# Three-Dimensional Wide-Band Electromagnetic Forward Modelling Using Potential Technique
## 1 Introduction  
The effectiveness and accuracy of the electromagnetic (EM) responses in heterogeneous media are vital for exploration and delineation of the geological formation [1]. The significance of the electromagnetic technique in geophysical perspective is the diverse application to a range of geological problems from shallow to deep investigation resulting from the sensitivity of the occurrence of fluids, mineralogy, etc. [2]. Numerical simulation has been a typical scheme for modeling and analyzing three-dimensional electromagnetic data [3]. Additionally, an efficient inversion of the electromagnetic data depends on the forward model algorithm. There are various numerical techniques used to solve the three-dimensional electromagnetic forward problems such as the differential equation (finite difference (FD), finite element (FE), finite volume (FV)) and the Integral equation (IE) method. The integral equation (IE) solutions avoid the discretization of the whole domain because only the anomalous domain is discretized [4; 5; 6]. As a result, the degree of freedom is minimal. Though, the system of linear equations is compact and dense. Several researchers have improved this method [7; 8; 9; 10]. However, for the differential equation such as the finite difference [11; 12; 13; 14], it has been extensively applied because of the simplicity of the approach [15]. It requires large discretization of the computational domain because it consider both the background and anomalous region. It allows for complex earth modelling. The widespread use of the finite element approach is due to the comprehensive mathematical basis [16; 17]. The finite elementscheme can be applied to both structured [18; 19; 20] and unstructured grids [21; 22]. The governing differential equation is formulated using the weighted residual. The Finite volume can be utilised on both structured [23] and unstructured form [24]. These unstructured grids are essential for accurate modelling of complex and irregular shaped geometry such as the bathymetry of the earth surface. In addition, the capacity to refine the grid locally in the zone of interest allows an efficient discretization of complex earth model [25; 26]. The discretized numerical solution of the Helmholtz equation resulted in continuity loss at low frequency for an iterative solver [27]. we can alleviate this problem using the divergence correction [27; 28]. On the other hand, we can implement the potential technique. Several investigation has adopted the use of potentials (magnetic vector \(\mathbf{A}\) and electric scalar \(\varphi\)) in 3-D forward simulation to solve specific geo-electromagnetic problems [29; 30; 31; 32]. Reference [33] applied the \(\mathbf{A}\)-\(\varphi\) disintegration of the electric field using the finite volume method to solve magnetotelluric problems. Reference [22] utilized the nodal based finite element approach of the \(\mathbf{A}\)-\(\varphi\) decomposition to simulate controlled source electromagnetic (CSEM) problems. The edge-based finite element technique was used to observe the (galvanic and inductive) effect of a controlled source electromagnetic forward model by [21]. The frequently used preconditioners such as incomplete lower-upper factorization (ILU) and Cholesky preconditioners that are nontrivial to adopt are necessary to improve the solution of linear systems [34]. However, such preconditioners are not efficient to some type of linear systems of equation. Thus, an efficient preconditioner is vital for broadband (plane-wave) electromagnetic applications as a result of developing an effective forward solver needed for the inversion routine. In this study, we present a wide-band plane wave electromagnetic modelling problem solved with finite volume method using potentials. We included the theoretical development in secondary formulation and applied the finite volume discretization. In addition, we describe the concept of the block ILU preconditioner.