ALBERT

Description: Plate-scale deformation is expected to impart seismic anisotropic fabrics on the lithosphere. Determination of the fast shear wave orientation ( ) and the delay time between the fast and slow split shear waves ( t ) via SKS splitting can help place spatial and temporal constraints on lithospheric deformation. The Canadian Appalachians experienced multiple episodes of deformation during the Phanerozoic: accretionary collisions during the Palaeozoic prior to the collision between Laurentia and Gondwana, and rifting related to the Mesozoic opening of the North Atlantic. However, the extent to which extensional events have overprinted older orogenic trends is uncertain. We address this issue through measurements of seismic anisotropy beneath the Canadian Appalachians, computing shear wave splitting parameters ( , t ) for new and existing seismic stations in Nova Scotia and New Brunswick. Average t values of 1.2 s, relatively short length scale (≥100 km) splitting parameter variations, and a lack of correlation with absolute plate motion direction and mantle flow models, demonstrate that fossil lithospheric anisotropic fabrics dominate our results. Most fast directions parallel Appalachian orogenic trends observed at the surface, while t values point towards coherent deformation of the crust and mantle lithosphere. Mesozoic rifting had minimal impact on our study area, except locally within the Bay of Fundy and in southern Nova Scotia, where fast directions are subparallel to the opening direction of Mesozoic rifting; associated t values of 〉1 s require an anisotropic layer that spans both the crust and mantle, meaning the formation of the Bay of Fundy was not merely a thin-skinned tectonic event.

Description: 〈span〉〈div〉SUMMARY〈/div〉We consider the case where the ‘solution’ to an inverse problem is an ensemble (e.g. drawn from the conditional probability density function $p( {{{\bf m}}|{{{\bf d}}^{obs}}} )$ of 〈span〉M〈/span〉 model parameters ${{\bf m}}$ given observed data ${{{\bf d}}^{obs}}$). Here we presume that the ${{\bf m}}$s have a natural ordering, say in position 〈span〉x〈/span〉, so that ‘resolution’ means the ability of the inverse problem to distinguish physically adjacent model parameters. The trade-off curve for resolution and variance is constructed using the following steps: (1) the single solution ${{{\bf m}}^{est}}$ and its covariance ${{{\bf C}}_m}$ are estimated as the ensemble mean and covariance; (2) the eigenvalue decomposition ${{{\bf C}}_m} = {\rm{\ }}{{{\bf V} {\boldsymbol \Lambda} }}{{{\bf V}}^{\rm{T}}}$ is computed and the submatrix ${{{\boldsymbol \Lambda }}^{( N )}}$ of the 〈span〉N〈/span〉 smallest eigenvalues, and submatrix ${{{\bf V}}^{( N )}}$of the 〈span〉N〈/span〉 corresponding eigenvectors, are formed; (3) the equation ${{{\boldsymbol \mu }}^{( N )}} = {{{\boldsymbol \Phi }}^{( N )}}\ {{\bf m}}$ with ${{{\boldsymbol \mu }}^{( N )}} = [ {{{{\bf V}}^{( N )}}} ]{\boldsymbol{\ }}{{{\bf m}}^{est}}$〈strong〉and〈/strong〉${{{\boldsymbol \Phi }}^{( N )}} = {[ {{{{\bf V}}^{( N )}}} ]^{\rm{T}}}$ is formed, as is its covariance ${{\bf C}}_\mu ^{( N )} = {{{\boldsymbol \Lambda }}^{( N )}}{\boldsymbol{\ }}$; (4) the equation is solved to yield a localized average ${\langle {{\bf m}} \rangle ^{( N )}} = \ {{{\boldsymbol \Phi }}^{ - g}}{{{\boldsymbol \mu }}^{( N )}}$, where ${{{\boldsymbol \Phi }}^{ - g}}$ is either the minimum length or Backus–Gilbert generalized inverse of ${{\boldsymbol \Phi }}$; (5) the resolution and covariance are computed as ${{{\bf R}}^{( N )}} = {{{\boldsymbol \Phi }}^{ - g}}{\boldsymbol{\ }}{{{\boldsymbol \Phi }}^{( N )}}$ and ${{\bf C}}_m^{( N )} = {{{\boldsymbol \Phi }}^{ - g}}{\boldsymbol{\ }}{{\bf C}}_\mu ^{( N )}{( {{{{\boldsymbol \Phi }}^{ - g}}} )^{\rm{T}}}$; (6) the spread ${K^{( N )}}$ of resolution and size ${J^{( N )}}\ $of covariance are computed using either the Dirichlet or Backus–Gilbert measures and (7) the process is repeated for $1 \le N \le M$ to build up the trade-off curve $K( J )$. We show that, in the Dirichlet case, ${K^{( N )}} = \ M - N$ and ${J^{( N )}} = \ {\rm{tr}}( {{{{\boldsymbol \Lambda }}^{( N )}}} )$. We also consider the case where the model parameters correspond to spline coefficients and a sequence ${y_i}( {{{\bf m}},{x_i}} )$ derived from these coefficients possesses natural ordering. Layered models are an example of such a parametrization. We construct the trade-off curve for ${{\bf y}}$ by converting each member of the ensemble from ${{\bf m}}$ to ${{\bf y}}$ and applying the above procedure to them. We demonstrate the method by applying it to several simple examples.〈/span〉

