A uniformly valid asymptotic representation of normal mode multiplet spectra on a laterally heterogeneous Earth

Summary. A new asymptotic formula is obtained for the spectrum of an isolated normal mode multiplet _nS_l or _nT_l , with n ≪ l , on a laterally heterogeneous Earth. The principal feature of this formula is that it is uniformly valid on the Earth's surface, including near the epicentre and its antipode. The formal conditions for its validity are that | δm / m ₀|≪ 1 and s _max≪ l ≪ s _min| δm / m ₀|^–1, where | δm / m ₀| is the relative magnitude of the lateral heterogeneity, and s _min and s _max are the minimum and maximum significant degrees in its spherical harmonic expansion. As well as providing a basis for the geographical interpretation of near-epicentral or near-antipodal long-period recordings, the new formula also unifies the asymptotic theory and adds insight into the phenomena which govern the details of multiplet spectra in general.     

Oxygen consumption in Macrotrachela musculosa and Trichotria truncata (Rotatoria) from the High Arctic

Oxygen consumption by rotifers Macrotrachela musculosa and Trichotria truncata from Spitsbergen tundra (77°N) was measured using the method of Cartesian divers. The metabolic rate of M. musculosa was: 0.205 10⁻³mm³ 0₂ per g 10⁻⁶ per hour at 2°C, 0.201 10⁻⁶mm³ at 6°C and 0.616 10⁻³mm³ 0₂ per g 10⁻⁶ per hour at 10°C. The metabolic rate of Trichotria truncata at 6° was 0.103 10⁻³mm³ per g 10⁻⁶ per hour. The relation between body weight and oxygen consumption by M. musculosa at 2°C is expressed with the equation R = 0.18W^0.67, with R – oxygen consumption in mm³10⁻³ per individual per hour and W – wet weight of an animal in g 10⁻⁶.     

The attenuation mechanism of seismic waves in northwestern Himalayas

We analysed local earthquake waveforms recorded on a broad-band seismic network in northwestern Himalayas to compute the intrinsic and scattered attenuation parameters from coda waves. Similar to other tectonically active and heterogeneous regions, attenuation-frequency relation for western Himalaya is   Q ⁻¹ _c = (113 ± 7)  f ^(1.01±0.05)  where   Q_c   is the coda Q parameter. Intrinsic  ( Q ⁻¹ _i )  and scattering  ( Q ⁻¹ _s )  attenuations was separated using   Q_c   and direct S -wave Q data  ( Q_d )  . It is observed that estimated   Q ⁻¹ _c   is close to   Q ⁻¹ _i   and both of them are much larger than   Q ⁻¹ _s   suggesting that coda decay is predominantly caused by intrinsic attenuation. At higher frequencies, both the attenuation parameters   Q_c   and,   Q_d   are similar indicating that coda is predominantly composed of back-scattered S waves at these frequencies.     

The traveltime perturbations for seismic body waves in factorized anisotropic inhomogeneous media

The traveltime perturbation equations for the quasi-compressional and the two quasi-shear waves propagating in a factorized anisotropic inhomogeneous (FAI) media are derived. The concept of FAI media simplifies considerably these equations. In the FAI medium, the density normalized elastic parameters a _ijkl ( X _i ) can be described by the relation a _ijkl ( X _i) = f ²( x _i ) A _ijkl, where A _ijkl are constants, independent of coordinates x _i and f ²( x _i) is a continuous smooth function of x _i . The types of anisotropy ( A _ijkl ) and inhomogeneity [ f ( x _i)] are not restricted. The traveltime perturbations of individual seismic body waves ( q ^P , qS 1 and qS 2) propagating in the FAI medium depend, of course, both on the structural pertubations [δ f ²( x _i)] and on the anisotropy perturbations (δ A _ijkl ), but both these effects are fully separated. The perturbation equations for the time delay between the two qS -waves propagating in the FAI medium are simplified even more. If the unperturbed (background) medium is isotropic, the perturbation of the time delay does not depend on the structural perturbations (δ f ²( x _i) at all. This striking result, valid of course only in the framework of first-order perturbation theory, will simplify considerably the interpretation of the time delay between the two split qS -waves in inhomogeneous anisotropic media. Numerical examples are presented.     

