= Detector-volume-averaging correction factor
{title2=$k_{\rm vol}$}
A <detector-volume-averaging correction factor> converts a specified spatially averaged <absorbed dose> into the point value associated with that measurement. It depends on the radiation field and the detector geometry. The IAEA defines this comparison in undisturbed <water>, with the detector absent, so the factor isolates <detector volume averaging> from perturbations caused by inserting an <ionization chamber>.<Dosimetry of Small Static Fields Used in External Beam Radiotherapy by the International Atomic Energy Agency>[§3.1.4, printed p. 37, definition and equation (5)]{ref}
For a normalized nonnegative spatial weighting function $w$, let
$$
\overline D(\mathbf x_0)=\int w(\mathbf u)D(\mathbf x_0+\mathbf u)\,d\mathbf u,
\qquad
k_{\rm vol}(\mathbf x_0)=\frac{D(\mathbf x_0)}{\overline D(\mathbf x_0)},
$$
assuming the denominator is positive. Multiplication by $k_{\rm vol}$ then recovers the point value \b[in this response model]. The averaging law is developed in <detector volume averaging>; the correction requires additional information about the spatial field, not just one scalar detector reading.
For an original exact example, take a uniformly sensitive one-dimensional detector of full width $a$ centred at a smooth peak. Over its footprint stipulate $D(x)=D_0(1-cx^2)$ with $c>0$ and $ca^2/4<1$. Direct integration gives
$$
\frac{\overline D(0)}{D_0}=1-\frac{ca^2}{12},\qquad
k_{\rm vol}(0)=\frac1{1-ca^2/12}.
$$
With $a=4\,\mathrm{mm}$ and $c=0.01\,\mathrm{mm^{-2}}$, the average is $0.986666\ldots D_0$ and $k_{\rm vol}=75/74\approx1.013514$. A small local under-response becomes a factor greater than one.
At a valley with $D(x)=D_0(1+cx^2)$ and the same parameters, the factor is instead $1/(1+ca^2/12)\approx0.986842$. Thus an averaging correction need not exceed one. A uniform field gives exactly one. Reusing a peak-derived factor at a different location can therefore introduce an error.
In a proposed characterization, obtain a sufficiently resolved beam profile, account for the profiling detector's own <spatial resolution>, and integrate that profile with the intended detector's geometric weighting. The IAEA's geometric models explicitly distinguish different weightings across the projected detector area.<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 weighting models]{ref} A factor derived this way addresses spatial averaging under the stated field conditions; it does not independently correct energy response, recombination or an erroneous <measurement calibration>.
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