CN103135140A - Computing method of center loop line transverse electric and magnetic field (TEM) whole period true resistivity without fringe effect - Google Patents

Abstract

The invention discloses a computing method of center loop line transverse electric and magnetic field (TEM) whole period true resistivity without fringe effect. The problem that analysis and solving of a central point out-field point are difficult is solved starting from a current-carrying point infinitesimal circular loop line electric field formula and according to the relationship among an electric field, a magnetic field and induced electromotive force; and according to relative content in electrical resistivity exploration, the computing problem containing Bessel functional integration is solved by a large amount of gradual models of a Bessel function. The method comprises the following steps: an induced electromotive analytical expression at any field point is obtained; an induced electromotive analytical expression of the single Bessel function at any field point is obtained; and the induced electromotive analytical expression at any field point is substituted to an inversion program so that the center loop line TEM true resistivity without the fringe effect is obtained. Influences by the fringe effect are eliminated fundamentally, correct judgment rate of an underground geologic structure is increased, the method can be used for processing and explaining center loop line TEM messages and explain accuracy is improved.

Description

True resistivity computing method of full phase of a kind of central loop TEM of non-flanged effect

Technical field

The invention belongs to the geophysical survey field, be specifically related to a kind of electricity and electromagnetic prospecting interpretation methods.。

Background technology

In order to improve the operating efficiency of central loop TEM (Transient Electro-Magnetic, TEM), in the open air in the exploration, with observation station from zone that central point has expanded center 1/3 to as shown in Figure 1.But the unevenness of field, central area forms edge effect as shown in Figure 2.Due to the apparent resistivity that has adopted the definition of central point formula, edge effect on apparent resistivity-depth section, formed relevant with transmitting loop, change with the irrelevant rhythm of geologic structure, cause the erroneous judgement to underground geologic structure.In order to address this problem, prior art will decide greatly the source loop line and center loop line theoretical formula is unified ^[1-2], acquisition remain apparent resistivity, and the theoretical formula of deciding greatly the source loop line derives from the dipole infinitesimal, the field point of center loop line is supposed to differ far away with dipole to the distance in source, error is still larger; For the calculating of true resistivity, prior art adopts the bearing calibration of measuring point playback to eliminate edge effect ^[3]But edge effect is the essence performance of Loop source field, and the method not only can not be eliminated edge effect, also can introduce new error.Erroneous judgement to underground geologic structure still exists.

Documents and list of references

[1] Li Jianping, Li Tonglin, Zhao Xuefeng, Liang Taimu. the research of layered medium arbitrary shape Loop source transient electromagnetic APPARENT RESISTIVITY. Advances in Geophysics, 2007,22 (6): 1777-1780

[2] Shi Xianxin, Yan Shu, Fu Junmei, Chen Mingsheng. the improvement of transient electromagnetic method center wire-retracting device interpretation methods. Chinese Journal of Geophysics, 2009,52 (7): 1931-1936

[3] http://www.phoenix-geophysics.com/

[4]Knight?J?H,Raiche?A?P.transient?electromagnetic?calculations?using?the?Gaver-Stehfest?inverse?Laplace?transform?method.Geophysics,1982,47(1):47-50

[5]Anderson?W?L.Numerical?integration?of?related?Hankel?transforms?of?order?0?and?1?by?adaptive?digital?filtering.Geophysics,1979,44(7):1287-1305.

[6]Koefoed?O,Ghoch?D?P，Polmen?G?J.Computation?of?type?curves?for?electromagnetic?depth?sounding?with?a?horizontal?transient?coil?by?means?of?a?digital?linear?filter.Geophysical?Prospecting,1972,20:406-420.

[7]Verma?R?K,Koefoed?O.A?note?on?the?linear?filter?method?of?computing?electromagnetic?sounding?curves.Geophysical?Prospecting,1973,21:70-76.

[8] Chen Mingsheng, Chen Leshou, Wang Tiansheng, Bai Gaixian. explain telluric electromagnetic sounding and electric sounding data with improved generalized inverse matrix method. Chinese Journal of Geophysics, 1983,26 (4): 390-400.

Summary of the invention

In order to overcome the defective of eliminating the edge effect method in prior art, the invention provides a kind of true resistivity computing method of full phase of central loop TEM of non-flanged effect, eliminate the erroneous judgement to underground geologic structure that is caused by edge effect.

