A consequence of using the simple dipole model to define the energy fraction is that it implies a specific theoretical difference ΔL between the event noise levels L_max and L_E. If the contour model is to be internally consistent, this needs to equal the difference of the values determined from the NPD curves. A problem is that the NPD data are derived from actual aircraft noise measurements — which do not necessarily accord with the simple theory. The theory therefore needs an added element of flexibility. But in principal the variables α₁ and α₂ are determined by geometry and aircraft speed — thus leaving no further degrees of freedom. A solution is provided by the concept of a scaled distance d_λ as follows.

The exposure level L_E,∞ as tabulated as a function of d_p in the ANP database for a reference speed V_ref, can be expressed as

$L_{\mathrm{E,\infty }}\left(V_{\mathrm{ref}}\right)=10\times \lg \left[\frac{\underset{-\infty }{\overset{\infty }{\int }}{p}^2\times \mathrm{dt}}{p_2^0\times t_{\mathrm{ref}}}\right]$

(E-10)

where p₀ is a standard reference pressure and t_ref is a reference time (= 1 s for SEL). For the actual speed V it becomes

$L_{\mathrm{E,\infty }}\left(V\right)=L_{\mathrm{E,\infty }}\left(V_{\mathrm{ref}}\right)+10\times \lg \left(\frac{V_{\mathrm{ref}}}{V}\right)$

(E-11)

Similarly the maximum event level L_max can be written

$L_{\max }=10\times \lg \left[\frac{p_2^p}{p_2^0}\right]$

(E-12)

For the dipole source, using equations E-8, E-11 and E-12, noting that (from equations E-2 and E-8) $\underset{-\infty }{\overset{\infty }{\int }}{p}^2\times \mathrm{dt}=\frac{\pi }{2}\times p_2^p\times \frac{d_p}{V}$, the difference ΔL can be written:

$\mathrm{\Delta L}=L_{\mathrm{E,\infty }}-L_{\max }=10\times \lg \left[\frac{V}{V_{\mathrm{ref}}}\times \left(\frac{\pi }{2}{p}_2^p\frac{d_p}{V}\right)\times \frac{1}{p_2^0\times t_{\mathrm{ref}}}\right]-10\times \lg \left[\frac{p_2^p}{p_2^0}\right]$

(E-13)

This can only be equated to the value of ΔL determined from the NPD data if the slant distance d_p used to calculate the energy fraction is substituted by a scaled distance d_λ given by

$d_{\lambda }=\frac{2}{\pi }\times V_{\mathrm{ref}}\times t_{\mathrm{ref}}\times {10}^{\left(L_{\mathrm{E,\infty }}-L_{\max }\right)∕10}$

(E-14a)

or

$d_{\lambda }=d_0\times {10}^{\left(L_{\mathrm{E,\infty }}-L_{\max }\right)∕10}$with$d_0=\frac{2}{\pi }\times V_{\mathrm{ref}}\times t_{\mathrm{ref}}$

(E-14b)

Replacing d_p by d_λ in equation E-5 and using the definition q = Vτ from Figure E-1 the parameters α₁ and α₂ in equation E-9 can be written (putting q = q₁ at the start-point and q – λ = q₂ at the endpoint of a flight path segment of length λ) as

${\alpha }_1=\frac{-q_1}{d_{\lambda }}$and${\alpha }_2=\frac{-q_1+\lambda }{d_{\lambda }}$

(E-15)

Having to replace the slant actual distance by scaled distance diminishes the simplicity of the fourth-power 90 degree dipole model. But as it is effectively calibrated in situ using data derived from measurements, the energy fraction algorithm can be regarded as semi-empirical rather than a pure theoretical.