Transient Electromagnetic Smoke Ring Due to a Grounded-Wire Source

The concept of a “smoke ring” in electromagnetic fields can be used to describe the spatial distribution and diffusion of electromagnetic fields with either frequency or time. This provides an intuitive basis for the qualitative interpretation of artificial-source electromagnetic measurement results. The existing discussions regarding this field of study have mainly focused on the smoke rings of a magnetic dipole or loop source. In the present work, smoke rings of other commonly used grounded-wire sources in the time domain are investigated. The results show that the smoke ring of grounded-wire sources is more complex than that of magnetic sources. A grounded-wire source can generate an induced current in both horizontal and vertical directions. The horizontal induced current includes positive and negative parts. Both parts of the horizontal induced current maximum diffuse downward perpendicular to the surface, but the diffusion velocity of the negative part is faster, approximately 5.6 times that of the positive part. The vertical induced current maximum diffuses at 45° from the grounded surface with a velocity approximately 5 times that of the positive horizontal induced current. The smoke ring of the horizontal induced current closely resembles a basin, while that of the vertical induced current is more similar to two separate cylinders. Finally, the imaging depth based on horizontal and vertical induced currents is obtained by the statistical relationship between the induced current maximums and the diffusion depth, which are equal to 1.4 times and 1.25 times the diffusion depth, respectively.


Introduction
The diffusion and distribution characteristics of the induced current excited by artificial transmitters are closely related to the electromagnetic fields on the surface. Understanding the diffusion of the induced current can help us more effectively interpret the observed electromagnetic response.
The diffusion of subsurface-induced current with time or frequency can be described by the concept of a ''smoke ring,'' which was first proposed by Nabighian (1979). He described the induced current generated by the loop source in a uniform half-space in terms of a conceptual smoke ring blown by the transmitter. The smoke ring gradually diffuses outward and downward with decreasing amplitude over time. In addition, the electromagnetic fields measured at the ground can be replaced by the response generated at the surface by the equivalent smoke rings. Hoversten and Morrison (1982) characterized the electric field distribution of various layered earth models with a repeated square wave system. They observed that the induced current excited by the loop source in a four-layer medium forms a single circular smoke ring that distorts at the layer boundaries. Oristaglio and Hohmann (1984) calculated the response of a 2D geoelectric model with a two-line source using the finite difference method and analyzed the influence of vertical conductive bodies on the diffusion of the electric field. Subsequently, Reid and Macnae (1998) introduced the smoke ring concept in the frequency domain. Their study illustrated that the smoke ring of the vertical magnetic dipole source in the frequency domain is present only in the in-phase component of the secondary electric field and bears characteristics similar to those of the smoke ring in the corresponding time domain. It has also been noted that the maximum current density exhibits outward radial motion at all times. Wang (2002) studied the smoke ring phenomenon in transverse anisotropic media and showed that the smoke ring is elliptic in transverse anisotropic media. As the anisotropy coefficient increases, the shape of the smoke ring is disrupted to the point of being indistinguishable. Yin and Hodges (2005) studied the electromagnetic diffusion of frequency-domain electromagnetic systems in homogeneous half-space media and then further extended the model to complex anisotropic and 2D and 3D models. Yin and Hodges (2007) multiplied the frequency-domain electromagnetic field with the time-harmonic factor e ixt and then presented 3D dynamic electromagnetic field isolines and vector diagrams. The induced current excited in the frequency domain was found to cause positive and negative wavefronts to propagate as the transmitting polarity changes. Yin et al. (2016) discussed the induced current of vertical and horizontal magnetic dipole sources in an airborne transient electromagnetic (TEM) system. They parametrized the imaging depth according to the relationship between the depth of the maximum induced current and diffusion depth. More recently, Smiarowski and Hodges (2021) studied the smoke ring of a homogeneous half-space under the excitation of a half-sine continuous waveform. The distribution of the current diagram of the continuous excitation waveform near the ground was found to be much denser than that of an off-time excitation system, thus suggesting that measurements during a continuous excitation on time are more sensitive to shallow targets.
All of the previous related studies have mainly focused on the smoke rings of magnetic dipole sources or loop sources; there have been few studies on the smoke rings of long grounded-wire sources, which are another important kind of source. Grounded-wire sources are widely used in EM methods, such as the frequency-domain controlled-source electromagnetic method (FCSEM) (Streich, 2016), long-offset transient electromagnetic method (LOTEM) (Strack, 1992), and short-offset transient electromagnetic method (SOTEM) (Xue et al., 2013). Compared with the horizontal or vertical magnetic dipole source, in which only the current flows along with the transmitting source [pure transverse electric (TE) or pure transverse magnetic (TM) field], the current direction is much more complex for a wireearth system, and both TE and TM polarization modes exist (Ward & Hohmann, 1988). Gunderson et al. (1986) illustrated, for the first time, the behavior of the transient fields generated by a grounded-wire source. They observed that the parallel electric field beneath the wire source is initially negative as a result of the ''return current'' flowing in the source before it is turned off. Over time, the field increases, becomes positive, reaches a maximum value, and then decreases. They also found that the grounded-source maximum moves downward more slowly than the line-source maximum. Chen et al. (2019) studied the diffusion and distribution characteristics of underground electromagnetic fields excited by a groundedwire source, thereby laying a foundation for surfaceto-borehole TEM detection using a grounded-wire source.
In the present article, we focus particularly on the smoke ring excited by a grounded-wire source. The behavior of the time-domain smoke ring is described by calculating the induced currents in uniform halfspace models and layered models. Moreover, the imaging depth for a grounded-wire source is confirmed according to the relationship between the diffusion depth and depth of the induced current maximum.

