Wave-wave interactions and deep ocean acoustics
Zachary Guralnik, William Farrell, John Bourdelais, Xavier Zabalgogeazcoa
aa r X i v : . [ phy s i c s . a o - ph ] M a r Wave-wave interactions and deep ocean acoustics
Z. Guralnik, a) J. Bourdelais, X. Zabalgogeazcoa, and W.E. Farrell b) Science Applications International Corporation 1710 SAIC Drive, McLean, VA 22102 (Dated: October 15, 2018)Deep ocean acoustics, in the absence of shipping and wildlife, is driven by surface processes. Bestunderstood is the signal generated by non-linear surface wave interactions, the Longuet-Higginsmechanism, which dominates from 0.1 to 10 Hz, and may be significant for another octave. For thissource, the spectral matrix of pressure and vector velocity is derived for points near the bottomof a deep ocean resting on an elastic half-space. In the absence of a bottom, the ratios of matrixelements are universal constants. Bottom effects vitiate the usual “standing wave approximation,”but a weaker form of the approximation is shown to hold, and this is used for numerical calculations.In the weak standing wave approximation, the ratios of matrix elements are independent of thesurface wave spectrum, but depend on frequency and the propagation environment. Data from theHawaii-2 Observatory are in excellent accord with the theory for frequencies between 0.1 and 1 Hz,less so at higher frequencies. Insensitivity of the spectral ratios to wind, and presumably waves, isindeed observed in the data. PACS numbers: (43.30.Nb (noise in water), 43.28.Py (Interaction of fluid motion and sound)
I. INTRODUCTION
Interest in deep water ambient noise has its originsin the 1930s, beginning with attempts to explain micro-seisms, persistent and ubiquitous low frequency groundoscillations. Their spectrum peaks near 0.17 Hz, roughlytwice the characteristic frequency of ocean swell, andthe amplitude correlates with storminess. The most vi-able explanation for this phenomena, based on the workof Miche, was originally proposed by Longuet-Higgins and extended by Hasselmann and Brekhovskikh. Thecause is the non-linear interaction of two oppositely trav-eling surface waves. Individually, the pressure attenu-ates exponentially with depth. However, wave pairs withnearly opposed wave vectors collide to excite an acousticwave at approximately twice the frequency which prop-agates downward with negligible attenuation. A reviewof the wave-wave mechanism and its seismic effects, to-gether with citations to much of the relevant literature,can be found in Kibblewhite and Wu. Here we give a unified exposition of the acoustic fieldradiated by non-linear surface wave interactions. Thepower spectral density matrix (PSDM), obtained fromautocorrelations and cross-correlations of pressure andvector velocity, is found both for an ocean of infinitedepth, and an ocean layer over an elastic half-space. Itis shown that the ratios of elements of the PSDM fordeep observations of the acoustic signal are nearly in-variant, depending on frequency and the propagationenvironment but not on the details of the wave spec-trum. While the wave spectrum displays dramatic dailyvariability that correlates with wind speed, the propaga-tion environment is essentially constant, aside from soundspeed variations due to seasonal fluctuations or internal a) Electronic address: [email protected] b) Electronic address: [email protected] waves. The variability of the wave spectrum manifests it-self in fluctuations in the amplitude of individual PSDMelements, but not in their ratios.The standing wave approximation is based on the factthat the sound speed is much greater than the phasevelocity of surface gravity or capillary waves. Conse-quently, only pairs of surface waves with almost exactlyequal frequency and almost exactly opposing propaga-tion directions (hence almost a standing wave) can gener-ate a non-evanescent acoustic wave, propagating withoutvertical attenuation (neglecting viscosity). For acousticwaves which generate microseisms, the constraint is evenmore stringent; not only must the acoustic wave prop-agate without vertical attenuation, it must propagatenearly vertically such that the horizontal component ofthe acoustic wave-number is sufficiently small to excitemicroseisms, which have characteristic wave-length ∼ ~k h ,the sum of surface wave vectors, in the region for whichthe acoustic wave propagates without vertical attenua-tion, | ~k h | < πf /c ( f is the acoustic frequency and c isthe sound speed). While ratios of PSDM elements are notuniversal constants in the “weak standing wave approxi-mation,” they will display little variability with changingsurface conditions, depending only on frequency and thepropagation environment but not on the details of theAcoustic radiation and wave-wave interactions 1urface wave spectrum.The theory is verified with data from the hydrophoneand a pair of three-component seismometers of theHawaii-2 Observatory (H2O), located at 5000 m depthnear 28N, 142W. Spectra and cross-spectra for all three-hour windows in a 50-day interval of year 2000 havebeen computed. Surface winds, according to the Eu-ropean Centre for Medium-Range Weather Forecasts(ECMWF), ranged between approximately 2 and 10m s − during the period, and the significant wave heightbetween 1.3 and 2.7 m. There is, indeed, little variationin the ratios of the PSDM elements compared with thevariations in the individual matrix elements; the separa-tion between spectra, in dB, is essentially constant. Thebest evidence is from coherency between pressure andvertical velocity, which is especially high for frequenciesless than 1 Hz. The diminution above 1 Hz is attributedto bottom effects not embraced by the simple model. II. ACOUSTIC RADIATION FROM NON-LINEAR WAVEINTERACTIONS
Starting with Longuet-Higgins, the most commonderivation of the acoustic signal radiated by wave-waveinteractions involves a perturbative solution of the hy-drodynamic equations. The expansion parameter char-acterizing the non-linearity is proportional to wave slope.The leading term is a superposition of plane surface grav-ity waves and the second-order term is an acoustic planewave. We give an immensely abbreviated review of thisapproach, closely following Kibblewhite and Wu, butextending the analysis with a unified treatment both ofpressure and vector velocity and simplifying it by con-sidering the bottom as an elastic half-space. Farrell andMunk (Ref. 7, Appendix A) give a brief review and ra-tionalization of the numerous solutions obtained sinceLonguet-Higgins. A. Perturbation equations
The relevant perturbative solution of the irrotationalhydrodynamic equations is expressed in terms of a veloc-ity potential φ ( ~x, t ) and a surface displacement ζ ( x, y, t ).The potential and displacement are expanded as φ = ǫφ + ǫ φ + · · · (1a) ζ = ǫζ + · · · (1b)The expansion parameter ǫ may be set to 1 at the end ofthe calculation.The first-order solution is taken to be an incompress-ible flow corresponding to a superposition of surface planewaves; φ ( ~x, t ) =1(2 π ) Z d~q − iσ ( ~q ) q ˜ ζ ( ~q ) exp( i~q · ~x h + qz − iσ ( q ) t )(2a) ζ ( x, y, t ) =1(2 π ) Z d~q ˜ ζ ( ~q ) exp( i~q · ~x h − iσ ( q ) t ) , (2b) where ~x h = ( x, y ) , ~q = ( q x , q y ), q = | ~q | and the dispersionrelation is σ = gq (cid:18) q q gc (cid:19) , q gc = ρgT (3)with surface tension T = .
