INDUCTIVE SOUNDING OF A STRATIFIED EARTH WITH TRANSITION LAYER RESTING ON DIPPING ANISOTROPIC BEDS *

The generalised three layer boundary value problem with a transition layer sand-witched between an isotropic overburden and dipping anisotropic substratum is discussed assuming that plane electro-magnetic waves are incident normally over the air-earth inter-face. The tangential electric (E_y) and magnetic (H_x) fields and the expression for surface-impedance (E_y/H_x) have been evaluated at the earth's surface. Through numerical analysis it is shown that changes in the values of the parameters m (coefficient of anisotropy), h (thickness of the transition layer), α (angle of inclination of the dipping beds), and b (conductivity ratio between substratum and upper layer) modify the amplitude and phase-variation curves (with skindepth) significantly.     

THE USE OF SEISMIC SHEAR WAVES AND COMPRESSIONAL WAVES FOR LITHOLOGICAL PROBLEMS OF SHALLOW SEDIMENTS*

From a great variety of in situ shear wave experiments, i.e., reflection, refraction and borehole surveys in the shallow sediments of the north German plains, several specific properties have been derived. Shear waves (S) differ from compressional waves (P) in that:

• 1 they are not affected by the degree of water saturation. Thus, they provide a better correlation between the velocity V_s and (solid) lithology;
• 2 they generally have lower frequencies, but shorter wavelength and, hence, a better resolution of thin layers;
• 3 they have lower absorption Q_s^?1 and hence a better penetration in partially saturated and gas-containing sediments than P-waves.

Correlations have been established between V_s and the confining pressure and between reduced V_s values and several lithological parameters like the grain size of sandy material. More lithological and hydrological information is obtained by using S- and P-wave surveys along the same profile. The best information on a sedimentological structure is obtained by the simultaneous observation of V_s, V_p, Q_s and Q_p.     

A multi‐azimuth seismic refraction study for a horizontal transverse isotropic medium: physical modelling results

To investigate the characteristics of the anisotropic stratum, a multi‐azimuth seismic refraction technique is proposed in this study since the travel time anomaly of the refraction wave induced by this anisotropic stratum will be large for a far offset receiver. To simplify the problem, a two‐layer (isotropy–horizontal transverse isotropy) model is considered. A new travel time equation of the refracted P‐wave propagation in this two‐layer model is derived, which is the function of the phase and group velocities of the horizontal transverse isotropic stratum. In addition, the measured refraction wave velocity in the physical model experiment is the group velocity. The isotropic intercept time equation of a refraction wave can be directly used to estimate the thickness of the top (isotropic) layer of the two‐layer model because the contrast between the phase and group velocities of the horizontal transverse isotropic medium is seldom greater than 10% in the Earth. If the contrast between the phase and group velocities of an anisotropic medium is small, the approximated travel time equation of a refraction wave is obtained. This equation is only dependent on the group velocity of the horizontal transverse isotropic stratum. The elastic constants A₁₁, A₁₃, and A₃₃ and the Thomsen anisotropic parameter ε of the horizontal transverse isotropic stratum can be estimated using this multi‐azimuth seismic refraction technique. Furthermore, under a condition of weak anisotropy, the Thomsen anisotropic parameter δ of the horizontal transverse isotropic stratum can be estimated by this technique as well.     

A MAN/COMPUTER INTERPRETATION SYSTEM FOR RESISTIVITY SOUNDINGS OVER A HORIZONTALLY STRAFIFIED EARTH*

The proposed system works as follows:

• 1 By a trial-and-error procedure using a graphic display terminal a geologically relevant layer sequence with parameters (ρ_j, d_j) is adjusted to yield roughly the measured curve.
• 2 The resulting layer sequence is used as starting model for an iterative least squares procedure with singular value decomposition. Minimization of the sum of the squares of the logarithmic differences between measured and calculated values with respect to the logarithms of the resistivities and thicknesses as parameters linearizes the problem to a great extent, with two important implications:

• a) a considerable increase in speed (the number of iterations goes down), thus making it cheap to achieve the optimum solution;
• b) the confidence surfaces in parameter space are well approximated by the hyper-ellipsoids defined by the eigenvalues and eigenvectors of the normal equations.

