Exploring a copula-based alternative to additive error models—for non-negative and autocorrelated time series in hydrology

Inaccurate description of uncertainty in the error models can cause biases in parameter estimation. When the parameters of the deterministic model and the error model are inferred jointly from the observations, the posterior converges to regions that reflect the processes in both high and low flows. If the nature of errors in low and high flows is different to the extent that the same error description cannot be used for both, biases in inference are introduced. In such cases, the parameter posterior will adjust to the region of the hydrograph with longer proportionate presence in the calibration time series. In this paper we demonstrate that the autoregressive order 1 (AR1) description of errors can lead to sub-optimally performing predictive models if the calibration period has substantial sections of inadequately modelled flows. Inference is performed within the Bayesian framework. We show this for a synthetic example as well as a case study. We also see that the predictive uncertainty bands that we get using the AR1 description can be overconfident and also admit negative values. To mitigate this, we analyze an alternative to additive error models. We use a distribution with a non-negative support, gamma in this study, reflecting the uncertainty in the system response at every time step. The gamma distribution is conditioned on the deterministic model output, which determines its mode and standard deviation. We capture autocorrelation in time using copulas. Given that copulas can capture dependence between different marginals, we use different specifications of the marginal distribution for high and low flows. The results show that 1) biases in parameter estimation can be reduced if a representative error description is attained using the flexibility of a copula-based likelihood. 2) The non-negative support allows to make more realistic uncertainty intervals for low flows. 3) However, the autocorrelation parameter in copulas severely interacts with the model and heteroscedasticity parameters. 4) While the formulation, in principle, should be of added value for parameter inference, in case of less informative priors, the flexibility of this description can produce non-robust inference.