Calculation of Induced Current
The calculation of the induced currents is based on 1D layered earth. Figure 1 shows a grounded-wire source (length is L) above layered earth with resistivities q i (i = 1, 2, …, N) and thicknesses h i (i = 1, 2, …, N). Next, a coordinate system is established with the x-and y-axes located at the earth surface and the z-axis positive downward. The source is laid along the x-axis. Assuming a harmonic time dependence e -ixt , the electric field in the frequency domain at any position is determined by the formulas given by Ward and Hohmann (1988): where x s m is the x coordinate of the grounded electrodes; r m is the distance between the receiver and grounded electrodes; r n is the distance between the observation point and segmented dipole source; and r TM and r TE are the reflection coefficients of the TM and TE modes, respectively. The specific formulas for r TM , r TE ,ŷ 0 ,ẑ 0 and u 0 are given by Ward and Hohmann (1988). Finally, J 1 ðkrÞ and J 0 ðkrÞ are the first-order and zero-order Bessel functions, respectively.
After the frequency-domain field has been obtained, it can be converted to the time domain through Fourier transform to obtain the step-off electromagnetic response at time t: Note that the smoke ring we study here is in fact the induced current density that distributes and diffuses underground in a characteristic manner. The induced current density can be calculated according to Ohm's law: The horizontal induced current density is the combination of current in the x and y directions, i.e., The vertical induced current density is calculated as follows: 3. Diffusion and Distribution of the Induced Current

Homogeneous Half-Space
The induced current density of a grounded-wire source, with a length of 200 m and a current of 1 A above a 100 X m uniform half-space, is calculated. The dimensions of the computation domain are 4 9 4 9 2 km in the x, y, and z directions, respectively, and the grid size is 20 9 20 9 20 m. The wire source is placed along the x-axis, and its midpoint is located at the origin of the coordinate system. Therefore, the amplitude of the electric field is symmetric in the four quadrants, and only 1/4 of the measurement points need to be calculated. The time gates of interest range from 0.01 ms to 100 ms, with 71 logarithmic equal-interval time channels.
Next, we successively analyze the distribution characteristics and diffusion process of the underground horizontal induced current density (Fig. 2) and vertical induced current density (Fig. 3). The xoy (top panel) and xoz (middle panel) cross-sections and 3D display (bottom panel) of the current density at 0.2 ms and 20 ms are respectively given in Figs. 2 and 3.
The horizontal induced current diffuses equally in the x and y directions. However, its maximum remains very close to the source throughout the entire measurement time; thus, its distribution in the xoy plane is approximately circular for a given time (top panels in Fig. 2). Vertically, the horizontal induced current consists of two parts: the upper and lower induced horizontal current. The two parts are separated by a prominent low-value band (middle panels of Fig. 2a). The upper horizontal current is Vol. 180, (2023) Transient Electromagnetic Smoke Ring Due to a Grounded-Wire Source positive and evenly distributed in both x and y directions (bottom panels of Fig. 2). The lower induced horizontal current is also called the ''return current'' (Gunderson et al., 1986;Um, 2005). It is negative, and its downward diffusion velocity is faster. In general, the return current can be observed at an earlier time only. The negative return current is the reason for the sign change in the horizontal magnetic field (Hy) observed at the surface (Chen, et al., 2019;Gunderson et al., 1986). The vertical induced current (Fig. 3) exhibits characteristics that are vastly different from those of the horizontal induced current. First, the vertical induced current diffuses downward and outward from the two grounded electrodes, while the diffusion directions on the two sides are opposite. Consequently, the density of the vertical induced current is zero on the midperpendicular (yoz section) of the source. Second, the diffusion of the vertical induced current is much faster than that of the horizontal induced current. As shown in Fig. 3b, the vertical induced current maximum has diffused far from the source at 20 ms. (We provide in a subsequent section quantitative descriptions of the diffusion velocity and diffusion path of the maximum vertical induced current.) Third, at a given time, the amplitude of the vertical induced current maximum is much smaller (only approximately 30%) than that of the horizontal induced current. However, at a given depth, the amplitude of the vertical current maximum is always larger than that of the horizontal current.

