Summary A new method for inverting P‐wave traveltimes for seismic anisotropy on a local scale is presented and tested. In this analysis, direction‐dependent seismic velocity is represented by a second‐ or fourth‐order Cartesian tensor, which is shown to be equivalent to decomposing a velocity surface using a basis set of Cartesian products of unit vectors. The new inversion method for P‐ and S‐wave anisotropy from traveltime data is based on the tensor decomposition. The formulation is formally derived from a Taylor series expansion of a continuously extended, 3‐D velocity function originally defined on the surface of the unit sphere. This approach allows us to solve a linear inversion instead of the standard non‐linear method. The resultant, linearized, fourth‐order traveltime equation is similar to a previous fourth‐order result (Chapman & Pratt 1992), although our representation offers a natural second‐order simpli‐fication. Conventional isotropic traveltime tomography is a special case of our tensorial representation of velocities. P‐wave velocity can be represented by a second‐order tensor (matrix) as a first approximation, although S‐wave traveltime tomography is intrinsically fourth order because of S‐wave solution duality. Differences between isotropic and anisotropic parametrizations are investigated when velocity is represented by a matrix A. The trade‐off between isotropy and anisotropy in practical tomography, which differs from the fundamental deficiency of anisotropic traveltime tomography (Mochizuki 1997), is shown to be ~ 1; that is, their effects are of the same order. We conclude that anisotropic considerations may be important in velocity inversions where ray coverage is less than optimal. On the other hand, when the ray directional coverage is complete and balanced, effects of anisotropy sum to zero and the isotropic part gives the result obtained from inverting for isotropic variations of velocity alone. Synthetic test data sets are inverted, demonstrating the effectiveness of the new inversion approach. When ray coverage is fairly complete, original anisotropy is well recovered, even with random noise introduced, although anisotropy ambiguities arise where ray coverage is limited. Random noise was found to be less important than ray directional coverage in anisotropic inversions.
A new inversion method for P wave anisotropy [Wu and Lees, 1999a] has been applied to high‐precision, microseismic traveltime data collected at Coso geothermal region, California. Direction‐dependent P wave velocity and thus its perturbation, are represented by a symmetric positive definite matrix A instead of a scalar. The resulting anisotropy distribution is used to estimate variations in crack density, stress distribution and permeability within the producing geothermal field. A circular dome‐like structure is observed at the southwestern part of the geothermal region southwest of Sugarloaf Mountain. Using a linear stress‐bulk modulus relationship, deviatoric stress is estimated to be 3–6 MPa at geothermal production depths (1–2 km), assuming all the anisotropy is related to stress. The stress field is compressional NNE‐SSW and dilational WNW‐ESE, coinciding with a previous, independent study using earthquake focal mechanisms. Following a theory on flat, elliptic cracks, residual crack density estimated from P anisotropy is ∼0.0078 assuming crack aspect ratios ≫ 1:60 and is ∼0.041 when crack aspect ratios are close to 1:60. Residual crack orientation distribution is related to velocity anisotropy. On the basis of anisotropic part of crack density distribution function, the anisotropic part of permeability distribution may be calculated by a statistical approach via simple parallel fluid flow along cracks.
Pulse width data are used to invert for attenuation structure in the Coso geothermal area, California. The dataset consists of pulse width measurements of 838 microseismic events recorded on a seismic array of 16 downhole stations between August 1993 and March 1994. The quality factor Q correlates well with surface geology and surface heat flow observations. A broad region of low Q (≈ 30 to 37) is located at 0.5 to 1.2 km in depth below Devil's Kitchen, Nicol Prospects, and Coso Hot Springs. A vertical, low Q (≈ 36 in contrast with surrounding rock of 80) region is interpreted as a channel through which hydrothermal energy is transported from depth to the surface. The location of the channel is between stations S1 and S4, and its dimension is about 1 km. At the deep end of the channel, a large, broad body of low Q is also located at 3 km in depth 2 to 4 km to the southwest of Nicol Prospects and Devil's Kitchen. Since it lies at the bottom of the target region and beyond the scope of seismicity, further research is needed to constrain its extent. Numerical modeling with a pseudospectral method is also done to investigate the applicability of the inversion scheme to fractured regions. A linear relationship between pulse width broadening and travel time is upheld, and the proportional constants are estimated.
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