用双曲正切和反正切函数拟合面波频散曲线的适应性分析

双曲正切函数和反正切函数的形态与面波频散曲线较相似,将其用于频散曲线拟合可减少数据噪音对反演稳定性的影响。这两类函数的曲线形态都是单调变化的,无法完全刻画某些特殊地层结构形成的频散曲线的复杂变化情形,如果用这两类曲线来拟合频散曲线则会存在本质上的模型误差。通过一系列正演模拟,分别运用这两类曲线拟合随层厚、P波速度、S波速度和密度变化模型的频散曲线,定量分析每个参数的变化对拟合误差的影响。结果表明,拟合结果随着P波速度、密度和层厚变化的分布很稳定,误差值保持在0.03~0.11。在S波速度随深度单调增加的情况下,拟合误差变化起伏比较大,误差范围为0.00~0.14,表明这两类三角函数模型的适应性会随着地层间S波速度差异的增大而降低。当模型中含有一定厚度的倒转低速层的时候,拟合误差可高达0.30,表明双曲正切函数和反正切函数频散曲线模型均不适用于拟合含强烈反差的低速倒转地层的频散曲线。实验发现,严重影响频散曲线拟合的误差突跳难题,可以通过多个随机种子的组合得到有效解决。

Adaptability Analysis of Dispersion Curve Fitting with Hyperbolic Tangent and Arctangent Functions

The shape of hyperbolic tangent and arctangent functions are similar with the dispersion curves of surface waves, hence it is used on dispersion curve fitting to reduce the influence of the noise. As both types of functions are monotonous curves, they are impossible to completely characterize the change of dispersion curves from some special stratigraphic structures. In this paper adaptability of these two types of curves was tested by modeling a series of geological models. The models were designed to characterize the effects on dispersion curves of layer thickness, P-wave velocity, S-wave velocity and density. The results show that the misfits of 0.03-0.11 are stable due to the changes of P-wave velocity, density and layer thickness. On the other hand, shear wave velocity variation plays a very significant role in the fitting quality. When S-wave velocity monotonously increases with depth, fluctuation of fitting errors is very significant, ranging from 0.00 to ~0.14. It means that the adaptability of these two kinds of trigonometric function models will decrease with the increase of S wave velocity variation. When a reversed layer is embedded in a geological structure, it can affect the fitting quality very significantly, with errors up to 0.30. This indicates that neither the hyperbolic tangent nor arctangent function model is suitable for fitting the dispersion curve from a geological structure embedded with reversed low velocity layers. Our experiment revealed that the error fluctuation problem, which seriously affects the fitting quality, can be effectively resolved by a combination of multi random seeds.