CN113219542A - Frequency domain electromagnetic inversion method based on improved damped least square method - Google Patents
Frequency domain electromagnetic inversion method based on improved damped least square method Download PDFInfo
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Abstract
The invention discloses a frequency domain electromagnetic inversion method based on an improved damping least square method, which is characterized in that the action of correction quantity on thickness parameters in each iteration is enhanced by adjusting the magnitude of each element of a Jacobian matrix in the iteration process, so that each parameter to be solved is converged near a preset model true value. The method improves the accuracy of the frequency domain electromagnetic inversion result.
Description
Technical Field
The invention belongs to a geophysical inversion technology, and particularly relates to a frequency domain electromagnetic inversion method based on an improved damped least square method.
Background
The frequency domain electromagnetic detection method is a mature geophysical exploration method, has the characteristics of non-plunging property, high resolution and the like, and is widely applied to the aspects of underground mineral deposit exploration, archaeological vestige excavation range defining, underground water exploration, urban underground supply and drainage pipeline detection, urban underground space exploration and the like. The problem of reconstructing the model from observed data by calculating the mathematical physical model parameters by a suitable method is the inverse problem, which is essentially an optimization problem. The frequency domain electromagnetic inversion problem is nonlinear, that is, there is no linear relation between the observation data and the model parameter to be solved, the nonlinear problem is generally approximately linearized, and the model parameter is solved by a linear inversion algorithm. The most common linear inversion algorithms such as conjugate gradient method and damped least square method have good effect in the field of frequency domain electromagnetic inversion.
The damped least squares algorithm was proposed by Marquardt in 1963, and is therefore also called a marquinter algorithm, mainly to solve the problem that the convergence of the traditional least squares algorithm is unstable. The method is effective in solving the frequency domain electromagnetic detection problem by applying the damped least square inversion algorithm, and the classical frequency domain electromagnetic detection instrument GEM-2 is integrated with the damped least square inversion algorithm for data processing and interpretation. In recent years, during inversion calculation and data interpretation, some improvements are made on a damped least square algorithm according to the characteristics of the algorithm and specific measurement area conditions, wherein transverse constraints and longitudinal constraints are most commonly added into an objective function, so that the transition of calculation results of adjacent measurement points is smoother and more practical. These methods blur the thickness information of subsurface anomalies in order to smooth the inversion results.
For underground space exploration, the purpose is to find out the depth information of an underground building body and the electromagnetic parameters of each medium layer, and the thickness of different medium layers is often the most important information. When the parameters to be solved simultaneously include the electromagnetic parameters and the thickness parameters of the underground medium layer, the correction quantity of the thickness parameters is very small in the iterative calculation process due to the fact that the electromagnetic parameters and the thickness parameters have large numerical difference, so that the thickness parameters cannot be accurately converged to the vicinity of the preset model true value, and accuracy of detection results is affected.
Disclosure of Invention
The invention aims to provide a frequency domain electromagnetic inversion method based on an improved damped least square method.
The technical scheme for realizing the purpose of the invention is as follows: a frequency domain electromagnetic inversion method based on an improved damping least square method comprises the following specific steps:
step 1: measuring a measured area by using a frequency domain electromagnetic detecting instrument, acquiring secondary field signals generated by an underground medium layer at different excitation frequencies as observation data, and determining an inversion target function according to the observation data;
step 2: setting an iteration initial value, a maximum allowable error, an initial damping factor and a maximum iteration number of inversion calculation, and calculating an initial Jacobian matrix according to the iteration initial value;
and 3, step 3: taking the numerical value of the 1 st row element in the initial Jacobian matrix as a reference, comparing the data of the rest rows with the numerical value of the 1 st row element to obtain the order of magnitude of each parameter to be solved,
and 4, step 4: adjusting the magnitude of each parameter in the iteration initial value according to the calculated magnitude of each parameter needing to be adjusted;
and 5, step 5: recalculating the Jacobian matrix;
and 6, step 6: solving an equation:
(Jk TJk+λkI)δk＝-Jk T·gk
wherein JkIs the Jacobian matrix, λ, during the kth iterationkIs damping factor of kth, I is unit diagonal matrix, deltakCorrection amount of model parameter for k time, gkIs the k-th residual vector;
and 7, step 7: iteratively correcting the model parameters and simultaneously calculating the current nonlinear factor;
and 8, step 8: judging whether the iteration times reach the maximum iteration times or whether the current residual vector is smaller than the maximum allowable error, if so, performing the step 9, otherwise, adjusting a damping factor according to the value of the current nonlinear factor, and jumping to the step 5;
step 9: and taking the currently obtained model parameters as results, and restoring the order of magnitude of the results according to the calculated order of magnitude of adjustment required by each parameter.
