A Metamodel-Based Analysis of the Sensitivity and Uncertainty of the Response of Chesapeake Bay Salinity and Circulation to Projected Climate Change

Abstract

Numerical models are often used to simulate estuarine physics and water quality under scenarios of future climate conditions. However, representing the wide range of uncertainty about future climate often requires an infeasible number of computationally expensive model simulations. Here, we develop and test a computationally inexpensive statistical model, or metamodel, as a surrogate for numerical model simulations. We show that a metamodel fit using only 12 numerical model simulations of Chesapeake Bay can accurately predict the early summer mean salinity, stratification, and circulation simulated by the numerical model given the input sea level, winter–spring streamflow, and tidal amplitude along the shelf. We then use this metamodel to simulate summer salinity and circulation under sampled probability distributions of projected future mean sea level, streamflow, and tidal amplitudes. The simulations from the metamodel show that future salinity, stratification, and circulation are all likely to be higher than present-day averages. We also use the metamodel to quantify how uncertainty about the model inputs transfers to uncertainty in the output and find that the model projections of salinity and stratification are highly sensitive to uncertainty about future tidal amplitudes along the shelf. This study shows that metamodels are a promising approach for robustly estimating the impacts of future climate change on estuaries.

Appendix A: Details of Gaussian Process Metamodel

We initially treat the model output as the sum of one or more trend terms and a zero-mean Gaussian process:

$$ \mathbf{Y}(\mathbf{x}) = f(\mathbf{x})^{\intercal} \upbeta + GP\left( 0, c(\mathbf{x}_{i}, \mathbf{x}_{j}) \right) $$

(7)

where, for an x consisting of n points in d-dimensional space, f(x)^⊺ is a n × p design matrix for the trend term(s) and β is a p × 1 vector of trend parameters. For a simple intercept only (constant mean, or flat trend), p = 1 and f(x)^⊺ would be a vector of n ones and β the intercept. For a linear trend, these terms are analogous to multiple linear regression, with p = 1 + d, f(x)^⊺ a matrix with rows consisting of a 1 followed by the d coordinates of one point, and β representing the intercept and a slope for each dimension.

The covariance function gives the covariance between the GP at two points x_i and x_j. Under the assumption that the model output is a relatively smooth function of its inputs (Roustant et al. 2012), we modeled the covariance with a squared exponential function:

$$ c(\mathbf{x}_{i}, \mathbf{x}_{j}) = \sigma^{2} {\prod}_{k=1}^{d} \exp \left( -\frac{\left( \mathbf{x}_{i,k} - \mathbf{x}_{j,k} \right)^{2}}{2 {\theta_{k}^{2}}} \right ) $$

(8)

Here, 𝜃_k functions as a length scale that adjusts the distance of the decay of the covariance between model results at different values of factor k, and σ² is a constant known as the process variance.

The separate terms in Eq. 7 can be combined into a single Gaussian process with non-zero mean, and, following Roustant et al. (2012), prediction of the numerical model output Ŷ at a new point x_∗ can be obtained from the expected value of the GP conditional on the n known values of the numerical model simulations Y at points x used to train the metamodel:

$$ E\left[\hat{Y}(\mathbf{x}_{*}) \right] = f(\mathbf{x}_{*})^{\intercal} \hat{\upbeta} + \mathbf{C}_{\mathbf{x}_{*}}^{\intercal} \mathbf{C}_{\mathbf{x}}^{-1} (\mathbf{Y} - \textbf{F} \hat{\upbeta}) $$

(9)

where f(x_∗)^⊺β̂ is the sum of the trend function(s) given estimated values of the coefficients β̂, Cx_∗⊺ is a 1 × n vector of the covariance between the output at the new point and the n training points, Cx− 1 is the inverse of the n × n covariance matrix of the training simulations, Y is a vector of the values of the numerical simulations used for training, and Fβ̂ is a vector of the values of the trend(s) at the training points. Eq. 9 shows that when numerical simulations are near the prediction point in parameter space, and therefore have high covariance, the deviation of the prediction from the trend will be influenced by the deviation of the nearby simulations from the trend. Far away from any numerical simulations used to fit the metamodel, the metamodel prediction will tend to revert towards the value from the trend functions only. Uncertainty about the outcome of the Gaussian process is also typically included when making predictions. See Roustant et al. (2012) for the formulation of the variance of the predicted values. Intuitively, variance is low near points where the numerical model has been run and is large at points far away from known model simulations.