Under quasi-static conditions, the receiver MN remains parallel to the current source AB, i.e., the angular difference between AB and MN is α = 0 ° . Set the harmonic factor e i ω t , where i = − 1 , ω = 2 π f is the angular frequency, and t is time. The analytical expressions for the horizontal components E x and E y in the laminar medium are written as (He, 2010b) E x = I d L 2 π 2 cos 2 ⁡ φ − 1 ∫ 0 ∞ i ω μ r R N m R N + m 1 J 1 m r d m − I d L ρ 1 cos 2 ⁡ φ 2 π ∫ 0 ∞ m m 1 R N * J 0 m r d m + I d L 2 π 2 cos 2 ⁡ φ − 1 ∫ 0 ∞ ρ 1 r m 1 R N * J 1 m r d m + I d L sin 2 ⁡ φ 2 π ∫ 0 ∞ i ω μ R N m R N + m 1 J 0 m r d m E y = I d L ⁡ sin ⁡ 2 φ 2 π ∫ 0 ∞ i ω μ r R N m R N + m 1 J 1 m r d m + I d L ⁡ sin ⁡ 2 φ 2 π ∫ 0 ∞ ρ 1 r m 1 R N * J 1 m r d m − I d L ρ 1 ⁡ sin ⁡ 2 φ 2 π ∫ 0 ∞ m m 1 R N * J 0 m r d m − I d L ⁡ sin ⁡ 2 φ 2 π ∫ 0 ∞ i ω μ R N m R N + m 1 J 0 m r d m (1)

Where I is Current; dL is electric dipole; r is distance between the receiving point and the center point of the electric dipole; J 1 and J 0 are 1st and 0th order Bessel functions respectively; R N is the first layer factor; R N * is the second layer factor. R N = c t h | m 1 h 1 + c t h − 1 | m 1 m 2 c t h { [ m 2 h 2 + c t h − 1 [ m 2 m 3 c t h [ m 3 h 3 + ⋅ ⋅ ⋅ + c t h − 1 m N − 1 m N ] } | | R N * = c t h m 1 h 1 + c t h − 1 σ 2 m 1 σ 1 m 2 c t h m 2 h 2 + ⋅ ⋅ ⋅ + c t h − 1 σ N m N − 1 σ N − 1 m N (2)

If the number of layers N = 1 , then the layer factors R N and R N * of Eq. 2 are both equal to 1, and Eq. 1 will be transformed into an expression for the field on the surface of a uniform earth (He, 2010a). E x = I ρ d L ρ 2 π r 3 3 cos 2 ⁡ φ − 2 + e − i k r 1 + i k r E y = I ρ d L ρ 2 π r 3 3 ⁡ cos φ ⁡ sin ⁡ φ (3)

The expressions of E _ E x wide field apparent resistivity ρ a (He, 2010b; He, 2019) are ρ a = K E − E x Δ V M N I 1 F E − E x i k r Δ V M N = M N ⋅ E x E x = I ρ d L ρ 2 π r 3 3 cos 2 ⁡ φ − 2 + e − i k r 1 + i k r K E − E x = 2 π r 3 / d L ⋅ M N (4)

Eq. 4, K E − E x is the device coefficient of the observation system; Δ V M N = M N ⋅ E x is the potential difference between the MN at the measurement end, the unit is V; M N is the distance between the MN at the measurement end, the unit is m; I is the intensity of the emission current, the unit is A; dL is the dipole moment length, in m; E x is the component of the uniform half-space electric field along the x-axis; k is the wave number; F E _ E x i k r is the electromagnetic response function, and the specific expressions are: F E − E x i k r = 3 cos 2 ⁡ φ − 2 + e − i k r 1 + i k r k = ω 2 μ ε − i μ ω / ρ (5)

In Eq. 5, φ is the angle between x-axis and radial vector r , μ is the magnetic permeability, ρ is the model resistivity of uniform half-space, ε is the dielectric constant. When the detection object is a non-uniform half-space, the wide field apparent resistivity at different measurement points and different frequencies can be obtained according to Eq. 4, which reflects the overall resistivity response caused by the entire subsurface medium.

The observation system is designed according to Eq. 4 in the field, and the electrical information of the observation point location is finally obtained. However, the field construction deployment is restricted by the terrain conditions, and it is difficult to keep the current source AB parallel to the receiving dipole MN, and there is always an azimuthal difference α between them, which will cause the wide field apparent resistivity calculation error and distortion of the geoelectric parameters if we continue to calculate the wide field apparent resistivity along the E _ E x direction.

