= Ruby pressure uncertainty worked example
{title2=$u_p^2=D^2u_a^2+a^2(u_\lambda^2+u_0^2+k_T^2u_T^2)$}
Construct a synthetic local calibration to show the calculation independently of the historical data. Choose reference pairs $(\Delta\lambda_i,p_i)=(1.00,2.69),(2.00,5.42),(3.00,8.09)$, with shifts in $\mathrm{nm}$ and <pressures> in $\mathrm{GPa}$. The reference zero is already subtracted. Treat the shifts as exact for this fit and give the three reference <pressures> equal weights. Minimizing the <ordinary least squares> objective
$$
S(a)=\sum_i(p_i-a\Delta\lambda_i)^2
$$
and setting its <derivative> to zero gives
$$
a=\frac{\sum_i\Delta\lambda_i p_i}{\sum_i(\Delta\lambda_i)^2}
=\frac{37.80}{14.00}=2.70\,\mathrm{GPa\,nm^{-1}}.
$$
The <regression residuals> are $(-0.01,0.02,-0.01)\,\mathrm{GPa}$. Their weighted moment $\sum_i\Delta\lambda_i(p_i-a\Delta\lambda_i)$ vanishes, as required by the <least-squares normal equations>. An unknown reference offset requires fitting an <intercept> instead; forcing it to zero can bias the slope. If shift errors matter, this elementary fit is insufficient.
\Image[/ruby-pressure-calibration-uncertainty.png]
{title=Synthetic <ruby pressure scale>: fitting the response and propagating <measurement uncertainty>}
{description=Original stipulated reference points and calculations, not measured historical calibration. The left panel shows the fitted slope and a chosen reading. The right panel separates independent contributions to <standard uncertainty>; bar lengths are standard contributions, which combine by sums of squares. The assigned slope <standard uncertainty> includes a hypothetical reference-scale contribution and is not estimated from the three residuals.}
Use the local measurement model
$$
p=aD,\qquad D=\lambda-\lambda_0-k_T(T-T_0).
$$
Here $p$ denotes <pressure> relative to the reference, $a$ is calibrated sensitivity, and $k_T$ is a local <wavelength> sensitivity to <temperature> at fixed <pressure>. Choose $a=2.70\,\mathrm{GPa\,nm^{-1}}$, $k_T=0.010\,\mathrm{nm\,K^{-1}}$ and a reading $\lambda-\lambda_0=2.00\,\mathrm{nm}$ at $T=T_0$. The inferred <pressure> is $5.40\,\mathrm{GPa}$. These coefficients and readings are stipulated, not measured <ruby> constants. With the same <pressure>, a $10\,\mathrm K$ increase in <temperature> would add $0.100\,\mathrm{nm}$ to the shift in this model; neglecting it would overestimate <pressure> by $0.270\,\mathrm{GPa}$.
Assign independent <standard uncertainties> $u_a=0.020\,\mathrm{GPa\,nm^{-1}}$, $u_\lambda=u_0=0.010\,\mathrm{nm}$ and $u_T=0.50\,\mathrm K$ for the <temperature> difference. Here $u_a$ is an independently assigned calibration uncertainty, not an estimate from the <regression residuals> above; treat $k_T$ as exact in this example. First-order <propagation of uncertainty> follows by differentiating the measurement model:
$$
dp=D\,da+a\,d\lambda-a\,d\lambda_0-a k_T\,d(T-T_0),
$$
$$
u_p^2=D^2u_a^2+a^2\left(u_\lambda^2+u_0^2+k_T^2u_T^2\right).
$$
The four standard contributions are $0.0400,0.0270,0.0270,0.0135\,\mathrm{GPa}$, giving $u_p=0.0569\,\mathrm{GPa}$. This is a <standard uncertainty>, not automatically a $95\%$ interval. Correlated reference and loaded <wavelength> readings require the corresponding <covariance> term; common wavelength-axis drift may cancel in their difference. Uncertain $k_T$, nonzero $T-T_0$, model inadequacy and unresolved <deviatoric stress> require a broader budget. Taking more optical readings reduces some random errors while leaving a common reference-scale error.
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