CN114779355A - Ground transient electromagnetic inversion method and device based on transmitting current full waveform - Google Patents

Ground transient electromagnetic inversion method and device based on transmitting current full waveform Download PDF

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CN114779355A
CN114779355A CN202210175778.0A CN202210175778A CN114779355A CN 114779355 A CN114779355 A CN 114779355A CN 202210175778 A CN202210175778 A CN 202210175778A CN 114779355 A CN114779355 A CN 114779355A
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李建慧
胡祥云
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China University of Geosciences
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Abstract

The invention provides a ground transient electromagnetic inversion method and a device based on a transmitting current full waveform, wherein the method comprises the following steps: and constructing a one-dimensional inversion target function based on a Tikhonov regularization mode, optimizing the inversion target function by adopting a Gauss-Newton method, and outputting a final inversion model. In the process of optimizing an inversion target function, expanding a basic waveform of a transmitting current for one-dimensional inversion of a ground transient electromagnetic method from a slope step waveform into a full waveform of the transmitting current containing turn-on time, stabilization time, turn-off time and a duty ratio, and incorporating a period number parameter into the basic waveform of the transmitting current. The strategy not only helps to improve the data processing interpretation precision, but also helps to perfect the instrument parameter design.

Description

Ground transient electromagnetic inversion method and device based on transmitting current full waveform
Technical Field
The invention relates to the field of geophysical exploration, in particular to a ground transient electromagnetic inversion method and device based on a transmitting current full waveform.
Background
The ground transient electromagnetic method mostly adopts a bipolar emission current waveform, namely, both positive power supply and negative power supply are carried out in one emission period. As shown in fig. 1, both the positive and negative supply half cycles include current on, settling and off phases, and a measurement phase. In a "bipolar" waveform, each turn-on and turn-off of the transmit current induces eddy currents in the subsurface medium, and the eddy currents generated by the previous turn-on or turn-off of the transmit current must partially cancel the eddy currents generated by the subsequent turn-off or turn-on. Under the influence of the phenomenon, the electromagnetic response amplitude measured in the two half periods of positive and negative power supply respectively has difference. In order to ensure that the absolute difference in the measured electromagnetic response over two successive supply half cycles is within a very small range, transient electromagnetic methods typically begin recording data after a number of transmission cycles. In addition, in order to reduce the influence of random interference, the transient electromagnetic method data measurement usually adopts a multi-superposition technology, namely, an electromagnetic signal with a specified transmission cycle number is required to be continuously observed. Therefore, the transient electromagnetic observation data is necessarily affected by the emission current waveform and the number of cycles thereof.
At present, the data processing software of the ground transient electromagnetic method adopts an oblique step current waveform, namely only the influence of turn-off time is considered, and the influences of stable time, turn-on time and cycle number in a transmitting current waveform are ignored. However, in the context of high conductivity surveys, the eddy current signals generated with current turn-on are strong, slow to decay, and not necessarily sufficiently decaying even for tens of milliseconds of settling time, but still partially cancel the post-turn-off transient electromagnetic response, and are particularly significant in the late phase. Therefore, the data processing method of the ground transient electromagnetic method only considering the influence of the turn-off time is not suitable for late data, and further improvement of data processing and interpretation precision is restricted.
Disclosure of Invention
The invention solves the main technical problem that the stability time and the opening time in the emission current waveform and the influence of the period number are also considered on the basis of considering the influence of the turn-off time so as to improve the one-dimensional inversion interpretation precision of the data of the ground transient electromagnetic method.
In order to achieve the above object, according to an aspect of the present invention, there is provided a ground transient electromagnetic inversion method based on a full waveform of a transmit current, comprising the steps of:
the method comprises the following steps: one-dimensional inversion target function is constructed based on Tikhonov regularization mode, and the inversion target function
Figure BDA0003519000860000021
Set as formula one:
Figure BDA0003519000860000022
wherein, | | Wd(dobs-dpre)||2Is a fitting difference function of the observed data and the forward model response, alpha | | Wm(m-m*)||2For the model regularization function, | | · | represents l2A norm; wherein, dobsIs an observation data vector; d is a radical ofpreA forward model response vector corresponding to the model parameter vector m, namely F (m), wherein F represents a one-dimensional forward function of the transient electromagnetic method; w is a group ofdWeighting the data by a matrix, here a diagonal matrix, WdThe element (b) is the reciprocal of the standard deviation of the observed data; alpha is a regularization factor; m is a unit of*Is a reference model containing prior information; wmAs a model roughness operator, WmIn the form of:
Figure BDA0003519000860000023
step two: optimizing an inversion target function by adopting a Gauss-Newton method, and outputting a final inversion model;
the process of optimizing the inversion objective function by using the gauss-newton method includes:
step S1: inputting observation data and working parameters of a transient electromagnetic method;
step S2: inputting an initial inversion model, a reference model and control parameters;
step S3: calculating the forward response of the transient electromagnetic method of the initial inversion model according to the working parameters;
step S3 includes:
s3.1: calculating the magnetic induction intensity impulse response or step response excited by the lower step current waveform;
s3.2: according to the magnetic induction intensity impulse response or the step response excited by the lower step current waveform, calculating the magnetic induction intensity impulse response or the step response excited by the emission current basic waveform by adopting a convolution algorithm;
the emission current basic waveforms are divided into two categories: the first category contains only off-times, i.e. ramp step waveforms; the second type includes a plurality of full waveforms which are different in the number of cycles and determined by the on-time, the settling time, the off-time, and the duty ratio, and full waveform parameters depending on the actual current waveform are input by step S1;
step S4: calculating a Jacobian matrix of a current inversion model (namely an inversion model obtained by the k-th inversion iteration) by adopting a perturbation method;
step S5: generating a normal equation of the Gauss-Newton method according to the observation data, the reference model, the control parameters and the Jacobian matrix, and solving the normal equation of the Gauss-Newton method to obtain a model updating amount;
step S6: updating the current inversion model according to the model updating amount, and calculating the forward response and the root mean square fitting difference of the updated inversion model;
step S7: and judging whether the current iteration times or the root-mean-square fitting difference meets the iteration termination condition in the control parameters, if so, stopping iteration and outputting a final inversion model, and otherwise, returning to the step S4 to start the next iteration.
