反射波分式展开时距方程及其精度分析

Fraction expansion time-distance equation of reflection wave and its accuracy analysis

从水平层状各向同性介质中反射波时距方程的泰勒级数展开式出发 ,推导出了时距方程的分式展开表达式 ,并给出了相应的系数表达式。针对 3个典型的速度模型 ,对双曲时距方程、泰勒级数展开四次时距方程和分式展开二次时距方程的精度进行了对比分析 ,结果表明 ,分式展开二次时距方程的精度最高。对泰勒级数展开六次时距方程、分式展开二次时距方程和分式展开四次时距方程的精度比较结果表明 ,分式展开二次方程的精度对模型速度结构依赖性最小 ,稳定性最好。对分式展开二次方程的参数进行了分析 ,分析中以垂向线性变化速度模型为例 ,得出了速度梯度值的解析表达式 ,得到了速度梯度值一般随着深度增加而逐渐减小的变化规律。由于分式展开二次时距方程具有高精度、高稳定性和参数意义明确等优点 ,可以应用到大炮检距地震资料处理中去。

Based on Taylor series expansion of time-distance equation of reflection wave in horizontal layered media, fraction expansion time-distance equation was derived. Then, hyperbolic time-distance equation, Taylor series expansion four-order time-distance equation and fraction expansion two-order time-distance equation were compared on three typical velocity models, which showed that fraction expansion two-order equation has the highest accuracy. Further, Taylor series expansion six-order equation, fraction expansion two-order equation and fraction expansion four-order equation were compared based on the same models, which indicated that fraction expansion two-order equation is characterized by the least dependence on models and the best stability. The signification of parameters of fraction expansion two-order equation was analyzed using vertically linear velocity model, expression of velocity gradient was derived, and a conclusion was drawn that velocity gradient generally increases with the increase of depth. With the advantage of higher accuracy, better stability and univocal parameters, fraction expansion two-order time-distance equation can be utilized to process seismic data of large offset.