波动方程反演中基于势函数的约束处理方法研究

地球物理反演问题很复杂，具有多解性，解决方法之一是在反演中引入先验信息，对解空间进行约束，增加反演过程的稳定性。首先对约束条件表示方法进行了讨论；然后论述了利用势函数描述约束条件的方法，即利用模糊数表示先验约束信息的不确定性，通过指数函数构造约束势函数，这种势函数能够描述地震资料反演中的等式约束、区间约束和方向约束；对地震资料波动方程反演中的区间约束、梯度约束和方向约束进行了详细讨论，分析了约束势函数的梯度特征，并将约束势函数结合到目标函数中；最后对梯度约束和方向约束势函数在应用中存在的问题进行了分析。梯度特簿分析和模型试验表明，约束势函数通过对解空间大小的约束，增大了有效信息在梯度场中的比重，提高了梯度计算的精度，增加了反演过程的稳定性，提高了收敛速度。

The potential-function restriction study on wave equation inversion.

To confine the extension of inversion results, increase the stability of seismic inversion procedure, a fuzzy expression is presented to express the uncertainty of transcendental restriction information; an exponential function is applied to construct a restriction potential function. The new restriction potential function woks like a punishing function, giving a value near zero when the value of restriction function is approaching the mid-point of the restriction range, while increasing the value rapidly when the value of restriction function goes beyond the restriction range. A set of potential templates is presented to simplify the construction of potential function, the range restriction, gradient restriction and direction restriction are studied in detail, the restriction potential function is integrated with the objective function. Problems raised in gradient and direction potential function application are also discussed. Gradient a-nalysis and modelling experiments show that restriction potential function can not only increase the share of valid information in gradient, improve the precision of gradient computation and the stability of inversion procedure, but accelerate the convergence as well.