Solution of the Inverse Problem of Subsurface Electric Exploration for Horizontally Stratified Medium

Abstract

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.

1. Introduction

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 [1].

There are many works devoted to the problems of reconstruction of conductivity 𝜎 and dielectric permittivity 𝜀 (or complex conductivity 𝜎−𝑖𝜔𝜀, or complex dielectric permittivity 𝜀+𝑖𝜎/𝜔), 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 𝜀+𝑖𝜔₀𝜎, the authors assume numerical algorithms for reconstruction either of two real functions 𝜎 and 𝜀 or of real and imaginary parts of correspondent complex function, considering that one of the functions 𝜎 or 𝜀 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 𝑤 by minimizing the functional 𝐽[𝑤], we go through the following steps: first, the functional 𝐽∶ℂ→ℝ and, secondly, by the definition of the gradient, increment of the functional should be presented in the form 𝛿𝐽[𝑤]=⟨𝐽^′[𝑤],𝛿𝑤⟩+𝑜(‖𝛿𝑤‖); that is, there should be a scalar product for two complex values ⟨⋅,⋅⟩∶ℂ×ℂ→ℝ.

In this paper the authors propose a numerical method of simultaneous definition of the conductivity 𝜎 and dielectric permittivity 𝜀 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—𝑁_𝑙-layered structure with interfaces 𝑧_𝑘 (𝑘=0,𝑁_𝑙), 𝑧₀=0; 𝑚-layer is the interval [𝑧_𝑚−1,𝑧_𝑚], the last 𝑁_𝑙+1 (underlying) layer is the half space [𝑧_𝑁𝑙,∞), and the air is the half space (−∞,0].

Electromagnetic properties of each layer are defined by the permittivity 𝜀₀𝜀, the conductivity 𝜎, and by magnetic permeability 𝜇₀𝜇, 𝜀₀=8,854⋅10⁻¹² F/m and 𝜇₀=4𝜋⋅10⁻⁷ H/m, in most cases the relative permittivity 𝜀 belongs to the interval [1;80] and relative magnetic permeability 𝜇=1. Since the medium is horizontally stratified, then 𝜀 and 𝜎 are piecewise-constant functions of the variable 𝑧 (𝑧∈(−∞,∞)).

Let the source of external current be a cable disposed on the height 𝑧_∗ parallel to the 𝑦-axis.

For the component 𝐸₂(𝑡,𝑥,𝑧) from the Maxwell equations finally, we can obtain the differential equation of the second order (see, e.g., [1]): 𝜀𝜕²𝐸₂𝜕𝑡²+𝜎𝜕𝐸₂=1𝜕𝑡𝜇₀𝜕²𝐸₂𝜕𝑥²+𝜕²𝐸₂𝜕𝑧². (1) The Fourier transform with respect to the horizontal variable 𝑥 and the time variable 𝑡 gives the following equation: 𝑢_𝑧𝑧−𝜆²−𝜔²𝜇₀𝜀₀𝜀+𝑖𝜔𝜇₀𝜎𝑢=0. (2) At points of discontinuity of the medium, we assume that the gluing conditions are as follows: [𝑢]_𝑧𝑘=0,𝑢_𝑧𝑧𝑘=0,𝑘=0,𝑁_𝑙. (3) Source concentrated at the point 𝑧_∗=0, which is equivalent to the gluing conditions at this point [𝑢]_𝑧∗=0,𝑢_𝑧𝑧∗=−𝑓(𝜔)𝜇₀. (4) We assume that we have the conditions of damping in infinity 𝑢⟶0(𝑧⟶±∞), (5) and, relative to the solution of the direct problem (2)–(5), the additional information is given 𝑢|_𝑧=0=𝑔(𝜔,𝜆). (6)

Here 𝜆 and 𝜔 are the Fourier parameters with respect to variables 𝑥 and 𝑡, respectively, [⋅]_𝑧 is the notation for gluing, that is, [𝑤]_𝑧=𝑤(𝑧+0)−𝑤(𝑧−0), and the bar over the complex value will denote the complex conjunction.

