## 1 Introduction  
The marine controlled-source electromagnetic method (MCSEM) uses controllable artificial field sources to transmit electromagnetic signals in the ocean and measures the electric or magnetic fields at a location far away from the field source to detect the electrical distribution of the media below the seabed. The resistance characteristics of oil and gas reservoirs are the most important properties, which can generate characteristic surface electromagnetic signals. In other words, MCSEM technology can distinguish underground oil and gas from other fluids. In theory, CSEM measurement is to record data with multiple power-receiver offsets, several different frequencies, and a certain power-receiver arrangement. The marine controlled-source electromagnetic (MCSEM) method has become a popular geophysical exploration tool for offshore hydrocarbon (HC) exploration [1; 2; 3]. During the last decade, marine CSEM has been widely used to reduce ambiguities in data interpretation and reduce the risk of exploration. With the rapid development of MCSEM exploration instruments, large-scale and high-quality data have been obtained, resulting in the need to develop high-precision MCSEM three-dimensional (3D) data interpretation tools. The 3D inversion algorithm technology of MCSEM is commonly used for data interpretation. Forward modeling is the basis of inversion, and high-precision forward modeling technology is key to achieving high-precision inversion.  
The integral equation (IE) [4; 5; 6], finite difference (FD) [7; 8; 9; 10; 11; 12], and finite element (FE) methods [13; 14; 15; 16; 17] are popular numerical techniques for electromagnetic (EM) field modeling. In the FE method, it is easier to use an unstructured mesh to create an irregular body, locally refine the interest region, and coarsen the boundary area of the model domain [15]. With this processing, numerical modeling becomes efficient and effective. Therefore, these advantages make the FE method more suitable for simulating the complex EM response of irregular models than FD and IE methods.  
Compared with the conventional finite element method, which is based on a node, the edge-based finite element method has the advantage that divergence-free conditions are automatically satisfied by appropriately selected basic functions [18]. In recent years, the vector finite element method based on edge elements has been used to solve the electromagnetic field problem, and is known as Maxwell's equations [19; 20]. It should be noted that to date, most formulations for the 3D CSEM problem have implemented the lowest order. In addition, some studies have used higher-order elements to solve three-dimensional forward modeling of CSEM, but these were all aimed at tetrahedral elements [14; 21]. To the best of our knowledge, no high-order hexahedron element is currently used for three-dimensional forward modeling of the controlled-source electromagnetic method. As we know, high-order edge elements can provide accurate solutions but result in an increase in unknown variables and lead to larger computing costs. Fortunately, we can improve the accuracy of the solution at a small computational cost by eliminating some variables when using higher-order elements [22; 23].  
In this study, we applied the unification method to the edge to reduce one edge variable on each side of the conventional serendipity and the Lagrange-type quadratic hexahedral element, resulting in two new types of second order edge elements. We also introduced a new vector FE method based on the proposed elements to solve the 3D marine CSEM modeling problem in an anisotropic medium. For a complex bathymetry simulation, we used a general hexahedral element to discretize the modeling domain. We validated our code by using several models in isotropic and anisotropic media.