Stacking for the frequencies and Qs of 0S0 and 1S0

Summary. Using nine IDA records for the Indonesian earthquake of 1977 August 19, we have formed an optimal linear combination of the records and have measured the frequency and Q of ₀ S ₀ and ₁ S ₀. The frequency was measured using the moment ratio method. The attenuation was measured by the minimum width method and by the time-lapse method. The frequency and attenuation were measured simultaneously by varying them to obtain a best fit to the data. A 2000-hr stack, the sum of nine individual records, for ₀ S ₀ gave a frequency of 0.814664 mHz±4 ppm. The values for the Q of ₀ S ₀ for the three different methods of measurement were 5600,5833 and 5700, respectively. The error in the estimates of Q ^-1 is about 5 per cent for the minimum power method. For ₁ S ₀ a 300-hr stack yielded a frequency of 1.63151 mHz±30 ppm. The values of Q for this mode were 1960, 1800 and 1850, respectively, with an error in Q ^-1 of about 12 per cent for the minimum power method.     

Approaches to Modelling the Surface Albedo of a High Arctic Glacier

Broadband surface albedo measurements, made during the summer melt season at three weather stations on John Evans Glacier (79°40 ' N, 74°00 ' W), varied strongly with the solar zenith angle ( θ _z ). Tests were carried out to assess the impact of diurnal variations in surface albedo on seasonal net shortwave radiation ( K ^* ) totals. Removing the diurnal signal from albedo measurements by daily averaging of hourly measurements, or by applying midday measurements to all hours of the day, changed K ^* by up to 16%. Ignoring measurements made at θ _z & 75°, to account for measurement (cosine) error at high θ _z , decreased K ^* by between 5 and 18%. Given the sensitivity of K ^* to diurnal patterns in surface albedo, experiments were carried out with two albedo models. One model accounted for albedo variations with θ _z and one did not. The model driven by θ _z , when implemented within a surface energy balance model for John Evans Glacier, produced better melt estimates. This suggests that diurnal variations in surface albedo should be accounted for in energy balance models of glacier melt.     

A note on the surface volume change of shallow earthquakes

Summary. This note presents an exact analytical formula for determining the magnitude of coseismic surface volume change (δ V ) of earthquake faults in a half-space. For a Poisson solid, the formula is remarkably simple; δ V = M _zz |8μ, where M _zz is one of the moment tensor elements of the source. Maximum δ V values derive from dip slip on faults plunging 45°. For these events, surface volume changes of 0.0001 and 4.3 km³ can be expected for magnitude 5 and 8 earthquakes respectively. All of the coseismic surface volume change is recovered in the interseismic period through relaxation of the Earth and rebound of the surface. A useful rule of thumb for estimating the magnitude of vertical rebound in 45° dip slip events is δ h _p=Δ s /24, where Δ s is the coseismic slip on the fault.     

Note on inversion of reflection data for layered media

Summary It is pointed out that the data on which the Marchenko inversion of the second-order equation for fluid media reconstructs the impedance are not all independent. In particular, the derivative Z' ₀ of the surface impedance is implied by the displacement velocity response at the surface. We also note that unless Z' ₀= 0 the zero-frequency reflection amplitude must always equal — 1.     