Preferably, in step S1, the observation data includes: observing magnetic induction intensity impulse response or step response normalized according to the equivalent area of the receiving probe and the intensity of the transmitted current corresponding to the time sequence and corresponding measurement standard deviation;
the working parameters comprise: the device comprises a working device, a transmitting loop size and position, a transmitting current waveform parameter, a measuring point position and an observation time sequence;
wherein the transmit current waveform parameters include: fundamental frequency, duty cycle, on time, settling time, and off time.
Preferably, in step S2, the initial inverse model is a one-dimensional layered medium, and the inputting the initial inverse model includes: inputting the number of layers of the layered medium, the thickness of each layer of medium and the conductivity;
inputting prior information in the form of inputting a reference model;
the control parameters include: target root mean square fitting difference, maximum iteration times, regularization factor initial value and updating parameters.
Preferably, step S31 includes:
a rectangular transmitting return line source laid on the surface of the horizontal layered medium is electrified with current with a certain frequency f, and the component B of the magnetic induction intensity along the z-axis direction in a frequency domain excited by a measuring point (x, y,0)z(f) Expressed as formula two:
Figure BDA0003519000860000031
wherein,
Figure BDA0003519000860000041
and
Figure BDA0003519000860000042
the expression is sequentially from formula three to formula six:
Figure BDA0003519000860000043
Figure BDA0003519000860000044
Figure BDA0003519000860000045
Figure BDA0003519000860000046
in the formula, ExAnd EyDenotes the electric field component in the x and y directions, respectively, I denotes the current intensity, W and L denote rectangular transmission loopsHalf length and half width of the source; z is a radical of formula0Satisfies z for the impedance ratio in vacuum0＝iωμ0I denotes a complex unit, ω is an angular frequency, ω ═ 2 π f, μ0Magnetic permeability in vacuum; r is a radical of hydrogenTEFor the reflection coefficient, the conductivity and thickness information of each layer is contained in the reflection coefficient; in the form of
Figure BDA0003519000860000047
Wherein k isxAnd kyIs the wave number in the Fourier transform; j is a unit of1A first class of Bessel function of order 1; r isL、r-L、rWAnd r-WRepresents the distance of the measuring point (x, y,0) to the four sides of the rectangular transmission loop, rL、r-L、rWAnd r-WAre respectively: r is a radical of hydrogenL＝[(x′-x)2+(L-y)2]1/2、r-L＝[(x′-x)2+(-L-y)2]1/2、rW＝[(W-x)2+(y′-y)2]1/2And r-W＝[(-W-x)2+(y′-y)2]1/2Wherein the coordinates of the four edges are sequentially expressed as (x ', L,0), (x', -L,0), (W, y ', 0) and (-W, y', 0);
respectively solving outer layer integrals and inner layer integrals in formulas from three to six by adopting 12-order Gauss-Legendre integrals and Hankel transformation, and obtaining the magnetic induction intensity of a frequency domain;
respectively carrying out sine or cosine transformation on the imaginary part of the magnetic induction intensity in the frequency domain to obtain the magnetic induction intensity pulse response excited by the lower step current waveform
Figure BDA0003519000860000048
Or step response bz(t), the expression is the formulas seven and eight:
Figure BDA0003519000860000049
Figure BDA0003519000860000051
preferably, step S32 includes:
and performing convolution operation on the calculated magnetic induction intensity impulse response or step response excited by the lower step current waveform and the differential of the current in any basic waveform to time, wherein the process is expressed as a formula nine:
Figure BDA0003519000860000052
in the formula, fwaveform(t) represents the magnetic induction impulse response or step response of the basic waveform excitation, fstepoff(t) represents the magnetic induction impulse response or step response of the step-down current waveform excitation,
Figure BDA0003519000860000053
represents the derivative of the current with respect to time t, τ represents the integral variable;
and solving a ninth formula by adopting a trapezoidal integral method to obtain the magnetic induction intensity impulse response or step response excited by the basic waveform.
Preferably, in step S4, the disturbance amount is determined according to the current inversion model mkThe parameters in (1) are determined.
Preferably, in step S5, for the inversion target function, the normal equation of the k +1 th iteration of gauss-newton method is the formula ten:
Figure BDA0003519000860000054
in the formula, δ mk+1For a backlog, i.e. model update, JkModel m is inverted at k-th time for forward function FkThe first derivative of (i.e. the current inverse model m)kThe first derivative at (i.e., the jacobian matrix), the superscript T denotes the transpose,
Figure BDA0003519000860000055
represents the kth inversion model mkIn response to the forward evolution of (c),i.e. F (m)k)；
Solving the formula ten by adopting a preconditioned conjugate gradient method to obtain delta mk+1(ii) a In the inversion, the regularization factor decreases proportionally with the number of inversion iterations.
Preferably, step S6 includes:
and updating the current inversion model by adopting a formula eleven according to the model updating amount:
mk+1＝mk+aδmk+1
where a is the model update step, δ mk+1Represents the model update quantity, mkRepresenting the inversion model obtained in the kth iteration of the inversion, mk+1Representing an inversion model obtained by inversion iteration at the k +1 th time, wherein k represents the inversion iteration times;
after the model is updated, the method in step S3 is used to calculate the forward response of the transient electromagnetic method of the updated inversion model, and the root mean square fitting difference x is calculated according to the formula twelvermsThe root mean square fitting difference is used to measure the fitness of the data:
Figure BDA0003519000860000061
wherein M is the number of data, smFor the m-th observation data
Figure BDA0003519000860000062
The corresponding standard deviation of the measured value of the standard deviation,
Figure BDA0003519000860000063
for an inverse model mk+1The mth data in the corresponding forward response.
