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.     

Exact and asymptotic synthetic multiplet spectra on an ellipsoidal earth

Summary. Exact spectra of a normal mode multiplet _nS_l or _n T_l on a non-rotating ellipsoidal earth are composed of l + 1 lines. Asymptotically, in the limit of large l , such spectra appear, however, as a single line or peak broadened by attenuation alone. The location within the multiplet of this single peak depends only upon the angle of inclination of the source-receiver great circle path to the axis of ellipsoidal symmetry. The appearance of a single peak is produced by the cancellation of nearby singlets. To assess the utility of the single peak approximation, exact and asymptotic synthetic spectra have been compared directly for a number of multiplets and for a variety of source-receiver configurations. Except in the immediate vicinity of the source and its antipode, the approximation appears to be satisfactory on an ellipsoidal earth down to about l ∼ 10. Additional studies will be required to determine the limits of validity on a laterally heterogeneous earth.     

The Influence of Surface Inhomogeneities On Deep Electromagnetic Soundings of the Earth

Summary. Asymptotic expressions for components of the electromagnetic field of a grounded electric dipole are considered for the model consisting of a thin surface-layer overlapping a stratified medium with a highly resistive screen on the roof. It is shown that the method of spatial derivatives makes it possible to obtain proper estimates of the impedance at distances of r ≥|λ₀| from the nearest edge of the surface anomaly (|λ₀| being the effective depth of the field penetration in the underlying section). the magnetotelluric methods allow one to obtain the true values of impedance, provided r ≥ max {|λ₀|, |/( S ⁻¹+ Z ⁻₀|^1/2} where S is the integrated conductivity of the surface layer, is the transverse resistance of the screen, and Z ⁻₀ is the Tikhonov—Cagniard impedance for the medium underlying the surface layer.     

The deformation and compaction of partial molten zones

Summary. The segregation of melt from a partially molten source region requires a corresponding deformation of the unmelted residue ('matrix'). The role of matrix deformation during melt segregation is examined using simple one-dimensional models, for which the deformation consists only of bulk compression or 'compaction'. In model I, a volume fraction φ₀ of ascending mantle material undergoes pressure-release melting at a depth z = 0 (localized melting). Compaction of the matrix occurs in a boundary layer whose thickness (reduced compaction length δ_R) is proportional to the square root of the matrix viscosity. In the Earth's mantle, δ_R∼ 10–100 m, indicating that compaction cannot be important over large distances. Model II examines the case in which melting occurs over a depth range of order h (distributed melting). In the limit h ≪δ_R, the solution is the same as for the case of localized melting, except in a 'melting layer' of thickness ∼ h near z = 0. In the more realistic limit h ≫δ_R, compaction makes a negligible contribution to the balance of forces associated with melt segregation. This result is also valid for the more general case of two-dimensional flow. Compaction is therefore likely to be of negligible importance in the Earth's mantle, with the consequence that melt segregation can be accurately described by Darcy's law.     

Rapid computation of magnetic anomalies with demagnetization included, for arbitrarily shaped magnetic bodies

Summary. The potential function ø for a magnetic body of susceptibility μ in a medium of susceptibility μ* satisfies the integral equation
Here Φ* is the potential function for the region without the heterogeneity and R is the distance from the point of observation to the point on the surface, s , of the body. δΦ /δn is the normal derivative, in the direction of the outward normal. The equation allows for the effects of demagnetization. For numerical purposes the surfaces can be divided into N facets over which δΦ/δ n is a constant. The unknown quantities δΦ/δn_j can be found from the system of equations defined by:
The prime on the summation sign denotes that the summation does not include the i th element. The magnetic field in the direction of the unit vector P( P ₁, P ₂, P₃ ) is given by:      

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.     

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.     

Estimation of the Bingham distribution function on nearly two-dimensional data sets

Summary. In cases where directional data, such as palaeomagnetic directions, lie nearly along a great circle, a good approximation to the maximum likelihood estimate of the intermediate concentration parameter k ₂ in the Bingham probability distribution is given by: 2( t ₂/ N ) – 1 = I ₁(1/2 k ₂)/ I ₀(1/2 k ₂), where t ₂ is the intermediate eigenvalue, N is the number of samples, and the I_i are the appropriate modified Bessel functions of the first kind. This estimate, the asymptotic limit as the smallest eigenvalue t ₁→ 0, corresponds to restricting all points to lie on a great circle. The limit is also useful as an endpoint for interpolation, especially since numerical calculation in this region is difficult.     

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.     

Average attenuation of 0.7–5.0 Hz Lg waves and magnitude scale determination for the region bounding the western branch of the East African Rift

The investigation of L g attenuation characteristics in the region bounding the western branch of the East African rift system using digital recordings from a seismic network located along the rift between Lake Rukwa and Lake Malawi is reported. A set of 24 recordings of L g waves from 12 regional earthquakes has been used for the determination of anelastic attenuation, Q _Lg , and regional body-wave magnitude, m _{b Lg} , scale. The events used have body-wave magnitudes, m _b , between 4.6 and 5.5, which have been determined teleseismically and listed in ISC bulletins. The data were time-domain displacement amplitudes measured at 10 different frequencies (0.7–5.0  Hz). Q _Lg and its frequency dependence, η , in the region can be represented in the form Q _Lg = (186.2 ± 6.5)  f  ^(0.78±0.05). This model is in agreement with models established in other active tectonic regions. The L g -wave-based magnitude formula for the region is given by m _{b Lg} = log   A + (3.76 ± 0.38)  log   D − (5.72 ± 1.06), where A is a half-peak-to-peak maximum amplitude of the 1  s L g wave amplitude in microns and D is the epicentral distance in kilometres. Magnitude results for the 12 regional earthquakes tested are in good agreement with the ISC body-wave magnitude scale.     