Since these are known from the singular value decomposition we do in fact know all possible solutions compatible with the measured curve and the geological concept.

• 3 It is possible to “freeze” any combination of parameters at predetermined values. Thus extra knowledge and/or hypotheses are easily incorporated and can be tested by rerunning step (2). The overall computing time for a practical case is of the order of 10 sec on a CDC 6400.

     

MEASURED UNDERWATER NEAR-FIELD E-PATTERNS OF A PULSED,HORIZONTAL DIPOLE ANTENNA IN AIR: COMPARISON WITH THE THEORY OF THE CONTINUOUS WAVE,INFINITESIMAL ELECTRIC DIPOLE1

At Delft Geotechnics the technique of ground-penetrating radar is in use for the detection of buried objects such as pipes. To enable us to give our ‘measurements in the field’ a more quantitative interpretation than can be deduced from these alone, a series of experiments has been started under well-defined conditions. A cylindrical vessel containing water simulates wet soil. Mounted horizontally above the water surface is a pulsed triangular half-wave dipole which is used as a transmitting antenna (TA). It has a carrier-frequency of about 160 MHz and a pulse repetition-frequency of about 50 kHz. A movable receiving dipole (‘probe’) in the water measures the transverse, mutually orthogonal E_φ,- and E_θ-components of the pulses as a function of probe-position (r, θ, φ) and of the height h of the TA above the water surface. When these pulses are Fourier-transformed, the transverse electric fields E_φ and E_θ at 200 MHz are obtained. The resulting field patterns are compared with computational results on the basis of the theory of the continuous wave, infinitesimal electric dipole (‘point dipole’). It can be concluded that:

• 1 Far-field conditions have not fully developed at a depth of about 2.50 m, the largest value of the radius r at which field patterns were measured, although it represents a distance of about 15 wavelengths.
• 2 The attenuation constant of the tapwater used, as deduced from E-field measurements for θ= 0, 2.50 m < r < 2.75 m, is slightly less than the value measured using a network analyser and air line combination, in agreement with (1).
• 3 E _φ field patterns calculated using the value of the conductivity σ corresponding to the former value of the attenuation constant agree reasonably well with the measured patterns for r≤ 2.50 m and for θ < 20° at all antenna heights considered. Calculated Eφ patterns do not agree so well with the measured patterns when h is close to zero. With increasing height the agreement inproves.
• 4 In accordance with the theory of the point-dipole, the angular distribution of the radiation patterns of the TA becomes wider as the frequency decreases.
• 5 The normalized underwater pulse-spectra shift to lower frequencies with increasing r. This can be explained since the attenuation constant of the water rises with rising frequency.

     

On conditions for the unique inversion of seismic travel-time curves

It is shown that when the travel-time curve of a refracted wave from a surface source is known and at least one of the following conditions is satisfied, i.e. when

1. the travel-time curve of a wave reflected from a horizontal interface lying below the deepest low velocity layer is known, or
2. the travel-time curve of a wave from a deep source situated below the deepest low velocity layer is known, or
3. the measureH(u)=mes {z∶z≥0,v ^?1 (z)≥u} is analytical in some segment [c, d], where \(0< c< d< \infty , c< a_n , H(a_n ) = \bar z_n ,\bar z_n\) is the depth of the lower end of the deepest low velocity layer and in the interval [c, ∞) an analytical functionH(u)) exists which providesH(u)≡H(u)) ifu∈[c, d], then (1) velocityv(z) outside the low velocity layers and (2) the measureH _k (u)=mes {z∶z∈L _k,v ^?1 (z)≥u} for each low velocity layerL _k,k=1, 2, ..., n, are defined unambiguously.