Three-Layer Model
In this section, the 1D conductor model and resistor model are considered to further understand the diffusion of the induced current. The background resistivity of the model is 100 X m, the thickness of the thin layer is 100 m, and the buried depth of the top interface is 500 m. The resistivity of the thin layer is 10 XÁm for the conductor model (H model) and 1000 X m for the resistor model (K model). The results are shown in Figs. 4 and 5.
According to the study results, we can draw the following conclusions: (1) The distribution of the current density is closely related to the layer resistivities.
(2) The current density is discontinuous across the boundaries.
(3) The current density decreases in the conductors but increases in the resistors, which is the same for both horizontal and vertical induced currents.
According to Kaufman and Keller (1983), the vertical electric field excited by an electric source leads to charge accumulation on both sides of the resistivity interface. The accumulated charge is electrically opposite on both sides of the thin layer. Therefore, such accumulated charge can be equivalent to numerous electric dipoles inside and perpendicular to the thin layer. When the thin layer is conductive, then the upper interface accumulates negative charges, whereas the lower interface accumulates positive charges, thereby generating the vertical electric field in the conductive body from the bottom to the top, which is exactly opposite to the direction of the background field. This results in a significant decrease in the field amplitude in the conductive body. In contrast, the field in the resistive layer is enhanced. The discontinuity of the vertical electric field at the electrical interface is of great use in practical applications, as it aids in accurately distinguishing different electrical layers (Chen et al., 2020).

Smoke Ring Effect
The distribution and diffusion of underground induced currents generated by the grounded-wire source are described above. To further study the smoke ring effect of the electromagnetic field, we focus on the diffusion locus and law of the horizontal and vertical induced current maximum. The reason for this is that the maximum induced current makes the greatest contribution to the electromagnetic fields observed on the ground, which can provide a basis upon which to analyze the electromagnetic response characteristics and imaging depth. We calculate the statistics of the positions (x and z) of the horizontal (positive and negative) and vertical induced current maximum with time. The model is consistent with Figs. 2 and 3, and the results are shown in Fig. 6. The locus of the horizontal induced current maximum is vertically downward along with the source, i.e., perpendicular to the ground. The vertical induced current maximum diffuses at an angle of approximately 45°to the ground. This is different from the case with a loop source, where the induced current maximum diffuses roughly at an angle of 30°to the ground (Nabighian, 1979). As shown in Fig. 6, the negative horizontal induced current maximum (ÀJ Hmax ) has the fastest diffusion velocity, followed by the vertical induced current maximum (J Vmax ), and the positive horizontal induced current maximum (þJ Hmax ) has the slowest diffusion velocity. The relationship between them is approximately ÀJ Hmax ¼ 5:6 þJ Hmax ð Þ and J Vmax ¼ 5 þJ Hmax ð Þ . For a loop source, the maximum of the induced current at a given moment is ring-like, and the ring gradually moves downward over time, while the size of the ring gradually increases (Nabighian, 1979;Yin et al., 2016). Next, we describe the diffusion patterns of horizontal and vertical induced currents excited by a grounded-wire source. We select 70% of the horizontal induced current density maximum and 90% of the vertical induced current density maximum as the thresholds to draw an isosurface map at 0.2 ms and 20 ms, respectively, as shown in Fig. 7. The reason for choosing a greater threshold for the vertical induced current is that it spreads more quickly. As shown in Fig. 7a, in the early time, the horizontal induced current has two separate smoke rings. The upper part is positive, and the shape is similar to a flat disc. In addition, the lower part is negative, the shape is approximately a sphere, and the field amplitude and size are both much smaller. As time increases, the upper ring gradually becomes larger, and the shape begins to resemble a basin, while the lower part moves deeper and gradually becomes invisible. The isosurface of the vertical induced current more closely resembles two cylinders, spreading down and out at a faster speed. Unlike the hollow smoke ring of the loop source, the smoke ring is solid for the electric source, whether horizontal or vertical.