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.     

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.     

Estimation of uncertainties in eigenspectral estimates from decaying geophysical time series

Summary. The response of many dynamical systems to an impulse is a linear combination of decaying cosines. The frequencies of the cosines have generally been estimated in geophysics by periodogram analysis and little formal indication of uncertainty has been provided. This work presents an estimation procedure by the methods of complex demodulation and nonlinear regression that specifically incorporates in the basic model the decaying aspect of the cosines (periodogram analysis does not). The use of plots of the instantaneous phase as a function of time is shown to greatly enhance resolution. Expressions for the variances of eigenfrequencies, amplitudes, phases and damping constants Q are derived by non-linear least-squares. The results are illustrated, for the problem of the free oscillations of the Earth, by computations with the record made at Trieste of the Chilean earthquake of 1960 May 22. Sample values are periods and standard errors of 737.79 ± 0.13 s, 506.25 ± 0.13 s and 429.60 ± 0.14 s for ₀ T ₈, ₀ T ₁₃ and ₀ T ₁₆ with Q values and standard errors of 200 ± 14, 230 ± 28 and 215 ± 30, respectively.     

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.     

A normal approximation to the Fisher distribution

Summary. A simple normal approximation is given for the joint probability density function of the polar coordinates (θ, φ) of a random vector following the Fisher distribution with arbitrary mean direction (θ₀, φ₀). Two important applications of this approximation are discussed, and conditions for the validity of the approximation are investigated and summarized.     

The solution of the inverse gravity potential problem for cylindrical bodies

Summary. The inverse gravity potential problem consists in the determination of the form and the density of the body by its exterior gravity potential. We describe two similar classes of bodies for which this problem has a unique constructive solution.
(1) The first class contains the cylindrical bodies with finite length, arbitrary form of section and ρ( R , ø, z) =ρ₁( z )ρ₂( R , ø) density distribution, where z is the cylindrical coordinate; R , ø are the polar coordinates in a section plane. This class is important for prospecting geophysics in that it allows us to determine in a unique and constructive way, the function ρ₁( R , ø), the length, form and orientation of the cylinder if we know the function ρ₁( z ) and the exterior potential. The classical moment problem of functions is the basis for the solution of this problem.
(2) The analogous problem for the class of the spherical cylinders, or bodies bounded by arbitrary similar sections of two different concentric spheres and the radial lateral surface, appears when bodies of planetary size are studied. (An example of these bodies would be the Moon mascons.) The density distribution of these cylinders is ρ(τ, θ, ø) =ρ₁(τ)ρ₂(θ, ø) where τ, θ, ø are the spherical coordinates. The function ρ₁(θ, ø), length and form of spherical sections can be uniquely determined by exterior potential if we know the function ρ₁(τ). We propose a new constructive method for harmonic continuation of the gravity potential into the region containing the perturbing masses for the solution of the problem.     

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.     

Odd zonal harmonics in the geopotential, from analysis of 28 satellite orbits

Summary. The geopotential is usually expressed as an infinite series of spherical harmonics, and the odd zonal harmonics are the terms independent of longitude and antisymmetric about the equator: they define the 'pear-shape' effect. The coefficients J ₃, J ₅, J ₇, … of these harmonics have been evaluated by analysing the variations in eccentricity of 28 satellite orbits from near-equatorial to polar. Most of the orbits from our previous determination in 1974 are used again, but three new orbits are added, including two at inclinations between 62° and 63°, which have been specially observed for more than five years by the Hewitt cameras. With the help of the new orbits and revised theory, we have obtained sets of J -coefficients with standard deviations about 40 per cent lower than before. A 9-coefficient set is chosen as representative, and is as follows (all × 10⁹): J ₃=– 2530 ± 4, J ₅=–245 ± 5, J ₇=–336 ± 6, J ₉=–90 ± 7, J ₁₁= 159 ± 9, J ₁₃=–158 ± 15, J ₁₅=– 20 ± 15, J ₁₇=– 236 ± 14, J ₁₉=– 27 ± 19. With this set of values, the pear-shape asymmetry of the geoid (north polar minus south polar radius) amounts to 45.1 m instead of the previous 44.7 m. The accuracy of the longitude-averaged geoid profile is estimated as 50 cm, except at latitudes above 86°. The geoid profile and predicted amplitude of the oscillation in eccentricity are compared with those from other sources.     

Focal mechanisms of recent earthquakes in the Southern Korean Peninsula

We evaluate the stress field in and around the southern Korean Peninsula with focal mechanism solutions, using the data collected from 71 earthquakes ( M_L = 1.9–5.2) between 1999 and 2004. For this, the hypocentres were relocated and well-constrained fault plane solutions were obtained from the data set of 1270 clear P -wave polarities and 46 SH / P amplitude ratios. The focal mechanism solutions indicate that the prevailing faulting types in South Korea are strike-slip-dominant-oblique-slip faultings with minor reverse-slip component. The maximum principal stresses (σ₁) estimated from fault-slip inversion analysis of the focal mechanism solutions show a similar orientation with E–W trend (269°–275°) and low-angle plunge (10°–25°) for all tectonic provinces in South Korea, consistent with the E–W trending maximum horizontal stress (σ_Hmax) of the Amurian microplate reported from in situ stress measurements and earthquake focal mechanisms. The directions of the intermediate (σ₂) and minimum (σ₃) principal stresses of the Gyeongsang Basin are, however, about 90 deg off from those of the other tectonic provinces on a common σ₂–σ₃ plane, suggesting a permutation of σ₂ and σ₃. Our results incorporated with those from the kinematic studies of the Quaternary faults imply that NNW- to NE-striking faults (dextral strike-slip or oblique-slip with a reverse-slip component) are highly likely to generate earthquakes in South Korea.