Estimating hydrologic model uncertainty in the presence of complex residual error structures

Abstract

Hydrologic models provide a comprehensive tool to estimate streamflow response to environmental variables. Yet, an incomplete understanding of physical processes and challenges associated with scaling processes to a river basin, introduces model uncertainty. Here, we apply generalized additive models of location, scale and shape (GAMLSS) to characterize this uncertainty in an Atlantic coastal plain watershed system. Specifically, we describe distributions of residual errors in a two-step procedure that includes model calibration of the soil and water assessment tool (SWAT) using a sequential Bayesian uncertainty algorithm, followed by time-series modeling of residual errors of simulated daily streamflow. SWAT identified dominant hydrological processes, performed best during moderately wet years, and exhibited less skill during times of extreme flow. Application of GAMLSS to model residuals efficiently produced a description of the error distribution parameters (mean, variance, skewness, and kurtosis), differentiating between upstream and downstream outlets of the watershed. Residual error distribution is better described by a non-parametric polynomial loess curve with a smooth transition from a Box–Cox t distribution upstream to a skew t type 3 distribution downstream. Overall, the fitted models show that low flow events more strongly influence the residual probability distribution, and error variance increases with streamflow discharge, indicating correlation and heteroscedasticity of residual errors. These results provide useful insights into the complexity of error behavior and highlight the value of using GAMLSS models to conduct Bayesian inference in the context of a regression model with unknown skewness and/or kurtosis.

Appendix A

1.1 Box–Cox t distribution

Let Y be a positive random variable (here observed streamflow time series) having a BCT distribution, denoted by BCT $(\mu ,\sigma ,\upsilon ,\tau )$, defined through the transformed random variable Z (Eq. A1), which is given by (Rigby and Stasinopoulos 2005a, b)

$$Z = \left\{ {\begin{array}{*{20}l} {\frac{1}{\sigma \upsilon }\left[ {\left( {\frac{Y}{\mu }} \right)^{\upsilon } - 1} \right],} \hfill & {if\,\, \upsilon \ne 0} \hfill \\ {\frac{1}{\sigma }\log \left( {\frac{Y}{\mu }} \right),} \hfill & {if\,\, \upsilon = 0} \hfill \\ \end{array} } \right.$$

(A1)

If Y > 0, where $\mu$ > 0 and $\sigma$ > 0, the random variable Z is then assumed to follow a t distribution with degrees of freedom, $\tau$ > 0, treated as a continuous parameter. From the probability density function of Y, a BCT $(\mu ,\sigma ,\upsilon ,\tau )$ random variable, is given by

$$f_{Y(y)} = f_{Z(z)} \left| {\frac{dz}{dy}} \right| = \frac{{y^{\upsilon - 1} }}{{\mu^{\upsilon } \sigma }}f_{Z(z)}$$

(A2)

where $f_{Z(z)}$ is the exact (truncated t) probability density function of Z.

Kurtosis parameter $\tau$ takes on values between − 3 and + 3 and determines the peakedness of the PDF, while skewness ($\upsilon$) parameters affects asymmetry ($\upsilon$ > 0; Schoups and Vrugt 2010), as illustrated in Fig. 1. The density is symmetric if $\upsilon$ = 0 and positively (negatively) skewed if $\upsilon$ > 1 ($\upsilon$ < 1).

1.2 Skew t type 3 distribution (ST3)

This is a “spliced-scale” distribution with PDF (see Fernandez et al. 1995; Rigby and Stasinopoulos 2005a, b), denoted by ST 3(μ, σ, ν, τ), defined by

$$f_{{Y(y\left| {\mu ,\sigma ,\upsilon ,\tau )} \right.}} = \frac{c}{\sigma }\left\{ {1 + \frac{{z^{2} }}{\tau }\left[ {\upsilon ^{2} I(y < \mu ) + \frac{1}{{\upsilon ^{2} }}I(y \ge \mu } \right]} \right\}$$

(A3)

For $- \infty < y < \infty ,$ where $- \infty < \mu < \infty ,\sigma > 0,\upsilon > 0,$ and $\tau > 0$, and where $z = (y - \mu )/\sigma$ and $c = 2\upsilon \left| {\left[ {\sigma (1 + \upsilon^{2} )B(\frac{1}{2},\frac{\tau }{2})\tau^{{\frac{1}{2}}} } \right]} \right.$, Fernandez and Steel (1998).

Note that μ is the mode of Y. The mean and variance of Y are given by $E(Y) = \mu + \sigma E(Z)$ and $Var(Y) = \sigma^{2} V(Z)$, where $E(z) = 2\tau^{{\frac{1}{2}}} (\upsilon^{2} - 1)/\left[ {(\tau - 1)B(\frac{1}{2},\frac{\tau }{2})\upsilon } \right]$ and $E(z^{2} ) = \tau (\upsilon^{3} + \frac{1}{{\upsilon^{3} }})/\left[ {(\tau - 2)(\upsilon + \frac{1}{\upsilon })} \right]$. See Fernandez and Steel (1998) for further information.