INTRODUCTION

Magnetotelluric sounding (MTS) is one of the most important methods of deep geophysical sounding to investigate the structure of the lithosphere by analyzing variations in the Earth’s natural electromagnetic field [1–6]. The theoretical and practical fundamentals of the MTS method were introduced in early 1950s by A.N. Tikhonov [1] and L. Cagniard [2]. Currently, the method is still being developed and is of great interest for solving both fundamental and applied geophysical problems. The development of theoretical foundations and methods for interpreting magnetotelluric data remains an important area of research.

One of the most well-studied classes of MTS problems is the class of forward one-dimensional problems of magnetotelluric sounding [3–6] based on the so-called Tikhonov–Cagniard physical model [1–6]. Such problems are solved by simulating the electromagnetic field components for a geological medium with known physical parameters.

In paper [7], the authors proposed and studied a difference scheme for solving a forward one-dimensional MTS problem obtained by a method of local integral equations (the basic formulas of the method are given in the next section). The difference schemes are constructed by using a piecewise constant interpolation of the specific electrical conductivity function \(\sigma(z)\) on the computational grid. Estimates of convergence of the approximate solution to the exact one for a class of quasi-uniform grids have been obtained in [7], but choosing the optimal, in a sense, version of such a grid is still a problem.

Among the physical and geological models for gradient geological media that have been poorly studied so far, the forward one-dimensional problems of magnetotelluric sounding characterized by a continuous one-dimensional specific electrical conductivity function \(\sigma\) are of considerable interest [3, 8]. Under certain conditions, at certain depths, this function may have zones of high gradients, which can cause considerable distortions in the resulting numerical solutions. For a more accurate piecewise constant approximation of the function \(\sigma(z)\), in the present paper it is proposed to use special adaptive grids.

There are various options of constructing grids adapted to the properties of the function being interpolated. Most often, the construction of an adaptive grid is considered as the problem of constructing a one-to-one mapping of a uniform computational grid into an adaptive one which allows an informative description of the function of interest. This approach, first proposed in [9], is based on an equidistant distribution of some auxiliary “weight” function which allows organizing the desired adaptive grid as follows: major nodes are localized in the zones where the values of the derivatives of the function of interest are large, and minor ones in the areas where the values of these derivatives are insignificant.

In the present paper, we propose another method of computational grid adaptation, which is based on a geometric principle of minimizing a functional of the defect of the areas described by plots of the initial function and its piecewise constant interpolant. Adaptive grids constructed in this way are used to solve the forward one-dimensional MTS problem for a gradient medium by using difference schemes obtained in [7].

DIFFERENCE SCHEMES FOR SOLVING THE FORWARD ONE-DIMENSIONAL PROBLEM OF MAGNETOTELLURIC SOUNDING OBTAINED BY A METHOD OF LOCAL INTEGRAL EQUATIONS

Paper [7] considers a matrix differential equation of the form

$$\dfrac{d\overrightarrow{U}(z)}{dz}=L(z)\overrightarrow{U}(z),\quad z\in(0,z_{\max}),$$
(1)

where \(\overrightarrow{U}(z)\) is the sought-for unknown two-component complex-valued vector function, \(L(z)\) is the variable operator of the equation defined by the matrix \(\begin{pmatrix}0&-\sigma(z)\\ i\omega\mu_{0}&0 \end{pmatrix}\), where \(\omega>0,~\mu_{0}>0\) are real constants, \(i\) is the imaginary unit, and \(\sigma(z)\) is a real function given on \(z\in(0,z_{\max})\). A method of local integral equations was used by the authors of [7] to construct a difference scheme for numerically solving the equation:

$$\overrightarrow{U}_{j+1}^{h}=e^{\varDelta z_{j}L_{j+1/2}}\overrightarrow{U}_{j}^{h},\quad j=1,2,\ldots,J-1,$$
(2)

