U1 



_J 



o 

 o 



-1 



SAMPLE 



20 



25 



10 



HOURS 



FIG. 3. Bermuda phase differences 6* before (top) and after adjustment (bottom). 



learn that sampling at ^-min intervals would have 

 avoided all ambiguities. As it is, the 5-min averaging 

 suppresses the high frequencies, and the phase ambig- 

 uities dominate the low frequencies. 



II. THE STATISTICAL MODEL 



We shall compare the observed data with a crude 

 statistical model of a multipath acoustic signal. In the 

 model, the components of the signal are 



x= 2^Xi 



n 



Y= JIy, 



(5) 



resulting from the superposition of n single-path com- 

 ponents 



Xj = fl, COS0,- , 

 y,=fl,sin(^j . 



(6) 



The amplitudes R^ and phases <^j of the single-path com- 

 ponents are independent random variables. We use the 

 notation ( ) to indicate a statistical average over all sin- 

 gle path signals simultaneously, while ( ), denotes an 

 average over the i?, and (pj in one single path. We 

 make three further assumptions concerning the single- 

 path signals: 



(1) Fluctuations in phase are more important than 

 fluctuations in amplitude, or in symbols (with dot ac- 

 cent designating d/dt) 



(R]),«{Rf,m, . (7) 



(2) The time scale (i^j)' for a phase 0j to change by 

 1 rad is on the average short compared with the time 

 scale ((f),/^i) for the phase variation to change direc- 

 tion, or in symbols 



{0!>i«<<^t>i 



(8) 



(3) Each phase velocity c^, is a Gaussian random vari- 

 able, so that 



m<=m^),f 



(9) 



In Appendix A it is verified that property (2) holds for 

 the single-path phases predicted by the ocean model 

 which we describe in Sec. IV. Properties (1) and (3) 

 also hold in the same model. We suppose that the fre- 



BERMUDA t.J 



iVv 



MID-STATION 



/ 



T 



20 cycles 



BERMUDA HIGHPASSED 

 . row 





5 10 15 20 25 



Days 

 FIG. 4. Bermuda and midstatlon phases before and after ad- 

 justment. 



237 



