Conventional diffraction tomography is based upon the Born approximation where the total field inside the object is substituted by the incident field by assuming that the relative dielectric constant of the object medium ε is close to that of the background medium $ε_b$ and satisfies $(\sqrt ε - \sqrt ε_b) D< λ/2$ where D is the size of the object and λ is the free space wavelength. Effects of attenuation of either the background and/or the object are studied by using the single frequency source.
Multi-frequency Born inversion may be used for the monostatic or multistatic diffraction tomography of a lossy object in the lossy background medium. It is shown here that the multi-frequency tomography faces difficulty, since the Fourier integral of the object function has its integrand as a function of frequency and has a complex transform variable which makes the integral diverge. If the background medium is slightly lossy, however, one may approximate the complex wave number in the background medium into the real part and the frequency-independent imaginary part. By choosing the real part of the wave number as the Fourier transform variable and taking the exponent having the imaginary part of the wave number as a part of integrand, one may show that the Fourier transform relates the scattered field and the object function having permittivity and conductivity. The permittivity and the conductivity terms are shown to be reconstructed here and its range of validity is shown. Numerical examples of reconstructing the lossy circular cylinder and the lossy L shaped cylinder in the lossy background medium are presented.
From the numerous numerical calculations, the validity of this type of Born inversion using multi-frequency signals for a lossy medium may be obtained. For the reconstruction of the relative dielectric constant, the close reconstruction is obtained if $ε/ε_b$ < 6.0$ and $\sqrt{ε}Dη(\frac{σ}{ε} -\frac{σ_b}{ε_b}) <1.2$, where η is 377 ohm and σ and $σ_b$ are the conductivity of the object and the background medium, respectively. For the close reconstruction of the conductivity profile, the condition becomes $\frac{ε}{ε_b}<1.5$ and $\sqrt{ε}Dη (\frac{σ}{ε} -\frac{σ_b}{ε_b})<1.2$. Improved Born inversion that corrects the length of the cross section from the conventional Born inversion is used here for the numerical calculations. By allowing the root mean square error of the reconstruction for a lossless medium, multi-frequency Born inversion reconstructs the dielectric cylinder of its contrast up to 1.4, while the improved Born Born inversion up to 8.0.
For the deep underground detection of fractures and inhomogeneous, cross borehole measurement is needed. For its reconstruction of the cross section, the measurement data may be obtained only along the vertical boreholes, which distorts the cross sectional image by elongating it along the horizontal direction. The reconstructed image may be corrected and the corrected image, in turn, provides estimated data along the horizontal plane which is impossible to obtain in the cross-borehole measurement. This makes the iterative reconstruction possible and improves the image quality, which is included.
Underground air tunnel of its diameter about 2 meters in the depth of 75 meters from the surface is measured in the configuration of the cross-borehole measurement. These measurement data are used for the reconstruction of the tunnel for which the developed algorithm for a lossy obstacle in the lossy background medium are used. The obtained image is impaired but recognizable, due to its large contrast where the measured relative dielectric constant and the conductivity of the background medium are, respectively, 7.7 and $1.74 × 10^{-3}[S/m]$ in average and continuous electromagnetic waves from 10 MHz to 80 MHz with the interval of 2 MHz are used for a 10 meter apart borehole.
