Research Article | Open Access
Solution of the Inverse Problem of Subsurface Electric Exploration for Horizontally Stratified Medium
In this paper, we offer the generalization of the known technique of the construction of the gradient of the residual functional based on the statement of the conjugate problem for the case when the unknown function is complex valued. The notion of the reference frequency of the medium is introduced. Knowing the value of the reference frequency lets us judge the possibility of simultaneous definition of the dielectric permittivity and conductivity.
The problem of definition of the dielectric permittivity and conductivity is the actual geophysical problem. Theoretical research of inverse problems of geoelectrics and some numerical examples of definition of these functions can be found in .
There are many works devoted to the problems of reconstruction of conductivity and dielectric permittivity (or complex conductivity , or complex dielectric permittivity ), e.g., in [2–19]). According to the rules, the inverse problem is solved in frequency domain; that is, the external current source is harmonic. Despite the quite obvious idea that we can search for one complex function , the authors assume numerical algorithms for reconstruction either of two real functions and or of real and imaginary parts of correspondent complex function, considering that one of the functions or is known. It leads to the fact that the resulting formulas are too complicated and inconvenient for analysis and implement on a computer. Apparently, another reason such numerical algorithms are offered is that when solving the inverse problem of finding a complex function by minimizing the functional , we go through the following steps: first, the functional and, secondly, by the definition of the gradient, increment of the functional should be presented in the form ; that is, there should be a scalar product for two complex values .
In this paper the authors propose a numerical method of simultaneous definition of the conductivity and dielectric permittivity considering the example of inverse problem of subsurface radiolocation. Instead of two unknown real functions, we consider one complex. The authors generalize the known technique of finding of the gradient of the residual functional, using the statement of the conjugate problem for the case when the unknown function is complex valued. The notion of the reference frequency of the medium is introduced, which helps to understand where these two functions can be determined simultaneously. Some test reconstructions for simulated data are offered.
2. Statement of the Inverse Problem
We consider the media—-layered structure with interfaces (), ; -layer is the interval , the last (underlying) layer is the half space , and the air is the half space .
Electromagnetic properties of each layer are defined by the permittivity , the conductivity , and by magnetic permeability , F/m and H/m, in most cases the relative permittivity belongs to the interval and relative magnetic permeability . Since the medium is horizontally stratified, then and are piecewise-constant functions of the variable ().
Let the source of external current be a cable disposed on the height parallel to the -axis.
For the component from the Maxwell equations finally, we can obtain the differential equation of the second order (see, e.g., ): The Fourier transform with respect to the horizontal variable and the time variable gives the following equation: At points of discontinuity of the medium, we assume that the gluing conditions are as follows: Source concentrated at the point , which is equivalent to the gluing conditions at this point We assume that we have the conditions of damping in infinity and, relative to the solution of the direct problem (2)–(5), the additional information is given
Here and are the Fourier parameters with respect to variables and , respectively, is the notation for gluing, that is, , and the bar over the complex value will denote the complex conjunction.
Introduce the notation
Fix some values of the angular frequency . In the inverse problem, we will recover the complex value () which is a piecewise-constant function since the functions and are piecewise constant.
Inverse problem (2)–(6) may be solved by minimizing the residual functionals as follows: (here are certain weight multipliers). Paying attention to that in the functional (9), we fix a value of the spatial frequency and prepare additional information (6) for different values of the angular frequency .
For minimization, we will use the gradient method, since the rate of convergence of such method is higher than that of the method that uses only the values of the functional; therefore, we need to get the gradient of the residual functional (9), which in turn requires the definition of the scalar multiplication of two complex numbers.
For two complex numbers and , we introduce the following relation: The relation (10) has all the properties of scalar multiplication. The proof of this is based on a geometric interpretation of complex numbers. Let and ; then, .
For (10) note useful relation
3. Gradient of the Residual Functional
First we note that where are the values of a piecewise constant function in the segment .
We obtain the expression of the gradient of the residual functional (9) by the statement of the conjugate problem
Let the value be incremented ; then, the function will get increment , which satisfies the following problem: (here for, simplicity, we set ).
In this case, the increment of the residual functional up the second order can be obtained as follows:
Let Then where
Therefore, the gradient of the residual functional will have the form
4. Analytic Formulas for the Key Expressions
In order to solve the direct problem (2)–(5) and conjugate problem (13), we will reduce the differential equation of the second order to the Riccati equation. This method was successfully used, for example, in [20–23].
For solving differential equation (2), we introduce the function as follows: which will satisfy the differential Riccati equation If we calculate from the right to the left, then the solution of (24) in each segment will be as follows: If we calculate from the left to the right, then (here , is the value of the piecewise constant function in the segment and ).