     

Styles of volcano-induced deformation: numerical models of substratum flexure, spreading and extrusion

The gravitational deformation of volcanoes is largely controlled by ductile layers of substrata. Using numerical finite-element modelling we investigate the role of ductile layer thickness and viscosity on such deformation. To characterise the deformation we introduce two dimensionless ratios; Π_a (volcano radius/ductile layer thickness) and Π_b (viscosity of ductile substratum/failure strength of volcano). We find that the volcanic edifice spreads laterally when underlain by thin ductile layers (Π_a>1), while thicker ductile layers lead to inward flexure (Π_a<1). The deformation style is related to the switch from predominantly horizontal to vertical flow in the ductile layer with increasing thickness (increasing Π_a). Structures produced by lateral spreading include concentric thrust belts around the volcano base and radial normal faulting in the cone itself. In contrast, flexure on thick ductile substrata leads to concentric normal faults around the base and compression in the cone. In addition, we show that lower viscosities in the ductile layer (low Π_b) lead to faster rates of movement, and also affect the deformation style. Considering a thin ductile layer, if viscosity is high compared to the failure strength of the volcano (high Π_b) then deformation is coupled and spreading is produced. However, if the viscosity is low (low Π_b) substratum is effectively decoupled from the volcano and extrudes from underneath it. In this latter case evidence is likely to be found for basement compression, but corresponding spreading features in the volcano will be absent, as the cone is subject to a compressive stress regime similar to that produced by flexure. At volcanoes where basement extrusion is operating, high volcano stresses and outward substratum movement may combine to produce catastrophic sector collapse. An analysis of deformation features at a volcano can provide information about the type of basement below it, a useful tool for remote sensing and planetary geology. Also, knowledge of substratum geology can be used to predict styles of deformation operating at volcanoes, where features have not yet become well developed, or are obscured.     

QUASI-ANALYTIC CONVOLUTION SOLUTION OF THE ELECTROMAGNETIC FIELD*

The objective of this study is to generate the separation-distance-domain (r-domain) transformation of the theoretically calculated wave number domain (m-domain) electromagnetic induction field component B_z(m, ω) of a stratified medium and to search for interpretive information which has been absent in the previously achieved numerical solutions of the problem. The r-domain kernel R?(r, ω) function defining the induction field appears to adequately reflect the layering and electrical properties of the medium if it is expressed as a function of the frequency if the source-receiver separation r is small with respect to the thickness of the first layer. However, exact values of the conductivity cannot be distinguished from those of the neighboring values unless a resistive basement layer is present. This feature is the result of the truncation in series representation of the kernel function R?(m, ω). However, this truncation is regarded as significant in the case of a conductive first layer. In m-domain static-zone studies, a conductive first layer slightly influences its r-domain correspondent. Although the computational cost of obtaining the kernel B(r, ω) by evaluation of the convolution in a cylindrical coordinate system is high, this semi-analytic solution is still superior to those based on the asymptotic assumptions.     

Migration velocity analysis for tilted transversely isotropic media

Tilted transversely isotropic formations cause serious imaging distortions in active tectonic areas (e.g., fold‐and‐thrust belts) and in subsalt exploration. Here, we introduce a methodology for P‐wave prestack depth imaging in tilted transversely isotropic media that properly accounts for the tilt of the symmetry axis as well as for spatial velocity variations. For purposes of migration velocity analysis, the model is divided into blocks with constant values of the anisotropy parameters ε and δ and linearly varying symmetry‐direction velocity V_P0 controlled by the vertical (k_z) and lateral (k_x) gradients. Since determination of tilt from P‐wave data is generally unstable, the symmetry axis is kept orthogonal to the reflectors in all trial velocity models. It is also assumed that the velocity V_P0 is either known at the top of each block or remains continuous in the vertical direction. The velocity analysis algorithm estimates the velocity gradients k_z and k_x and the anisotropy parameters ε and δ in the layer‐stripping mode using a generalized version of the method introduced by Sarkar and Tsvankin for factorized transverse isotropy with a vertical symmetry axis. Synthetic tests for several models typical in exploration (a syncline, uptilted shale layers near a salt dome and a bending shale layer) confirm that if the symmetry‐axis direction is fixed and V_P0 is known, the parameters k_z, k_x, ε and δ can be resolved from reflection data. It should be emphasized that estimation of ε in tilted transversely isotropic media requires using nonhyperbolic moveout for long offsets reaching at least twice the reflector depth. We also demonstrate that application of processing algorithms designed for a vertical symmetry axis to data from tilted transversely isotropic media may lead to significant misfocusing of reflectors and errors in parameter estimation, even when the tilt is moderate (30°). The ability of our velocity analysis algorithm to separate the anisotropy parameters from the velocity gradients can be also used in lithology discrimination and geologic interpretation of seismic data in complex areas.     