074 N m − . q gc is the wavenumber of the gravity-capillary transition.At next order in the ǫ expansion, the Navier-Stokesequations yield an acoustic wave equation and a surfaceboundary condition, with source terms dependent on thefirst-order solution; (cid:18) ~ ∇ − c ∂ ∂t (cid:19) φ = 1 c ∂∂t (cid:16) ( ~ ∇ φ ) (cid:17) (4a) (cid:18) ∂ ∂t + g ∂∂z (cid:19) φ (cid:12)(cid:12)(cid:12)(cid:12) z =0 = − ∂∂t (cid:16) ( ~ ∇ φ ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) z =0 (4b)where c is the (constant) sound speed. B. Acoustic source
In terms of the Fourier transform of the second-orderacoustic potential,˜ φ ( ω, ~k h , z ) = Z dtd~x h φ ( t, ~x h , z ) e iωt − i~k h · ~x h , (5)equations (4a) and (4b) become (cid:18) d dz + ω c − k h (cid:19) ˜ φ = ˜ S ( ω, ~k h , z ) (6a) (cid:18) ddz − ω g (cid:19) ˜ φ (cid:12)(cid:12)(cid:12)(cid:12) z =0 = − c g ˜ S ( ω, ~k h , , (6b)where the source term is [cf. Ref. 6, (4.15)]˜ S ( ω, ~k h , z ) ≡ Z dtd~x h (cid:18) c ∂∂t (cid:16) ( ~ ∇ φ ) (cid:17)(cid:19) e ( − iωt + i~k h · ~x h ) . (7)Using (2a),˜ S ( ω, ~k h , z ) = iω πc Z d~q d~q ′ σ ( q ) σ ( q ′ ) (cid:18) − ~q · ~q ′ qq ′ (cid:19) ˜ ζ ( ~q )˜ ζ ( ~q ′ ) δ ( ω − σ ( q ) − σ ( q ′ )) δ ( ~k h − ~q − ~q ′ ) e ( q + q ′ ) z . (8)The ~q ′ integral is immediately evaluated because of thetwo-dimensional delta function. This gives˜ S ( ω, ~k h , z ) = (9) iω πc Z d~q σ ( q ) σ ( | ~k h − ~q | ) − ~q · ( ~k h − ~q ) q | ~k h − ~q | ! ˜ ζ ( ~q )˜ ζ ( ~k h − ~q ) δ ( ω − σ ( q ) − σ ( | ~k h − ~q | )) e ( q + | ~k h − ~q | ) z . Acoustic radiation and wave-wave interactions 2 . Solving for the acoustic velocity potential
In terms of a Green’s function satisfying (cid:18) d dz + ω c − k h (cid:19) G ω,~k h ( z, z ′ ) = − πδ ( z − z ′ ) (10a) (cid:18) ddz − ω g (cid:19) G ω,~k h ( z, z ′ ) (cid:12)(cid:12)(cid:12)(cid:12) z ′ =0 = 0 , (10b)(cf. Ref. 6, Eqns. 4.30, 4.31) the solution of (6a) and(6b) is, via Green’s theorem,˜ φ ( ω, ~k h , z ) = 14 π c g ˜ S ( ω, ~k h , G ω,~k h ( z, − π Z −∞ dz ′ ˜ S ( ω, ~k h , z ′ ) G ω,~k h ( z, z ′ ) . (11)The second(bulk) term on the right-hand side of (11) issmaller than the first (surface) term by a factor of order gL/c , where L is a the characteristic depth scale of thesurface waves, and may therefore be neglected, giving˜ φ ( ω, ~k h , z ) ≈ π c g ˜ S ( ω, ~k h , G ω,~k h ( z, . (12) III. THE POWER SPECTRAL DENSITY MATRIX
We introduce the 4-vector( v , v , v , v ) ≡ ( Pρc , v x , v y , v z ) , (13)with P pressure and velocity components v x , v y in thehorizontal, v z in the vertical. The PSDM is defined by M µ,ν ( ω ) ≡ π Z ∞−∞ dτ e iωτ h v µ ( t ) v ν ( t + τ ) i , (14)where hi denotes averaging over time. The 10 indepen-dent elements of (14) are evaluated by means of the far-field velocity potential φ . With the notation( X , X , X , X ) = ( ct, x, y, z ) , (15)one can write v µ = ∂∂X µ φ . (16)Evaluation of (14) is thus a slight extension of the calcu-lation which yields the spectrum of the velocity potential, F φ = 12 π Z ∞−∞ dτ e − iωτ h φ (0 , ~x h , z ) φ ∗ ( τ, ~x h , z ) i , (17)normalized such that R ∞−∞ dωF φ ( ω ) = h φ i .Using (12) one obtains h φ (0 , ~x h , z ) φ ∗ ( τ, ~x h , z ) i = 1(2 π ) π Z dωd~k h dω ′ d~k ′ h c g h ˜ S ( ω, ~k h ,
0) ˜ S ∗ ( ω ′ , ~k ′ h , i G ( ω, ~k h , z ) ˜ G ∗ ( ω ′ , ~k ′ h , z ) e iω ′ τ , (18) where the source ˜ S ( ω, ~k h ,
0) is given in (9).The wave elevation spectrum is F ζ ( ~q ) ≡ π ) Z d~x h h ζ (0) ζ ∗ ( ~x ) i e i~q · ~x h , (19)again normalized such that R d~q F ζ ( ~q ) = h ζ i . Equiva-lently, h ζ ( ~q ) ζ ∗ ( ~q ′ ) i = (2 π ) F ζ ( ~q ) δ ( ~q − ~q ′ ).Introducing the source spectrum (9) into (18), us-ing the elevation spectrum (19), and assuming Gaussianstatistics, we get h φ (0 , ~x h , z ) φ ∗ ( τ, ~x h , z ) i = 116 π Z dωd~k h ω g | ˜ G ( ω, ~k h , z ) | Z d~q σ ( ~q ) σ ( ~k h − ~q ) − ~q · ( ~k h − ~q ) q | ~k h − ~q | ! F ζ ( ~q ) F ζ ( ~k h − ~q ) δ ( ω − σ ( q ) − σ ( | ~k h − ~q | )) e iωτ . (20)Note that one must be careful not to omit a combinatoricfactor of 2, which arises because h ζ ( ~q ) ζ ( ~k h − ~q ) ζ ∗ ( ~q ′ ) ζ ∗ ( ~k ′ h − ~q ′ ) i = h ζ ( ~q ) ζ ∗ ( ~q ′ ) ih ζ ( ~k h − ~q ) ζ ∗ ( ~k ′ h − ~q ′ ) i + h ζ ( ~q ) ζ ∗ ( ~k ′ h − ~q ′ ) ih ζ ( ~k h − ~q ) ζ ∗ ( ~q ′ ) i + · · · The remaining terms, indicated by · · · , do not contribute.We introduce the functionΣ( ω, ~k h ) ≡ π ω g Z d~q σ ( ~q ) σ ( ~k h − ~q ) − ~q · ( ~k h − ~q ) q | ~k h − ~q | ! F ζ ( ~q ) F ζ ( ~k h − ~q ) δ (cid:16) ω − σ ( q ) − σ ( | ~k h − ~q | ) (cid:17) , (21)which depends on the surface wave statistics and the dis-persion relation. With this definition, the autocorrelationof the second-order velocity potential can be written h φ (0 , ~x h , z ) φ ∗ ( τ, ~x h , z ) i = Z dωd~k h | ˜ G ( ω, ~k h , z ) | Σ( ω, ~k h ) e iωτ (22a)= Z dωe iωτ F φ . (22b)Because of the second equivalence, which is the inverseof (17), it immediately follows that F φ ( ω ) = Z d~k h | ˜ G ( ω, ~k h , z ) | Σ( ω, ~k h ) . (23)Taking the appropriate derivatives, via equation (16),Acoustic radiation and wave-wave interactions 3e get M , = ω c Z d~k h | G ω,~k h ( z, | Σ( ω, ~k h ) M i,j = Z d~k h k h,i k h,j | G ω,~k h ( z, | Σ( ω, ~k h ) M ,i = − ωc Z d~k h k h,i | G ω,~k h ( z, | Σ( ω, ~k h ) M i, = i Z d~k h k h,i G ω,~k h ( z, ∂∂z G ∗ ω,~k h ( z, ω, ~k h ) M , = Z d~k h ∂∂z G ω,~k h ( z, ∂∂z G ∗ ω,~k h ( z, ω, ~k h ) M , = − i ωc Z d~k h G ω,~k h ( z, ∂∂z G ∗ ω,~k h ( z, ω, ~k h )The following definitions lead to a more compact no-tation ג = | G ω,~k h ( z, | Σ( ω, ~k h ) (24a) ג = G ω,~k h ( z, ∂∂z G ∗ ω,~k h ( z, ω, ~k h ) (24b) ג = ∂∂z G ω,~k h ( z, ∂∂z G ∗ ω,~k h ( z, ω, ~k h ) (24c) ג = ג ( ~k h = 0) (24d)With these, the PSDM may be written M , = ω c Z d~k h ג (25a) M i,j = Z d~k h k h,i k h,j ג ( i, j = 1 ,