Explicit, approximate expressions for the resolution and a posteriori covariance of massive tomographic systems

We present an approximate method to estimate the resolution, covariance and correlation matrix for linear tomographic systems Ax = b that are too large to be solved by singular value decomposition. An explicit expression for the approximate inverse matrix A ⁻ is found using one-step backprojections on the Penrose condition AA ⁻≈ I , from which we calculate the statistical properties of the solution. The computation of A ⁻ can easily be parallelized, each column being constructed independently.
The method is validated on small systems for which the exact covariance can still be computed with singular value decomposition. Though A ⁻ is not accurate enough to actually compute the solution x , the qualitative agreement obtained for resolution and covariance is sufficient for many purposes, such as rough assessment of model precision or the reparametrization of the model by the grouping of correlating parameters. We present an example for the computation of the complete covariance matrix of a very large (69 043 × 9610) system with 5.9 × 10⁶ non-zero elements in A . Computation time is proportional to the number of non-zero elements in A . If the correlation matrix is computed for the purpose of reparametrization by combining highly correlating unknowns x _i , a further gain in efficiency can be obtained by neglecting the small elements in A , but a more accurate estimation of the correlation requires a full treatment of even the smaller A _ij . We finally develop a formalism to compute a damped version of A ⁻.     

Run-up from impact tsunami

We report on calculations of the on-shore run-up of waves that might be generated by the impact of subkilometre asteroids into the deep ocean. The calculations were done with the COULWAVE code, which models the propagation and shore-interaction of non-linear moderate- to long-wavelength waves  ( kh < π)  using the extended Boussinesq approximation. We carried out run-up calculations for several different situations: (1) laboratory-scale monochromatic wave trains onto simple slopes; (2) 10–100 m monochromatic wave trains onto simple slopes; (3) 10–100 m monochromatic wave trains onto a compound slope representing a typical bathymetric profile of the Pacific coast of North America; (4) time-variable scaled trains generated by the collapse of an impact cavity in deep water onto simple slopes and (5) full-amplitude trains onto the Pacific coast profile. For the last case, we also investigated the effects of bottom friction on the run-up. For all cases, we compare our results with the so-called 'Irribaren scaling': The relative run-up   R / H ₀=ξ= s ( H ₀/ L ₀)^−1/2  , where the run-up is   R , H ₀  is the deep-water waveheight, L ₀ is the deep-water wavelength, s is the slope and ξ is a dimensionless quantity known as the Irribaren number. Our results suggest that Irribaren scaling breaks down for shallow slopes   s ≤ 0.01  when  ξ < 0.1 − 0.2  , below which   R / H ₀  is approximately constant. This regime corresponds to steep waves and very shallow slopes, which are the most relevant for impact tsunami, but also the most difficult to access experimentally.     

Regional variations of heat flow differences with depth in Alberta, Canada

Summary. Differences between estimated average heat flow values for the Mesozoic and Cenozoic formations ( Q ₁) and estimated average heat flow values for the Palaeozoic formations below the erosional unconformity ( Q ₂) are calculated for the Alberta part of the western Canadian sedimentary basin. Significant heat flow differences exist for these two intervals and the map of Δ Q = Q ₁– Q ₂ shows that Q ₂ is generally greater than Q ₁ in the western and south-western part of Alberta, while in the northern part of the province Q ₂ is generally less than Q ₁. The regional variations of Δ Q are large, with standard deviation of 26 mW m⁻² and average value –13.5 mW m⁻². A regional trend of Δ Q correlates with topographic relief and the hydraulic head variations in the basin. It is shown that there is a heat flow increase with depth in water recharge areas and a decrease in heat flow with depth in the low topographic elevation water discharge areas when comparing the average heat flow in Mesozoic + Cenozoic and Palaeozoic formations.     

Some comments on the descriptions of the polarization states of waves

Summary. Procedures are developed for specifying the polarization characteristics of n -dimensional waves, and in particular three-dimensional waves of geophysical interest. We show that when a wave is in a pure state or is totally polarized, all the polarization information can be represented by a single vector u in an n -dimensional unitary space. Simple measures of the degree of polarization of the wave are constructed from the characteristic equation of the spectral matrix S . These measures are functions only of the scalar invariants of S and consequently S need not be diagonalized. If S represents a purely polarized wave, the unitary vector which contains the polarization information about the wave can be obtained directly from S using any 2 n – 1 equations of n ² possible equations. By multiplying by a phase-factor this unitary vector can be written in the form u = r ₁+ i r ₂ where r ₁ and r ₂ are orthogonal vectors in a real space. For an elliptically polarized wave, r ₁ and r ₂ locate the major and minor axes of the ellipse, and the ellipticity is given by the ratio of their magnitudes. The polarization parameters of ULF magnetic waves at the Earth's surface are computed from one set of five equations ( n = 3) and compared with parameters calculated using established techniques.     

