基于Huber函数的频率域全波形反演

Full waveform inversion in frequency domain based on Huber function

全波形反演利用波形整体特征,是一种高分辨率的成像方法。然而在实际应用中,该方法受到各种噪声的制约。传统的全波形反演采用L2范数准则来构建目标函数,受噪声影响大。L1范数相对于L2范数来说抗噪性强,但数据误差接近于零时求取的目标函数梯度会出现奇点而导致反演不稳定。为此,将L1范数和L2范数结合,引入了复数形式的Huber函数准则来建立目标函数,在数据误差较小时采用L2范数,在误差较大时采用L1范数。在此基础上,推导出新的Huber目标函数的梯度表达式,并利用预梯度法进行全波形反演。通过在数值模型的合成记录上分别加入随机脉冲噪声、连续噪声以及高斯噪声来验证本方法的正确性。结果表明:相对于传统的L2目标函数,Huber目标函数既具备抗噪性,又有稳定性,即使在噪声的影响下也能得到较好的反演结果。对于随机脉冲噪声和连续噪声,Huber函数的抗噪性要优于L2范数,而对于高斯噪声则两者差别不大。

Full waveform inversion is a high-resolution imaging method by using the overall waveform characteristics.It is limited by many kinds of noises when applied to the real seismic data.The L2 norm criterion is used to construct the target function in traditional full waveform inversion.In terms of anti-noise,the L1 norm is more robust than the L2 norm.However,the gradient of target function based on L1 norm has a singularity and are unstable,when the residual becomes zero.To achieve robustness and stability in waveform inversion,a new complex-valued Huber function by combining the L1 norm with L2 norm is constructed.The Huber function uses the L1 norm when the data error is large and the L2 norm for small data error.On the basis of this,the gradient formula of the Huber function is derived,and the preconditioned gradient method is used for full waveform inversion.The new algorithm is proved by synthetic seismic data with the random spike noise,the coherent noise and the Gaussian noise.Numerical tests results show that the Huber function is more robust and stable than L2 norm even with the noises.In terms of anti-random spike noise and anti-coherent noise, the Huber function is better than L2 norm compared with the leastsquare method.Both of these two methods could get similar results for the Gaussian noise.