Preferably, the inverted objective function is determined from the observation data
Comprises the following steps:
wherein M represents the number of observed data, fm(x) Is the forward response function at the mth frequency, dmFor electromagnetic observation data corresponding to the mth frequency point, F is a frequency domain electromagnetic method positive operator of the uniform layered medium, and x is a model parameter to be solved, called a parameter vector:
x＝[x1，x2，x3，...，x3N-1]T
＝[σ1，σ2，...σN，μ1，...，μN，h1，...hN-1]T
wherein sigmaiDenotes the conductivity, μ, of the i-th layer mediumiDenotes the permeability, h, of the i-th layer mediumiThe thickness of the ith layer of medium is shown, and N represents the number of layers of the underground medium layer.
Preferably, the initial Jacobian matrix is:
in the formula (f)mIs a forward response function at the mth frequency, wherein M is more than or equal to 1 and less than or equal to M, xjJ is more than or equal to 1 and less than or equal to 3N-1;
preferably, the order of magnitude that each parameter needs to be adjusted includes an order of magnitude that the conductivity parameter to be solved needs to be adjusted, an order of magnitude that the permeability parameter to be solved needs to be adjusted, and an order of magnitude that the thickness parameter to be solved needs to be adjusted.
Preferably, the order of magnitude of the adjustment required is:
nc＝[nc1，...nci，...，ncN]
ns＝[ns1，...nsi，...，nsN]
nh＝[nh1，...nhi，...，nhN-1]
the subscript i represents the ith layer of medium, nc represents the order of magnitude that the conductivity parameter sigma to be solved needs to be adjusted, ns represents the order of magnitude that the permeability parameter mu to be solved needs to be adjusted, nh represents the order of magnitude that the thickness parameter h to be solved needs to be adjusted, wherein l is less than or equal to i and less than or equal to N;
the order of magnitude adjusting method comprises the following steps:
wherein j is more than 1 and is not more than N;
wherein j is more than N and less than or equal to 2N;
wherein J is more than 2N and less than or equal to 3N-1, J0(1) is the 1 st column element in the initial Jacobian matrix, J0And (j) is the j-th column element in the initial Jacobian matrix.
Preferably, according to the calculated order of magnitude that each parameter needs to be adjusted, a specific method for adjusting the order of magnitude of each parameter in the iteration initial value is as follows:
wherein sigmaiRepresenting electricity of the i-th layer mediumConductivity, μiDenotes the permeability, h, of the i-th layer mediumiRepresents the thickness of the ith layer of medium, and has 1-N, nciRepresenting the conductivity parameter σ to be solvediOrder of magnitude of adjustment, nsiRepresenting the permeability parameter mu to be solvediThe order of magnitude of the adjustment, nhiRepresenting the thickness parameter h to be solvediThe order of magnitude of the adjustment required;
preferably, the specific method for iteratively correcting the model parameters comprises the following steps:
x(k+1)＝x(k)+δk
current non-linearity factor r(k)The method specifically comprises the following steps:
wherein
In the formula (I), the compound is shown in the specification,
is the value of the objective function for the kth iteration, JkIs the Jacobian matrix, λ, during the kth iterationkIs damping factor of kth, I is unit diagonal matrix, deltakCorrection amount of model parameter for k time, gkIs the k-th residual vector;
preferably, according to the current non-linearity factor r(k)Is used for adjusting the damping factor lambdakThe specific method comprises the following steps:
if r is(k)< 0.25, then λk+1＝10λkIf 0.25 < r(k)< 0.75, then λk+1＝λkIf 0.75 < r(k)Then λk+1＝0.1λk。
Compared with the prior art, the invention has the following remarkable advantages: by adjusting the magnitude of each element of the Jacobian matrix in the iterative process, the electromagnetic parameters and the thickness parameters with larger numerical value difference can be converged near the true value, and the accuracy of the inversion result is improved; the thickness is used as an important parameter for underground space detection, the accuracy of an inversion result of the parameter is improved, and the drawing imaging of an underground space structure is facilitated.
