# Section 2 Theory

The stream temperature model is a nested hierarchical Bayesian model that predicts daily stream temperature based on catchment characteristics and climate conditions. An early version of this model can be found in Letcher et al. (2016).

Daily mean stream temperature for each catchment is assumed to be a normally distributed random variable:

\[t_{h,c,y,d} \sim \mathcal{N}(\mu_{h,c,y,d},\sigma_{[t]})\]

where \(t_{h,c,y,d}\) is the mean stream temperature on day \(d\) within year \(y\) for catchment \(c\), which is located within HUC8 \(h\). This random variable is normally distributed with an expected mean \(\mu_{h,c,y,d}\) and standard deviation \(\sigma_{[t]}\).

The expected mean is computed as:

\[ \mu_{h,c,y,d} = \left \{ \begin{array}{l l} \omega_{h,c,y,d} + \delta_{h}(t_{h,c,y,d-1} - \omega_{h,c,y,d-1}) & \quad \text{for } t_{h,c,y,d-1} \text{ is real} \\ \omega_{h,c,y,d} & \quad \text{for } t_{h,c,y,d-1} \text{ is not real} \end{array} \right. \]

where \(\delta_h\) is an autoregressive [AR(1)] coefficient and \(\omega_{h,c,y,d}\) is the expected temperature before accounting for temporal autocorrelation in the error structure.

The expected temperature is computed as a linear equation with four sets of terms:

\[\omega_{h,c,y,d} = X_{[0]} B_{[0]} + X_{h,c} B_{h,c} + X_{h} B_{h} + X_{y} B_{y}\]

where

- \(B_{[0]}\) is a vector of fixed effect coefficients
- \(B_{h,c}\) is a vector of random effect coefficients for catchment \(c\)
- \(B_{h}\) is a vector of random effect coefficients for HUC \(h\)
- \(B_{y}\) is a vector of random effect coefficients for year \(y\)

Each of these vectors is multiplied by a corresponding matrix containing the corresponding predictor values (\(X\)) of each catchment \(c\) (located within HUC \(h\)) and on each day \(d\) (within year \(y\)).

## 2.1 Fixed Effects

The fixed effects are shared among all catchments within the model domain. They include the following terms:

Variable | Description |
---|---|

`intercept` |
Intercept |

`AreaSqKM` |
Total Drainage Area (km2) |

`impoundArea` |
Impounded Drainage Area (km2) |

`agriculture` |
Agricultural Land Cover (%) |

`devel_hi` |
High Development Land Cover (%) |

`forest` |
Riparian (200 ft Buffer) Forest Cover (%) |

`prcp2` |
2-day Precipitation (mm) |

`prcp30` |
30-day Precipitation (mm) |

The fixed effects also include the following interaction terms.

Interaction Term | Description |
---|---|

`prcp2.da` |
2-day Precipitaation x Drainage Area |

`prcp30.da` |
30-day Precipitaation x Drainage Area |

`airTemp.da` |
Air Temperature x Total Drainage Area |

`airTemp.impoundArea` |
Air Temperature x Impounded Drainage Area |

`airTemp.agriculture` |
Air Temperature x Agricultural Land Cover |

`airTemp.forest` |
Air Temperature x Riparian (200 ft Buffer) Forest Cover |

`airTemp.devel_hi` |
Air Temperature x High Development Land Cover |

`airTemp.prcp2` |
Air Temperature x 2-day Precipitation |

`airTemp.prcp30` |
Air Temperature x 30-day Precipitation |

`airTemp.prcp2.da` |
Air Temperature x 2-day Precipitation x Drainage Area |

`airTemp.prcp30.da` |
Air Temperature x 30-day Precipitation x Drainage Area |

## 2.2 Catchment Random Effects

The random effects for each catchment (\(c\)) include the following variables:

Variable | Description |
---|---|

`intercept` |
Intercept |

`airTemp` |
Air Temperature (degC) |

`temp7p` |
7-day Mean Air Temperature (degC) |

## 2.3 HUC Random Effects

The random effects for each HUC (\(h\)) include the following variables:

Variable | Description |
---|---|

`intercept` |
Intercept |

`airTemp` |
Air Temperature (degC) |

`temp7p` |
7-day Mean Air Temperature (degC) |

## 2.4 Year Random Effects

The random effects for each year (\(y\)) include the following variables:

Variable | Description |
---|---|

`intercept` |
Intercept |

### References

Letcher, Benjamin H., Daniel J. Hocking, Kyle O’Neil, Andrew R. Whiteley, Keith H. Nislow, and Matthew J. O’Donnell. 2016. “A Hierarchical Model of Daily Stream Temperature Using Air-Water Temperature Synchronization, Autocorrelation, and Time Lags.” *PeerJ* 4: e1727. https://doi.org/10.7717/peerj.1727.