= Joint radius-setting and log-ratio bounds
\b[Joint radius-setting and log-ratio bounds] retain the allowed relationship between the radius-setting parameter $h$ and measured log-ratio $D$ when inferring an optical radius ratio. Separate marginal intervals form a rectangle that may contain combinations excluded by shared calibration information. Propagating that rectangle can therefore exaggerate the feasible radius range.
For $h>0$ and $|D|<4h$, define
$$
z=-\frac{\tanh(D/4)}{\tanh h},\qquad
\rho(D,h)=\sqrt{\frac{1+z}{1-z}}.
$$
For a stated joint set $\mathcal F$, the compatible ratios are $\{\rho(D,h):(D,h)\in\mathcal F,\ |D|<4h\}$. This set definition precedes any approximation. A deterministic set is not automatically a statistical <confidence region>; a probability interpretation needs the corresponding input model.
An original exactly linked calibration example has $0.6\le h\le0.8$ and $D=2h$. The marginal intervals are $D\in[1.2,1.6]$ and $h\in[0.6,0.8]$, but their extreme values cannot be chosen independently. Writing $t=\tanh(h/2)$ and using $\tanh h=2t/(1+t^2)$ gives
$$
\rho^2=\frac{1-t^2}{3+t^2}.
$$
Thus the joint model gives approximately $0.5216\le\rho\le0.5447$. Treating the two marginal intervals as an independent rectangle instead allows approximately $0.4139\le\rho\le0.6246$. The wider result contains parameter combinations that violate $D=2h$. All values describe a stipulated calibration relation, not observed optical data.
For a positive-$D$ rectangle wholly inside the inversion range, monotonicity places the minimum ratio at $(D_U,h_L)$ and maximum at $(D_L,h_U)$. That shortcut is valid for the rectangle, but those corners may be absent from the actual joint set. With nonlinear dependencies, optimize over the stated joint set or parameterize it by the genuinely independent calibration inputs.
The <feasible-set boundary in overlap inversion> also matters. If part of a joint set crosses $|D|=4h$, some allowed observations have no finite positive solution in the assumed model. Report that incompatibility and examine the measurement model; silently discarding those points changes the set or probability distribution. JCGM 101 distinguishes input assignment, propagation and output summaries, and specifies conditions for its Monte Carlo approach.<Propagation of distributions using a Monte Carlo method by JCGM>[§§5.1–5.3 and 5.10]{ref}
Experimentally, retain common calibration records when deriving both $h$ and $D$. A shared scale or fitted parameter should be varied consistently through both calculations. Independent draws from their marginals erase that information. Conversely, imposing a convenient correlation without physical calibration evidence can create an unjustifiably narrow range. <Interval propagation for bounded overlap data> is a numerical step conditional on this joint experimental model.
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