where \(\bigl{\{}z_{j}\bigr{\}}_{j=1}^{J}\) are the nodes of an arbitrary nonuniform computational grid on the interval \([0,z_{\max}]\); \(\varDelta z_{j}=z_{j+1}-z_{j}\) are the grid cell sizes; \(\overrightarrow{U}_{j}^{h}\approx\overrightarrow{U}(z_{j})\) are the values of an approximate solution to Eq. (1) at the grid points; \(\sigma_{j+1/2}=\frac{\sigma(z_{j})+\sigma(z_{j+1})}{2}\); \(k_{j+1/2}=(1-i)\sqrt{\frac{\omega\mu_{0}\sigma_{j+1/2}}{2}}\); \(e^{\varDelta z_{j}L_{j+1/2}}=\begin{pmatrix}\cosh\bigl{(}\varDelta z_{j}k_{j+1/2}\bigr{)}&-\frac{\sigma_{j+1/2}}{k_{j+1/2}}\sinh\bigl{(}\varDelta z_{j}k_{j+1/2}\bigr{)}\\[5pt] \frac{i\omega\mu_{0}}{k_{j+1/2}}\sinh\bigl{(}\varDelta z_{j}k_{j+1/2}\bigr{)}&\cosh\bigl{(}\varDelta z_{j}k_{j+1/2}\bigr{)} \end{pmatrix}\) is a matrix exponent.

In [7], in addition to the basic difference scheme (2), formulas of a natural interpolation of the resulting approximate solution obtained by using a method of local integral equations are also proposed. Thus, at the assumption that we know the grid function \(\overrightarrow{U}^{h}=\bigl{\{}\overrightarrow{U}_{j}^{h}\bigr{\}}_{j=1}^{J}\) found as an approximate solution to the problem (1) by using the difference scheme (2), in [7] it is proposed to consider, as the interpolant, a continuous vector function \(\overrightarrow{U}^{h}(z)\) whose values at the internal points of each grid cell are determined by the formulas

$$\overrightarrow{U}^{h}(z)=e^{(z-z_{j})L_{j+1/2}}\overrightarrow{U}_{j}^{h},\quad z\in[z_{j},z_{j+1}],\ \ j=1,2,\ldots,J-1,$$
(3)

where

$$e^{(z-z_{j})L_{j+1/2}}=\begin{pmatrix}\cosh\bigl{[}(z-z_{j})k_{j+1/2}\bigr{]}&{\displaystyle -\frac{\sigma_{j+1/2}}{k_{j+1/2}}\sinh\bigl{[}(z-z_{j})k_{j+1/2}\bigr{]}}\\[6pt] {\displaystyle \frac{i\omega\mu_{0}}{k_{j+1/2}}\sinh\bigl{[}(z-z_{j})k_{j+1/2}\bigr{]}}&\cosh\bigl{[}(z-z_{j})k_{j+1/2}\bigr{]} \end{pmatrix}.$$

If the vector \(\begin{pmatrix}H(z)\\ E(z) \end{pmatrix}\) is used as the vector function \(\overrightarrow{U}(z)\) in the matrix differential equation (1), this equation, rewritten in component form, is

$$\left\{ \begin{array}{lcl} \dfrac{dH(z)}{dz}=-\sigma(z)E(z),\\[10pt] \dfrac{dE(z)}{dz}=i\omega\mu_{0}H(z), \end{array}\right.\quad z\in(0,z_{\max}).$$
(4)

Considering the physical meaning of the functions and constants of the system (4), we realize that it corresponds to a mathematical model describing the behavior of the electromagnetic field in a one-dimensional geological medium [1–6]: \(z\in(0,z_{\max})\) is a spatial coordinate along which the properties of the medium change (the Oz-axis directed deep into the Earth); \(H(z)\) and \(E(z)\) are the mutually orthogonal complex-valued components of the magnetic and electric field strength, respectively; \(\sigma(z)\) the specific electrical conductivity of the medium; \(\mu_{0}=4\pi\,10^{-7}\) the magnetic constant; and \(\omega\) the electromagnetic field frequency.

