Inversion Of Airborne Time-domain Electromagnetic Data Based On Bayesian Theory

Airborne electromagnetic method(AEM),as a cost-effective means for the exploration of large areas,has the effect that is difficult to achieve with the general exploration methods in complex areas where it is difficult to carry out the ground exploration operations,such as Gobi,Seas,High mountains and Forest.With the improvement of AEM hardware system and computer performance,an inversion interpretation method with higher precision is urgently in demand.The traditional inversion methods based on deterministic theory take the inversion parameter as a deterministic value,and the obtained inversion result is a single optimal solution,but they can not provide the uncertainty information of inversion parameters,and can be easily trapped at local minima.The Bayesian inversion methods based on statistical theory regard the inversion parameter as a random variable,and the obtained inversion result is not a certain value,but the probability distribution of the parameter.According to the probability distribution of the parameters after the inversion,the Bayesian inversion method can not only obtain the best solution,but also get the uncertainty information of the inversion parameters,so that the inversion results can be effectively evaluated.In addition,when both the likelihood function and prior information in the Bayesian inversion method are Gaussian distribution,the Bayesian inversion method can be simplified to the Tikhonov regularization inversion method which is the mainstream practical method in the inversion interpretation of AEM data.Therefore,studying the Bayesian method of AEM data can achieve more sufficient and effective quantitative analysis and evaluation of the inversion results.In order to obtain more accurate inversion results,and uncertainty information of inversion parameters,this paper studies the inversion methods of airborne time-domain electromagnetic method(ATEM)based on Bayesian theory,including the influence of initial model for inversion,regularization inversion method,fixed-dimensional Bayesian inversion method and trans-dimensional Bayesian inversion method.At first,we discussed the theoretical basis of forward and inverse of ATEM.For forward modeling,the analytical solutions and numerical solutions of frequency domain and time domain in one-dimensional forward modeling were compared by using the uniform half-space model.Then,we used the uniform half-space model and the layered model to verify the consistency of the one-dimensional and 2.5-dimensional forward modeling results.The accuracy and consistency of the forward modeling results provide accurate and reliable simulation data for the subsequent research on inversion methods.For the inversion problem,we analyzed the basic solving methods of inverse problem,and the relationship between the deterministic regularization inversion method and the Bayesian inversion method.It provides a correct theoretical framework for the subsequent research on the inversion method.To solve the inversion problem,an initial model is usually needed.The quality of the initial model directly affects the inversion results,which make the selection of the initial model is difficult.However,conductivity-depth imaging(CDI)does not require an initial model and is a fast approximation interpretation approach of AEM data.In order to analyze the influence of the initial model,this paper discussed the effects of the uniform half-space model and CDI result as initial models on the inversion results of the traditional damped eigenparameter inversion method and the Occam inversion method.The results show that taking CDI result as initial model can improve the precision of the inversion results and speed up the convergence rate of the inversion process.At the same time,it can solve the difficult problem of initial model selection.Therefore,in the research of the inversion method in this paper,we not only use the result of CDI as the reference model for regularization inversion method,but also use it as the initial state of Markov chain in the Bayesian inversion method.For regularization inversion,we presented a combining regularization inversion method that can invert all ATEM soundings independently.The inversion method applies three regularization constraint terms simultaneously,that is,the conductivitydepth imaging result is used as the reference model for inversion,and the inversion result of the previous sounding is used as the lateral constraint of the current sounding inversion,and the roughness of the vertical direction of the model is used as a vertical smooth constraint.First,the combining regularization strategy was tested on synthetic data to analyze the influence of the first and second derivatives of vertical model roughness and that of the weight factors of the three regularization terms.Then,the practicability of the combining regularization inversion method was verified by field data.Inversion results show that taking the CDI result as the reference model of inversion can obtain better results than that of taking the uniform half-space model as the reference model,at the same time,lateral and vertical constraints can ensure the continuity in the horizontal direction and the smoothness in the vertical direction of inversion results.So this combining regularization inversion strategy is a practical inversion method of ATEM data.For the fixed-dimensional Bayesian inversion,in this paper,an improved Markov chain Monte Carlo(MCMC)method was proposed.This method forms a combining model sampling update method which combines the advantages of the blockwise updating and componentwise updating sampling.In addition,the CDI result is used as the initial state of Markov chain.Firstly,the effect of the combining model sampling update method was tested by the modeling data of a three-layer model,and the effects of the random initial state and the CDI result as the initial state were discussed.The results show that this improved MCMC method can effectively shorten the burn-in period of Markov chain and improve the overall efficiency of model sampling after the burn-in period;It is better to use the result of CDI as the initial state than the random initial state,which can avoid the situation that the initial state is too different from the real model,resulting in the non-convergence of Markov chain and the waste of sampling.Secondly,the modeling data of the three-layer model was used to analyze how to improve the acceptance rate of the model sampling based on data error and proposal distribution.It is pointed out that setting a suitable data error or adopting a random mixture of Uniform and Gaussian distribution can effectively improve the acceptance rate.Then,the dimension of the model in the fixed-dimensional Bayesian inversion was discussed.It is concluded that the overall effect of the inversion is better when the dimension of the model parameters is close to the real situation.Finally,the improved MCMC method is further verified by two-dimensional layered model data with Gaussian noise and field data.The experiments revealed that,in the model sampling results obtained by the improved MCMC method,the model parameters estimated by the mean and median are relatively stable and reasonable.And compared with the results obtained by the combining regularization constraint inversion method,the results obtained by the improved MCMC method show more clear demarcation between different media layers,and also have better lateral continuity.As for the trans-dimensional Bayesian inversion,we proposed an improved Reversible jump MCMC method,which not only includes the improvement measures of the MCMC method mentioned above,but also automatically adjusts the sampling step length in the sampling process through the data fitting error of each sampling model,so as to avoid the influence of unreasonable sampling step.At the same time,when the new model parameters are generated,it is recommended to select the corresponding update status by different probability ratios.In addition,the height of transmitter and data error are also taken as inversion parameters to be estimated.The improved Reversible jump MCMC method was tested by the forward modeling data of onedimensional three-layer and four-layer models,and two-dimensional layered model,and the field data,respectively.The results show that compared with the random initial state and initial state with the best half-space model,the acceptance rate of model sampling is higher when the CDI result is used as the initial state,and the sampling step adjusted automatically can further improve the acceptance rate of the model.In summary,compared with the combining regularization constraint inversion method and the fixed-dimensional Bayesian inversion method,the inversion results obtained by the trans-dimensional Bayesian inversion method is better,more versatile and applicable.