JOURNAL OFGEOPHYSICAL RESEARCH, VOL.96, NO. C2,PAGES 2593-2597, FEBRUARY 15,1991 The Detection of Coastal-Trapped Waves JOHN W. HAINES, l KEITH R. THOMPSON, ANDDOUG P. WIENS Department ofOceanography, Dalhousie University, Halifax, Nova Scotia, Canada We outline a simple method for estimating thecross-spectral matrix of coastal-trapped wave amplitudes, A, from a set ofoceanographic observations. Specifically, we propose that A may be estimated by (M'M)-IM'OM(M'M) -1 where aprime denotes conjugate transpose, fJ is the sample cross-spectral matrix of observations and M is a matrix which has the spatial form ofthe waves for columns. In general, M willbe complex and frequency-dependent. Wediscuss the bias of this estimator and show how toestimate the variance ofthe power and cross spectra of wave amplitudes. Wealso outline anadhoc scheme for assessing the predictiv•e skill of thecoastal trapped wave representation and finally give some advice on how to interpret A.Although the method is presented in thecontext 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. 1. INTRODUCTION Theoreticians interested in ocean circulation often express the flow field as a linear combination of prescribed spatial modes andthen determine the time-varying coefficients from the forcing and the initialconditions. The modes usually correspond to the eigenfunctions of an idealized, linear modelof the real ocean. This technique hasproved useful in the study of deepocean, shelf and nearshore circulation wherethe eigenfunctions include vertical normal modes, coastal-trapped waves andedge waves [e.g.,LeBlond and Mysak, 1978]. Observationalists alsouse linear combinations of modes to reduce large multivariate data sets down to more physically meaningful indices such as coastal-trapped wave amplitudes [e.g., Freeland etal., 1986]. If the modes are real-valued and independent of frequency, their amplitudes are usually esti- mated by least squares fitting themodes to observations in the time domain. If the modes are complex and frequency- dependent, several techniques areavailable to theobserva- tionalist, and, in general, they will attribute different ener- gies to each mode and different coherences between pairs of modal amplitudes. In this paperwe present a method for estimating the cross-spectral matrix of modal amplitudes, henceforth re- ferred to as A. The expression for A is simple anddepends only on the cross-spectral matrix ofobservations (U)and the modes (M), both of which may becomplex and a function of frequency. The diagonal elements of A are the spectral densitiesof the modal amplitudes; they give the energy associated with each mode. The normalized off-diagonal elements of A give the phase lagand coherence, and hence thestrength of thecoupling, between themodal amplitudes. To keep thediscussion focused, wepresent the method in a shelf circulation context where the modes correspond to coastal-trapped waves. However, the method may be ap- plied to any linear system where themodes are known and thecross-spectral matrix of modal amplitudes is required. 1Nowat the Center for Coastal Geology, U.S. Geological Survey, Saint Petersburg,Florida. Copyright 1991 by theAmerican Geophysical Union. Paper number90JC02218. 0148-0227/91/90J C-02218505.00 There are similarities betweenour methodand that devel- oped by Freeland et al. [1986] in their search for coastal- trapped waves off the southeast coast of Australia. In essence, Freeland et al. complex demodulate the observa- tions at a given frequency and then calculate timeseries of modal amplitudes. These complex series are then combined and time-averaged to give a cross-spectral matrix of modal amplitudes. In our method thetime averaging is "built in" and this simplifies the estimation of A. Given thesimple form of A, it is straightforward to test its sensitivity to changes in array configuration (by modifying M) or instrument performance (by postulating different forms for U). Thuswe anticipate that our method may be useful in thedesign of observing arrays. It is also easy to test theperformance of different models (i.e., combinations of modes) ontheobservations, after they have been collected. In addition to A, we also define a cross-spectral matrixof residuals, R. It too canbe easily calculated fromM andU. This matrix is of special interestto the oceanographer because it describes motions which cannotbe represented by coastal-trapped waves. In the following sections we outline our methodfor estimating A. We stress thatthemethod is simple to under- stand, and use, as illustrated by therecent paper of Middle- ton and Wright [thisissue] on the generation andpropaga- tion of coastal-trapped waves on the Labrador Shelf. 2. PHYSICAL BACKGROUND TO THE PROBLEM In a seminal paper, Gill andSchumann [1974] used shelf waves to explain how a narrowcontinental shelfsea re- sponds to time-varying windstress. Their model has been extended over theyears to include stratification and friction [e.g., Clarke and VanGorder, 1986]. Themodern theory is now a useful tool for understanding and modeling large- scale, wind-driven circulation on the continental shelf. As part of a study of nearshore and shelf circulation, we have developed a new statistical method for detecting trapped waves. In order to motivate thediscussion of this method we will briefly review theGill andSchumann theory. Consider a narrow shelf with the x axis pointingseaward and they axis aligned withthecoast. Assume thebathyme- try and wind donot change in thealongshore direction and thereis no friction. If the long-wave and rigid lid approxi- 2593