The system (4) with the initial conditions

$$\left\{ \!\!\begin{array}{lcl} H(0)=H_{0},\\[4pt] E(0)=E_{0}, \end{array}\right.\text{ or in matrix form }\overrightarrow{U}(0)=\overrightarrow{U}_{1}^{h}=\begin{pmatrix}H_{0}\\[4pt] E_{0} \end{pmatrix},$$
(5)

where \(H_{0}\) and \(E_{0}\) are given complex constants, is the forward one-dimensional magnetotelluric sounding problem or the so-called Tikhonov–Cagniard problem [1–6]. The most common method of analyzing solutions to forward MTS problems is to study the properties of the magnetotelluric impedance function (Tikhonov–Cagniard impedance), which is defined in the one-dimensional case as follows:

$$Z(z)={\displaystyle \frac{E(z)}{H(z)}.}$$
(6)

Therefore, in many studies dealing with approaches to solving the forward one-dimensional MTS problem, the impedance is calculated but not the functions \(E(z)\) and \(H(z)\) themselves. However, efficient calculation of these functions in one-dimensional media is of interest and practical importance, for example, for determining the boundary values of components of the electromagnetic field in two-dimensional media [4, 6, 10].

Note that an estimate of convergence of the approximate solution of the problem (4), (5) obtained by formula (2) to its exact solution is obtained in [7] as follows:

$$\max\Big{\{}\big{|}H(z_{j})-H_{j}^{h}\big{|},\big{|}E(z_{j})-E_{j}^{h}\big{|}\Big{\}}\leqslant K\,(\varDelta z)^{2},\quad j=1,2,\ldots,J,$$

where \(H(z_{j}),~E(z_{j})\) and \(H_{j}^{h},~E_{j}^{h}\) are the values of the exact and approximate solutions of the system (4) at the grid nodes and \(K\) is a constant independent of \(\varDelta z\equiv\max\limits _{1\leqslant j\leqslant J-1}\varDelta z_{j}\). That is, the difference scheme (2) has convergence rate to the exact solution of the problem of second order with respect to the grid size.

A similar estimate has also been obtained for the interpolant (3):

$$\max\Big{\{}\big{|}H(z)-H^{h}(z)\big{|},\big{|}E(z)-E^{h}(z)\big{|}\Big{\}}\leqslant K\,(\varDelta z)^{2},\quad0\leqslant z\leqslant z_{\max}.$$

Here \(H(z),E(z)\) and \(H^{h}(z),E^{h}(z)\) are the exact and interpolated approximate solutions of the system (4), \(K\) is a constant independent of \(\varDelta z\equiv\max\limits _{1\leqslant j\leqslant J-1}\varDelta z_{j}\). That is, the interpolant obtained by formula (3) has second-order convergence rate as \(\varDelta z\to0\).

The above numerical methods have been implemented inMT1Dhlm software and tested on various problems [11, 12].

ALGORITHM FOR CONSTRUCTING AN ADAPTIVE COMPUTATIONAL GRID BASED ON OPTIMIZATION OF A PIECEWISE CONSTANT INTERPOLANT

Consider an arbitrary nonuniform grid \(\bigl{\{}z_{j}\bigr{\}}_{j=1}^{J}\) on the interval \([a,b\,]\):

$$a=z_{1}<z_{2}<\dots<z_{J-1}<z_{J}=b.$$
(7)

Let \(f(z)\) be a sufficiently smooth function defined on \([a,b\,]\). To approximate this function on the chosen grid (7), we use a piecewise constant function

$$f(z,\theta)\equiv\left\{ \begin{array}{lcl} f_{j}(\theta)\equiv(1-\theta)f(z_{j})+\theta f(z_{j+1}),\\[1ex] z\in[z_{j},z_{j+1}],~j=1,2,\ldots,J-1, \end{array}\right.$$
(8)

where \(\theta\in[0,1]\) is an arbitrary parameter. Then from formula (8) we have the equation

