基于对数目标函数的跨孔雷达频域波形反演

Frequency domain waveform inversion of cross-hole GPR data based on a logarithmic objective function

波形反演在探地雷达领域的应用已有十余年历史,但绝大部分算例属于时间域波形反演.频率域波形反演由于能够灵活地选择迭代频率并可以使用不同类型的目标函数,因而更加多样化.本文的频率域波形反演基于时间域有限差分(FDTD)法,采用对数目标函数,可在每一次迭代过程中同时或者单独反演介电常数和电导率.文中详细推导了频率域波形反演的理论公式,给出对数目标函数下的梯度表达式,并使用离散傅氏变换(DFT)实现数据的时频变换,能够有效地减少大模型反演的内存需求.在后向残场源的时频域转换过程中,提出仅使用以当前频点为中心的一个窄带数据,可以消除高频无用信号的干扰,获得可靠的反演结果.为加速收敛,采用每迭代十次则反演频率跳跃一定频带宽度的反演策略.实验证明适当的频率跳跃能够在不降低分辨率的基础上有效地提高反演效率.通过两组不同情形下合成数据反演的分析对比,证明基于对数目标函数的波形反演结果准确可靠.最后,将该方法应用到一组实际数据,得到较好的反演结果.

Ground penetrating radar（GPR）tomography plays an important role in geology,hydrogeology and engineering investigations.Traditional tomography（i.e.,the first-arrival times and maximum first-cycle amplitudes）based on ray theory cannot provide high-resolution images because only a fraction of the information contained in the radar data is used in the inversion.In recent years,waveform inversion is one of the biggest concerns because it can provide sub-wavelength resolution images.Waveform inversion has been applied in GPR over ten years,but most of the results are computed in the time domain.In the frequency domain,the choice of inverted frequencies is flexible and different types of objective functions can be used,therefore the results of frequency-domain waveform inversion are more diversified than that of the time domain.In this paper,waveform inversion is implemented by means of a finite-difference time-domainsolution of Maxwell′s equations and a logarithmic objective function is applied.Permittivity and conductivity can be updated simultaneously or separately at each iterative step.The derivation process of the formulas is described in detail and we show the specific expression of gradient under the logarithmic objective function.It is important to note that the gradient formula of the frequency domain waveform inversion is different from the gradient formula of the time domain waveform inversion.The reason is that the cost function is essentially different.Discrete Fourier transform（DFT）is applied to transform time domain data into the frequency domain,which only increases a few calculations in the inversion.The method can greatly reduce memory requirement when the inverted model is on a large scale.When transforming the back-residual source,we present that only a narrow-band data whose center is the current frequency is used.The method can effectively reduce the interference of high frequency information,so reliable inversion results can be obtained.In order to accelerate the convergence,a special frequency strategy is applied.The inversion frequency skips a number of bandwidth after every ten times iterative step.Results turn out that the strategy can efficiently improve the inversion efficiency and does not influence the resolution.To verify the effect of our algorithm,we test it on two different synthetic data.Then we compare with the results of conventional waveform inversion：（1）Two small cylindrical bodies of permittivity in a layered medium.（2）A layered medium with multiple embedded cylindrical inclusions of permittivity and conductivity.Finally,we apply the logarithmic algorithm to field data.The results on synthetic data show that the logarithmic waveform inversion is better than the conventional waveform inversion.When permittivity and conductivity are inverted at the same time,the results of logarithmic waveform inversion can accurately reconstruct both shape and location of the inclusions.This is because that the amplitude of the residual is normalized by the amplitude of the modeled wavefield.The inversion results of the field data also prove the practical value of logarithmic waveform inversion.