1; 1 ,
2; 2 ,
2) (25b) M ,j = − ωc Z d~k h k h,j ג ( j = 1 ,
2) (25c) M i, = i Z d~k h k h,i ג ( i = 1 ,
2) (25d) M , = Z d~k h ג (25e) M , = − i ωc Z d~k h ג (25f) IV. THE STANDING WAVE APPROXIMATION
The collision of two surface waves with horizon-tal wave-numbers ~q , ~q , and corresponding frequen-cies σ ( q ) , σ ( q ), yields an acoustic wave with fre-quency ω = σ ( q ) + σ ( q ) and horizontal wave-number ~k h = ~q + ~q . Thus, the vertical wave-number of theacoustic wave is γ = q ( ω/c ) − ~k h . (26)For un-attenuated propagation to the bottom, γ must bereal, or k ≡ | ~k h | < ω/c . This constraint can be re-writtenas σ ( ~q ) + σ ( ~q ) | ~q + ~q | > c . (27) Given the disparity between the surface wave phasevelocity and the much larger acoustic phase velocity, σ ( q ) /q << c , the constraint (27) implies that ~q ≈ − ~q .The allowed deviation from the standing wave-case, ~q = − ~q , expressed as a variation in the relative wave-length ∆ λ/λ or of the propagation angle ∆ θ of one of theopposing surface waves, is of order c ζ /c , where c ζ is thesurface wave phase velocity. Wave spectra are essentiallyconstant over such small variations of angle and wave-length, which are too small to resolve experimentally.Thus, at sufficient depth, the integral over the hori-zontal wave number ~k h in (25a - 25f) can be restricted to k h < ω/c , and the source term Σ( ω, ~k h ) can be replacedwith its value at ~k h = 0. In the absence of a bottom, thesquared amplitude of the Green’s function | G ω,~k h ( z, | can also be replaced with its value at ~k h = 0 for thefollowing reason.For a bottomless ocean, the Green’s function is thesolution of (10a) with the surface boundary condition(10b). For z < G ω,~k h ( z,
0) = − πgiγg + ω e − iγz (28)(cf. Ref. 6, Section 4.2.3). In the region k h < ω/c forwhich there is un-attenuated propagation to the bottom, | G ω,~k h ( z, | = 16 π ω g + ω c − k h = 16 π g ω (cid:20) O ( g ω c ) (cid:21) . (29)All the dependence of | G ω,k h | on k h is in the sub-leading term, which is negligible since g/ ( ωc ) is extremelysmall (about 10 − at 1 Hz). This is also of order c ζ /c ,since for gravity waves the phase velocity is c ζ = g/σ and, for a standing wave, ω = 2 σ . Note that the samesmall parameter, g/ωc , determines the dominance of thesurface term over the bulk term in (11). The char-acteristic penetration depth of surface gravity waves is L = 1 /q = g/σ = 4 g/ω , such that the small parameterdetermining the dominance of the surface term over thebulk term, gL/c , is proportional to ( g/ωc ) .The far-field spectrum is generally computed by re-stricting the integration region to k h < ω/c and replac-ing both source and propagation terms, Σ ω,~k h and G ω,~k h ,respectively, by their values at ~k h = 0. This is known asthe standing wave approximation. In light of the abovediscussion, the standing wave approximation is a verygood approximation in the absence of a bottom, or for aperfectly absorbing bottom, with errors proportional tothe small parameter c ζ /c .To make contact with well-known results, we evalu-ate M , in the standing wave approximation. Using ג ,Acoustic radiation and wave-wave interactions 424d), M , = ω c Z π dθ Z ω/c dk k ג = πω c | G ω,~k h ( z, | Σ( ω, ~k h = 0)= π ω c Z d~q F ζ ( ~q ) F ζ ( − ~q ) δ ( ω − σ ( q )) . (30)The wave spectrum is factored in the usual way, so F ζ ( ~q ) = F ζ ( q ) H ( θ, q ), where R dθ H ( θ, q ) = 1, and I ( q ) ≡ Z π dθH ( θ, q ) H ( θ + π, q ) . (31)With these definitions, equation (30) becomes M , ( ω ) = π ω c qv F ζ ( q ) I ( q ) , (32)where ω = 2 σ ( q ), and v is the group velocity, ∂σ/∂q .The pressure spectrum is then F P ( ω ) = ρ c M , = π (cid:16) ρc (cid:17) ω qv F ζ ( q ) I ( q ) . (33)This result is twice the generally adopted formula (Ref.7, Appendix A), indicating a discrepancy between our ap-proach and some previous derivations of the source term.The difference, within the spread of prior derivations, hasno influence on the subsequent analysis of the standingwave approximation and the power spectral density ma-trix.The implications of the standing wave approximationupon the elements of the PSDM, other than M , , havenot been considered previously. V. RATIOS OF PSDM ELEMENTS IN THE STANDINGWAVE APPROXIMATIONA. Perturbation theory computation
For a perfectly absorbing bottom and surface wavepairs satisfying k h < ω/c , the wave interaction sourceterm Σ( ω, ~k h ) and the squared amplitude of the Green’sfunction | G ω,~k h ( z, | are very well approximated bytheir values at ~k h = 0. Thus, the PSDM elements (25a -25f) become M , = ω c ג Z d~k h (34a) M i,j = ג Z d~k h k h,i k h,j (34b) M ,j = − ωc ג Z d~k h k h,j (34c) M i, = − ג Z d~k h k h,i r ω c − k h (34d) M , = ג Z d~k h (cid:18) ω c − k h (cid:19) (34e) M , = ωc ג Z d~k h r ω c − k h , (34f) and all integrals have the limits k h < ω/c .The integrals over ~k h are trivial, giving M = π ω c ג . (35)The ratios of PSDM elements are universal constants,so we define the ratio matrix, R = M µ,ν M , = . (36) R is independent of both frequency and the ocean wavespectrum. We see also that the spectrum of vertical ve-locity is 3 dB greater than either of the horizontals. Fur-thermore, the normalized pressure is exactly equal to thesum of the three velocity spectra on the diagonal. Thisis characteristic of any homogeneous acoustic field in theocean. Due to rotational invariance, all off-diagonal compo-nents vanish except for r , and its mirror image. Sen-sor calibration determines the accuracy to which the el-ements of R can be estimated. A better metric is thesquared coherency. This will be zero for all off-diagonalelements except r , . For this term, χ , = r , /r , = 89 . The full cross-correlation matrix is χ = r , r , r , r , r , r , r , . r , r , r , r , r , r , r , . . r , r , r , r , . . . r , . (37) B. PSDM elements from incoherent dipoles
