cross -spectral values (at a given frequency) are ordered in terms of gage 

 separation distance (called lag space) , a pattern emerges that is interpreted 

 for the directional distribution of all the waves at a given frequency. Since 

 there are 28 frequencies (in this analysis), there are 28 sets of cross- 

 spectral patterns to interpret. The pattern for any one frequency is 

 considered independent of the pattern at any other frequency. 

 Estimation of directional spectra 



111. Conversion of cross -spectral patterns in lag space to directional 

 spectra is done with a statistical method known as Iterative Maximum Likeli- 

 hood Estimation (IMLE) The IMLE method is a more accurate adaptation of a 

 simpler method called Maximum Likelihood Estimation (MLE) . Both methods are 

 alternatives to the more conventional Fourier transformation of directional 

 spectral wave numbers (the spatial equivalent of frequency transformation) . 

 Either MLE or IMLE is necessary because the gage array cannot be made long 

 enough to sample adequately the longer wind waves . 



112. In the MLE method, an empirical function is found to represent a 

 directional spectrum that is nearly consistent with the observed cross - 

 spectral patterns. The empirical function is found from a weighted sum of the 

 cross-spectral estimates. The weights are required to minimize the effects of 

 noise in cross spectra and give an optimum response in the vicinity of each 

 discrete set of viewing directions. This scheme was first proposed by Capon, 

 Greenfield, and Kolker (1967) for the analysis of seismic data. Davis and 

 Regier (1977) describe the theory and application of the method to ocean wave 

 data obtained from spatial arrays. A further application of the method to 

 data from heave-pitch-roll buoys is derived by Oltman-Shay and Guza (1984). 

 Details of the computational algorithms are given in these references. 



113. The MLE method has two properties that make it less than optimum. 

 One is that directional spectra from the MLE algorithm frequently do not 

 recover observed cross spectra upon inversion of the algorithm. The other is 

 that formal estimates of error (confidence intervals) for MLE spectral es- 

 timates are computationally prohibitive for routine use and not necessarily 

 reliable. 



114. In the first problem, Davis and Regier (1977) noted that if the 

 MLE directional spectral estimate is inverted to obtain the cross spectrum, 

 the observed cross -spectral matrix is not recovered exactly. They performed 

 tests with simulated data in a six-gage array. They found that energy 



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