CN100573195C - Geophysical exploration method and equipment - Google Patents

Abstract

The invention discloses a kind of geophysical exploration method and equipment, based on the physical characteristics of the earth and being closely connected of geology, utilize standing wave to determine to survey the bottom boundary degree of depth of object, utilize standing electromagnetic wave exploring equipment of the present invention, reach " magnetic signal receiver " by " electric signal receiver " and receive electromagnetic wave, determine standing electromagnetic wave antinode frequency by data recording and processor, survey the known electric magnetic parameter of object again according to input, calculate and the output data result.The present invention has made full use of natural energy resources, and is economical and practical, and rotating detection process is simple to operate, and cost is low.

Description

Geophysical exploration method and equipment

Technical field

The standing wave method geophysical survey, the physical characteristics and the geology that are based on the earth are closely connected, so can find out the problem that geology need be found out with the method for geophysical survey.The standing wave method geophysical survey is a kind of brand-new method in many geophysical exploration methods.

Background technology

Materials on the earth have various physical characteristicss, as density, elasticity, magnetic field intensity, resistivity etc.Same the physical index of different materials on the earth, often difference to some extent again.According to these difference, with the method for geophysical survey, reach the geologic prospecting purpose, then produced gravity prospecting, seismic prospecting, magnetometer survey, the geophysical survey method that resistivity prospecting etc. are many has formed a kind of specialty of maturation in the world.

Existing many geophysical survey methods respectively have length: the method that has is utilized natural field source, and instrument is light, and equipment simply is its advantage, but the quantitative property of achievement data is then slightly inadequate; The method data acquisition amount that has is big, and achievement data is directly perceived, and geological effect is good, but mechanism of exploration troop is huge, and personnel are numerous, technical sophistication, and apparatus expensive is invested very huge.

Standing wave is that frequency is identical, the direction of propagation relatively and the ripple that two train waves of row interfere the back to form.Its waveform was constantly pushed ahead when ripple was propagated in medium, so claim the row ripple; Above-mentioned two row interfere the back waveform not pushed ahead, so claim standing wave.Amplitude is that zero point is called node, and the amplitude maximum is called antinode.The vibration phase of node both sides is opposite.Distance between adjacent two node or antinode all is a half wavelength.Be expert in the ripple energy with wave propagation constantly to front transfer, its average energy current density is non-vanishing; But the average energy current density of standing wave equals zero, and energy can only operation back and forth between node and antinode.

Measure the distance of two adjacent wave internodes and just can measure wavelength.Various musical instruments comprise stringed musical instrument, wind instrument and percussion instrument, all are sounding owing to produce standing wave.For obtaining the strongest standing wave, the length L of string or inner air tube post must equal the integral multiple of half-wavelength.

Yet,, in the practice of geophysical survey, do not see that but it relates to although standing wave all has argumentation in the vibration of the physical textbook of institution of higher education and ripple one chapter.

Summary of the invention

Technical matters to be solved by this invention is to provide a kind of geophysical exploration method and equipment, is closely connected based on the physical characteristics and the geology of the earth, just can utilize standing wave to determine to survey the bottom boundary degree of depth of object.

For solving the problems of the technologies described above, the invention provides a kind of geophysical exploration method, utilize standing wave to determine to survey the bottom boundary degree of depth of object, comprise the steps:

(1) measuring point is determined at the interface on the detection object;

(2) receive electromagnetic wave at measuring point, and according to the electromagnetic wave that receives, determine standing electromagnetic wave antinode frequency, wherein, described standing electromagnetic wave is the standing wave that is formed with the reflection wave interference that produces through the reflection of detection object bottom boundary by the incident wave that electromagnetic wave source produces;

(3) according to the antinode frequency of described standing electromagnetic wave, and the known electric magnetic parameter of surveying object, determine to survey the bottom boundary degree of depth of object.

Wherein, described detection object is the stratum.

Wherein, described electromagnetic wave source comprises: the electromagnetic wave of solar radiation, solar wind impact electromagnetic wave, the electromagnetic wave of " Van Allen radiation belt " radiation and/or the electromagnetic wave that the thunderstorm discharge forms that the magnetic field of the earth produces.

Wherein, described step (3) comprising: at n ₂/ μ _R2Greater than n ₃/ μ _R3The area, according to formula:

$h=\mathrm{\π}/{\left(\frac{{\mathrm{\ω}}^2{\mathrm{\ϵ}}_2{\mathrm{\μ}}_2}{2}\right)}^{1/2}{\{{[1+{\left(\frac{{\mathrm{\σ}}_2}{\mathrm{\ω}{\mathrm{\ϵ}}_2}\right)}^2]}^{1/2}+1\}}^{1/2}$

Determine the stratum bottom boundary degree of depth, wherein:

N is the refractive index of medium, μ _rIt is the relative permeability of medium; Footnote 2 and 3 is represented the up and down of bottom boundary respectively;

H is the stratum bottom boundary degree of depth, and ω is an electric field standing wave antinode circular frequency, ε ₂Be the specific inductive capacity on stratum, μ ₂Be the magnetoconductivity on stratum, σ ₂It is the conductivity on stratum.

Wherein, described step (3) comprising: at n ₂/ μ _R2Greater than n ₃/ μ _R3The area, according to formula:

$h=\mathrm{\π}/{\left({2\mathrm{\ω}}^2{\mathrm{\ϵ}}_2{\mathrm{\μ}}_2\right)}^{1/2}{\{{[1+{\left(\frac{{\mathrm{\σ}}_2}{\mathrm{\ω}{\mathrm{\ϵ}}_2}\right)}^2]}^{1/2}+1\}}^{1/2}$

Determine the stratum bottom boundary degree of depth, wherein:

N is the refractive index of medium, μ _rIt is the relative permeability of medium; Footnote 2 and 3 is represented the up and down of bottom boundary respectively;

H is the stratum bottom boundary degree of depth, and ω is a magnetic field standing wave antinode circular frequency, ε ₂Be the specific inductive capacity on stratum, μ ₂Be the magnetoconductivity on stratum, σ ₂It is the conductivity on stratum.

Wherein, described step (3) comprising: at n ₂/ μ _R2Less than n ₃/ μ _R3The area, according to formula:

$h=\mathrm{\π}/{\left(\frac{{\mathrm{\ω}}^2{\mathrm{\ϵ}}_2{\mathrm{\μ}}_2}{2}\right)}^{1/2}{\{{[1+{\left(\frac{{\mathrm{\σ}}_2}{\mathrm{\ω}{\mathrm{\ϵ}}_2}\right)}^2]}^{1/2}+1\}}^{1/2}$

Determine the stratum bottom boundary degree of depth, wherein:

N is the refractive index of medium, μ _rIt is the relative permeability of medium; Footnote 2 and 3 is represented the up and down of bottom boundary respectively;

H is the stratum bottom boundary degree of depth, and ω is a magnetic field standing wave antinode circular frequency, ε ₂Be the specific inductive capacity on stratum, μ ₂Be the magnetoconductivity on stratum, σ ₂It is the conductivity on stratum.

Wherein, described step (3) comprising: at n ₂/ μ _R2Less than n ₃/ μ _R3The area, according to formula:

$h=\mathrm{\π}/{\left({2\mathrm{\ω}}^2{\mathrm{\ϵ}}_2{\mathrm{\μ}}_2\right)}^{1/2}{\{{[1+{\left(\frac{{\mathrm{\σ}}_2}{\mathrm{\ω}{\mathrm{\ϵ}}_2}\right)}^2]}^{1/2}+1\}}^{1/2}$

Determine the stratum bottom boundary degree of depth, wherein:

N is the refractive index of medium, μ _rIt is the relative permeability of medium; Footnote 2 and 3 is represented the up and down of bottom boundary respectively;

H is the stratum bottom boundary degree of depth, and ω is an electric field standing wave antinode circular frequency, ε ₂Be the specific inductive capacity on stratum, μ ₂Be the magnetoconductivity on stratum, σ ₂It is the conductivity on stratum.

The present invention also provides a kind of geophysical prospecting equipment, utilizes standing wave to determine to survey the bottom boundary degree of depth of object, comprising:

The electromagnetic signal receiving trap is used for receiving electromagnetic wave at measuring point;

The electromagnetic parameter input media is used to import the known electric magnetic parameter of surveying object; And

Data recording and processor are used for determining standing electromagnetic wave antinode frequency according to the electromagnetic wave that receives, and according to the known electric magnetic parameter of the detection object of described input, determine to survey the bottom boundary degree of depth of object,

Wherein, described standing electromagnetic wave is the standing wave that is formed with the reflection wave interference that produces through the reflection of detection object bottom boundary by the incident wave that electromagnetic wave source produces.

Wherein, described electromagnetic signal receiving trap comprises:

The electric signal reception amplifier is used to receive electric field signal;

The magnetic signal reception amplifier is used to receive field signal.

