Interpretation of multipath scintillations Eleuthera to 

 Bermuda in terms of internal waves and tides* 



Freeman Dyson 



The Inslitule for Advanced Study. School of Natural Sciences. Princeton. New Jersey 08540 



Walter Munk and Bernard Zetler 



Institute of Geophysics and Planetary Physics. Scripps Institution of Oceanography. La Jolla. California 92093 

 (Received 10 December 1975) 



Rate-of-phase and intensity spectra due to time-varying multipath interference depend essentially on a 

 single parameter v' which can be interpreted as the mean-square rate-of-phase for any typical single path. 

 MIMl 406-Hz phase and intensities are consistent with v ' — 270 and 357 sec for Eleuthera to Bermuda and 

 Eleuthera to midstation transmissions, respectively, compared to 192 and 286 sec from a ray-geometnc 

 calculation using an internal wave mode! based on oceanographic observations. Internal tides play a 

 significant but not dominant role. 



Subject Classification: 143] 30,20,(43] 30.35. 



INTRODUCTION 



The purpose of this paper is to compare some statis- 

 tical properties of random vector fields with measured 

 cw transmissions of MIMl' between Eleuthera (Baha- 

 mas) and Bermuda. (Among the previous analyses we 

 refer particularly to the work of Clark; Dyer; DeFer- 

 rari; and Jacobsen.^) The observational material, gen- 

 erously made available to us by John G. Clark, con- 

 sists of intensity /(/) (decibels, arbitrary reference) 

 and phase <J>(/) (in cycles), as presented by the top two 

 curves of Fig. 1, a selected portion is shown point by 

 point in Fig. 2. The observed acoustic pressure fluc- 

 tuations (frequency (j = 406 Hz) relative to some (refer- 

 ence) scale />o can be written 



pU)/Po = x(t) cosa/-f r(/)sin(7/, 



where x, y are slowly varying (compared to o) ampli- 

 tudes. The original measurements consist of the 5- 

 min averages* of amplitudes 



1 r " 

 X(t)=— ^ xU)di = R(t)cos<p (t) , 



YU) = R{t)sin<pU) , 



which are related to the plotted time series (as fur- 

 nished to us) according to 



/= 20 logfl , 



<^ = (p/2Tr. 



We have then reconverted to 



X=10"2''cos2;r$, 



K=10''20sin27r* . 



For statistical theory, it is convenient to refer to 



I n2 1"10 r 



'=^"^=-10-^ ■ 



A suitable intensity reference is 

 ■ o = ln(fl^or/o=10 log(fl^ . 



(1) 



(2) 



(3) 



(4) 



I. MEASURED VARIANCE IN MULTIPATH PHASE 

 AND INTENSITY 



Only fractional cycles are measured, and there is an 

 ambiguity concerning the integer number of cycles. 

 Normally this can be resolved by the continuity of the 

 time series. Phase difference over the sampling in- 

 terval a/ = 5" has an rms value of 6* = 0. 24 cycles, at 

 Bermuda, and the (Gaussian) probability for |6$| to 

 exceed ^ cycle is 4% (8% were observed). A restric- 

 tion to |6$| s ^ cycle (which can be attained by adding 

 and subtracting Integer cycles) is not realistic. We 

 have edited the observations to remove phase "kinks," 

 replacing the reported value * by 4>± 1, $± 2, . . . cy- 

 cles when required to make the adjusted phase differ- 

 ence 6<i>„ = *„^.i -$„ subject to the restriction 



IS^'n-KS'J'^i+S^n-i)! si cycles. 



This is essentially placing an upper limit on second dif- 

 ferences in phase; 5% of the Bermuda observations and 

 ,1.5% for the midstation were adjusted accordingly. 

 Figure 3 shows a sample of 6$ before and after adjust- 

 ment, and Fig. 4 the reconstituted <!> = 2)6*. Midstation 

 phases are not severely altered by phase adjustments. 

 At Bermuda the low (week-to-week) frequencies bear 

 no resemblance to the midstation trend and are con- 

 siderably altered by the phase adjustment; however, 

 the high-passed records (tidal frequencies and higher) 

 are not significantly altered. We conclude that sam- 

 pling was adequate for midstation phases and high-fre- 

 quency Bermuda phases, but that sampling was not ade- 

 quate to obtain low -frequency trends at Bermuda. Ad- 

 justed mean-square phases and phase differences are 

 given in Table I. 



Multipath intensities are characterized by occasional 

 deep fades (Figs. 1 and 2). They are censored for a 

 subsequent analysis of fade statistics, by replacing the 

 recorded values of / by /q - /" whenever I<Io — F, but 

 otherwise leave / unchanged; censored X and Y are sub- 

 sequently computed according to Eq. (3). Accordingly 

 the three columns in Table I refer to the removal of 

 fades exceeding F=°o, 20, 10 dB (the first column then 



235 



