Electromagnetic field of electric dipole field source on horizontal layered earth

It is inconvenient to use a pair of vectors H and E when solving the boundary value problem of electromagnetic field. If a vector bit is introduced, the unknown number will be reduced in the solution process. The vector bit A caused by current source is introduced as follows: Starting from ▽ h = 0, using the identity ▽ × a ≡ 0, we can make

H=▽×A( 1？ 3? 208)

Yes (1? 3? 208) type (1? 2? 1a) type:

ψ×E = Iψμψ×A

▽×(E-IπμA)= 0( 1？ 3? 209)

This formula points out that the vector in parentheses can be expressed by the gradient of any scalar, that is, taking

e-IπμA =-▽U

e = IπμA-▽U( 1？ 3? 2 10)

Where u is the scalar potential of the electromagnetic field. In DC electric field, because ω=0, E =-U.

Considering (1? 3? 208) and (1? 3? 2 10), will (1? 2? 2) Type is written as:

▽×▽×A =-Iωω*(IωμA-▽U)

▽▽A-▽2A =ω2ε*μA+I▽ω*▽U

Where ε*=ε+i is the complex dielectric constant. Group items with gradients and make them zero:

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▽2A=k2A( 1？ 3? 2 12)

formula

k2=-ω2ε*μ=-iωσμ-ω2εμ

This is the Helmholtz equation of vector potential. Yes (1? 3? 2 1 1) into (1? 3? 2 10), so:

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In this way, we only need to solve the vector potential equation once (1? 3? 2 12), passed (1? 3? 208) and (1? 3? 2 13), the magnetic field and electric field can be obtained respectively. These three equations form a set of equations:

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If the magnetic excitation source (such as magnetic dipole, ungrounded loop, etc.). ), the ground will produce eddy current. Characterized in that

▽ E=0

So you can also introduce the vector bit A* of the magnetic source, that is, take

E=▽×A*( 1？ 3? 2 15)

Through similar derivation, the corresponding equation can be obtained:

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By Maxwell equations (1? 3? 2 14) and (1? 3? 2 16), it is not difficult to see that the electromagnetic analogy relationship from the power supply equation to the magnetic source equation or the opposite is

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Therefore, in many cases, it is not necessary to solve the positive solutions of power supply and magnetic source separately, but to write another kind of solution directly by analogy.

Starting from the boundary conditions of harmonic field, consider (1? 3? 208) and (1? 3? 2 13) can write the boundary condition represented by vector bit a:

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The boundary condition of A* can be directly obtained by analogy from the above formula. At this time, A* becomes a, and-ε * becomes μ.

Electromagnetic field of (1) harmonic electric dipole field source

In this paper, the electromagnetic response of horizontal electric dipole field source on horizontal layered earth is discussed. Let the law of field source change be E-I Ω t, and the equation of electric dipole field source to be solved is:

▽2A-k2A=0( 1？ 3? 2 19)

Vector bit a has only Ax and Az components, while Ay=0. The field component is determined by the following factors:

H=▽×A

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Solve (1? 3? 2 19) type. The final result of the equatorial dipole device has the following form:

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In the above formula, PE = I AB is the current dipole moment, u 1=,

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The sum of the functions R 1 is called the frequency characteristic function. In DC detection, this function is not only a function of geoelectric parameters ρi and hi, but also a function of geometric factor λ. In magnetotelluric sounding, ω is used instead of λ, but in frequency sounding, it is a function of both * * *, that is, it is also a function of frequency and geometric factor λ.

(2) Electromagnetic field of step electric dipole field source

According to the solution of electromagnetic field in frequency domain, the corresponding transient field expression can be derived, and various mathematical processing methods can be used. At present, the widely used method is the spectral method based on inverse Fourier transform. Using this method, the expression of electromagnetic field component or apparent resistivity in frequency domain can be directly converted into the expression in time domain.

For equatorial devices, there are:

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B=B2=σ 1μ0ωr2/2 is called normalized frequency.

T=2t/σ 1μ0r2 is called normalization time.

here

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Binling

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