Description: 〈span〉〈div〉Summary〈/div〉We consider the case where the ‘solution’ to an inverse problem is an ensemble (e.g. drawn from the conditional probability density function $p( {{{\bf m}}{\rm{|}}{{{\bf d}}^{obs}}} )$ of 〈span〉M〈/span〉 model parameters ${{\bf m}}$ given observed data ${{{\bf d}}^{obs}}$). Here we presume that the ${{\bf m}}$s have a natural ordering, say in position 〈span〉x〈/span〉, so that ‘resolution’ means the ability of the inverse problem to distinguish physically adjacent model parameters. The trade-off curve for resolution and variance is constructed using the following steps: (1) the single solution ${{{\bf m}}^{est}}$ and its covariance ${{{\bf C}}_m}$ are estimated as the ensemble mean and covariance; (2) the eigenvalue decomposition ${{{\bf C}}_m} = {\rm{\ }}{{\bf V\Lambda }}{{{\bf V}}^{\rm{T}}}$ is computed and the submatrix ${{{\bf \Lambda }}^{( N )}}$ of the 〈span〉N〈/span〉 smallest eigenvalues, and submatrix ${{{\bf V}}^{( N )}}$of the 〈span〉N〈/span〉 corresponding eigenvectors, are formed; (3) the equation ${{{\bf \mu }}^{( N )}} = {{{\bf \Phi }}^{( N )}}\ {{\bf m}}$ with ${{{\bf \mu }}^{( N )}} = [ {{{{\bf V}}^{( N )}}} ]{\boldsymbol{\ }}{{{\bf m}}^{est}}$〈strong〉and〈/strong〉${{{\bf \Phi }}^{( N )}} = {[ {{{{\bf V}}^{( N )}}} ]^{\rm{T}}}{\boldsymbol{\ }}$ is formed, as is its covariance ${{\bf C}}_\mu ^{( N )} = {{{\bf \Lambda }}^{( N )}}{\boldsymbol{\ }}$; (4) the equation is solved to yield a localized average ${{{\bf m}}^{( N )}} = \ {{{\bf \Phi }}^{ - g}}{{{\bf \mu }}^{( N )}}$, where ${{{\bf \Phi }}^{ - g}}$ is either the minimum length or Backus-Gilbert generalized inverse of ${{\bf \Phi }}$; (5) the resolution and covariance are computed as ${{{\bf R}}^{( N )}} = {{{\bf \Phi }}^{ - g}}{\boldsymbol{\ }}{{{\bf \Phi }}^{( N )}}$ and ${{\bf C}}_m^{( N )} = {{{\bf \Phi }}^{ - g}}{\boldsymbol{\ }}{{\bf C}}_\mu ^{( N )}{( {{{{\bf \Phi }}^{ - g}}} )^{\rm{T}}}$; (6) the spread ${K^{( N )}}$ of resolution and size ${J^{( N )}}\ $of covariance are computed using either the Dirichlet or Backus-Gilbert measures; and (7) the process is repeated for $1 \le N \le M$ to build up the trade-off curve $K( J )$. We show that, in the Dirichlet case, ${K^{( N )}} = \ M - N$ and ${J^{( N )}} = \ {\rm{tr}}( {{{{\bf \Lambda }}^{( N )}}} )$. We also consider the case where the model parameters correspond to spline coefficients and a sequence ${y_i}( {{{\bf m}},{x_i}} )$ derived from these coefficients possesses natural ordering. Layered models are an example of such a parameterization. We construct the trade-off curve for ${{\bf y}}$ by converting each member of the ensemble from ${{\bf m}}$ to ${{\bf y}}$ and applying the above procedure to them. We demonstrate the method by applying it to several simple examples.〈/span〉