For solving Riccati equation (24), we will do our recurrent calculations from the layer up to the layer moving to the point , where the source is disposed.
The condition of damping in infinity (5) enables Due to conditions (6), we obtain the gluing conditions Therefore, we may set and begin the recurrent calculation of from the right to the left by formula Thus we get .
Analogously, taking into account damping in minus infinity (5) we get Since the source is disposed in the half space , we may immediately take
Gluing condition (28) in the point allows us to determine
Integrating (23) in the interval , we obtain
The conjugate problem (13) is similar to direct problem (2)–(5); therefore, it can be solved similarly. Moreover, since the damping conditions in infinity are the same, the solution of the Riccati equation introduced for the function will coincide with for all . Therefore, we have and in each interval .
5. Numerical Experiment
5.1. Reference Frequency of the Medium
First we note, that (see, e.g., formulas (25) and (35)) where That is, the variation of the solution of direct problem (2)–(5) depends directly on how the function depends on variations of the functions and in the layers.
Evidently, the greatest influence on change in the value is rendered by variations and , when
We put , whence we get This means that we know for certain medium the mean values of dielectric permittivity and conductivity. Then we derive and fix the value of reference frequency of the medium (43).
5.2. Dependence of the Properties of the Functional from the Value
For numerical experiments below we choose model shown in Table 1.
We set the mean values at and , and then the reference angular frequency is . The capacity of the skin-layer, from here, is
Fixing the values and (), we will change the values of dielectric permittivity and conductivity of the first layer in the segment for different values of , where and we will observe the behavior of the value which is one of the components of the residual functional (9). Results of numerical experiment are presented in Figure 1.
We note that first, if decreases with respect to the reference frequency of the medium , then the residual functional loses sensitivity when varies. If increases, then the functional loses the sensitivity when varies. Secondly, the more the difference between and the less the sensitivity of the residual functional to the change of desired unknown values, since it becomes more standing.
If and () vary, the behavior of will be similar; however, if the layer is lower, so is the sensitivity of the functional.
Thus, the assumption that the value of the angular frequency must by such that is verified (see (42)).
In addition, we clarify the known geophysical condition of quasistationarity of the electromagnetic field: . The value of notation “” is lax. Proceeding from the numerical experiment, we may account the quasistationary approximation to be suitable if .
5.3. Dependence of the Properties of the Residual Functional from the Value
For the model 1 (see Table 1), the numerical experiment similar to those in the previous section was developed. The values of the dielectric permittivity and conductivity of the first layer were changing (in above mentioned segments) and the values of the residual functional were derived. The result is shown in Figure 2.
The tendency is seen as the greater the less is the sensitivity of residual functional to variations of the unknown parameters. The assumption that is verified.
For model 1 we obtain . It is seen that with increasing at first the residual functional loses the sensitivity to the variations of and then to variations of . When , the sensitivity of the residual functional rapidly decreases.
6. Numerical Examples of Solution of the Inverse Problem
In order to test the operability of the proposed numerical algorithm, we carry out a number of reconstruction of electromagnetic properties of the medium, using simulated data. In order to obtain the additional information (6) we, first, solve the direct problem (2)–(5). Then we add the random value with this form where is a random value from the unit circle and is the percent of the introduced error. In Figure 3 we see the example of using the additional information with this error.
For each model of the medium the coordinate of the boundary of the last layer coincides with the value of capacity of the skin-layer .
The parameters used in the construction of the residual functional are collected in Table 6.
In each layer, the initial approximations were and .
For minimizing of the residual functional the conjugate gradient method modified for complex values used
The results of restoration of piecewise constant functions and are shown in Figure 4.
Remember that we used the value of the reference angular frequency such that the condition is satisfied. Recovery experiment shows that the less the value of , the more difficult restoring of the functions and is, since small values of in the differential equation (2) render little impact on changing values of the solution of the equation. The consequence of this is the large “flatness” of the residual functional and its low sensitivity to variations of and . Note: .
In this paper, the authors suggested the generalization of the known technique of constructing the gradient of the residual functional with the use of the statement of the conjugate problem, when the unknown function is complex.
The numerical examples had shown that the conjugate gradient method in the case of complex valued gradient of the residual functional and the known function is applicable and we can find the minimum of the functional.
Efficiency rate of the method is confirmed by examples of simultaneous reconstruction of the dielectric permittivity and conductivity on synthetic data with introduced random error.
The work was supported by the project of Ministry of Education and Science of the Republic of Kazakhstan (Grant 1173/GF2 (N. 378 from 04.02.2013)), Grant N. 773 from 01.10.2013, SB RAS and NAS of Ukraine (Project 12-2013), and RFBR (Grant 12-01-00773).
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