| In geophysical exploration,the direct current(DC)resistivity method and the timedomain induced polarization(IP)method have become key techniques due to their unique detection capabilities and application values,playing critical roles in revealing underground geological structures and clarifying material distributions.The DC resistivity method,based on the resistivity differences of underground media,is applied across various scales of subterranean spaces,significantly contributing to mineral resource exploration,engineering surveys,and hydrological and environmental geology.On the other hand,the time-domain IP method leverages the differences in induced polarization responses of underground media under electric fields,offering vital information on the electrochemical properties of these media,proving especially useful in searching for metallic,non-metallic minerals,and exploring water resources.Research and application of three-dimensional DC resistivity and time-domain IP methods encounter numerous challenges.Complex terrain conditions significantly impact the inversion issues associated with both methods.Under a stable current source,irregular terrain features can cause complex alterations in the propagation paths of underground currents and the distribution of electric potentials.This leads to false anomalies when inverting data from undulating terrain areas using flat terrain models,and traditional terrain correction techniques often fail to meet the exploration accuracy requirements in complex three-dimensional terrains.Thus,directly simulating the undulations of complex terrains through structured or unstructured grid models to achieve three-dimensional terrain-inclusive forward and inversion modeling has become a focal point of this study.In the domain of forward and inversion modeling for the time-domain IP method,especially in more mature commercial software,the majority of algorithms are based on Seigel’s linearization theory due to their fast computation speed and high inversion efficiency.However,this approach also has significant disadvantages.Inversion methods based on Seigel’s linearization theory can exhibit substantial deviations in areas with high polarization anomalies due to an unavoidable approximation process inherent in Seigel’s theory.The occurrence of negative polarization anomalies in the inversion results of the time-domain IP method also impacts its inversion accuracy.Thus,seeking a precise inversion method for the time-domain IP method to achieve higher inversion accuracy has become one of the critical issues addressed in this paper.Reducing the non-uniqueness of geophysical inversion has always been an objective in the ongoing development and update of geophysical inversion techniques.One cause of the non-uniqueness issue in geophysical inversion is the limitations of information or data obtained by a single method.DC resistivity exploration and timedomain IP exploration focus on two independent physical properties: resistivity and polarization rate,respectively.These two exploration methods are often conducted simultaneously in practical exploration operations,sharing a set of observation equipment for data collection.This enables joint inversion of these two methods at lower costs and higher efficiency,thus enhancing the accuracy of inversion.Therefore,how to conduct joint inversion of the DC resistivity method and the time-domain IP method is also a key issue addressed in this paper.To address these issues,this paper reviews the research on three-dimensional terrain-inclusive DC resistivity and time-domain IP forward and inversion modeling by scholars both domestically and internationally.Through derivation and analysis,this paper asserts that an unavoidable approximation process exists in the polarization rate forward and inversion modeling based on Seigel’s linearization theory,and simulation experiment results have verified that this approximation can lead to errors in polarization rate forward modeling.To reduce the approximation process,this paper proposes three-dimensional terrain-inclusive time-domain IP forward and inversion modeling algorithms based on deformed hexahedral grids and unstructured tetrahedral grids.The algorithms directly calculate the polarization rate through two resistivity forward or inversion modeling processes,avoiding the errors of indirect calculations.On the basis of the deformed hexahedral grid algorithm,a joint inversion algorithm for three-dimensional terrain-inclusive DC resistivity and time-domain IP data based on cross-gradient structural constraints is proposed.Additionally,for unstructured tetrahedral grids,a joint inversion algorithm for three-dimensional terrain-inclusive DC resistivity and time-domain IP data based on four-direction cosine dot product gradient structural constraints is introduced.In the study of three-dimensional terrain-inclusive DC resistivity forward and inversion modeling based on deformed hexahedral grids,this paper establishes a mapping relationship between actual arbitrary hexahedral grid elements and constructed parametric elements through parametric transformation.This approach utilizes the flexibility of arbitrary hexahedral grids to fit complex undulating terrains and reduces the complexity of calculations through the regularity of parametric elements,thereby achieving three-dimensional terrain-inclusive DC resistivity forward and inversion modeling.Based on this,using an exact inversion strategy for timedomain induced polarization,three-dimensional terrain-inclusive time-domain induced polarization forward and inversion modeling was realized.Model test results show that the aforementioned algorithms can significantly reduce the impact of undulating terrain on three-dimensional DC resistivity inversion and time-domain induced polarization inversion.Additionally,the exact inversion of time-domain induced polarization is actually composed of two direct current resistivity inversions based on different observation data,so structural constraints can be applied in these two direct current resistivity inversions to achieve joint inversion.Utilizing the regular arrangement of arbitrary hexahedral grid elements,cross-gradient structural constraints are applied between arbitrary hexahedral grid elements,ultimately achieving a joint inversion of three-dimensional terrain-inclusive DC resistivity and time-domain induced polarization data based on deformed hexahedral grids.Model test results indicate that the joint inversion algorithm proposed in this paper reduces the influence of nonuniqueness in inversion and improves inversion accuracy.Based on unstructured tetrahedral grids,this paper uses the same exact inversion method as that for arbitrary hexahedral time-domain induced polarization inversion to achieve three-dimensional terrain-inclusive time-domain induced polarization inversion based on unstructured tetrahedral grids.However,in the joint inversion of unstructured tetrahedral grids,due to the irregular arrangement of unstructured tetrahedral grid elements,structural constraint methods based on Cartesian coordinates may inevitably be affected by calculation errors.To address this,a four-direction cosine dot product gradient structural constraint method based on the direction axes between the centroids of adjacent tetrahedral faces is introduced,using the positional relationships between tetrahedra for direct calculation,avoiding indirect calculations of Cartesian axial gradients,thereby reducing calculation errors and achieving a joint inversion of three-dimensional terrain-inclusive DC resistivity and time-domain induced polarization data based on unstructured tetrahedral grids.Model test results show that the joint inversion algorithm applied in this paper achieves more consistent inversion results,reducing the non-uniqueness of inversion to a certain extent.Finally,the joint inversion algorithms based on deformed hexahedral grids and unstructured tetrahedral grids proposed in this paper are applied to the processing of actual measurement data from the upstream lead-zinc mine area of the Xia Galaiaoyi River in the Huzhong area of Heilongjiang Province,China.Using the observed elevation points to establish the research area’s deformed hexahedral grid model and unstructured tetrahedral grid model,joint inversion calculations are then performed.Through analysis and comparison of the joint inversion results,it is believed that the joint inversion algorithms proposed in this paper have certain effectiveness and practicality in actual applications. |
