= Detector volume averaging
<Detector volume averaging> is the spatial averaging of a nonuniform radiation field over a detector's finite sensitive region. A reading associated with the detector centre need not equal the local <absorbed dose> at that point. This limits <spatial resolution> even if the detector position and electronic readout are exact.
For an idealized linear response, let $D(\mathbf x)$ be the <absorbed dose> field in the undisturbed medium and let $w(\mathbf u)\ge0$ be a normalized spatial weighting function, with $\int w(\mathbf u)\,d\mathbf u=1$. Model the spatially averaged quantity by
$$
\overline D(\mathbf x_0)=\int w(\mathbf u)D(\mathbf x_0+\mathbf u)\,d\mathbf u.
$$
The model separates averaging from the change in the radiation field caused by inserting the detector. A real <ionization chamber> can require both effects to be considered. The IAEA definition compares a point dose with the average over the detector's sensitive volume, both evaluated in water with the detector absent.<Dosimetry of Small Static Fields Used in External Beam Radiotherapy by the International Atomic Energy Agency>[§3.1.4, printed p. 37, definition and spatial weighting in equation (5)]{ref}
For an original one-dimensional example, suppose the detector averages uniformly over a full width $a>0$. Then
$$
\overline D(x)=\frac1a\int_{x-a/2}^{x+a/2}D(s)\,ds.
$$
An ideal sharp edge $D(x)=D_0H(x)$, where $H$ is the <Heaviside step function>, is recorded as
$$
\frac{\overline D(x)}{D_0}=\begin{cases}
0&x\le-a/2,\\
\frac12+\frac{x}{a}&-a/2<x<a/2,\\
1&x\ge a/2.
\end{cases}
$$
Thus a $4\,\mathrm{mm}$ detector gives a $2.4\,\mathrm{mm}$ distance between the 20-percent and 80-percent response points, although the stipulated field edge has zero transition width. A $1\,\mathrm{mm}$ detector gives $0.6\,\mathrm{mm}$ in the same model.
\Image[/detector-volume-averaging-step.png]
{title=Uniform finite-detector averaging turns a sharp radiation-field edge into a ramp. Original model calculation; the indicated 2.4 mm width belongs to the 4 mm detector, not to an intrinsic field transition.}
For a smooth field, a <Taylor expansion> under the symmetric uniform weighting gives $\overline D(x)=D(x)+a^2D''(x)/24+O(a^4)$. A constant slope averages to its centre value; curvature produces the leading local discrepancy. A <detector-volume-averaging correction factor> therefore depends on the field shape and detector geometry, rather than only on the detector's nominal volume.
A proposed apparatus test scans a stable radiation field with detectors of different effective widths or orientations, keeping depth and irradiation geometry fixed. Changes in the recorded edge width test spatial response, while a uniform region checks relative sensitivity. The IAEA's geometric weighting examples show why a projected detector area may need nonuniform weighting; finer scan steps alone cannot remove averaging within each reading.<Dosimetry of Small Static Fields Used in External Beam Radiotherapy by the International Atomic Energy Agency>[Appendix I.2.3, printed pp. 147–150, equations (52)–(53) and geometric weighting models]{ref}
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