Rezistif yüzeyli bir tabakada gömülü silindirik cisimlerin belirlenmesi
Başlık çevirisi mevcut değil.
- Tez No: 75385
- Danışmanlar: PROF. DR. İBRAHİM AKDUMAN
- Tez Türü: Yüksek Lisans
- Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
- Anahtar Kelimeler: Silindirik cisim, Öz direnç, Cylindrical body, Resistivity
- Yıl: 1998
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Elektronik ve Haberleşme Mühendisliği Ana Bilim Dalı
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
Bu çalışmada, bir tabaka içindeki gömülü silindirik cisimlere ilişkin bir ters saçılma problemi incelenmiştir. Problemde; tabakanın bir yüzünün kalınlığı ihmal edilebilecek kadar küçük homojen rezistif bir yüzey ile kaplı olduğu düşünülmüştür. Problem önce Green fonksiyonu yardımıyla formüle edilerek bir integral denkleme indirgenmiş, daha sonra bu integral denklem bir maske operatörü tanımlanarak Born yaklaşıklığı altında genelleştirilmiş ART (Algebraic Reconstruction Technique) yöntemiyle çözülmüştür.
Özet (Çeviri)
I. INTRODUCTION This work addresses an inverse scattering problem which aims to determine the geometrical as well as the physical properties of a cylindrical body buried in a layered half-space whose layers are separated by resistive planes. The resistive boundaries lead to a mathematical problem which is not solvable by the classical techniques based on the Fourier transform. To overcome this dificulty, the resulting system of operator equations is solved by using a generalization of the algebraic reconstruction technique. The waves reflected from the resistive boundaries cause the data collected by measurements to contain more information about the object to be determined. Due to large domain of applications such as non-destructive probing, geophysical prospecting, determination of underground tunnels and pipelines etc. the invers scattering problems related to the buried bodies has a particular importance in the inverse scattering theory. In that problems the unknown body is buried, in general, in a known limited host medium whose constitutive parameters differ from those of the surrounding infinite space. Then one tries to determine the geometrical (location and shape) as well as physical properties (constitutive parameters) of the body through the measurements of the scattered field outside the host medium. If the host medium is of finite size, then one can collect maximum data from all the points located around the body. On the contrary, when the host medium is of infinite extend, one has to confine oneself to limited data which can be collected only from accessible points. The cases of bodies buried in a half-space or in a slab are simple example of this kind of problems. The objective of this work is to solve an inverse scattering problem connected with bodies buried in a layered half-space whose layers are separated by resistive planes. The problem may be arisen, for example, if one tries to recover underwater objects lying in the sea or lake from the field measurements performed at the surface of the water or inside the atmosphere. In such a case the bottom surface of the water reservoir is modelled by a resistive boundary condition. For the sake of simplicity we restrict ourselves to case of three layers and cylindrical bodies. We assume also that the layers are non-magnetic and non-conducting simple mediums having different constant permittivities. The material of the cylindrical body is supposed to be local, instantaneous, linear, isotropic and non magnetic, i.e.: its dielectric permittivetiy e(x) and conductivity o(x) are scalafunctions of the space co-ordinates in R2. The problem then consists of finding the functions e(x) and o(x) by using the data collected through measurements along a line in the half-space not containing the body. The body will be illuminated by a plane wave from the same region. 2 FORMULATION OF THE PROBLEM The geometry of the problem is illustrated in figure 1. In this configuration the half-space x2>0, the layer -d<x2 the half-space x2<-d are supposed to be filled with simple nonconducting and nonmagnetic materials having known permittivities e0, e, and e2, respectively. The layer is bounded by a homogeneous resistive plane (whose surface resistivity is R) at x2=-d and contains an ifinitely long nonmagnetic cylindrical body, say D, which is located parallel to the Ox3 axis. The body can also be composed of several disjoint subcylinders. The dielectric permittivity e(x) and the conductivity o(x) of the body are the functions of the coordinates x1 and x2 while