Time Domain And Frequency Domain Waveform Inversion Of Crosshole Radar Data And Comparative Research

Ground penetrating radar(GPR) tomography play an important role in geology,engineering geology and hydrogeology. But traditional tomographic techniques(such as first-arrival times and maximum first-cycle amplitudes) based on ray theory cannot provide high resolution images because only a frection of the information contained in the radar data is used in the inversion. In recent years, waveform inversion is one of the most famous methods because it can provide sub-wavelength resolution results.Crosshole radar, as a highly efficient near-surface geophysical technology, can provide high resolution images between the two bore-holes. So we can get accurate and high-resolution subsurface dielectric information by combining waveform inversion and crosshole radar data.Because of the great amount of calculation, the gradient methods(such as steepest descent method and conjugate gradient method) are used to obtain the optimal solution of waveform inversion. These methods require that the initial models must be close to the real models to avoid converging into local minimum. Usually, the initial models are obtained by ray theory. The speed results from first-arrival times can be transformed into the relative permittivity and attenuation tomography has conductivity. This means that we have to extract information of first-arrival times and attenuation before the waveform inversion. In the processing of field data, the signal-to-noise ratio is usually not high. It need a lot of time to manually extract these information. Laplace domain waveform inversion is a new way to obtain initial models.Firstly, we give the detailed formula of gradient and step length of crosshole radar waveform inversion in the Laplace domain. The electric field value in Laplace domain equal to the zero frequency component of damping electric field, so the Laplace domain waveform inversion can get long wavelength scale results. Because the finite difference time domain method(FDTD) is used in the modeling process, the inverse of the real valued Laplace transform is an illed-posed problem. In this paper,the back propagation field is obtained by backward propagating the given sources.The amplitude of these given sources equal to the size of the Laplace domain residual.Then the choosen of damping constant, iterative strategy and source wavelet estimation are analyzed. Note that, the difference between the gradients of permittivity and conductivity is only a fixed ratio(the value of damping constant) in the Laplace domain waveform inversion. In the inversion process, the gradient of permittivity and conductivity are exactly the same. We propose that the permittivity and conductivity should be cascaded updated in the Laplace domain inversion to avoid the failure of convergence. The data used in the Laplace domain waveform inversion is the same as that of the time and frequency domain waveform inversion,so there is no need to do extra processing of the data. In this paper, the results of Laplace domain waveform inversion are used as the initial models of the synthetic data inversion in the time domain and frequency domain.Then, we obtain the gradient from the derivative of object function in the time domain waveform inversion, and the introduce of virtual source vector make the algorithm easy to undenstand. Permittivity and conductivity are updated by using the conjugate gradient method. The forward modeling use the high order FDTD method based on CPML absorbing boundary. In order to improve the convergence and stability of the inversion, the logarithm of permittivity and conductivity is applied. In the inversion of field data, the 3D correction of field data and the estimation of source wavelet are important. The results of synthetic data turn out the 3D correction is right.And two examples of field data are shown.In the waveform inversion of field data, the source wavelet is unknown.Normally, the source wavelet can be estimated by using a deconvolution method. In the method, the source wavelet is a new unknown parameter added in the inversion and updated with iteration. When the results of inversion are same to the true models,the estimated source wavelet is same to the true source wavelet. The method is useful in the inversion of synthetic data, but it doesn’t perform well in the field data. Lots of intervention is needed to choose the best source wavelet after the estimation. The reason may be that the subsurface medium of field data is highly complex and the signal-to-noise ratio of the field data is low. We realize a source-independent time-domain waveform inversion. Firstly, the observed wavefields are convolved with a reference trace of the modeled wavefield, then the modeled wavefields are convolved with a reference trace of observed wavefield. A new object function isbased on the convolved wavefields. In theory, no matter which source wavelet is used in the forward modeling. The source wavelet of the observed and the modeled wavefields are equally convolved with both terms in the object function, so the effect of the source wavelet is removed. Another important feature of this object function is that the modeled wavefields act as a filter of the field wavefield based on the frequency range of the source used for modeling. So it is very easy to employ a multiscale strategy.Waveform inversion has been applied in GPR over ten years, but most of the results are computed in the time domain. In 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 time domain. The frequency domain waveform inversion is implemented by means of a FDTD solution 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 is different from the gradient formula of the time domain. The reason is that the cost function is essentially different. Discrete Fourier Transform(DFT) is applied to transform the data from time domain into frequency domain, which only increases a few calculations in the inversion. The method can greatly reduce the memory requirements when the inverted model is in a large scale. When transform 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 influence of high frequency information, so reliable inversion results can be obtained. In order to accelerate the convergence, a special frequency stagecy is applied. The inversion frequency skips 10 MHz after every ten times iterations step. Results turn out that the strategy can efficiently improve the inversion efficiency and does not influence the resolution.Alao, we realize a source-independent frequency domain waveform inversion. In frequency domain, there are two kind of source-independent algorithms: one based on the deconvolution and one based on the convolution approach. In the deconvolution-based approach, the wavefields are normalized by a reference wavefield and expressed as a ratio of Green’s functions in which the source functiongets canceled. On the other hand, the convolution-based approach for the source-independent waveform inversion in wich the wavefields are multiplied by a cross-reference wavefield. In the paper, we propose a new frequency domain source-independent objective function based on the logarithemic wavefields. The new objective function can be regarded as both the deconvolution and the convolution approach, and can be simplified into a simple form which only contains addition and subtraction.Inversion results of simple models in the time and frequency domain are used to verify the methods’ s validity. Then, inversion results of complex models turn out the capability of small size anomaly bodies in the complex environment. In the end,inversion results of field data are shown.This paper show the results of waveform inversion in the Laplace domain, time domain and frequency domain. In order to study the behavior of different types of waveform inversion, we calculate the two-dimensional image of different objective functions based on a specific model. And a comparison between the behavior of objective functions for different waveform inversion is given in detail. In the last part,the inversion results of same models are compared.