Wherein, described detection object is the stratum; Described data recording and processor are at n ₂/ μ _R2Greater than n ₃/ μ _R3The area, according to formula:

$h=\mathrm{\π}/{\left(\frac{{\mathrm{\ω}}^2{\mathrm{\ϵ}}_2{\mathrm{\μ}}_2}{2}\right)}^{1/2}{\{{[1+{\left(\frac{{\mathrm{\σ}}_2}{\mathrm{\ω}{\mathrm{\ϵ}}_2}\right)}^2]}^{1/2}+1\}}^{1/2}$

Determine the stratum bottom boundary degree of depth, wherein:

N is the refractive index of medium, μ _rIt is the relative permeability of medium; Footnote 2 and 3 is represented the up and down of bottom boundary respectively;

H is the stratum bottom boundary degree of depth, and ω is an electric field standing wave antinode circular frequency, ε ₂Be the specific inductive capacity on stratum, μ ₂Be the magnetoconductivity on stratum, σ ₂It is the conductivity on stratum.

Wherein, described detection object is the stratum; Described data recording and processor are at n ₂/ μ _R2Greater than n ₃/ μ _R3The area, according to formula:

$h=\mathrm{\π}/{\left({2\mathrm{\ω}}^2{\mathrm{\ϵ}}_2{\mathrm{\μ}}_2\right)}^{1/2}{\{{[1+{\left(\frac{{\mathrm{\σ}}_2}{\mathrm{\ω}{\mathrm{\ϵ}}_2}\right)}^2]}^{1/2}+1\}}^{1/2}$

Determine the stratum bottom boundary degree of depth, wherein:

N is the refractive index of medium, μ _rIt is the relative permeability of medium; Footnote 2 and 3 is represented the up and down of bottom boundary respectively;

H is the stratum bottom boundary degree of depth, and ω is a magnetic field standing wave antinode circular frequency, ε ₂Be the specific inductive capacity on stratum, μ ₂Be the magnetoconductivity on stratum, σ ₂It is the conductivity on stratum.

Wherein, described detection object is the stratum; Described data recording and processor are at n ₂/ μ _R2Less than n ₃/ μ _R3The area, according to formula:

$h=\mathrm{\π}/{\left(\frac{{\mathrm{\ω}}^2{\mathrm{\ϵ}}_2{\mathrm{\μ}}_2}{2}\right)}^{1/2}{\{{[1+{\left(\frac{{\mathrm{\σ}}_2}{\mathrm{\ω}{\mathrm{\ϵ}}_2}\right)}^2]}^{1/2}+1\}}^{1/2}$

Determine the stratum bottom boundary degree of depth, wherein:

N is the refractive index of medium, μ _rIt is the relative permeability of medium; Footnote 2 and 3 is represented the up and down of bottom boundary respectively;

H is the stratum bottom boundary degree of depth, and ω is a magnetic field standing wave antinode circular frequency, ε ₂Be the specific inductive capacity on stratum, μ ₂Be the magnetoconductivity on stratum, σ ₂It is the conductivity on stratum.

Wherein, described detection object is the stratum; Described data recording and processor are at n ₂/ μ _R2Less than n ₃/ μ _R3The area, according to formula:

$h=\mathrm{\π}/{\left({2\mathrm{\ω}}^2{\mathrm{\ϵ}}_2{\mathrm{\μ}}_2\right)}^{1/2}{\{{[1+{\left(\frac{{\mathrm{\σ}}_2}{\mathrm{\ω}{\mathrm{\ϵ}}_2}\right)}^2]}^{1/2}+1\}}^{1/2}$

Determine the stratum bottom boundary degree of depth, wherein:

N is the refractive index of medium, μ _rIt is the relative permeability of medium; Footnote 2 and 3 is represented the up and down of bottom boundary respectively;

H is the stratum bottom boundary degree of depth, and ω is an electric field standing wave antinode circular frequency, ε ₂Be the specific inductive capacity on stratum, μ ₂Be the magnetoconductivity on stratum, σ ₂It is the conductivity on stratum.

The present invention also provides a kind of geophysical exploration method, utilizes standing wave to determine to survey the bottom boundary degree of depth of object, comprises the steps:

(1) measuring point is determined at the interface on the detection object;

(2) in measuring point emission and reception elastic wave, determine that the measuring point place states the antinode frequency of elasticity standing wave, wherein, described elasticity standing wave is the standing wave that is formed with the reflection wave interference that produces through the reflection of detection object bottom boundary by the incident wave that motor oscillating produces as focus;

(3) according to the antinode frequency of described elasticity standing wave, and survey the velocity of wave that the object surface records, determining to survey the bottom boundary degree of depth of object.

Wherein, described detection object is a bored concrete pile, and the bottom boundary degree of depth of described detection object is that bored concrete pile is long.

Wherein, described elastic wave drives the electromagnetic vibrator vibration by electrical oscillator and produces, and enters bored concrete pile and propagate by being arranged in shot point emission on the grout pile top; By continuously changing the frequency of electrical oscillator, when acceptance point used prison ripple device to monitor the wave amplitude maximum, the oscillation frequency of reading on electrical oscillator was exactly the frequency of standing wave antinode.

Wherein, in the described step (3), determine that according to formula: l=u/4f bored concrete pile is long, wherein:

L is that bored concrete pile is long, and f is the frequency of standing wave antinode, and u is a velocity of wave, and u=s/t, s are that shot point is to the distance between the acceptance point when measuring velocity of wave, and t is the travel-time of elastic wave between s.

The present invention has following characteristics:

1. utilize natural energy resources, do not need the artificial energy source, or with light motor oscillating as focus, economical and practical.2. data are specifically directly perceived, directly provide the geological interface buried depth.3. instrument is light, equipment is simple.4. rotating detection process is simple to operate, only needs energized, the automatic output data of instrument.5. staffing is few, a machine one people.6. reduced investment.These characteristics are that traditional geophysical prospecting method can not have concurrently.

Description of drawings

Fig. 1 is the bed boundary synoptic diagram;

Fig. 2 is electromagnetic reflection and refraction synoptic diagram;

Fig. 3 is that incident electromagnetic wave and reflection electromagnetic wave are at the electric field at last interface and the polar plot in magnetic field;

Fig. 4 is according to the described standing electromagnetic wave exploring equipment of embodiment of the invention block diagram;

Fig. 5 is according to the described elasticity standing wave of embodiment of the invention detecting devices synoptic diagram.

Embodiment

Standing electromagnetic wave method geophysical survey in the standing wave method geophysical survey provided by the invention is to utilize natural energy resources, comprises with 1, the electromagnetic wave of solar radiation; 2, solar wind impacts the electromagnetic wave that the magnetic field of the earth produces; 3, the electromagnetic wave of " Van Allen radiation belt " radiation; Or 4, the electromagnetic wave that forms of thunderstorm discharge, as the energy.Elasticity standing wave method geophysical survey then is to make focus with the vibration that light motor focus produces.

When ripple is propagated in the stratum or in the bored concrete pile, reflected by bottom boundary.Incident wave and reflection wave interference produce standing wave.As wavelength X and ground bed thickness h, or wavelength X and the long l of bored concrete pile satisfy when necessarily concern (h=λ/2, l=λ/4), on ground or push up standing wave formation antinode.On ground or stake top record the frequency f of antinode, calculate velocity of wave u, just can calculate bed thickness h=u/2f, or a long l=u/4f.The electric wave abdomen and the magnetic wave abdomen of standing wave can not come across same frequency.According to being electric wave abdomen or magnetic wave abdomen, judge n ₂/ μ _R2And n ₃/ μ _R3Relative size (n is the refractive index of medium, μ _rIt is the relative permeability of medium; Footnote 2 and 3 is represented the up and down of bottom boundary respectively), thus the geological property and the physical attribute on judgement stratum, both sides, interface.According to a large amount of, large-area bottom boundary buried depth (bed thickness h) data, draw the contour map of bottom boundary, reach the purpose of seeking structure.Perhaps according to the l value, determined whether broken pile, quality problems such as folder mud exist.