Obtain the key of the central loop TEM true resistivity of non-flanged effect: the first, the analytical expression of any point induced electromotive force V (t) in the acquisition loop line.But due to the unevenness of field distribution, the induced electromotive force except central point can not pass through around receiving coil electric field E _θIntegration obtain; The second, in the central loop TEM theoretical formula with the numerical integration computational problem of two Bessel functions of multi-form appearance.

In order to solve above technical matters, the technical solution adopted in the present invention is as follows.

True resistivity computing method of full phase of a kind of central loop TEM of non-flanged effect comprise the following steps:

Step 1 obtains a central loop TEM point vertical magnetic field analytical expression arbitrarily

In cylindrical-coordinate system, when the loop line mid point overlaps with true origin, central loop TEM electric field E on Earth Surface _θThe frequency field expression formula be

$E_{\mathrm{\θ}}(r,\mathrm{\ω})=-\mathrm{j\ω}{\mathrm{\μ}}_{0}I\left(\mathrm{\ω}\right)a\underset{0}{\overset{\∞}{\∫}}{R}_n(\mathrm{\λ},\mathrm{\ω})J_1\left(\mathrm{\λa}\right)J_1\left(\mathrm{\λr}\right)\mathrm{d\λ}---\left(1\right)$

In formula, r is a bit to the true origin distance on ground; ω=2 π f are circular frequency, and wherein f is frequency; μ ₀=4 π * 10 ^-7H/m is non magnetic the earth magnetic permeability; I is transmitter current, and a is the transmitting loop radius; J ₁Be 1 rank Bessel function, R _nTotal reflectance on the stratiform Earth Surface; λ is the integration variable of Hankel conversion;

Utilize the Maxwell vorticity equation

$\▿\×E=-\mathrm{j\ω}{\mathrm{\μ}}_{0}H---\left(2\right)$

Wherein E is electric field intensity, and H is magnetic field intensity.Because electric field only has the θ component and is only the function of r, therefore the magnetic field H of following vertical component is arranged _z

$H_z=\frac{1}{r}\frac{\∂}{\∂r}\left(r{E}_{\mathrm{\θ}}\right)=\frac{I\left(\mathrm{\ω}\right)a}{r}\underset{0}{\overset{\∞}{\∫}}{R}_n(\mathrm{\λ},\mathrm{\ω})J_1\left(\mathrm{\λa}\right)[\mathrm{\λ}{J}_0\left(\mathrm{\λr}\right)-\frac{1}{r}{J}_1\left(\mathrm{\λr}\right)]\mathrm{d\λ}---\left(3\right)$

J in formula (3) ₀It is 0 rank Bessel function.Contain two Bessel functions in analytic formula (3), also need further to change into single Bessel function, could use existing filter factor scheduling algorithm and try to achieve principal value of integral.

Step 2 obtains the induced electromotive force analytical expression of a single Bessel function of point arbitrarily

Large transmitting loop for the a=600m ~ 800m that generally uses utilizes the gradual of Bessel function

$J_1\left(x\right)\≈\sqrt{\frac{2}{\mathrm{\πx}}}\cos (x-\frac{3\mathrm{\π}}{4})(x\→\∞)---\left(4\right)$

With formula (4) substitution formula (3)

$h_z\left(t\right)\≈\frac{I\left(\mathrm{\ω}\right)a}{r}\sqrt{\frac{2}{\mathrm{\πa}}}\underset{0}{\overset{\∞}{\∫}}{R}_n(\mathrm{\λ},\mathrm{\ω})\sqrt{\frac{1}{\mathrm{\λ}}}\cos (\mathrm{\λa}-\frac{3\mathrm{\π}}{4})[\mathrm{\λ}{J}_0\left(\mathrm{\λr}\right)-\frac{1}{r}{J}_1\left(\mathrm{\λr}\right)]\mathrm{d\λ}---\left(5\right)$

Formula (5) is done contrary Laplace conversion, obtain the time domain form

Actual measurement induced electromotive force V (t) and h _z(t) pass is

$V\left(t\right)=\frac{\∂}{\∂t}{h}_z\left(t\right)---\left(7\right)$

The induced electromotive force analytical expression of a single Bessel function of point is with getting arbitrarily after formula (6) substitution formula (7)

Step 3 in formula (8) substitution inversion program, namely obtains the full phase true resistivity of the central loop TEM of non-flanged effect.

