CN114460654B - Semi-aviation transient electromagnetic data inversion method and device based on L1L2 mixed norm - Google Patents
Semi-aviation transient electromagnetic data inversion method and device based on L1L2 mixed norm Download PDFInfo
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Abstract
The invention discloses a semi-aviation transient electromagnetic data inversion method based on L1L2 mixed norm, which adopts ground to transmit transient electromagnetic signals and utilizes an unmanned aerial vehicle carrying and receiving device to obtain electromagnetic response data, and comprises the following steps: collecting electromagnetic response data of a region to be detected by using a receiving device; constructing a stratum model of a region to be detected, and presetting an iteration termination condition; obtaining a transverse constraint weighting matrix corresponding to an L2 norm constraint term of the electromagnetic response data; solving or updating a longitudinal constraint matrix corresponding to the L1 norm constraint term of the electromagnetic response data; forming a target function by the data fitting gradient term, the L1 norm constraint term and the L2 norm constraint term, and solving an expansion formula after the incremental derivative of the stratum model is equal to zero; and (4) calculating the increment of the stratum model by using an expansion formula through inversion calculation, correcting the stratum model, and obtaining an inversion imaging image after the iteration termination condition is reached.
Description
Technical Field
The invention relates to the technical field of geophysical aviation electromagnetic exploration, in particular to a semi-aviation transient electromagnetic data inversion method and device based on an L1L2 mixed norm.
Background
At present, the geophysical electromagnetic exploration in the prior art mostly adopts an aeroelectromagnetic method, a helicopter or a fixed-wing airplane is adopted to carry a launching and observing system, and the system cannot be applied to scenes such as urban underground space exploration, ground geological survey and the like. In addition, the aeroelectromagnetic method in the prior art adopts a one-dimensional single-point inversion algorithm which has no restriction in the transverse direction, and the continuity of the section resistivity is poor when the data noise is strong; regularization constraint is generally adopted on longitudinal resistivity constraint, and the regularization constraint has the main defect that a model is too smooth and cannot effectively depict electrical abrupt interface information.
For example, the invention patent of china with the patent application number of "202110820825.8" and the name of "a semi-aviation transient electromagnetic conductivity-depth imaging method and device" simulates an electromagnetic response query database conforming to the actual situation in advance according to the actual situation, and then searches the detected electromagnetic response data in the database without performing iterative computation for many times like inversion, so that the semi-aviation transient electromagnetic method can perform rapid imaging and rapidly obtain a preliminary imaging result and an inversion initial model.
For another Chinese invention patent with the patent application number of '202010751396.9' and the name of 'an unmanned aerial vehicle semi-aviation time domain electromagnetic detection data analysis and interpretation method', the method comprises the following steps: a. preprocessing the measuring line data to eliminate motion noise; b. carrying out spectrum analysis and digital filtering processing on the secondary field data; c. forming data to be imaged or interpreted in an inversion; d. and performing one-dimensional fast inversion on the secondary field data after superposition and channel extraction, establishing an underground electrical structure profile, and performing geological interpretation on the inversion imaging result by combining geological data to form a comprehensive interpretation result.
The above technique has the following problems: the above techniques have the disadvantages that the rapid imaging result is rough, and the formation resistivity value and the layer interface information cannot be accurately reflected; for the data containing noise, because the single-point inversion does not consider the influence of the stratum information of the adjacent points of the measuring points, the inversion result is transversely discontinuous and takes the shape of a strip.
Therefore, a semi-aviation transient electromagnetic data inversion method based on the L1L2 mixed norm, which is simple in logic, accurate and reliable, is urgently needed to be provided.