The inverse problem is to find the piecewise-constant functions 𝜀 and 𝜎, if for solution of the direct problem (2)–(5), the additional information (6) is known.

Introduce the notation 𝑘²=𝜆²−𝜘,𝜘=𝜔²𝜇₀𝜀₀𝜀−𝑖𝜔𝜇₀𝜎. (7)

Fix some values of the angular frequency 𝜔₀. In the inverse problem, we will recover the complex value 𝜘₀ (𝜘₀=𝜔²₀𝜇₀𝜀₀𝜀−𝑖𝜔₀𝜇₀𝜎) which is a piecewise-constant function since the functions 𝜀 and 𝜎 are piecewise constant.

As is easy to see, if we find 𝜘₀ as a solution of the inverse problems (2)–(6), we immediately recover the functions 𝜀 and 𝜎 as follows: 𝜘𝜀=Re₀𝜔²₀𝜇₀𝜀₀𝜘,𝜎=−Im₀𝜔₀𝜇₀. (8)

Inverse problem (2)–(6) may be solved by minimizing the residual functionals as follows: 𝐽𝜘₀=_𝜔ℎ_𝜔𝑢0,𝜔,𝜆₀−𝑔𝜔,𝜆₀², (9) (here ℎ_𝜔 are certain weight multipliers). Paying attention to that in the functional (9), we fix a value of the spatial frequency 𝜆₀ and prepare additional information (6) for different values of the angular frequency 𝜔.

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 𝑤₁ and 𝑤₂, we introduce the following relation: 𝑤₁,𝑤₂=12𝑤₁𝑤₂+𝑤₁𝑤₂. (10) The relation (10) has all the properties of scalar multiplication. The proof of this is based on a geometric interpretation of complex numbers. Let 𝑤₁=(𝛼₁,𝛽₁) and 𝑤₂=(𝛼₂,𝛽₂); then, ⟨𝑤₁,𝑤₂⟩=𝛼₁𝛼₂+𝛽₁𝛽₂.

For (10) note useful relation 𝑤₁,𝑤₂=Re𝑤₁𝑤₂=Re𝑤₁𝑤₂. (11)

3. Gradient of the Residual Functional

First we note that 𝐽𝜘₀=𝐽𝜘¹₀,…,𝜘^𝑚₀,…,𝜘^𝑁𝑙₀, (12) where 𝜘^𝑚₀ are the values of a piecewise constant function 𝜘₀ in the segment [𝑧_𝑚−1,𝑧_𝑚].

We obtain the expression of the gradient of the residual functional (9) by the statement of the conjugate problem 𝜓_𝑧𝑧−𝜆²𝜓−𝜘𝜓=0,_𝑧𝑘=0,𝜓_𝑧𝑧𝑘=0,𝑘=1,𝑁_𝑙,𝜓_𝑧∗=0,𝜓_𝑧𝑧∗=−2𝑢0,𝜔₀,𝜆−𝑔𝜔₀,𝜓⟶0(𝑧⟶±∞). (13)

Let the value 𝜘 be incremented 𝛿𝜘; then, the function 𝑢 will get increment 𝛿𝑢, which satisfies the following problem: 𝛿𝑢_𝑧𝑧−𝑘²[]𝛿𝑢+𝑢𝛿𝜘=0,𝛿𝑢_𝑧𝑘=0,𝛿𝑢_𝑧𝑧𝑘=0,𝑘=−1,𝑁_𝑙,𝑢⟶0(𝑧⟶±∞) (14) (here for, simplicity, we set 𝑧₋₁=𝑧_∗).