Description: Information on the time intervals between large earthquakes is now available for several fault segments along plate boundaries in Japan, Alaska, California, Cascadia, and Turkey. When dates in a sequence are known historically, as along much of the Nankai trough, they provide information on the natural (intrinsic) variability of the rupture process. Most sets of repeat times, however, are dominated by paleoseismic determinations of dates of older large earthquakes, which contain measurement uncertainties in addition to intrinsic variability. A Bayesian technique along with prior information on measurement uncertainties is used to make maximum-likelihood estimates of intrinsic repeat time and its normalized standard deviation, the coefficient of variation (CV). It is these intrinsic parameters and their uncertainties that are most useful for understanding the mechanics of earthquakes and for prediction for timescales of a few decades. Our estimates of intrinsic CV are small, 0 to 0.25, for several very active fault segments where deformation is relatively simple, large events do not appear to be missing in historic and paleoseismic records, and data are available at or near major asperities and away from the ends of rupture zones. CV is larger for regions of multibranched faulting, overlapping slip near the ends of rupture zones and for data from uplifted terraces at subduction zones. A Poisson process is an inferior characterization of all of the 11 segments we examined. Scenarios used by recent working groups that assume either Poissonian behavior or renewal processes with CV of 0.5+ or -0.2 for the most active fault segments in the San Francisco Bay area likely lead to incorrect 30-year probability estimates. The Hayward fault and perhaps the Peninsular segment of the San Andreas fault in the San Francisco Bay area appear to be advanced in their buildup of stress that will be released in future large earthquakes. Multibranched faulting may account for why the predicted Tokai earthquake in Japan has not occurred as of 2006. Parkfield earthquakes from 1857 to 2004 were characterized by the largest uncertainty of the sequences we studied, CV = 0.37, which may account for the failure of past predictions. The large CV for Parkfield fits our hypothesis that relatively weak fault segments are characterized by more irregular earthquake recurrence. Paleoseismic data from coastal sites along the Cascadia subduction zone are characterized by CVs of about 0.3.

Description: Notwithstanding the commonly held wisdom that "you can't determine the absolute location of earthquakes using the double-difference method", you can We present a way of visualizing double-difference data, and use it to show how differential arrival-time data can, in principle, be used to determine the absolute locations of earthquakes. We then analyze the differential form of Geiger's Method, which is the basis of many double-difference earthquake-location algorithms, and show that it can be used to make estimates of the absolute location of earthquake sources. Finally, we examine absolute-location error in one earthquake-location scenario, using Monte Carlo simulations that include both measurement error and velocity model error, and show that the double-difference method produces absolute locations with errors that are comparable in magnitude, or even less, than traditional methods. The improvement in absolute locations arises from exactly the same, and often-cited, reasons that the double-difference method yields superior relative locations: observations of differential travel times determined via cross correlation have a much smaller error than observations of absolute travel times determined via phase picking; and predictions of differential travel times are less sensitive to unmodeled near-surface heterogeneity than are predictions of absolute travel times. Absolute earthquake locations that are already routinely produced by most implementations of the double-difference method have a better accuracy than has been credited.

Description: Seismic coda is composed of scattered wavelets originated from various heterogeneities. The phase composition of regional seismic coda still remains unknown, despite its use for several decades. This is caused partly because the ray paths of scattered wavelets in coda are not on the great-circle path between a source and receiver. We examine the constituent original phases of regional coda with the help of a source-array analysis. A set of uniform sources that are essential for a source-array analysis is organized with underground nuclear explosions. Strong Rg-origin energy is observed in the coda at frequencies of 0.2-0.8 Hz, and it lasts more than 700 sec until the end of records. The coherent energy in the coda reduces with frequency. It constitutes about 20% of the total coda energy at frequencies of 0.2-0.4 Hz, and 12% at frequencies of 0.4-0.8 Hz. The other 80% of coda energy in 0.2-0.8 Hz is mixed with complex phases from various untraceable origins. The Rg energy is the most influential component in the construction of low-frequency regional coda. On the other hand, the coda at higher frequencies, 0.8-3.2 Hz, is observed to be mixed with complex phases that cause the wave field to be diffused. The observation of Rg-origin energy at the regional coda suggests that scattered energy from phase coupling of Rg is not significant compared to Rg-to-Rg scattered energy.

Description: When earthquakes are used as sources in velocity tomography, the unknown velocity structure and the unknown hypocentral parameters (that is, source location and origin time) must be simultaneously estimated during the imaging process. This coupling allows the two sets of unknowns to trade off, and increases the degree of nonuniqueness of the resulting tomographic image above what would have been present had the hypocentral parameters been precisely known. We analyze, in detail, the nonuniqueness associated with unknown origin time, which we argue is often a more important source of nonuniqueness than is unknown location. While this type of nonuniqueness has long been understood to be a problem in teleseimic tomography, we show here that it is of equal importance in all coupled problems. We provide a practical method for calculating null solutions and calculate them for several commonly encountered experimental geometries. We also show that the attenuation tomography possesses a mathematically identical nonuniqueness, with unknown source amplitude being the analogue to unknown origin time.