Disclosure of Invention
Aiming at the problems, the invention aims to provide a semi-aviation transient electromagnetic data inversion method based on L1L2 mixed norm, and the technical scheme adopted by the invention is as follows:
the invention provides a semi-aviation transient electromagnetic data inversion method based on L1L2 mixed norm, which adopts a ground transmitted transient electromagnetic signal and utilizes an unmanned aerial vehicle carrying and receiving device to obtain electromagnetic response data, and comprises the following steps:
acquiring electromagnetic response data of a region to be detected by using a receiving device;
constructing a stratum model of a region to be detected, and presetting iteration termination conditions;
obtaining a transverse constraint weighting matrix corresponding to an L2 norm constraint term of the electromagnetic response data;
solving or updating a longitudinal constraint matrix corresponding to the L1 norm constraint term of the electromagnetic response data;
forming a target function by the data fitting gradient term, the L1 norm constraint term and the L2 norm constraint term, and solving an expansion formula after the incremental derivative of the stratum model is equal to zero;
and (5) calculating by using an expansion formula in an inversion way to obtain the increment of the stratum model and correct the stratum model, and obtaining an inversion imaging image after an iteration termination condition is reached.
In a second aspect, the present invention provides a semi-airborne transient electromagnetic data inversion apparatus, comprising:
the electromagnetic signal excitation module is arranged in a geological region to be detected and used for exciting and transmitting a transient electromagnetic signal;
the electromagnetic signal receiving device is coupled with the electromagnetic signal excitation module, carried on the unmanned aerial vehicle and used for collecting electromagnetic response data;
the stratum model building module is used for obtaining stratum information and building a stratum model;
and the stratum model correction module is connected with the stratum model construction module, corrects by adopting a data fitting gradient term, an L1 norm constraint term and an L2 norm constraint term, and performs inversion by using the corrected stratum model.
In a third aspect, the present invention provides a computer readable storage medium storing a computer program which, when executed by a processor, performs the steps of a method for semi-airborne transient electromagnetic data inversion based on L1L2 mixed norms.
Compared with the prior art, the invention has the following beneficial effects:
(1) The invention adopts the ground to transmit transient electromagnetic signals, and utilizes the unmanned aerial vehicle carrying receiving device to obtain electromagnetic response data, and has the advantages of higher precision, convenient implementation, lower cost and good safety; the method also has the advantages of high exploration speed and capability of exploring across obstacles. The invention has wide application prospect in the fields of urban underground space detection, ground geological survey, mineral resource exploration and environment monitoring.
(2) The invention considers that the longitudinal space constraint (regular term) of the resistivity distribution of the measuring points adopts L1 norm, and the inversion method of L2 norm is adopted in the transverse space constraint. The target function of the method comprises two constraint terms of L1 norm and L2 norm, and the problems of mutual constraint of parameters between layers in the vertical direction of the stratum, namely longitudinal constraint and transverse constraint of the stratum between adjacent measuring points are considered respectively, and compared with the result of a general method, the inversion result is more continuous in the transverse direction of the stratum, and the identification of the stratum in the longitudinal direction is more accurate.
(3) According to the method, the weight of the constraint term is adjusted by adopting the self-adaptive regularization factor, so that the fitting precision of inversion is higher, and the iteration is more stable.
(4) The method is stable in inversion iteration of the data containing the constraint noise, can clearly distinguish the electrical interface of the stratum, and makes up the defects of the current semi-aviation time domain electromagnetic detection data inversion interpretation system.
In conclusion, the method has the advantages of simple logic, accuracy, reliability, high detection efficiency and the like, and has high practical value and popularization value in the technical field of geophysical aviation electromagnetic exploration.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings needed to be used in the embodiments will be briefly described below, it should be understood that the following drawings only illustrate some embodiments of the present invention, and therefore should not be considered as limiting the scope of protection, and it is obvious for those skilled in the art that other related drawings can be obtained according to these drawings without inventive efforts.
FIG. 1 is a logic flow diagram of the present invention.
FIG. 2 is the inversion result of groundwater low resistivity anomaly in a certain area in the invention.
FIG. 3 is a diagram comparing a theoretical model and an inversion result in a certain area.
FIG. 4 is an inversion imaging diagram of measured data in a certain area.
Detailed Description
To further clarify the objects, technical solutions and advantages of the present application, the present invention will be further described with reference to the accompanying drawings and examples, and embodiments of the present invention include, but are not limited to, the following examples. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present application.