Below in conjunction with accompanying drawing, introduce the standing wave method geophysical survey in detail:

1. standing wave method geophysical survey

True origin is fixed on the bottom boundary of horizontal layered stratum ground floor, and the ground floor bed thickness is h, with Fig. 1 signal, is atmosphere more than the stratum, and upper and lower interface is parallel.One row circular frequency is ω (ω=2 π f), and wavelength is that the ripple of λ is propagated at the interface vertically downward by last interface, and at the interface last, the equation of incident wave is

${\mathrm{<figure-callout\; id="1"\; label="E\; i"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">E</figure-callout>}}_i=E_{0\mathrm{<figure-callout\; id="1"\; label="i"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">i</figure-callout>}1}\cos (\mathrm{\&omega;t}+\frac{2\mathrm{\&pi;h}}{\mathrm{\&lambda;}})---(1-1)$

After the bottom boundary reflection, reflection wave E _R1Get back to the interface, the equation of reflection wave is

${\mathrm{<figure-callout\; id="1"\; label="E\; r"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">E</figure-callout>}}_r=E_{0r1}{e}^{-2\mathrm{\&alpha;h}}\cos (\mathrm{\&omega;t}-\frac{2\mathrm{\&pi;h}}{\mathrm{\&lambda;}})$

E in the formula ^{-2 α h}Decay factor when being ripple process 2h distance, α claims attenuation coefficient.If the wave amplitude ratio of two ripples

E _0r1/E _0i1＝p ₂ (1-2)

Then

${\mathrm{<figure-callout\; id="1"\; label="E\; r"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">E</figure-callout>}}_r=E_{0i1}{p}_2{e}^{-2\mathrm{\&alpha;h}}\cos (\mathrm{\&omega;t}-\frac{2\mathrm{\&pi;h}}{\mathrm{\&lambda;}})---(1-3)$

p ₂Footnote 2 expression bottom boundarys (the 2nd interface).At last E at the interface _I1With E _R1Interfere to produce and close vibration

${\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">E</figure-callout>}}_1={\mathrm{<figure-callout\; id="1"\; label="E\; i"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">E</figure-callout>}}_i+{\mathrm{<figure-callout\; id="1"\; label="E\; r"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">E</figure-callout>}}_r$

$=E_{0i1}\{\cos (\mathrm{\&omega;t}+\frac{2\mathrm{\&pi;h}}{\mathrm{\&lambda;}})+p_2{e}^{-2\mathrm{\&alpha;h}}\cos (\mathrm{\&omega;t}-\frac{2\mathrm{\&pi;h}}{\mathrm{\&lambda;}})\}---(1-4)$

Through simple calculations,

${\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">E</figure-callout>}}_1=E_{0i1}\{(1+p_2{e}^{-2\mathrm{\&alpha;h}})\cos \frac{2\mathrm{\&pi;h}}{\mathrm{\&lambda;}}\cos \mathrm{\&omega;t}-(1-p_2{e}^{-2\mathrm{\&alpha;h}})\sin \frac{2\mathrm{\&pi;h}}{\mathrm{\&lambda;}}\sin \mathrm{\&omega;t}\}$

First of following formula be with

Be the cosine term of amplitude, second be with

Be the sine term of amplitude, then close vibration and be two amplitude is different, frequency is identical and phase differential is vibration synthetic of pi/2

E ₁＝E ₀₁cos(ωt-θ)

In the formula

$\mathrm{\&theta;}={\tan }^{-1}\frac{(1-p_2{e}^{-2\mathrm{\&alpha;h}})\sin \frac{2\mathrm{\&pi;h}}{\mathrm{\&lambda;}}}{(1+p_2{e}^{-2\mathrm{\&alpha;h}})\cos \frac{2\mathrm{\&pi;h}}{\mathrm{\&lambda;}}}$

$E_{01}=E_{0i1}{\{{(1+p_2{e}^{-2\mathrm{\&alpha;h}})}^2{\cos }^2\frac{2\mathrm{\&pi;h}}{\mathrm{\&lambda;}}+{(1-p_2{e}^{-2\mathrm{\&alpha;h}})}^2{\sin }^2\frac{2\mathrm{\&pi;h}}{\mathrm{\&lambda;}}\}}^{1/2}$

Because cos ²β=(1+cos2 β)/2 and sin ²β=(1-cos2 β)/2, following formula becomes after arrangement

${\mathrm{<figure-callout\; id="1"\; label="E\; o"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">E</figure-callout>}}_o=E_{\mathrm{oi}1}{\{1+p_2^2{e}^{-4\mathrm{\&alpha;h}}+2p_2{e}^{-2\mathrm{\&alpha;h}}\cos \frac{4\mathrm{\&pi;h}}{\mathrm{\&lambda;}}\}}^{1/2}---(1-5)$

Work as p ₂For on the occasion of, and satisfy condition

$\frac{4\mathrm{\&pi;h}}{\mathrm{\&lambda;}}=2\mathrm{k\&pi;},$ k＝1，2，3，… (1-6)

Then

$\cos \frac{4\mathrm{\&pi;h}}{\mathrm{\&lambda;}}=1,$ $h=\frac{\mathrm{k\&lambda;}}{2}---(1-7)$

Amplitude has maximum value

E ₀₁＝E _0i1(1+p ₂e ^-2αh) (1-8)

If p ₂Be negative value, get its absolute value p ₂=-| p ₂|, (1-5) formula becomes

$E_{01}=E_{0i1}{\{1+{\left|p_2\right|}^2{e}^{-4\mathrm{\&alpha;h}}-2|p_2\left|e^{-2\mathrm{\&alpha;h}}\cos \frac{4\mathrm{\&pi;h}}{\mathrm{\&lambda;}}\right\}}^{1/2}---(1-9)$

When satisfying condition $\frac{4\mathrm{\&pi;h}}{{\mathrm{\&lambda;}}^{\&prime;}}=(2k^{\&prime;}+1)\mathrm{\&pi;},$ k′＝0，1，2，…… (1-10)

Then $\cos \frac{4\mathrm{\&pi;h}}{{\mathrm{\&lambda;}}^{\&prime;}}=-1,h=\frac{(2k^{\&prime;}+1)}{4}{\mathrm{\&lambda;}}^{\&prime;}---(1-11)$

(λ, ω are corresponding to the k value, and λ ', ω ' are corresponding to k ' value) (1-9) formula has maximum value

E ₀₁＝E _0i1(1+|p ₂|e ^-2αh) (1-12)

Because for the h value of same measuring point, (1-7) and (1-11) two formulas are equal, so can get (getting k=1, k '=0)

ω ₁＝4ω ₀’ (1-12)′

That is to say: under the constant situation of bottom boundary buried depth h, p ₂For on the occasion of the time, the circular frequency ω of the standing wave antinode at interface on the stratum ₁Be p ₂Circular frequency ω during for negative value ₀' four times [see (2-26) and (2-27) two formulas].

Reflection wave arrives the upper bound, stratum face, is subjected to the reflection downwards once more of stratum-air interface again, claims current reflection wave to be the incident wave E second time _I2

$E_{i2}=E_{0i2}\cos (\mathrm{\&omega;t}+\frac{2\mathrm{\&pi;h}}{\mathrm{\&lambda;}})$

Wave amplitude E _0i2Be E _R1Wave amplitude E _0i1p ₂e ^{-2 α h}The reflection at interface on the process and getting, so

E _0i2＝E _0i1p ₂e ^-2αh·p ₁

In the formula

${\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">p</figure-callout>}}_1=\frac{E_{0i2}}{E_{0i1}{p}_2{e}^{-2\mathrm{\&alpha;h}}}---(1-13)$

So

$E_{i2}=E_{0i1}{p}_1{p}_2{e}^{-2\mathrm{\&alpha;h}}\cos (\mathrm{\&omega;t}+\frac{2\mathrm{\&pi;h}}{\mathrm{\&lambda;}})$

p ₁Also can just can bear, here, tentatively establish it and be just (proof is seen below), i.e. E _0i2With E _0i1Same-phase.

Reflexing to the wave amplitude at interface for the second time from following interface, is E _I2Wave amplitude again through following boundary reflection, and get by way of the decay of 2h, so

$E_{r2}=E_{0i1}{p}_1{p_2}^2{e}^{-4\mathrm{\&alpha;h}}\cos (\mathrm{\&omega;t}-\frac{2\mathrm{\&pi;h}}{\mathrm{\&lambda;}})$

E _I2With E _R2Form standing wave E ₂

$E_2=E_{i2}+E_{r2}$

$=E_{0i1}{p}_1{p}_2{e}^{-2\mathrm{\&alpha;h}}\cos (\mathrm{\&omega;t}+\frac{2\mathrm{\&pi;h}}{\mathrm{\&lambda;}})+E_{0i1}{p}_1{p_2}^2{e}^{-4\mathrm{\&alpha;h}}\cos (\mathrm{\&omega;t}-\frac{2\mathrm{\&pi;h}}{\mathrm{\&lambda;}})$

$=E_{0i1}{p}_1{p}_2{e}^{-2\mathrm{\&alpha;h}}\{\cos (\mathrm{\&omega;t}+\frac{2\mathrm{\&pi;h}}{\mathrm{\&lambda;}})+p_2{e}^{-2\mathrm{\&alpha;h}}\cos (\mathrm{\&omega;t}-\frac{2\mathrm{\&pi;h}}{\mathrm{\&lambda;}})\}$

Content in the following formula brace is identical with (1-4) formula, therefore to calculate E ₁Same steps as is calculated E ₂Amplitude E ₀₂When satisfying condition (1-6) formula

E ₀₂＝E _0i1p ₁p ₂e ^-2αh(1+p ₂e ^-2αh) (1-14)

(1-8) formula of summary reaches (1-14) formula, and summarizes the later standing wave wave amplitude that repeatedly reflects to form thus

E ₀₃＝E _0i1p ₁ ²p ₂ ²e ^-4αh(1+p ₂e ^-2αh)

..............