Preferably, step S7 includes:
judging whether the root mean square fitting difference is less than or equal to a target fitting difference or reaches a set iteration number, if so, stopping iteration and outputting a final inversion model; otherwise, the next iteration is started.
According to another aspect of the invention, there is provided a ground transient electromagnetic inversion device based on a full waveform of a transmitting current, comprising:
an inversion target function construction module used for constructing a one-dimensional inversion target function based on a Tikhonov regularization mode, wherein the inversion target function
Figure BDA0003519000860000064
The formula is set as formula one:
Figure BDA0003519000860000065
wherein, | | Wd(dobs-dpre)||2Is a fitting difference function of the observed data and the forward model response, alpha | | | Wm(m-m*)||2For the model regularization function, | | | | - | represents l2A norm; wherein, dobsIs an observation data vector; dpreA forward model response vector corresponding to the model parameter vector m, namely F (m), wherein F represents a one-dimensional forward function of the transient electromagnetic method; w is a group ofdWeighting the data by a matrix, here a diagonal matrix, WdThe element (b) is the reciprocal of the standard deviation of the observed data; alpha is a regularization factor; m is a unit of*Is a reference model containing prior information; wmAs a model roughness operator, WmIs of the form:
Figure BDA0003519000860000071
the inversion target function optimization module is used for optimizing the inversion target function by adopting a Gaussian Newton method and outputting a final inversion model;
the inversion objective function optimization module comprises:
the first input unit is used for inputting observation data and working parameters of a transient electromagnetic method;
the second input unit is used for inputting the initial inversion model, the reference model and the control parameters;
the forward response calculation unit is used for calculating the forward response of the transient electromagnetic method of the initial inversion model according to the working parameters;
wherein the forward response calculation unit includes:
the first calculating unit is used for calculating the magnetic induction intensity impulse response or step response excited by the step current waveform;
the second calculation unit is used for calculating the magnetic induction impulse response or the step response excited by the basic waveform of the emission current by adopting a convolution algorithm according to the magnetic induction impulse response or the step response excited by the lower step current waveform;
the emission current basic waveforms are divided into two categories: the first category contains only off-times, i.e. ramp step waveforms; the second type comprises a plurality of full waveforms which are different in cycle number and determined by turn-on time, stabilization time, turn-off time and duty ratio, full waveform parameters depend on actual current waveforms and are input by the first input unit;
the Jacobian matrix calculating unit is used for calculating a Jacobian matrix of the current inversion model by adopting a disturbance method;
the normal equation generating and solving unit is used for generating a normal equation of the Gauss-Newton method according to the observation data, the reference model, the control parameters and the Jacobian matrix, and solving the normal equation of the Gauss-Newton method to obtain a model updating amount;
the iteration updating unit is used for updating the current inversion model according to the model updating amount and calculating the forward response and the root mean square fitting difference of the updated inversion model;
and the iteration termination unit is used for judging whether the current iteration times or the root-mean-square fitting difference meets the iteration termination condition in the control parameters, if so, stopping iteration and outputting a final inversion model, and if not, returning to the Jacobian matrix calculation unit to start the next iteration.
The technical scheme provided by the invention has the following beneficial effects:
the method comprises the steps of expanding a basic transmitting current waveform for data processing of the ground transient electromagnetic method from an oblique step waveform into a full transmitting current waveform containing turn-on time, stabilization time, turn-off time and duty ratio, and incorporating a period number parameter into the basic transmitting current waveform. The strategy not only helps to improve the data processing interpretation precision, but also helps to perfect the instrument parameter design.
Drawings
The invention will be further described with reference to the accompanying drawings and examples, in which:
FIG. 1 is a schematic diagram of a single cycle "bipolar" waveform;
FIG. 2 is a flow chart of a ground transient electromagnetic inversion method based on a full waveform of a transmitting current in embodiment 1 and embodiment 2;
FIG. 3 is a schematic diagram of basic waveforms of emission current in examples 1 and 2, wherein FIG. 3(a) waveform 1 (prior art method), FIG. 3(b) waveform 2 (invention), FIG. 3(c) waveform 3 (invention), FIG. 3(d) waveform 4 (invention);
FIG. 4 is a schematic view of a model of a layered medium in examples 1 and 2;
FIG. 5 is a geoelectric model obtained based on the 4 basic waveform one-dimensional inversion of FIG. 3 in example 1, wherein FIG. 5(a) waveform 1, FIG. 5(b) waveform 2, FIG. 5(c) waveform 3, and FIG. 5(d) waveform 4;
FIG. 6 is a graph of field observation data (magnetic induction step response) in example 2;
FIG. 7 is a geoelectric model obtained based on the 4 fundamental waveform one-dimensional inversion of FIG. 3 for the observed data of FIG. 6;
fig. 8 is a structural diagram of a ground transient electromagnetic inversion apparatus based on a full waveform of a transmitting current according to the present invention.
Detailed Description
For a more clear understanding of the technical features, objects, and effects of the present invention, embodiments of the present invention will now be described in detail with reference to the accompanying drawings.
