*3.3. Simple Regression Model*

Water agency analysts have long recognized the inverse relationship between flow and EC. This relationship was utilized for many years in applications of the previous USBR water supply allocation models for the federal service area within the San Joaquin Valley to estimate New Melones reservoir releases for water quality. However, the poor performance of these models for estimating EC at low-flow conditions, based on simple regression relationships, was one of the reasons a data-driven flow and salinity mass balance approach was adopted for the state-federal California (Water Allocation) Simulation Model (CALSIM) model that replaced the previous models. A re-examination of the flow–EC relationship [29] suggested a new approach using the rate of change of salinity that was found to be approximately proportional to the rate of change (or gradient) of the measured flow in the SJR. This new algorithm was not as susceptible to low-flow conditions as the prior approach.

The flow gradient was calculated as follows:

$$\mathbf{Q\_{grad}} = (\mathbf{Q\_t} - \mathbf{Q\_{(t-1)}}) / \mathbf{Q\_{(t-1)}}$$

where Qt is the flow at time t, and Q(t−1) is the flow at the previous time step.

A The salinity gradient was calculated in a similar fashion. Further analysis of daily flow and salinity data of the SJR at Vernalis for the period 2000–2018 showed that a clear linear regression relationship exists between flow and salinity gradients. After removing one percent of the outliers from the plot of flow and salinity gradients using daily data for the 2000–2018 time period, the resulting regression equation of flow and salinity relationship at Vernalis became (Lu et al., 2019):

$$\text{EC}\_{\text{grad}} = -0.5396 \times \text{Q}\_{\text{grad}} + 0.0038 \text{ J}$$

or

$$\left[ \left( \text{EC} \right)\_{\text{t}} - \left[ \text{EC}\_{\text{(t-1)}} \right) / \left[ \text{EC} \right]\_{\text{(t-1)}} \right] = -0.5396 \times \text{Qt} - \text{Q}\_{\text{(t-1)}} / \left( \text{Q}\_{\text{(t-1)}} + 0.0038 \right)$$

Using this relationship, the salinity forecast (measured as EC) at time step t can be determined as follows:

$$\text{[EC]}\_{\text{t}} = \text{[EC]}\_{\text{(t-1)}} - \text{[0.5396} \times (\text{Q}\_{\text{t}} - \text{Q}\_{\text{(t-1)}})/\text{Q}\_{\text{(t-1)}} + 0.0038] \times \text{[EC]}\_{\text{(t-1)}}.$$

This equation was initially applied to daily Vernalis flow and salinity data (Figure 4) for the period 2000–2018 to generate six-day model-based forecasts that were compared to historical data. The correlation coefficients for the relationship between the six-day forecasted salinity and observed flow ranged from 0.8780 to 0.9787. The same regression method was then applied to the upstream Crows Landing compliance monitoring station, resulting in the following equation for forecasting the SJR salinity concentration downstream of that location.

$$\text{[EC]}\_{\text{t}} = \text{[EC]}\_{\text{(t-1)}} + \text{[}-0.4413 \times (\text{Q}\_{\text{t}} - \text{Q}\_{\text{(t-1)}})/\text{Q}\_{\text{(t-1)}} + 0.0036] \times \text{[EC]}\_{\text{(t-1)}}$$

The correlation coefficients of the relationship of observed flow and the six-day forecasted salinity concentration ranged from 0.9831 to 0.9154.

**Figure 4.** Flow and EC observations at Vernalis compliance monitoring station on the SJR for the period 2000–2018.
