, 14)
  for(i in 1:nYear){
    bAbundanceYear[i] ~ dnorm(0, sAbundanceYear^-2)
    bPeakTimingYear[i] ~ dnorm(0, sPeakTiming^-2)
  }

  bSD ~ dnorm(0, 10^-2)
  bSDAbundance ~ dnorm(0, 5^-2)
  for(i in 1:length(Count)){

    log(eAbundance[i]) <- bAbundance + bAbundanceYear[Year[i]]
    ePeakTiming[i] <- bPeakTiming + bPeakTimingYear[Year[i]]
    log(eDuration[i]) <- bDuration

    eArrived[i] <- pnorm(Dayte[i], (ePeakTiming[i] - bResidenceTime/2), eDuration[i]^-2)
    eDeparted[i] <- pnorm(Dayte[i], (ePeakTiming[i] + bResidenceTime/2), eDuration[i]^-2)
    eSpawners[i] <- (eArrived[i] - eDeparted[i]) * eAbundance[i]

    eEfficiency[i] <- bEfficiency

    log(eSD[i]) <- bSD + bSDAbundance * eAbundance[i]

    Count[i] ~ dnorm(eSpawners[i] * eEfficiency[i], eSD[i]^-2)
  }
..

Template 2. Model definition.

Results

Tables

Observer Efficiency

Table 1. Parameter descriptions.

Parameter Description
Aerial[i] Aerial count for ith survey
bEfficiency eEfficiency intercept
eAerial[i] Expected aerial count for ith survey
Ground[i] Ground count for ith survey
sDispersion SD of overdispersion

Table 2. Model coefficients.

term estimate sd zscore lower upper pvalue
bEfficiency 1.3987374 0.3823003 3.832617 0.8919527 2.397768 7e-04
sDispersion 0.8690359 0.1721166 5.164291 0.6171624 1.276703 7e-04

Table 3. Model summary.

n K nchains niters nthin ess rhat converged
15 2 3 500 200 195 1.02 TRUE

Table 4.

Dayte Year Channel Ground Aerial
1972-09-24 2008 6.9R 1484 1710
1972-09-22 2009 3.5R 45 75
1972-10-02 2009 3.5R 101 100
1972-09-22 2009 8.2L 75 172
1972-10-02 2009 8.2L 35 15
1972-10-09 2009 8.2L 299 10
1972-10-06 2010 3.5R 284 822
1972-09-29 2011 3.5R 2518 5720
1972-09-29 2011 6.9R 490 230
1972-09-27 2014 3.5R 7 20
1972-09-27 2014 8.2L 285 294
1972-09-23 2016 3.5R 3 2
1972-09-23 2016 CAL 67 61
1972-09-29 2016 CAL 30 29
1972-09-22 2017 CAL 4 2

Channel Count

Table 5. Number of observed Kokanee distributed in the LDR main stem and side channels for 2017.

Date Channel Migrating Holding Spawning
2017-09-22 Main 0 0 175
2017-09-22 Side 0 0 68
2017-09-29 Main 0 0 1105
2017-09-29 Side 0 0 20
2017-10-11 Main 0 0 603
2017-10-11 Side 0 0 0
2017-10-23 Main 0 0 66
2017-10-23 Side 0 0 0

Area-Under-The-Curve

Table 6. Parameter descriptions.

Parameter Description
bAbundance log(eAbundance) intercept
bAbundanceYear[i] Effect of ith year on log(eAbundance)
bDuration eDuration intercept
bEfficiency eEfficiency intercept
bPeakTiming ePeakTiming intercept
bPeakTimingYear[i] Effect of ith year on ePeakTiming
bResidenceTime Spawner residence time
Count[i] Spawner count for ith survey
Dayte[i] Centred day of the year for ith survey
eAbundance[i] Expected annual spawner abundance for ith survey
eCount[i] Expected spawner count for ith survey
eDuration[i] Expected SD of the duration of spawner arrival timing
eEfficiency[i] Expected aerial observer efficiency for ith survey
ePeakTiming[i] Expected timing of annual peak spawner abundance for ith survey
eSpawners[i] Expected number of spawners for ith survey
FullYear[i] Indicates whether seven or more surveys in ith year
sAbundanceYear SD of effect of year on log(eAbundance)
sCount SD of residual variation in eCount
sPeakTiming SD of effect of year on ePeakTiming

Table 7. Model coefficients.

term estimate sd zscore lower upper pvalue
bAbundance 9.0683836 0.5110704 17.750724 8.1233442 10.1389831 7e-04
bDuration 1.6892354 0.1562009 10.753932 1.3445456 1.9681027 7e-04
bEfficiency 1.5184787 0.3228957 4.745053 0.9537047 2.1960404 7e-04
bPeakTiming 5.4369435 1.3516356 3.995801 2.6696997 8.0230040 7e-04
bResidenceTime 10.8699687 1.9879654 5.404758 7.2094767 13.8366737 7e-04
bSD 6.3971805 0.4339157 14.629159 5.3688834 7.0451946 7e-04
bSDAbundance 0.0000905 0.0000541 1.899899 0.0000367 0.0002467 7e-04
sAbundanceYear 1.1937338 0.4130019 3.096462 0.7339040 2.3227421 7e-04
sPeakTiming 3.2113385 1.4890858 2.288338 1.0493521 6.9305643 7e-04

Table 8. Model summary.

n K nchains niters nthin ess rhat converged
55 9 3 500 100 246 1.022 TRUE

Figures

Environmental

figures/environmental/discharge.png
Figure 1. Mean daily discharge at DRL by spawn year.
figures/environmental/temperature.png
Figure 2. Mean daily water temperature at DRL by spawn year.