E _0n＝E _0i1p ₁ ^n-1p ₂ ^n-1e ^-2(n-1)αh(1+p ₂e ^-2αh)

These wave amplitude additions, draw and repeatedly reflect the total wave amplitude equation of standing wave

E ₀＝E ₀₁+E ₀₂+E ₀₃+……+E _0n

＝E _0i1(1+p ₂e ^-2αh)+E _0i1p ₁p ₂e ^-2αh(1+p ₂e ^-2αh)+E _0i1p ₁ ²p ₂ ²e ^-4αh(1+p ₂e ^-2αh)

+……+E _0i1p ₁ ^n-1p ₂ ^n-1e ^-2(n-1)αh(1+p ₂e ^-2αh)

＝E _0i1(1+p ₂e ^-2αh){1+p ₁p ₂e ^-2αh+p ₁ ²p ₂ ²e ^-4αh+……+p ₁ ^n-1p ₂ ^n-1e ^-2(n-1)αh}

Be that a common ratio is p in the following formula brace ₁p ₂e ^{-2 α h}Geometric progression, value of series $S=\frac{1-{\left(p_1{p}_2{e}^{-2\mathrm{\&alpha;h}}\right)}^n}{1-p_1{p}_2{e}^{-2\mathrm{\&alpha;h}}}$ When n was big, second of molecule went to zero, and draws the total wave amplitude of standing wave

${\mathrm{<figure-callout\; id="0"\; label="E"\; filenames="C2007101790290028H1.png"\; state="\{\{state\}\}">E</figure-callout>}}_0=\frac{E_{\mathrm{oi}1}(1+p_2{e}^{-2\mathrm{\&alpha;h}})}{1-p_1{p}_2{e}^{-2\mathrm{\&alpha;h}}}---(1-15)$

Work as p ₂For on the occasion of the time, E ₀It is the standing wave antinode; p ₂During for negative value, E ₀It is the standing wave node.

Work as p ₁Leveled off to 1 o'clock, the ratio that is drawn antinode and node by (1-15) formula is

So antinode can recognize.

Above discussion was not made any regulation to the attribute of plane wave, so above-mentioned conclusion is applicable to any plane ripple, no matter it is electromagnetism or flexible.

2. standing electromagnetic wave method geophysical survey

2.1 electromagnetic wave source:

One of energy that generates electromagnetic waves is from the sun, comprises the electromagnetic wave of solar radiation, and solar wind impacts the electromagnetic wave that the magnetic field of the earth produces.

The magnetic field of the earth towards sun one side, is compressed by solar wind; The one side of the back of the body sun is then by " extension ".If this regional magnetoconductivity is μ.The terrestrial magnetic field

Fluctuating, fluctuation in compression and compression process just have an amount from mathematics

There are it and electric field

Curl relevant, magnetic induction density $\stackrel{\&RightArrow;}{B}=\mathrm{\&mu;}\stackrel{\&RightArrow;}{H}.$ Thereby can derive $\&dtri;\&times;\stackrel{\&RightArrow;}{E}=-\&PartialD;\stackrel{\&RightArrow;}{B}/\&PartialD;t$

Can derive from following formula: $\&dtri;\&CenterDot;\stackrel{\&RightArrow;}{B}=0$

More than two formulas be exactly first pair of Maxwell equation.

Directed movement by charged particle and particle provides to Maxwell equation for another.If electric density is ρ _e, have: $\&dtri;\&CenterDot;\stackrel{\&RightArrow;}{E}={\mathrm{\&rho;}}_e/\mathrm{\&epsiv;}$

ε is a specific inductive capacity.The directed movement of charged particle has formed current density, J and displacement current density

So: $\&dtri;\&times;\stackrel{\&RightArrow;}{B}-\mathrm{\&epsiv;\&mu;}\&PartialD;\stackrel{\&RightArrow;}{E}/\&PartialD;t=\mathrm{\&mu;J}$

This two pairs of Maxwell equations have been arranged, and two of the electromagnetic wave source of the directive earth has just produced.

The charged particle band that capture the terrestrial magnetic field is called the radiation belt of the earth, also cries " Van Allen radiation belt ".These high energy particles in certain zone, are gone out electromagnetic wave along for the helical movement and continuous eradiation of the magnetic line of force by terrestrial magnetic field detention under geomagnetic field action.This is the 3rd wave source that generates electromagnetic waves.

The 4th electromagnetic wave source is the thunderstorm discharge.

These electromagnetic wave sources are with various frequency radiated electromagnetic waves.Electromagnetic wave is once generation, just at the earth's surface and between the ionosphere, does repeatedly reflection back and forth, is the basis of standing electromagnetic wave method geophysical survey, also is humanly for the first time to make the energy with solar wind and universe high energy particle.

2.2 electromagnetic wave propagation:

The earth is a conductor, the plane electric wave

The equation of axially propagating down along Z in conducting medium is

${\&dtri;}^2\stackrel{\&RightArrow;}{E}-\mathrm{\&epsiv;\&mu;}\frac{{\&PartialD;}^2\stackrel{\&RightArrow;}{E}}{\&PartialD;t^2}-\mathrm{\&sigma;\&mu;}\frac{\&PartialD;\stackrel{\&RightArrow;}{E}}{\&PartialD;t}=0---(2-1)$

ε is that specific inductive capacity, μ are that magnetoconductivity, σ are conductivity in the formula.

Magnetic wave Have and electric wave identical differential equation on expression-form.The equation that magnetic wave and electric wave interrelate is

$\&dtri;\&times;\stackrel{\&RightArrow;}{E}+\mathrm{j\&omega;\&mu;}\stackrel{\&RightArrow;}{H}=0---(2-2)$

The electric wave that (2-1) formula is described is write as exponential form and is exactly

$\stackrel{\&RightArrow;}{E}=E_0{e}^{j(\mathrm{\&omega;t}-\mathrm{kZ})}\stackrel{\&RightArrow;}{i}---(2-3)$

It is the axial unit vector of x.Solve by (2-3), (2-1) formula

k ²＝ω ²εμ(1-jσ/ωε) (2-4)

Wave number k is a plural number

k＝k _r-jk _i (2-5)

Separate (2-4), (2-5)

$k_r={\left(\frac{{\mathrm{\&omega;}}^2\mathrm{\&epsiv;\&mu;}}{2}\right)}^{1/2}{\{{[1+{\left(\frac{\mathrm{\&sigma;}}{\mathrm{\&omega;\&epsiv;}}\right)}^2]}^{1/2}+1\}}^{1/2}---(2-6)$

$k_i={\left(\frac{{\mathrm{\&omega;}}^2\mathrm{\&epsiv;\&mu;}}{2}\right)}^{1/2}{\{{[1+{\left(\frac{\mathrm{\&sigma;}}{\mathrm{\&omega;\&epsiv;}}\right)}^2]}^{1/2}-1\}}^{1/2}---(2-7)$

The utmost point formula of k is

$k={\left({\mathrm{\&omega;}}^2\mathrm{\&epsiv;\&mu;}\right)}^{1/2}{[1+{\left(\frac{\mathrm{\&sigma;}}{\mathrm{\&omega;\&epsiv;}}\right)}^2]}^{1/4}{e}^{-\mathrm{j\&theta;}}---(2-8)$

θ＝tan ^-1k _i/k _r

Reaching (2-8) by (2-2), (2-3), formula solves magnetic field

$\stackrel{\&RightArrow;}{H}={\left(\frac{\mathrm{\&epsiv;}}{\mathrm{\&mu;}}\right)}^{1/2}{[1+{\left(\frac{\mathrm{\&sigma;}}{\mathrm{\&omega;\&epsiv;}}\right)}^2]}^{1/4}{E}_0{e}^{-k_{i}Z}{e}^{j(\mathrm{\&omega;t}-k_{r}Z-\mathrm{\&theta;})}\stackrel{\&RightArrow;}{j}$

$=H_0{e}^{-\mathrm{kiZ}}{e}^{j(\mathrm{\&omega;t}-k_{r}Z-\mathrm{\&theta;})}\stackrel{\&RightArrow;}{j}---(2-9)$

The polarization direction in magnetic field

Polarization direction perpendicular to electric field

And the direction of propagation

And hysteresis electric field one phase angle theta.

In nonconductor, medium conductivity σ=0, (2-5) imaginary term in the formula are zero,

k＝ω(εμ) ^1/2 (2-13)

But generally speaking, be plural number as (2-5) formula, (2-6), (2-7) be respectively its real part and imaginary part to the earth material k of conduction.

Real part k _r=2 π/λ (2-14)

Imaginary part k _i, be a function that wave amplitude is decayed with the increase of propagation distance.When propagation distance increases,

To decay, and when the Z value reaches a particular values, order this Z value and be δ,

δ＝1/k _i (2-15)

This moment, amplitude fading arrived the 1/e of former amplitude.Amount δ be " attenuation distance ", is also referred to as " skin depth ", meaning promptly pass through δ apart from after, electromagnetic amplitude only is 0.368 times of original wave amplitude.