In this case, the increment of the residual functional up the second order can be obtained as follows: 𝛿𝐽𝜘₀=Re_𝜆2ℎ_𝜆𝑢−𝑔𝛿𝑢=−Re_𝜆ℎ_𝜆𝜓_𝑧0𝛿𝑢. (15)

Given statements of problems (14), (13), and the equality ^∞_−∞𝑑𝑑𝑧𝛿𝑢𝜓_𝑧𝑑𝑧=−𝛿𝑢|_𝑧=𝑧0𝜓_𝑧𝑧0, (16) we get 𝛿𝐽𝜘₀=Re_𝜆ℎ_𝜆^𝑧𝑁𝑙_𝑧0𝑢𝜓𝛿𝜘𝑑𝑥. (17) Since in each layer [𝑧_𝑚−1,𝑧_𝑚], the function 𝜘 is constant and takes the value 𝜘^𝑚, we can write 𝛿𝐽𝜘₀=^𝑁𝑙_𝑚=1Re_𝜔ℎ_𝜔^𝑧𝑚_𝑧𝑚−1𝑢𝜓𝑑𝑧⋅𝛿𝜘^𝑚. (18)

Let 𝑁_𝑚=^𝑧𝑚_𝑧𝑚−1𝑢𝜓𝑑𝑧. (19) Then Re𝑁_𝑚𝛿𝜘^𝑚=Re𝑁_𝑚⋅𝜔²𝜇₀𝜀₀𝛿𝜀_𝑚+Im𝑁_𝑚⋅𝜔𝜇₀𝛿𝜎_𝑚=Re𝑁_𝑚⋅𝜔²𝜔²₀⋅𝜔²₀𝜇₀𝜀₀𝛿𝜀_𝑚+Im𝑁_𝑚⋅𝜔𝜔₀⋅𝜔₀𝜇₀𝛿𝜎_𝑚=Re𝑁_𝑚𝛿𝜘^𝑚₀, (20) where 𝑁_𝑚=Re𝑁_𝑚𝜔²𝜔²₀+𝑖Im𝑁_𝑚𝜔𝜔₀. (21)

Therefore, the gradient of the residual functional 𝐽[𝜘₀] will have the form 𝐽^′𝜘₀=𝑗₁,…,𝑗_𝑚,…,𝑗_𝑁𝑙,𝑗_𝑚=_𝜔ℎ_𝜔𝑁_𝑚. (22)

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 𝑠 as follows: 𝑢_𝑧(𝑧)=𝑠(𝑧)𝑢(𝑧), (23) which will satisfy the differential Riccati equation 𝑠^′+𝑠²=𝑘². (24) If we calculate from the right to the left, then the solution of (24) in each segment [𝑧_𝑚−1,𝑧_𝑚] will be as follows: 𝑠(𝑧)=𝑘_𝑚𝑠^𝑚+𝑘_𝑚𝑒^{2𝑘𝑚(𝑧−𝑧𝑚)}+𝑠^𝑚−𝑘_𝑚𝑠^𝑚+𝑘_𝑚𝑒^{2𝑘𝑚(𝑧−𝑧𝑚)}−𝑠^𝑚−𝑘_𝑚. (25) If we calculate from the left to the right, then 𝑠(𝑧)=𝑘_𝑚𝑠^𝑚−1+𝑘_𝑚+𝑠^𝑚−1−𝑘_𝑚𝑒^{2𝑘𝑚(𝑧𝑚−1−𝑧)}𝑠^𝑚−1+𝑘_𝑚−𝑠^𝑚−1−𝑘_𝑚𝑒^{2𝑘𝑚(𝑧𝑚−1−𝑧)} (26) (here 𝑠^𝑚=𝑠|_{𝑧=𝑧𝑚}, 𝑘_𝑚 is the value of the piecewise constant function 𝑘 in the segment [𝑧_𝑚−1,𝑧_𝑚] and Re[𝑘_𝑚]≥0).

For solving Riccati equation (24), we will do our recurrent calculations from the layer up to the layer moving to the point 𝑧_∗, where the source is disposed.

