ABSTRACT Acoustic inversion in one‐dimension gives impedance as a function of travel time. Inverting the reflection response is a linear problem. Recursive methods, from top to bottom or vice versa, are known and use a fundamental wave field that is computed from the reflection response. An integral over the solution to the Marchenko equation, on the other hand, retrieves the impedance at any vertical travel time instant. It is a non‐recursive method, but requires the zero‐frequency value of the reflection response. These methods use the same fundamental wave field in different ways. Combining the two methods leads to a non‐recursive scheme that works with finite‐frequency bandwidth. This can be used for target‐oriented inversion. When a reflection response is available along a line over a horizontally layered medium, the thickness and wave velocity of any layer can be obtained together with the velocity of an adjacent layer and the density ratio of the two layers. Statistical analysis over 1000 noise realizations shows that the forward recursive method and the Marchenko‐type method perform well on computed noisy data.

中文翻译：

---

教程：声学反射响应的统一一维反演

摘要 一维声学反演给出了作为传播时间函数的阻抗。反转反射响应是一个线性问题。从上到下或反之亦然的递归方法是已知的，并且使用从反射响应计算的基波场。另一方面，对 Marchenko 方程的解进行积分，可以在任何垂直传播时间瞬间检索阻抗。它是一种非递归方法，但需要反射响应的零频率值。这些方法以不同的方式使用相同的基波场。将这两种方法结合起来可以得到一个在有限频率带宽下工作的非递归方案。这可用于面向目标的反演。当在水平分层介质上的一条线上有反射响应时，任何层的厚度和波速都可以与相邻层的速度和两层的密度比一起获得。对 1000 多个噪声实现的统计分析表明，前向递归方法和 Marchenko 型方法在计算的噪声数据上表现良好。