The pressure field from a homogeneous surface layer ofincoherent and vertically oriented dipoles is P ( ~x h , z, t ) = Z dω Z d~x ′ h e − iωt F ω ( ~x ′ h ) ∂ z (cid:18) e i ωc R R (cid:19) (38)where F is the source amplitude, the subscript h indicateshorizontal coordinates, and R = q ( ~x h − ~x ′ h ) + z . (39)The more general case, allowing for multiple reflectionsat the surface and bottom, is given by Hughes. The scaled pressure spectrum, M , , is M , ( ω ) = 1( ρc ) Z d~x ′ h D ω ( ~x ′ h ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ z (cid:18) e i ωc R R (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≈ ρc ) Z d~x ′ h D ω ( ~x ′ h ) (cid:16) ωc zR (cid:17) R (40)Acoustic radiation and wave-wave interactions 5here D ω ( ~x ′ h ) is obtained from the dipole spectrum, h F ω ( ~x ) F ′ ω ( ~x ′ ) i = D ω ( ~x ) δ ωω ′ δ ( ~x − ~x ′ ), and we haveassumed that we are at sufficient depth z so that ω/c >> R − .Since the dipole distribution is assumed to be homoge-nous, D ω is independent of ~x ′ h . To obtain expressionssimilar to those in (25a), we make a change of variablesfrom space to wave-number coordinates, ~x ′ h → ~k h , where ~k h ≡ ωc ~x ′ h − ~x h p z + ( ~x ′ h − ~x h ) (41)or ~x ′ h − ~x h = z ~k h q ω c − ~k h . (42)Then (40) becomes M , ( ω ) = 1( ρc ) D ω Z | ~k h | < ωc d~k h . (43)Using P = ρ ˙ φ and v i = ∂φ/∂x i , we obtain the rest of thePSDM elements: M , ( ω ) = 1( ρc ) D ω Z d~k h (44a) M i,j ( ω ) = c ω ρc ) D ω Z d~k h k h,i k h,j (44b) M ,i ( ω ) = − cω ρc ) D ω Z d~k h k h,i (44c) M i, ( ω ) = − c ω ρc ) D ω Z d~k h k h,i r ω c − k h (44d) M , ( ω ) = c ω ρc ) D ω Z d~k h (cid:18) ω c − k h (cid:19) (44e) M , ( ω ) = cω ρc ) D ω Z d~k h r ω c − k h , (44f)and each integral has the limits k h < ω/c This agrees precisely with the result (34a)–(34f) ob-tained using perturbative non-linear wave interactiontheory in the standing wave approximation, providedthat the surface dipole spectrum is related to the wavespectrum by D ω = ρ ω ג . (45) VI. BOTTOM EFFECTS
In general, the effect of the bottom is a modification ofthe Green’s function appearing in (25a-25f). The stand-ing wave approximation which leads to (35) is dependenton the squared amplitude of the Green’s function of theoperator ˆ L = d dz + ω c − k h (46)being a nearly constant function of k h in the region k h < ω/c . The fact that this holds in the bottomless case is a lucky accident. In general, the Green’s function isnon-analytic at k h = ω/c , which is the transition pointfrom vertical propagation to attenuation, so there is noreason to expect its squared amplitude to be constant in aneighborhood of this point. Upon inclusion of a bottom,the squared amplitude of the Green’s function becomesa rapidly varying function on the interval k h = [0 , ω/c ].The location of the peaks of this function are determinedby solutions of the eigenvalue equation (cid:18) d dz + ω c − k h,l (cid:19) φ = 0 , (47)subject to the appropriate top and bottom boundary con-ditions. On the real interval k h = [0 , ω/c ], the Green’sfunction has narrow peaks centered about real k h,l cor-responding to normal modes and wider peaks centeredabout the real part of complex k h,l corresponding to leakymodes (see, for example, Ref. 11, Section 2.1). A. Green’s function for a layer over a half-space
We wish to obtain the Green’s function G ω,~k h ( z, z s )satisfying (cid:18) ∂ z + ω c − k h (cid:19) G ω,~k h ( z, z s ) = δ ( z − z s ) , (48)subject to the surface boundary condition (for the caseof gravity waves), (cid:18) ∂ z − ω g (cid:19) G (cid:12)(cid:12)(cid:12)(cid:12) z =0 = 0 (49)and the mixed Dirichlet-Neuman (impedance) bottomboundary condition, (cid:16) G + h ( ~k h ) ∂ z G (cid:17)(cid:12)(cid:12)(cid:12) z = z bottom = 0 . (50)The function h ( ~k h ) depends on the characteristics of theregion below z bottom ≡ d .Assuming constant sound speed c in the water columnabove z bottom , and taking z s = 0 (the surface), the solu-tion is G ω,~k h ( z,