The condition of damping in infinity (5) enables 𝑠(𝑧)=−𝑘_𝑁𝑙+1,𝑧∈𝑧_𝑁𝑙,∞. (27) Due to conditions (6), we obtain the gluing conditions [𝑠]_𝑧𝑘=0. (28) Therefore, we may set 𝑠^𝑁𝑙=−𝑘_𝑁𝑙+1 (29) and begin the recurrent calculation of 𝑠^𝑚 from the right to the left by formula 𝑠^𝑚−1=𝑘_𝑚𝑠^𝑚+𝑘_𝑚𝑒^{−2𝑘𝑚ℎ𝑚}+𝑠^𝑚−𝑘_𝑚𝑠^𝑚+𝑘_𝑚𝑒^{−2𝑘𝑚ℎ𝑚}−𝑠^𝑚−𝑘_𝑚,ℎ_𝑚=𝑧_𝑚−𝑧_𝑚−1. (30) Thus we get 𝑠^∗=𝑠|_{𝑧=𝑧∗+0}.

Analogously, taking into account damping in minus infinity (5) we get 𝑠(𝑧)=𝑘₀,𝑧∈∞,𝑧_∗. (31) Since the source is disposed in the half space (−∞,0], we may immediately take 𝑠|_{𝑧=𝑧∗−0}=𝑘₀. (32)

Gluing condition (28) in the point 𝑧=𝑧_∗ allows us to determine 𝑢|_𝑧=𝑧∗≡𝑢^∗𝑓(𝜔)𝜇=−⁰𝑠^∗−𝑘₀. (33)

Integrating (23) in the interval [𝑧_∗,0], we obtain 𝑢|_𝑧=0≡𝑢⁰=𝑢^∗2𝑘₀𝑒^{−𝑘0𝑧∗}𝑠⁰+𝑘₀𝑒^{−2𝑘0𝑧∗}−𝑠⁰−𝑘₀. (34)

Further, integrating in each interval [𝑧_𝑚−1,𝑧_𝑚] (23), we derive the solution of the problem (2)–(5) in this interval 𝑢(𝑧)=𝑢^𝑚−1𝑙(𝑧),𝑙(𝑧)=𝑒^{𝑘𝑚(𝑧𝑚−1−𝑧)}𝑠^𝑚+𝑘_𝑚𝑒^{2𝑘𝑚(𝑧−𝑧𝑚)}−𝑠^𝑚−𝑘_𝑚𝑠^𝑚+𝑘_𝑚𝑒^{−2𝑘𝑚ℎ𝑚}−𝑠^𝑚−𝑘_𝑚. (35)

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 𝜓 will coincide with 𝑠(𝑧) for all 𝑧∈[0,∞). Therefore, we have 𝜓_𝑧=0≡𝜓⁰2=−𝑢(0,𝜔,𝜆)−𝑔(𝜔,𝜆)𝑠⁰−𝑘₀ (36) and in each interval [𝑧_𝑚−1,𝑧_𝑚]. 𝜓(𝑧)=𝜓^𝑚−1𝑙(𝑧). (37)

Taking into account (35) and (37), we can obtain the formulas for the gradient components of 𝐽_𝜔[𝜘₀] (𝛼=𝜆,𝜔); that is, calculate the integral ^𝑧𝑚_𝑧𝑚−1𝑢𝑢𝜓𝑑𝑧=^𝑚−1𝜓^𝑚−1𝑅^𝑚𝑠^𝑚+𝑘_𝑚𝑒^{−2𝑘𝑚ℎ𝑚}−𝑠^𝑚−𝑘_𝑚², (38) where 𝑅^𝑚=1−𝑒^{−2𝑘𝑚ℎ𝑚}2𝑘_𝑚×𝑠^𝑚+𝑘_𝑚²𝑒^{−2𝑘𝑚ℎ𝑚}+𝑠^𝑚−𝑘_𝑚²−2ℎ_𝑚𝑠^𝑚+𝑘_𝑚𝑠^𝑚−𝑘_𝑚𝑒^{−2𝑘𝑚ℎ𝑚}. (39)