Magnetotelluric response of a uniformly stratified earth containing a magnetized layer

Summary The magnetotelluric (MT) response is studied of a uniformly stratified earth which contains a magnetized layer. The impedance as a function of the layer parameters (resistivity, ρ permeability, μ and thickness h ) is discussed. The MT response from a layer (μ, ρ, h ) is equivalent to that from a layer (μ/μ_r=μ₀, μ_rρ, μ_r h ) where μ_r is the relative permeability of the layer. Thus the effect of a magnetized layer is to make it apppear μ_r times more resistive and μ_r times thicker than an unmagnetized equivalent layer. Master curves of apparent resistivity and phase are computed for three-layer models with varying permeability associated with varying resistivity in each layer. An example of MT field data is presented in which the most reasonable interpretation is that a magnetized layer exists beneath the observatory site.     

Seismic sources with observable glut moments of spatial degree two

Let ζ_Λ and r _Λ. be the hypocentral position and time of an extended indigenous seismic source. Backus showed that the force moment tensors of the source, Γ^{( m +1, n )}(ζ_Λ, r _Λ), determine and are determined by the motion which the source produces. For small m + n , only the long-period motion is relevant. The glut moment tensor Λ^{( m,n )} (ζ_Λ, r _Λ.) can be calculated uniquely from γ^{( m +1, n )}(ζ_Λ r _Λ) only if m = 0 or m = 1. The tensor G =Λ^(2,0)(ζ_Λ) gives the spatial variance tensor W_Λ of the source, and W_Λ. roughly describes the size, shape and orientation of the source region. Therefore the failure of the observed F =Γ^(3,0)(ζ_Λ) to determine G uniquely is of seismological interest. In the present paper we show that F determines G uniquely if we assume the source to be a simple straight line source (SSLS) or an ideal fault in an isotropic medium with isotropic prestress (IFIMIP). We give tests on F which determine whether it can come from a SSLS, from an IFIMIP or from a simple plane surface source (SPSS). If we assume the source to be a SPSS then knowing F and the fault plane determines G to within an unknown scalar multiple of a certain tensor tangent to the fault plane. Moreover F determines the fault plane uniquely unless F can come from a SSLS. If it can, then F determines this virtual source line uniquely, and F permits the fault plane to be any plane containing the virtual source line.     

On the linear relation between mb and Ms for discrimination between explosions and earthquakes

Summary. The statistical capability of the m _b: M _s discriminant for the discrimination of earthquake and explosion populations is examined by application of discriminant functions to a group of 83 explosions and 72 earthquakes in Eurasia. Equations are derived for the probability that an event is an earthquake or an explosion. The positive sign of DIS in the decision index equation, DIS _i = 34.3383 – 11.9569 mb _t + 7.1161 M _si , indicates that the event i is an earthquake. Its negative sign indicates that event i is an explosion. The probability of correct classification for an event, P _i , is related to its DIS _i value, by P _i = [1-exp (DIS _i )]⁻¹, where a large, positive DIS indicates a high probability that an event is an earthquake and a large, negative DIS indicates a high probability that an event is an explosion. The discrimination line M _s = 1.680 m _b– 4.825, or m _b= 0.595 M _s+ 2.872 very successfully separates the explosion population from the earthquake population. The points on this line have an equal chance of being an earthquake or an explosion; moreover, for any event, the distance parallel to the M _s-axis from the point representing that event in the m _b: M _s plane to this line is a measure of the probability for the correct classification of that event.     

