CHAPTER 1 Wavefields

The theory of univariate stochastic processes [1, 2] forms the backbone of the analysis and processing of single sensor data. Stochastic processes X(t) are parameterized by a single free variable t , which is a timelike variable. By performing some appropriate processing on samples from realizations of the process X(t), one hopes to be able to infer physical properties about the source of the signal, about the transmission medium, or both. To aid in the development of processing techniques, it is customary to seek simplifying assumptions. A standard assumption is that of stationarity [3], which allows for drastic simplifications of moment functions and related quantities. Needless to say, the standard assumptions that we apply are often questionable for real-world situations. However, our problems of interest in the physical world are often parameterized in four-dimensional space–time coordinates rather than in time alone. Examples of such are abundant; just think about any wave or fluctuation phenomena in physical space [4–7]. For engineering applications, the propagation and space–time distribution of radio waves is an important example [8–11], as is the propagation and space–time distribution of acoustic waves in audio applications [12]. In the geosciences and in oil-related prospecting and research, space–time statistics enter at all levels [13–15]. Such phenomena necessitate a generalization of the theory of univariate stochastic processes to a theory of multivariate stochastic wavefields [11, 16–18]. An appropriate notation for a space–time wavefield could then be X(r, t), where r is a three-dimensional position vector in some preferred coordinate system. One would think that the generalization of stochastic processes to stochastic wavefields should be straightforward. On the surface, it may seem so, but in reality, it is not that simple. The reason is that for wave or fluctuation phenomena in physical media, the supporting medium imposes severe constraints on the waves and fluctuations that can be supported. This is the ubiquitous phenomenon of wave dispersion, which implies that the medium only allows certain combinations of frequencies and wavenumbers (spatial vector frequencies) to exist [19, 20]. While dispersion is well understood for classical deterministic space–time systems, it is poorly understood for stochastic systems.