its magnetic permeability is /^. The problem is then to recover the functions e(x) and c(x) from the field measurements performed in the region x2>0; more precisely, on a certain line x2=l. To this end, one illuminates the body from the accessible region mX2 ea^o.o=0 A. İ observation line eu\xoto=0 resistive plane e u cr=0 -d Figure 1. Geometry of the problem x2>0 by a plane wave (incident wave) whose electric field vector El is parallel to the Ox3 axis, namely E\x)={Q,O,u\x)) (2a)with UHX)^-**^*^^, ^e(p>ie) (2b) where is the incidence angle (see figure 1) and k^ is the wavenumber of the region x2>0. In such a case, the scattered field, say uD(x), which is defined by u(x)=u0(x)+uD(x), where u0(x) is the total field when the body D were absent, satisfies ^D+k\x^uD= -kx v(x)u(x) (2c) and uD(xv -d+0) -uD(xv -d-0) =0 (2d) du du m\x0 dx. £.(xv-d+0)--^(xv-d-o)=-^uD(xv-d+0) dx. R (2e) In the sense of distributions under the radiation condition. In (2.a,c), the functionfc2(;c2) 2 2 2 is equal to k^, kt and k^ in the regions x2>0, -d<x2 and x2<-d, respectively while the function v(x) is so-called object function and is defined by the relation e(x2) (2f) where e(^)=< e0, x2>0 ?j, -d<x 2<0, x2>-d (2g) In the following analysis we will try to recover the function v(x) from the scattered filed measurements. By considering the Green's Function of the related problem, say G(x;y), the equation (2c) can be reduced to the solution of the following Fredholm integral equation under the Born approximation; »d(*) =fci2/G(x;^“o(>')vO')^>' (2h)3 THE FIELD u0(x) For the plane wave given by (2a,b) the field «0(x) can be expressed as uQ(x)= -ikffic^cos^^x^a^ T4e -iki(xxoos^l *x^n^ ; x.<-d where rp r2, T3 and 74 are the coefficients to be determined through the same boundary conditions written for u0(x). After some stranghtforward computations we get *0sin4>0(l +Me ^^^ ^sin^l -Me -°M**«> (3b) where A =*0sin +Me ”^i***») -^sin^l -Me“^l (3c) and M=- sin^j i sin(|)2 I sine)), sinek - + i+ £ /? z, z, (3d)4 AN EXPLICIT EXPRESSION OF G(x;y) Consider now the Green function G(x,y) related to (2.c-e). By definition it satisfies AG+kz(x2)G=-Ö(x-y) (4a) and G(xv-d+0;y)-G(xv-d-0;y) (4b) dG BG 'M V-n ^-{xv -d+0;y) -^-(xv -d-0;y)=-pG(xv -d+0;y) (4c) ax2 ox2 K under the appropriate boundary condition. In order to find a suitable expression for G, we consider its Fourier transform with respect to xx, say G(y,x2;y), defined as follows; Ğ(y,x2;y)=fG(xl,x2;y)e'İVXldxl (4d) Then the transformation of (4a) written for y2e(-d,Q) results in solving the following differantial equation under the appropriate boundary and radiaton conditions: ^-Yfc=0 ; x2>0 dxl ¦^-Y?G=-e'vyi8(x2-y2) ; x2e(-d,0) (4e) dxl vG 2a n ^ J - --Y2G=0 ; x2<-d dx* where Yo> Yi and Y2 are square root functions Yon/*H^ Yı^^î Y2=1pI? (4f) defined in the complex v -plane such thatY0(0)=-*V Yi(0)=-*p y2(0)=-ik2 (4g) Now its required to know the explicit expression of G for ^>0 only. After some stranghtforward manipulations one gets the following; G^yd^Ş^^^ JoVyA Ya J e ne 7(rt ; XjX), -d<y2 (4h) where A^, Av A4 are all known coefficients. An explicit expression of G(x,y) can then be obtained through the inverse transform integral, namely; Gix,y)=^-fĞiv^y2)ei}>Xidy, (4İ) 5. AN ART TYPE SOLUTION In the theory we astablished the equation (2h) is first reduced to the solution of a system of linear operator eqations in the form (Arfix^aJxJ, n=\X-N (5a) with 1 -fci (5b) and k2”1 -itf (Mv)j +-J2(Mv)2+-^(M v)3 +-2-52 (Mv)4 ^ * h 3 L^ e ldv (5c)where An is the linear operator corresponding the incidence angle 4>=(J>n> n=l,...N. In these expressions L1, L2, T2n, T3n, are all known functions The functions (Mv)n appearing in (5c) is the two-dimensional Fourier transform of (Mv)(x) evaluated at different spectral points where M is the masking operator defined by (v(;t) ; -d<x2 (Mv)(x)= (5d) İ0 ; otherwise, Notice that in order to solve the above system of linear equations we have to introduce such a masking operator. The function ûD(y,t) in (2h) is the Fourier transform of the scattered field measured on the line x2=l. Then the system of operator equations (5a) can be solved through the Kaczmarz iterative method. This interesting iterative technique connects the value of v(x) obtained after (j + l)'th iteration with that of the j'th iteration by using N number of operational process, namely writes: VoCxWCc) M ? 1 e fc* v“(x)=v”Ax)+ f P!(vxla"(v)-i4 v,(v)l 2wJfe?4 ^vw L J (5e) xeKv+k^^dv,«=1,...^ ^\x)=v^x) where Pn and F are known functions.
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