波动方程反演中基于加权的混合目标函数研究

基于 L_2范数的目标函数要求原始数据噪声遵从 Gauss 分布,因此对大的数据异常点过分敏感。为了解决这一问题,利用反正切函数作为加权因子构造了反正切混合目标函数,并利用模型试验进行了验证。简要阐述了最优化目标函数的构造方法；给出了利用反正切函数构造混合目标函数的基本原理,并对目标函数的微分特征和概率特征进行了分析；利用模型论述了反正切混合目标函数中控制参数μ和σ的意义,μ控制反正切函数变化的速度,σ控制数据误差项被处理或加权的方式。设计了一维层状模型,对比了反正切混合目标函数和其它混合目标函数的反演效果。反正切混合目标函数的模型试验表明,对于小数据误差项用近似 L_2范数的方式处理,对于较大的数据误差项用近似 L_1范数的方式处理,可以抑制虚假异常点对反演的影响。对反正切混合目标函数的梯度特征分析表明,反正切加权因子加大了有效模型参数误差梯度分量的权重,从而加速了迭代收敛过程。分析还表明反正切目标函数能够适应 Gauss 和 Cauchy 等不同概率密度分布的数据误差,因此,提高了目标函数的适应能力,扩大了波动方程反演的处理范围。

The weighted hybrid objective function in wave equation inversion.

The objective function based on norm,more sensitive to singular data points,requires the wild data to meet Gaussian distribution.To eliminate the limit,a hybrid ob jective function weighted by anti-tangent function is con- structed,which behaves like an objective function of norm for small data points and like an objective function of norm for large data points,restraining the influence of false data points on inversion procedure.The study of the gradient of anti-tangent-based objective function shows that the anti- tangent weighting factor increases the weights of the gradi- ent of valid model difference and accelerates the inversion procedure.The significance and selection method of control parameters in hybrid objective function,determining how the data difference are treated,are also discussed through model experiments and probability study.The paper also demonstrates that the anti-tangent-based objective function is suitable for wild data sets with different probability dis- tribution,e.g.Gaussian distribution,exponential distribu- tion etc.,enhancing the adjustability of the objective func- tion and extending the range in which wave equation inver- sion works well.