$$f(z)-f_{j}(\theta)=(1-\theta)\int\limits _{z_{j}}^{z}f'(s)ds-\theta\int\limits _{z}^{z_{j+1}}f'(s)ds,\quad z\in[z_{j},z_{j+1}],~j=1,\ldots,J-1,$$
(9)

which can be used to obtain an estimate characterizing the defect of areas on the interval \([a,b\,]\) when the function \(f(z)\) is approximated by formula (8):

$$\big{\|}f-f({\boldsymbol{\cdot}},\theta)\big{\|}_{L_{1}}\leqslant\sum_{j=1}^{J-1}\int\limits _{z_{j}}^{z_{j+1}}\big{|}\theta(s-z_{j})+(1-\theta)(z_{j+1}-s)\big{|}\,\big{|}f'(s)\big{|}ds\equiv\varphi(\vec{z},\theta),$$
(10)

where \(\vec{z}=(z_{1},z_{2},\ldots,z_{J-1},z_{J})\) is a vector whose components are the nodes of the computational grid (7).

An adaptive computational grid is constructed by solving the following optimization problem:

$$\varphi(\vec{z},\theta)\rightarrow\min\limits _{\vec{z}}~(\text{the parameter}~\theta\in[0,1]~\text{is fixed}).$$
(11)

The solution to problem (11) can be reduced to the solution of the system of nonlinear equations

$$\left\{ \begin{array}{l} \!\!F_{1}(\vec{z}\,)\equiv z_{1}-a=0,\\[1ex] \!\!F_{j}(\vec{z}\,)=0,\quad j=2,\ldots,J-1,\\[1ex] \!\!F_{J}(\vec{z}\,)\equiv z_{J}-b=0, \end{array}\right.$$
(12)

where

$$F_{j}(\vec{z})=\frac{\partial\varphi(\vec{z},\theta)}{\partial z_{j}},\quad j=2,\ldots,J-1.$$
(13)

The system (12) can be solved, for instance, by Newton’s iterative method:

$$\left\{ \begin{array}{l} \!\!\vec{z}^{\ (0)}~~\text{is set},\\[1ex] \!\!F'\big{(}\vec{z}^{\ (k-1)}\big{)}\,\big{(}\vec{z}^{\ (k)}-\vec{z}^{\ (k-1)}\big{)}=-F\big{(}\vec{z}^{\ (k-1)}\big{)},\quad k=1,2,\ldots, \end{array}\right.$$
(14)

where \(k\) is the iteration number, \(\vec{z}^{\ (k)}\equiv\big{(}z_{1}^{(k)},z_{2}^{(k)},\ldots,z_{J}^{(k)}\big{)}\), \(F\big{(}\vec{z}^{\ (k)}\big{)}=\Big{(}F_{1}\big{(}\vec{z}^{\ (k)}\big{)},F_{2}\big{(}\vec{z}^{\ (k)}\big{)},\ldots,\) \(F_{J}\big{(}\vec{z}^{\ (k)}\big{)}\Big{)},\) \(F'\big{(}\vec{z}^{\ (k)}\big{)}=\biggl{(}\frac{\partial F_{i}\big{(}\vec{z}^{\ (k)}\big{)}}{\partial z_{j}}\biggr{)}_{i,j=1}^{J}\) is the Jacobi matrix.

In (14), let us replace the vector \(\vec{z}\) by a vector \(\vec{v}\) with components

$$v_{j}^{(k)}=z_{j}^{(k)}-z_{j}^{(k-1)},\quad j=1,\ldots,J,$$

and use some special approximations (described in detail in [13]) to calculate the components of the vector function \(F\big{(}\vec{z}\,\big{)}\) and the Jacobi matrix \(F'\big{(}\vec{z\,}\big{)}\). Then at the \(k\)-th step of the iteration the process (14) is reduced to solving the system of linear algebraic equations