Example 1
As shown in fig. 1, the present embodiment provides a method and an apparatus for inverting semi-aviation transient electromagnetic data based on an L1L2 mixed norm, specifically, the apparatus of the present embodiment includes: the device comprises an electromagnetic signal excitation module, an electromagnetic signal receiving device, a stratum model construction module and a stratum model correction module; the electromagnetic signal excitation module is installed in a geological region to be detected and used for exciting and transmitting transient electromagnetic signals. In addition, the electromagnetic signal receiving device is coupled with the electromagnetic signal excitation module, is carried on the unmanned aerial vehicle and acquires electromagnetic response data; the stratum model building module is used for obtaining stratum information and building a stratum model; and the stratum model correction module is connected with the stratum model construction module, corrects by adopting a data fitting gradient term, an L1 norm constraint term and an L2 norm constraint term, and performs inversion by using the corrected stratum model.
In this embodiment, the semi-aviation transient electromagnetic data inversion method includes the following steps:
the method comprises the following steps of firstly, acquiring electromagnetic response data of a region to be detected by using a receiving device, wherein the method further comprises the following basic information: receiving coil area, line source length, measuring point number of a single measuring line, total measuring line number, current peak value, track pumping time number and track pumping time; and sequentially acquiring the x and y coordinates, the elevation and the response value corresponding to each time channel of all the measuring points participating in the operation.
And secondly, constructing a stratum model of the area to be detected according to the actual workplace, and presetting iteration termination conditions. The parameters of the stratum model comprise the total stratum number, the total stratum thickness, the initial stratum thickness and the resistivity of any layer. In addition, the iteration termination condition includes a maximum number of iterations and a minimum fitting error.
Thirdly, solving a transverse constraint weighting matrix corresponding to the L2 norm constraint term of the electromagnetic response data; solving or updating a longitudinal constraint matrix corresponding to the L1 norm constraint term of the electromagnetic response data; forming a target function by using the data fitting gradient term, the L1 norm constraint term and the L2 norm constraint term, carrying out derivation on the model increment to be equal to zero, then developing a formula, carrying out inversion calculation by using the developed formula to obtain the model increment, correcting the model, and obtaining an inversion imaging image after an iteration termination condition is reached
Specifically, the method comprises the following steps:
(1) Forward response computation
In semi-aviation transient electromagnetic exploration, a receiving coil mainly receives a magnetic field H z Component, forward response in this example is based on analyzing the horizontal laminar vertical component H z ：
And performing frequency-time conversion to obtain the change rate of the magnetic field strength with respect to time:
obtaining the induced electromotive force, wherein the expression is as follows:
in the formula, H z (w) a frequency domain magnetic field response (A/m) representing the vertical component; w represents the sampling angular frequency (rad/s); i represents the current intensity (a); setting the center of the wire source to be positioned at the origin of coordinates, and extending to L and-L along two sides of the x axis; r represents an offset; r is TE Represents the reflection coefficient in the TE mode; k is a radical of 0 Representing the wave number of the air medium; z represents a coilA reception height (m); j. the design is a square 1 The first-order Bessel function is different from the previous Jacobian matrix; λ represents an integral variable;
in the formula, reHz (w) j ) Representing in Hz (w) j ) The real part of (a); n represents the number of coil turns; s represents the effective area (m) of the coil 2 )；μ 0 Represents the magnetic permeability (4. Pi. Times.10) under vacuum -7 H/m)。
Using cosine transform to respond to H in frequency domain z (w) conversion to a time domain response V z (t)。
(2) Inversion imaging method
The basic idea of the semi-aviation transient electromagnetic data inversion method based on the mixed norm is to divide underground media into a multilayer layered structure, add the thickness of each layer into a model constraint function, namely a roughness matrix, and obtain the resistivity of each layer through calculation so as to construct an underground geoelectric structure. The overall objective function can be summarized as:
is unfolded into
In the formula: phi (m) represents the overall objective function; λ represents a regularization factor; phi is a unit of m (m) representing a model constraint objective function; phi is a unit of d (m) represents an observed data objective function; m represents the model vector and can be expressed as formula (3); σ represents the resistivity.