Drawings
FIG. 1 is a flow chart of a frequency domain electromagnetic inversion method based on an improved damped least squares method.
Fig. 2 is a result obtained by inversion calculation using a classical damped least squares algorithm, wherein the abscissa is the depth of the dielectric layer, and the ordinate is the conductivity of the dielectric at the corresponding depth.
Fig. 3 is a result obtained by inversion calculation using a classical damped least squares algorithm, where the abscissa is the depth of the dielectric layer and the ordinate is the magnetic permeability of the dielectric at the corresponding depth.
Fig. 4 is a result obtained by inversion calculation using an improved damped least squares algorithm, where the abscissa is the depth of the dielectric layer and the ordinate is the conductivity of the dielectric at the corresponding depth.
Fig. 5 is a result obtained by performing inversion calculation using an improved damped least squares algorithm, where the abscissa is the depth of the dielectric layer, and the ordinate is the magnetic permeability of the dielectric at the corresponding depth.
Detailed Description
As shown in fig. 1, a frequency domain electromagnetic inversion method based on an improved damped least square method enhances the effect of a correction amount on a thickness parameter during each iteration by adjusting the magnitude of each element of a jacobian matrix in the iteration process, so that each parameter to be solved converges to a vicinity of a preset model true value, thereby improving the accuracy of a frequency domain electromagnetic inversion result, and specifically comprises the following steps:
step 1: measuring the measured area by using a frequency domain electromagnetic detecting instrument, acquiring secondary field signals generated by the underground medium layer at different excitation frequencies, and taking the secondary field signals as observation data d obtained by an experimental instrumentobsBased on the observed data dobsDetermining an inverted objective function:
wherein M represents the number of observed data, fm(x) Is the forward response function at the mth frequency, dmFor electromagnetic observation data corresponding to the mth frequency point, F is a frequency domain electromagnetic method positive operator of the uniform layered medium, and x is a model parameter to be solved, called a parameter vector:
x＝[x1，x2，x3，...，x3N-1]T
＝[σ1，σ2，...σN，μ1，...，μN，h1，...hN-1]T
wherein sigmaiDenotes the conductivity, μ, of the i-th layer mediumiDenotes the permeability, h, of the i-th layer mediumiThe thickness of the ith layer of medium is shown, N is the number of layers of the underground medium layer, and the total number of the parameters is 3N-1 by considering the parameters of the conductivity, the permeability and the thickness of the underground medium layer.
Step 2: setting an initial iteration value x of inversion calculation(0)I.e. the parameter vector of the model, where the superscript (0) represents the initial iteration calculation, the maximum allowable error epsilon, the initial damping factor lambda are set0And a maximum number of iterations n;
according to the initial value x of the iteration(0)Computing an initial Jacobian matrix J0，
And 3, step 3: with an initial Jacobian matrix J0The numerical value of the 1 st row element is taken as a reference, and the data of the rest rows and J0Comparing 1 to obtain the order of magnitude that each parameter to be solved needs to be adjusted, representing the order of magnitude that the conductivity parameter sigma to be solved needs to be adjusted by nc, representing the order of magnitude that the permeability parameter mu to be solved needs to be adjusted, and representing the order of magnitude that the thickness parameter h to be solved needs to be adjusted by nh;
because the convergence conditions of the parameters in the iterative process are different, the electromagnetic parameters and the thickness parameters of different dielectric layers need to be adjusted in different orders of magnitude, namely
nc＝[nc1，...nci，...，ncN]
ns＝[ns1，...nsi，...，nsN]
nh＝[nh1，...nhi，...，nhN-1]
Wherein subscript i represents the ith layer of media;
the order of magnitude adjusting method comprises the following steps:
wherein j is more than 1 and is not more than N;
wherein j is more than N and less than or equal to 2N;
wherein j is more than 2N and less than or equal to 3N-1.