Example 1: theoretical model
And constructing a one-dimensional inversion target function based on a Tikhonov regularization mode, and optimizing the one-dimensional inversion target function by adopting a Gauss-Newton method. Inverting an objective function
Figure BDA0003519000860000091
The setting is as follows:
Figure BDA0003519000860000092
in the formula, the first term is a fitting difference function of observation data and forward modeling response, the second term is a modeling regularization function, and | | · | | | represents l2And (4) norm. Wherein, dobsIs an observation data vector; dpreA forward model response vector corresponding to the model parameter vector m, namely F (m), wherein F represents a one-dimensional forward function of the transient electromagnetic method; w is a group ofdWeighting the data by a matrix, here a diagonal matrix, WdThe element (d) is the reciprocal of the standard deviation of the observed data; alpha is a regularization factor; m is*For reference to the model, a priori information is included. WmIs a model roughness operator of the form:
Figure BDA0003519000860000093
as shown in fig. 3, in the inversion of this embodiment, 4 basic waveforms are sequentially used for the emission current basic waveform, and 10 inversion calculations are performed for each waveform.
As shown in fig. 2, each inversion calculation process includes the following steps:
step S1: inputting observation data and working parameters of a transient electromagnetic method;
when a theoretical model is adopted, the observation data comprises two parts: namely the forward response (generated by the method in step S3) and random noise for the particular earth model and operating parameters.
The geoelectrical model and operating parameters are as follows: the geoelectricity model is a uniform half space of 5 omega m; the emission source is a square loop with the side length of 100m and is laid on the surface of the uniform half space; the device type is a central loop device; the emission current is 1A; the emission current basic waveform is a bipolar current waveform (duty ratio is 50%) which is adopted by a Canadian PROTEM instrument and emits fundamental frequency of 25Hz, and the corresponding time sequence comprises 30 moments and ranges from 6.8 mu s to 6978 mu s. The corresponding period of the 25Hz emission fundamental frequency is 0.04s, namely the period of 0.25 is 10ms, and the turn-on time, the stabilization time and the turn-off time are set to be 1ms, 8.99ms and 0.01ms in sequence. The forward response is an induced electromotive force (magnetic induction intensity impulse response) generated by excitation of the basic waveform of 8.25 cycles.
The random noise is generated by the following method: first, gaussian noise is generated and then multiplied by the standard deviation to be used as random noise. Standard deviation set to a time instant forward response
Figure BDA0003519000860000094
0.5-3% (see table 1 for details).
The forward response is summed with the random noise as the observed data for inversion.
TABLE 1 theoretical model forward response standard deviation setup table
Figure BDA0003519000860000101
Step S2: inputting an initial inversion model, a reference model and control parameters;
the initial inversion model is 39 layers of laminated media, the resistivity of each layer of the laminated media is 20 ohm m, the thickness is gradually increased from 5m to 40m (the 1 st to 10 th layers are 5m, the 11 th to 25 th layers are 10m, the 26 th to 35 th layers are 20m, the 36 th to 38 th layers are 40m, and the 39 th layer of the laminated media are uniform half-spaces), and the thickness of the layers is constant in the inversion. There is no reference model. The target root mean square fit difference is 1 and the maximum number of iterations is 30. Initial value alpha of regularization factor0360000, the update scheme is alphak+1＝αkAnd k is the inversion iteration number at 1.5.
Step S3: according to the working parameters of the transient electromagnetic method input in the step S1, the method in the step S3 is adopted to calculate the magnetic induction intensity impulse response under the excitation of the basic waveform
Figure BDA0003519000860000102
The step of step S3 is distinguished therefromThe key of the inversion method is that the full waveform influence of the emission current is included in the forward response calculation. Step S3 can be divided into two steps, firstly, the magnetic induction intensity impulse response excited by the step-down current waveform is calculated
Figure BDA0003519000860000103
Then, the convolution algorithm is adopted to calculate the magnetic induction intensity impulse response excited by the emission current basic waveform (see figure 3)
Figure BDA0003519000860000104
In FIG. 3, the waveform 1 in FIG. 3(a) corresponds to a ramp waveform adopted by the conventional data processing method, and the waveforms in FIGS. 3(b), 3(c) and 3(d) correspond to the emission current basic waveforms of the present invention, respectively, 2-4. In fig. 3(b), the basic waveform is a forward power supply emission current waveform, i.e., 0.25 cycles; the basic waveform in fig. 3(c) is added by one cycle, i.e., 1.25 cycles, to the waveform shown in fig. 3 (b); the basic waveform in fig. 3(d) is added by two cycles, i.e., 2.25 cycles, to the waveform shown in fig. 3 (b). The emission current waveform parameters (fundamental frequency, duty cycle, on-time, settling time, and off-time) thereof are determined by the input values in step S1. In some embodiments, the number of basic waveforms and the number of periods of the emission current of the present invention can be adjusted according to actual needs.
A rectangular transmitting return line source laid on the surface of the horizontal layered medium is electrified with current with a certain frequency f, and a magnetic induction intensity z component B in a frequency domain excited by a measuring point (x, y,0)z(f) The expression is as follows:
Figure BDA0003519000860000111
wherein,
Figure BDA0003519000860000112
and
Figure BDA0003519000860000113
the expression is as follows in sequence:
Figure BDA0003519000860000114
Figure BDA0003519000860000115
Figure BDA0003519000860000116
Figure BDA0003519000860000117
in the formula, I represents the current intensity, and W and L represent the half length and half width of the rectangular emission loop. z is a radical of0＝iωμ0I denotes a complex unit, ω is an angular frequency, ω ═ 2 π f, μ0Is the permeability in vacuum. r isTEFor the reflection coefficient, the conductivity and thickness information of each layer is included in the reflection coefficient. In the form of
Figure BDA0003519000860000118
Wherein k isxAnd kyIs the wave number in the Fourier transform; j. the design is a square1A first class of Bessel function of order 1; r isL、r-L、rWAnd r-WRepresents the distance from the measuring point (x, y,0) to the four sides of the rectangular emission loop, rL、r-L、rWAnd r-WThe expression of (a) is:
rL＝[(x′-x)2+(L-y)2]1/2、r-L＝[(x′-x)2+(-L-y)2]1/2、rW＝[(W-x)2+(y′-y)2]1/2and r-W＝[(-W-x)2+(y′-y)2]1/2Wherein the four-edge coordinates are sequentially represented as (x ', L,0), (x', -L,0), (W, y ', 0), and (-W, y', 0).
