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.     

Rayleigh-wave amplitudes from earthquakes in the range 0–150°

Summary . Vertical component Rayleigh-wave amplitudes from 1461 shallow earthquakes recorded in the distance range 0–150° are analysed to separate the effects of earthquake size, epicentral distance (Δ) and recording station.
The estimated decay of amplitude with distance has the form of a theoretical curve for the decay of Rayleigh waves with distance if the assumption is made that the decay due to dispersion for the data analysed is that of an Airy phase. Writing the decay due to anelastic attenuation as exp (- k Δ), k is estimated to be 0.676/rad over the whole range of distance. If the distance effects are represented by a straight line of the form h log Δ+ constant, h is estimated to be 1.15. The calibration function for computing M _s derived from the estimated distance effects is very similar to that of Marshall & Basham.
Station effects on Rayleigh-wave amplitudes though statistically significant are small, and can probably be ignored in the computation of M _s.
Comparing the estimated surface-wave magnitudes (earthquake size) obtained in this study with the long and short period body-wave magnitudes ( m ^LP_b and m ^SP_b respectively) obtained by Booth, Marshall & Young for the same earthquake shows that m ^LP_b is about equal to M _s over the magnitude range of interest (˜4.0–7.0). The m ^LP_b and M_s relationship shows that the greater the long-period energy radiated by an earthquake the smaller proportionately is the short-period energy.     

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.     

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.     

Relocation of M ≥ 2 events of the 1989 Dôbi seismic sequence in Afar: evidence for earthquake migration

The Ethiopian side of central Afar was struck in August 1989 by the largest seismic sequence (three 6.1 ≤ M _s ≤ 6.3 events, 15 with M _s or m _b ≥ 5.0) since that of Serdo in 1969. Using the Djibouti seismological network, we relocated 297 of the events of that sequence. As most of the large events took place outside the network, we assessed the accuracy and stability of earthquake relocations by using three different velocity models and two relocation codes to try to relate individual shocks to distinct faults and surface breaks. A majority of the events apparently occurred underneath the floor of the Dôbi graben, an area about 45  km long and 15  km wide, rupturing boundary and inner floor faults, in agreement with the surface cracks and scarps that we mapped in the area. The relocation shows that the principal events propagated about 50  km northwestwards along the graben in the first 40  hr. A day and a half after the beginning of the sequence, smaller events ( M ≤ 4) started to propagate more than 55  km eastwards, towards Asal Lake. Using two three-component stations installed near the Ethiopian border, we could determine reliable depths for 21 events. The depths are compatible with a seismogenic crust about 14  km thick in the Dôbi and Hanle graben area. Although the Dôbi sequence ruptured about 50  km of the fault array extending from Serdo to Asal, where the regional stress was released by earthquakes in 1969 and 1978, respectively, a seismic gap about 50  km long still subsists along the northern part of the Gaggade region (Der'êla half-graben).     

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.     

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 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.     

Multifractal analysis of spatial distribution of microearthquakes in the Kanto region

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The spatial distribution of earthquakes is a fractal, which is characterized by a fractal dimension. However, if a spatial distribution has a heterogeneous fractal structure, a single value of fractal dimension [e.g. D_o (capacity dimension) or D ₂ (correlation dimension)] is not enough to characterize it. From a multifractal viewpoint, we analysed the spatial distribution of microearthquakes in the Kanto region by using a local density function. Generalized dimensions, D_q , of the spatial distribution were calculated from the slopes of generalized correlation integrals, C_q(r) versus distance r , on a log-log plot, examining the self-similarity of the spatial distribution of microearthquakes. Self-similar structures are held well at scales from 1.26 to 12.6 km. Our results suggest that the spatial distribution of microearthquakes in the Kanto region is not a homogeneous fractal structure but a heterogeneous one with generalized dimensions D₂ = 2.2 ≤ D ₃≤…≤ D _∞= 1.7. The value of D _∞, the lower limit of fractal dimension, is the fractal dimension of the most intensive clustering in the heterogeneous fractal set. The fractal dimension of the most intensive clustering of microearthquakes in the Kanto region is 1.7.     

Is the seismic moment–frequency relation universal?

We evaluate the seismic moment–frequency relation for the Harvard catalogue in the period 1977–1994. This catalogue is composed of about 12 000 earthquakes. After selection of events in terms of depth and energy, we retain about 8000 data points. We estimate two parameters of the seismic moment distribution: the power exponent β and the cut-off value M _m . The method used is a least-squares linear fit on a log–log scale performed over a range selected on the basis of the standard deviation from the histogram. The analysis is carried out for different subdivisions of the Earth in square grids of different sizes. Neither parameter exhibits a dependence on cell size, suggesting the universality of their values and the interpretation of the existence of a cut-off as a finite size effect linked to a finite catalogue length. The variations of the parameters are investigated as a function of time (duration of the catalogue) and versus the number of events used for building up the distribution. Again, β and M _m do not depend on time, but M _m depends on the number of events, reaching a stable value for N ≈ 1000. The only significant change in the parameters is observed for different values of M _0upper in the catalogue, revealing the existence of universality classes.     

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.     

