The detection of coastal-trapped waves

We outline a simple method for estimating the cross-spectral matrix of coastal-trapped wave amplitudes,A, from a set of oceanographic observations. Specifically, we propose that A may be estimated by (M'M)^-1M'UM(M'M)^-1 where a prime denote conjugate transpose, U is the sample cross-spectral matrix of observations and M is a matrix which has the spatial form of the waves for columns. In general, M will be complex and frequency-dependent. We discuss the bias of this estimator and show how to estimate the variance of the power and cross spectra of wave amplitudes. We also outline an ad hoc scheme for assessing the predictive skill of the coastal trapped wave representation and finally give some advice on how to interpret A. Although the method is presented in the context of shelf circulation and coastal trapped waves, it may be applied to any linear system where the spatial forms of the waves are known and the cross-spectral matrix of their amplitudes is required.