And respectively solving the outer layer integral and the inner layer integral in the formulas (4) to (7) by adopting 12-order Gauss-Legendre integral and Hankel transformation, and obtaining the magnetic induction intensity in the frequency domain.
Respectively carrying out sine transformation on the frequency domain magnetic induction imaginary parts to obtain the magnetic induction impulse response excited by the step current waveform
Figure BDA0003519000860000121
The expression is as follows:
Figure BDA0003519000860000122
and solving the electromagnetic field excited by the basic waveform of the emission current by adopting a convolution algorithm. In the convolution algorithm, first, the magnetic induction intensity impulse response (f) excited by the step-down current waveform is calculated according to the formula (3)stepoff) Then convolution operation is carried out on the impulse response and the current-to-time differential (dI/dt) in a certain basic waveform, and the magnetic induction intensity impulse response (f) excited by the basic waveform can be obtainedwaveform). This process can be represented by the following formula:
Figure BDA0003519000860000123
equation (8) is solved by the trapezoidal integration method.
Step S4: and calculating the Jacobian matrix of the current inversion model by adopting a perturbation method. In the embodiment, the number of the media is 39, and because the thickness of the layer is fixed, the resistivity of each layer of the media needs to be disturbed one by one, and a complete Jacobian matrix can be obtained after 39 times of disturbance. The disturbance amount is 10% of a certain parameter of the current inversion model. In some embodiments, the number of layers of the layered medium may be selected according to actual needs, and the corresponding disturbance times and disturbance amount may be adjusted accordingly.
Step S5: and substituting step S3 and step S4 to generate a forward response and a Jacobian matrix respectively, and substituting the current regularization factor to generate a normal equation of the Gaussian Newton method. Solving normal equation by adopting a preconditioned conjugate gradient method, wherein the termination condition is that the iteration number reaches 200 or the relative residual error is reachedLess than or equal to 10-6。
Step S6: and calculating the model updating amount and the root mean square fitting difference, wherein the model updating step size is set to be 1.
Step S7: and judging termination conditions. In this theoretical model study, the root mean square fitting difference was less than the target value within 10 inversion iterations.
The inversion procedure described above was run 10 times. The observed data used for the inversion contains random noise, so the earth model obtained for each inversion is slightly different. The inversion result is shown in fig. 5, in which the solid line represents the theoretical model and the dashed line represents the inversion model, and when the waveform 1 (existing method) is adopted, most of the inversion geoelectric model is obviously different from the theoretical model (see fig. 5 (a)); with waveforms 2, 3 and 4 (invention), most of the inverted earth electrical models were consistent with the theoretical model (see fig. 5(b), 5(c) and 5 (d)). Therefore, compared with the existing data processing method based on the inclined step current waveform, the ground transient electromagnetic inversion method based on the full waveform of the emission current provided by the invention has obvious advantages, and the expected target is realized.
Example 2: data observed in the field
Step S1: inputting observation data and working parameters of a transient electromagnetic method;
the observation data acquisition place is a certain tidal flat in the east sea of China, the region is in an alternate sea and land environment, shallow strata are generally full of seawater, the resistivity is only 0.1-0.8 ohm m, and the resistivity of deep strata is slightly increased. The observation data is the magnetic induction step response of a measuring point and is collected by a superconducting quantum interference device (SQUID) produced by Crone instruments of Canada. The emission source is a rectangular loop with the side length of 400m multiplied by 200m, the center coordinate of the loop is (0,0,0) m, and the measuring point coordinate is (-100,0,0) m. The emission current was 25A; the emission current basic waveform is a bipolar current waveform (50% of duty ratio) which is adopted by a Canada Crone Pulse EM instrument and emits a fundamental frequency of 0.833333Hz, and the corresponding time sequence comprises 45 moments in the range of-0.15 ms-222.7 ms. The corresponding period of the 0.833333Hz emission fundamental frequency is 1.2s, namely the period of 0.25 is 300ms, and the turn-on time, the stabilization time and the turn-off time are set to be 0.5ms, 299.5ms and 1ms in sequence. In the inversion, only the observed data at time 2 to 45 are used, see fig. 6. Because the observed data collected by the clone Pulse EM instrument does not contain standard deviation, the standard deviation is matched for the actually measured data according to the characteristics of strong early signal, high signal-to-noise ratio, weak late signal and low signal-to-noise ratio of the transient electromagnetic method, and the details are shown in Table 2.
TABLE 2 Standard deviation setting table for field observation data
Figure BDA0003519000860000131
Step S2: inputting an initial inversion model, a reference model and control parameters;
the initial inversion model is 46 layers of laminated media, the resistivity of each layer of the laminated media is 1 omega m, the thickness is gradually increased from 10m to 40m (the thickness of the 1 st to 20 th layers is 10m, the thickness of the 21 st to 30 th layers is 20m, the thickness of the 31 st to 45 th layers is 40m, and the thickness of the 46 th layer of the laminated media is uniform half space), and the thickness of the layers is constant in the inversion. There is no reference model. The target root mean square fit difference is 1 and the maximum number of iterations is 30. Initial value alpha of regularization factor0360000, the update scheme is alphak+1＝αkAnd k is the inversion iteration number at 1.5.