2.3 electromagnetic reflection and refraction:

Reflection and refraction will take place in the electromagnetic wave of propagating in discontinuous media in the both sides, interface.Set two kinds of linearities, evenly, the interphase between isotropic medium, be unlimited thin, infinitely-great plane.The refractive index of interphase media of both sides is respectively n ₁And n ₂, relative permeability μ _R1And μ _R2Defining the plane that is incorporated into ray, reflected ray and refracted ray again is plane of incidence, establishes this plane and XZ planes overlapping (see figure 2), in Fig. 2, incident wave is arranged on plane of incidence Reflection wave

And refraction wave

Incident angle equals reflection angle

θ _i＝θ _r (2-16)

The incident wave of Si Naier refraction law statement and the relation between the refraction wave are

$\frac{\sin {\mathrm{\&theta;}}_i}{{\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">u</figure-callout>}}_1}=\frac{\sin {\mathrm{\&theta;}}_t}{u_2}$

Perhaps, because wave number k _rEqual ω/u, then

k _r1sinθ _i＝k _r2sinθ _t (2-17)

Also can be write as following formula

$\frac{\sin {\mathrm{\&theta;}}_t}{\sin {\mathrm{\&theta;}}_i}=\frac{n_1}{n_2}---(2-18)$

But, at the both sides, interface of atmosphere and ground material formation, the refractive index n of atmosphere ₁Refractive index n much smaller than the earth material ₂Therefore, enter the plane electromagnetic wave on stratum by atmosphere with various angles, by (2-18) formula, mostly can be vertically or the near vertical earth-layer propagation.This makes and might the electromagnetic wave that enters the stratum from atmosphere, be used as vertical incidence and handle under many circumstances.So θ _iAnd θ _tEqual zero, plane of incidence becomes uncertain.

With reference to (2-3) formula, represent incident, reflection and the plane of refraction ripple of electric field respectively with Ei, Er, Et, under the situation by the incident of SEQUENCING VERTICAL stratum, they are

Ei＝E _0iexpj{ωt+k ₁Z} (2-19)

Er＝E _0rexpj{ωt-k ₁Z} (2-20)

Et＝E _0texpj{ωt+k ₂Z} (2-21)

Relation between their wave amplitudes can draw according to the Fei Nieer formula

$\frac{E_{0r}}{E_{0i}}=\frac{\frac{{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">n</figure-callout>}}_1}{{\mathrm{\&mu;}}_{r1}}-\frac{n_2}{{\mathrm{\&mu;}}_{r2}}}{\frac{{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">n</figure-callout>}}_1}{{\mathrm{\&mu;}}_{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">r</figure-callout>}1}}+\frac{n_2}{{\mathrm{\&mu;}}_{r2}}}---(2-22)$

$\frac{E_{0t}}{E_{0i}}=\frac{2\frac{{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">n</figure-callout>}}_1}{{\mathrm{\&mu;}}_{r1}}}{\frac{{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">n</figure-callout>}}_1}{{\mathrm{\&mu;}}_{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">r</figure-callout>}1}}+\frac{n_2}{{\mathrm{\&mu;}}_{r2}}}---(2-23)$

For the refraction wave E that enters the stratum _t, by (2-23) formula

$\frac{E_{0t}}{E_{0i}}=\frac{2}{1+n_2/{\mathrm{\&mu;}}_{r2}}={\mathrm{<figure-callout\; id="1"\; label="q"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">q</figure-callout>}}_1$

q ₁The last interface on footnote 1 expression stratum.As can be seen, q ₁Always positive number this means, on interphase, transmitted wave is always consistent with the incident wave phase place.Get refraction wave E _tExpression formula

E _t＝E _oiq ₁exp{j(ωt+k _r2Z)+k _iZ}

This is a ripple that enters the stratum through interfacial refraction on the stratum.But for the stratum, it but is an incident wave.Therefore, when the electromagnetic wave physical phenomenon is discussed in the stratum afterwards, just this ripple E _t, be called incident wave E _I1Its wave amplitude:

E _0i1＝E _0iq ₁

This ripple form at interface on the stratum is

${\mathrm{<figure-callout\; id="1"\; label="E\; i"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">E</figure-callout>}}_i=E_{0i}{\mathrm{<figure-callout\; id="1"\; label="q"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">q</figure-callout>}}_1\mathrm{expj}(\mathrm{\&omega;t}+\frac{2\mathrm{\&pi;h}}{\mathrm{\&lambda;}})---(2-24)$

Here, still adopt the coordinate system of Fig. 1.Ripple arrives goes up the interface, does not propagate this decay factor as yet in the stratum $e^{-k_{i}Z}=1.$

This ripple moves on, and will be subjected to the reflection of bottom boundary.Material refractive index under the bottom boundary is n ₃, relative permeability is u _R3The Fei Nieer formula of bottom boundary reflection is

$\frac{E_{0r}}{E_{0i}{\mathrm{<figure-callout\; id="1"\; label="q"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">q</figure-callout>}}_1}=\frac{\frac{n_2}{{\mathrm{\&mu;}}_{r2}}-\frac{n_3}{{\mathrm{\&mu;}}_{r3}}}{\frac{n_2}{{\mathrm{\&mu;}}_{r2}}+\frac{n_3}{{\mathrm{\&mu;}}_{r3}}}$

$=p_2$

p ₂The following interface on footnote 2 expression stratum.So the equation that reflection wave arrives when going up the interface is

E _r1＝E _0iq ₁p ₂expj{ωt-k ₂Z}

＝E _0iq ₁p ₂exp{j(ωt-2πh/λ)-2k _ih} (2-25)

Ratio p ₂Can be positive also can bearing, it depends on n ₂/ μ _R2With n ₃/ μ _R3Relative size, if n ₂/ μ _R2＞n ₃/ μ _R3, then reflection wave phase differential to incident wave on following interface is zero; Work as n ₂/ μ _R2＜n ₃/ μ _R3, then be π.

In above discussion, wave equation is write as exponential form, be for mathematical convenience.If get the real part of index, (2-24), just two formulas are the same with (1-1) with (1-3) for (2-25) two formulas, just will change Z into h, and α is changed into k _iSo, discussed in " standing wave method geophysical survey " and conclusion, can both be used in this section.

But, also need to prove the p of (1-13) formula ₁For just.According to (2-22) formula, the refractive index n of air ₂With relative permeability μ _R2Be 1, the relative permeability μ of the earth material _R1Modal value also be 1, and the refractive index n of the earth material ₁Much larger than n ₂So,

$\frac{E_{0i2}}{E_{0\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">r</figure-callout>}1}}=\frac{E_{0i2}}{E_{0i}{q}_1{p}_2{e}^{-2\mathrm{\&alpha;h}}}=\frac{\frac{{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">n</figure-callout>}}_1}{{\mathrm{\&mu;}}_{r1}}-\frac{n_2}{{\mathrm{\&mu;}}_{r2}}}{\frac{{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">n</figure-callout>}}_1}{{\mathrm{\&mu;}}_{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">r</figure-callout>}1}}+\frac{n_2}{{\mathrm{\&mu;}}_{r2}}}={\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">p</figure-callout>}}_1\text{>0}$

Promptly by the incident wave E of the earth air interface reflections in the stratum _I2Wave amplitude E _0i2, with E _R1Wave amplitude E _0iq ₁p ₂e ^{-2 α h}In the ground same-phase.

Through above discussion, just can write out total wave amplitude (1-15) formula of electric field standing wave

${\mathrm{<figure-callout\; id="0"\; label="E"\; filenames="C2007101790290028H1.png"\; state="\{\{state\}\}">E</figure-callout>}}_0=\frac{E_{0i}{q}_1(1+p_2{e}^{-2k_{i}h})}{1-p_1{p}_2{e}^{-2\mathrm{kih}}}$

Compare with (1-15) formula, just E _0i1=E _0iq ₁,

Work as p ₂Be timing, E ₀Be electric field standing wave antinode, p ₂When negative, E ₀It is node.

Can be with the way of calculating the electric field standing wave, from (2-9) formula calculating magnetic field standing wave wave amplitude.But well-known is that the electromagnetic wave propagation direction is

It is the electric field polarization direction

With the magnetic field polarised direction

Press the cross product of right-handed helix rule.Promptly

${\stackrel{\&RightArrow;}{i}}_E\&times;{\stackrel{\&RightArrow;}{j}}_H={\stackrel{\&RightArrow;}{k}}_Z$

Represent last interface (ground) (see figure 3) on stratum with paper, electromagnetic wave in the incident during interface, will have electric field as shown in Figure 3 from the top down

Vector and magnetic field

If power on a formation antinode at last interface, then reflect electric vector and incident electric vector same-phase.And the reflected electromagnetic direction of wave travel is opposite with the incident electromagnetic wave direction, so the magnetic vector of reflection wave

Certainty and incident magnetic vector

Difference π on phase place.Therefore, at last interface, the standing wave in magnetic field forms node.Learn thus that also at last interface, if the electric field standing wave is a node, then the magnetic field standing wave must be antinode.This just gives the possibility of a kind of selection of people: at n ₂/ μ _R2＞n ₃/ μ _R3Regional p ₂For on the occasion of, the electric field standing wave is ω in circular frequency ₁The time on ground antinode is arranged, then the magnetic field standing wave is a node.Antinode appears in the magnetic field standing wave on ground frequency is ω ₀'=ω ₁/ 4[sees (1-12) ' formula], at ω ₀' on the frequency, the electric field standing wave is a node.In like manner, at n ₂/ μ _R2＜n ₃/ μ _R3, p ₂Be the area of negative value, magnetic field standing wave circular frequency is ω ₁The time on ground antinode is arranged, then the electric field standing wave is a node.Antinode appears in the electric field standing wave on ground frequency is ω ₀'=ω ₁/ 4.Frequency relation according to electric field and the appearance of magnetic field antinode for judging the relativeness of upper and lower ground interlayer rerum natura, provides a kind of foundation that can be for reference.