Earthquake intensity attenuation pattern in the United States

Summary. Attenuation of earthquake intensities with epicentral distance was studied by analysing the intensity data for 39 earthquakes in the United States. Attenuation of MM intensity ( I ) with distance (Δ km) follows a simple relation of the type log I = log I ₀ - m Δ, where I ₀ is the intensity at the epicentre and m is a constant. Slope m is found to be inversely proportional to the square of the focal depth. Intensity attenuation pattern in the United States in general can be represented by a unified relation I/I ₀= exp [-(0.8999/ h ²+ 0.0014)Δ] where 16km ≤ h ≤ 60km. Intensities were calculated with the help of this equation and a good agreement with the observed intensities were found. A comparative study has also been made between the attenuation relations applicable to India and the United States.     

Surface motion of a fluid planet induced by impacts

In order to approximate the free-surface motion of an Earth-sized planet subjected to a giant impact, we have described the excitation of body and surface waves in a spherical compressible fluid planet without gravity or intrinsic material attenuation for a buried explosion source. Using the mode summation method, we obtained an analytical solution for the surface motion of the fluid planet in terms of an infinite series involving the products of spherical Bessel functions and Legendre polynomials. We established a closed form expression for the mode summation excitation coefficient for a spherical buried explosion source, and then calculated the surface motion for different spherical explosion source radii a (for cases of   a / R = 0.001  to 0.035, R is the radius of the Earth) We also studied the effect of placing the explosion source at different radii r ₀ (for cases of   r ₀/ R = 0.90  to 0.96) from the centre of the planet. The amplitude of the quasi-surface waves depends substantially on a / R , and slightly on   r ₀/ R   . For example, in our base-line case,   a / R = 0.03, r ₀/ R = 0.96  , the free-surface velocity above the source is 0.26 c , whereas antipodal to the source, the peak free surface velocity is 0.19 c . Here c is the acoustic velocity of the fluid planet. These results can then be applied to studies of atmosphere erosion via blow-off caused by asteroid impacts.     

The tidal dynamics of the Irish and Celtic Seas

Summary. Current meter data collected over periods of more than 14 day from the Irish and Celtic Seas are harmonically analysed and presented in maps of tidal stream information. Making use of the analysed current data, and by constructing time series of frictional and inertial stresses which are also harmonically analysed, harmonic constituents of the surface tidal slopes at current meter stations are obtained. Using these with data collected from offshore tide gauges, and in conjunction with coastal tide data, cotidal maps are drawn with some confidence for M ₂, S ₂, O ₁ and K ₁, the M ₂ chart resolving the discrepancy which exists between the different charts of the Celtic Sea already produced. Cotidal maps for M ₃ and M ₄ are also presented.
The mean over a tidal cycle of the energy flux for M ₂, S ₂ and O ₁ is also presented in the form of the total energy flux in these constituents which crosses different sectional lines. A flux of 44 × 10⁶ kW is observed to enter the Celtic Sea from the Atlantic and this is compared with previous estimates. An energy budget is also performed for M ₂, including all the effects of astronomical forcing and Earth tides to enable comparison to be made between the true energy inflow and the estimated frictional dissipation. Finally, comparison is made between the mean of the instantaneous energy flux and the sum of the energy fluxes associated with the major harmonics.     

Lateral variation and frequency dependence of coda-Q in the southern part of Iberia

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A method based on the coda attenuation law: Q = Q ₀( f/f ₀)^v leads to the determination of the lateral variation of coda- Q in the southern part of the Iberian Peninsula using seismograms belonging to the seismological network of the Cartuja Observatory, located in Granada. The lateral variation of Q ₀ ( Q value corresponding to a reference frequency f ₀ of 1 Hz) and its frequency dependence for the 1 to 5 Hz frequency range are, in general, in agreement with coda- Q values for frequencies less than about 1 Hz, previously determined in the region under study.
To determine the coda- Q values analytical functions have been used to fit the magnification curves of the vertical component short-period seismographs belonging to the Cartuja network. The problem is solved by using least-squares techniques and non-linear inversion. The determined coda- Q ₀ values and its frequency dependence correlate well with several known geophysical parameters in the southern part of the Iberian Peninsula.