Linear and nonlinear response to parameter variations in a mesoscale model

It is widely recognized that ensemble prediction system (EPS) skill can improve when considering model error in the system design. Typical approaches include varying physical parameterization schemes or entire modeling systems within an EPS, and including stochastic terms in the dynamical equations. But perhaps the simplest approach to accounting for uncertainty in a model is to perturb inherently uncertain parameters within sub-grid parameterization schemes. Although it almost certainly cannot introduce the modes of variability produced by some other methods, its simplicity alone suggests that its effect on prediction skill and variability deserves quantitative scrutiny. In this work we seek to merely understand how perturbations to uncertain parameter manifest in a mesoscale model, and evaluate their potential for use in EPSs or data assimilation systems that can exploit ensemble covariances and linear or nonlinear responses.

A set of four parameters are varied, corresponding to one each in cumulus, cloud microphysics, boundary-layer turbulence, and radiation schemes within the Weather Research and Forecast (WRF) mesoscale numerical weather prediction model. Parameters are drawn only once from distributions intended to capture the uncertainty estimated by experts and reported in the literature. Each set of parameters is drawn with a latin hypercube sampling technique that ensures the parameter sets are independent and fill the four-dimensional space spanned by the parameters. The parameter sets are then fixed and an ensemble of 10 members uses them for approximately 30 ensemble forecasts that are also subject to initial-condition and land-surface uncertainty.

Linear response and ensemble sensitivity are quantified with simple lagged-regression techniques. We show that linearity is intermittent and time-dependent, but that a measurable signal is evident. Only one of the parameter perturbation sets leads to noticeable changes in model bias. The response to each parameter is different in magnitude and direction in the model state space; it does not simply follow a common sensitivity pattern. We deduce a nonlinear response from lack of measurable linear sensitivity and also by noting departures from statistical normality in the response.

We argue that a linear relationship between parameters and predictions is potentially useful for parameter estimation via data assimilation. But because the response is spatially and temporally variable, global parameters such as those perturbed here are inappropriate. One implication for ensemble data assimilation, for example, is that the number of ensemble members may need to be large because parameters need to be estimated as a function of time and space, and number of parameters needed may approach the number of modes in the linear response. We also argue that nonlinear response is useful for ensemble prediction, and although perturbing parameters does not produce as much spread as other techniques, simple calibration may prove fruitful.