Observer Efficiency

figures/efficiency/ground.png
Figure 3. Aerial versus ground kokanee spawner counts (on log10 scales) and predicted ground count (with 95% CRIs) by year and channel.

Area-Under-The-Curve

figures/auc/spawners.png
Figure 4. Kokanee spawner aerial counts with predicted aerial counts by date and year. Blue points indicate missing visibility values.
figures/auc/abundance.png
Figure 5. Predicted total kokanee spawner abundance by year (with 95% CIs).
figures/auc/count.png
Figure 6. Number of migrating, holding and spawning Kokanee enumerated in the Lower Duncan River in 2017.
figures/auc/timing.png
Figure 7. Predicted kokanee start, peak and end spawn timing by year (with 95% CIs).
figures/auc/emergence.png
Figure 8. Predicted kokanee start, peak and end emergence timing by year (with 95% CIs).
figures/auc/discharge.png
Figure 9. Predicted kokanee peak spawn timing by mean September discharge (with 95% CIs).
figures/auc/temperature.png
Figure 10. Predicted kokanee peak spawn timing by mean September water temperature (with 95% CIs).

Abundance

figures/abundance/duncan.png
Figure 11. Estimated abundance of Kokanee spawning in the Duncan River by year.
figures/abundance/abundance.png
Figure 12. Estimated abundance of Kokanee spawning by system and year.
figures/abundance/proportion.png
Figure 13. Estimated percent of Kokanee spawning by system and year.
figures/abundance/total.png
Figure 14. Estimated total abundance of Kokanee spawning (excluding the Upper Duncan) by year.

Window

figures/window/close.png
Figure 15. Mean hourly water temperature and discharge at DRL during the 2017 surveys. The dotted lines indicate the timing of the aerial surveys.
figures/window/window.png
Figure 16. Mean hourly water temperature and discharge at DRL in 2017. The dotted lines indicate the timing of the aerial surveys.

Acknowledgements

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

References

Acara, A.H. 1970. “The Meadow Creek Spawning Channel. Unpublished Report.” Victoria, BC: Fish; Wildlife Branch.

Bradford, Michael J, Josh Korman, and Paul S Higgins. 2005. “Using Confidence Intervals to Estimate the Response of Salmon Populations (Oncorhynchus Spp.) to Experimental Habitat Alterations.” Canadian Journal of Fisheries and Aquatic Sciences 62 (12): 2716–26. https://doi.org/10.1139/f05-179.

Hilborn, Ray, Brian G Bue, and Samuel Sharr. 1999. “Estimating Spawning Escapements from Periodic Counts: A Comparison of Methods.” Canadian Journal of Fisheries and Aquatic Sciences 56 (5): 888–96. https://doi.org/10.1139/f99-013.

Kery, Marc, and Michael Schaub. 2011. Bayesian Population Analysis Using WinBUGS : A Hierarchical Perspective. Boston: Academic Press. http://www.vogelwarte.ch/bpa.html.

McElreath, Richard. 2016. Statistical Rethinking: A Bayesian Course with Examples in R and Stan. Chapman & Hall/CRC Texts in Statistical Science Series 122. Boca Raton: CRC Press/Taylor & Francis Group.

Morbey, Y. E., and R. C. Ydenberg. 2003. “Timing Games in the Reproductive Phenology of Female Pacific Salmon (Oncorhynchus Spp.).” The American Naturalist 161 (2): 284–98. http://www.jstor.org/stable/10.1086/345785.

Plummer, Martyn. 2015. “JAGS Version 4.0.1 User Manual.” http://sourceforge.net/projects/mcmc-jags/files/Manuals/4.x/.

R Core Team. 2015. “R: A Language and Environment for Statistical Computing.” Vienna, Austria: R Foundation for Statistical Computing. http://www.R-project.org/.

———. 2016. “R: A Language and Environment for Statistical Computing.” Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.

Su, Zhenming, Milo D. Adkison, and Benjamin W. Van Alen. 2001. “A Hierarchical Bayesian Model for Estimating Historical Salmon Escapement and Escapement Timing.” Canadian Journal of Fisheries and Aquatic Sciences 58 (8): 1648–62. https://doi.org/10.1139/cjfas-58-8-1648.

Wickham, Hadley. 2014. “Tidy Data.” Journal of Statistical Software 59 (10): 1–22. http://www.jstatsoft.org/v59/i10.