$$\left\{ \begin{array}{l} \!\!B_{1}^{(k-1)}v_{1}^{(k)}-C_{1}^{(k-1)}v_{2}^{(k)}=g_{1}^{(k-1)},\\[2ex] \!\!-A_{j}^{(k-1)}v_{j-1}^{(k)}+B_{j}^{(k-1)}v_{j}^{(k)}-C_{j}^{(k-1)}v_{j+1}^{(k)}=g_{j}^{(k-1)},\quad j=2,\ldots,J-1,\\[2.5ex] \!\!-A_{J}^{(k-1)}v_{J-1}^{(k)}+B_{J}^{(k-1)}v_{J}^{(k)}=g_{J}^{(k-1)}, \end{array}\right.$$
(15)

whose coefficients are determined as follows:

$$\begin{aligned} &&B_{1}^{(k-1)}=1,\qquad C_{1}^{(k-1)}=0,\qquad g_{1}^{(k-1)}=0,\\[1ex] &&A_{J}^{(k-1)}=0,\qquad B_{J}^{(k-1)}=1,\qquad g_{J}^{(k-1)}=0;\\[1ex] &&A_{j}^{(k-1)}=(1-\theta)w_{j-1}^{(k-1)}+\theta w_{j}^{(k-1)},\\[1ex] &&C_{j}^{(k-1)}=(1-\theta)w_{j}^{(k-1)}+\theta w_{j+1}^{(k-1)},\\[1ex] &&B_{j}^{(k-1)}=3w_{j}^{(k-1)}-\theta w_{j-1}^{(k-1)}-(1-\theta)w_{j+1}^{(k-1)},\\[1ex] &&g_{j}^{(k-1)}=\Big{(}z_{j+1}^{(k-1)}-2z_{j}^{(k-1)}+z_{j-1}^{(k-1)}\Big{)}w_{j}^{(k-1)}+\frac{\theta}{2}\Big{(}z_{j+1}^{(k-1)}-z_{j}^{(k-1)}\Big{)}\Big{(}w_{j+1}^{(k-1)}-w_{j}^{(k-1)}\Big{)}\\[1ex] &&\hspace{15mm}+\frac{1-\theta}{2}\Big{(}z_{j}^{(k-1)}-z_{j-1}^{(k-1)}\Big{)}\Big{(}w_{j}^{(k-1)}-w_{j-1}^{(k-1)}\Big{)},\quad j=2,\ldots,J-1,\end{aligned}$$

where we use the following auxiliary functions \(w_{j}\):

$$\begin{aligned} &&w_{1}=\biggl{|}\frac{z_{3}-z_{1}}{(z_{3}-z_{2})(z_{2}-z_{1})}\bigl{(}f(z_{2})-f(z_{1})\bigr{)}-\frac{z_{2}-z_{1}}{(z_{3}-z_{2})(z_{3}-z_{1})}\bigl{(}f(z_{3})-f(z_{1})\bigr{)}\biggr{|},\\[1ex] &&w_{j}=\biggl{|}\frac{z_{j}-z_{j-1}}{(z_{j+1}-z_{j-1})(z_{j+1}-z_{j})}\bigl{(}f(z_{j+1})-f(z_{j})\bigr{)}\\ &&\hspace{11mm}+\frac{z_{j+1}-z_{j}}{(z_{j+1}-z_{j-1})(z_{j}-z_{j-1})}\bigl{(}f(z_{j})-f(z_{j-1})\bigr{)}\biggr{|},\quad j=2,\ldots,J-1,\\[1ex] &&w_{J}=\biggl{|}\frac{z_{J}-z_{J-2}}{(z_{J-1}-z_{J-2})(z_{J}-z_{J-1})}\bigl{(}f(z_{J})-f(z_{J-1})\bigr{)}\\ &&\hspace{11mm}-\frac{z_{J}-z_{J-1}}{(z_{J-1}-z_{J-2})(z_{J}-z_{J-2})}\bigl{(}f(z_{J})-f(z_{J-2})\bigr{)}\biggr{|}.\end{aligned}$$

In the present paper, we use the double sweep method to solve the system (15) [14]. For regularization, to obtain strict diagonal dominance, the diagonal coefficients \(B_{j}^{(k-1)}\) are formed as follows:

$$B_{j}^{(k-1)}=\max\biggl{(}A_{j}^{(k-1)}+C_{j}^{(k-1)}+E,~3w_{j}^{(k-1)}-\theta w_{j-1}^{(k-1)}-(1-\theta)w_{j+1}^{(k-1)}\biggr{)},$$

where \(E>0\) is a sufficiently small positive number.