2.4 the explanation of standing electromagnetic wave method geophysical survey data

The interpretation work of standing electromagnetic wave method geophysical survey will be got back to (1-7) formula: h=λ/2

With multiply by π, substitution again (2-14) and (2-6) two formulas must with the molecule denominator

$h=\mathrm{\&pi;\&lambda;}/2\mathrm{\&pi;}=\mathrm{\&pi;}/k_{r2}$

$=\mathrm{\&pi;}/{\left(\frac{{\mathrm{\&omega;}}^2{\mathrm{\&epsiv;}}_2{\mathrm{\&mu;}}_2}{2}\right)}^{1/2}{\{{[1+{\left(\frac{{\mathrm{\&sigma;}}_2}{{\mathrm{\&omega;\&epsiv;}}_2}\right)}^2]}^{1/2}+1\}}^{1/2}---(2-26)$

Perhaps get back to (1-11) formula h=λ '/4

$h=\mathrm{\&pi;}/{\left(2{\mathrm{\&omega;}}^2{\mathrm{\&epsiv;}}_2{\mathrm{\&mu;}}_2\right)}^{1/2}{\{{[1+{\left(\frac{{\mathrm{\&sigma;}}_2}{{\mathrm{\&omega;\&epsiv;}}_2}\right)}^2]}^{1/2}+1\}}^{1/2}---(2-27)$

Adopt (2-26) still (2-27) formula, this will be decided by the standing wave antinode frequency in electric field and magnetic field [(1-12) ' formula] actually.This is the net result of standing electromagnetic wave method geophysical survey.It connects the electromagnetic parameter on zone thickness h and stratum.The magnetic permeability mu on stratum ₂And conductivity ₂, can collect from magnetic method geophysical survey and electrical method geophysical survey department.This also is the geophysical survey customary way.Frequency f is that instrument records, ε ₂Can survey, again according to (2-26) formula or (2-27) formula solve bed thickness h.Learn ε if having no way of ₂, then can ε ₀For ε ₂Certainly this can produce error, still, for an area, is a systematic error.After the drilling data correction, systematic error is soluble.

2.5 attenuation of Electromagnetic under the maximum conditions

(2-6) and (2-7) formula has shown k _rWith k _iAll relevant with ω ε/σ, this ratio also has the people that it is defined as the Q of medium, to nonconductor, and conduction current density σ E=0, Q → ∞.But to the earth material, modal value σ＞10 of conductivity ^-4Mho/rice (is equivalent to electricalresistivity＜10 ⁴Ohm meter), get ε=ε ₀=8.85 * 10 ^-12Farad/rice, in a very big band frequency scope, the Q value is all little a lot of than 1.So (2-6), become in (2-7) two formula braces

{(1+1/Q ²) ^1/2±1} ^1/2≈(1/Q) ^1/2{1+Q ²/2±Q} ^1/2

≈(1/Q) ^1/2(1±Q/2)

≈(1/Q) ^1/2 (2-28)

Work as condition

Q＝ωε/σ≤1/50

The error of (2-28) formula is less than 1% during establishment.This inequality means that conduction current density σ E need compare displacement current density at least

Big 50 times.

Q than 1 little a lot of situation under, below relation is set up

k _i＝k _r＝(ωμσ/2) ^1/2＝2π/λ

And

k＝(ωμσ/2) ^1/2(1-j)

At this moment, when electromagnetic wave propagation distance Z equals a wavelength, decay factor

$e^{-k_{i}Z}=e^{-2\mathrm{\&pi;}}=1.87\&times;{10}^{-3}$

This conclusion shows: very less than the area of ω ε, electromagnetic wave will be subjected to serious decay at electricalresistivity=1/ σ, use standing electromagnetic wave method geophysical survey and solve geological problem, will can not receive Expected Results.

3. elasticity standing wave method geophysical survey:

The elasticity standing wave method is used for geophysical survey, is to produce the plane wave of enough area to one of requirement of wave source; Two, ripple will have continuity in time; Three, the frequency continuous variable.It is very difficult to satisfy these three conditions simultaneously, is feasible but be used for the piling quality detection.Make one output energy little, the continuously adjustable focus of frequency, be the thing of accomplishing easily.As for plane wave, if the length of rod-like element much larger than diameter, the wavefront almost plane ripple of the ripple of then in bar, propagating.So on the bored concrete pile top, the equation of elastic wave is exactly

$\mathrm{Ai}=A_{0i}\cos (\mathrm{\&omega;t}+\frac{2\mathrm{\&pi;l}}{\mathrm{\&lambda;}})---(3-1)$

Here, with A _0iExpression elastic wave A _iAmplitude, l be the stake long.

In elastic wave theory, have and the consistent in form wave amplitude ratio of Fei Nieer formula

$\frac{A_{0r}}{A_{0i}}=\frac{{\mathrm{\&rho;}}_2{u}_2-{\mathrm{\&rho;}}_1{\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">u</figure-callout>}}_1}{{\mathrm{\&rho;}}_2{u}_2+{\mathrm{\&rho;}}_1{\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">u</figure-callout>}}_1}=p_2---(3-2)$

ρ is a density of medium in the formula, the velocity of wave u of the outer material of stake (soil) ₂And density p ₂Always less than the speed u of concrete ₁And density p ₁So, p ₂Total is negative value.

By the reflection wave to the stake top at the bottom of the stake be

Ar＝A _0re ^-2αlcos(ωt-2πl/λ)

＝A _0ip ₂e ^-2αlcos(ωt-2πl/λ) (3-3)

(3-1), (3-3) two formulas with (1-1), (1-3) two formulas relatively, just bed thickness h has been changed into the long l of stake.On the stake top, vibration A _iWith A _rInterfere, (1-5) formula is just got back in the calculating of standing wave wave amplitude

$A_0=A_{0i}{\{1+{p_2}^2{e}^{-4\mathrm{\&alpha;l}}+2p_2{e}^{-2\mathrm{\&alpha;l}}\cos \frac{4\mathrm{\&pi;l}}{\mathrm{\&lambda;}}\}}^{1/2}$

p ₂Be negative value, with absolute value representation:

p ₂=-| p ₂So |

$A_0=A_{0i}{\{1+{\left|p_2\right|}^2{e}^{-4\mathrm{\&alpha;l}}-2|p_2\left|e^{-2\mathrm{\&alpha;l}}\cos \frac{4\mathrm{\&pi;l}}{\mathrm{\&lambda;}}\right\}}^{1/2}$

If

$\frac{4\mathrm{\&pi;l}}{\mathrm{\&lambda;}}=(2k^{\&prime;}+1)\mathrm{\&pi;},$ k′0，1，2...... (3-4)

Then

$\cos \frac{4\mathrm{\&pi;l}}{\mathrm{\&lambda;}}=-1,$ $l=\frac{2k^{\&prime;}+1}{4}\mathrm{\&lambda;}---(3-5)$

The wave amplitude of this moment

A ₀＝A _0i{(1+|p ₂| ²e ^-4αl)+2|p ₂|e ^-2αl} ^1/2

＝A _0i(1+|p ₂|e ^-2αl) (3-6)

Be that maximum value is the standing wave antinode.Continuously change the transmission frequency of Vib., if velocity of wave u is a constant, will be in k '=0,1,2...... etc. have located relation:

f ₀＝f ₁/3＝f ₂/5...... (3-7)

And find maximum amplitude at these frequency places.It is long to calculate stake according to (3-5) formula:

l＝λ ₀/4

＝u/4f ₀ (3-8)

λ ₀Be that frequency is f ₀The time wavelength.Velocity of wave can directly record from the stake top.Judged whether quality accidents such as broken pile, folder mud again according to the long l of stake, can also judge the total quality of stake according to velocity of wave.

4. the instrument of standing wave method geophysical survey, work is carried out and achievement:

With reference to figure 4, according to embodiments of the invention, the standing electromagnetic wave exploring equipment is made of two passages." electric signal receiver " reaches " magnetic signal receiver " and receives electric field signal and field signal respectively." electric signal receiver " is made up of plane-parallel capacitor, is filled with the dielectric that specific inductive capacity is ε in the capacitor, to reduce output impedance." magnetic signal receiver " is made up of multiturn coil.The voltage amplification multiplying power k of signal amplifier _v〉=110 decibels.Wherein, electrical signal paths and magnetic signal passage constitute the electromagnetic signal receiving trap jointly, and signal is sent into data recording and processor.Electromagnetic parameter input media (not shown) is one of content of data recording and processor, and the known electric magnetic parameter of object is surveyed in input, the specific inductive capacity on stratum for example, the magnetoconductivity on stratum, the conductivity on stratum etc.; The signal of sending here by data recording and processor for recording electricity, two passages of magnetic draws standing electromagnetic wave antinode frequency again, and the electromagnetic parameter that the electromagnetic wave antinode frequency determined and input are set is in conjunction with handling, store, and the data result of output after calculating.