In order to eliminate the effect caused by the difference α in azimuth between AB and MN. In this paper, the formula for calculating E_E _MN wide field apparent resistivity at any azimuth is derived. Instead of calculating the apparent resistivity along the E _ E x direction, this method calculates the apparent resistivity along the MN direction at the measurement end. The advantage of this method is that the field measurement does not require the current source AB to be strictly parallel to the lateral or tangential direction to obtain the true resistivity parameters.

Let the electromagnetic field excited by the electric dipole source be located in the coordinate system xyz, the electric dipole source AB is parallel to the x-axis, and the angle between the receiving dipole MN and AB is α (Figure 1).

The relationship between E M N and E x , E y is obtained as follows. E M N = E x ⁡ cos α + E y ⁡ sin ⁡ α (6)

According to Eq. 6, the expression for the electric field of E M N is as follows. E M N = cos α I d L 2 π 2 cos 2 ⁡ φ − 1 ∫ 0 ∞ i ω μ r R N m R N + m 1 J 1 m r d m + 2 cos 2 ⁡ φ I d L 2 π ∫ 0 ∞ ρ 1 r m 1 R N * J 1 m r d m − 2 I d L 2 π ∫ 0 ∞ ρ 1 r m 1 R N * J 1 m r d m − I d L ρ 1 cos 2 ⁡ φ 2 π ∫ 0 ∞ m m 1 R N * J 0 m r d m + sin ⁡ α I d L ⁡ sin ⁡ 2 φ 2 π ∫ 0 ∞ i ω μ r R N m R N + m 1 J 1 m r d m + I d L ⁡ sin ⁡ 2 φ 2 π ∫ 0 ∞ ρ 1 r m 1 R N * J 1 m r d m − I d L ρ 1 ⁡ sin ⁡ 2 φ 2 π ∫ 0 ∞ m m 1 R N * J 0 m r d m − I d L ⁡ sin ⁡ 2 φ 2 π ∫ 0 ∞ i ω μ R N m R N + m 1 J 0 m r d m (7)

The E _ E M N electric field expression is defined by Eq. 7.

If the number of layers N = 1, both R N and R N * are equal to 1. The above equations will be transformed into the expressions for the field on a uniform earth obtained in the previous section. E M N = I ρ d L 2 π r 3 cos α 1 + e − i k r 1 + i k r + 3 ⁡ sin ⁡ φ ⁡ sin α − φ (8)

Substituting Eq. 8 into Eq. 4, the accurate expression of the wide field apparent resistivity at the measurement of MN can be obtained ρ a = K E − E x Δ V M N I 1 F E − E x i k r Δ V M N = M N ⋅ E M N E M N = I ρ d L 2 π r 3 cos α 1 + e − i k r 1 + i k r + 3 ⁡ sin ⁡ φ ⁡ sin α − φ F E − E x i k r = cos α 1 + e − i k r 1 + i k r + 3 ⁡ sin ⁡ φ ⁡ sin α − φ (9)

In Eq. 9: when α = 0 ° , E x = E M N , the correct geoelectric parameters under the observation station can be obtained either by using Eq. 9 or Eq. 4; when α ≠ 0 ° , E x ≠ E M N , the correct geoelectric parameters under the MN at the measurement end can be obtained by iterative calculation of Eq. 9. The wide field apparent resistivity parameters along the MN direction of the receiving dipole can be obtained by iteration or inverse spline difference.

When discussing the evaluation of the error caused by the azimuthal angle difference α between the current source dipole AB and the receiving dipole MN, the degree of separation between the observed change value caused by α and the background must be determined as a criterion, and the relative error between the apparent resistivity value of α ≠ 0 ° , and the apparent resistance value of α = 0 ° , under the condition of the same frequency point is taken as a criterion in the paper. e r r o r = ρ a j − ρ 0 j ρ 0 i × 100 % ; j = 1,2 , ⋅ ⋅ ⋅ , n (10)

In Eq. 10: j is the frequency number, n is the number of frequencies, ρ a j is the wide field apparent resistivity value corresponding to the jth frequency at α ≠ 0 ° , and ρ 0 j is the wide field apparent resistivity value corresponding to the jth frequency at α = 0 ° .

In order to analyze the effect of α on the observed parameters and further verify the correctness of E _ E M N method in this paper. Different experimental work were carried out without obvious electromagnetic humanistic interference, in which α was 0°, 1°, 3°, 5°, 10°, and 15° respectively. The field experiment scheme is shown in Figure 4: only the position of pole B is changed, while other experimental parameters remain unchanged. When α = 0 ° , it indicates that the parallel observation position of the transmitting source AB is parallel to that of the receiving end MN, and there is no azimuth Angle difference.

This field using pseudo-random 7-frequency wave signal, that is, a simultaneous transmission and reception from the underground 7 frequency signals. The parameters of this experiment: current transmitting source AB = 1 km, receiving end MN = 100 m, transmitting current I = 80 A, keeping the intensity of transmitting current constant, transmitting frequency range 8,192∼0.01 Hz, total 54 frequency points.

