Studying nonlinear and potentially chaotic phenomena in geophysics from measured signals is problematic when system noise interferes with the dynamic processes that one is trying to infer. In such circumstances, a framework for statistical hypothesis testing is necessary but the nonlinear nature of the phenomena studied makes the formulation of standard hypothesis tests, such as analysis of variance, problematic as they are based on underlying linear, Gaussian assumptions. One approach to this problem is the method of surrogate data, which is the technique explained in this paper. In particular, we focus on (i) hypothesis testing for nonlinearity by generating linearized surrogates as a null hypothesis, (ii) a variant of this that is perhaps more appropriate for image data where structural nonlinearities are common and should be retained in the surrogates, and (iii) gradual reconstruction where we systematically constrain the surrogates until there is no significant difference between data and surrogates and use this to understand geophysical processes. In addition to time series of sunspot activity, solutions to the Lorenz equations, and spatial maps of enstrophy in a turbulent channel flow, two examples are considered in detail. The first concerns gradual wavelet reconstruction testing of the significance of a specific vortical flow structure from turbulence time series acquired at a point. In the second, the degree of nonlinearity in the spatial profiles of river curvature is shown to be affected by the occurrence of meander cutoff processes but in a more complex fashion than previously envisaged. Plain Language Summary Complexity and nonlinearity in physics and the geosciences are at the heart of understanding a great number of processes. It can therefore be difficult to infer exactly what is happening in a system due to this complexity. This paper examines various methods for generating surrogate data for testing hypotheses about nonlinearity in data. That is, the surrogates provide the null model and one looks to see if the real data has a value significantly different to the surrogates. Examples are given for how a variable changes in time, how several variables measured at the same point change in time, and how the spatial values for a variable change. In particular, we consider a method called gradual wavelet reconstruction where we can systematically change the properties of the surrogates until there is no longer a difference between data and surrogates and use this as a way to understand geophysical processes. Examples of the use of this technique are given for understanding scour caused by turbulent flows transporting sediment and how river meandering dynamics effect the curvature of the river channel.
展开▼