Azimuthal resistivity soundings over a steeply dipping anisotropic formation: A case history in central Tunisia

Azimuthal Resistivity Soundings (ARS), using the so-called “Arrow-type array” as proposed by Bolshakov et al. were carried out in Central Tunisia, together with azimuthal resistivity tomography, because of the known anisotropic behaviour of the nearly vertical formations.First, the developments designed by Bolshakov et al. are reviewed: they deal with the separation between the effects of anisotropy and of heterogeneities, the design of the Arrow-type array and the introduction of the azimuthal spectral analysis.Second, the main methodological results obtained near Gouazine Lake are presented: (1) the clear effect of a quasi-vertical contact and (2) the characterisation of the anisotropic substratum below a thin superficial layer in one site close to the axis of the valley: the strike direction (α = 50°N), and a rather high anisotropy coefficient (λ ≈ 4) are determined.And lastly two directions for further developments are suggested.     

Inversion of Travel Times in Weakly Anisotropic Rock Samples

Based on the perturbation theory, inversion formulae for travel time of qP and qS waves in arbitrary weak anisotropy media are presented. The inversion formulae are linear expressions of elastic parameters expressed in terms of weak anisotropy (WA) parameters. The formulae of qS₁ and qS₂ waves have the same form and they can be used without identifying which wave is considered. A synthetic experiment similar to the measurement of rock sample in the laboratory is carried out to illustrate the efficiency of the presented inversion formulae. Two data sets for qP wave travel time from rock samples in the laboratory are inverted and 15 WA parameters are obtained.     

A laboratory study of seismic velocity and attenuation anisotropy in near-surface sedimentary rocks

The laboratory ultrasonic pulse‐echo method was used to collect accurate P‐ and S‐wave velocity (±0.3%) and attenuation (±10%) data at differential pressures of 5–50 MPa on water‐saturated core samples of sandstone, limestone and siltstone that were cut parallel and perpendicular to the vertical borehole axis. The results, when expressed in terms of the P‐ and S‐wave velocity and attenuation anisotropy parameters for weakly transversely isotropic media (ɛ, γ, ɛ_Q, γ_Q) show complex variations with pressure and lithology. In general, attenuation anisotropy is stronger and more sensitive to pressure changes than velocity anisotropy, regardless of lithology. Anisotropy is greatest (over 20% for velocity, over 70% for attenuation) in rocks with visible clay/organic matter laminations in hand specimens. Pressure sensitivities are attributed to the opening of microcracks with decreasing pressure. Changes in magnitude of velocity and attenuation anisotropy with effective pressure show similar trends, although they can show different signs (positive or negative values of ɛ, ɛ_Q, γ, γ_Q). We conclude that attenuation anisotropy in particular could prove useful to seismic monitoring of reservoir pressure changes if frequency‐dependent effects can be quantified and modelled.     

On Rossby wave propagation in a meridionally stratified channel

Rossby wave propagation in the presence of a nonseparable Brunt-Väisälä frequency,N(y,z), and the associated geostrophic zonal flow,U(y,z), is examined in this paper. The usual quasi-geostrophic potential vorticity equation only includes vertical variations in Brunt-Väisälä frequency (i.e.N(z)). We derive a linearised quasi-geostrophic potential vorticity equation which explicitly includesN(y, z), where variations inN may occur on the internal Rossby radius length scale. A mixed layer distribution that monotonically deepens in the poleward direction leads to a nonseparableN(y,z). The resulting meridional pressure gradient is balanced by an eastward zonal geostrophic flow.By assuming mixed layer depth changes occur slowly, relative to a typical horizontal wavelength of a Rossby wave, a local analysis is presented. The Rossby wave is found to have a strongly modulated meridional wavenumber,l, with amplitude proportional to |l|^–1/2. To elucidate whether the modulations of the Rossby wave are caused by the horizontal variations inN orU we also consider the cases where eitherN orU vary horizontally. Mixed layer depth changes lead to largestl where the mixed layer is deepest, whereasl is reduced in magnitude whereU is nonzero. When bothU(y,z) andN(y,z) are present, the two effects compete with one another, the outcome determined by the size of |c|/U _max, wherec is the Rossby wave phase speed. Finally, the slowly varying assumption required for the analytical approach is removed by employing a numerical model. The numerical model is suitable for studying Rossby wave propagation in a rectangular zonal channel with generalN(y, z) andU(y, z).     

