76 IEEE JOURNAL ON zyxwvuts OCEANIC ENGINEERING, VOL. OE-4, NO. 3, JULY 1979 z . Estimation of Scalar Ocean Wave Spectra by the Maximum Entropy Method zyx SVERRE HOLM AND JENS M. HOVEM Abstract-This paper describes the estimationof ocean wave spectra with the maximum entropy method. This method is based on an auto- gressive model for the time series of zyxwvutsr sea surface elevation, and givesan all-pole estimate. The wave series are also prefiltered to introduce one zero in the spectral estimate. The methods are successfully applied to estimate the spectra of some typical wave series, requiring between 10 and 30 parameters. Our comparison with conventional spectral estimates also show that the new methods give good estimates of the spectral moments, that is important because the moments are parameters in a statistical descrip- tion of the sea surface. The algorithms involved are well suited for zyxwvut real- time estimation o f high-resolutionwave spectra. D INTRODUCTION ESCFUPTION of ocean waves is of vital importance for any engineering activityin the ocean environment,and therefore a considerable effort is spent on the collection and analysis of wave data. Considering the importance and the effort involved it therefore seems natural also to investigate alternative methods for collection and processing wave data. With the large amounts of data involved, it is also worthwhile to investigate if it is possible to reduce this amount without reducing the information contentsignificantly. The purpose of this paper is to present the results of an attempt to apply parametric methods to the problem of esti- mating scalar ocean wave spectra. The method used is known as the maximum entropy method and is based on an auto- regressive model forthe direct or prefiltered time series of wave heights. This method has been used by statisticians for a long time to characterize time series. Its ability to give high- resolution spectral estimates, however, has only been explored in the last decade or so. This application is often attributed to Burg [3] . Several papers have described various applications of the maximum entropy method. Among them are the analysis of seismic signals zyxwvutsrqpo [6] , speech [7], and EEG signals. Compared to the conventional periodogram and correlogram spectral estimates, themaximum entropy method generally gives higher resolution and a description of the spectrum with fewer parameters [SI. This paper first gives a brief outline of the theory behind the maximum entropy method and the conventional methods. To gain more insight into the properties of the methods the underlying autoregressive and moving average models are discussed. Manuscript received December 7, 1978; revised April 12, 1979. This work was supported by the Royal Norwegian Council for Scientificand Industrial Research. The authors are with the Electronics Research Laboratory, Nor- wegian Institute of Technology, N-7034 Trondheim-NTH, Norway. In the second part examples of different wave series are analyzed. The spectra are found using the autoregressive model on either the raw time series or the time series prefiltered with one moving average parameter. These spectral estimates are thencomparedto conventional smoothed periodogram esti- mates. The first 4 moments of the spectra have also been calculated. Finally, a system for real-time processing and data reduction with a microprocessor in the wave-measuring buoy is proposed. THE MAXIMUM ENTROPY PRINCIPLE The power spectrum of a stationary process is the discrete Fourier transform of its autocorrelation function as given by m fYf) = T R(kT) exp (-21~jkTf) (1) z k=--m where Tis the sampling interval. It is useful to compare various spectral estimates by looking at the implied assumptions about the autocorrelation function. In the correlogram method the infinite sum in (1) is trun- cated. To smooth out some of the effects of this the zy q esti- mates of the autocorrelation function are often multiplied by a window function w(k). The new estimate is given by 9 s(f) = T I+~T)w(~) exp (-21~jk~). zy (2) k =- 9 Another way to overcome the effects of truncation would be to extrapolate the autocorrelation function from p estimated values, where p often is less than q. The two concepts are illustrated inFig. 1. In this figure the amplitude of the windowed estimate decreases gradually to zero as the lag reaches 4. The extrapolated estimate, however, is nonzero beyond both p and q. Maximum entropy spectral analysis is one way to achieve this extrapolation according to the following criteria: 1) Thespectralestimate shall agree with the p estimated values of the autocorrelation function. 2) The estimate shall be based on the fewest possible assumptions about the values of the autocorrelation function beyond lag p (maximizing the entropy). When treating these criteria mathematically the result is an all-pole spectral estimate [4] bn2T 0364-9059/79/0700-0076$00.75 0 1979 IEEE