Cross-gradient Joint Inversion Of Ground-penetrating Radar Data And Seismic Data Based On The Full Waveform Inversion

Full waveform inversion(FWI)is a high-resolution imaging tool in the geophysical methods,which is widely used for seismic exploration and ground-penetrating radar(GPR)in recent years.FWI belongs to a least-square optimization problem that the best optimized model can be obtained by continually minimizing the misfit between the observed and simulated data.FWI has a higher resolution than the ray-based tomography technique.Tomography images the velocity profile or the attenuation profile of the subsurface medium using the traveltime or amplitude of the data.The resolution of tomography is approximately the width of the first Fresnel zone.In contrast,FWI can take full use of the waveform by the consideration of both traveltime and amplitude of data.Therefore,the resolution of FWI approximately ranges from one-half to one-third of the wavelength.When the scale of target much less than the dominant wavelength of the signature,the target only can be imaged using the FWI methods.FWI is developeded in the seismic exploration and then widely used to image the velocity structure of medium in the deep area or crosshole.In recent years,FWI becomes more and more important in the large-scale 3D seismic explorations,which is benefited from the developing of computing technology and optimization methods.Meanwhile,there are more and more small-scale applications using the FWI method,such as civil engineering and environment monitoring.Meanwhile,FWI also gets rapidly development for the processing of GPR data.GRP FWI is widely used in hydrocarbon exploration to recover the electric properties of medium,i.e.permittivity and conductivity,such that other attributes of the medium(i.e.porosity and water saturation)can be evaluated.Joint inversion is an effective way to reduce the uncertainty of geophysical inverse problem.It is critical for the integrated interpretation of different geophysical datasets.Using the petro-physical empirical relationship or structural similarity between models,joint inversion combines multiple parameters into the inverse process and improves the quality of comprehensive interpretation of the study area.However,the resolution of joint inversion is limited by the imaging methods used in the joint procedure.Joint inversion of seismic and GPR datasets is usually implemented based on the tomography technique before,therefore the resolution of joint inversion will be improved if the tomography technique is replaced by the FWI methods.A joint inversion algorithm for the combination of the acoustic FWI and GPR FWI is presented in this paper.A joint objective function including individual misfit function of both acoustic FWI and GPR FWI as well as several cross-gradient functions is established,aiming to simultaneously invert the elastic and electric properties of medium(i.e.P-wave velocity,permittivity and conductivity).Joint inversion procedure is implemented in the frequency domain for the calculation of several frequencies.The simulation of acoustic wave and electromagnetic wave is performed using the finite-difference frequency-domain method.The optimization of joint inverse problem is solved by the truncated Newton method which can consider the Hessian matrix in the FWI problem.The corresponding wavelength of acoustic wave and electromagnetic wave in each coupled frequencies are required to be similar in the joint inversion procedure.The final updated models of P-wave velocity,permittivity and conductivity,which are sequentially inverted in the current frequencies,are used to be the initial models in the next frequency.The forward problems of the acoustic wave and electromagnetic wave are necessary to be solved firstly for the realization of the 2D joint FWI algorithm.The acoustic wave is simulated with the scalar wave equation for a homogeneous isotropic medium.The electromagnetic wave is simulated with the transverse electric mode(TE)of electromagnetic wave through restricting the Maxwell's equations to a 2D geometry.A 9-point finite-difference frequency-domain method is used to discretize the wave equations.The linear equations can be expressed in a matrix form.The perfectly matched layer absorbing boundary condition is used to absorb the reflections.The acoustic wave and electromagnetic wave is solved using lower-upper triangular layer decomposition of equations.In this paper,several numerical examples are used to show the effect of joint FWI approach,including three crosshole datasets and three on-ground datasets that generated with simple models or realistic models.The accuracy of inverted models in the individual and joint inversions is quantitatively compared using the root mean square(rms)error of models.The convergence cures of inversions and the variation of cross gradient terms are plotted to show the convergence speed and improvement of model structural similarities.The results show that the structural similarity between models is significantly enhanced in the joint inversion,resulting in all models forming a uniform structure.In this process,the uncertain structures in each model are reduced.Meanwhile,the accuracy of inverted models,which is evaluated by rms error of models,is also obviously improved with the variation of structures.Besides the structure,the artifacts in the conductivity model are also significantly reduced.