The scattering of shear-waves in the crust

The two major sources of scattering for shear-waves in the crust, interactions with the topography at the surface and the effective anisotropy of aligned cracks throughout the rockmass, introduce first-order changes to the shear-wave particle-motion. At the surface, shear-waves are scattered by the topography within a wavelength or two of the recording site so that, unless the effective incidence angle is less than the critical angle sin^–1 V _S/V _P, the recorded waveforms may bear little relationship to the waveforms of the incident wave. Within the rockmass, shear-waves are scattered by extensive-dilatancy anisotropy (EDA), the distribution of stress-aligned fluid-filled cracks, microcracks, and preferentially oriented pore-space pervading most rocks in the crust. Analysis of this shear-wave splitting yields new information about the internal structure of thein situ rockmass which is not otherwise available.     

Estimating the elastic parameters of anisotropic media using a joint inversion of P-wave and SV-wave traveltime error

A method is presented to estimate the elastic parameters and thickness of media that are locally laterally homogeneous using P‐wave and vertically polarized shear‐wave (SV‐wave) data. This method is a ‘layer‐stripping’ technique, and it uses many aspects of common focal point (CFP) technology. For each layer, a focusing operator is computed using a model of the elastic parameters with which a CFP gather can be constructed using the seismic data. Assuming local homogeneity, the resulting differential time shifts (DTSs) represent error in the model due to anisotropy and error in thickness. In the (τ?p) domain, DTSs are traveltimes Δτ that connect error in layer thickness z, vertical slowness q, and ray parameter p. Series expansion is used to linearize Δτ with respect to error in the elastic parameters and thickness, and least‐squares inversion is used to update the model. For stability, joint inversion of P and SV data is employed and, as pure SV data are relatively rare, the use of mode‐converted (PSV) data to represent SV in the joint inversion is proposed. Analytic and synthetic examples are used to demonstrate the utility and practicality of this inversion.     

Peak surface strains during strong earthquake motion

Peak amplitudes of surface strains during strong earthquake ground motion can be approximated by ε = Aν_max/β₁, where ν_max is the corresponding peak particle velocity, β₁ is the velocity of shear waves in the surface layer, and A is a site specific scaling function. In a 50 m thick layer with shear wave velocity β₁ 300 m/s, A 0·4 for the radial strain ε_rr, A 0·2 for the tangential strain ε_rθ, and A 1·0 for the vertical strain, ε_z. These results are site specific and representative of strike slip faulting and of soil in Westmoreland, in Imperial Valley, California. Similar equations can be derived for other sites with known shear wave velocity profile versus depth.     

A compact representation of magnetotelluric responses for two-layer models

In this paper the locations where ρ_app = ρ₁ and ? = π/4 and where these parameters reach an extreme value in two-layer magnetotelluric (MT) sounding curves are summarized in an extremely compact form. The key parameters over two-layer models with conductivities σ₁, σ₂ and upper layer thickness h are the real S and α, where S is the conductivity contrast and α is the distance between the observation site and the conductivity interface, normalized to the half skindepth in the first layer. If the impedance components, various resistivity definitions ( ρ_{Re Z}, ρ_{Im Z} and ρ_|Z|, based on different parts of the complex impedance Z ) and the magnetotelluric phase ? are derived as a function of S and α, then the conditions for the apparent resistivity ρ_app and the phase ? are that they either satisfy ρ_app = ρ₁ and ? = π/4 or attain extreme values which can be given in terms of simple algebraic equations between S and α. All equations are valid for observation sites at any depth 0 ≤ z ≤ h in the first layer. The set of equations, presented in a tabular form, may make it possible to determine a layer boundary from the short period part of the sounding curves, in particular the ρ_{Re Z} and the ?_MT curves.     

