The conventional full-waveform inversion (FWI) often minimizes the objective function using some local optimization algorithms. As a result, when the initial model is not good enough, the inversion process will drop into a local minimum. The low-frequency components contained in seismic data are of vital importance for reducing the initial model dependence and mitigating the cycle-skipping phenomenon of FWI. In this research, a frequency extension method using the nth power operation is proposed, which compresses the seismic data in time domain and extends their frequency band. Based on this, we construct a new objective function using the nth power wavefield and derive the corresponding gradient formula. The new objective function shows better property to overcome local minimum than the conventional one. When conduct inversion, we can invert from high-order to low-order successively, which is a new multiscale strategy. Since seismic data is more sensitive to source wavelet errors after high-order operation, we make the method more robust by proposing a source-independent method to mitigate the effects of source wavelet inaccuracy. After that, we extend the proposed method to encoded multisource waveform inversion. The numerical examples on the Marmousi model demonstrate that the proposed method can effectively mitigate the cycle-skipping of FWI, and it also has good anti-noise property.

传统的全波形反演（FWI）通常使用一些局部优化算法来最小化目标函数。结果，当初始模型不够好时，反演过程将降至局部最小值。地震数据中包含的低频成分对于降低初始模型依赖性和减轻FWI的跳频现象至关重要。在这项研究中，提出了一种使用第n次幂运算的频率扩展方法，该方法在时域内压缩地震数据并扩展其频带。基于此，我们使用n构造一个新的目标函数。功率波场并推导相应的梯度公式。新的目标函数比常规函数具有更好的克服局部极小值的性能。当进行反演时，我们可以从高阶到低阶依次反转，这是一种新的多尺度策略。由于高阶操作后地震数据对源子波误差更敏感，因此我们通过提出一种独立于源的方法来减轻源子波不准确性的影响，使该方法更加稳健。之后，我们将提出的方法扩展到编码多源波形反演。Marmousi模型的数值算例表明，该方法可以有效减轻FWI的周期跳跃，并且具有良好的抗噪性能。