# 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:

$\omega_{h,c,y,d} = X_{} B_{} + X_{h,c} B_{h,c} + X_{h} B_{h} + X_{y} B_{y}$

where

• $$B_{}$$ 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. doi:10.7717/peerj.1727.