Storing the obtained nc, ns and nh as fixed values in an inversion calculation program;
and 4, step 4: according to the calculated nc, ns and nh, adjusting the magnitude of each parameter in the iteration initial value, namely adjusting the magnitude of each parameter in the iteration initial value
Simultaneously adjusting the magnitude of each parameter in the forward calculation function, namely, in the process of calculating the forward calculation function F, adjusting the conductivity parameter sigma of the ith layer of mediumiMultiplication by
Power of the magnetic permeability muiMultiplication by
Power, thickness parameter h for i-th layer mediumiMultiplication by
The power;
and 5, step 5: recalculating Jacobian matrix Jk：
Computing residual vector gk：
gk＝dobs-F(x(k))
The subscript k represents the kth calculation, and the superscript k represents the parameter vector obtained by the kth updating;
and 6, step 6: solving equations
(Jk TJk+λkI)δk＝-Jk T·gk
And 7, step 7: model parameter correction by iteration
x(k+1)＝x(k)+δk
Simultaneously calculating the current non-linear factor r(k)：
Wherein
And 8, step 8: judging whether the iteration number k reaches the maximum iteration number n or not, or judging whether the current residual vector g reaches the maximum iteration number n or notkWhether or not it is less than the maximum allowable errorThe difference epsilon, if yes, step 9 is carried out, otherwise, the current nonlinear factor r is used as the basis(k)Is used for adjusting the damping factor lambdakIf r is(k)< 0.25, then λk+1＝10λkIf 0.25 < r(k)< 0.75, then λk+1＝λkIf 0.75 < r(k)Then λk+1＝0.1λkAnd jumping to the step 5;
step 9: with x(k+1)As a result of the iteration, x is recovered according to the calculated nc, ns and nh(k+1)Of order of magnitude, i.e.
x(k+1)＝[σ1·nc1，...，σN·ncN，μ1·ns1，...，μN·nsN，h1·nh1，...，hN-1·nhN-1]
Outputting final model parameters in an inversion procedure
After k times of iterative computation, the conductivity, permeability and thickness parameters of the underground medium layer to be solved are corrected, and the actual conditions of the measuring area are better met.
The invention adopts an improved damping least square inversion algorithm to adjust the magnitude of each element in the Jacobian matrix so as to enhance the correction quantity delta in each iteration processkFor thickness parameter hiThe specific improvement method comprises the following steps:
will iterate the initial value x(0)Conductivity parameter σ of the intermediate layeriMultiplication by
Power, permeability parameter μiMultiplication by
Thickness parameter hiMultiplication by
Wherein nci、nsi、nhiEach integer is greater than 0, and i is 1, 2,. N;
in order to balance the results of the frequency domain response, correspondingly, in the calculation of the forward function F, the conductivity parameter σ for the i-th layer mediumiMultiplication by
Power of the magnetic permeability muiMultiplication by
Power, thickness parameter h for i-th layer mediumiMultiplication by
The power;
the adjusted iteration initial value x(0)Substituting into a classical damped least square algorithm for calculation to obtain an inversion result x;
conducting inversion calculation to obtain conductivity parameter sigma of each layer in parameter vector xiMultiplication by
Permeability parameter muiMultiplication by
Thickness parameter hiRiding device
And output as a final result.
The idea of adjusting the magnitude of each element in the Jacobian matrix is to perform the electromagnetic inversion process in the whole frequency domain only once initially, and the adjusted magnitude cannot change along with the change of the iteration times in the continuous iteration process.
