Detection of Local Inhomogeneities Against a Background of Distributed Ones Using Radiation Transmission Methods
Abstract Generated abstract
This paper analyzes the detection of localized inclusions in heterogeneous media by radiation transmission methods when random distributed inhomogeneities produce background flux fluctuations. It models transmitted intensity in terms of attenuation coefficients, filling density, fluctuation variance, autocorrelation, buildup factor, and inclusion geometry, treating defect detection as a signal-to-noise problem. The authors derive expressions for the minimum detectable inclusion thickness and cross-sectional area under amplitude analysis, averaging over radiograms, differential measurement schemes, and integral functionals constructed from detector readings. The proposed averaging and functional methods are presented as ways to reduce the effect of structural fluctuations and improve localization and sizing of inclusions, although detectability is poorer when total flux is used because scattered radiation reduces relative fluctuations.
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Detection of Local Inhomogeneities against a Background of Distributed Ones by Radiation Transmission Methods
V. A. Vorob’ev, V. K. Kivray, A. M. Molbin
Presented by the joint seminar of the DSM and MRD sectors of the Research Institute of EI
In a certain homogeneous medium with attenuation coefficient \(\mu_0\), inhomogeneous inclusions are distributed that have a different attenuation coefficient, \(\mu_1\). The inhomogeneity of such a structure is characterized by the coefficient of linear homogeneity \(\nu\) with statistical parameters: \(\bar{\nu}\) — the density of filling of the structure with inhomogeneous inclusions, \(\sigma\) — the root-mean-square fluctuation of the coefficient of linear homogeneity, \(K(r)\) — the autocorrelation function of this coefficient. By the coefficient of linear homogeneity here is meant the ratio of the path of the radiation flux through the inhomogeneous inclusions to its entire path in the given structure [1].
The magnitude of the flux at each point of the detector is determined by the expression
\[ I(x)=I_0 e^{-\mu_0 H}\cdot e^{-(\mu_1-\mu_0)\nu(x)H}+I_p, \tag{1} \]
where the inhomogeneity of the scattered-radiation flux \(I_p\) may be neglected [2]. Here \(I_0\) is the initial radiation flux, and \(H\) is the thickness of the barrier.
The mean value of the flux is determined as
\[ \bar{I}=I_0 B e^{-\mu_0 H}\cdot e^{-(\mu_1-\mu_0)\bar{\nu}H} = I_0 B e^{-\bar{\mu}_{\mathrm{e}}H}, \tag{2} \]
where \(B\) is the value of the buildup factor, and
\[ \overline{\mu}_{э}=\mu_1\overline{v}+\mu_0(1-\overline{v}) \tag{3} \]
acts as the attenuation coefficient of the absorber as a whole, including the inhomogeneities. The root-mean-square fluctuation of the flux is determined by the expression
\[ \sigma(I)=(\mu_1-\mu_0)H\frac{\sigma}{B}\,\overline{I} \tag{4} \]
on the assumption that the fluctuations of the radiation scattered beyond the barrier are small in comparison with the fluctuations of the primary radiation.
The presence in the structure of an additional local inclusion of thickness \(h(x)\) in the direction of flux propagation and with attenuation coefficient \(\mu_2\) leads to a change in the flux beyond this inhomogeneity according to the expression
\[ I(x)=I_0 e^{-\mu_0H} e^{-(\mu_1-\mu_0)v(x)H} e^{-h(x)[(\mu_2-\mu_0)-(\mu_1-\mu_0)v(x)]} \tag{5} \]
The problem of detecting this local inhomogeneity is similar to the problem of extracting signals from noise, where the change in flux \(\Delta I\) due to the local inclusion may be treated as a signal against the background of a random flux, which may be regarded as noise; that is, the detection of such an inhomogeneity is determined by the so-called signal-to-noise ratio, i.e.,
\[ \Delta I=\mathfrak{x}\sigma(I), \]
where \(\mathfrak{x}\) is a multiplier determined by the confidence coefficient; it is usually taken equal to three (the three-sigma rule).
It follows that, in amplitude analysis of fluxes, the minimum detectable inhomogeneity is determined by the expression
\[ \frac{h}{H} = \mathfrak{x} \left| \frac{\mu_2-\mu_0}{\mu_1-\mu_0} -\overline{v} \right|^{-1} \sigma\frac{\xi}{B}, \tag{6} \]
where the coefficient \(\xi\) is determined by the transverse dimensions of the local inhomogeneity (its extent) and by the depth at which it is embedded. Thus, if the local inclusion has a considerable extent, or if it is located near the detector, \(\xi\to 1\); if, however, the inhomogeneity lies sufficiently deep and has an insignificant extent, then the multiplier \(\xi/B\to 1\).
If the transverse dimensions of the local inhomogeneity exceed
the integral of the autocorrelation dependence of the coefficient of linear homogeneity; using various methods of averaging the fluctuations, one can improve the detectability of a local inhomogeneity [3].
One of these methods reduces to calculating, from radiogram data, the total attenuation of the flux over the entire length of the radiogram (or its mean value over this radiogram). The cross-sectional area of a local inclusion is determined from the expression
\[ S = L\, \frac{ \mu\left[(\mu_1-\mu_0)\bar{\nu}+\mu_0\right]+\ln\frac{\bar{I}}{I_0}-\ln B }{ \mu_2-\mu_0-(\mu_1-\mu_0)\bar{\nu} }, \tag{7} \]
if there is an assumption concerning the attenuation coefficient of the local inclusion, \(\mu_2\) (\(L\) is the length of the radiogram). Otherwise, the specimen can be transilluminated in two directions and a system of analogous equations can be solved for the attenuation coefficient of the inclusion and its volume.