Imaging Depth
Imaging is a simple and effective technique for processing time-domain electromagnetic data, particularly when using the airborne or semiairborne transient electromagnetic method, which involves a large amount of data. The steps of the method are to take the resistivity of the homogeneous half-space model that produces the same response as the apparent resistivity and then approximate half of the diffusion depth (d TD ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2qt=l 0 p ) as the imaging depth to construct the apparent resistivity-depth profile (Fullagar, 1989;Huang & Rudd, 2008;Wolfgram & Karlik, 1995). Yin et al. (2016) proved the basis for selecting the imaging depth of airborne transient electromagnetic data by constructing a relationship between the depth of the induced current maximum of the magnetic source and diffusion depth. There have been relatively few studies on imaging technology for grounded-wire sources. Macnae and Lamontagne (1987) and Eaton and Hohmann (1989) proposed an imaging technique based on equating the response measured on the ground at each delay time to the magnetic field of images of the source. One of the most important steps in this process is to determine the depth of the image source. In general, the depth of the induced current maximum is considered the imaging depth since it is closely related to the location of the smoke ring or the ''footprint'' (used with airborne electromagnetic data) that makes the greatest contribution to the ground electromagnetic fields.
As mentioned above, the horizontal induced current generated by a grounded-wire source consists of a positive part and a negative part. Therefore, we first need to decide which part of the maximum value is larger. We calculate the decay curves of the underground horizontal electric field (Ex) at different given depths, as shown in Fig. 8. The depth refers to the position directly below the source, where Ey = 0. The horizontal induced current is generated only by the parallel electric field Ex, which simplifies the discussion. The model is a homogeneous half-space with a resistivity of 100 X m, and the length of the source is 1 km. In Fig. 8, the dashed lines represent negative responses, while the solid lines represent positive responses. Notably, Ex has two maximums for a given depth. The first is a negative maximum (hollow black circle) at an earlier time, and the other is a positive maximum (solid black circle) at a later time. The absolute value of the negative maximum is always greater than that of the positive maximum. In other words, the negative maximum of the horizontal induced current at a given depth contributes more to the electromagnetic response observed on the surface than the positive part.
Therefore, the negative horizontal induced current maximums and vertical induced current maximum are considered. The relationship between the imaging depth and diffusion depth corresponding to these two induced currents is calculated and shown in Fig. 9. This relationship is statistically obtained according to the depth (d i ) of the maximum underground induced current and the diffusion depth d TD at the corresponding time in a uniform half-space model with different resistivity. The corresponding resistivities of the homogeneous half-space are 10 XÁm, 100 XÁm and 1000 XÁm.
The results show that the depths of both induced current maximums maintain a linear relationship with the diffusion depth d TD . It is calculated that this linear coefficient is approximately 1.4 for the negative horizontal induced current maximum and 1.25 for the vertical induced current maximum, as shown in Formula (7): The imaging depth obtained based on the horizontal induced current shown in Formula (7) is inconsistent with the image depth given by Eaton and Hohmann (1989) and Strack (1992), which is Since previous studies considered only the vertical magnetic field or its time derivative observed on the ground, only the imaging depth based on the horizontal induced current can be considered. When other components, such as horizontal electric fields or horizontal magnetic fields, are observed, the imaging depth based on the vertical induced current needs to be considered. The imaging depth we give here based on the vertical induced current enables these components to be imaged.

Conclusion
The smoke ring effect of the grounded-wire source was analyzed based on the uniform half-space and layered models. Compared with magnetic sources, the smoke ring effect of grounded-wire sources is more complex. It consists of induced currents in both horizontal and vertical directions, in which the horizontal induced current includes positive and negative parts. The negative part of the horizontal induced current diffuses the fastest, followed by the vertical induced current, and the positive part of the horizontal induced current is the slowest. The relationships between them are approximately ÀJ Hmax ¼ 5:6 þJ Hmax ð Þand J Vmax ¼ 5 þJ Hmax ð Þ . The total horizontal induced current maximum diffuses vertically downward at the center of the source. Moreover, the vertical induced current maximum diffuses downward and outward from the two electrodes of the source and along the direction of 45°w ith the ground. The smoke ring of the horizontal induced current closely resembles a solid cycloidal basin, while that of the vertical induced current is more similar to two separate solid cylinders. This is quite different from the smoke ring of the loop source, which has the shape of a hollow circle.
The imaging depths based on horizontal and vertical induced current maximums were calculated according to the statistical relationship between the diffusion depth and the depth of the induced current maximums. The results show that the imaging depth based on the horizontal induced current is approximately 1.4 times the diffusion depth, while the imaging depth based on the vertical induced current is approximately 1.25 times the diffusion depth. This provides a basis for resistivity-depth imaging based on different observed electromagnetic field components.