With reference to figure 5, according to embodiments of the invention, elasticity standing wave detecting devices comprises that mainly an elasticity standing wave antinode frequency determines device, forms the elastic wave radiating portion by electric signal oscillator, electromagnetic vibrator; Form signal receive section by receiver and prison ripple device.These four fractions all are in linear working state.The electrical oscillator output driving current drives the frequency vibration that Vib. is pressed drive current.Prison ripple device monitors the wave amplitude that receiver is received, with the signal determining to be received standing wave antinode whether.The standing wave frequency is read by electrical oscillator.By continuously changing the frequency of electrical oscillator, when acceptance point used prison ripple device to monitor the wave amplitude maximum, the oscillation frequency of reading on electrical oscillator was exactly the frequency of standing wave antinode.

In addition, elasticity standing wave detecting devices also comprises a velocity of wave measurement mechanism (not shown) and the long calculation element (not shown) of stake, wherein, the velocity of wave measurement mechanism according to described launching site and acceptance point apart from s, and the travel-time t of elastic wave between s, measure and obtain velocity of wave u; The long calculation element of stake determines that according to formula: l=u/4f bored concrete pile is long, wherein:

L is that bored concrete pile is long, and f is the frequency of standing wave antinode, and u is a velocity of wave, u=s/t, s be shot point to the distance between the acceptance point, t is the travel-time of elastic wave between s.

The present invention in the specific implementation, by operating personnel one people, take " the standing electromagnetic wave exploring equipment " one, in the zone of needs explorations, by the measuring point pointwise testing that arranges in advance.On measuring point, connect the power supply of instrument, to instrument input measuring point numbering and necessary electromagnetic parameter, press play button, instrument is promptly worked voluntarily, receives the standing electromagnetic wave on the measuring point.Monitor the instrument working condition by operating personnel.After operating personnel judge the information of having received necessity, promptly shut down, this point data is stored in the machine, finishes this work, is transferred to next measuring point.

In the present invention, operating personnel are to necessary electromagnetic parameter μ, σ, the ε of instrument input, and then the data of instrument storage are this measuring points, the buried depth h at interface under the stratum, otherwise, be standing wave antinode frequency.

Behind the same day end-of-job, the total data intersection number by the buried depth of data preparation personnel according to interface under the stratum of each measuring point, is drawn interface contour map (structural map) under the stratum according to the arrangement personnel, and this is the achievement data.

About computing formula:

$h=\mathrm{\&pi;}/{\left(\frac{{\mathrm{\&omega;}}^2\mathrm{\&epsiv;\&mu;}}{2}\right)}^{1/2}{\{{[1+{\left(\frac{\mathrm{\&sigma;}}{\mathrm{\&omega;\&epsiv;}}\right)}^2]}^{1/2}+1\}}^{1/2}---{\left(7\right)}^{\&prime;}$

With

$h=\mathrm{\&pi;}/{\left(2{\mathrm{\&omega;}}^2\mathrm{\&epsiv;\&mu;}\right)}^{1/2}{\{{[1+{\left(\frac{\mathrm{\&sigma;}}{\mathrm{\&omega;}\mathrm{\&epsiv;}}\right)}^2]}^{1/2}+1\}}^{1/2}---{\left(8\right)}^{\&prime;}$

The problem how to use can be summarized as:

At p ₂Value is positive area, and electric wave abdomen frequency is on the ω point, and with equation (7) ' formula, magnetic wave abdomen frequency is used equation (8) ' formula when asking the h value on ω ' when asking the h value.At p ₂＜0 area, the frequency of magnetic wave abdomen on the ω point, ask the h value with equation (7) ', the electric wave width of cloth becomes the Frequency point of standing wave abdomen at ω ', asks h with equation (8) ' and ω＞ω '.

For example in the somewhere, the frequency of receiving electric standing wave abdomen is f =2.5 * 10 ³Hz, magnetic wave abdomen frequency f _Magnetic=10 ⁴Hz.And these regional conductivity=2.86 * 10 ^-4Mho/rice, specific inductive capacity are ε=8.85 * 10 ^-12Farad/rice, magnetic permeability mu=12.75 * 10 ^-7Henry/rice.

The first step: compare f With f _MagneticIs who big? because ω＞ω ', bigger frequency is selected for use

$h=\mathrm{\&pi;}/{\left(\frac{{\mathrm{\&omega;}}^2\mathrm{\&epsiv;\&mu;}}{2}\right)}^{1/2}{\{{[1+{\left(\frac{\mathrm{\&sigma;}}{\mathrm{\&omega;\&epsiv;}}\right)}^2]}^{1/2}+1\}}^{1/2}$

Less frequency is selected for use

$h=\mathrm{\&pi;}/{\left(2{\mathrm{\&omega;}}^2\mathrm{\&epsiv;\&mu;}\right)}^{1/2}{\{{[1+{\left(\frac{\mathrm{\&sigma;}}{\mathrm{\&omega;\&epsiv;}}\right)}^2]}^{1/2}+1\}}^{1/2}$

Obvious in this example f _Magnetic＞f So, with f _MagneticIn substitution (7) ' formula:

$=1/2f_H{(56.4\&times;{10}^{-19})}^{1/2}{\{{[1+{\left(\frac{5143880}{f_H}\right)}^2]}^{1/2}+1\}}^{1/2}$

${\underset{\&CenterDot;}{\&CenterDot;\&CenterDot;}\left(\frac{5143880}{f_H}\right)}^2>>1,$

So

$h=1/2f_H\&times;2.37\&times;{10}^{-9}{\{\frac{5143880}{f_H}+1\}}^{1/2}$

$=1/4.74\&times;{10}^{-9}\&times;\frac{2268}{{{\mathrm{<figure-callout\; id="1"\; label="f\; H"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">f</figure-callout>}}_H}^{1/2}}{f}_H=1/10750.3{\mathrm{<figure-callout\; id="1"\; label="f\; H"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">f</figure-callout>}}_H^{}\&times;{10}^{-9}$

$=1/10750.3\&times;{10}^2\&times;{10}^{-9}=9.30\&times;{10}^2$

Electricity consumption standing wave abdomen frequency is asked h

$=1/2f_E{(2\&times;8.85\&times;{10}^{-12}\&times;12.75\&times;{10}^{-7})}^{1/2}{\{{[1+{\left(\frac{2.86\&times;{10}^{-4}}{2\mathrm{\&pi;}\&times;8.85\&times;{10}^{-12}{f}_E}\right)}^2]}^{1/2}+1\}}^{1/2}$

$=1/2\&times;{(225.7\&times;{10}^{-19})}^{1/2}{f}_E{\left\{\frac{5143880}{f_E}\right\}}^{1/2}$

$=1/2\&times;4.75\&times;{10}^{-9}\&times;2268{{\mathrm{<figure-callout\; id="1"\; label="f\; E"\; filenames="C2007101790290029H1.png"\; state="\{\{state\}\}">f</figure-callout>}}_E}^{1/2}$

$=1/21546\&times;5\&times;10\&times;{10}^{-9}=1/1077300\&times;{10}^{-9}$

It is have an appointment 0.2% the error of calculation of reflecting surface buried depth that electric standing wave abdomen frequency and magnetic standing wave abdomen frequency meter are calculated.

And " elasticity standing wave method geophysical prospecting equipment " is bonded at Vib. and receiver on the pouring pile head when concrete the use.After the energized, instrument is in running order: electrical oscillator drives the electromagnetic vibrator vibrations, produces elastic wave.Size by the prison ripple device prison oscillography width of cloth.Operating personnel continuously change the frequency of electrical oscillator, and when prison ripple device was discovered the wave amplitude maximum, the concussion frequency of reading on electrical oscillator was exactly the frequency f of standing wave antinode.According to the velocity of wave u that in stake, records, long again by (3-8) calculating stake.

In addition, need to prove, measure the instrument of bored concrete pile elastic wave velocity u, the commodity of multiple model are arranged in this industry, just be to use existing goods Instrument measuring elastic wave velocity u in the embodiment of the invention.

Claims (8)

1, a kind of geophysical exploration method utilizes standing wave to determine to survey the bottom boundary degree of depth of object, comprises the steps:

(1) measuring point is determined at the interface on the detection object;

(2) receive electromagnetic wave at measuring point, and according to the electromagnetic wave that receives, determine standing electromagnetic wave antinode frequency, wherein, described standing electromagnetic wave is the standing wave that is formed with the reflection wave interference that produces through the reflection of detection object bottom boundary by the incident wave that electromagnetic wave source produces;

(3) according to the antinode frequency of described standing electromagnetic wave, and the known electric magnetic parameter of surveying object, determine to survey the bottom boundary degree of depth of object,

Wherein,

Described detection object is the stratum,

Described electromagnetic wave source comprises: the electromagnetic wave of solar radiation, solar wind impact electromagnetic wave, the electromagnetic wave of " Van Allen radiation belt " radiation and/or the electromagnetic wave that the thunderstorm discharge forms that the magnetic field of the earth produces,

Described step (3) comprising: at n ₂/ μ _R2Greater than n ₃/ μ _R3The area, according to formula:

$h=\mathrm{\&pi;}/{\left(\frac{{\mathrm{\&omega;}}^2{\mathrm{\&epsiv;}}_2{\mathrm{\&mu;}}_2}{2}\right)}^{1/2}{\{{[1+{\left(\frac{{\mathrm{\&sigma;}}_2}{\mathrm{\&omega;}{\mathrm{\&epsiv;}}_2}\right)}^2]}^{1/2}+1\}}^{1/2}$

Determine the stratum bottom boundary degree of depth, wherein:

N is the refractive index of medium, μ _rIt is the relative permeability of medium; Footnote 2 and 3 is represented the up and down of bottom boundary respectively;

H is the stratum bottom boundary degree of depth, and ω is an electric field standing wave antinode circular frequency, ε ₂Be the specific inductive capacity on stratum, μ ₂Be the magnetoconductivity on stratum, σ ₂It is the conductivity on stratum.

