We present an analytical formulation of the correlation coefficient of the electromagnetic fields scattered at near-specular direction by a rough or gently undulating surface and measured at two spatially separated positions occupied by a moving receiver at slightly different times. This allows us obtaining an explicit expression of the correlation time of the received signal in terms of radar and surface parameters. This work provides a contribution to the discussion, currently ongoing in the Global Navigation Satellite System Reflectometry (GNSS-R) scientific community, about the behavior of received signal fluctuations, especially when surface profile variations are such that the scattering is neither coherent nor completely incoherent. The scattering surface is here modeled as randomly rough, and the Kirchhoff approximation (KA) or the first-order small slope approximation (SSA1) is employed to compute the scattered field. In fact, the expression of the correlation coefficient is the same for both approximations. The obtained closed-form expression shows that as the surface correlation length increases, the degree of coherence smoothly increases from the value obtained with the expression already available in the literature for very rough surfaces to a value close to unity for gently undulating surfaces. The obtained behavior of correlation time as a function of surface parameters, system resolution, and observation geometry is in agreement with numerical simulations available in the literature. In general, obtained analytical results are in agreement with the observed behavior of GNSS-R signals over flat land surfaces.

中文翻译：

---

GNSS-R 信号相关性的分析公式

我们提出了一种电磁场相关系数的分析公式，该电磁场在近镜面方向被粗糙或轻微起伏的表面散射，并在移动接收器在稍微不同的时间占据的两个空间分离位置进行测量。这使我们能够根据雷达和表面参数获得接收信号的相关时间的明确表达。这项工作为全球导航卫星系统反射计 (GNSS-R) 科学界目前正在进行的关于接收信号波动行为的讨论做出了贡献，特别是当表面轮廓变化导致散射既不相干也不完全不相干时. 散射表面在这里被建模为随机粗糙，并采用基尔霍夫近似（KA）或一阶小斜率近似（SSA1）来计算散射场。事实上，两种近似的相关系数的表达式是相同的。所获得的封闭式表达式表明，随着表面相关长度的增加，相干度从文献中已经可用于非常粗糙表面的表达式获得的值平滑地增加到接近于平缓起伏表面的统一的值。获得的相关时间作为表面参数、系统分辨率和观测几何的函数的行为与文献中可用的数值模拟一致。一般来说，获得的分析结果与在平坦地面上观测到的 GNSS-R 信号行为一致。