The evaluation of above-mentioned formula (8) utilizes Differential Properties, G-S algorithm, the filter factor algorithm of Laplace.

Described filter factor algorithm calculates 1 rank Bessel functions for 47 filter factors that Koefoed etc. provides, and 51 filter factors that provide with Verma calculate 0 rank Bessel functions; Perhaps utilize 441 filter factors that Anderson provides to calculate 1 rank and 0 rank Bessel function.

The present invention has beneficial effect.Non-flanged effect central loop TEM disclosed by the invention true resistivity solution of full phase from the circular loop line electric field formula of current-carrying point infinitesimal, has overcome the inherent error of dipole infinitesimal.According to the relation between electric field, magnetic field, induced electromotive force, solved central point outfield point Analytical Solution hard problem; According to the relative concept in electric resistivity exploration, use the gradual of large argument Bessel function, solved the computational problem that contains the Bessel functional integration, fundamentally eliminated edge effect; By induced electromotive force formula (8) the substitution inversion program with any the single Bessel function of point of the present invention, namely obtain full phase true resistivity and the thickness of layered earth.Improved the accuracy of central loop TEM Underground geologic structure.Full phase true resistivity can reflect subsurface geologic structures better; The theoretical formula of analytical form is provided for the TEM response investigations of arbitrfary point in loop line.

Description of drawings

Fig. 1 is the distribution schematic diagram of measuring point in the actual exploration of central loop TEM, and the rectangular box in figure is transmitting loop, and cross mark is measuring point.

Fig. 2 is that the measured data with edge effect is surveyed road figure, observation duration 30ms, 20 roads, time road, and rear 4 roads disturb larger, have only got front 16 roads, transmitting loop 600m * 600m, survey district, center 200m * 200m.

Fig. 3 is the central loop TEM resistivity-depth section comparison diagram of non-flanged effect and calibration edge effect, wherein (a) is that the Canadian Phoenix V8 of company instrument standard configuration software calculates, as to still have the edge effect impact after proofreading and correct resistivity-depth section figure, is (b) the full phase central loop TEM true resistivity-depth section figure of non-flanged effect of the present invention.

Embodiment

Below in conjunction with accompanying drawing, specific embodiments of the present invention is described in further detail.

The measured data that obtains in the coalfield-hydrogeology exploration is as example, and transmitting loop 600m * 600m observes in the central area of 200m * 200m therein and carrying out.In Fig. 3, (a) is resistivity-depth section that the Canadian phoenix V8 of company instrument standard configuration calibration edge effect software calculates, and can find out that the correction of adopting the measuring point playback do not eliminate the impact of edge effect.Same measured data is calculated as follows with the full phase true resistivity method of non-flanged effect:

Need to record the attaching relation of each measuring point and transmitting loop during the exploration construction, and arrive the position of transmitting loop mid point.Consider that in practice of construction, transmitting loop is generally square, the emission radius in formula (8) is converted by following formula

$a=\frac{L}{\sqrt{\mathrm{\π}}}---\left(9\right)$

In formula, L is the length of side of square transmitting loop.The span of formula (8) midfield point r is in the scope at transmitting loop middle part 1/3rd, as shown in Figure 1.

Use Differential Properties, the G-S algorithm of Laplace ^[4]441 filter factors that the filter factor algorithm adopts Anderson to provide ^[5]Above-mentioned formula (8) is programmed, then the improved generalized inverse matrix of substitution together with as shown in Figure 2 measured data ^[8]Inversion program is carried out Inversion Calculation, obtains full phase true resistivity and zone thickness, with the resistivity-depth section of the mapping softwares such as Surfer generation as shown in (b) in Fig. 3.The inverting initial parameter is provided with several different methods, and the present embodiment adopts is homogeneous half space ground electric model, ground resistivity ρ ₁With apparent resistivity formula estimation in early stage or late period, the stratum number of plies and time road number be all mutually 16 with existing correction as shown in (a) in Fig. 3 after still have edge effect to affect true resistivity-depth section ^[3]Compare, the present invention has eliminated the impact of edge effect.

