[13] We modeled the medium under the array by a set of horizontal layers underlain by a homogeneous half-space. The model parameters m included P and S wave velocities, density, and layer thicknesses. The model was constructed through the minimization of the functional
A synthetic receiver function Q_{ syn}( m, t) was calculated by the formula [Kind et al., 1995]
The response components of the layered medium H_{Q} ( m, w) and H_{L} ( m, w) were calculated by the Thomson-Haskell method [Haskell, 1962]. ( R(w) is the surface wave dispersion curve, and w_{a} are empirically chosen weights.)
[14] The minimization problem formulated above is strongly nonlinear and ill-conditioned. The traditional method used for solving problems of this type is based on the regularization method [Tikhonov and Arsenin, 1979]. The minimization reduces to the solution of a system of linear equations that can be found very rapidly. However, the model obtained from an implementation of this method [Kosarev et al., 1987] is essentially dependent on the initial approximation whose choice is generally complicated.
[15] Statistical (nongradient) methods (primarily, the Monte Carlo method) "scan" the entire space of parameters in the process of solution and are free from the necessity of specifying the initial approximation. However, the direct application of statistical methods to the search for the solution in a multidimensional space involves time consuming computations, which makes the practical use of these methods unrealistic. A possible way of solving this problem is to seek the solution in three stages: identification of one or more promising regions in the initial space of potential models, careful examination of each of these regions, and "fine adjustment" of the best solution inferred at the preceding stage. Such an approach is realized in the modern modification of the simulated annealing (SA) method [Ingber, 1989]. The software implementation of this method reduces the time of the search for the solution to an acceptable value (no more than one hour on a standard PC for the present case).
[16] To invert data, we used a model consisting of 11 layers on a half-space. Velocities of S waves in the layers and the half-space and thicknesses of some layers were varied, whereas the ratio V_{p}/V_{s} was fixed (1.73 in the crust and 1.8 in the mantle) and the density was determined by the Birch formula. The chosen time interval of the receiver function minimization ( - 2 s to 8 s) bounds the overall depth of the model by about 70-80 km. Deeper layers have no effect on the receiver function. However, in order to use traveltimes of waves converted at the 410-km boundary, the velocity structure should be specified to a depth of 410 km. For this purpose, our current 11-layer model (more specifically, the line defining the velocity in the half-space) was continued downward until its intersection with the IASP91 standard model of the Earth. IASP91 velocities were used for the calculation of tps410 in the depth interval from the intersection point to 410 km.
Figure 4 |
Figure 5 |
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