M＝[σ 11 ,σ 12 …σ 1p …σ i1 ,σ i2 ,…σ ip …σ n1 ,σ n2 …σ np ] T (6)
Equation (5) to the right where the first term data fits, F (M) represents the forward response of model vector M, W d Typically a main diagonal matrix whose elements are the number of observationsAccording to the reciprocal of the noise; the second term is a model vertical constraint term also called a regular penalty term, W m For the vertical constraint matrix of the model, the constraint term of the model adopted in this embodiment is:
namely a unit first-order difference operator; the third term is a model transverse constraint term, and the constraint terms of four adjacent measuring points, namely a front measuring point, a rear measuring point, a left measuring point, a right measuring point, a left measuring point and a right measuring point, W are considered in the model transverse constraint term h，i For the transverse constraint matrix, q represents the number of constraint items, and also adopts a first-order difference operator to represent as follows:
due to the fact that the L1 norm has the situation of being incorgable, iteration is unstable, and the slowest descending direction of the regular term cannot be obtained. A small value xi may be taken and equation (5) rewritten as follows:
using an iterative reweighted least squares method (IRLS), the L1 regularization term can be rewritten:
the present embodiment improves on this algorithm by expressing the L1 regularization term as:
wherein, V is a diagonal weighting matrix, and the elements on the main diagonal are:
where V represents a diagonal weighting matrix, x i The element xi representing the weighted model vector represents a small value; x T Representing a transpose of a weighted model vector matrix; x represents a weighted model vector;
representing a roughness matrix transpose of a longitudinal constraint term; w m A longitudinal constraint roughness matrix is represented.
The weighting matrix D in the L2 norm constraint term is expanded as:
wherein r represents the diagonal elements of the weighting matrix of the lateral constraint term; n represents the nth measuring point; p represents the p-th adjacent point of the measuring point n.
Different diagonal weighting elements can be considered according to actual conditions, and distance weighting is adopted
Where x and y are the relative coordinates of the calculated measured point to the adjacent points involved in the calculation, where the distance between the two points calculated is inverted. If weighting is not taken into account, r i The diagonal element is taken as 1. Final calculation
Terms and add to the objective function.
Regarding the model correction amount Δ m k And (3) obtaining a derivation, and enabling the derived expression to be equal to 0 to obtain a solving formula of the model correction quantity:
the first term on the right side is a data fitting gradient term, the second term is an L1 norm vertical space constraint term, and the third term is an L2 norm transverse space constraint term (regularization gradient term). In addition, the air conditioner is provided with a fan,
representing a Jacobian matrix transpose;
representing data fitting item roughness matrix transposition; w is a group of d Representing a data fitting term roughness matrix; d obs Representing the measured response data; f (M) k ) Representing a model vector forward calculation item; lambda 1 Representing a first regularization factor;
representing a longitudinal constraint item roughness matrix transpose; v k Representing a longitudinal constraint term weighting matrix; w m Representing a longitudinal constraint term roughness matrix; m k Representing a model vector; lambda 2 Representing a second regularization factor;
representing a transverse constraint roughness matrix transposition; d i Representing a lateral constraint weighting matrix; w h,i The laterally constrained roughness matrix is represented.
Fourthly, adjusting the weight of the L1 norm constraint term and the L2 norm constraint term in the inversion process by adopting a self-adaptive regularization factor; and carrying out inversion by using the corrected stratum model, and obtaining an inversion imaging image.
The embodiment provides that the regularization factor adopts an adaptive adjustment method, a data gradient term is used as a numerator, and five model constraint gradient terms are used as denominators. Using a parameter a to adjust lambda separately 1 And λ 2 The weight in the objective function is used for adjusting the weight problem of the longitudinal constraint term and the transverse constraint term in the iterative process, and the specific formula is as follows:
wherein a represents a weight proportion parameter;
representing fitted data gradient terms;
representing a longitudinal constraint gradient term;
representing the transverse constraint gradient term.
Example 2
The present embodiment provides a computer-readable storage medium storing a computer program which, when executed by a processor, implements the steps of the semi-airborne transient electromagnetic data inversion method of embodiment 1.
