## 1 Introduction  
Ground-airborne frequency-domain electromagnetic method (GAFEM) is one of the important geophysical exploration methods, which has high efficiency and is especially suitable for geological surveys under complex terrain conditions [1; 2; 3; 4]. In frequency-domain electromagnetic interpretation, the received electromagnetic response is typically transformed into apparent resistivity, apparent depth, and other parameters that can characterize the subsurface electrical distribution. This enables the generation of visual cross-sectional images depicting the electrical properties of the subsurface medium and provides a foundation for subsequent interpretation work [2; 5]. At present, the definition of the apparent resistivity in GAFEM is based on electromagnetic fields generated by a horizontal electric field source on the surface of a homogeneous half-space, and it only requires the \(z\)-component of the magnetic field [2]. Based on the monotonic relationship between the \(z\)-component magnetic field and resistivity, an iteration-matching procedure to calculate the apparent resistivity, based on the quasistatic response for a homogeneous half-space, was developed. Currently, the calculation of apparent resistivity involves magnetic fields. However, the magnetic field is sensitive to the interface where the resistivity varies, so the recognition resolution of the magnetic field is ideal for one-dimensional objects, but limited for three-dimensional objects. To enhance the resolution of targets, [PERSON] et al. inverse the Hessian estimated by the BFGS method and realize a significant improvement of the inversion convergence and inversion results. It reduces the number of iterations by 2 to 4, but still needs 50 iterations [6]. [PERSON] et al. present an improved CG algorithm that needs fewer iterations. However, the method has poor imaging ability for multiple object cases [7]. [PERSON] et al. propose a full magnetic gradient orthonormal basis function method to effectively characterize magnetic anomaly of magnetic targets, and the SNR of the method is improved greatly. However, seven orthonormal functions are required to realize the method, which is tedious [8]. [PERSON]'s research demonstrates that the magnetic field gradient is more sensitive to the center of three-dimensional objects, thereby enhancing the recognition resolution, and realizing rapid imaging of space magnetic gradient anomaly [9]. Nevertheless, there is a lack of information regarding the calculation of the apparent resistivity derived from the magnetic field gradients. In this study, we investigate a conversion method for converting the apparent resistivity from magnetic field gradients, and compare it with the apparent resistivity calculated directly from the magnetic field. Due to the monotonic relationship between the magnetic field gradient and underground resistivity, the iteration-matching procedure can be utilized for calculating apparent resistivity theoretically. However, the calculated apparent resistivity by the electromagnetic field gradient of the uniform earth has a background value [10], which needs to be corrected to avoid introducing false anomalies and affecting the resolution of anomalous objects. Since the background apparent resistivity is closely related to the background resistivity of the model, it is difficult to obtain the background apparent resistivity when solving an unknown model. Therefore, it is essential to establish a correlation between the magnetic field gradient and the apparent resistivity after correcting, i.e., apparent resistivity variation. In GAFEM, the relationship between resistivity and magnetic field response is implicit, so the iteration-matching algorithm is predominantly employed for the translation process [2]. The magnetic field undergoes a gradient calculation to derive the magnetic field gradient, which exhibits a more implicit relationship with resistivity variation.