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A List of Symbols

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A List of Symbols A List of Symbols The following list of mathematical symbols includes most, but not all of those used in the main text and appendices. The arguments of functions are not shown in this list, as different arguments are used in different contexts. In many cases, the same symbol is used for more than one parameter, but only if the parameter appears in well-separated portions of the monograph. In other cases, the same parameter is represented by two different symbols, adhering to differences in acoustic and geoacoustic notation. Definitions are not given here for some of the symbols that appear only once, in which case the reader should be able to find the definition in nearby text. ± A A diagonal matrix with elements taken from Aα A Area A˚ Angstroms (10−10 m) Aall Generalization of Aww to the poroelastic case Awp Analogous to Aww, but for scattering into sediment Aww Function appearing in perturbation and small-slope approximations ± Aα A transform used in perturbation theory a Pore size, radius, coefficient in modified Archie’s law, parameter in bulk density profile function ap Complex ratio of compressional wave speed to water sound speed aq Complex ratio of wave speed to water sound speed for wave type q at Complex ratio of shear wave speed to water sound speed aα Amplitude of zeroth-order compressional wave aρ Ratio of sediment and water mass densities Bs Interface slope covariance matrix B Interface relief covariance, matrix in poroelastic roughness scattering BPP Covariance of random, complex, pressure time series Bα A wave vector function used in perturbation theory Bαβ Volume fluctuation covariance B˜αβ Estimated volume fluctuation covariance b Azimuthally averaged directivity bi Source complex directivity function in wave vector space br Receiver complex directivity function bx Source complex directivity function C Modulus in Biot theory, a factor in the T-matrix second moment Ch Structure constant for interface relief C0 Gas thermal conductivity cp Speed (complex) for compressional wave cq Speed (complex) for wave type q cqphase Phase speed (real) for wave type q ct Speed (complex) for shear wave cw Sound speed in water c0 Wood’s value for compressional wave speed, reference wave speed D Two-dimensional difference coordinate, a matrix appearing in perturbation theory D Matrix appearing in model for volume scattering in elastic media, magnitude of D, water depth Db Bioturbation mixing coefficient (diffusivity) Dmn Elements of matrix, D Dr Receiver directivity function in dB, relative density of sediment Dx Source directivity function in dB D1 A function appearing in volume perturbation theory D2 A function appearing in volume perturbation theory d Transducer aperture size, cross-range, sediment grain diameter, dimension of ensonified region, inner diameter of core liner, depth below water–sediment interface di Range from source to scattering patch dn Thickness of nth seafloor layer ds Range from scattering patch to receiver d0 Wave orbital diameter at the seafloor E A diagonal matrix appearing in perturbation theory E(n) Rectangular matrices appearing in poroelastic scattering theory ± eh Horizontal polarization vector for shear wave ± eq Unit vector in direction of propagation for wave type q ± ev Vertical polarization vector for shear wave ex Unit vector in x-direction (similarly for y and z) e Sediment void ratio (not to be confused with base of natural logarithms), specific modulus F Fourier transform of f Ft Proportional to Fourier transform of ∇×f F Flow correction factor in Biot theory, Fourier transform of interface relief function (f), formation factor, spatial frequency n Fn Fourier transform of f Fp Proportional to Fourier transform of ∇·f f Force per unit volume, first-order vector source term f Frequency, seafloor interface relief function fc Biot’s characteristic frequency fi Components of f Gpw Green’s function, fluid sediment to water Gwp Green’s function, water to fluid sediment Gww Green’s function, water to water Gα Green’s functions for unbounded, homogeneous elastic medium G0 Green’s function for unbounded, homogeneous fluid g Acceleration of gravity, dipole Green’s function, parameter in Crowther’s false-alarm model H Modulus in Biot theory, transfer function, height, hydraulic tortuosity Hp Fourier transform of material impulse response function Ht Fourier transform of material impulse response function H Real part of transfer function H Imaginary part of transfer function h RMS interface roughness, impulse response, specific density hp Material impulse response function ht Material impulse response function h0 Ripple RMS relief, ripple peak-to-trough amplitude h Inverse Fourier transform of H h Inverse Fourier transform of iH I Intensity I Magnitude of I, electric