Step S3: according to the transient electromagnetic method working parameters input in the step S1, calculating the step response b of the magnetic induction intensity under the excitation of the basic waveform by adopting the method in the step S3z(t)。
Step S4: and calculating the Jacobian matrix of the current inversion model by adopting a perturbation method. In the embodiment, the dielectric layers are 46 layers, and because the thickness of the dielectric layers is fixed, the resistivity of each layer of dielectric needs to be disturbed one by one, and a complete Jacobian matrix can be obtained after the disturbance for 46 times.
Step S5: and substituting the forward response and the Jacobian matrix generated in the step S3 and the step S4 respectively, and substituting the current regularization factor to generate a normal equation of the Gaussian Newton method. Solving normal equation by adopting a preconditioned conjugate gradient method, wherein the termination condition is that the iteration number reaches 200 times or the relative residual error is less than or equal to 10-6。
Step S6: and calculating the model updating amount and the root mean square fitting difference, wherein the model updating step size is set to be 1.
Step S7: and judging a termination condition. In this theoretical model study, the root mean square fitting difference was less than the target value within 10 inversion iterations.
The inversion result is shown in fig. 7, the inversion result obtained by using the waveform 1 (the existing method) is obviously different from the inversion result obtained by using the waveforms 2, 3 and 4 (the present invention) in the underground range of 100 to 1000m, and the inversion result obtained by using the waveforms 2, 3 and 4 (the present invention) is more consistent with the real geological condition. Therefore, the ground transient electromagnetic inversion method based on the full waveform of the transmitting current provided by the invention achieves the expected target.
As shown in fig. 8, the present embodiment further provides a ground transient electromagnetic inversion apparatus based on a full waveform of a transmit current, including:
an inversion target function construction module 01 for constructing a one-dimensional inversion target function based on a Tikhonov regularization mode, and inverting the target function
Figure BDA0003519000860000141
Set as formula one:
Figure BDA0003519000860000142
wherein, | | Wd(dobs-dpre)||2Is a fitting difference function of the observed data and the forward model response, alpha | | | Wm(m-m*)||2For the model regularization function, | | · | represents l2A norm; wherein, dobsIs an observation data vector; dpreA forward model response vector corresponding to the model parameter vector m, namely F (m), wherein F represents a one-dimensional forward function of the transient electromagnetic method; wdWeighting the data by a matrix, here a diagonal matrix, WdThe element (b) is the reciprocal of the standard deviation of the observed data; alpha is a regularization factor; m is a unit of*Is a reference model containing prior information; wmAs model roughness operator, WmIs of the form:
Figure BDA0003519000860000151
the inversion target function optimization module 02 is used for optimizing the inversion target function by adopting a Gaussian Newton method and outputting a final inversion model;
the inversion objective function optimization module 02 includes:
the first input unit 021 is used for inputting observation data and working parameters of a transient electromagnetic method;
a second input unit 022 for inputting the initial inversion model, the reference model and the control parameters;
a forward response calculation unit 023, configured to calculate a transient electromagnetic forward response of the initial inversion model according to the input working parameters;
the forward response calculation unit 023 includes:
a first calculating unit 0231, configured to calculate a magnetic induction pulse response or a step response excited by the next step current waveform;
a second calculating unit 0232, configured to calculate, according to the magnetic induction impulse response or the step response excited by the next step current waveform, the magnetic induction impulse response or the step response excited by the emission current basic waveform by using a convolution algorithm;
the emission current basic waveforms are divided into two categories: the first category contains only off-times, i.e. ramp step waveforms; the second class includes a plurality of full waveforms which are different in the number of periods and determined by on-time, settling time, off-time, and duty ratio, full waveform parameters depending on an actual current waveform, inputted by the first input unit 021;
a Jacobian matrix calculation unit 024 for calculating a Jacobian matrix of the current inversion model by using a perturbation method;
the normal equation generating and solving unit 025 is used for generating a normal equation of the gauss-newton method according to the input observation data, the reference model, the control parameters and the calculated Jacobian matrix, and solving the normal equation of the gauss-newton method to obtain a model updating amount;
an iteration updating unit 026, configured to update the current inversion model according to the model update amount, and calculate a forward response and a root mean square fitting difference of the updated inversion model;
and the iteration termination unit 027 is configured to determine whether the current iteration number or the root-mean-square fitting difference meets an iteration termination condition in the control parameter, stop the iteration and output the final inversion model if the current iteration number or the root-mean-square fitting difference meets the iteration termination condition, and return to the jacobian matrix calculation unit 024 to start the next iteration if the current iteration number or the root-mean-square fitting difference does not meet the iteration termination condition.
It should be noted that, in this document, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or system that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or system. Without further limitation, an element defined by the phrase "comprising an … …" does not exclude the presence of other identical elements in the process, method, article, or system in which the element is included.
The above-mentioned serial numbers of the embodiments of the present invention are merely for description and do not represent the merits of the embodiments. In the unit claims enumerating several means, several of these means can be embodied by one and the same item of hardware. The use of the words first, second, third and the like do not denote any order, but rather the words first, second and the like may be interpreted as indicating any order.
The above description is only a preferred embodiment of the present invention, and is not intended to limit the scope of the present invention, and all the equivalent structures or equivalent processes performed by the present invention or directly or indirectly applied to other related technical fields are included in the scope of the present invention.