The cross-sectional area of the inclusion can also be obtained from the expression
\[ S = \frac{ \ln \frac{\bar{I}_1}{\bar{I}_2} }{ \left(\frac{1}{L_2}-\frac{1}{L_1}\right) \left[\mu_2-\mu_0-(\mu_1-\mu_0)\bar{\nu}\right] }, \tag{8} \]
if the mean fluxes \(\bar{I}_1\) and \(\bar{I}_2\) are calculated from one and the same radiogram, but using different segments, \(L_1\) and \(L_2\).
When differential measurement schemes are used, that is, when the mean fluxes are calculated over two adjacent portions of the radiogram, the cross-sectional area of the inclusion is determined by the expression
\[ S = L\, \frac{ \ln \frac{\bar{I}_1}{\bar{I}_2} }{ \mu_2-\mu_0-(\mu_1-\mu_0)\bar{\nu} }. \tag{9} \]
The averaging method described above is equally applicable both in radiometry and in radiography, where instead of the mean attenuation of the flux one calculates the mean value of detector readings, which are functionally related to the magnitude of the flux. In the expressions for determining the sizes of the detected inclusions, additional coefficients may enter; their presence can be determined experimentally.
The minimum thickness of detectable inclusions is determined by the expression
\[ \frac{h}{H} = 2\frac{\sqrt{\ell \tau}}{\ell} \left|\frac{\mu_2-\mu_0}{\mu_1-\mu_0}\right|^{-1} \frac{\sigma}{B} + \frac{L}{\ell}\, \frac{(\mu_1-\mu_0)\Delta \nu+\Delta \mu\,\bar{\nu}} {\mu_2-\mu_0-(\mu_1-\mu_0)\bar{\nu}} . \tag{10} \]
Here \(\ell\) is the extent of the inclusion, and \(\tau\) is the interval of the principal autocorrelation dependence of the coefficient of linear homogeneity; \(\Delta \nu\) and \(\Delta \mu\) are the errors in determining the filling density and the attenuation coefficients. The dependence of detectability on the depth at which the inclusion occurs is insignificant, while with an increase in thickness the relative detectability improves only because of the increase in the extent of the detectable inclusions, since their absolute detectability by radiation methods, generally speaking, decreases.
To improve sensitivity, another method of ordering fluctuations may be used, based on constructing, from detector readings, special curves of functionals that have a specific form in the presence of a foreign local inclusion. The proposed method requires that the local inclusion fit completely within the length of the radiogram; it is still better if the length of the section of the shadow image under consideration is 2–5 times longer than the extent of the local inhomogeneity. Here it is proposed to construct, from the radiogram data, the functional
\[ F(x)=\int_0^x\left[\ln I(x)-\frac{1}{L}\int_0^L \ln I(x)\,dx\right]dx . \tag{11} \]
The curve of this functional has a clearly pronounced minimum and maximum at the boundaries of the local inclusion, with a transition through zero above it. At the boundaries of the radiogram this curve starts from zero. The magnitude of the inclusion is determined by the swing of the functional curve \(\Delta F\) and is found from the expression
\[ S= \frac{\Delta F}{1-\frac{\ell}{L}}\, \frac{1}{\mu_2-\mu_0-(\mu_1-\mu_0)\bar{\nu}} . \tag{12} \]
The minimum width of the detectable local inhomogeneity is determined by the expression
\[ \frac{h}{H} = 2\sqrt{\frac{\ell \tau}{(L-\ell)\ell}}\, \left|\frac{\mu_2-\mu_0}{\mu_1-\mu_0}-\bar{\nu}\right|^{-1} \frac{\sigma}{3}. \tag{13} \]
The dependence of detectability on the depths of occurrence of local inclusions is still less than with the first method of averaging fluctuations, since here, in practice, longer radiograms are used, exceeding the extent of the detected inhomogeneities by a factor of 2–5.
When using the described method of averaging fluctuations, one may construct another functional
\[ \psi(y)=\int_0^y \left[I(x)-\frac{1}{L}\int_0^L I(x)\,dx\right]dx, \tag{14} \]
which has the same features of the curve as functional (11). The cross-sectional area is determined by an expression analogous to (12):
\[ S=I_0 e^{\mu_0 H} e^{(\mu_1-\mu_2)\bar{y}H}\, \frac{\Delta\varphi}{1-\frac{l}{L}}\, \frac{1}{(\mu_2-\mu_0)-(\mu_1-\mu_0)\bar{y}} . \tag{15} \]
That is, the cross section of the local inclusion is determined by the magnitude of the swing of the functional, the general coefficient in which can be determined experimentally.
The physical meaning of functional (14) consists in the fact that here the value of the flux at each point is compared with the mean flux over the entire radiogram; in this case, as a rule, this difference is of one sign behind the local inhomogeneity and of the opposite sign in sections of the radiogram without it.
Generally speaking, instead of the quantities \(\ln I(x)\) and \(I(x)\), in functionals (11) and (14) one may substitute any other readings taken from the detectors that are functionally related to the flux magnitude. The detectability of local inclusions does not change as a result, and their cross section is determined, as before, by the magnitude of the swing of the corresponding functional curves.
Both methods of averaging fluctuations can be used in detecting local inhomogeneities both in the primary and in the total flux; however, their detectability in the total flux is poorer than in the primary flux owing to the additional reduction of the relative flux fluctuations by scattered radiation.
Literature
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