## 1 Introduction  
Electrical exploration is one of the main methods of geophysical exploration. The resistivity method and TDIP are two branch methods of electrical exploration. The resistivity method studies the differences in electrical conductivity (expressed as resistivity) between different underground rocks, while TDIP studies the difference in the IP effect (expressed as polarizability) of electric field changes in a certain time span under the excitation of the stable current of different rocks and ore, and the polarizability of the medium is expressed through the ratio between the potential measured just after canceling the external field (secondary potential) and the potential measured just before canceling the applied field (primary potential). With the innovation of methods and technologies, the resistivity method and TDIP have been widely used in geological structure exploration, environmental pollution investigation, geological disaster warning and other fields [1; 2; 3]. With the continuous development of separate method inversion algorithms and the advancement of computing science, the accuracy of geophysical inversion has been increasingly scrutinized strictly. [PERSON] and [PERSON] proposed the integration of various geophysical data for joint inversion [4]. After more than 40 years of development, joint inversion hasbeen shown to play a vital role in suppressing the ambiguity of geophysical inversion and improving the accuracy of inversion. Joint inversion has become an important quantitative interpretation method in comprehensive geophysical research, and has promising developmental and application prospects [5]. Joint inversion refers to the simultaneous inversion of a variety of geophysical data, using different physical parameters to comprehensively interpret the same target and finally obtaining an inverted model that represents the actual geological situation more closely. Joint inversion is mainly divided into two categories: physical coupling and structural coupling. Joint inversion based on physical property coupling takes advantage of the connections between different known geophysical parameters (velocity, density, resistivity) or deduced parameters (saturation, permeability, reservoir porosity) and unknown geophysical parameters, and strives to strengthen the linear or nonlinear correlations of the physical properties to be inverted by incorporating all kinds of known physical parameters into an objective function. Joint inversion based on the coupling of physical properties has been applied in practice [6; 7]. However, it is a complicated and time-consuming process to obtain prior information on rock physical properties, and the multimodal nature of rock physical data makes it almost impossible to accurately determine the spatial applicability of these rock physical relationships in complicated underground geophysical cases [8]. Although physical coupling applies stronger constraints than structural coupling, the inversion results will be undermined if the physical constraint conditions of rock and the true geological conditions of the study area are not well matched [9]. Structural coupling joint inversion is based on the premise that the underground rock mass structures and boundaries of different physical properties are identical or partially identical. The coupling is derived from the spatial structure similarity of different physical properties; therefore, the prior information of certain known physical properties is no longer mandatory, which makes it more applicable in practice. [PERSON] and [PERSON] proposed a scheme of joint inversion with the introduction of the cross-gradient constraint and carried out a study on the two-dimensional cross-gradient joint inversion of seismic travel time data and the resistivity data [10; 11]. The core concept is to minimize the data misfit and the cross gradient of the models of different physical properties at the same time. When the cross gradient of the models is close to zero, the two models are structurally similar.