Estimating Loss of Chinook Salmon and Central Valley Steelhead at the Central Valley Project and State Water Project

Operations of the Central Valley Project (CVP) and the State Water Project (SWP) are hypothesized to have significant impacts on special status species populations. Both projects have salvage facilities that remove fish from diverted water and return them alive to the west delta. However, mortality occurs between the time fish enter the pre-screen space and when they are released in the west delta. This mortality is referred to as “loss”. The National Marine Fisheries Service 2009 Biological Opinion (BiOp) on the long-term operation of the CVP and SWP contains a Reasonable and Prudent Alternative (RPA) that includes actions that modify project operations based on specific levels of loss per-acre-feet diverted. The BiOp also directed the Bureau of Reclamation to pursue an alternative calculation of loss that addresses issues with the original calculation. An alternative calculation was developed by Jahn (2011) and a sensitivity analysis of the calculation was performed by Teply and Ceder (2013). Both the alternative calculation and sensitivity analysis were reviewed by the Independent Review Panel (IRP) on the long-term operations BiOp and reported in Anderson et al. (2013). The IRP identified several issues with the alternative loss calculation of Jahn (2011) and recommended that the closed-form error propagation approach be replaced and improved by a random variable approach. Anderson et al. (2013) reasoned that a random variable approach would provide a better accounting of variability in model parameters and estimate loss on a time scale that is more relevant to the salvage and loss process. Additionally, the IRP recommended using an alternate method for estimating loss when salvage is zero.

We developed a Partially Observed Markov Process (POMP) model for estimating loss that addresses all of these issues and provides a flexible framework for estimating loss at both facilities. POMP models (a.k.a. state-space models) are a general class of mathematical models that can include random parameters and are used to analyze time series of data that have been measured incompletely and with error (Ionides et al. 2006, King et al. 2015b). Such approaches are of fundamental use in fisheries biology and management, where time series of population sizes or harvest quantities are estimated via subsampling methods (Hilborn and Mangel 1997).

The POMP model framework provides a flexible and powerful modeling approach (King et al. 2015a) that allows us to mechanistically describe the processes of fish passage and sample collection at both facilities. Recent mathematical and computational developments allow us to obtain robust estimates of model likelihood, despite systems being non-linear, non-normal, time-series, and only partially observed (Ionides et al. 2006, Andrieu et al. 2010, Ionides et al. 2015, King et al. 2015b). Given the utility of model likelihood for a wide range of Frequentist and Bayesian inferential approaches (Pawitan 2001, Burnham and Anderson 2002), the POMP approach forms the foundation of a robust yet evolvable model of fish salvage.

This report details our derivation of a general POMP model of fish passage that can be used for any species or race of fish at either facility. In the model, each facility is treated mechanistically as a series of components (e.g., pre-screen, primary louvers), reflecting their physical structures. Fish passage through each facility is governed by rates at which an individual survives and passes through each component. These rates are key parameters that not only describe the system but also are fundamental to our estimation of loss from sub-sampled salvage. We treat fish as discrete entities that move through distinct compartments in the facility in continuous time by formulating passage specifically using a compartment model (Godfrey 1983) enacted as a Continuous Time Markov Process (CTMP; Allen 2011). Samples are “taken” in the model by tracking the fish that are passed into salvage tanks that are counted at specific, but flexible intervals. This generates model predictions that are distributed appropriately for our observations (discrete number of fish counted in a variable amount of time, using a variable fraction of the water), which facilitates robust statistical inference even when no fish are counted in a sample.

To provide a mathematical background for interested readers, we derive the basic concepts of POMP, compartment, and CTMP models, with special reference to our statistical and inferential needs, before developing our specific model. By constructing our models from a general starting point, we demonstrate the flexibility in specific application, thereby highlighting the evolvability of our models. As additional data (from, e.g., mark-recapture studies) become available, they can easily be integrated into the presented models without needing to redevelop the model’s general structure or inference machinery. In addition, the model we develop here can be used as a roadmap to design future experimental research about fish passage.

We show how, using the POMP model we derive here and based on sound statistical inference (Pawitan 2001, Burnham and Anderson 2002, Ionides et al. 2006, Ionides et al. 2015, King et al. 2015b), one can make estimates of entrainment, loss, and salvage at timescales relevant for management decisions (i.e., daily), including the probability that certain triggers have been reached. In addition, we provide starting-point parameterizations for the model for Chinook and Steelhead at CVP and SWP, based on previous reports (CDWR 1973, Skinner 1974, CDWR 1986, Karp et al. 1995, Gingras 1997, Clark et al. 2009, Jahn 2011, Teply and Ceder 2013). We use these parameters to show how the model can be used to estimate loss (when salvage is zero or non-zero), quantify uncertainty associated with parameter variability, and quantify the probability that a threshold loss value had been exceeded.

Given the mathematical and computational complexities of the model derived here, we recognize that most end-users of the loss calculation may not have the time, interest, or skill necessary to write, edit, run, and evaluate the code. Therefore, in addition to the mathematical description of our model, we provide here (see Appendix A6) a draft of its coded application, using the language R (R Core Team 2015) and relying on the pomp package (King et al. 2015b). However, we also recognize that most end-users still may not have the time, interest, or skill necessary to execute code within the R console. To that end, we have developed a user-friendly web application with a graphic user interface (GUI) to facilitate application of the tool in real time by end-users. We include here a live version of our application for evaluation along with this report.

The POMP model we develop and apply here provides a flexible framework for estimating loss at the CVP and SWP facilities. The model is able to address issues with the Jahn (2011) calculation and additional recommendations made in the IRP report (Anderson et al. 2013), including estimation of loss when zero salvage is observed, more realistic accounting of uncertainty in model parameters, and estimation of loss at the time scale over which it occurs. However, construction of our model highlighted several issues with accuracy and precision of estimation given the available data and the reporting of daily values from a continuous process.

Estimation of loss from salvage requires robust estimates of the survival rates for each species through the compartments of the facilities. The available estimates from previous experiments encompass a considerable range of values (CDWR 1973, Skinner 1974, CDWR 1986, Karp et al. 1995, Gingras 1997, Clark et al. 2009, Jahn 2011, Teply and Ceder 2013). When the variability in all these parameters is integrated into a model, the confidence intervals become so wide that their utility for evaluating if an RPA trigger has been reached becomes compromised. This would be true for any model formulation that simultaneously accounts for variability in all model parameters while modeling passage in a statistically robust fashion. We highlight specific issues with particular parameters and in general across the delta and conclude with recommendations for future avenues of experimental and modeling research and development that will improve the accuracy and precision of loss estimates at the CVP and SWP facilities.