## 4 Conclusions  
In this paper, we examine the effect of the quasi-static approximation on the forward modeling of EM diffusion, based on the electric-field formulation. This is carried out by comparing the numerical modeling results from cases using the quasi-static approximation and using the complete wavenumber. For the quasi-static approximation, the conductivity in the air is set to be small values; for cases with the complete wavenumber, zero conductivity is used for the air. Several simple models are designed to verify the numerical solution and study how the assigned conductivity in the air affects the modeling accuracy. One flat model and two models with topography are then designed to examine the effect of the quasi-static approximation on the EM modeling results.  
The tests on simple models indicate that our numerical solution with the complete wavenumber is accurate. The quasi-static approximation in the air affects the accuracy of EM simulation based on electric-field formulation. The extent of this impact on the modeling results depends on the conductivity used for the air and the conductivity of the earth, as well as the operating frequency. For a given earth conductivity, a saturation air conductivity (a further decrease in conductivity cannot further improve the modeling accuracy) exists. A decrease in the conductivity of the earth can result in a lower upper frequency boundary above which large errors will occur and the quasi-static approximation is violated. While the flat complex model exhibits similar behavior as the two simple models, two models with uneven surface act in a different way. For the two models with undulating surfaces, the accuracy of the numerical results for EM fields deteriorates substantially at much lower frequencies (as low as several hundred Hz), most likely due to the violation of the quasi-static approximation in the air. However,  
Figure 11: Comparison plots of the apparent resistivity (**a,c**) and phase (**b,d**) from the case based on the quasi-static approximation (upward-pointing triangle lines)) with \(10^{-6}\) S/m for air conductivity, and the one with the complete wavenumber (circle lines) at frequency of \(5\times 10^{2}\) Hz for the steep hill model. The relative errors are referenced to the results based on the use of the complete wavenumber.  
the apparent resistivity and phase results are generally accurate at the quasi-static approximation at frequencies considered in this paper. For EM explorations in uneven topographical areas based on the interpretation of EM fields, the quasi-static approximation is likely to be problematic at frequencies as low as several hundred Hz. In such applications, the use of the complete wavenumber may be required.  
Conceptualization, R.G.; methodology, J.L. and R.L.; software, Y.X. and Z.L.; validation, R.L. and C.C.; formal analysis, J.L.; investigation, C.C.; resources, J.L. and R.G.; data curation, Y.X. and Z.L.; writing--original draft preparation, Y.X.; writing--review and editing, R.G. and Z.L.; visualization, R.L.; supervision, J.L.; project administration, J.L.; funding acquisition, J.L. and Y.X.; All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China under Grant 41674079, Grant 41674080, and Grant 41404061, and in part by the Project of Innovation-Driven Plant from Central South University under Grant 2018 zzts698, and in part by the project of Hunan Provincial Natural Resources Department under Grant 2018-04.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.