The effect of load shape on the deflection of thin elastic plates

Summary. I derive formulae which permit the comparison of the amplitude and node offset of a distributed load with an equivalent point load on a one-dimensional elastic plate with one free end. The first and second moments of the load about the equivalent point load are the most important parameters affecting amplitude and node offset. When the equivalent point load is placed at the distributed load's centre of mass, we find approximately: Here A' and A are the amplitudes of the damped sinusoids which characterize the deflection of the distributed and point loads, Δ l is the distance between corresponding nodes of the deflection, k is the plate's characteristic wave- number, and m ₂ is the load's normalized second moment. The parameter s depends on the distance from the free end to the load, and varies from zero to one. When the load is near the free end of the plate, and the equivalent point load is placed at the free end, the corresponding formulae are: Here m ₁ is the load's normalized first moment about the free end of the plate.     

A comment on depths of sources of anomalies in the Earth's gravity field

Summary. One method to determine the depths of sources of anomalies in the Earth's gravity field is to plot log [(2 n + 1) σ _n ] versus n where σ _n is the n th term in the amplitude spectrum of the Earth's gravitational potential. This procedure assumes that the amplitude spectrum of the anomalous density variations does not vary with n. Such an assumption may not apply to the Earth.     

Palaeomagnetism of the Fongen-Hyllingen gabbro complex, southern Scandinavian Caledonides; plate rotation or polar shift?

Summary. Palaeomagnetic results are presented from the c . 160 km² Caledonian synorogenic layered Fongen-Hyllingen gabbro complex (of probable late Silurian age) located about 75 km SE of Trondheim, Norway, in the allochthonous Seve-Kdli Nappe Complex. A total of 80 oriented samples from eight sites in the northern part of the gabbro were investigated. After detailed af demagnetization two stable high coercivity components emerge: one with a well defined NW direction with D =325°, I =−21° (α₉₅=8°, N =8), and another, less well defined, probably younger, SW direction with D = 237°, I = 6° (α₉₅= 9°, N = 8). Correction for dip of these two directions gives D = 329°, I =−7° (α₉₅= 10°) and D = 238°, I =−11° (α₉₅= 12°), respectively. The corresponding pole positions are P ₁ : 19° N, 225° E and P ₂: 19° S, 308° E, respectively. The reversed pole -P ₂ of the SW direction lies close to other NW European palaeomagnetic poles of Caledonian, Upper Silurian-Lower Devonian age. However, the dominant pole PI is far away from these, and could be due to a late Caledonian geomagnetic excursion of considerable duration; or it could record a c . 90° rotation around a vertical axis of a crustal block within the Scandinavian Caledonides. Block rotation could have been related to nappe translation, although geological observations do not at present appear to support the occurrence of such an event.     

Multiple scattering and energy transfer of seismic waves – separation of scattering effect from intrinsic attenuation – I. Theoretical modelling

Summary. In order to separate the scattering effect from the intrinsic attenuation, we need a multiple scattering model for seismic wave propagation in random heterogeneous media. In this paper, we apply radiative transfer theory to seismic wave propagation and formulate in the frequency domain the energy density distribution in space for a point source. We consider the cases of isotropic scattering and strong forward scattering. Some numerical examples are shown. It is seen that the energy density–distance curves have quite different shapes depending on the values of medium seismic albedo B ₀=η_s/(η_s+η_a) where η_s is the scattering coefficient and η_a is the absorption coefficient of the medium. For a high albedo ( B > 0.5) medium, the energy–distance curve is of arch shape and the position of the peak is a function of the extinction coefficient of the medium η_e=η_s+η_a. Therefore it is possible to separate the scattering effect and the absorption based on the measured energy density distribution curves.     

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.     

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.     

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.     

On the nature of regional seismic phases–II. On the influence of structural barriers

The blockage of the L _g wave by crustal barriers such as continental margins and graben structures has long been recognized as providing a very useful tool for mapping large-scale lateral crustal variations along the propagation path. Numerical simulation of L _g -wave propagation in complex anelastic media using the pseudospectral method provides insight into the nature of the propagation process using both snapshots of the wavefield and synthetic seismograms. A variety of 2-D structures have been investigated, including the influence of sediments, crustal thickness and attenuation.
Thick sedimentary basins covering a graben structure can have a major influence, since they remove L _g energy by generating P conversion and scattering–the principal mechanisms for strong L _g attenuation across a graben. The reduction of the L _g energy is reinforced by anelastic attenuation in the sediments as well as the influence of the gradually thinning crustal waveguide associated with an elevated Moho.
The extinction of L _g in a sequence of explosions fired across the central graben of the North Sea can be simulated by numerical calculations for the structure derived from refraction experiments.     

Analysis of palaeomagnetic inclination data

Summary The analysis of palaeomagnetic data where only inclinations are available is considered. Maximum likelihood estimates for the mean inclination I ₀ and Fisher's precision parameter k are derived. It is shown that they are in all cases biased although the bias is small for low inclinations. The case of steep inclinations and small values of k is examined and it is shown that in this region I ₀ and k are not separable as distinct variables, because the lack of declination information in this region leads to fundamental ambiguities. Unbiased estimates for I ₀ and (1/ k ) are derived for the case where the portion of the distribution folded about the vertical is insubstantial. A worked example of the method, with calculation of confidence limits, is appended.