# The Generalized Skin Depth for Polarized Porous Media Based on the Cole-Cole Model
## 1 Introduction  
In practical work, the calculation of the detection depth of the electromagnetic (EM) method is very important for survey design and data interpretation. However, due to the influence of instrument accuracy and noise, the actual detection depth is limited [1]. In the field of EM exploration, the skin depth is usually used to calculate the exploration depth, which is an important parameter for EM data interpretation and imaging [2, 3].  
At present, research on skin depth is based mainly on the plane wave skin depth formula. [PERSON] (1990) proposed the concept of effective skin depth for artificial-source EM methods and discussed the effective skin depth of vertical dipole sources and horizontal dipole sources under different parameters [4]. It was pointed out that the effective skin depth is affected by the emission frequency, conductivity, and size of the emitter source. [PERSON] (2002) studied the effective skin depth of horizontal electric dipoles and large horizontal loop sources using the concept of plane wave skin depth [5]. It was found that the skin depth of the local source is closely related to the EM response. The skin depth of the local source can be used as the optimum parameters in the interpretation of EM measurement designand data solution. [PERSON] (2004) studied the skin depth for an airborne electromagnet, summarized the influence of altitude on this skin depth, and pointed out that the dipole skin depth was much smaller than that of the plane wave skin depth at low frequencies of less than 50 Hz [6]. [PERSON] (2014) calculated the effective skin depths of different EM components and studied their variations with different parameters using the expression of a horizontal ground electric dipole magnetic field in a uniform half space, comparing the field with the skin depth of a plane wave and calculating the effect of the offset on the effective skin depth [7]. [PERSON] (2014) used a large loop transient electromagnet (TEM) to study the detection depth of a small loop system and a large loop and pointed out that the detection depths of the two systems are similar [8]. [PERSON] (2017) studied the relationship between the electrical source offset and detection depth in a transient EM method [9]. It was noted that the maximum detection depth can be obtained when the offset is equal to 0.7-1 times the depth of the target body.  
Many scholars have studied the detection depth of EM methods, and most studies have been based on the classical formula of skin depth; however, the effect of induced polarization (IP) has not been considered. In fact, the structure of underground media is very complex. Especially in the detection of underground porous polarized media or metallic ore, the classical definition of skin depth is no longer applicable. Therefore, it is of great significance to redefine the generalized skin depth based on a fractional conductivity model that can satisfy the propagation law of an underground EM wave.  
The IP effect is related to conductivity and frequency and its dispersion characteristics can be expressed by the Debye model and the Cole-Cole model [10; 11]. [PERSON] (2009) proposed a generalized Cole-Cole model [12]. [PERSON] (2018) proposed new equations for both electric conductivity and normalized chargeability [13].  
In this paper, we take the classic Cole-Cole model as an example. Based on the Cole-Cole conductivity model of underground polarized media and the principle of plane wave propagation, a generalized skin depth formula is deduced to calculate the skin depth for polarized media, and the characteristics of skin depth under the polarized media model are analyzed. Particle swarm optimization (PSO) is used to extract polarization parameters and image the mineral resources with IP effects. The purpose of this paper is to improve the interpretation accuracy of underground polarized media and to illustrate the application of the proposed method in survey design and data interpretation.