Table 1. Relative errors for the function \(H(z)\) obtained at different values of the parameters of the problem (18) and the computational grid, %
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Table 2. Relative errors for the function \(E(z)\) obtained at different values of the parameters of the problem (18) and the computational grid, %
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Table 3. Relative errors for the function \(Z(z)\) obtained at different values of the parameters of the problem (18) and the computational grid, %
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Fig. 1
figure 1

Function \(\sigma(z)\) and its grid approximations for variant no. 2 of combinations of parameters in Tables 1–3 (upper curve); nodes of uniform and adaptive computational grids, respectively (lower curve).

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Fig. 2
figure 2

Real and imaginary parts of the function \(H(z)\) for variant no. 2 of combinations of parameters in Tables 1–3.

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Fig. 3
figure 3

Function \(\sigma(z)\) and its grid approximations for variant no. 5 of combinations of parameters in Tables 1–3 (upper curve); nodes of uniform and adaptive computational grids, respectively (lower curve).

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Fig. 4
figure 4

Real and imaginary parts of the function \(H(z)\) for variant no. 5 of combinations of parameters in Tables 1–3.

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Fig. 5
figure 5

Function \(\sigma(z)\) and its grid approximations for variant no. 9 of combinations of parameters in Tables 1–3 (upper curve); nodes of uniform and adaptive computational grids, respectively (lower curve).

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Fig. 6
figure 6

Real and imaginary parts of the function \(H(z)\) for variant no. 9 of combinations of parameters in Tables 1–3.

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As an initial approximation for the iterative process (14), we use a grid that is uniform on \([a,b\,]\):

$$z_{j}^{(0)}=a+\frac{j-1}{J-1}\bigl{(}b-a\bigr{)},\quad j=1,\ldots,J,$$

and as a stopping criterion of the iterative process (14) we use the condition that the inequality

$$\big{\|}\vec{v}\,\big{\|}_{\infty}=\max\limits _{1\leqslant j\leqslant J}{|}v_{j}{|}<\delta$$
(16)

with a sufficiently small parameter \(\delta=O\left(\big{(}J-1\big{)}^{-3}\right)\) given beforehand is satisfied.

ANALYTICAL SOLUTION OF THE FORWARD ONE-DIMENSIONAL MTS PROBLEM FOR THE KATO–KIKUCHI MODEL (TEST PROBLEM)

A well-known example of a gradient function for the specific electric conductivity \(\sigma\) used in MTS problems is a power function in the Kato–Kikuchi model [3]:

$$\sigma(z)=\sigma_{0}\bigl{(}1+pz\bigr{)}^{-2},$$
(17)

where \(\sigma_{0}\) and \(p\) are some positive numbers. At certain values of \(\sigma_{0}\) and \(p\) this function forms a boundary layer at the point \(z=0\). Note that in the case of a function given as (17), an analytical solution of the corresponding forward one-dimensional MTS problem [3, 11, 12] is known. This is an advantage of this model in numerical experiments on testing the efficiency of the difference scheme (2).