Claims (10)

1. A ground transient electromagnetic inversion method based on a transmitting current full waveform is characterized by comprising the following steps:
the method comprises the following steps: constructing a one-dimensional inversion target function based on a Tikhonov regularization mode, wherein the inversion target function
Figure FDA0003519000850000011
Set as formula one:
Figure FDA0003519000850000012
wherein, | | Wd(dobs-dpre)||2Is a fitting difference function of the observed data and the forward model response, alpha | | | Wm(m-m*)||2For the model regularization function, | | | | - | represents l2A norm; wherein d isobsIs an observation data vector; dpreA forward model response vector corresponding to the model parameter vector m, namely F (m), wherein F represents a one-dimensional forward function of the transient electromagnetic method; wdWeighting the data by a matrix, here a diagonal matrix, WdThe element (d) is the reciprocal of the standard deviation of the observed data; alpha is a regularization factor; m is*Is a reference model containing prior information; wmAs a model roughness operator, WmIn the form of:
Figure FDA0003519000850000013
step two: optimizing the inversion target function by adopting a Gauss-Newton method, and outputting a final inversion model;
the process of optimizing the inverted objective function using gauss-newton method comprises:
step S1: inputting observation data and working parameters of a transient electromagnetic method;
step S2: inputting an initial inversion model, a reference model and control parameters;
step S3: calculating a transient electromagnetic forward response of the initial inversion model according to the working parameters;
step S3 includes:
s3.1: calculating the magnetic induction intensity impulse response or step response excited by the lower step current waveform;
s3.2: according to the magnetic induction intensity impulse response or the step response excited by the lower step current waveform, calculating the magnetic induction intensity impulse response or the step response excited by the emission current basic waveform by adopting a convolution algorithm;
the emission current basic waveforms are divided into two categories: the first category contains only off-times, i.e. ramp step waveforms; the second class includes a plurality of full waveforms which are different in the number of cycles and determined by the on-time, the settling time, the off-time, and the duty ratio, and full waveform parameters depending on the actual current waveform are input by step S1;
step S4: calculating a Jacobian matrix of the current inversion model by adopting a disturbance method;
step S5: generating a normal equation of the Gauss-Newton method according to the observation data, the reference model, the control parameters and the Jacobian matrix, and solving the normal equation of the Gauss-Newton method to obtain a model updating amount;
step S6: updating the current inversion model according to the model updating amount, and calculating a forward response and a root mean square fitting difference of the updated inversion model;
step S7: and judging whether the current iteration times or the root-mean-square fitting difference meets the iteration termination condition in the control parameters, if so, stopping iteration and outputting a final inversion model, and otherwise, returning to the step S4 to start the next iteration.
2. The method for inversion of ground transient electromagnetic method based on full waveform of emission current as claimed in claim 1, wherein in step S1, the observation data comprises: observing magnetic induction intensity impulse response or step response normalized according to the equivalent area of the receiving probe and the intensity of the transmitted current corresponding to the time sequence and corresponding measurement standard deviation;
the working parameters comprise: the device comprises a working device, a transmitting loop size and position, a transmitting current waveform parameter, a measuring point position and an observation time sequence;
wherein the transmit current waveform parameters include: fundamental frequency, duty cycle, on time, settling time, and off time.
3. The inversion method of ground transient electromagnetic method based on the full waveform of the emission current as claimed in claim 1, wherein in step S2, the initial inversion model is a one-dimensional layered medium, and the inputting the initial inversion model comprises: inputting the number of layers of the layered medium, the thickness of each layer of medium and the conductivity;
inputting prior information in the form of inputting a reference model;
the control parameters include: target root mean square fitting difference, maximum iteration times, regularization factor initial value and updating parameters.
4. The ground transient electromagnetic inversion method based on the full waveform of the transmitted current as claimed in claim 1, wherein the step S31 comprises:
a rectangular transmitting return line source laid on the surface of the horizontal layered medium is electrified with current with a certain frequency f, and the component B of the magnetic induction intensity along the z-axis direction in a frequency domain excited by a measuring point (x, y,0)z(f) Expressed as formula two:
Figure FDA0003519000850000031
wherein,
Figure FDA0003519000850000032
and
Figure FDA0003519000850000033
the expression is sequentially from formula three to formula six:
Figure FDA0003519000850000034
Figure FDA0003519000850000035
Figure FDA0003519000850000036
Figure FDA0003519000850000037
in the formula, ExAnd EyRespectively, the electric field components along the x-axis and the y-axis, I represents the current intensity, and W and L represent the half length and the half width of the rectangular emission return line source; z is a radical of formula0Satisfies z for the impedance ratio in vacuum0＝iωμ0I denotes a complex unit, ω is an angular frequency, ω ═ 2 π f, μ0Magnetic permeability in vacuum; r isTEFor the reflection coefficient, the conductivity and thickness information of each layer is contained in the reflection coefficient; in the form of
Figure FDA0003519000850000038
Wherein k isxAnd kyIs the wave number in the Fourier transform; j. the design is a square1A first class of Bessel function of order 1; r is a radical of hydrogenL、r-L、rWAnd r-WRepresents the distance from the measuring point (x, y,0) to the four sides of the rectangular emission loop, rL、r-L、rWAnd r-WAre respectively:
rL＝[(x′-x)2+(L-y)2]1/2、r-L＝[(x′-x)2+(-L-y)2]1/2、rW＝[(W-x)2+(y′-y)2]1/2and r-W＝[(-W-x)2+(y′-y)2]1/2Wherein the coordinates of the four edges are sequentially expressed as (x ', L,0), (x', -L,0), (W, y ', 0) and (-W, y', 0);
respectively solving outer layer integrals and inner layer integrals in formulas from three to six by adopting 12-order Gauss-Legendre integrals and Hankel transformation, and obtaining the magnetic induction intensity of a frequency domain;
respectively carrying out sine or cosine transformation on the imaginary part of the magnetic induction intensity of the frequency domain to obtainMagnetic induction impulse response excited by stepped-down current waveform
Figure FDA0003519000850000039
Or step response bz(t), the expression is the formulas seven and eight:
Figure FDA0003519000850000041
Figure FDA0003519000850000042
5. the ground transient electromagnetic inversion method based on the full waveform of the transmitted current as claimed in claim 1, wherein the step S32 comprises:
and performing convolution operation on the calculated magnetic induction intensity impulse response or step response excited by the lower step current waveform and the differential of the current in any basic waveform to time, wherein the process is expressed as a formula nine:
Figure FDA0003519000850000043
in the formula, fwaveform(t) represents the magnetic induction impulse response or step response of the basic waveform excitation, fstepoff(t) represents the magnetic induction impulse response or step response of the step down current waveform excitation,
Figure FDA0003519000850000044
represents the derivative of the current with respect to time t, τ represents the integral variable;
and solving a ninth formula by adopting a trapezoidal integral method to obtain the magnetic induction intensity impulse response or step response excited by the basic waveform.