0) = g (cid:16) e − iγ ( d + z ) + R ( ω, ~k h ) e iγ ( d + z ) (cid:17) ( ω + iγg ) e − iγd + R ( ω, ~k h )( ω − iγg ) e iγd , (51)where γ ≡ r ω c − ~k h , (52)and R is the reflection coefficient; R = M − M + 1 (53)Acoustic radiation and wave-wave interactions 6ith M = Z b /Z , and the top and bottom layerimpedances are defined by Z = ρωγ (54a) Z b = iρωh ( ~k h ) . (54b)The function h ( ~k h ) entering the bottom boundarycondition (50) is obtained from the matching conditionacross the top–bottom interface or across multiple inter-faces if the bottom is layered. For a fluid half-space bot-tom, one matches the vertical velocity v z on either sideof the interface, while for a solid half-space bottom, onematches v z and the components of the stress tensor con-taining a vertical index, τ iz . This has been done beforein numerous places (see for example Ref. 12). With-out derivation, the result for a fluid layer over an elastichalf-space is given below.For the fluid half-space with density ρ b and soundspeed c b , h ( ~k h ) = − i ρ b ργ b , γ b ≡ s ω c b − ~k h . (55)For the elastic half-space with density ρ b , shear wavevelocity c s , and compression wave velocity c p , h ( ~k h ) = i c s ω ρ " ~k h − γ s γ p (cid:16) (2 µ + λ ) γ p + λ~k h (cid:17) − µ~k h γ s γ p ≡ s ω c p − ~k h , γ s ≡ s ω c s − ~k h (56)where µ and λ are the Lam´e parameters, related to theshear and compression wave velocities by c p = s λ + 2 µρ b c s = r µρ b . (57) B. Numerical example
As a specific example, Green’s function for the modelgiven in Table I was calculated for an acoustic fre-quency of 3 Hz (Fig. 1). A power loss of 6 dB perbounce, independent of angle, is assumed for k h > k h, crit ( k h, crit = ω/c s = . TABLE I. Model used for calculating power spectrum matrixOcean layer Half-space ρ (kg m − ) 1000 2000 c p (m s − ) 1500 4400 c s (m s − ) 2200thickness (m) 5000 ∞ Wave number at 3 Hz, radians m -1 | G | - FIG. 1. The squared amplitude of the Green’s function G ω,k h (0 , z bot ) on the interval k h = [0 , ω/c ] for a fluid layerover an elastic half-space at 3 Hz. Red shows the result fora bottomless ocean, which is very nearly flat on this interval.The cutoff wavenumber, ω/ . k h, crit = ω/ . × . The figure shows pronounced narrow peaks,corresponding to normal modes on the interval k h = [ k h, crit , ω/c ] where the reflection coefficienthas modulus nearly equal to 1 and acoustic energy isnot lost by propagation through the bottom layer. Thecritical wave number is ω/c s where c s is the shear wavevelocity in the half-space. A small power loss per bottombounce is incorporated to give the normal modes k h,l asmall imaginary component such that these peaks havefinite height.There are also a series of wider peaks on the inter-val k h = [0 , k h, crit ], corresponding to leaky modes. Ageneral feature of bottom effects is that the acoustic sig-nal increases over the bottomless case. Moreover, thePSDM ratios can be expected to differ from (36) dueto oscillations of the squared amplitude of the Green’sfunction which violate the standing wave approxima-tion; it is no longer possible to make the replacement G ω,~k h → G ω,~k h =0 inside the integrals in (25a). VII. THE WEAK STANDING WAVE APPROXIMATION
Although the squared amplitude of the Green’s func-tion is not flat on the interval k h = [0 , ω/c ] when bot-tom effects are considered, the surface wave spectrumremains unaltered by bottom effects, except possibly forthe organ-pipe modes. With this limitation, one canmake the replacementΣ( ω, ~k h ) → Σ( ω, ~k h = 0) , (58)in (25a-25f), which we call the “weak standing wave”approximation. Since the wave spectrum, Σ, is evaluatedAcoustic radiation and wave-wave interactions 7t ~k h = 0, it can be taken outside those ~k h integrals,giving M µ,ν = Σ( ω, ~k h = 0)Ξ µ,ν (59)whereΞ , = ω c Z d~k h | G ω,~k h ( z, | (60a)Ξ i,j = Z d~k h k h,i k h,j | G ω,~k h ( z, | (60b)Ξ ,j = − ωc Z d~k h k h,j | G ω,~k h ( z, | (60c)Ξ i, = i Z d~k h k h,i G ω,~k h ( z, ∂∂z G ∗ ω,~k h ( z,
0) (60d)Ξ , = Z d~k h ∂∂z G ω,~k h ( z, ∂∂z G ∗ ω,~k h ( z,
0) (60e)Ξ , = − i ωc Z d~k h G ω,~k h ( z, ∂∂z G ∗ ω,~k h ( z, , (60f)and each integral has the limits k h < ω/c .The term that depends on the wave spectrum, Σ, can-cels in the ratios of PSDM elements, which therefore de-pend solely on Ξ µ,ν . In the absence of signal other thanthat generated by surface wave interactions, similar argu-ments imply that the dependence on the wave spectrumalso cancels in other ratios of deep ocean acoustic corre-lation functions.Thus we propose that the PSDM element ratios in deepwater for the signal generated by wave interactions, whilenot universal constants, are independent of the details ofthe wave spectrum. They will depend solely on frequencyand the propagation environment, which together deter-mine the form of the Green’s function G ω,~k h . A. Pressure near the bottom
The effect of the bottom on acoustic pressure is ex-pressed as the ratio of the M , ( z ) matrix elements. Us-ing the weak standing wave approximation, (60a), therelative effect of the bottom on pressure is given by M b , ( z ) M , ( z ) = Ξ b , ( z )Ξ , ( z ) (61)where () b indicates there is a bottom under the modelocean.Evaluating (61) for two depths, we find the solid bot-tom raises the pressure at the bottom by an average of2.3 dB, and at 500 m off the bottom the increase is 1.6dB (Fig. 2). These values are averages over frequenciesless than 15 Hz, but the figure shows the smoothed spec-trum is flat. Kibblewhite and Wu, using a more detailedelastic model, found the bottom lifted the pressure spec-trum by 3 dB (Ref. 6, p. 116 and Fig. 7.8). The spectraplotted in the figure are equivalent to the function B in-troduced by Farrell and Munk to express the effect of thebottom on the spectrum of acoustic pressure arising fromwave-wave interactions on the ocean surface [Ref. 7, (4)]. P r ess u r e s p ec t r u m r a t i o Acoustic frequency, Hz
FIG. 2. The effect of the solid half-space under the oceanis a uniform increase in bottom pressure by the factor 1.66(black), or 2.3 dB. The pressure increase is slightly less, 1.46,500 m above the bottom (red).