Consider the forward one-dimensional magnetotelluric sounding problem for the case of a vertical-gradient geological medium described by the Kato–Kikuchi power model [3]. A mathematical model of this problem is the system of equations (4) (or, equivalently, the matrix equation (1)) with the initial conditions (5) (in this case the specific electrical conductivity function \(\sigma\) is defined by formula (17)):

$$\left\{ \begin{array}{lcl} \dfrac{dH(z)}{dz}=-\sigma_{0}(1+pz)^{-2}E(z),\\[10pt] \dfrac{dE(z)}{dz}=i\omega\mu_{0}H(z), \end{array}\right.\ \ z\in(0,z_{\max}),\qquad\left\{ \begin{array}{lcl} H(0)=H_{0},\\[1ex] E(0)=E_{0}, \end{array}\right.$$
(18)

where \(z\) is the spatial coordinate; \(H(z)\) and \(E(z)\) are unknown functions of the magnetic and electric fields; \(\sigma_{0}\) and \(p\) are positive real numbers; \(i\) the imaginary unit; \(\mu_{0}=4\pi\,10^{-7}\) the magnetic constant; and \(\omega\) the electromagnetic field frequency.

It is shown in [11] that the exact solution to this problem is

$$E(z)=E_{0}(1+pz)^{\nu+\frac{1}{2}},\quad H(z)=H_{0}(1+pz)^{\nu-\frac{1}{2}},\quad\nu=\sqrt{\frac{1}{4}+\frac{k_{0}^{2}}{p^{2}}},\quad z\in[0,z_{\max}],$$
(19)

where \(k_{0}^{2}=-i\omega\mu_{0}\sigma_{0}\). For the problem (18) to be solvable, the complex constants \(H_{0}\) and \(E_{0}\) are interrelated as \(p\Big{(}\frac{1}{2}-\nu\Big{)}H_{0}=\sigma_{0}E_{0}\). The magnetotelluric impedance (6) in this case is described by the function

$$Z(z)=\frac{E_{0}}{H_{0}}(1+pz),\quad z\in[0,z_{\max}].$$
(20)

NUMERICAL SOLUTION OF THE FORWARD ONE-DIMENSIONAL MTS PROBLEM BY USING ADAPTIVE COMPUTATIONAL GRIDS

In this section we present the results of numerical experiments to solve the forward one-dimensional problem of magnetotelluric sounding (18) by using the difference scheme (2) with uniform and adaptive computational grids. The main purpose of the calculations is to analyze the efficiency of using adaptive grids in numerically solving the Tikhonov–Cagniard problem in the case of a gradient medium (the Kato–Kikuchi model) for which corresponding exact solutions to this problem (19), (20) are known.

The calculations have been made in MatLab in matrix form for various sets of parameters of the problem itself and the computational grid. For this, a software program MT1Dhlm has been improvedFootnote 1 by implementing an algorithm for constructing adaptive computational grids.

The impedance function \(Z(z)\) is approximated as a ratio of the corresponding approximate functions \(E(z)\) and \(H(z)\) by formula (6).

The accuracy of the approximate solutions obtained is estimated by using two types of errors:

$$ rel_{-}error_{-}max = {\displaystyle \frac{\big{\|}(Y)^{h}-Y^{h}\big{\|}_{\infty}}{\big{\|}(Y)^{h}\big{\|}_{\infty}}\,100\%} $$
(21)

and

$$ rel_{-}error_{-}L_{1} = {\displaystyle \frac{\sum\limits _{j=1}^{J-1}(z_{j+1}-z_{j})\biggl{|}{\displaystyle \frac{(Y)_{j+1}^{h}+(Y)_{j}^{h}}{2}-{\displaystyle \frac{Y_{j+1}^{h}+Y_{j}^{h}}{2}\biggr{|}}}}{\sum\limits _{j=1}^{J-1}(z_{j+1}-z_{j})\biggl{|}{\displaystyle \frac{(Y)_{j+1}^{h}+(Y)_{j}^{h}}{2}\biggr{|}}}\,100\%,} $$
(22)

where \((Y)_{j}^{h}\) is the exact solution of the problem (18) at the \(j\)th grid node, \(Y_{j}^{h}\) is the approximate solution of (18) at the \(j\)th node, \(\big{\|}(Y)^{h}-Y^{h}\big{\|}_{\infty}=\max\limits _{1\leqslant j\leqslant J}\big{|}(Y)_{j}^{h}-Y_{j}^{h}\big{|},\) \(\big{\|}(Y)^{h}\big{\|}_{\infty}=\max\limits _{1\leqslant j\leqslant J}\big{|}(Y)_{j}^{h}\big{|}\). The solutions obtained on uniform and adaptive grids are estimated separately. To compare the results obtained on uniform and adaptive grids, a reference uniform grid with a larger number of nodes is used. On this grid, corresponding interpolants are constructed by using formula (3) for the approximate solutions obtained on the basic uniform and adaptive grids.