current IK Kirchhoff integral IOI Index of impedance J Electrical current density K Two-dimensional horizontal wave vector Ki Two-dimensional horizontal incident wave vector Ks Two-dimensional horizontal scattered wave vector K0 Two-dimensional horizontal wave vector for ripple K Magnitude of K, hydraulic conductivity KG Graphic kurtosis Kb Bulk modulus Kb Real part of bulk modulus Kb Negative of imaginary part of bulk modulus Keff Effective modulus Kf Bulk modulus of (drained) sediment frame Kg Bulk modulus of sediment grain material Ki Magnitude of Ki Ks Magnitude of Ks Kw Bulk modulus of water Kx x-component of K (similarly for Ky) K0 Magnitude of K0, coefficient of earth pressure, spectral cutoff wave number k Three-dimensional wave vector ki Incident three-dimensional wave vector kp Three-dimensional compressional wave vector kpi Incident, down-going compressional wave vector (α) kpi Incident, up- and down-going compressional wave vectors kps Scattered, down-going compressional wave vector (α) kps Scattered, up- and down-going compressional wave vectors ± kq Up-going and down-going wave vectors for wave type q kr Reflected three-dimensional wave vector ks Scattered three-dimensional wave vector kt Three-dimensional shear wave vector k Wavenumber, attenuation factor (dB m−1 kHz−1) kb Thermal conductivity kp Compressional wavenumber kp Real part of compressional wavenumber kp Imaginary part of compressional wavenumber kq Wavenumber for wave type q ks Shear wave attenuation factor, sediment thermal conductivity kt Shear wavenumber kw Acoustic wavenumber in water L Bottom loss in dB, correlation length of interface roughness, length parameter in Crowther’s false-alarm model, length Lb Depth of mixing by benthic fauna Lc Low-frequency cutoff length in spectrum, correlation length, cross-range dimension of ensonified region Ld Down-range dimension of ensonified region Lf Correlation length of surface field M A matrix appearing in perturbation theory M Modulus in Biot theory, mass of sediment sample Md Graphic median grain size Mz Graphic mean grain size m Exponent in Archie’s law, strain-hardening index N Normal to rough interface (not a unit vector) N Bubble size distribution Nj Components of N n Unit normal to rough interface n Refractive index, strain-hardening index n Real part of n n Imaginary part of n P Acoustic pressure field (complex amplitude), absolute water pressure (including atmospheric pressure), normalized vertical wave vector component PA Atmospheric pressure Peff Effective pressure Pf Pore fluid pressure Pfa Probability of false alarm Pi Incident pressure (complex amplitude) Pm Probability associated with mth component in Rayleigh-mixture PDF Pr Reflected pressure (complex amplitude) Ps Scattered pressure (complex amplitude) Psurf Effective surface field P0 Hydrostatic pressure, zeroth-order acoustic pressure field, amplitude of plane pressure wave, vertical component of overburden pressure P1 First-order acoustic pressure field P+ Upward-diffracted acoustic pressure field P− Downward-diffracted acoustic pressure field P (n) Square matrices appearing in poroelastic scattering theory PL Propagation loss p Time-domain pressure field, a function appearing in the volume perturbation approximation, probability, probability parameter in Crowther’s false-alarm model pd Probability density for natural logarithm of (real) envelope pref Reference pressure (usually 1 μPa) ps Probability density for scattering cross section px Probability density for (real) envelope py Probability density for squared envelope pys Joint probability density for squared envelope and scattering cross section pβ Probability density appearing in Crowther’s false-alarm model Q Heat flux, a matrix appearing in small-roughness perturbation theory Q(n) Column matrices appearing in poroelastic scattering theory Qp Quality factor for compressional wave Qq Quality factor for wave type q R Two-dimensional horizontal position vector R Two-dimensional horizontal position vector Ri Horizontal coordinates of source Rs Horizontal coordinates of receiver R Magnitude of R R Magnitude of R RS Receiver sensitivity r Three-dimensional position vector r Three-dimensional position vector ri Position vector for source rs Position vector for receiver r Magnitude of r, range rref Reference length (usually 1 m) rs Range from scattering patch (at origin) to receiver rt Range corresponding to travel time, t r1 Range from source to reflection point r2 Range from reflection point to receiver r2 Coefficient of determination S Structure function for interface relief, first-order scalar source term, salinity, degree of sediment saturation Sb Bottom scattering strength SkI Inclusive graphic skewness Sp Specific heat at constant pressure, pore water salinity SL Source level s Interface slope (vector or column matrix), a column matrix
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