5. Numerical Experiment

5.1. Reference Frequency of the Medium 𝜔₀

First we note, that (see, e.g., formulas (25) and (35)) 𝑢=𝑢(𝑘), (40) where 𝑘²=𝜆²−𝜔²𝜇₀𝜀₀𝜀−𝑖𝜔𝜇₀𝜎,𝑘=⁴𝜆²−𝜔²𝜇₀𝜀₀𝜀²+𝜔𝜇₀𝜎²𝑒^𝑖𝛽/2,𝛽=arctg𝜔𝜇₀𝜎𝜆²−𝜔²𝜇₀𝜀₀𝜀,𝜆²>𝜔²𝜇₀𝜀₀𝜋𝜀,2,𝜆²=𝜔²𝜇₀𝜀₀𝜀,𝜋+arctg𝜔𝜇₀𝜎𝜆²−𝜔²𝜇₀𝜀₀𝜀,𝜆²<𝜔²𝜇₀𝜀₀𝜀. (41) That is, the variation of the solution of direct problem (2)–(5) depends directly on how the function 𝑘 depends on variations of the functions 𝜀 and 𝜎 in the layers.

Evidently, the greatest influence on change in the value 𝑘 is rendered by variations 𝜀 and 𝜎, when 𝜔²𝜇₀𝜀₀𝜀≈𝜔𝜇₀𝜎,𝜆²∼𝜔²𝜇₀𝜀₀𝜀. (42)

We put 𝜔²𝜇₀𝜀₀𝜀=𝜔𝜇₀𝜎, whence we get 𝜔₀=𝜎𝜀₀𝜀. (43) 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 𝐽[𝜘] from the Value 𝜔

For numerical experiments below we choose model shown in Table 1.

Table 1 

Model of the medium 1.

We set the mean values at 𝜀=20 and 𝜎=0,020, and then the reference angular frequency is 𝜔₀=1.12⋅10⁸. The capacity of the skin-layer, from here, is ℎ_𝑠=2𝜔₀𝜇₀𝜎≈0.84(𝑚). (44)

Fixing the values 𝜀_𝑚 and 𝜎_𝑚 (𝑚=2,5), we will change the values of dielectric permittivity and conductivity of the first layer in the segment 𝜀₁∈[1.0,40.0],𝜎₁∈[0.01,0.03], (45) for different values of 𝜔=𝑟𝜔₀, where 𝑟={1/10,1/7,1/5,1/2,1,2,5,7,10} and we will observe the behavior of the value Δ_𝜔=𝑢0,𝜔,𝜆₀−𝑔𝜔,𝜆₀², (46) which is one of the components of the residual functional (9). Results of numerical experiment are presented in Figure 1.

Figure 1 

The behavior of Δ𝜔 for various values 𝜔.

We note that first, if 𝜔 decreases with respect to the reference frequency of the medium 𝜔₀, then the residual functional loses sensitivity when 𝜀₁ varies. If 𝜔 increases, then the functional loses the sensitivity when 𝜎₁ varies. Secondly, the more the difference between 𝜔 and 𝜔₀ the less the sensitivity of the residual functional 𝐽[𝜘] to the change of desired unknown values, since it becomes more standing.

If 𝜀_𝑚 and 𝜎_𝑚 (𝑚=2,5) vary, the behavior of Δ_𝜔 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 𝜔 must by such that 𝜔²𝜇₀𝜀₀𝜀≈𝜔𝜇₀𝜎 (47) is verified (see (42)).

In addition, we clarify the known geophysical condition of quasistationarity of the electromagnetic field: 𝜔𝜀₀𝜀≪𝜎. The value of notation “≪” is lax. Proceeding from the numerical experiment, we may account the quasistationary approximation to be suitable if 𝜔𝜀₀𝜀≤𝜎/10.