Note that in [7] when constructing difference schemes for solving the system (4) it is possible to perform a piecewise constant approximation of the specific electrical conductivity function \(\sigma(z)\) by using a value of this function inside each grid cell corresponding to half the sum of its values at the ends of this cell. Therefore, logically, when constructing adaptive grids and corresponding piecewise constant interpolants in the test problem under consideration, we use a value of the parameter \(\theta=\frac{1}{2}\).

A series of numerical experiments have been performed with different values of the initial conditions (5). In the present paper, we show the results obtained at an arbitrary value of \(H_{0}=100+100i\). Let us fix \(\delta=0.01\) as the value of the parameter in the stopping criterion of the iterative process for finding the adaptive grid nodes in formula (16) and 1001 as the number of the reference uniform grid nodes.

Tables 1–3 show relative errors for the functions \(H(z)\), \(E(z)\), and \(Z(z)\), respectively, obtained at different values of the parameters of the problem (18) and the computational grid (in the tables UG is uniform grid, and AG is adaptive grid). The results of the numerical experiments have shown that the errors of the interpolants on any grids do not exceed the errors of the corresponding grid functions. Therefore, we do not present tables of errors for the interpolants.

Figures 1, 3, and 5 show examples of gradient functions \(\sigma(z)\) (curve 1) and their approximants constructed using piecewise constant interpolation (curve 2 for uniform grids, curve 3 for adaptive grids). A visual analysis of the curves shows a clear advantage of adaptive grids for the piecewise constant interpolation of the functions \(\sigma(z)\): in contrast to the uniform grids, the density of adaptive grid nodes increases in the boundary layer zone of the function and allows its more accurate approximation in this zone. Figures 2, 4, and 6, using the function \(H(z)\) as an example, provide a comparison of the results of numerically solving the problem (18) with corresponding exact solutions for different \(\sigma(z)\) on different grids (see figure captions): curve 1 shows the known analytical solutions (19); curve 2 and \(\blacksquare\) the approximate solution and the interpolant obtained by using the difference scheme (2) and the natural interpolation formula (3) on the uniform computational grid; curve 3 and \(\bullet\)the approximate solution and the interpolant obtained by using the difference scheme (2) and the natural interpolation formula (3) on the adaptive computational grid.

It follows from the above tables of errors and the graphical examples that in the simulation of the electromagnetic field components by using the difference scheme (2), the approximate solution converges to the exact one, which is due to the increased number of the computational grid nodes. This takes place both for the uniform and adaptive grids. The efficiency of using adaptive computational grids (compared to uniform ones) in solving the forward one-dimensional magnetotelluric sounding problem for the case of a gradient medium is obvious. This has been clearly manifested in the numerical experiments with a relatively small number of nodes and a sufficiently large value of \(z_{\max}\).

CONCLUSIONS

A method of increasing the accuracy of numerical simulation for the electromagnetic field components when solving the forward one-dimensional problem of magnetotelluric sounding for the case of a gradient geological medium was considered in this paper. This was done by using difference schemes obtained by a method of local integral equations and piecewise constant interpolants for a specific electrical conductivity function. It has been shown that the accuracy of the thus obtained approximate solutions of the problem can be increased by using special computational grids adapted to the properties of this function. The results of numerical experiments on solving the Tikhonov–Cagniard problem for the Kato-Kikuchi model confirmed the efficiency of the proposed adaptive computational grids. The accuracy of simulating the electromagnetic field components increased by 1–3 orders of magnitude.

CONFLICT OF INTEREST

The authors of this work declare that they have no conflicts of interest.