Example 3
As shown in fig. 2 to fig. 3, the present embodiment takes inversion of the underground low-resistance model data containing 5% of noise as an example:
the semi-aviation transient electromagnetic technology of the unmanned aerial vehicle is applied to simulation search of underground water, a 10 omega-m underground water low-resistance layer (with the thickness of 60 m) is arranged in a high-resistance layer (with the thickness of 240 m) with the background resistivity of 100 omega-m, the thickness of each layer is arranged by taking 1.05 as an exponential growth factor in inversion, and an initial model of the inversion is a uniform half space of 50 omega-m. The inversion results are shown in fig. 2. The inversion result is divided into three layers of high-low-high on the whole, wherein the high resistance is distributed between 100 omega m and 110 omega m, the resistivity in inversion is about 10 omega m, the inversion is very close to a real model, the semi-aviation transient electromagnetic inversion method of the unmanned aerial vehicle can effectively invert low resistance abnormity, the inversion of data containing noise is stable, and the electric interface is clearly depicted. This indicates that semi-airborne transient electromagnetic is sensitive to low-resistance anomalous body reactions.
Example 4
As shown in fig. 4, in this embodiment, after analyzing the data measured in some place and simply processing the flight detection data, a uniform half-space is used as an initial model, and the inversion results of a plurality of lines are shown in the figure. The overall resistivity is expressed by a low-high-low distribution trend, the overall inversion effect is good, the inversion speed of the method is higher than that of one-dimensional single-point inversion, the continuity of the inversion result is good, the electrical interface is clearly depicted, and the inversion iteration process of the noise-containing data is more stable.
The above-described embodiments are only preferred embodiments of the present invention, and are not intended to limit the scope of the present invention, but all changes that can be made by applying the principles of the present invention and performing non-inventive work on the basis of the principles shall fall within the scope of the present invention.
Claims (9)
1. The semi-aviation transient electromagnetic data inversion method based on the L1L2 mixed norm adopts a ground transmitted transient electromagnetic signal and utilizes an unmanned aerial vehicle carrying and receiving device to obtain electromagnetic response data, and is characterized by comprising the following steps of:
collecting electromagnetic response data of a region to be detected by using a receiving device;
constructing a stratum model of a region to be detected, and presetting an iteration termination condition;
obtaining a transverse constraint weighting matrix corresponding to an L2 norm constraint term of the electromagnetic response data;
solving or updating a longitudinal constraint matrix corresponding to the L1 norm constraint term of the electromagnetic response data;
forming a target function by the data fitting gradient term, the L1 norm constraint term and the L2 norm constraint term, and solving an expansion formula after the incremental derivation of the stratum model is equal to zero;
obtaining stratum model increment and correcting the stratum model by using an expansion formula for inversion calculation, and obtaining an inversion imaging image after an iteration termination condition is reached; adjusting the weights of the L1 norm constraint term and the L2 norm constraint term in the inversion process by adopting an adaptive regularization factor;
fitting a gradient term, an L1 norm constraint term and an L2 norm constraint term according to the data to form an objective function, performing model increment derivation equal to zero and expanding, and then correcting the stratum model, wherein the expression is as follows:
is unfolded into
Wherein,
representing the fitting of the data to the gradient term,
represents the L1 norm constraint term;
represents the L2 norm constraint term;
representing a Jacobian matrix transpose;
representing data fitting item roughness matrix transposition; w d Representing a data fitting term roughness matrix; d is a radical of obs Representing measured response data; f (M) k ) Forward calculation terms representing model vectors; lambda 1 Representing a first regularization factor;
a roughness matrix representing longitudinal constraint terms; v k Representing a longitudinal constraint weighting matrix; w m A roughness matrix representing longitudinal constraint terms; m k Representing a model vector; lambda 2 Representing a second regularization factor;
representing a transverse constraint term roughness matrix; d i A weighting matrix representing the laterally constrained terms; w is a group of h,i Roughness moment representing transverse constraint termArraying; m represents a model vector; f (M) represents the forward response of the model vector M; q represents the number of constraint terms.