THE VERTICAL MAGNETIC FIELD OF AN ALTERNATING CURRENT DIPOLE FOR HORIZONTALLY STRATIFIED MEDIA *

For the computation of the vertical component H_z of the magnetic field of a horizontal A.C. dipole lying on the earth's surface, a recurrence formula is presented for a horizontally stratified half space, to obtain the (n+ 1)-layer case from the w-layer case. By means of several computed diagrams for the two-layer case, H_z can be determined for different ratios of conductivity of the subsoil and that of the overburden. Thereby the distance from the dipole as well as the layer thickness h are expressed in terms of the wave length A of a plain wave in the overburden. Assuming a sufficiently large conductivity difference, the results show that evidence about the subsurface conditions can be obtained if the distance between the measuring coil and the dipole is of the order of A/3, and if the thickness h of the layer varies within the range A/100 < h < A/6. As an example for the 3-layer case, a nonconducting intermediate layer is assumed.     

Non-hyperbolic moveout inversion of wide-azimuth P-wave data for orthorhombic media

The azimuthally varying non‐hyperbolic moveout of P‐waves in orthorhombic media can provide valuable information for characterization of fractured reservoirs and seismic processing. Here, we present a technique to invert long‐spread, wide‐azimuth P‐wave data for the orientation of the vertical symmetry planes and five key moveout parameters: the symmetry‐plane NMO velocities, V⁽¹⁾_nmo and V⁽²⁾_nmo , and the anellipticity parameters, η⁽¹⁾, η⁽²⁾ and η⁽³⁾ . The inversion algorithm is based on a coherence operator that computes the semblance for the full range of offsets and azimuths using a generalized version of the Alkhalifah–Tsvankin non‐hyperbolic moveout equation. The moveout equation provides a close approximation to the reflection traveltimes in layered anisotropic media with a uniform orientation of the vertical symmetry planes. Numerical tests on noise‐contaminated data for a single orthorhombic layer show that the best‐constrained parameters are the azimuth ? of one of the symmetry planes and the velocities V⁽¹⁾_nmo and V⁽²⁾_nmo , while the resolution in η⁽¹⁾ and η⁽²⁾ is somewhat compromised by the trade‐off between the quadratic and quartic moveout terms. The largest uncertainty is observed in the parameter η⁽³⁾ , which influences only long‐spread moveout in off‐symmetry directions. For stratified orthorhombic models with depth‐dependent symmetry‐plane azimuths, the moveout equation has to be modified by allowing the orientation of the effective NMO ellipse to differ from the principal azimuthal direction of the effective quartic moveout term. The algorithm was successfully tested on wide‐azimuth P‐wave reflections recorded at the Weyburn Field in Canada. Taking azimuthal anisotropy into account increased the semblance values for most long‐offset reflection events in the overburden, which indicates that fracturing is not limited to the reservoir level. The inverted symmetry‐plane directions are close to the azimuths of the off‐trend fracture sets determined from borehole data and shear‐wave splitting analysis. The effective moveout parameters estimated by our algorithm provide input for P‐wave time imaging and geometrical‐spreading correction in layered orthorhombic media.     

The influence of dry and water saturated cracks on seismic velocities of crustal rocks - A comparison of experimental data with theoretical model

The pressure dependence of P- and S-wave velocities, velocity anisotropy, shear wave splitting and crack-porosity has been investigated in a number of samples from different crustal rock types for dry and wet (water saturated) conditions. At atmospheric pressure, P-wave velocities of the saturated, low-porosity rocks (< 1%) are significantly higher than in dry rocks, whereas the differences for S-wave velocities are less pronounced. The effect of intercrystalline fluids on seismic properties at increased pressure conditions is particularly reflected by the variation of the Poisson's ratio because P-wave velocities are more sensitive to fluids than S-wave velocities in the low-porosity rocks. Based on the experimental data, the respective crack-density parameter (), which is a measure of the number of flat cracks per volume unit contained within the background medium (crack-free matrix), has been calculated for dry and saturated conditions. There is a good correlation between the calculated crack-densities and crack-porosities derived from the experimentally determined volumetric strain curves. The shear wave velocity data, along with the shear wave polarisation referred to a orthogonal reference system, have been used to derive the spatial orientation of effective oriented cracks within a foliated biotite gneiss. The experimental data are in reasonable agreement with the self consistent model of O'Connell and Budiansky (1974). Taking the various lithologies into account, it is clear from the present study, that combined seismic measurements ofV _p andV _s , using theV _p V _s -ratio, may give evidence for fluids on grain boundaries and, in addition, may provide an estimate on the in-situ crack-densities.