Polarization Migration of Multi-component Seismic Data for Survey in the Tunnel of Mountain Cities

With the development of transportation in the west of China, tunnel construction in mountain cities is becoming very important and widespread. Tunneling safety in the tunnels is usually controlled by faults, and the advanced prediction of faults by seismic detection method has become a research hotspot in the field of engineering geophysics. Unlike seismic exploration on the ground, the sources and receivers are not properly arranged due to the limitation of the tunnel detection space, so as to cause migration artifacts problem in the process of advanced migration imaging. The problem results in inaccurate imaging of the faults. To solve the problem, this paper proposed a new polarization migration method. The method makes use of the polarization characteristic of three-component seismic signals. The principal polarization direction is calculated by Hilbert transform and complex covariance matrix analysis. A weighted function of the principal polarization direction factor is incorporated into the migration calculation. To verify the effectiveness of the polarization migration method, this paper carries out numerical simulations. Test results demonstrate that the artifacts are eliminated by the polarization migration, and occurrence parameters of faults, such as dip and trend are calculated accurately. The field detection case shows that seismic advanced prediction which is based on polarization migration provided parameters of faults in the front of the tunnel face with 100m, and the distance error is less than 2m, and the dip error is less than 3°, which ensures efficient and safe construction of tunneling.

IMPROPER INTERSECTIONS OF KUDLA–RAPOPORT DIVISORS AND EISENSTEIN SERIES

We consider a certain family of Kudla–Rapoport cycles on an integral model of a Shimura variety attached to a unitary group of signature (1, 1), and prove that the arithmetic degrees of these cycles are Fourier coefficients of the central derivative of an Eisenstein series of genus 2. The integral model in question parameterizes abelian surfaces equipped with a non-principal polarization and an action of an imaginary quadratic number ring, and in this setting the cycles are degenerate: they may contain components of positive dimension. This result can be viewed as confirmation, in the degenerate setting and for dimension 2, of conjectures of Kudla and Kudla–Rapoport that predict relations between the intersection numbers of special cycles and the Fourier coefficients of automorphic forms.

Computing separable isogenies in quasi-optimal time

Let $A$ be an abelian variety of dimension $g$ together with a principal polarization ${\it\phi}:A\rightarrow \hat{A}$ defined over a field $k$. Let $\ell$ be an odd integer prime to the characteristic of $k$ and let $K$ be a subgroup of $A[\ell ]$ which is maximal isotropic for the Riemann form associated to ${\it\phi}$. We suppose that $K$ is defined over $k$ and let $B=A/K$ be the quotient abelian variety together with a polarization compatible with ${\it\phi}$. Then $B$, as a polarized abelian variety, and the isogeny $f:A\rightarrow B$ are also defined over $k$. In this paper, we describe an algorithm that takes as input a theta null point of $A$ and a polynomial system defining $K$ and outputs a theta null point of $B$ as well as formulas for the isogeny $f$. We obtain a complexity of $\tilde{O} (\ell ^{(rg)/2})$ operations in $k$ where $r=2$ (respectively, $r=4$) if $\ell$ is a sum of two (respectively, four) squares which constitutes an improvement over the algorithm described in Cosset and Robert (Math. Comput. (2013) accepted for publication). We note that the algorithm is quasi-optimal if $\ell$ is a sum of two squares since its complexity is quasi-linear in the degree of $f$.