B. Model spectra on the ocean floor
We have calculated all elements of Ξ µν for the sim-ple half-space model (see Table 1). For an observationpoint at the bottom but in the water, the matrix elementsshown in Fig. 3 are obtained. Smoothing the elementsover plotted band, 0-15 Hz, gives the ratio matrix, R ( z bot ) = − . . i. .
23 0 0 . . .
23 0 . . . . . (62)When smoothed over twice the bandwidth the results areessentially the same. If the bottom loss is halved, to 3dB, the changes in the matrix are small. Element r , is reduced by 10%, the other two diagonal elements areraised by 10%, and the magnitude of r , becomes 25%less.The principal effect of the bottom is to reduce the ver-tical velocity, relative to pressure, by about 40% with re-spect to the case of the bottomless ocean. There are twoconsequences: both r , and r , are smaller by aboutthis amount. In addition, the coherency between pres-sure and vertical velocity is less: χ , = .
58, vs. 0.89 forthe bottomless ocean. The diagonal sum of R is 1.75,the difference from 2 reflecting the inhomogeneity of theacoustic field near the bottom. C. Model spectra near the bottom
As the observation point rises to 500 m above the bot-tom, 10% of the water depth, the spectra evolve as shownin Fig. 4. Smoothing the spectra as before gives the fol-Acoustic radiation and wave-wave interactions 8 requency, Hz r r -90018090 D eg r ee s r magnitude and phase CBA
FIG. 3. PSDM element ratios r µ,ν = Ξ µ,ν / Ξ , (black) for apoint just above the bottom for a 5,000 m water layer over anelastic half-space (Table I). The short red (blue) line segmentsare the ratios for a bottomless ocean. The red segments termi-nate at 3 Hz, the reference frequency of the Green’s functionshown in Fig. 1. lowing ratio matrix: R ( . bot ) = . . i. .
27 0 0 . . .
27 0 . . . . . (63)The vertical velocity element, r , , has nearly revertedto the bottomless value (36), but, surprisingly, the r , element is 25% less. The diagonal sum is 2.03, reflectingthe near homogeneity of the acoustic field. VIII. EVIDENCE IN DATA FROM STATION H2O
The theory is applied to bottom data from sta-tion H2O, located at 5,000 meters in a thin-sedimentarea of the Pacific, mid-way between Hawaii and Cal-ifornia. The instrumentation consists of a buried(0.5 m) Guralp seismometer (Guralp System Limited’sCMG-3) and Geospace geophone (GTC, Inc.’s GeospaceTechnologies TM HS-1), and a hydrophone a little off thebottom. The two velocity sensors differ in the way themotion of the inertial mass is sensed: the Guralp is bet-ter at very low frequencies, the Geospace at very high,but they are comparable over our analysis band. In ad-dition, assimilated surface winds for the H2O locationwere provided by the European Centre for Medium-rangeWeather Forecasts from the ERA-interim (ECMWF Re-analysis) results. C r o ss s p ec t r u m ph ase S p ec t r a l r a t i o S p ec t r a l r a t i o Frequency, Hz S p ec t r a l r a t i o r magnituder r r phase CBAD
FIG. 4. PSDM element ratios (smoothed over 0.3 Hz) atthe ocean bottom (black, recapitulating Fig. 3) and 500 me-ters above the bottom (red). For the diagonal elements andfrequencies towards the right, the red spectra approach thebottomless ocean result, denoted by the red line segments onthe right margin (panels A, B). The off-diagonal element doesnot (panels C, D).
H2O data (hydrophone, channel HDH; Geospace,channels EHZ, EH1, EH2; Guralp, channels HHZ, HL1,HL2), obtained from the Incorporated Research Institu-tions for Seismology (IRIS) Data Management Center,have been studied for years 2000-2002, inclusive. Spectrawere calculated for three-hour windows to a resolution of0.1 Hz, giving about 2000 equivalent degrees of freedom.The spectra have been examined for the whole interval,but this discussion is restricted to days 200-250 in year2000. Spectra were calibrated according to the nominaltransfer functions on file at the IRIS Data ManagementCenter. Various small inaccuracies were discovered andcorrected. For example, there were discrepancies at thetimes the gain was changed. The spectra for the verticalcomponent of the Geospace geophone were raised by 3dB. This makes them consistent with spectra from theAcoustic radiation and wave-wave interactions 9ertical component of the Guralp seismometer, is com-patible with the instrument noise model, and corrects ademonstrable error in its nominal generator constant.
A. Bottom acoustics and surface winds
It is generally observed that bottom acoustics, fromfrequencies less than 1 Hz to frequencies above 30, arehighly correlated with surface wind. Typical are theprofiles of the acoustic power at 0.5 Hz (Fig. 5) and 1.76Hz (Fig. 6). The lower frequency was chosen becausethe coherency between pressure and vertical velocity isgreatest. The higher was selected to be on a relativeextremum of the coherency and away from any sedimentresonance.The vertical scales in Fig. 5, Panel A, have been ad-justed so that the variation in wind (black) is about aslarge as the variation in acoustics (colored). In Panel Aof Fig 6, the wind has about half the range of the acous-tics. Comparing axes, the slope in log acoustic power isroughly 2 . / (m s − ) at these frequencies. It has pre-viously been shown (Ref. 7, Fig 2) that between 6 and30 Hz the slope of the spectrum is in the range 2.7 - 2.9dB / (m s − ).In both figures, the flatness of the components of thevelocity ratios (Panels B, C) shows that scaling vectorvelocity by normalized pressure is effective at reducing oreven eliminating the correlation with surface wind. Thisis as expected from the foregoing theory, to the extentthe wind is a proxy for the waves. B. Off-diagonal matrix elements
The off-diagonal elements of the PSDM are the mostdiagnostic of the wave-wave acoustic field because, ex-pressed as squared coherency (see Eq. 37), they are in-dependent of sensor calibration. The element χ , is theonly off-diagonal element that does not vanish for theelementary models we are testing.The spectrum of χ , (Fig. 7) shows a profound dis-continuity at 1.03 Hz, which is just below the gravestsediment resonance (see Fig. 10). The two effects arepresumably related. For lower frequencies the measuredcoherency is even higher than the model (0.8 vs 0.58,see Eqs. 37 and 62). The phase of the cross-spectrum(Fig 7, Panel B) in the high-coherency band is about − ◦ , far from the model expectation of 104 ◦ (62). How-ever, the phase of the cross-spectrum depends on thephase response of both instruments, and, as explained inthe Discussion, there are questions about the hydrophonetransfer function.At higher frequencies the coherency wobbles around0.1-0.2, and there are large swings in phase. There ispronounced rippling in the coherency, with a periodicityof approximately 6 cycles/Hz. These are likely organpipe modes, a possibility previously considered but notembraced. In theory, the other five off-diagonal elements vanish;in practice, they nearly do so for f < E C M W F U , m s - Year 2000 day
215 220 225 230 235 240 V e l o c i t y S p ec t r u m , d B -130-120-100-140-110-90-28-30-26 R a t i o S p ec t r u m , d B -28-30-26 R a t i o S p ec t r u m , d B
215 220 225 230 235 240215 220 225 230 235 240
CBA
FIG. 5. Profile of acoustic power at 0.5 Hz for Guralp datawith dash red pressure (scaled by ( ρc ) ), and red, blue, greendenoting the vertical and two horizontal components of veloc-ity (A). ECMWF wind (black) is referenced to the right axis.Panels B and C are the the three ratios of velocity to scaledpressure for the two instruments. Blanks indicate sectionswhere there were no data or where the value was dropped ona first difference screen. The plus symbols at the bottom ofeach panel denote windows at days 218.125 ( U = 3 .