6. The base station as claimed in claim 1The ground transient electromagnetic inversion method of the full-wave form of the emission current is characterized in that in the step S4, the disturbance quantity is according to the current inversion model mkThe parameter determination in (1).
7. The inversion method of the ground transient electromagnetic method based on the full waveform of the emission current as claimed in claim 1, wherein in step S5, for the inversion target function, the normal equation of the k +1 th iteration of the gauss-newton method is the formula ten:
Figure FDA0003519000850000045
in the formula, δ mk+1For the quantity to be sought, i.e. the model update quantity, JkModel m is inverted at k-th time for forward function FkThe first derivative of (i.e. the current inverse model m)kThe first derivative at (i.e., the jacobian matrix, the superscript T denotes transposition,
Figure FDA0003519000850000046
representing the k-th inversion model mkForward response of (1), i.e. F (m)k)；
Solving the equation ten by adopting a preconditioned conjugate gradient method to obtain delta mk+1(ii) a In the inversion, the regularization factor decreases proportionally with the number of inversion iterations.
8. The inversion method of ground transient electromagnetic method based on full waveform of emission current as claimed in claim 1, wherein step S6 includes:
and updating the current inversion model by adopting a formula eleven according to the model updating amount:
mk+1＝mk+aδmk+1
where a is the model update step, δ mk+1Represents the model update quantity, mkRepresenting the inversion model obtained in the kth iteration of the inversion, mk+1Representing an inversion model obtained by the k +1 th inversion iteration, wherein k represents the inversion iteration times;
after the model is updated, the method in step S3 is used to calculate the forward response of the transient electromagnetic method of the updated inversion model, and the root mean square fitting difference x is calculated according to the formula twelvermsThe root mean square fitting difference is used to measure the fitness of the data:
Figure FDA0003519000850000051
wherein M is the number of data, smFor the m-th observed data
Figure FDA0003519000850000052
The corresponding standard deviation of the measured value of the standard deviation,
Figure FDA0003519000850000053
for an inverse model mk+1The mth data in the corresponding forward response.
9. The inversion method of ground transient electromagnetic method based on full waveform of emission current as claimed in claim 1, wherein step S7 includes:
judging whether the root mean square fitting difference is less than or equal to a target fitting difference or reaches a set iteration number, if so, stopping iteration and outputting a final inversion model; otherwise, starting the next iteration.
10. A ground transient electromagnetic inversion device based on a transmitting current full waveform is characterized by comprising:
an inversion target function construction module used for constructing a one-dimensional inversion target function based on a Tikhonov regularization mode, wherein the inversion target function
Figure FDA0003519000850000054
Set as formula one:
Figure FDA0003519000850000055
wherein, | | Wd(dobs-dpre)||2Is a fitting difference function of the observed data and the forward model response, alpha | | | Wm(m-m*)||2For the model regularization function, | | · | represents l2A norm; wherein, dobsIs an observation data vector; dpreA forward model response vector corresponding to the model parameter vector m, namely F (m), wherein F represents a one-dimensional forward function of the transient electromagnetic method; w is a group ofdWeighting the data by a matrix, here a diagonal matrix, WdThe element (d) is the reciprocal of the standard deviation of the observed data; alpha is a regularization factor; m is*Is a reference model containing prior information; wmAs a model roughness operator, WmIn the form of:
Figure FDA0003519000850000061
the inversion target function optimization module is used for optimizing the inversion target function by adopting a Gaussian Newton method and outputting a final inversion model;
the inversion objective function optimization module comprises:
the first input unit is used for inputting observation data and working parameters of a transient electromagnetic method;
the second input unit is used for inputting the initial inversion model, the reference model and the control parameters;
the forward response calculation unit is used for calculating the forward response of the transient electromagnetic method of the initial inversion model according to the working parameters;
wherein the forward response calculation unit includes:
the first calculating unit is used for calculating the magnetic induction intensity impulse response or step response excited by the step current waveform;
the second calculation unit is used for calculating the magnetic induction intensity impulse response or the step response excited by the emission current basic waveform by adopting a convolution algorithm according to the magnetic induction intensity impulse response or the step response excited by the lower step current waveform;
the emission current basic waveforms are divided into two categories: the first category contains only off-times, i.e. ramp step waveforms; the second type comprises a plurality of full waveforms which are different in cycle number and determined by turn-on time, stabilization time, turn-off time and duty ratio, full waveform parameters depend on actual current waveforms and are input by the first input unit;
the Jacobian matrix calculating unit is used for calculating a Jacobian matrix of the current inversion model by adopting a disturbance method;
the normal equation generating and solving unit is used for generating a normal equation of the Gauss-Newton method according to the observation data, the reference model, the control parameters and the Jacobian matrix, and solving the normal equation of the Gauss-Newton method to obtain a model updating amount;
the iteration updating unit is used for updating the current inversion model according to the model updating amount and calculating the forward response and the root mean square fitting difference of the updated inversion model;
and the iteration termination unit is used for judging whether the current iteration times or the root-mean-square fitting difference meets the iteration termination condition in the control parameters, if so, stopping iteration and outputting a final inversion model, and if not, returning to the Jacobian matrix calculation unit to start the next iteration.
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