5.3. Dependence of the Properties of the Residual Functional from the Value 𝜆

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 𝐽[𝜘₀] were derived. The result is shown in Figure 2.

Figure 2 

The behavior of the residual functional 𝐽[𝜘0] for various values of 𝜆. 450 values of 𝜔 were taken from the segment [𝜔0/10;10𝜔0] with constant step.

The tendency is seen as the greater 𝜆 the less is the sensitivity of residual functional to variations of the unknown parameters. The assumption that 𝜆²∼𝜔²𝜇₀𝜀₀𝜀 (48) is verified.

For model 1 we obtain 𝜔²𝜇₀𝜀₀𝜀≈2.84. It is seen that with increasing 𝜆 at first the residual functional 𝐽[𝜘₀] loses the sensitivity to the variations of 𝜎₁ and then to variations of 𝜀₁. When 𝜆²≥2𝜔²𝜇₀𝜀₀𝜀, 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 𝑔𝜔,𝜆₀=𝑔𝜔,𝜆₀𝑃1+𝜉,100 (49) where 𝜉 is a random value from the unit circle and 𝑃 is the percent of the introduced error. In Figure 3 we see the example of using the additional information with this error.

Figure 3 

The example of additional information 𝑔(𝜔,𝜆) given with the error 𝑃=20% (𝜔∈[𝜔0/50,𝜔0/10], 𝜆=5⋅10−1); the exact values of real and imaginary parts of 𝑔(𝜔,𝜆) are shown by the smooth line.

In addition to the model of the medium 1 in Table 1 were selected four more models (see Tables 2, 3, 4, and 5).

Table 2 

Model of the medium 2.

Table 3 

Model of the medium 3.

Table 4 

Model of the medium 4.

Table 5 

Model of the medium 5.

For each model of the medium the coordinate of the boundary of the last layer 𝑧_𝑁𝑙 coincides with the value of capacity of the skin-layer ℎ_𝑠.

The parameters used in the construction of the residual functional 𝐽[𝜘₀] are collected in Table 6.

Table 6 

In each layer, the initial approximations were 𝜀_𝑁𝑙 and 𝜎_𝑁𝑙.

For minimizing of the residual functional 𝐽[𝜘₀] the conjugate gradient method modified for complex values used 𝜘₀^𝑛+1=𝜘^𝑛₀−𝛼_𝑛𝑝_𝑛,𝑝_𝑛=𝐽^′𝜘^𝑛₀−𝛽_𝑛𝑝_𝑛−1,𝑝₀=𝐽^′𝜘⁰₀,𝛽_𝑛𝐽=−^′𝜘^𝑛₀²𝐽^′𝜘₀^𝑛−12,𝛼_𝑛=argmin_𝛼>0𝐽𝜘^𝑛₀−𝛼𝑝_𝑛. (50)

The results of restoration of piecewise constant functions 𝜀 and 𝜎 are shown in Figure 4.

(a)

(b)

(c)

(d)

(e)

Figure 4 

The examples or recovery 𝜀 and 𝜎: (a) model of the medium 1, (b) model of the medium 2, (c) model of the medium 3, (d) model of the medium 4, and (e) model of the medium 5. The exact solution is shown by the continuous line. The recovered solution is shown by the dotted line. For all the examples 𝑃=20%.

Remember that we used the value of the reference angular frequency 𝜔₀ such that the condition 𝜔²₀𝜇₀𝜀₀𝜀≈𝜔₀𝜇₀𝜎 is satisfied. Recovery experiment shows that the less the value of 𝜔𝜇₀𝜎, the more difficult restoring of the functions 𝜀 and 𝜎 is, since small values of 𝑘² 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 𝜀 and 𝜎. Note: 𝜔₀𝜇₀𝜎=2/ℎ²_𝑠.

7. Conclusions

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.

Acknowledgments

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).