4.2.1. Model Calibration

The resulting pre-development relationship between Delta outflow and Central Valley runoff, both expressed in terms of annual flows, is displayed in Figure 6. Equation (2) fitting parameters and regression statistics are summarized in Table 5.

**Figure 6.** Pre-development annual outflow–runoff relationship (Model 2). See Equation (2) and Table 4 for model fitting parameters.

**Table 5.** Model fitting parameters and regression statistics for Equation (2): relationships between Delta outflow and Central Valley runoff under pre-development (Model 2) and contemporary (Model 3) conditions. Flow units are in BCM per year.


(1) The contemporary outflow–runoff relationship for WYs 1945–2018 was adjusted using Equation (4) to reflect significant increases in water use in the Central Valley and Delta following construction of Shasta Dam in WY 1944. See Table 6.

**Table 6.** Model fitting parameters and regression statistics for Equation (4): adjusted relationships between Delta outflow and Central Valley runoff under WYs 1945–2018 contemporary (Model 3) conditions.


Historical annual Delta outflow was correlated with annual Central Valley runoff (as measured by the 8RI) over a subset of the contemporary period spanning WYs 1912– 1944; the resulting model fit is displayed in Supplementary Material D (see Figure D-1). Equation (2) fitting parameters and regression statistics are summarized in Table 5. Model residuals, reported as predicted minus observed, are plotted as a time series in Figure 7a

for the full contemporary period spanning WYs 1912–2018. This figure clearly shows that Equation (2) increasingly over-estimates Delta outflow over time following WY 1944, signifying a decreasing trend in Delta outflow relative to the 8RI. Equation (2) residuals were de-trended through the following re-formulation (see Figure D-2 in Supplementary Material D):

$$\text{Delta Outflow} = \mathbf{a}\_1 \times 8 \mathbf{R} \mathbf{I}^{\text{cz}} \times \left\{ 1 - \left[ \mathbf{a}\_8 \times \left( \text{Water } \mathbf{Y} \mathbf{r} - 1944 \right)^2 + \mathbf{a}\_9 \times \left( \text{Water } \mathbf{Y} \mathbf{r} - 1944 \right) + \mathbf{a}\_{10} \right] \right\} \tag{4}$$

where α8, α9, and α<sup>10</sup> are dimensionless fitting parameters. Hutton et al. [101] observed that Delta outflow trends, when normalized to the 8RI, were different between low and high runoff years. In high runoff years, they observed a decreasing trend in normalized outflow. However, they reported that:

**Figure 7.** Time series of model residuals associated with contemporary annual runoff–outflow relationship (Model 3). Residuals from Equation (2) applied to the entire WYs 1912–2018 contemporary period are shown in (**a**). De-trended model residuals from Equation (4) applied to WYs 1945–2018 are shown in (**b**).

In drier years, the downward trend in normalized Delta outflow appears to have been curbed (and possibly reversed) over the last few decades due to more restrictive water management (i.e., lower normalized Delta exports) in the estuary and a leveling of water use in the upstream watershed.

Following the observations of [101], Equation (4) was independently calibrated for low runoff years (8RI < 24.6 BCM/yr) and high runoff years (8RI > 24.6 BCM/yr) with a combined standard error of 3.0 BCM. Fitting parameters and regression statistics are provided in Table 6. Model residuals, reported as predicted minus observed, are plotted as a time series in Figure 7b. This figure shows no apparent time trend in the de-trended model residuals. The tree-ring reconstructions of Central Valley runoff (i.e., the 8RI) were used in conjunction with Equations (2) and (4) to estimate contemporary annual Delta outflow volumes for each time series beginning in WY 1912.
