基于Hinge损失函数的垂向全变差约束全波形反演

全波形反演可以为叠前深度偏移成像提供更高精度的速度模型，但该方法具有较强的非线性，对初始速度模型的依赖性较强，尤其是在实际应用中，地质条件复杂多变，速度变化不连续，增加了反演非线性程度，常常使反演陷入局部极小值，影响反演的精度.全变差约束在图像去噪领域应用广泛，属于非光滑约束，在去噪过程中能有效的保留图像的不连续界面和边缘信息.本文提出基于Hinge损失函数的垂向全变差约束全波形反演方法，在全变差约束的基础上，利用Hinge损失函数控制模型的更新方向，并使用原-对偶混合梯度算法进行求解，给出这一优化问题的迭代格式，有效提高了对地下不连续界面的重构精度，同时也降低反演对初始速度模型的依赖程度.数值算例证明：与常规全波形反演方法相比，基于全变差约束的全波形反演方法可以有效的重构速度模型中的不连续界面，尤其对高速体边缘的重构效果更明显，但该方法对初始速度模型的依赖性仍然较强；基于Hinge损失函数的垂向全变差约束全波形反演方法降低了对初始速度模型的依赖程度，可以从一个较差的初始模型通过循环迭代的方式最终得到同样精确的速度模型，较好的重构了高速体边缘和不连续界面.

Full-waveform inversion with a vertical total variation constraint based on the Hinge loss function

Full waveform inversion can provide a high-precision velocity model for pre-stack depth migration imaging. However, this method has a non-linearity and dependence on the initial velocity model. Especially under the complex conditions of highly variable geology and discontinuous velocity changes, the degree of its nonlinearity increases, leading to an inversion trap in a local minima, lowering the accuracy of inversion.#br#The total variation regularization method, a kind of unsmooth constraints which is widely used in image denoising, can preserve discontinuity interfaces and edges of the image during the denoising. In this paper we propose a full waveform inversion using the vertical total variation constraint based on the hinge loss function. By virtue of this constraint, we use the hinge loss function to control the model's updating direction, and get the iteration format of the optimization problem with the primal-dual hybrid gradient algorithm. This method improves the reconstruction accuracy of underground discontinuous interfaces effectively, and reduces the degree of dependence on the initial velocity model.#br#Numerical examples demonstrate that this new full waveform inversion method is effective. Compared with the conventional full waveform inversion method, it can reconstruct the discontinuous interface effectively based on total variation constraint. Especially the reconstructing effect of the high-speed body edge is more obvious. Although the dependence of the initial velocity model of this method remains strong, It can reduce the degree of dependence on the initial velocity model and get a same accurate velocity model from a poor initial model eventually by means of cyclic iteration. In particular the high-speed body edge and discontinuous interfaces can be reconstructed accurately.