2, the method for claim 1 is characterized in that, described step (3) comprising: at n ₂/ μ _R2Greater than n ₃/ μ _R3The area, according to formula:

$h=\mathrm{\&pi;}/{\left(2{\mathrm{\&omega;}}^2{\mathrm{\&epsiv;}}_2{\mathrm{\&mu;}}_2\right)}^{1/2}{\{{[1+{\left(\frac{{\mathrm{\&sigma;}}_2}{\mathrm{\&omega;}{\mathrm{\&epsiv;}}_2}\right)}^2]}^{1/2}+1\}}^{1/2}$

Determine the stratum bottom boundary degree of depth, wherein:

N is the refractive index of medium, μ _rIt is the relative permeability of medium; Footnote 2 and 3 is represented the up and down of bottom boundary respectively;

H is the stratum bottom boundary degree of depth, and ω is a magnetic field standing wave antinode circular frequency, ε ₂Be the specific inductive capacity on stratum, μ ₂Be the magnetoconductivity on stratum, σ ₂It is the conductivity on stratum.

3, the method for claim 1 is characterized in that, described step (3) comprising: at n ₂/ μ _R2Less than n ₃/ μ _R3The area, according to formula:

$h=\mathrm{\&pi;}/{\left(\frac{{\mathrm{\&omega;}}^2{\mathrm{\&epsiv;}}_2{\mathrm{\&mu;}}_2}{2}\right)}^{1/2}{\{{[1+{\left(\frac{{\mathrm{\&sigma;}}_2}{\mathrm{\&omega;}{\mathrm{\&epsiv;}}_2}\right)}^2]}^{1/2}+1\}}^{1/2}$

Determine the stratum bottom boundary degree of depth, wherein:

N is the refractive index of medium, μ _rIt is the relative permeability of medium; Footnote 2 and 3 is represented the up and down of bottom boundary respectively;

H is the stratum bottom boundary degree of depth, and ω is a magnetic field standing wave antinode circular frequency, ε ₂Be the specific inductive capacity on stratum, μ ₂Be the magnetoconductivity on stratum, σ ₂It is the conductivity on stratum.

4, the method for claim 1 is characterized in that, described step (3) comprising: at n ₂/ μ _R2Less than n ₃/ μ _R3The area, according to formula:

$h=\mathrm{\&pi;}/{\left(2{\mathrm{\&omega;}}^2{\mathrm{\&epsiv;}}_2{\mathrm{\&mu;}}_2\right)}^{1/2}{\{{[1+{\left(\frac{{\mathrm{\&sigma;}}_2}{\mathrm{\&omega;}{\mathrm{\&epsiv;}}_2}\right)}^2]}^{1/2}+1\}}^{1/2}$

Determine the stratum bottom boundary degree of depth, wherein:

N is the refractive index of medium, μ _rIt is the relative permeability of medium; Footnote 2 and 3 is represented the up and down of bottom boundary respectively;

H is the stratum bottom boundary degree of depth, and ω is an electric field standing wave antinode circular frequency, ε ₂Be the specific inductive capacity on stratum, μ ₂Be the magnetoconductivity on stratum, σ ₂It is the conductivity on stratum.

5, a kind of geophysical prospecting equipment utilizes standing wave to determine to survey the bottom boundary degree of depth of object, comprising:

The electromagnetic signal receiving trap is used for receiving electromagnetic wave at measuring point;

The electromagnetic parameter input media is used to import the known electric magnetic parameter of surveying object; And

Data recording and processor are used for determining standing electromagnetic wave antinode frequency according to the electromagnetic wave that receives, and according to the known electric magnetic parameter of the detection object of described input, determine to survey the bottom boundary degree of depth of object,

Wherein, described standing electromagnetic wave is the standing wave that is formed with the reflection wave interference that produces through the reflection of detection object bottom boundary by the incident wave that electromagnetic wave source produces,

And described electromagnetic signal receiving trap comprises:

The electric signal reception amplifier is used to receive electric field signal;

The magnetic signal reception amplifier is used to receive field signal,

Described detection object is the stratum; Described data recording and processor are at n ₂/ μ _R2Greater than n ₃/ μ _R3The area, according to formula:

$h=\mathrm{\&pi;}/{\left(\frac{{\mathrm{\&omega;}}^2{\mathrm{\&epsiv;}}_2{\mathrm{\&mu;}}_2}{2}\right)}^{1/2}{\{{[1+{\left(\frac{{\mathrm{\&sigma;}}_2}{\mathrm{\&omega;}{\mathrm{\&epsiv;}}_2}\right)}^2]}^{1/2}+1\}}^{1/2}$

Determine the stratum bottom boundary degree of depth, wherein:

N is the refractive index of medium, μ _rIt is the relative permeability of medium; Footnote 2 and 3 is represented the up and down of bottom boundary respectively;

H is the stratum bottom boundary degree of depth, and ω is an electric field standing wave antinode circular frequency, ε ₂Be the specific inductive capacity on stratum, μ ₂Be the magnetoconductivity on stratum, σ ₂It is the conductivity on stratum.

6, equipment as claimed in claim 5 is characterized in that, described detection object is the stratum; Described data recording and processor are at n ₂/ μ _R2Greater than n ₃/ μ _R3The area, according to formula:

$h=\mathrm{\&pi;}/{\left(2{\mathrm{\&omega;}}^2{\mathrm{\&epsiv;}}_2{\mathrm{\&mu;}}_2\right)}^{1/2}{\{{[1+{\left(\frac{{\mathrm{\&sigma;}}_2}{\mathrm{\&omega;}{\mathrm{\&epsiv;}}_2}\right)}^2]}^{1/2}+1\}}^{1/2}$

Determine the stratum bottom boundary degree of depth, wherein:

N is the refractive index of medium, μ _rIt is the relative permeability of medium; Footnote 2 and 3 is represented the up and down of bottom boundary respectively;

H is the stratum bottom boundary degree of depth, and ω is a magnetic field standing wave antinode circular frequency, ε ₂Be the specific inductive capacity on stratum, μ ₂Be the magnetoconductivity on stratum, σ ₂It is the conductivity on stratum.

7, equipment as claimed in claim 5 is characterized in that, described detection object is the stratum; Described data recording and processor are at n ₂/ μ _R2Less than n ₃/ μ _R3The area, according to formula:

$h=\mathrm{\&pi;}/{\left(\frac{{\mathrm{\&omega;}}^2{\mathrm{\&epsiv;}}_2{\mathrm{\&mu;}}_2}{2}\right)}^{1/2}{\{{[1+{\left(\frac{{\mathrm{\&sigma;}}_2}{\mathrm{\&omega;}{\mathrm{\&epsiv;}}_2}\right)}^2]}^{1/2}+1\}}^{1/2}$

Determine the stratum bottom boundary degree of depth, wherein:

N is the refractive index of medium, μ _rIt is the relative permeability of medium; Footnote 2 and 3 is represented the up and down of bottom boundary respectively;

H is the stratum bottom boundary degree of depth, and ω is a magnetic field standing wave antinode circular frequency, ε ₂Be the specific inductive capacity on stratum, μ ₂Be the magnetoconductivity on stratum, σ ₂It is the conductivity on stratum.

8, equipment as claimed in claim 5 is characterized in that, described detection object is the stratum; Described data recording and processor are at n ₂/ μ _R2Less than n ₃/ μ _R3The area, according to formula:

$h=\mathrm{\&pi;}/{\left(2{\mathrm{\&omega;}}^2{\mathrm{\&epsiv;}}_2{\mathrm{\&mu;}}_2\right)}^{1/2}{\{{[1+{\left(\frac{{\mathrm{\&sigma;}}_2}{\mathrm{\&omega;}{\mathrm{\&epsiv;}}_2}\right)}^2]}^{1/2}+1\}}^{1/2}$

Determine the stratum bottom boundary degree of depth, wherein:

N is the refractive index of medium, μ _rIt is the relative permeability of medium; Footnote 2 and 3 is represented the up and down of bottom boundary respectively;

H is the stratum bottom boundary degree of depth, and ω is an electric field standing wave antinode circular frequency, ε ₂Be the specific inductive capacity on stratum, μ ₂Be the magnetoconductivity on stratum, σ ₂It is the conductivity on stratum.

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