2. The L1L2 mixed norm based semi-aviation transient electromagnetic data inversion method of claim 1, further comprising:
acquiring basic information of the receiving device; the basic information comprises the area of a receiving coil, the flying height, the number of track extraction time and the track extraction time;
acquiring basic information of a transmitting device; the basic information of the emitting device comprises a line source length and a current peak value;
and acquiring corresponding values corresponding to the number of measurement points of a single measurement line in the area to be measured, the total number of measurement lines, the coordinates of any measurement point, the offset distance, the elevation and the time channel.
3. The L1L2 mixed norm-based semi-aviation transient electromagnetic data inversion method as recited in claim 1, wherein the iteration termination condition comprises a maximum number of iterations and a minimum fitting error.
4. The method for semi-aeronautical transient electromagnetic data inversion based on the L1L2 mixed norm according to claim 1, 2 or 3, characterized in that the parameters of the formation model include total number of formation layers, total formation thickness, starting formation thickness, resistivity of any layer.
5. The method for inverting semi-aviation transient electromagnetic data based on L1L2 mixed norm as claimed in claim 1, 2 or 3 wherein the expression of the transverse constraint weighting matrix is:
wherein r represents the diagonal element of the transverse constraint weighting matrix, n represents the nth measuring point, and p represents the p adjacent point of the n measuring point.
6. The L1L2 mixed-norm based semi-aviation transient electromagnetic data inversion method as claimed in claim 5, wherein the expression of the regular term of the L1 norm is:
wherein V represents a diagonal weighting matrix; x is the number of i Representing a small value of model vector element xi; x T Representing a transpose of a weighted model matrix; x weights the model matrix;
transposing the roughness matrix; w is a group of m Representing a roughness matrix; m represents a model vector;
the main diagonal elements of the diagonal weighting matrix are:
7. the semi-aviation transient electromagnetic data inversion method based on the L1L2 mixed norm as claimed in claim 1, characterized in that the weights of the L1 norm constraint term and the L2 norm constraint term in the inversion process are adjusted by using an adaptive regularization factor, and the expression is as follows:
wherein, a represents a weight proportion parameter;
representing fitted data gradient terms;
representing longitudinal constrained gradientsAn item;
representing the transverse constraint gradient term.
8. A semi-airborne transient electromagnetic data inversion apparatus, comprising:
the electromagnetic signal excitation module is arranged in a geological region to be detected and used for exciting and transmitting transient electromagnetic signals;
the electromagnetic signal receiving device is coupled with the electromagnetic signal excitation module, carried on the unmanned aerial vehicle and used for collecting electromagnetic response data;
the stratum model building module is used for obtaining stratum information and building a stratum model;
the stratum model correction module is connected with the stratum model construction module, corrects the stratum model by adopting a data fitting gradient term, an L1 norm constraint term and an L2 norm constraint term, and performs inversion by using the corrected stratum model; adjusting the weight of the L1 norm constraint term and the L2 norm constraint term in the inversion process by adopting a self-adaptive regularization factor;
fitting a gradient term, an L1 norm constraint term and an L2 norm constraint term according to the data to form an objective function, solving the incremental derivative of the model to be equal to zero, expanding, and then correcting the stratum model, wherein the expression is as follows:
is unfolded into
Wherein,
representing the fitting of the data to the gradient term,
represents the L1 norm constraint term;
represents the L2 norm constraint term;
representing a Jacobian matrix transpose;
representing data fitting item roughness matrix transposition; w is a group of d Representing a data fitting item roughness matrix; d obs Representing measured response data; f (M) k ) Forward calculation terms representing model vectors; lambda [ alpha ] 1 Representing a first regularization factor;
a roughness matrix representing longitudinal constraint terms; v k Representing a longitudinal constraint weighting matrix; w is a group of m A roughness matrix representing longitudinal constraint terms; m is a group of k Representing a model vector; lambda [ alpha ] 2 Representing a second regularization factor;
representing a transverse constraint term roughness matrix; d i A weighting matrix representing the laterally constrained terms; w h,i A roughness matrix representing lateral constraint terms; m represents a model vector; f (M) represents the forward response of the model vector M; q represents the number of constraint terms.
9. A computer readable storage medium storing a computer program, wherein the computer program, when executed by a processor, performs the steps of the L1L2 mixed norm based semi-airborne transient electromagnetic data inversion method of any one of claims 1 to 7.
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