3) and232.125 ( U = 10 .
1) for which detailed results are presentedbelow. large at higher frequencies (Fig. 8). The spike in χ , near 2.5 Hz corresponds to one of the P-SV modes in thesediments (see Fig. 10).Dividing the velocity spectra by the (scaled) pressure(Figs 5 and 6, panels B and C) is effective at obliteratingthe wind correlation. Thus, it is not surprising that thecross-spectrum of pressure and vertical velocity is equallyquiescent (Fig. 9). Another perspective on the residualwind signature is visible in comparing the pairs of spectraplotted in Figs. 7 and 8. Red is the coherency under astrong wind, and black dash under a weak wind. Thepairs of curves are virtually indistinguishable.Acoustic radiation and wave-wave interactions 10 E C M W F U , m s - Year 2000 day
215 220 225 230 235 240 V e l o c i t y S p ec t r u m , d B -145-165-155-135
215 220 225 230 235 240215 220 225 230 235 240
CBA R a t i o S p ec t r u m , d B R a t i o S p ec t r u m , d B FIG. 6. Profiles of acoustic power at 1.76 Hz for Guralp data(colored) and ECMWF wind (black) (A). Panels B and Care the the three ratios of velocity to scaled pressure for thetwo instruments as in Fig. 5. See its caption for additionaldetails. Note that the acoustics in Panel A more closely followthe dips in the wind than is the case at the lower frequencyof Fig. 5.
C. Diagonal matrix elements
The power spectra of the velocity components downthe diagonal of the PSDM (Fig. 10) are relatively flat forfrequencies above 1 Hz but fall precipitously below that.Excluding sediment resonances, the spectra are in ac-cord with the spot measurements obtained from smooth-ing the results at 0.5 Hz and 1.76 Hz displayed in Figs. 5and 6, respectively, and listed in Table II. The steep dropbelow 1 Hz we tentatively attribute to a hydrophone cali-bration error (see Discussion). The bumps on the spectraof both components at 1 Hz and 2.5 Hz are attributedto P-SV sediment resonances, as is the broader peak at4.25 Hz for r , (blue). χ P h ase , d e g r ees -180-90 Frequency, Hz , FIG. 7. Pressure and vertical velocity are highly correlated forfrequencies less than 1 Hz (A), with phase angle of − ◦ at 0.5Hz (B, Table II). The coherency at higher frequencies, thoughless, is still significant, but because of the lower coherency,the phase fluctuates more. These Guralp exemplars includea high-wind window (red, day 232.125, U=10.1 m s − ) and alow-wind window (black dash, day 218.125, U=3.3 m s − ). D. Summary
The model yields a PSDM ratio matrix for which allelements are independent of both frequency and wavespectrum. This is true whether they are expressed innatural units or as squared coherency.The measured PSDM ratio matrix is, indeed, insensi-tive to wind, which we take as a proxy for waves, butstrongly depends on frequency. There is a sharp tran-sition at 1 Hz. For lower frequencies, the coherency ofthe off-diagonal elements is closer to the model than forhigher. A sharp transition in the diagonal matrix ele-ments occurs at this same critical frequency. Below it,the ratio elements plunge by 30 dB, which is entirelycaused but the sharp rise in the pressure spectrum. Al-though we have suggested this is a calibration matter, thecorrespondence in frequency of the two effects suggeststhey are linked in the physics. Above the transition fre-quency, the diagonal elements are flatter, which is morein accord with the model. The variations are within 5 dB.Peaks of up to 10 dB, which are attributed to P-SV res-onances in the sediments, are discounted. However, thelevels are much larger than the model results, an effectalso attributed to energy trapped in the sediments.The important numeral results are highlighted in Ta-ble II. For six representative matrix elements (column1), this shows the model values (columns 2) and the ob-Acoustic radiation and wave-wave interactions 11
Frequency, Hz χ , χ , AB FIG. 8. For the same windows displayed in Fig. 7, the co-herency between pressure and horizontal velocity (A) and be-tween vertical and horizontal velocity (B) is low everywhere,but especially for frequencies less than 1 Hz, where χ , ishighest (see Table II). Year 2000 day χ FIG. 9. The coherency between pressure and vertical velocity(Geospace sensor, three representative frequencies) is virtu-ally constant, irrespective of wind speed. For 0.5 Hz, there isa slight dimple near day 218 when the wind was low. Resultsfor 1.67 Hz around day 235 are unusual, as is the scatter be-tween days 240 and 245 at 1.76 Hz. (Gaps are due to missingdata or spectra dropped on a first difference criterion.) served values (columns 3-6) for both sensors and the tworepresentative frequencies.For the diagonal elements (top rows), the values for0.5 Hz are orders of magnitude smaller than the model,and the values at 1.76 Hz (and above) are 10 to 40 timeslarger. The off-diagonal elements (bottom three rows)fit the model better at 0.5 Hz than at 1.76, althoughin neither case has the result been smoothed over theprominent ripples.The frequency dependence observed in the coherencyof the off-diagonal matrix elements is inconsistent withthe universal constants predicted by the standing wave
Frequency, Hz S p ec t r a l r a t i o , d B FIG. 10. Guralp matrix elements r , (blue) and r , (red)for day 232.125 ( U = 10 . − ). The adjacent black dashcurves are the spectra for day 218.125 ( U = 3 . − ). Theblack bars are averages of the power at 1.76 Hz, exemplifiedin Panel B of Fig. 6. The short bars on the right are matrixvalues from (62). Spectra for the Geospace instrument aresimilar.TABLE II. Selected elements of the spectral ratio matrix at0.5 and 1.76 Hz for both H2O seismic sensors. The 12 valuesfor the diagonal elements r i, j were calculated by averagingthe spectrum over the 50 days, for which shorter segmentsare plotted in Figs. 5 and 6. The values of the off-diagonalelements were calculated by similar smoothing in time at theappropriate frequencies (for the Guralp, see Figs. 7 and 8).Theory 0.5 Hz 1.76 Hz(62) Guralp Geospace Guralp Geospace r , . × − . × − r , . × − . × − r , . × − . × − r , < . < . < . < . r , − . . i . − . i . − . i . − . i . − . ir , < . < . < . < . approximation in the case of an ocean layer resting on anelastic half-space. Thus, other effects, possibly bottomscattering, are influencing the wave-generated sound atthis site. On the other hand, the insensitivity to wind,as predicted by the weak standing wave approximation,is strongly upheld by these data. IX. DISCUSSION
We have shown that the theoretical predictions of thestanding wave approximation are extremely strong, yield-ing PSDM ratios which are universal constants. Whileit is possible that these PSDM ratios are observed undercertain conditions, such as a high loss bottom, they arecertainly not observed in the H2O data.Acoustic radiation and wave-wave interactions 12ottom effects vitiate the standing wave approxima-tion, leaving in place a weaker version of the approxi-mation, the predictions of which are observed at H2O. Inparticular, the PSDM ratios depend on frequency, but areinsensitive to changes in the surface conditions. Directcomparison of the PSDM ratios to an elastic half-spacemodel show less agreement. This is due, presumably,to the oversimplification of the bottom and propagationconditions in the model, as well as to instrument calibra-tion error.
