Thus, equations 8, 9 and 10 hold for mean values of roll angle and relative flow 

 velocity, since recordings of roll angle and flow velocity are assumed to be samples 

 from the same type stationary Gaussian random process. 



B . Digital Estimation of Power Spectra 



The practical problem of estimating energy spectra has received widespread 

 attention and can be viewed from two distinct but practically equivalent points of 

 view. One method of estimating the energy spectrum of a stationary Gaussian random 

 process is to sample the continuous (analog) recording of the process at discrete inter- 

 vals of time, obtain an estimate of the autocorrelation function, and then obtain raw 

 or unsmoothed estimates of the energy spectrum by Fourier transformation of the auto- 

 correlation function. The power spectrum Is then obtained by suitably smoothing the 

 raw or uncorrected spectral estimates. Since the study of such processes is the study 

 of the sample variance of the process, It Is essential that confidence intervals be given 

 for each spectrum . The theory and procedures of digital estimation of power spectra 

 are given in References 1, 3, and 6. 



If the process can be "quantized" or digitized at the transducer or in the recording 

 process, much of the manual data reduction can be eliminated. For example, the Sonic 

 Surface Scanner measures instantaneous height of the sea surface above the transducers 

 located on the hovering submarine's deck, and these data are recorded directly in 

 digital form on magnetic tape. These values are played back, put on punched cards, 

 and programmed through a high speed computer to obtain the estimated spectra of the 

 surface wave heights. The method of recording and playback is outlined briefly in 

 Appendix A. 



As an example, consider Figure 4, which shows the power spectrum obtained from a 

 one-half hour recording from transducer No. 1, located on the submarine's bow (Fig. 1). 

 The dashed lines indicate the 90-percent confidence intervals (See Section C). For 

 comparison, the spectrum of a simultaneous one-half hour recording from transducer No. 

 8 (located near the stern) is shown in Figure 5 with its 90-percent confidence intervals. 

 Figure 6 shows the good agreement between the two measured spectra. 



Amplitudes of submarine motion during this recording were low. The heave accel- 

 eration recording, for example, was practically flat. However, at higher sea states and 

 lower wave frequencies, a submarine will move appreciably in response to surface wave 

 motion. For example, heave displacement would be in phase with a pure swell, whose 

 relative heading was 90-degrees, with the result that the fraction of the true recorded 

 wave amplitude decreases with decreasing frequency (See Appendix A for a discussion of 

 expected measurement errors). In general, surface wave spectra estimated from record- 

 ings of the Sonic Surface Scanner, will have to be corrected for the hovering submarine's 

 motion. Furthermore, if the submarine is underway, an additional correction will be 



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