Examples
Taking a single-point model of three layers of media as an example, the accuracy of the improved inversion algorithm is verified through simulation calculation, wherein table 1 shows the electromagnetic parameters and the thickness parameters of the preset single-point model. The parameters in table 1 are used as the true values of the model, and the error of the inversion results before and after the improvement of the algorithm is compared.
TABLE 1 Preset electromagnetic and thickness parameters of Single Point model
Fig. 2 and 3 are inversion results obtained using a classical damped least squares algorithm. From the results shown in the figure, the calculated conductivity, permeability and thickness of the dielectric layer have larger deviation from the true value of the preset model. The relative error between the conductivity of the second layer medium and the true value reaches 44%, and the magnetic conductivity inversion result of the second and third layers media deviates from the preset true value of the model seriously; in addition, the thickness inversion result of the second layer is 4.99m, which is 2.5 times of the true value, and the inversion result is basically the same as the set initial iteration value, which indicates that the thickness inversion result stays near the initial iteration value without accurate convergence. From the overall result, the classical damped least square algorithm has the problems that the correction quantity of the thickness result is too small to be converged near the true value of the preset model in the frequency domain electromagnetic inversion calculation, and the deviation of the electromagnetic parameter inversion result of a part of the dielectric layer is large.
Fig. 4 and 5 are inversion results obtained by using the modified damped least squares algorithm. As can be seen from the results shown in the figure, the inversion result is basically consistent with the true value of the preset model, the maximum error is the conductivity result of the second layer of medium, the relative error is about 9.6%, the relative error of the permeability result of the second layer of medium is about 8.72%, and the relative error of the inversion result of the electromagnetic parameters is greatly reduced compared with the result obtained by adopting the classical damped least square algorithm; in addition, the thickness inversion result of each dielectric layer is basically consistent with the preset model parameters, the error is very small, and the inversion result can be obviously improved by adopting the improved damped least square algorithm.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and all equivalent modifications made within the spirit and principle of the present invention are included in the scope of the present invention.
Claims (8)
1. A frequency domain electromagnetic inversion method based on an improved damping least square method is characterized by comprising the following specific steps:
step 1: measuring a measured area by using a frequency domain electromagnetic detecting instrument, acquiring secondary field signals generated by an underground medium layer at different excitation frequencies as observation data, and determining an inversion target function according to the observation data;
step 2: setting an iteration initial value, a maximum allowable error, an initial damping factor and a maximum iteration number of inversion calculation, and calculating an initial Jacobian matrix according to the iteration initial value;
and 3, step 3: taking the numerical value of the 1 st row element in the initial Jacobian matrix as a reference, and comparing the data of the rest rows with the numerical value of the 1 st row element to obtain the order of magnitude of each parameter to be solved, which needs to be adjusted;
and 4, step 4: adjusting the magnitude of each parameter in the iteration initial value according to the calculated magnitude of each parameter needing to be adjusted;
and 5, step 5: recalculating the Jacobian matrix;
and 6, step 6: solving an equation:
(Jk TJk+λkI)δk＝-Jk T·gk
wherein JkIs the Jacobian matrix, λ, during the kth iterationkIs damping factor of kth, I is unit diagonal matrix, deltakCorrection amount of model parameter for k time, gkIs the k-th residual vector;
and 7, step 7: iteratively correcting the model parameters and simultaneously calculating the current nonlinear factor;
and 8, step 8: judging whether the iteration times reach the maximum iteration times or whether the current residual vector is smaller than the maximum allowable error, if so, performing the step 9, otherwise, adjusting a damping factor according to the value of the current nonlinear factor, and jumping to the step 5;
step 9: and taking the currently obtained model parameters as results, and restoring the order of magnitude of the results according to the calculated order of magnitude of adjustment required by each parameter.