A. Bottom interaction
The primitive model of the ocean’s bottom replicatesthe essential features of more elaborate approaches. Pres-sure is increased a few dB because of the bottom, andthis increase is uniform over the band 1-30 Hz, at least.The augmentation diminishes slightly as the observationdepth lessens. Given a quantitative estimate for theinfluence of the bottom, overhead wave properties canbe inferred from (corrected) deep pressure observationsthrough application of (33), although there is a factor 2discrepancy between this result and some other theories.The model of the ocean bottom will need to be extendedto account for the layer of ocean sediments before velocityobservations can be used for the same purpose.Pressure and vertical velocity are most coherent forfrequencies below 1 Hz; indeed, the observed coherenceexceeds the model result. The coherence falls above 1 Hz.The sediments overlying the basement were not modeled,but the decrease in coherence is attributed to P-SV en-ergy trapped in this layer. This same effect can explainthe weak but significant rise at 1 Hz in the coherence be-tween horizontal velocity and both pressure and verticalvelocity.We view the peaks in the velocity spectra as evidence ofvertically polarized shear waves, naturally excited whena pressure wave in an acoustic medium is incident on anelastic medium. The extremely low shear velocity of thesurficial sediments has two effects: the transmitted SVrays are nearly perpendicular to the boundary and theyare strongly polarized in the horizontal direction. Theresonance occurs when the rays are efficiently reflectedby internally layering in the sediments. Zeldenrust andStephen applied the theory of Godin and Chapman to interpret the resonances at H2O as evidence of a chertlayer approximately 13 m below the bottom, about inthe middle of the 30 m of sediments. Scattering is afurther complication. Any zones in the elastic mediumwith strong impedance contrasts and rough boundarieswill scatter both SV and SH energy. B. Influence of the ocean’s sound speed profile
Taking the ocean to have constant sound speed ignoresthe refraction due to the sound channel (e.g. Ref. 7, Fig.3). However, for a surface layer of incoherent dipoles,the bottom signal is dominated by the source region withdiameter six times the water depth (Ref 13, Eq. 7).
C. Inference of the wind-wave spectrum
Bottom acoustic observations are beginning to be usedto estimate the spectrum of ocean surface waves. Todo this, within the framework of acoustic radiation fromwave-wave interactions, two corrections are necessary.Allowance must be made for the bottom interaction andthe overlap integral.The correction of the pressure spectrum for bottominteraction appears to be straightforward. The bottomelevates the pressure a few dB at the bottom, with aslight decrease moving up the water column away fromthe bottom. The bottom effect, in fact, is smaller thanthe factor-of-two discrepancy between our derivation andsome previous results.It is just as significant that the bottom correction doesnot depend on frequency. Thus, the theory for the bot-tomless ocean can be used to infer the slope of the wavespectrum from the slope of the pressure spectrum withno further corrections.Spectra of bottom velocity at this site are contami-nated by sediment effects for f ' D. Sensor calibration
1. Hydrophone
When the nominal transfer function of the H2O HDHhydrophone is adopted, spectral ratios of velocity to(scaled) pressure, drop precipitously for frequencies lessthan 1 Hz (Fig. 10). At 0.5 Hz, the observed spectralratios for both seismic sensors are more than 100 timessmaller than theory (Table II).There is more evidence that the nominal gain of the hy-drophone is too high at low frequencies. Velocity spectraat low frequencies have been calculated from ECMWF di-rectional wave models, and they are in reasonable agree-ment with the H2O seismic observations (Ref. 7, Fig.4). Pressure spectra calculated from the same modelsare orders of magnitude less than observations.In addition, the (scaled) pressure spectrum at 0.5 Hzvaries between -95 dB and -105 dB at H2O, dependingon overhead wind (e.g. Figs. 5, 6). Observations atthe Aloha Cabled Observatory, scaled similarly, rangebetween -113 and -128 dB, some 20 dB less.
2. Seismometers
The similarity between the spectra of all six seismicchannels at low frequency is in accord with the model(cf. panels AA in Figs. 5, 6, and Fig. 10) and indicatesrelative calibrations accurate to a dB (once spectra of theGeospace vertical have been lifted 3 dB). However, theAcoustic radiation and wave-wave interactions 13hase of the cross-spectrum between pressure and verti-cal velocity differs for the two instruments for frequenciesless than 1 Hz. Sediment effects preclude applying thesechecks at higher frequencies (e.g. panels BB in Figs. 5,6).
Acknowledgments
Walter Munk has been a constant inspiration, and wehave greatly benefited from innumerable conversationswith Chuck Spofford and Brian Sperry. We again ac-knowledge the extraordinary accomplishment of F. Duen-nebier and the entire H2O team for installation and oper-ation of the H2O system and thank the IRIS Data Man-agement Center for data curation. We are grateful toJean Bidlot at the European Center for Medium-RangeWeather Forecasts for several custom data sets extractedfrom the ECMWF database. This work was partiallysupported by the Office of Naval Research. H. Miche, “Movements ondulatoires de la mer en pro-fondeur constants ou decrissant”, Ann. des Fonts etChaussees , pages TBD (1944). M. S. Longuet-Higgins, “A theory of microseisms”, Philos.Trans. Roy. Soc. London , 1–35 (1950). K. Hasselmann, “A statistical analysis of the generation ofmicroseisms”, Rev. Geoph. , 177–210 (1963). L. M. Brekhovskikh, “Underwater sound waves generatedby surface waves in the ocean”, Izv. Acad. Sci. USSR, At-mos. Oceanic Phys. , 970–980 (1966). L. M. Brekhovskikh, “Generation of sound waves in a liq-uid by surface waves”, Sov. Phys. Acoust. , 323–350(1967). A. C. Kibblewhite and C. Y. Wu,
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