Polarization analysis and polarization filtering of three-component signals with the time–frequency S transform

From basic Fourier theory, a one-component signal can be expressed as a superposition of sinusoidal oscillations in time, with the Fourier amplitude and phase spectra describing the contribution of each sinusoid to the total signal. By extension, three-component signals can be thought of as superpositions of sinusoids oscillating in the x -, y -, and z -directions, which, when considered one frequency at a time, trace out elliptical motion in three-space. Thus the total three-component signal can be thought of as a superposition of ellipses. The information contained in the Fourier spectra of the x -, y -, and z -components of the signal can then be re-expressed as Fourier spectra of the elements of these ellipses, namely: the lengths of their semi-major and semi-minor axes, the strike and dip of each ellipse plane, the pitch of the major axis, and the phase of the particle motion at each frequency. The same type of reasoning can be used with windowed Fourier transforms (such as the S transform), to give time-varying spectra of the elliptical elements. These can be used to design signal-adaptive polarization filters that reject signal components with specific polarization properties. Filters of this type are not restricted to reducing the whole amplitude of any particular ellipse; for example, the 'linear' part of the ellipse can be retained while the 'circular' part is rejected. This paper describes the mathematics behind this technique, and presents three examples: an earthquake seismogram that is first separated into linear and circular parts, and is later filtered specifically to remove the Rayleigh wave; and two shot gathers, to which similar Rayleigh-wave filters have been applied on a trace-by-trace basis.     

Determination of the polynomial moments of the seismic moment rate density distribution with positivity constraints

We present a new formulation of the inverse problem of determining the temporal and spatial power moments of the seismic moment rate density distribution, in which its positivity is enforced through a set of linear conditions. To test and demonstrate the method, we apply it to artificial data for the great 1994 deep Bolivian earthquake. We use two different kinds of faulting models to generate the artificial data. One is the Haskell-type of faulting model. The other consists of a collection of a few isolated points releasing moment on a fault, as was proposed in recent studies of this earthquake. The positions of 13 teleseismic stations for which P - and SH -wave data are actually available for this earthquake are used. The numerical experiments illustrate the importance of the positivity constraints without which incorrect solutions are obtained. We also show that the Green functions associated with the problem must be approximated with a low approximation error to obtain reliable solutions. This is achieved by using a more uniform approximation than Taylor's series. We also find that it is necessary to use relatively long-period data first to obtain the low- (0th and 1st) degree moments. Using the insight obtained into the size and duration of the process from the first-degree moments, we can decrease the integration region, substitute these low-degree moments into the problem and use higher-frequency data to find the higher-power moments, so as to obtain more reliable estimates of the spatial and temporal source dimensions. At the higher frequencies, it is necessary to divide the region in which we approximate the Green functions into small pieces and approximate the Green functions separately in each piece to achieve a low approximation error. A derivation showing that the mixed spatio-temporal moments of second degree represent the average speeds of the centroids in the corresponding direction is given.     

Joint inversion of receiver function and surface wave dispersion observations

We implement a method to invert jointly teleseismic P wave receiver functions and surface wave group and phase velocities for a mutually consistent estimate of earth structure. Receiver functions are primarily sensitive to shear wave velocity contrasts and vertical traveltimes, and surface wave dispersion measurements are sensitive to vertical shear wave velocity averages. Their combination may bridge resolution gaps associated with each individual data set. We formulate a linearized shear velocity inversion that is solved using a damped leastsquares scheme that incorporates a priori smoothness constraints for velocities in adjacent layers. The data sets are equalized for the number of data points and physical units in the inversion process. The combination of information produces a relatively simple model with a minimal number of sharp velocity contrasts. We illustrate the approach using noisefree and realistic noise simulations and conclude with an inversion of observations from the Saudi Arabian Shield. Inversion results for station SODA, located in the Arabian Shield, include a crust with a sharp gradient near the surface (shear velocity changing from 1.8 to 3.5 km s¹ in 3 km) underlain by a 5kmthick layer with a shear velocity of 3.5 km s¹ and a 27kmthick layer with a shear velocity of 3.8 km s¹, and an upper mantle with an average shear velocity of 4.7 km s¹. The crustmantle transition has a significant gradient, with velocity values varying from 3.8 to 4.7 km s¹ between 35 and 40 km depth. Our results are compatible with independent inversions for crustal structure using refraction data.     

Using evidence of non-linear induced polarization for detecting extended ore mineralizations

The discrimination between electrolytic and electronic conductors is highly relevant to geological modelling as it allows conclusions to be drawn about the formation and mineral composition of rocks. The induced polarization (IP) method, which compares the electric current injected into the ground with the corresponding earth potential differences can be used for this purpose.
  This paper describes a new method based on the theory that non-linear electrochemical processes on the surface of electronic conductors are responsible for non-linear IP (NLIP) phenomena. This results in multiples of the fundamental frequency being observed in the telluric voltage spectra when a monochromatic current signal is fed into the ground. The non-linearity of the current–voltage characteristic is most effectively described by a spectral method.
  A laboratory experiment was carried out, using an electrolytic trough with a small graphite cylinder serving as an electronic conductor, which clearly demonstrated the validity of the method. A field experiment was undertaken at a borehole of approximately 450  m depth, located in the transition zone of the Tepla-Barrandium and Moldanubicum in East Bavaria. A sinusoidal current was injected into the ground using a logging tool at depths varying between 150 and 450  m. The corresponding potential differences were simultaneously observed along a profile on the surface. Field and laboratory results show a striking similarity. It can be concluded that an extensive electronic conductor—probably graphite—is steeply dipping southwards meeting the borehole at approximately 310  m depth.     

Surface-wave polarization data and global anisotropic structure

In the past few years, seismic tomography has begun to provide detailed images of seismic velocity in the Earth's interior which, for the first time, give direct observational constraints on the mechanisms of heat and mass transfer. The study of surface waves has led to quite detailed maps of upper-mantle structure, and the current global models agree reasonably well down to wavelengths of approximately 2000 km. Usually, the models contain only elastic isotropic structure, which provides an excellent fit to the data in most cases. For example, the variance reduction for minor and major arc phase data in the frequency range 7–15 mHz is typically 65–92 per cent and the data are fit to within 1–2 standard deviations. The fit to great-circle phase data, which are not subject to bias from unknown source or instrument effects, is even better. However, there is clear evidence for seismic anisotropy in various places on the globe. This study demonstrates how much (or little) the fit to the data is improved by including anisotropy in the modelling process. It also illuminates some of the trade-offs between isotropic and anisotropic structure and gives an estimate of how much bias is introduced by neglecting anisotropy. Finally, we show that the addition of polarization data has the potential for improving recovery of anisotropic structure by diminishing the trade-offs between isotropic and anisotropic effects.     

Tomographic inversion using ℓ1-norm regularization of wavelet coefficients

We propose the use of ℓ₁ regularization in a wavelet basis for the solution of linearized seismic tomography problems   A m = d   , allowing for the possibility of sharp discontinuities superimposed on a smoothly varying background. An iterative method is used to find a sparse solution m that contains no more fine-scale structure than is necessary to fit the data d to within its assigned errors.