基于横向导数的走时计算方法及其在叠前时间偏移中的应用

The travel time calculation method via lateral derivative of velocity and its application in pre-stack time migration

（中国科学院地质与地球物理研究所，北京100029）

Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China

在Kirchhoff积分叠前偏移方法中，需要反复使用两点之间的射线走时，现有走时计算方法的计算量（或存储量）受到了计算条件的限制。针对此，基于时间空间域到频率波数域和向量场到指数流形上的正反变换，提出了计算单程波算子旁轴走时的简便公式，将走时表示成空间变量(地面点到地下相点的水平距离)的多项式,将频率波数域单平方根算子表示成波数的多项式,运用Lie代数积分、指数映射和鞍点法将走时多项式的系数与单平方根算子的系数联系起来，运用单平方根算子的系数计算走时多项式的系数。该展开式与通常的旁轴展开式相比，增加了关于空间坐标的非对称项。在进行Kirchhoff积分叠前偏移时，先将走时多项式的系数计算好并存储起来（这些系数大约10个）；在处理地震数据时，再利用这些系数“现算”走时，用于偏移成像。给出了Lie代数积分多项式、指数映射多项式计算的Magnus方法，该方法利用根树结构从低阶展开计算高阶展开；利用数值模拟方法，将有限差分法的结果与该公式（非对称）和对称公式的结果进行了对比，结果表明其比对称的走时公式有更高的精度。

In pre-stack time migration by Kirchhoff integral, the travel time of the two points are repeatedly used. However, the computation amount (or storage cost) of conventional travel time computation method is limited by computer conditions. In order to solve the problem，We proposed a simple formula for computing paraxial travel time of single-way wave operator. The formula is based on the forward and inverse transform between time-space domain to frequency-wavenumber domain and from vector field to exponential manifold. The travel time are expressed as polynomials of the horizontal offset between the two points, and the single-square-root operator in frequency-wavenumber domain are expressed as polynomials of wavenumber. Coefficients of travel time polynomials and that of single-square-root operator are related each other and calculated by Lie algebraic integrand, exponential map and the saddle-point method. Comparing to conventional expansion, our paraxial travel time expansion increases asymmetric terms of the horizontal offset in the expression. In Kirchhoff migration, the coefficients of travel time polynomials are calculated and stored before seismic data input (number of the coefficients are about 10). Travel times are calculated via the coefficients during Kirchhoff stacking of seismic data. A Magnus method calculated from Lie algebraic integrand polynomials, exponent mapping polynomials is given, which uses the tree-root scheme to compute high order terms based on lower order terms in the expansion method. The numerical comparison between our formulas, the results of finite difference method, symmetry formula and our formula are compared by numerical modeling. The results show that our method has higher accuracy than symmetry travel time formula.