Claims (4)

1. true resistivity computing method of full phase of the central loop TEM of a non-flanged effect is characterized in that comprising the following steps:

Step 1 obtains a central loop TEM point vertical magnetic field analytical expression arbitrarily

In cylindrical-coordinate system, when the loop line mid point overlaps with true origin, central loop TEM electric field E on Earth Surface _θThe frequency field expression formula be

$E_{\mathrm{\θ}}(r,\mathrm{\ω})=-\mathrm{j\ω}{\mathrm{\μ}}_{0}I\left(\mathrm{\ω}\right)a\underset{0}{\overset{\∞}{\∫}}{R}_n(\mathrm{\λ},\mathrm{\ω})J_1\left(\mathrm{\λa}\right)J_1\left(\mathrm{\λr}\right)\mathrm{d\λ}---\left(1\right)$

In formula, r is a bit to the true origin distance on ground; ω=2 π f are circular frequency, and wherein f is frequency; μ ₀=4 π * 10 ^-7H/m is non magnetic the earth magnetic permeability; I is transmitter current, and a is the transmitting loop radius; J ₁Be 1 rank Bessel function, R _nTotal reflectance on the stratiform Earth Surface;

Utilize the Maxwell vorticity equation

$\▿\×E=-\mathrm{j\ω}{\mathrm{\μ}}_{0}H---\left(2\right)$

Get the magnetic field H of vertical component _z

$H_z=\frac{1}{r}\frac{\∂}{\∂r}\left(r{E}_{\mathrm{\θ}}\right)=\frac{I\left(\mathrm{\ω}\right)a}{r}\underset{0}{\overset{\∞}{\∫}}{R}_n(\mathrm{\λ},\mathrm{\ω})J_1\left(\mathrm{\λa}\right)[\mathrm{\λ}{J}_0\left(\mathrm{\λr}\right)-\frac{1}{r}{J}_1\left(\mathrm{\λr}\right)]\mathrm{d\λ}---\left(3\right)$

J in formula (3) ₀It is 0 rank Bessel function;

Step 2 obtains the induced electromotive force analytical expression of a single Bessel function of point arbitrarily

Large transmitting loop for the a=600m ~ 800m that generally uses utilizes the gradual of Bessel function

$J_1\left(x\right)\≈\sqrt{\frac{2}{\mathrm{\πx}}}\cos (x-\frac{3\mathrm{\π}}{4})(x\→\∞)---\left(4\right)$

With formula (4) substitution formula (3)

$h_z\left(t\right)\≈\frac{I\left(\mathrm{\ω}\right)a}{r}\sqrt{\frac{2}{\mathrm{\πa}}}\underset{0}{\overset{\∞}{\∫}}{R}_n(\mathrm{\λ},\mathrm{\ω})\sqrt{\frac{1}{\mathrm{\λ}}}\cos (\mathrm{\λa}-\frac{3\mathrm{\π}}{4})[\mathrm{\λ}{J}_0\left(\mathrm{\λr}\right)-\frac{1}{r}{J}_1\left(\mathrm{\λr}\right)]\mathrm{d\λ}---\left(5\right)$

Formula (5) is done contrary Laplace conversion, obtain the time domain form

Actual measurement induced electromotive force V (t) and h _z(t) pass is

$V\left(t\right)=\frac{\∂}{\∂t}{h}_z\left(t\right)---\left(7\right)$

The induced electromotive force analytical expression of a single Bessel function of point arbitrarily will be got after formula (6) substitution formula (7)

Step 3 in formula (8) substitution inversion program, namely obtains the full phase true resistivity of the central loop TEM of non-flanged effect.

2. true resistivity computing method of full phase of the central loop TEM of a non-flanged effect as claimed in claim 1, is characterized in that in described step 2, and the evaluation of formula (8) utilizes the Differential Properties of Laplace, G-S algorithm, filter factor algorithm.

3. true resistivity computing method of full phase of the central loop TEM of a non-flanged effect as claimed in claim 2, it is characterized in that, described filter factor algorithm calculates 1 rank Bessel functions for 47 filter factors that Koefoed etc. provides, and 51 filter factors that provide with Verma calculate 0 rank Bessel functions.

4. true resistivity computing method of full phase of the central loop TEM of a non-flanged effect as claimed in claim 2, is characterized in that, 441 filter factors that described filter factor algorithm provides for Anderson.

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