## 1 Introduction  
Transient electromagnetic (TEM) methods can detect a pure secondary field after the cutoff of the primary field, and they have been widely used in land applications [1; 2; 3; 4; 5]. In recent decades, with rapid ocean development and utilization, greater attention has been given to the application of TEM methods in the marine field [6; 7; 8; 9; 10; 11]. To overcome the masking effects of seawater, most studies have adopted the towed configuration [10; 11; 12]. However, this configuration has low work efficiency and high acquisition cost. In addition, many shallow areas are inaccessible for exploration ships because of rocks or the depth of seawater. For configurations above the sea surface, the airborne TEM method is considered a promising candidate because it is convenient and highly efficient [1; 2; 13; 14; 15; 16]. In practice, experiments conducted at sea need to consider not only the complicated environment but also other factors such as manpower and costs. Thus, numerical design studies and accurate date interpretation need to be reinforced before launching a sea-going measurement. Some existing interpretation methods for land applications can be transferred directly to marine TEM methods. However, the seawater layer causes these to be different from land TEM methods in terms of sounding ability, device design, and numerical modeling. Therefore, some issues pertaining to the airborne TEM method in marine investigations need to be further studied for it to become a feasible method of analysis. An accurate numerical method is necessary in marine TEM methods. The method of 2.5D modeling, i.e., a 2D geoelectric model with a 3D loop source, is often used because it is closer to the practical geological conditions than either 1D or 2D models, and has a lower computation cost than 3D models. Moreover, many marine activities such as surveys of seafloor topography and subsea targets, can be viewed as 2.5D models [17; 18; 19; 20]. Analysis by 2.5D forward modeling is based on the background/anomalous-filed algorithm. In order to eliminate the time and strike components, it is usually manipulated in the Laplace and Fourier domains and calculated using the finite element method (FEM) [21; 22; 23; 24]. The inverse Laplace and Fourier transforms are used to recover the solution in the time domain. Each step in the modeling inevitably introduces some calculation errors. As the Fourier inversion can be evaluated accurately by the sine and cosine transforms [25; 26], the Laplace inversion becomes a key factor that influences the accuracy of the marine TEM response. There are many algorithms for computing the inverse Laplace transform [27; 28; 29], but a favorable one should balance the computational load and calculation accuracy. In land investigations, for the airborne TEM methods with a step waveform source, the Gaver-Stehfest (G-S) algorithm is widely used to compute the inverse Laplace transform because it is a pure real number operation and can be implemented simply and easily [22; 30]. However, in most airborne TEM systems, commonly used current waveforms include trapezoidal [31; 32; 33], half-sine [34; 35], and triangular waves [15]; these waveforms have a considerable influence on the TEM response. It has been demonstrated that the G-S algorithm for a complex wave yields large errors, especially in the late observation time. As seawater is a high-conductivity medium, the masking effect causes EM waves to yield a lower penetration depth and attenuate quickly in seawater. The response of the target in the sea is low, and sampling it for analysis takes a long time. In addition, the accuracy of the G-S algorithm is closely related to the problem and the summation terms, which means it has poor robustness. Therefore, an alternative algorithm matched with the current waveform is necessary for the inverse Laplace transform in marine TEM calculations. In this study, 2.5D forward modeling in marine TEM methods was investigated using the FEM. The trapezoidal wave was selected as the source waveform.