2. The method of claim 1, wherein the inverted objective function is determined from observation data
Comprises the following steps:
wherein M represents the number of observed data, fm(x) Is the forward response function at the mth frequency, dmFor electromagnetic observation data corresponding to the mth frequency point, F is a frequency domain electromagnetic method positive operator of the uniform layered medium, and x is a model parameter to be solved, called a parameter vector:
x＝[x1，x2，x3，…，x3N-1]T
＝[σ1，σ2，…σN，μ1，…，μN，h1，…hN-1]T
wherein sigmaiDenotes the conductivity, μ, of the i-th layer mediumiDenotes the permeability, h, of the i-th layer mediumiThe thickness of the ith layer of medium is shown, N represents the number of layers of the underground medium layer, and i is more than or equal to 1 and less than or equal to N.
3. The method of claim 1, wherein the initial Jacobian matrix J is used as a basis for frequency domain electromagnetic inversion based on the modified damped least squares method0Comprises the following steps:
in the formula (f)mIs a forward response function at the mth frequency, wherein M is more than or equal to 1 and less than or equal to M, xjJ is more than or equal to 1 and less than or equal to 3N-1.
4. The method for frequency domain electromagnetic inversion based on the improved damped least squares method as claimed in claim 1, wherein the order of magnitude that each parameter needs to be adjusted includes an order of magnitude that a conductivity parameter to be solved needs to be adjusted, an order of magnitude that a permeability parameter to be solved needs to be adjusted, and an order of magnitude that a thickness parameter to be solved needs to be adjusted.
5. The frequency domain electromagnetic inversion method based on the improved damped least squares method as claimed in claim 1, wherein the required adjustment orders of magnitude are respectively:
nc＝[nc1，…nci，…，ncN]
ns＝[ns1，…nsi，…，nsN]
nh＝[nh1，…nhi，…，nhN-1]
the subscript i represents the ith layer of medium, nc represents the order of magnitude that the conductivity parameter sigma to be solved needs to be adjusted, ns represents the order of magnitude that the permeability parameter mu to be solved needs to be adjusted, and nh represents the order of magnitude that the thickness parameter h to be solved needs to be adjusted;
the order of magnitude adjusting method comprises the following steps:
wherein 1< j is not more than N;
wherein N < j is less than or equal to 2N;
wherein 2N is<j≤3N-1，J0(1) is the 1 st column element in the initial Jacobian matrix, J0And (j) is the j-th column element in the initial Jacobian matrix.
6. The frequency domain electromagnetic inversion method based on the improved damped least squares method as claimed in claim 1, wherein the specific method for adjusting the magnitude of each parameter in the initial iteration value according to the calculated magnitude of each parameter to be adjusted is as follows:
wherein sigmaiDenotes the conductivity, μ, of the i-th layer mediumiDenotes the permeability, h, of the i-th layer mediumiRepresents the thickness of the ith layer of medium, and has 1-N, nciRepresenting the conductivity parameter σ to be solvediOrder of magnitude of adjustment, nsiRepresenting the permeability parameter mu to be solvediThe order of magnitude of the adjustment, nhiRepresenting the thickness parameter h to be solvediThe order of magnitude of the adjustment is required.
7. The frequency domain electromagnetic inversion method based on the improved damped least squares method as claimed in claim 1, wherein the specific method for model parameter modification by iteration is:
x(k+1)＝x(k)+δk
current non-linearity factor r(k)The method specifically comprises the following steps:
wherein
In the formula (I), the compound is shown in the specification,
is the value of the objective function for the kth iteration, JkIs the Jacobian matrix, λ, during the kth iterationkIs damping factor of kth, I is unit diagonal matrix, deltakCorrection amount of model parameter for k time, gkIs the k-th residual vector;
8. the method for frequency domain electromagnetic inversion based on the improved damped least squares method as claimed in claim 1, wherein the current non-linear factor r is determined according to(k)Is used for adjusting the damping factor lambdakThe specific method comprises the following steps:
if r is(k)<0.25, then lambdak+1＝10λkIf 0.25<r(k)<0.75, then lambdak+1＝λkIf 0.75<r(k)Then λk+1＝0.1λk。
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