To eliminate the effect of current, the observed potential difference data were normalized by current and the “frequency-electric field” curves were plotted along the MN direction for different α conditions (Figure 5). From the analysis of Figure 5, the difference of electric field caused by different α is small, and it is difficult to discern the influence of α on the observation results intuitively. In the later analysis, the influence of α on the observation results is analyzed from the apparent resistivity parameter, and the validity and correctness of the calculation formula proposed in this paper are verified.

Figure 5 shows the “frequency-apparent resistivity” graph and “frequency-relative error” graph for different azimuthal differences without considering the effect of azimuthal difference α , but directly calculated by inverse spline numerically from Eq. 4. The analysis of the “frequency-apparent resistivity” graph and the relative error graph in Figure 6 is shown as follows.

(1) When α = 1 ° , the corresponding “frequency-apparent resistivity” curve is the same as the “frequency-apparent resistivity” curve with α = 0 ° , and there is no obvious separation, and the relative error of the corresponding apparent resistivity is ≤5%, while the relative error of the apparent resistivity of individual frequency points is >5%. The relative error of the apparent resistivity at individual frequency points is >5%. It means that the azimuthal difference between the current source AB and the receiver MN α ≤ 1 ° has a small effect on the wide field apparent resistivity.

From the above analysis, it can be concluded that: in the range of α ≤ 1 ° , the azimuthal difference α has a small effect on the apparent resistivity parameters; when α ≤ 5 ° , the azimuthal difference aa mainly affects the apparent resistivity parameters in the middle and low frequency bands; when α = 10 ° ∼ 15 ° , the azimuthal difference aa has a large effect on the apparent resistivity parameters in the whole frequency band, and the more serious the distortion of the high-middle frequency data, the more obvious the false anomaly caused, and even causes the interpretation of the parameter distortion. If the relative error of the apparent resistivity value ≤10% is considered reasonable, it must be ensured that α ≤ 3 ° , but the field construction conditions are restricted by the terrain, and it is often difficult to achieve the azimuthal difference α ≤ 3 ° between the current source AB and the receiver MN, so the angle correction must be made to the observed data to eliminate the influence brought by α and improve the interpretation accuracy of the apparent resistivity parameters.

Figure 7 shows the “frequency-apparent resistivity” curve and the relative error curve calculated and plotted according to the arbitrary orientation E _ E M N of Eq. 9. The analysis of the “frequency-apparent resistivity” curve and the relative error graph in Figure 7 is shown as follows.

(1) When α = 1 ° , the “frequency-apparent resistivity” curve of Figure 7 and the “frequency-apparent resistivity” curve of α = 0 ° do not show significant separation and approximately coincide, and the relative error of the apparent resistivity at each frequency point is ≤1%, which is reduced from 8% to 1% compared with the calculation of Figure 6.

The analysis results of Figures 6, 7 show that: firstly, the maximum relative error of the wide area apparent resistivity obtained by the arbitrary azimuthal wide area electromagnetic method E _ E M N is reduced from 25% to 4% when α ≤ 5 ° , which makes the apparent resistivity parameter closer to the real underground geoelectric information; secondly, the relative error of the wide area apparent resistivity obtained by the arbitrary azimuthal wide area electromagnetic method E _ E M N is reduced from 270% to less than 10% when α = 10 ° and 15 ° . Thirdly, if the azimuth angle difference α is larger, the effect of using E _ E M N method of arbitrary azimuth wide field electromagnetic method on eliminating α is more obvious, which reflects the real resistivity value of the subsurface and improves the accuracy of the wide field apparent resistivity effectively.

In this section, we take α = 15 ° as an example, and perform single-point inversions of E _ E x and E _ E M N apparent resistivity respectively, and further analyze the effect of azimuthal difference α on the interpretation parameters by combining the electric logging data of Ning 227 borehole. One-dimensional continuum imaging was performed on the processed data, and the relevant parameters of inversion were as follows: inversion depth was 4.5 km; The number of iterations in the inversion process was 20; The horizontal and depth resolution were 1; The regularization parameter was 5; The fitting error was 0.02. Figure 8 shows the bathymetric curves of “frequency-apparent resistivity” obtained from different calculation methods of E _ E x and E _ E M N , and the single-point inversion curves are shown in Figure 9.

(1) From Figure 8 E _ E x and E _ E M N “frequency-apparent resistivity” curve analysis, it can be seen that when the azimuth angle difference is 15°, the corresponding “frequency-apparent resistivity” sounding curves of different calculation methods of the same measurement point are different, and the two curves are completely separated, and the curve change pattern is also different, which means that the azimuth angle difference α has a greater influence on the apparent resistivity parameter.