The first-order statistical moment of the seismic moment tensor

Summary. If a complex earthquake is assumed to be a set of individual, randomly oriented elementary pure double couple sources, the solution for the seismic moment of the complex event projected on the mean trend of the fault will perforce be comprised of sources of both double couple and compensated linear vector dipole (CLVD) types. We investigate the statistical properties of these two components of seismic sources in terms of the invariants of the seismic moment tensor of a realistic set of synthetic earthquakes. It is very likely that the size of the CLVD component is two to three orders of magnitude smaller than that of the double couple component.     

Cluster analysis of seismic moment tensor orientations

This paper demonstrates that well-known methods of cluster analysis and multivariate data analysis are useful for geodynamic interpretation of seismic moment tensors. To use these methods, moment tensors are expressed as vectors in a 6-D space. These are vectors in a rigorous sense, rather than an arbitrary set of ordered numbers, because a dot product can be defined that is independent of the coordinate system. In this vector space, non-isotropic moment tensors are a 5-D linear subspace and normalized moment tensors are unit vectors, or points on a unit sphere. Distance along a great circle of the unit sphere satisfies reasonable requirements for any measure of the difference between normalized moment tensors. In regions with a few isolated sets of orientations, cluster analysis based on the great circle distance identifies the same groups of earthquakes that a seismologist would. Figures based on principal component analysis and discriminant analysis illustrate orientation clustering better than equal area projections of moment tensor principal axes. In one case where clusters have been claimed to exist, orientations appear to be continuously distributed and no evidence is found for separate populations of moment tensors.     

The two-point correlation function of the seismic moment tensor

Summary. We use the invariants of the two-point correlation function of the seismic moment to investigate the degree of irregularity of an earthquake fault, i.e. to study the rapidity with which a complex fault changes its direction of orientation. The two-point correlation function is a fourth-order tensor which has three scalar invariants in the isotropic case. Although the accuracy of present-day catalogues of fault plane solutions is rather low for our purpose, nevertheless the invariants of these correlation tensors confirm the generally     

Bias in the estimate of seismic moment tensor by the linear inversion method

Summary. We investigate the effects of various sources of error on the estimation of the seismic moment tensor using a linear least squares inversion on surface wave complex spectra. A series of numerical experiments involving synthetic data subjected to controlled error contamination are used to demonstrate the effects. Random errors are seen to enter additively or multiplicitively into the complex spectra. We show that random additive errors due to background recording noise do not pose difficulties for recovering reliable estimates of the moment tensor. On the other hand, multiplicative errors from a variety of sources, such as focusing, multipathing, or epicentre mislocation, may lead to significant overestimation or underestimation of the tensor elements and in general cause the estimates to be less reliable.     

Autoregressive estimation of complex eigenfrequencies in low frequency seismic spectra

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The fact that a seismogram can be represented as an autoregressive (AR) time series and the use of Prony's method enable us to obtain the complex frequencies from the poles of the corresponding AR process. A frequency-domain formulation which employs a time-domain tapering technique is devised so that spectral peaks can be analysed individually, or in small groups. Statistical analysis of the estimators and several examples from two recent earthquakes illustrate the method.     

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.     

Centroid moment tensor inversion of low-frequency seismic spectra using Green's functions for aspherical earth models

We present a new method for centroid moment tensor (CMT) inversion, in which we employ the Green's function computed for aspherical earth models using the Direct Solution Method. We apply this method to CMT inversion of low-frequency seismic spectra for the 1994 Bolivia and 1996 Flores Sea deep earthquakes. The estimated centroid locations agree well with those obtained by multiple-shock analyses using body-wave data. This shows that it is possible to obtain reliable CMT solutions by analyses of low-frequency seismic spectra using accurate Green's functions computed for present 3-D earth models.     

An approach to reserve estimation enhanced with 3-D seismic data

Reserve estimation for hydrocarbon reservoirs can be improved by incorporating values extracted from three-dimensional (3-D) seismic data with those obtained from more conventional data sources of data, such as drill-core and well-log data. An example of this improved method is illustrated by an application to the QW pool located in the Buohaiwan Basin in eastern China. Parameter values extracted from 3-D seismic data extend the knowledge about the spatial distributions of such reservoir parameters as net thickness, porosity, and oil saturation. To assist in the extraction of these values, different pattern-recognition techniques can be applied. The results that are obtained by this method offer a more reliable and more credible approach to reserve estimation and can be applied at every stage of resource extraction from exploration to development.